Finite Group Databasebeta

← #33144155

Subgroups — #33144155

Order 18688.

706 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614946C211yesmaximal, characteristic, complemented
2#22612904C211yesmaximal, characteristic
3#22611315C211yesmaximal, characteristic
4#22611301C211yesmaximal, characteristic
5#22611075C211yesmaximal, characteristic
6#22611065C211yesmaximal, characteristic
7#22610887C211yesmaximal, characteristic, complemented
8#22610881C211yesmaximal, characteristic, complemented
9#22610699C211yesmaximal, characteristic, complemented
10#22610649C211yesmaximal, characteristic, complemented
11#22610526C211yesmaximal, characteristic, complemented
12#22610519C211yesmaximal, characteristic, complemented
13#22610062C211yesmaximal, characteristic, complemented
14#22609941C211yesmaximal, characteristic
15C4×#102×C73C211yesmaximal, characteristic
16C2×#14556590C2211yescharacteristic
17C2×#14556582C2211yescharacteristic
18C2×#14556524C2211yescharacteristic
19C2×#14556522C2211yescharacteristic
20C2×C4⋊#10596831C2211yescharacteristic
21C2×#14556520C2211yescharacteristic
22C2×#14556520C2211yescharacteristic
23C2×#14556512C2211yescharacteristic
24C2×#14556512C2211yescharacteristic
25#17988519—21no—
26#17988508—21no—
27#17988493—21no—
28#17988490C2211yescharacteristic
29#17988490C2211yescharacteristic
30#17988486C2211yescharacteristic
31#17988486C2211yescharacteristic
32#17988485—21no—
33#17988480—21no—
34#17988480—21no—
35#17988453C2211yescharacteristic
36#17988453C2211yescharacteristic
37#17988450—21no—
38C4⋊C4×Dic73C2211yescharacteristic
39C4⋊C4×Dic73C2211yescharacteristic
40C4×C4⋊Dic73C2211yescharacteristic
41C4×C22⋊Dic73—21no—
42#17988350C2211yescharacteristic
43#17988350C2211yescharacteristic
44#17988350C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988349C2211yescharacteristic
47#17988349C2211yescharacteristic
48#17988349C2211yescharacteristic
49#17988349C2211yescharacteristic
50#17988341C2211yescharacteristic
51#17988341C2211yescharacteristic
52C2×C4×D4×C73C2211yescharacteristic
53C2×#102×C73C2211yescharacteristic
54C2×#102×C73C2211yescharacteristic
55C2×C4×C8×C73C2211yescharacteristic
56C4×C4⋊C4×C73C2211yescharacteristic
57#340×C73C2211yescharacteristic
58#340×C73C2211yescharacteristic
59C22×Dic146—21no—
60C22×#10596831C2311yescharacteristic
61C22×#10596831C2311yescharacteristic
62#14556590—22no—
63#14556590—41no—
64#14556582—22no—
65#14556582—41no—
66C2×C4⋊Dic73—21no—
67C2×C4⋊Dic73—21no—
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73D411yescharacteristic
70C2×C4⋊Dic73D411yescharacteristic
71C2×C4⋊Dic73C2311yescharacteristic
72C2×C4⋊Dic73C2311yescharacteristic
73C2×C4⋊Dic73C2311yescharacteristic
74C2×C4⋊Dic73C2311yescharacteristic
