Finite Group Databasebeta

← #29696190

Subgroups — #29696190

Order 15616.

1106 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840473C211yesmaximal, characteristic
2M16×D244C211yesmaximal, characteristic
3#21454554C211yesmaximal, characteristic
4#21446809C211yesmaximal, characteristic, complemented
5#21446286C212yesmaximal, complemented
6#21446286C212yesmaximal, complemented
7#21445898C212yesmaximal, complemented
8#21445812C211yesmaximal, characteristic, complemented
9C4×#14519695C211yesmaximal, characteristic, complemented
10#343×D61C212yesmaximal, complemented
11C61×#1997C211yesmaximal, characteristic, complemented
12C2×M16×D61C2212yes—
13C2×M16×D61C2212yes—
14#16840473—21no—
15#16840473—21no—
16#16840473—22no—
17#16840473—22no—
18#16840473—21no—
19#16840473—21no—
20C2×#14519768C412yes—
21C2×#14519757C2212yes—
22C2×#14519753C414yes—
23C2×C4×Dic122C2211yescharacteristic
24C2×C4×D244C2211yescharacteristic
25C2×#14519739C412yes—
26C2×C4⋊C4×D61C2212yes—
27#16839841—21no—
28#16839839—21no—
29#16839835—22no—
30#16839832—22no—
31C8×D244—21no—
32#16839827—21no—
33#16839786C2211yescharacteristic
34C2×#14519695C2211yescharacteristic
35C2×#14519695C2211yescharacteristic
36C2×#14519695C2212yes—
37#16839725C2212yes—
38#16839725C2212yes—
39C4×#11758194C2211yescharacteristic
40#16839531C2212yes—
41#16839531C2212yes—
42#16839530C2211yescharacteristic
43#16839530C2212yes—
44#16839530C2211yescharacteristic
45C2×C4⋊Q8×C61C2211yescharacteristic
46C61×#423C2211yescharacteristic
47C4×M16×C61C2211yescharacteristic
48C61×#343C2212yes—
49C61×#343C2212yes—
50C22×Dic122C2311yescharacteristic
51C22×D244C2×C411yescharacteristic
52C22×C4×D61C2×C414yes—
53C22×C4×D61C2312yes—
54C2×#11758194C2311yescharacteristic
55C2×#11758194C2312yes—
56C2×#11758194C2311yescharacteristic
57C2×#11758189—22no—
58C2×#11758189—22no—
59M16×D61—24no—
60M16×D61—24no—
61M16×D61—42no—
62M16×D61—42no—
63C2×C8×D61—22no—
64C2×C8×D61—22no—
65#14519768—24no—
66#14519768—22no—
67#14519768—22no—
68#14519757—22no—
69#14519757—22no—
70#14519757—22no—
71#14519757—22no—
72#14519753—24no—
73#14519753—24no—
74#14519753—24no—
75#14519753—24no—
76C4×Dic122—21no—
77C4×Dic122—21no—
78C4×Dic122—21no—
79C4×Dic122—21no—
80C4×D244D411yescharacteristic
81C4×D244D411yescharacteristic
82C4×D244D412yes—
83C4×D244D412yes—
84C4×D244D411yescharacteristic
85C4×D244D411yescharacteristic
86#14519739—24no—
87#14519739—22no—
88#14519739—22no—
89C2×C4⋊Dic61C2×C414yes—
90C2×C4⋊Dic61C2×C411yescharacteristic
91C2×C4⋊Dic61C2311yescharacteristic
92C2×#11758196C2×C414yes—
93C2×#11758196C2×C412yes—
94C4⋊C4×D61—22no—
95C4⋊C4×D61—22no—
96C4⋊C4×D61—22no—
97C4⋊C4×D61—22no—
98C2×#11758198C2×C414yes—
99C2×#11758198C2×C412yes—
100C2×C4×Dic61C2×C412yes—
101C4⋊#92418—21no—
