Finite Group Databasebeta

← #33093462

Subgroups — #33093462

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×#17988714C211yesmaximal, characteristic, complemented
2C2×M16×Dic73C211yesmaximal, characteristic
3C4×#14556568C211yesmaximal, characteristic
4#22610869C211yesmaximal, characteristic, complemented
5#22610651C211yesmaximal, characteristic, complemented
6#22610489C211yesmaximal, characteristic, complemented
7#22610485C211yesmaximal, characteristic, complemented
8#22610114C211yesmaximal, characteristic
9#22609909C211yesmaximal, characteristic
10#22609808C211yesmaximal, characteristic, complemented
11#22609804C211yesmaximal, characteristic, complemented
12#22609792C211yesmaximal, characteristic, complemented
13#22609717C211yesmaximal, characteristic
14#22609715C211yesmaximal, characteristic
15C73×#1851C211yesmaximal, characteristic
16C22×#11924012C2211yescharacteristic
17C2×#14556568C411yescharacteristic
18C2×C4×Dic146C2211yescharacteristic
19C22×C4×Dic73C2211yescharacteristic
20#17988714—24no—
21C2×#14556520C2211yescharacteristic
22C2×#14556520C2211yescharacteristic
23C2×#14556520C2211yescharacteristic
24C2×#14556520C2211yescharacteristic
25C2×C22⋊#10596831C2211yescharacteristic
26M16×Dic73—24no—
27C2×#14556502C2211yescharacteristic
28C2×#14556501C2211yescharacteristic
29C2×#14556501C2211yescharacteristic
30C2×#14556501C2211yescharacteristic
31C2×C4×#10596831C2211yescharacteristic
32C2×C8×Dic73C2211yescharacteristic
33C2×C8×Dic73C2211yescharacteristic
34C2×C8×Dic73C2211yescharacteristic
35#17988491C411yescharacteristic
36#17988484C411yescharacteristic
37#17988474C411yescharacteristic
38#17988470C411yescharacteristic
39#17988462C411yescharacteristic
40#17988459C411yescharacteristic
41#17988445C411yescharacteristic
42C4×#11924007C2211yescharacteristic
43C4×#11924007C2211yescharacteristic
44C4×C4⋊Dic73C2211yescharacteristic
45C4×C22⋊Dic73C2211yescharacteristic
46#17988342C2211yescharacteristic
47#17988342C2211yescharacteristic
48#17988341C2211yescharacteristic
49#17988341C2211yescharacteristic
50#17988341C2211yescharacteristic
51#17988341C2211yescharacteristic
52#17988341C2211yescharacteristic
53#17988341C2211yescharacteristic
54C22×M16×C73C2211yescharacteristic
55C2×C4⋊C8×C73C2211yescharacteristic
56C2×C4⋊C8×C73C2211yescharacteristic
57C2×C22⋊C8×C73C2211yescharacteristic
58C4×C22⋊C4×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60#336×C73C2211yescharacteristic
61C22×Dic146C2×C411yescharacteristic
62C23×Dic73C2×C411yescharacteristic
63C23×Dic73C2×C411yescharacteristic
64C2×#11924012—22no—
65C2×#11924012—22no—
66C2×#11924012Q814yescomplemented
67C22×#10596831C2311yescharacteristic
68C22×#10596831C2311yescharacteristic
69C22×#10596831C2311yescharacteristic
70C22×#10596831C2311yescharacteristic
71#14556568—24no—
72C4×Dic146—22no—
73C4×Dic146—22no—
74C2×C4⋊Dic73C2×C411yescharacteristic
75C2×C4⋊Dic73C2×C411yescharacteristic
76C2×C4⋊Dic73C2×C411yescharacteristic
77C2×#11924007C2×C411yescharacteristic
78C2×#11924007C2×C411yescharacteristic
79C2×#11924007C2×C411yescharacteristic
80C2×#11924007C2×C411yescharacteristic
81C2×#11924007C2×C411yescharacteristic
82C2×#11924007C2×C411yescharacteristic
83C2×C22⋊Dic73C2×C411yescharacteristic
84C2×C22⋊Dic73C2×C411yescharacteristic
85C2×C4×Dic73—22no—
86C2×C4×Dic73—22no—