75C2×#11924007—21no—
76C2×#11924007—21no—
77C2×C22⋊Dic73—21no—
78C2×C22⋊Dic73—21no—
79C2×C22⋊Dic73—21no—
80C2×C22⋊Dic73—21no—
81C2×C4×Dic73—21no—
82C2×C4×Dic73—21no—
83C2×C4×Dic73C2311yescharacteristic
84C2×C4×Dic73C2311yescharacteristic
85#14556524—22no—
86#14556522—22no—
87C4⋊#10596831—22no—
88#14556520—41no—
89#14556520—41no—
90#14556512—41no—
91#14556512—41no—
92#14556498—21no—
93#14556498—21no—
94#14556498—21no—
95#14556498—21no—
96#14556498—21no—
97#14556498—21no—
98#14556498—21no—
99#14556498—21no—
100#14556498D411yescharacteristic
101#14556498D411yescharacteristic
102C22×D4×C73C2311yescharacteristic
103C23×C4×C73—21no—
104C22×C8×C73C2311yescharacteristic
105C22×C8×C73C2311yescharacteristic
106C4×D4×C73—22no—
107C4×D4×C73Q814yescomplemented
108C2×C4⋊C4×C73C2311yescharacteristic
109C2×C4⋊C4×C73C2311yescharacteristic
110C2×C4⋊C4×C73C2311yescharacteristic
111C2×C22⋊C4×C73—21no—
112C2×C42×C73—21no—
113C2×C42×C73C2311yescharacteristic
114#102×C73—22no—
115#102×C73—22no—
116C4×C8×C73—22no—
117#95×C73—21no—
118C2×Dic146—42no—
119C2×Dic146—42no—
120C2×Dic146—81no—
121C22×Dic73—21no—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C22×Dic73—21no—
125C22×Dic73C2×D411yescharacteristic
126C22×Dic73C2×D411yescharacteristic
127C22×Dic73C2×D411yescharacteristic
128C22×Dic73C2×D411yescharacteristic
129C2×#10596831—21no—
130C2×#10596831—21no—
131C2×#10596831—21no—
132C2×#10596831—21no—
133C2×#10596831—21no—
134C2×#10596831—21no—
135C4⋊Dic73—22no—
136C4⋊Dic73—21no—
137C4⋊Dic73—21no—
138C4⋊Dic73—21no—
139C4⋊Dic73—21no—
140C4⋊Dic73—21no—
141C4⋊Dic73—21no—
142C4⋊Dic73—21no—
143C4⋊Dic73—21no—
144C4⋊Dic73—41no—
145C4⋊Dic73—42no—
146C4⋊Dic73—42no—
147C4⋊Dic73—81no—
148#11924007—42no—
149#11924007—42no—
150C22⋊Dic73—42no—
151C22⋊Dic73—42no—
152C22⋊Dic73—81no—
153C22⋊Dic73—81no—
154C4×Dic73—21no—
155C4×Dic73—21no—
156C4×Dic73—21no—
157C4×Dic73—21no—
158C4×Dic73—81no—
159C4×Dic73—81no—
160C24×C73—21no—
161C2×D4×C73—21no—
162C2×D4×C73—21no—
163C2×D4×C73—21no—
164C2×D4×C73—21no—
165C2×D4×C73—21no—
166C2×D4×C73C2×Q812yes—
167C22×C4×C73—21no—
168C22×C4×C73—21no—
169C22×C4×C73—21no—
170C22×C4×C73—24no—
171C22×C4×C73—81no—
172C22×C4×C73—81no—
173C22×C4×C73C2×D411yescharacteristic
174C22×C4×C73C2×D411yescharacteristic
175C22×C4×C73C2411yescharacteristic
176C2×C8×C73—21no—
177C2×C8×C73—21no—
178C2×C8×C73—21no—
179C2×C8×C73—21no—
180C2×C8×C73—21no—
181C2×C8×C73—21no—
182C4⋊C4×C73—21no—
183C4⋊C4×C73—21no—
184C4⋊C4×C73—21no—
185C4⋊C4×C73C2×Q812yes—
186C4⋊C4×C73#4112yes—
187C4⋊C4×C73#4112yes—
188C22⋊C4×C73—24no—
189C42×C73—81no—
190C42×C73C2×Q812yes—
191C42×C73#4112yes—
192C23×C73—22no—
193C23×C73—41no—
194C23×C73—41no—
195C23×C73—41no—
196C23×C73—41no—
197C23×C73—41no—
198C23×C73—41no—