102C4⋊#92418—21no—
103#14519695—22no—
104#14519695—22no—
105#14519695—42no—
106#14519689—22no—
107#14519689—22no—
108#14519689—22no—
109#14519689—22no—
110#14519688—21no—
111C4×#92418—21no—
112C22×Q8×C61C2×C412yes—
113C2×M16×C61C2311yescharacteristic
114C2×M16×C61C2312yes—
115C2×M16×C61C2311yescharacteristic
116C4⋊Q8×C61—21no—
117C4⋊Q8×C61—21no—
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61—21no—
120C2×C4⋊C4×C61C2×C412yes—
121C2×C4⋊C4×C61C2312yes—
122C2×C42×C61C2311yescharacteristic
123C4⋊C8×C61—21no—
124C4⋊C8×C61—21no—
125#97×C61—21no—
126C4×C8×C61—21no—
127C23×D61C22×C412yes—
128C2×Q8×C61—22no—
129C2×Q8×C61—22no—
130C2×Q8×C61—82no—
131C22×C4×C61C22×C411yescharacteristic
132C22×C4×C61C22×C414yes—
133C22×C4×C61C22×C411yescharacteristic
134C22×C4×C61C2411yescharacteristic
135C22×Dic61C22×C412yes—
136C22×Dic61C22×C412yes—
137C22×Dic61C22×C412yes—
138C2×D244—24no—
139C2×D244C22⋊C412yes—
140C2×D244C22⋊C411yescharacteristic
141C2×D244C22⋊C411yescharacteristic
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—24no—
147C2×C4×D61—24no—
148C2×C4×D61—84no—
149C2×C4×D61C2×D412yes—
150C2×C4×D61C2×D412yes—
151C2×C4×D61C2×D412yes—
152C2×C4×D61C2×D412yes—
153C2×Dic122—21no—
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159M16×C61—21no—
160M16×C61—21no—
161M16×C61—22no—
162M16×C61—22no—
163M16×C61—21no—
164M16×C61#4112yes—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—22no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—22no—
172C42×C61C2×D411yescharacteristic
173C42×C61C2×D411yescharacteristic
174C42×C61C2×D411yescharacteristic
175C42×C61C2×D411yescharacteristic
176#11758198—24no—
177#11758198—24no—
178#11758198C22⋊C412yes—
179#11758198C22⋊C412yes—
180#11758198C22⋊C412yes—
181#11758198C22⋊C412yes—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—24no—
185C4⋊Dic61—24no—
186C4⋊Dic61C22⋊C412yes—
187C4⋊Dic61C22⋊C411yescharacteristic
188C4⋊Dic61C22⋊C411yescharacteristic
189#11758196—22no—
190#11758196—22no—
191#11758196—24no—
192#11758196—24no—
193C4×Dic61—22no—
194C4×Dic61—22no—
195#11758194—21no—
196#11758194—21no—
197#11758194—22no—
198#11758194—22no—
199#11758194—21no—
200#11758194#4112yes—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—22no—
204#11758189—44no—
205#11758189—44no—
206C8×D61—44no—
207C8×D61—44no—
208C23×C61C23×C411yescharacteristic
209C22×D61—22no—
210C22×D61—22no—
211C22×D61—24no—
212C22×D61—42no—
213C22×D61C2×C22⋊C412yes—
214C22×D61C2×C22⋊C412yes—
215Q8×C61—82no—
216Q8×C61—82no—
217C2×C4×C61—22no—
218C2×C4×C61—24no—
219C2×C4×C61—41no—
220C2×C4×C61—44no—
221C2×C4×C61C2×C22⋊C411yescharacteristic
222C2×C4×C61C2×C22⋊C411yescharacteristic
223C2×C4×C61C2×C22⋊C411yescharacteristic
224C2×C4×C61C2×C22⋊C411yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61H(5,𝔽2)11yescharacteristic
