Finite Group Databasebeta

← #33093129

Subgroups — #33093129

Order 18688.

976 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988921C211yesmaximal, characteristic
2#22612469C211yesmaximal, characteristic
3C4×C4⋊C4×D73C211yesmaximal, characteristic
4#22610569C211yesmaximal, characteristic, complemented
5#22610524C211yesmaximal, characteristic, complemented
6#22610523C211yesmaximal, characteristic, complemented
7#22610235C211yesmaximal, characteristic
8#22610117C211yesmaximal, characteristic
9#22610020C211yesmaximal, characteristic
10#107×Dic73C211yesmaximal, characteristic, complemented
11#22609823C211yesmaximal, characteristic, complemented
12#103×Dic73C211yesmaximal, characteristic, complemented
13#103×Dic73C211yesmaximal, characteristic, complemented
14#22609688C211yesmaximal, characteristic
15C73×#1928C211yesmaximal, characteristic, complemented
16C2×#14556567C2211yescharacteristic
17C2×Q8×Dic73C2211yescharacteristic
18C2×C4⋊C4×D73C2211yescharacteristic
19C2×C4⋊C4×D73C2211yescharacteristic
20C2×C42×D73—21no—
21C2×C42×D73C2211yescharacteristic
22#17988921C418yescomplemented
23C2×#14556525C2211yescharacteristic
24C2×#14556524C2211yescharacteristic
25C2×#14556524C2211yescharacteristic
26C2×#14556515C2211yescharacteristic
27C2×#14556515C2211yescharacteristic
28C2×#14556503C2211yescharacteristic
29C2×#14556503C2211yescharacteristic
30C2×C8×Dic73C2211yescharacteristic
31C2×C8×Dic73C2211yescharacteristic
32#17988485C2211yescharacteristic
33#17988474—21no—
34#17988461—21no—
35#17988444C2211yescharacteristic
36C4⋊C4×Dic73C2211yescharacteristic
37C4⋊C4×Dic73C2211yescharacteristic
38C4⋊C4×Dic73C2211yescharacteristic
39C4×#11924007—21no—
40C4×C4⋊Dic73C2211yescharacteristic
41#17988438—21no—
42#17988438—21no—
43#17988434—21no—
44#95×D73—21no—
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988349C2211yescharacteristic
51#17988349C2211yescharacteristic
52#17988342C2211yescharacteristic
53C2×#107×C73C2211yescharacteristic
54C2×#103×C73C2211yescharacteristic
55C2×#103×C73C2211yescharacteristic
56#391×C73C2211yescharacteristic
57C4×C4⋊C4×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59#336×C73C2211yescharacteristic
60C22×C4×D73—21no—
61C22×C4×D73—21no—
62C22×C4×D73D411yescharacteristic
63C22×C4×D73D411yescharacteristic
64C22×C4×D73C2311yescharacteristic
65C22×#10596831C2311yescharacteristic
66C22×#10596831C2311yescharacteristic
67#14556567—22no—
68#14556567C2×C414yes—
69Q8×Dic73—22no—
70Q8×Dic73—22no—
71C2×C4⋊Dic73—21no—
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×C4⋊Dic73C2311yescharacteristic
76C2×#11924007—21no—
77C2×#11924007—21no—
78C2×#11924007—21no—
79C2×#11924007—21no—
80C4⋊C4×D73—22no—
81C4⋊C4×D73—22no—
82C4⋊C4×D73—22no—
83C4⋊C4×D73C2×C414yes—
84C2×#11924005—21no—
85C2×#11924005—21no—
86C2×C4×Dic73—21no—
87C2×C4×Dic73—21no—
88C2×C4×Dic73—21no—
89C2×C4×Dic73—21no—
90C2×C4×Dic73—21no—
91C2×C4×Dic73D411yescharacteristic
92C2×C4×Dic73D411yescharacteristic