199C23×C73C22×D411yescharacteristic
200D4×C73—22no—
201D4×C73—22no—
202D4×C73—41no—
203D4×C73—41no—
204C2×C4×C73—21no—
205C2×C4×C73—21no—
206C2×C4×C73—21no—
207C2×C4×C73—21no—
208C2×C4×C73—22no—
209C2×C4×C73—22no—
210C2×C4×C73—22no—
211C2×C4×C73—41no—
212C2×C4×C73—41no—
213C2×C4×C73—41no—
214C2×C4×C73—41no—
215C2×C4×C73—41no—
216C2×C4×C73—41no—
217C2×C4×C73—84no—
218C2×C4×C73C22⋊Q812yes—
219C2×C4×C73C22⋊Q812yes—
220C2×C4×C73C22×Q811yescharacteristic
221C2×C4×C73C2×#4111yescharacteristic
222C2×C4×C73C2×#4111yescharacteristic
223C2×C4×C73C2×#4111yescharacteristic
224C2×C4×C73H(5,𝔽2)11yescharacteristic
225C2×C4×C732−1+411yescharacteristic
226C2×Dic73—21no—
227C2×Dic73—21no—
228C2×Dic73—21no—
229C2×Dic73—21no—
230C2×Dic73—21no—
231C2×Dic73—21no—
232C2×Dic73—21no—
233C2×Dic73—21no—
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—41no—
238C2×Dic73—41no—
239C2×Dic73—41no—
240C2×Dic73—41no—
241C2×Dic73—41no—
242C2×Dic73—41no—
243C2×Dic73—41no—
244C2×Dic73—41no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73#13512yes—
250Dic146—84no—
251C8×C73—41no—
252C8×C73—41no—
253#10596831—41no—
254#10596831—41no—
255C22×C73—42no—
256C22×C73—42no—
257C22×C73—42no—
258C22×C73—81no—
259C22×C73—81no—
260C22×C73—81no—
261C22×C73—81no—
262C22×C73C2×C22⋊Q811yescharacteristic
263C22×C73#54611yescharacteristic
264C22×C73#54811yescharacteristic
265C22×C73C2×#13511yescharacteristic
266C22×C73#57511yescharacteristic
267C22×C73#57611yescharacteristic
268C22×C73#57711yescharacteristic
269C4×C73—22no—
270C4×C73—22no—
271C4×C73—42no—
272C4×C73—42no—
273C4×C73—42no—
274C4×C73#53311yescharacteristic
275C4×C73#54411yescharacteristic
276C4×C73#55511yescharacteristic
277C4×C73#55611yescharacteristic
278Dic73—22no—
279Dic73—41no—
280Dic73—41no—
281Dic73—41no—
282Dic73—81no—
283Dic73—81no—
284Dic73—81no—
285Dic73—81no—
286#26028—731nomaximal, Sylow
287C2×C73—82no—
288C2×C73#243911yescharacteristic
289C2×C73#307611yescharacteristic
290C2×C73#308111yescharacteristic
291C2×C73#329711yescharacteristic
292C2×C73#332611yescharacteristic
293C2×C73#336511yescharacteristic
294C2×C73#337011yescharacteristic
295#2550—731no—
296#2357—731no—
297#2067—731no—
298#2065—731no—
299#2025—731no—
300#2023—731no—
301#1989—731no—
302#1988—731no—
303#1949—731no—
304#1939—731no—
305#1921—731no—
306#1920—731no—
307#1835—731no—
308#1804—731no—
309C4×#102D7311yescharacteristic
310C73#2602811yescharacteristic, complemented, Sylow
311C2×C4⋊Q8—731no—
312C2×#125—731no—
313C2×C4×D4C2×D7311yescharacteristic
314C2×#107—731no—
315C2×#106—731no—
316C2×C4⋊C8—731no—
317C2×C4⋊C8—731no—
318C2×C4⋊C8—731no—
319C2×#102—731no—
320C2×#102—731no—
321C2×#102C2×D7311yescharacteristic
322C2×#102C2×D7311yescharacteristic
323C2×C4×C8C2×D7311yescharacteristic