87C2×C4×Dic73C2×C411yescharacteristic
88C2×C4×Dic73C2×C411yescharacteristic
89C2×C4×Dic73C2311yescharacteristic
90C2×C4×Dic73C2311yescharacteristic
91C2×C4×Dic73C2311yescharacteristic
92C2×C4×Dic73C2311yescharacteristic
93#14556520—22no—
94#14556520—22no—
95#14556520—22no—
96#14556520—22no—
97C22⋊#10596831—22no—
98#14556502—22no—
99#14556501—22no—
100#14556501—22no—
101#14556501D414yes—
102C4×#10596831—22no—
103C8×Dic73—22no—
104C8×Dic73—22no—
105C8×Dic73—22no—
106#14556498C2×C411yescharacteristic
107#14556498C2×C411yescharacteristic
108#14556498C2×C411yescharacteristic
109#14556498C2×C411yescharacteristic
110#14556498C2×C411yescharacteristic
111#14556498C2×C411yescharacteristic
112#14556498C2×C411yescharacteristic
113#14556498C2×C411yescharacteristic
114C23×C4×C73C2311yescharacteristic
115C2×M16×C73—22no—
116C2×M16×C73—22no—
117C2×M16×C73—22no—
118C22×C8×C73C2311yescharacteristic
119C22×C8×C73C2311yescharacteristic
120C22×C8×C73C2311yescharacteristic
121C22×C8×C73C2311yescharacteristic
122C2×C22⋊C4×C73C2×C411yescharacteristic
123C2×C22⋊C4×C73C2×C411yescharacteristic
124C2×C42×C73C2311yescharacteristic
125C2×C42×C73C2311yescharacteristic
126C4⋊C8×C73—22no—
127C4⋊C8×C73—22no—
128C22⋊C8×C73—22no—
129#95×C73C2×C411yescharacteristic
130#95×C73C2×C411yescharacteristic
131C2×Dic146—22no—
132C2×Dic146—22no—
133C2×Dic146—41no—
134C22×Dic73—22no—
135C22×Dic73—22no—
136C22×Dic73—22no—
137C22×Dic73—22no—
138C22×Dic73—41no—
139C22×Dic73—41no—
140C22×Dic73C22×C411yescharacteristic
141C22×Dic73C22×C411yescharacteristic
142C22×Dic73C22×C411yescharacteristic
143C22×Dic73C22×C411yescharacteristic
144C22×Dic73C22×C411yescharacteristic
145C22×Dic73C22×C411yescharacteristic
146C22×Dic73C22×C411yescharacteristic
147C22×Dic73C22×C411yescharacteristic
148#11924012—28no—
149C2×#10596831—21no—
150C2×#10596831—21no—
151C2×#10596831—21no—
152C2×#10596831—21no—
153C2×#10596831—21no—
154C2×#10596831—21no—
155C2×#10596831—21no—
156C2×#10596831—21no—
157C2×#10596831C2×D412yes—
158C2×#10596831C2×Q812yes—
159C2×#10596831C2×Q812yes—
160C2×#10596831#4112yes—
161C4⋊Dic73—22no—
162C4⋊Dic73—22no—
163C4⋊Dic73—22no—
164#11924007—22no—
165#11924007—22no—
166#11924007—22no—
167#11924007—22no—
168#11924007—41no—
169#11924007M1614yescomplemented
170C22⋊Dic73—22no—
171C22⋊Dic73—41no—
172C4×Dic73—21no—
173C4×Dic73—21no—
174C4×Dic73—21no—
175C4×Dic73—21no—
176C4×Dic73—21no—
177C4×Dic73—21no—
178C4×Dic73—22no—
179C4×Dic73—22no—
180C4×Dic73C2×D412yes—
181C4×Dic73#4112yes—
182C24×C73C22×C411yescharacteristic
183C22×C4×C73—21no—
184C22×C4×C73—21no—
185C22×C4×C73—21no—
186C22×C4×C73—21no—
187C22×C4×C73C22×C411yescharacteristic
188C22×C4×C73C22×C411yescharacteristic
189C22×C4×C73C22×C411yescharacteristic
190C22×C4×C73C22×C411yescharacteristic
191C22×C4×C73C22×C411yescharacteristic
192C22×C4×C73C2×Q812yes—
193C22×C4×C73#4112yes—
194C22×C4×C73C2411yescharacteristic
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×C73—21no—
203C2×C8×C73—21no—
204C2×C8×C73—21no—
205C2×C8×C73—21no—
206C2×C8×C73C2×D412yes—
207C2×C8×C73#4112yes—
208C22⋊C4×C73—22no—
209C22⋊C4×C73—41no—
210C42×C73—21no—
211C42×C73—21no—
212C42×C73—21no—
213C42×C73—21no—
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—22no—
227C2×C4×C73—22no—
228C2×C4×C73—22no—
229C2×C4×C73—22no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73C4×Q812yes—