93C2×C4×Dic73C2311yescharacteristic
94C42×D73—22no—
95C42×D73—22no—
96C42×D73—44no—
97#14556525C2×C414yescomplemented
98#14556524—22no—
99#14556524C2×C414yescomplemented
100#14556515—22no—
101#14556515C2×C414yescomplemented
102#14556503—22no—
103#14556503C2×C414yes—
104C8×Dic73—22no—
105C8×Dic73—22no—
106#14556498—21no—
107#14556498—21no—
108#14556498—21no—
109#14556498—21no—
110#14556498—21no—
111#14556498—21no—
112C22×Q8×C73C2311yescharacteristic
113C22×C8×C73C2311yescharacteristic
114C22×C8×C73C2311yescharacteristic
115C2×C4⋊C4×C73C2311yescharacteristic
116C2×C4⋊C4×C73C2311yescharacteristic
117C2×C4⋊C4×C73C2311yescharacteristic
118C2×C42×C73—21no—
119C2×C42×C73C2311yescharacteristic
120#107×C73—22no—
121#103×C73—22no—
122#103×C73C2×C414yescomplemented
123#95×C73—21no—
124#95×C73—21no—
125#95×C73—21no—
126C23×D73C2×D411yescharacteristic
127C22×Dic73—21no—
128C22×Dic73—21no—
129C22×Dic73—21no—
130C22×Dic73—21no—
131C22×Dic73C2×D411yescharacteristic
132C22×Dic73C2×D411yescharacteristic
133C22×Dic73C2×D411yescharacteristic
134C2×C4×D73—21no—
135C2×C4×D73—21no—
136C2×C4×D73—21no—
137C2×C4×D73—21no—
138C2×C4×D73—22no—
139C2×C4×D73—22no—
140C2×C4×D73—24no—
141C2×C4×D73—24no—
142C2×C4×D73—22no—
143C2×C4×D73—22no—
144C2×C4×D73—41no—
145C2×C4×D73—42no—
146C2×C4×D73—42no—
147C2×C4×D73—42no—
148C2×C4×D73—42no—
149C2×C4×D73Q1614yes—
150C2×C4×D73C22×C412yes—
151C2×C4×D73#4112yes—
152C2×#10596831—21no—
153C2×#10596831—21no—
154C2×#10596831—21no—
155C2×#10596831—21no—
156C2×#10596831—21no—
157C2×#10596831C22×C412yes—
158C4⋊Dic73—21no—
159C4⋊Dic73—21no—
160C4⋊Dic73—21no—
161C4⋊Dic73—21no—
162C4⋊Dic73—21no—
163C4⋊Dic73—42no—
164C4⋊Dic73C22×C412yes—
165C4⋊Dic73C22×C412yes—
166C4⋊Dic73#4112yes—
167#11924007—24no—
168#11924007—24no—
169#11924007—24no—
170#11924007—42no—
171#11924005—24no—
172#11924005—42no—
173C4×Dic73—22no—
174C4×Dic73—22no—
175C4×Dic73—42no—
176C4×Dic73—42no—
177C4×Dic73—42no—
178C4×Dic73—42no—
179C4×Dic73—42no—
180C4×Dic73#4112yes—
181C4×Dic73#4112yes—
182C2×Q8×C73—21no—
183C2×Q8×C73—21no—
184C2×Q8×C73—21no—
185C2×Q8×C73—21no—
186C2×Q8×C73—21no—
187C2×Q8×C73C22×C412yes—
188C22×C4×C73—21no—
189C22×C4×C73—21no—
190C22×C4×C73—21no—
191C22×C4×C73—21no—
192C22×C4×C73C2×D411yescharacteristic
193C22×C4×C73C2×D411yescharacteristic
194C22×C4×C73C2411yescharacteristic
195C2×C8×C73—21no—
196C2×C8×C73—21no—
197C2×C8×C73—21no—
198C2×C8×C73—21no—
199C2×C8×C73—21no—
200C2×C8×C73C22×C412yes—
201C4⋊C4×C73—21no—
202C4⋊C4×C73—21no—
203C4⋊C4×C73—21no—
204C4⋊C4×C73—21no—
205C4⋊C4×C73C22×C412yes—
206C4⋊C4×C73#4112yes—
207C42×C73—21no—
208C42×C73—21no—
209C42×C73—42no—
210C23×C73C22×D411yescharacteristic
211C22×D73—21no—
212C22×D73—21no—
213C22×D73—21no—
214C22×D73—21no—
215C22×D73C4×D412yes—
216C22×D73C2×Q1612yes—
217C22×D73#13512yes—
218Q8×C73—22no—
219Q8×C73—22no—
220Q8×C73—41no—
221Q8×C73—41no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—22no—
225C2×C4×C73—22no—