324#398—1461no—
325#395—1461no—
326#391—1461no—
327#390—731no—
328#390—731no—
329#389—731no—
330#389—731no—
331#389—1461no—
332#388—1461no—
333#388—1461no—
334#382—731no—
335#382—731no—
336#381—1461no—
337C4×C4⋊C4—731no—
338C4×C4⋊C4—731no—
339C4×C4⋊C4—731no—
340C4×C4⋊C4C2×D7311yescharacteristic
341C4×C22⋊C4—1461no—
342#340—731no—
343#340—731no—
344#340—731no—
345#340—731no—
346#340—731no—
347#340—731no—
348#340—731no—
349#340—731no—
350#340C2×D7311yescharacteristic
351#340C2×D7311yescharacteristic
352#336—731no—
353#336—731no—
354C22×Q8—1461no—
355C22×D4C22×D7311yescharacteristic
356C23×C4—21no—
357C22×C8—731no—
358C22×C8—731no—
359C22×C8C22×D7311yescharacteristic
360C22×C8C22×D7311yescharacteristic
361C4⋊Q8—2921no—
362C4⋊Q8—1462no—
363#125—2921no—
364#125—1462no—
365C4×D4—22no—
366C4×D4Dic14614yescomplemented
367C2×C4⋊C4—731no—
368C2×C4⋊C4—731no—
369C2×C4⋊C4—731no—
370C2×C4⋊C4—731no—
371C2×C4⋊C4—731no—
372C2×C4⋊C4—731no—
373C2×C4⋊C4—1461no—
374C2×C4⋊C4—1461no—
375C2×C4⋊C4—1461no—
376C2×C4⋊C4—1461no—
377C2×C4⋊C4—1461no—
378C2×C4⋊C4C22×D7311yescharacteristic
379C2×C4⋊C4C22×D7311yescharacteristic
380C2×C4⋊C4C22×D7311yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—1461no—
383C2×C22⋊C4—1461no—
384C2×C22⋊C4—1461no—
385C2×C22⋊C4—1461no—
386C2×C42—21no—
387C2×C42—731no—
388C2×C42—731no—
389C2×C42—1461no—
390C2×C42—1461no—
391C2×C42C22×D7311yescharacteristic
392#107—1462no—
393#106—1462no—
394C4⋊C8—2921no—
395C4⋊C8—2921no—
396C4⋊C8—1462no—
397#102—22no—
398#102—22no—
399#102—2921no—
400#102—2921no—
401C4×C8—22no—
402#95—21no—
403#95—731no—
404#95—731no—
405#95—1461no—
406#95—1461no—
407#95—1461no—
408#95—1461no—
409#95—1461no—
410#95—1461no—
411#95—1461no—
412#95—1461no—
413C24—21no—
414C2×Q8—2922no—
415C2×Q8—2922no—
416C2×Q8—5841no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4C2×Dic14612yes—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—731no—
430C22×C4—731no—
431C22×C4—731no—
432C22×C4—731no—
433C22×C4—1461no—
434C22×C4—1461no—
435C22×C4—1461no—
436C22×C4—1461no—
437C22×C4D4×D7311yescharacteristic
438C22×C4D4×D7311yescharacteristic
439C22×C4C23×D7311yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—1461no—
447C2×C8—1461no—
448C2×C8—1461no—
449C2×C8—1461no—
450C2×C8—1461no—
451C2×C8—1461no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—2922no—
456C4⋊C4—2922no—
457C4⋊C4—2922no—
458C4⋊C4—2921no—
459C4⋊C4—2922no—
460C4⋊C4—5841no—
461C4⋊C4—1462no—
462C4⋊C4—1461no—
463C4⋊C4—1461no—
464C4⋊C4—1461no—
465C4⋊C4—1461no—
466C4⋊C4—1461no—
467C4⋊C4—1461no—
468C4⋊C4—1461no—
469C4⋊C4—1461no—
470C4⋊C4C2×Dic14612yes—
471C4⋊C4#1192402812yes—
472C4⋊C4#1192402812yes—
473C22⋊C4—24no—
474C22⋊C4—2922no—
475C22⋊C4—2922no—
476C22⋊C4—5841no—
