Finite Group Databasebeta

← #33093457

Subgroups — #33093457

Order 18688.

886 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988715C211yesmaximal, characteristic, complemented
2C4×#14556570C211yesmaximal, characteristic
3#22610871C211yesmaximal, characteristic, complemented
4#22610654C211yesmaximal, characteristic, complemented
5#22610488C211yesmaximal, characteristic, complemented
6#22610488C211yesmaximal, characteristic, complemented
7#22610114C211yesmaximal, characteristic
8#22610114C211yesmaximal, characteristic
9#22609911C211yesmaximal, characteristic
10#22609805C211yesmaximal, characteristic, complemented
11#22609803C211yesmaximal, characteristic, complemented
12C8×#11924007C211yesmaximal, characteristic, complemented
13#22609712C211yesmaximal, characteristic
14#22609712C211yesmaximal, characteristic
15C2×C4×M16×C73C211yesmaximal, characteristic
16C2×#14556570C411yescharacteristic
17C2×C4×Dic146C2211yescharacteristic
18#17988715—24no—
19C2×C4⋊#10596831C2211yescharacteristic
20C2×C4⋊#10596831C2211yescharacteristic
21C2×#14556520C2211yescharacteristic
22C2×#14556520C2211yescharacteristic
23C2×#14556520C2211yescharacteristic
24C2×#14556520C2211yescharacteristic
25C2×C22⋊#10596831C2211yescharacteristic
26C2×C22⋊#10596831C2211yescharacteristic
27C2×#14556501C2211yescharacteristic
28C2×#14556501C2211yescharacteristic
29C2×C8×Dic73C2211yescharacteristic
30C2×C8×Dic73C2211yescharacteristic
31#17988493C411yescharacteristic
32#17988485C411yescharacteristic
33#17988471C411yescharacteristic
34#17988471C411yescharacteristic
35#17988460C411yescharacteristic
36#17988460C411yescharacteristic
37#17988447C411yescharacteristic
38C4×#11924007C2211yescharacteristic
39C4×#11924007C2211yescharacteristic
40C4×C4⋊Dic73C2211yescharacteristic
41C4×C22⋊Dic73C2211yescharacteristic
42C4×C22⋊Dic73C2211yescharacteristic
43#17988342C2211yescharacteristic
44#17988342C2211yescharacteristic
45#17988341C2211yescharacteristic
46#17988341C2211yescharacteristic
47#17988341C2211yescharacteristic
48#17988341C2211yescharacteristic
49#17988341C2211yescharacteristic
50#17988341C2211yescharacteristic
51#17988341C2211yescharacteristic
52#17988341C2211yescharacteristic
53C22×M16×C73C2211yescharacteristic
54C22×M16×C73C2211yescharacteristic
55C22×C42×C73C2211yescharacteristic
56C4×M16×C73—24no—
57C2×#97×C73C2211yescharacteristic
58C2×#97×C73C2211yescharacteristic
59C2×C4×C8×C73C2211yescharacteristic
60C2×C4×C8×C73C2211yescharacteristic
61C22×Dic146C2×C411yescharacteristic
62C22×#10596831C2311yescharacteristic
63C22×#10596831C2311yescharacteristic
64C22×#10596831C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66#14556570—24no—
67C4×Dic146—22no—
68C4×Dic146—22no—
69C2×C4⋊Dic73C2×C411yescharacteristic
70C2×C4⋊Dic73C2×C411yescharacteristic
71C2×C4⋊Dic73C2×C411yescharacteristic
72C2×#11924007C2×C411yescharacteristic
73C2×#11924007C2×C411yescharacteristic
74C2×#11924007C2×C411yescharacteristic
75C2×#11924007C2×C411yescharacteristic
76C2×#11924007C2×C411yescharacteristic
77C2×#11924007C2×C411yescharacteristic
78C2×C22⋊Dic73C2×C411yescharacteristic
79C2×C22⋊Dic73C2×C411yescharacteristic