226C2×C4×C73—21no—
227C2×C4×C73—22no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—41no—
231C2×C4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73—41no—
238C2×C4×C73—41no—
239C2×C4×C73C2×Q1612yes—
240C2×C4×C73C23×C411yescharacteristic
241C2×C4×C73C2×#4111yescharacteristic
242C2×C4×C73C2×#4111yescharacteristic
243C2×C4×C73C2×#4111yescharacteristic
244C2×C4×C73C2×#4111yescharacteristic
245C2×C4×C73C2×#4111yescharacteristic
246C2×Dic73—21no—
247C2×Dic73—21no—
248C2×Dic73—21no—
249C2×Dic73—21no—
250C2×Dic73—22no—
251C2×Dic73—22no—
252C2×Dic73—22no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73C4×D412yes—
263C2×Dic73#12312yes—
264C2×Dic73#12312yes—
265C2×Dic73C2×Q1612yes—
266C2×Dic73AGL(1,ℤ/8)12yes—
267C4×D73—22no—
268C4×D73—22no—
269C4×D73—22no—
270C4×D73—22no—
271C4×D73—28no—
272C4×D73—44no—
273C4×D73—48no—
274C4×D73—48no—
275C8×C73—22no—
276C8×C73—41no—
277#10596831—22no—
278#10596831—41no—
279C22×C73C2×C4×D411yescharacteristic
280C22×C73C2×#12311yescharacteristic
281C22×C73#52511yescharacteristic
282C22×C73C22×Q1611yescharacteristic
283C22×C73C2×#13511yescharacteristic
284C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
285C22×C73#57511yescharacteristic
286C2×D73—22no—
287C2×D73—22no—
288C2×D73—22no—
289C2×D73—22no—
290C2×D73—22no—
291C2×D73—22no—
292C2×D73C4×Q1614yes—
293C4×C73—22no—
294C4×C73—41no—
295C4×C73—42no—
296C4×C73—42no—
297C4×C73—42no—
298C4×C73—81no—
299C4×C73C4×#4111yescharacteristic
300C4×C73C4×#4111yescharacteristic
301C4×C73#52911yescharacteristic
302C4×C73#52911yescharacteristic
303Dic73—22no—
304Dic73—22no—
305Dic73—42no—
306Dic73—42no—
307Dic73—42no—
308Dic73—81no—
309Dic73#48414yescomplemented
310C4×#484—731nomaximal, Sylow
311C2×C73C4×#12311yescharacteristic
312C2×C73C2×C4×Q1611yescharacteristic
313C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
314C2×C73C2×#48411yescharacteristic
315C2×C73#309111yescharacteristic
316C2×C73#310411yescharacteristic
317C2×C73#310511yescharacteristic
318D73—24no—
319C2×#484—731no—
320C4×C22⋊Q8—731no—
321C2×C4×C4⋊C4—731no—
322#1928—731no—
323#1928D7311yescharacteristic, complemented
324#1921—731no—
325#1921—731no—
326#1872—731no—
327#1852—731no—
328#1824—731no—
329C4×#107—731no—
330C4×#107—731no—
331C4×#103—731no—
332C4×#103—731no—
333C4×C22⋊C8—731no—
334C73C4×#48411yescharacteristic, complemented, Sylow
335C2×C22⋊Q8—731no—
336C2×C4×Q8—731no—
337C22×C4⋊C4—731no—
338C22×C4⋊C4—731no—
339C22×C42—731no—
340C22×C42—1461no—
341#484—738no—
342C2×#107—731no—
343C2×#107—731no—
344C2×#107—731no—
345C2×#107C2×D7311yescharacteristic
346C2×#103—731no—
347C2×#103—731no—
348C2×#103C2×D7311yescharacteristic
349C2×#103C2×D7311yescharacteristic
350C2×C22⋊C8—731no—
351C2×C22⋊C8—731no—
352C2×C4×C8—731no—
353C2×C4×C8—731no—
354#391C2×D7311yescharacteristic
355#389—731no—
356#387—1461no—
357#385—1461no—
358#380—731no—
359C4×C4⋊C4—731no—
360C4×C4⋊C4—731no—
361C4×C4⋊C4—731no—
362C4×C4⋊C4—731no—