240C2×C4×C73C2×M1612yes—
241C2×C4×C73C22×D411yescharacteristic
242C2×C4×C73C22×Q811yescharacteristic
243C2×C4×C73C2×#4111yescharacteristic
244C2×C4×C73C2×#4111yescharacteristic
245C2×C4×C73C2×#4111yescharacteristic
246C2×C4×C73C2×#4111yescharacteristic
247C2×Dic73—21no—
248C2×Dic73—21no—
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—44no—
267C2×Dic73—44no—
268C2×Dic73C4×D412yes—
269C2×Dic73C4×D412yes—
270C2×Dic73C2×M1612yes—
271C2×Dic73C2×M1612yes—
272C2×Dic73#13112yes—
273Dic146—44no—
274C8×C73—22no—
275C8×C73—22no—
276C8×C73—41no—
277C8×C73—41no—
278#10596831—22no—
279#10596831—41no—
280#10596831—41no—
281#10596831C22⋊Q814yescomplemented
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—41no—
299C4×C73—41no—
300C4×C73—41no—
301C4×C73C2×C22⋊Q811yescharacteristic
302C4×C73#52511yescharacteristic
303C4×C73#52911yescharacteristic
304C4×C73#53311yescharacteristic
305Dic73—22no—
306Dic73—22no—
307Dic73—22no—
308Dic73—22no—
309Dic73—22no—
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×M16—731no—
324C4×C22⋊Q8—731no—
325#1985—731no—
326#1940—731no—
327#1916—731no—
328#1916—731no—
329#1851—731no—
330#1851D7311yescharacteristic
331#1797—731no—
332#1771—731no—
333#1771—731no—
334C8×C4⋊C4—731no—
335#1753—731no—
336#1753—731no—
337C73#2160011yescharacteristic, complemented, Sylow
338C22×M16—731no—
339C22×M16C2×D7311yescharacteristic
340C2×C22⋊Q8—731no—
341C2×C4×Q8—731no—
342C22×C42—731no—
343#446—1464no—
344C2×C4⋊C8—731no—
345C2×C4⋊C8—731no—
346C2×C4⋊C8—731no—
347C2×C4⋊C8—731no—
348C2×C4⋊C8C2×D7311yescharacteristic
349C2×C4⋊C8C2×D7311yescharacteristic
350C2×C22⋊C8—731no—
351C2×C22⋊C8C2×D7311yescharacteristic
352C4×M16—1464no—
353C2×#97—731no—
354C2×#97—731no—
355C2×#97—731no—
356C2×#97—731no—
357C2×C4×C8—731no—
358C2×C4×C8—731no—
359C2×C4×C8—731no—
360C2×C4×C8—731no—
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⋊C4C2×D7311yescharacteristic
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#336C2×D7311yescharacteristic
382#336C2×D7311yescharacteristic
383C22×Q8—731no—
384C23×C4—731no—
385C23×C4—731no—
386C23×C4C22×D7311yescharacteristic
387C2×M16—22no—
388C2×M16—22no—
389C2×M16—22no—
390C2×M16—734no—
391C2×M16—1462no—
392C2×M16—1462no—
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⋊C4C4×D7311yescharacteristic
416C2×C22⋊C4C4×D7311yescharacteristic
417C2×C42—731no—
418C2×C42—731no—
419C2×C42—731no—
420C2×C42—731no—
421C2×C42—731no—
422C2×C42—731no—
423C2×C42—1462no—
424C2×C42—1462no—
425C2×C42C22×D7311yescharacteristic
426C2×C42C22×D7311yescharacteristic
427C4⋊C8—22no—
428C4⋊C8—22no—
429C4⋊C8—1462no—
430C4⋊C8—1462no—
431C4⋊C8—1462no—
432C4⋊C8—1462no—
433C22⋊C8—22no—
434C22⋊C8—1462no—
435#97—734no—
436#97—1462no—
437#97—1462no—
438#97—1462no—
439C4×C8—1462no—
440C4×C8—1462no—
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#95C4×D7311yescharacteristic
452#95C4×D7311yescharacteristic
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—2921no—
462C22×C4—2921no—
463C22×C4—731no—
464C22×C4—731no—
465C22×C4—731no—
466C22×C4—731no—
467C22×C4—731no—
468C22×C4—731no—
469C22×C4—731no—
470C22×C4—731no—
471C22×C4—1462no—
472C22×C4—1462no—
473C22×C4—1462no—
474C22×C4—1462no—
475C22×C4C2×C4×D7311yescharacteristic
476C22×C4C2×C4×D7311yescharacteristic
477C22×C4C2×C4×D7311yescharacteristic