80C2×C22⋊Dic73C2×C411yescharacteristic
81C2×C22⋊Dic73C2×C411yescharacteristic
82C2×C4×Dic73C2311yescharacteristic
83C2×C4×Dic73C2311yescharacteristic
84C2×C4×Dic73C2311yescharacteristic
85C2×C4×Dic73C2311yescharacteristic
86C4⋊#10596831—22no—
87C4⋊#10596831—22no—
88#14556520—22no—
89#14556520—22no—
90#14556520—22no—
91#14556520—22no—
92C22⋊#10596831—22no—
93C22⋊#10596831—22no—
94#14556501—22no—
95#14556501—22no—
96C8×Dic73—22no—
97C8×Dic73—22no—
98#14556498C2×C411yescharacteristic
99#14556498C2×C411yescharacteristic
100#14556498C2×C411yescharacteristic
101#14556498C2×C411yescharacteristic
102#14556498C2×C411yescharacteristic
103#14556498C2×C411yescharacteristic
104#14556498C2×C411yescharacteristic
105#14556498C2×C411yescharacteristic
106#14556498C2×C411yescharacteristic
107#14556498C2×C411yescharacteristic
108C23×C4×C73C2×C411yescharacteristic
109C23×C4×C73C2×C411yescharacteristic
110C23×C4×C73C2311yescharacteristic
111C2×M16×C73—22no—
112C2×M16×C73—22no—
113C2×M16×C73—22no—
114C2×M16×C73—22no—
115C2×M16×C73—22no—
116C2×M16×C73Q814yescomplemented
117C22×C8×C73C2311yescharacteristic
118C22×C8×C73C2311yescharacteristic
119C22×C8×C73C2311yescharacteristic
120C22×C8×C73C2311yescharacteristic
121C2×C42×C73—22no—
122C2×C42×C73—22no—
123C2×C42×C73C2×C411yescharacteristic
124C2×C42×C73C2×C411yescharacteristic
125C2×C42×C73C2311yescharacteristic
126C2×C42×C73C2311yescharacteristic
127#97×C73—22no—
128#97×C73D414yes—
129C4×C8×C73—22no—
130C4×C8×C73—22no—
131C2×Dic146—22no—
132C2×Dic146—22no—
133C2×Dic146—41no—
134C22×Dic73C22×C411yescharacteristic
135C22×Dic73C22×C411yescharacteristic
136C22×Dic73C22×C411yescharacteristic
137C22×Dic73C22×C411yescharacteristic
138C22×Dic73C22×C411yescharacteristic
139C22×Dic73C22×C411yescharacteristic
140C22×Dic73C22×C411yescharacteristic
141C22×Dic73C22×C411yescharacteristic
142C2×#10596831—21no—
143C2×#10596831—21no—
144C2×#10596831—21no—
145C2×#10596831—21no—
146C2×#10596831—21no—
147C2×#10596831—21no—
148C2×#10596831—21no—
149C2×#10596831—21no—
150C2×#10596831—21no—
151C2×#10596831—21no—
152C2×#10596831—21no—
153C2×#10596831—21no—
154C4⋊Dic73—22no—
155C4⋊Dic73—22no—
156C4⋊Dic73—41no—
157#11924007—22no—
158#11924007—22no—
159#11924007—22no—
160#11924007—22no—
161#11924007—22no—
162#11924007M1614yescomplemented
163C22⋊Dic73—22no—
164C22⋊Dic73—22no—
165C22⋊Dic73—41no—
166C22⋊Dic73—41no—
167C4×Dic73—21no—
168C4×Dic73—21no—
169C4×Dic73—21no—
170C4×Dic73—21no—
171C4×Dic73—21no—
172C4×Dic73—21no—
173C4×Dic73—21no—
174C4×Dic73—21no—
175C24×C73C22×C411yescharacteristic
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73—22no—
181C22×C4×C73—22no—
182C22×C4×C73—22no—
183C22×C4×C73—22no—
184C22×C4×C73—41no—
185C22×C4×C73—41no—
186C22×C4×C73C22×C411yescharacteristic
187C22×C4×C73C22×C411yescharacteristic
188C22×C4×C73C22×C411yescharacteristic
189C22×C4×C73C22×C411yescharacteristic
190C22×C4×C73C22×C411yescharacteristic
191C22×C4×C73C2×Q812yes—
192C22×C4×C73#4112yes—
193C22×C4×C73C2411yescharacteristic
194M16×C73—28no—
195M16×C73—28no—
196C2×C8×C73—21no—
197C2×C8×C73—21no—
198C2×C8×C73—21no—
199C2×C8×C73—21no—
200C2×C8×C73—21no—
201C2×C8×C73—21no—