477C22⋊C4—5841no—
478C42—81no—
479C42—5841no—
480C42—5841no—
481C42—1461no—
482C42—1461no—
483C42—1461no—
484C42—1461no—
485C42C2×Dic14612yes—
486C42#1192402712yes—
487C23—22no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23C2×D4×D7311yescharacteristic
495Q8—5844no—
496D4—22no—
497D4—22no—
498D4—41no—
499D4—41no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—84no—
514C2×C4—2921no—
515C2×C4—2921no—
516C2×C4—2921no—
517C2×C4—2921no—
518C2×C4—2921no—
519C2×C4—2921no—
520C2×C4—2921no—
521C2×C4—2921no—
522C2×C4—2921no—
523C2×C4—2921no—
524C2×C4—2921no—
525C2×C4—2921no—
526C2×C4—732no—
527C2×C4—1461no—
528C2×C4—1461no—
529C2×C4—1461no—
530C2×C4—1461no—
531C2×C4—1461no—
532C2×C4—1461no—
533C2×C4—1461no—
534C2×C4—1461no—
535C2×C4—1461no—
536C2×C4—1461no—
537C2×C4—1461no—
538C2×C4#1455656812yes—
539C2×C4#1455656812yes—
540C2×C4C22×Dic14611yescharacteristic
541C2×C4C2×#1192402711yescharacteristic
542C2×C4C2×#1192402811yescharacteristic
543C2×C4C2×#1192402811yescharacteristic
544C2×C4#1455663711yescharacteristic
545C2×C4#1455663811yescharacteristic
546C8—41no—
547C8—41no—
548C8—2921no—
549C8—2921no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C2×#1455656811yescharacteristic
558C22#1798923311yescharacteristic
559C22#1798924611yescharacteristic
560C22#135×D7311yescharacteristic
561C22#1798938411yescharacteristic
562C22#1798938811yescharacteristic
563C22#1798939311yescharacteristic
564C4—22no—
565C4—22no—
566C4—42no—
567C4—42no—
568C4—42no—
569C4—2921no—
570C4—2921no—
571C4—2921no—
572C4—5841no—
573C4—5841no—
574C4—5841no—
575C4—5841no—
576C4—1462no—
577C4#1798914411yescharacteristic
578C4#1798921511yescharacteristic
579C4#1798930211yescharacteristic
580C4#1798930711yescharacteristic
581C2—82no—
582C2#2261362211yescharacteristic
583C2#2261971511yescharacteristic
584C2#2261975811yescharacteristic
585C2#2262178611yescharacteristic
586C2#2262218911yescharacteristic
587C2#2262262811yescharacteristic
588C2#2262268011yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC4×#102×C73C2
socleC23×C73C22×D4
prime radicalC22×D4×C73C23
hypercenterC4×#102D73
hypocenterC73#26028
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
1. #22614946C2semidirect
7. #22610887C2semidirect
8. #22610881C2semidirect
9. #22610699C2semidirect
10. #22610649C2semidirect
11. #22610526C2semidirect
12. #22610519C2semidirect
13. #22610062C2semidirect
107. C4×D4×C73Q8semidirect
310. C73#26028semidirect
366. C4×D4Dic146semidirect

Maximal subgroups

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

subgroupindex
#226149462
#226129042
#226113152
#226113012
#226110752
#226110652
#226108872
#226108812
#226106992
#226106492
#226105262
#226105192
#226100622
#226099412
C4×#102×C732
#2602873