478C22×C4C2×C4×D7311yescharacteristic
479C22×C4C2×C4×D7311yescharacteristic
480C22×C4Q8×D7312yes—
481C22×C4#1192402712yes—
482C22×C4C23×D7311yescharacteristic
483M16—28no—
484M16—1468no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—732no—
496C2×C8—732no—
497C2×C8—732no—
498C2×C8—732no—
499C2×C8—1461no—
500C2×C8—1461no—
501C2×C8—1461no—
502C2×C8—1461no—
503C2×C8—1461no—
504C2×C8—1461no—
505C2×C8—1461no—
506C2×C8—1461no—
507C2×C8D4×D7312yes—
508C2×C8#1192402712yes—
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—22no—
519C22⋊C4—41no—
520C22⋊C4—2921no—
521C22⋊C4—1462no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—732no—
527C42—732no—
528C42—1462no—
529C42—1462no—
530C42—1461no—
531C42—1461no—
532C42—1461no—
533C42—1461no—
534C42—1461no—
535C42—1461no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23#1455655612yes—
543C23C22×C4×D7311yescharacteristic
544Q8—2924no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—2924no—
563C2×C4—2924no—
564C2×C4—732no—
565C2×C4—732no—
566C2×C4—732no—
567C2×C4—732no—
568C2×C4—732no—
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×C4—1461no—
586C2×C4—1461no—
587C2×C4—1461no—
588C2×C4#1455655612yes—
589C2×C4M16×D7312yes—
590C2×C4C2×D4×D7311yescharacteristic
591C2×C4C2×Q8×D7311yescharacteristic
592C2×C4#41×D7311yescharacteristic
593C2×C4#41×D7311yescharacteristic
594C2×C4C2×#1192402711yescharacteristic
595C2×C4C2×#1192402711yescharacteristic
596C8—22no—
597C8—22no—
598C8—41no—
599C8—41no—
600C8—2921no—
601C8—2921no—
602C8—734no—
603C8—1462no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22C4×D4×D7311yescharacteristic
612C22C2×#1455655611yescharacteristic
613C22C4×#1192402711yescharacteristic
614C22C2×M16×D7311yescharacteristic
615C22#131×D7311yescharacteristic
616C22#1798935211yescharacteristic
617C22#1798935311yescharacteristic
618C4—22no—
619C4—22no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—2921no—
624C4—2921no—
625C4—2921no—
626C4—1462no—
627C4—1462no—
628C4—1462no—
629C4—1462no—
630C4—1462no—
631C4C22⋊Q8×D7311yescharacteristic
632C4#1798909311yescharacteristic
633C4#1798911411yescharacteristic
634C4#1798913911yescharacteristic
635C2—42no—
636C2#2261246311yescharacteristic
637C2#2261866011yescharacteristic
638C2#2261866411yescharacteristic
639C2#2261887711yescharacteristic
640C2#2261888211yescharacteristic
641C2#2261889411yescharacteristic
642C2#2261891011yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D73
commutator, socleC23×C73C23×C4
FittingC73×#1851C2
prime radicalC24×C73C22×C4
cosocleC22×C4×C73C24
hypercenter#1851D73
hypocenterC73#21600
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ #1851

Complements

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

NH ≅ G/Ntype
1. C2×#17988714C2semidirect
4. #22610869C2semidirect
5. #22610651C2semidirect
6. #22610489C2semidirect
7. #22610485C2semidirect
10. #22609808C2semidirect
11. #22609804C2semidirect
12. #22609792C2semidirect
66. C2×#11924012Q8semidirect
169. #11924007M16semidirect
281. #10596831C22⋊Q8semidirect
337. C73#21600semidirect

Maximal subgroups

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

subgroupindex
C2×#179887142
C2×M16×Dic732
C4×#145565682
#226108692
#226106512
#226104892
#226104852
#226101142
#226099092
#226098082
#226098042
#226097922
#226097172
#226097152
C73×#18512
#2160073