202C2×C8×C73C2×D412yes—
203C2×C8×C73C2×D412yes—
204C2×C8×C73C2×Q812yes—
205C2×C8×C73C2×Q812yes—
206C2×C8×C73#4112yes—
207C2×C8×C73#4112yes—
208C42×C73—22no—
209C42×C73—22no—
210C42×C73—21no—
211C42×C73—21no—
212C42×C73C2×D412yes—
213C42×C73#4112yes—
214C23×C73—21no—
215C23×C73—21no—
216C23×C73—21no—
217C23×C73—21no—
218C23×C73—21no—
219C23×C73—21no—
220C23×C73C4×Q812yes—
221C23×C73C23×C411yescharacteristic
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—22no—
233C2×C4×C73—22no—
234C2×C4×C73—22no—
235C2×C4×C73—22no—
236C2×C4×C73—44no—
237C2×C4×C73—44no—
238C2×C4×C73C4×D412yes—
239C2×C4×C73C4×D412yes—
240C2×C4×C73C4×Q812yes—
241C2×C4×C73C2×M1612yes—
242C2×C4×C73#13112yes—
243C2×C4×C73C22×D411yescharacteristic
244C2×C4×C73C22×Q811yescharacteristic
245C2×C4×C73C2×#4111yescharacteristic
246C2×C4×C73C2×#4111yescharacteristic
247C2×C4×C73C2×#4111yescharacteristic
248C2×C4×C73C2×#4111yescharacteristic
249C2×Dic73—21no—
250C2×Dic73—21no—
251C2×Dic73—21no—
252C2×Dic73—21no—
253C2×Dic73—21no—
254C2×Dic73—21no—
255C2×Dic73—21no—
256C2×Dic73—21no—
257C2×Dic73—21no—
258C2×Dic73—21no—
259C2×Dic73—21no—
260C2×Dic73—21no—
261C2×Dic73—21no—
262C2×Dic73—21no—
263C2×Dic73—21no—
264C2×Dic73—21no—
265C2×Dic73—21no—
266C2×Dic73—21no—
267C2×Dic73—21no—
268C2×Dic73—21no—
269C2×Dic73—21no—
270C2×Dic73—21no—
271C2×Dic73C2×M1612yes—
272C2×Dic73C2×M1612yes—
273Dic146—44no—
274C8×C73—22no—
275C8×C73—22no—
276C8×C73—22no—
277C8×C73C22⋊Q814yescomplemented
278#10596831—41no—
279#10596831—41no—
280#10596831—41no—
281#10596831—41no—
282C22×C73—22no—
283C22×C73—22no—
284C22×C73—22no—
285C22×C73—41no—
286C22×C73—41no—
287C22×C73—41no—
288C22×C73—41no—
289C22×C73C2×C4×D411yescharacteristic
290C22×C73C2×C4×Q811yescharacteristic
291C22×C73C4×#4111yescharacteristic
292C22×C73C22×M1611yescharacteristic
293C22×C73C2×#13111yescharacteristic
294C22×C73#56811yescharacteristic
295C22×C73#56811yescharacteristic
296C4×C73—22no—
297C4×C73—22no—
298C4×C73—22no—
299C4×C73—22no—
300C4×C73—22no—
301C4×C73C2×C22⋊Q811yescharacteristic
302C4×C73#52511yescharacteristic
303C4×C73#52911yescharacteristic
304C4×C73#53311yescharacteristic
305Dic73—22no—
306Dic73—22no—
307Dic73—41no—
308Dic73—41no—
309Dic73—41no—
310Dic73—41no—
311Dic73—41no—
312Dic73—41no—
313#21600—731nomaximal, Sylow
314C2×C73—42no—
315C2×C73C4×C22⋊Q811yescharacteristic
316C2×C73#293311yescharacteristic
317C2×C73D4×M1611yescharacteristic
318C2×C73#296211yescharacteristic
319C2×C73Q8×M1611yescharacteristic
320C2×C73#296511yescharacteristic
321C2×C73#296711yescharacteristic
322C2×#446—731no—
323C2×C4×M16D7311yescharacteristic
324C4×C22⋊Q8—731no—
325#1985—731no—
326#1940—731no—
327#1916—731no—
328#1916—731no—
329#1851—731no—
330#1851—731no—
331#1797—731no—
332#1771—731no—
333#1771—731no—
334C8×C4⋊C4—731no—
335#1753—731no—
336#1753—731no—
337C73#2160011yescharacteristic, complemented, Sylow
338C22×M16C2×D7311yescharacteristic
339C22×M16C2×D7311yescharacteristic
340C2×C22⋊Q8—731no—
341C2×C4×Q8—731no—
342C22×C42C2×D7311yescharacteristic
343#446—1464no—
344C2×C4⋊C8—731no—
345C2×C4⋊C8—731no—
346C2×C4⋊C8—731no—
347C2×C4⋊C8—731no—
348C2×C4⋊C8—731no—