363C4×C4⋊C4—1461no—
364C4×C4⋊C4—1461no—
365C4×C4⋊C4—1461no—
366C4×C4⋊C4C2×D7311yescharacteristic
367C4×C22⋊C4—1461no—
368C2×#95—1461no—
369#340—731no—
370#340—731no—
371#340—731no—
372#340—731no—
373#340—731no—
374#340—731no—
375#340—731no—
376#340C2×D7311yescharacteristic
377#336—731no—
378#336C2×D7311yescharacteristic
379C22×Q8C22×D7311yescharacteristic
380C23×C4—731no—
381C23×C4—731no—
382C23×C4—731no—
383C23×C4—1461no—
384C23×C4—1461no—
385C22×C8—731no—
386C22×C8—731no—
387C22×C8C22×D7311yescharacteristic
388C22×C8C22×D7311yescharacteristic
389C22⋊Q8—734no—
390C22⋊Q8—1462no—
391C4×Q8—1462no—
392C4×Q8—1462no—
393C2×C4⋊C4—731no—
394C2×C4⋊C4—731no—
395C2×C4⋊C4—731no—
396C2×C4⋊C4—734no—
397C2×C4⋊C4—731no—
398C2×C4⋊C4—1462no—
399C2×C4⋊C4—1462no—
400C2×C4⋊C4—1461no—
401C2×C4⋊C4—1461no—
402C2×C4⋊C4—1461no—
403C2×C4⋊C4—1461no—
404C2×C4⋊C4—1461no—
405C2×C4⋊C4—1462no—
406C2×C4⋊C4C22×D7311yescharacteristic
407C2×C4⋊C4C22×D7311yescharacteristic
408C2×C4⋊C4C22×D7311yescharacteristic
409C2×C22⋊C4—1461no—
410C2×C22⋊C4—1461no—
411C2×C42—21no—
412C2×C42—2924no—
413C2×C42—731no—
414C2×C42—731no—
415C2×C42—731no—
416C2×C42—1462no—
417C2×C42—1462no—
418C2×C42—1461no—
419C2×C42—1461no—
420C2×C42—1461no—
421C2×C42—1461no—
422C2×C42—1461no—
423C2×C42C22×D7311yescharacteristic
424#107—22no—
425#107—734no—
426#107—734no—
427#107—1462no—
428#103—22no—
429#103—734no—
430#103—1462no—
431#103C4×D7314yescomplemented
432C22⋊C8—734no—
433C22⋊C8—1462no—
434C4×C8—1462no—
435C4×C8—1462no—
436#95—21no—
437#95—21no—
438#95—21no—
439#95—1461no—
440#95—1461no—
441#95—1461no—
442#95—1461no—
443#95—1461no—
444#95—1461no—
445C24—731no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8C2×C4×D7312yes—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—2921no—
457C22×C4—2922no—
458C22×C4—2922no—
459C22×C4—2922no—
460C22×C4—2922no—
461C22×C4—731no—
462C22×C4—731no—
463C22×C4—731no—
464C22×C4—734no—
465C22×C4—732no—
466C22×C4—732no—
467C22×C4—1462no—
468C22×C4—1464no—
469C22×C4—1462no—
470C22×C4—1464no—
471C22×C4—1462no—
472C22×C4—1462no—
473C22×C4—1461no—
474C22×C4—1461no—
475C22×C4—1461no—
476C22×C4—1461no—
477C22×C4—1461no—
478C22×C4—1461no—
479C22×C4—1461no—
480C22×C4—1461no—
481C22×C4D4×D7311yescharacteristic
482C22×C4D4×D7311yescharacteristic
483C22×C4C23×D7311yescharacteristic
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—732no—
490C2×C8—1461no—
491C2×C8—1461no—
492C2×C8—1461no—
493C2×C8—1461no—
494C2×C8—1461no—
495C2×C8C2×C4×D7312yes—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—2922no—
501C4⋊C4—2922no—
502C4⋊C4—732no—
503C4⋊C4—732no—
504C4⋊C4—732no—
505C4⋊C4—1464no—
506C4⋊C4—1464no—
507C4⋊C4—1464no—
508C4⋊C4—1461no—
509C4⋊C4—1461no—
510C4⋊C4—1461no—
511C4⋊C4—1461no—
512C4⋊C4—1461no—
513C4⋊C4C2×C4×D7312yes—
514C4⋊C4#1192402712yes—
515C22⋊C4—2922no—
516C22⋊C4—1464no—
517C42—21no—
518C42—21no—
519C42—42no—
520C42—2922no—
521C42—2922no—
522C42—2922no—
523C42—2922no—
524C42—2922no—
525C42—732no—