349C2×C4⋊C8—731no—
350C2×C22⋊C8—731no—
351C2×C22⋊C8—731no—
352C4×M16—24no—
353C2×#97—731no—
354C2×#97—731no—
355C2×#97C2×D7311yescharacteristic
356C2×#97C2×D7311yescharacteristic
357C2×C4×C8—731no—
358C2×C4×C8—731no—
359C2×C4×C8C2×D7311yescharacteristic
360C2×C4×C8C2×D7311yescharacteristic
361#391—731no—
362#389—731no—
363#387—731no—
364#387—731no—
365#385—731no—
366#385—731no—
367#380—731no—
368C4×C4⋊C4—731no—
369C4×C4⋊C4—731no—
370C4×C4⋊C4—731no—
371C4×C22⋊C4—731no—
372C4×C22⋊C4—731no—
373#336—731no—
374#336—731no—
375#336—731no—
376#336—731no—
377#336—731no—
378#336—731no—
379#336—731no—
380#336—731no—
381#336—731no—
382#336—731no—
383C22×Q8—731no—
384C23×C4C4×D7311yescharacteristic
385C23×C4C4×D7311yescharacteristic
386C23×C4C22×D7311yescharacteristic
387C2×M16—22no—
388C2×M16—22no—
389C2×M16—22no—
390C2×M16—22no—
391C2×M16—22no—
392C2×M16Dic14614yescomplemented
393C22×C8—731no—
394C22×C8—731no—
395C22×C8—731no—
396C22×C8—731no—
397C22×C8C22×D7311yescharacteristic
398C22×C8C22×D7311yescharacteristic
399C22×C8C22×D7311yescharacteristic
400C22×C8C22×D7311yescharacteristic
401C22⋊Q8—1464no—
402C4×Q8—1462no—
403C4×Q8—1462no—
404C2×C4⋊C4—731no—
405C2×C4⋊C4—731no—
406C2×C4⋊C4—731no—
407C2×C4⋊C4—731no—
408C2×C4⋊C4—731no—
409C2×C4⋊C4—731no—
410C2×C4⋊C4—731no—
411C2×C4⋊C4—731no—
412C2×C4⋊C4—731no—
413C2×C22⋊C4—731no—
414C2×C22⋊C4—731no—
415C2×C22⋊C4—731no—
416C2×C22⋊C4—731no—
417C2×C42—22no—
418C2×C42—22no—
419C2×C42—731no—
420C2×C42—731no—
421C2×C42—731no—
422C2×C42—731no—
423C2×C42C4×D7311yescharacteristic
424C2×C42C4×D7311yescharacteristic
425C2×C42C22×D7311yescharacteristic
426C2×C42C22×D7311yescharacteristic
427C4⋊C8—1462no—
428C4⋊C8—1462no—
429C4⋊C8—1462no—
430C4⋊C8—1462no—
431C4⋊C8—1462no—
432C4⋊C8—1462no—
433C22⋊C8—1462no—
434C22⋊C8—1462no—
435#97—22no—
436#97—1462no—
437#97—1462no—
438#97C22⋊D7314yes—
439C4×C8—22no—
440C4×C8—22no—
441C4×C8—1462no—
442C4×C8—1462no—
443#95—731no—
444#95—731no—
445#95—731no—
446#95—731no—
447#95—731no—
448#95—731no—
449#95—731no—
450#95—731no—
451#95—731no—
452#95—731no—
453C24C2×C4×D7311yescharacteristic
454C2×Q8—2921no—
455C2×Q8—1462no—
456C2×Q8—1462no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—731no—
468C22×C4—731no—
469C22×C4—731no—
470C22×C4—731no—
471C22×C4—731no—
472C22×C4—731no—
473C22×C4—731no—
474C22×C4—731no—
475C22×C4C2×C4×D7311yescharacteristic
476C22×C4C2×C4×D7311yescharacteristic
477C22×C4C2×C4×D7311yescharacteristic
478C22×C4C2×C4×D7311yescharacteristic
479C22×C4C2×C4×D7311yescharacteristic
480C22×C4C2×Dic14612yes—
481C22×C4#1192402812yes—
482C22×C4C23×D7311yescharacteristic
483M16—28no—
484M16—28no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—1461no—
492C2×C8—1461no—
493C2×C8—1461no—
494C2×C8—1461no—
495C2×C8—1461no—
496C2×C8—1461no—
497C2×C8—1461no—
498C2×C8—1461no—
499C2×C8—1461no—
500C2×C8—1461no—
501C2×C8—1461no—
502C2×C8—1461no—
503C2×C8C2×C22⋊D7312yes—
504C2×C8C2×C22⋊D7312yes—
505C2×C8C2×Dic14612yes—
506C2×C8C2×Dic14612yes—
507C2×C8#1192402812yes—
508C2×C8#1192402812yes—
509C4⋊C4—2921no—
510C4⋊C4—734no—
511C4⋊C4—1462no—
512C4⋊C4—1462no—
513C4⋊C4—1462no—
514C4⋊C4—1462no—
515C4⋊C4—1462no—
516C4⋊C4—1462no—
517C4⋊C4—1462no—
518C22⋊C4—2921no—
519C22⋊C4—2921no—
520C22⋊C4—1462no—
521C22⋊C4—1462no—
522C42—22no—
523C42—22no—
524C42—21no—
525C42—21no—
526C42—1461no—
527C42—1461no—
528C42—1461no—
529C42—1461no—
530C42—1461no—
531C42—1461no—
532C42—1461no—
533C42—1461no—
534C42C2×C22⋊D7312yes—
535C42#1192402812yes—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23C4×Dic14612yes—
543C23C22×C4×D7311yescharacteristic
544Q8—2924no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—732no—
562C2×C4—732no—
563C2×C4—1461no—
564C2×C4—1461no—
565C2×C4—1461no—
566C2×C4—1461no—
567C2×C4—1461no—
568C2×C4—1461no—
569C2×C4—1461no—
570C2×C4—1461no—
571C2×C4—1461no—
572C2×C4—1461no—
573C2×C4—1461no—
574C2×C4—1461no—
575C2×C4—1461no—
576C2×C4—1461no—
577C2×C4—1461no—
578C2×C4—1461no—
579C2×C4—1461no—
580C2×C4—1461no—
581C2×C4—1461no—
582C2×C4—1461no—
583C2×C4—1461no—
584C2×C4—1461no—
585C2×C4C4×C22⋊D7312yes—
586C2×C4C4×C22⋊D7312yes—
587C2×C4C4×Dic14612yes—
588C2×C4M16×D7312yes—
589C2×C4#1455659912yes—
590C2×C4C22×C22⋊D7311yescharacteristic
591C2×C4C22×Dic14611yescharacteristic
592C2×C4C2×#1192402811yescharacteristic
593C2×C4C2×#1192402811yescharacteristic
594C2×C4C2×#1192402811yescharacteristic
595C2×C4C2×#1192402811yescharacteristic
596C8—22no—
597C8—22no—
598C8—22no—
599C8—2921no—
600C8—2921no—
601C8—2921no—
602C8—2921no—
603C8#1455657014yescomplemented
604C22—22no—
605C22—22no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22C2×C4×C22⋊D7311yescharacteristic
612C22C2×C4×Dic14611yescharacteristic
613C22C4×#1192402811yescharacteristic
614C22C2×M16×D7311yescharacteristic
615C22C2×#1455659911yescharacteristic
616C22#1798935311yescharacteristic
617C22#1798935311yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—22no—
623C4—2921no—
624C4—2921no—
625C4—2921no—
626C4—2921no—
627C4—2921no—
628C4—2921no—
629C4—1462no—
630C4—1462no—
631C4C2×#1455657011yescharacteristic
632C4#1798909811yescharacteristic
633C4#1798912111yescharacteristic
634C4#1798914411yescharacteristic
635C2—42no—
636C2C4×#1455657011yescharacteristic
637C2#2261865611yescharacteristic
638C2M16×C22⋊D7311yescharacteristic
639C2#2261887811yescharacteristic
640C2M16×Dic14611yescharacteristic
641C2#2261889311yescharacteristic
642C2#2261891611yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D73
commutator, socleC23×C73C23×C4
FittingC2×C4×M16×C73C2
prime radicalC24×C73C22×C4
cosocleC22×C4×C73C24
hypercenterC2×C4×M16D73
hypocenterC73#21600
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ C2×C4×M16

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#17988715C2semidirect
3. #22610871C2semidirect
4. #22610654C2semidirect
5. #22610488C2semidirect
6. #22610488C2semidirect
10. #22609805C2semidirect
11. #22609803C2semidirect
12. C8×#11924007C2semidirect
116. C2×M16×C73Q8semidirect
162. #11924007M16semidirect
277. C8×C73C22⋊Q8semidirect
337. C73#21600semidirect
392. C2×M16Dic146semidirect
603. C8#14556570semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#179887152
C4×#145565702
#226108712
#226106542
#226104882
#226101142
#226099112
#226098052
#226098032
C8×#119240072
#226097122
C2×C4×M16×C732
#2160073