526C42—732no—
527C42—1462no—
528C42—1462no—
529C23—732no—
530C23—732no—
531C23—732no—
532C23—1461no—
533C23—1461no—
534C23—1461no—
535C23—1461no—
536C23C2×D4×D7311yescharacteristic
537Q8—22no—
538Q8—22no—
539Q8—41no—
540Q8—41no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—2928no—
559C2×C4—2928no—
560C2×C4—2924no—
561C2×C4—2921no—
562C2×C4—2921no—
563C2×C4—2921no—
564C2×C4—2921no—
565C2×C4—2921no—
566C2×C4—2921no—
567C2×C4—2921no—
568C2×C4—2921no—
569C2×C4—2921no—
570C2×C4—732no—
571C2×C4—732no—
572C2×C4—732no—
573C2×C4—732no—
574C2×C4—732no—
575C2×C4—1468no—
576C2×C4—1462no—
577C2×C4—1462no—
578C2×C4—1462no—
579C2×C4—1461no—
580C2×C4—1461no—
581C2×C4—1461no—
582C2×C4—1461no—
583C2×C4—1462no—
584C2×C4—1462no—
585C2×C4—1462no—
586C2×C4—1462no—
587C2×C4Q16×D7312yes—
588C2×C4C22×C4×D7311yescharacteristic
589C2×C4#41×D7311yescharacteristic
590C2×C4#41×D7311yescharacteristic
591C2×C4#41×D7311yescharacteristic
592C2×C4#41×D7311yescharacteristic
593C2×C4C2×#1192402711yescharacteristic
594C8—22no—
595C8—41no—
596C8—2921no—
597C8—1462no—
598C22—734no—
599C22—1462no—
600C22—1462no—
601C22—1462no—
602C22—1462no—
603C22—1462no—
604C22—1462no—
605C22C4×D4×D7311yescharacteristic
606C22#123×D7311yescharacteristic
607C22#1798909711yescharacteristic
608C22C2×Q16×D7311yescharacteristic
609C22#135×D7311yescharacteristic
610C22AGL(1,ℤ/8)×D7311yescharacteristic
611C22#1798938611yescharacteristic
612C4—22no—
613C4—42no—
614C4—42no—
615C4—41no—
616C4—42no—
617C4—81no—
618C4—2922no—
619C4—2922no—
620C4—2922no—
621C4—5841no—
622C4—734no—
623C4—1462no—
624C4—1462no—
625C4C4×#1192402711yescharacteristic
626C4#1798905511yescharacteristic
627C4#1798911611yescharacteristic
628C4#1798912011yescharacteristic
629C2—1464no—
630C2#2261245811yescharacteristic
631C2C4×Q16×D7311yescharacteristic
632C2#2261873611yescharacteristic
633C2#484×D7311yescharacteristic
634C2#2261984911yescharacteristic
635C2#2261996911yescharacteristic
636C2#2261997711yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1928C2
socleC23×C73C22×D4
prime radicalC23×D73C2×D4
cosocleC22×C4×C73C24
hypercenter#1928D73
hypocenterC73C4×#484
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1928

Complements

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

NH ≅ G/Ntype
4. #22610569C2semidirect
5. #22610524C2semidirect
6. #22610523C2semidirect
10. #107×Dic73C2semidirect
11. #22609823C2semidirect
12. #103×Dic73C2semidirect
13. #103×Dic73C2semidirect
15. C73×#1928C2semidirect
22. #17988921C4semidirect
97. #14556525C2×C4semidirect
99. #14556524C2×C4semidirect
101. #14556515C2×C4semidirect
122. #103×C73C2×C4semidirect
309. Dic73#484semidirect
323. #1928D73semidirect
334. C73C4×#484semidirect
431. #103C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179889212
#226124692
C4×C4⋊C4×D732
#226105692
#226105242
#226105232
#226102352
#226101172
#226100202
#107×Dic732
#226098232
#103×Dic732
#226096882
C73×#19282
C4×#48473