230C2×C4×C612−1+411yescharacteristic
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—42no—
238C2×Dic61C2×C22⋊C412yes—
239C2×Dic61C2×C22⋊C412yes—
240D244—24no—
241D244—22no—
242D244#10114yescomplemented
243C4×D61—22no—
244C4×D61—22no—
245C4×D61—22no—
246C4×D61—22no—
247C4×D61—22no—
248C4×D61—22no—
249C4×D61—22no—
250C4×D61—22no—
251C4×D61—84no—
252C4×D61—84no—
253Dic122—41no—
254Dic122—41no—
255Dic122—41no—
256Dic122—41no—
257C8×C61—22no—
258C8×C61—42no—
259C8×C61—41no—
260#92418—22no—
261#92418—42no—
262#92418—41no—
263#22481—611nomaximal, Sylow
264C22×C61—21no—
265C22×C61—21no—
266C22×C61C22×C22⋊C411yescharacteristic
267C22×C61C4×#4111yescharacteristic
268C22×C61#52011yescharacteristic
269C2×D61—22no—
270C2×D61—22no—
271C2×D61—22no—
272C2×D61—22no—
273C2×D61—22no—
274C2×D61—44no—
275C2×D61—44no—
276C2×D61C2×#10114yes—
277C4×C61—21no—
278C4×C61—21no—
279C4×C61—21no—
280C4×C61—44no—
281C4×C61C2×#10112yes—
282C4×C61#54611yescharacteristic
283C4×C61#54611yescharacteristic
284C4×C61#54711yescharacteristic
285C4×C61#54711yescharacteristic
286Dic61—22no—
287Dic61—22no—
288Dic61—42no—
289Dic61—42no—
290C2×#554—611no—
291D4×M16—611no—
292D4×M16—611no—
293#1997—611no—
294#1997D6111yescharacteristic, complemented
295#1901—612no—
296#1901—612no—
297#1821—612no—
298#1801—611no—
299C4×#101—611no—
300C2×#343—612no—
301C2×C61—21no—
302C2×C61—21no—
303C2×C61#233911yescharacteristic
304C2×C61C22×#10111yescharacteristic
305C2×C61#288811yescharacteristic
306D61—24no—
307D61—42no—
308C22×M16—612no—
309C22×M16—612no—
310#554—1221no—
311#554—1221no—
312#554—1222no—
313#554—1222no—
314#554—1221no—
315#554—1221no—
316C2×C4⋊Q8C2×D6111yescharacteristic
317C2×#125—612no—
318C2×C22⋊Q8—614no—
319C2×C22⋊Q8—612no—
320C2×C4×Q8—611no—
321C2×C4×D4—612no—
322C2×C4×D4—611no—
323C22×C4⋊C4—612no—
324#436—1221no—
325#436—1221no—
326#435—1222no—
327#435—1222no—
328C8×D4—1221no—
329C8×D4—1221no—
330#423—611no—
331#423C2×D6111yescharacteristic
332C2×#101—611no—
333C2×#101—611no—
334C2×#101—612no—
335#407—612no—
336#407—612no—
337C4×M16—611no—
338C4×M16C2×D6111yescharacteristic
339#343—612no—
340#343—611no—
341#343—611no—
342#343—612no—
343#343—612no—
344#343C2×D6112yes—
345#343C2×D6112yes—
346C61#2248111yescharacteristic, complemented, Sylow
347C22×Q8—611no—
348C22×Q8C4×D6112yes—
349C22×D4—611no—
350C23×C4—612no—
351C23×C4—614no—
352C2×M16—611no—
353C2×M16—612no—
354C2×M16—611no—
355C2×M16—1222no—
356C2×M16—1222no—
357C2×M16—1224no—
358C2×M16—1224no—
359C2×M16—2442no—
360C2×M16—2442no—
361C2×M16C22×D6112yes—
362C2×M16C22×D6111yescharacteristic
363C2×M16C22×D6111yescharacteristic
364C22×C8—1222no—
365C22×C8—1222no—
366C4⋊Q8—21no—
367C4⋊Q8—21no—
368C4⋊Q8—21no—
369C4⋊Q8—21no—
370#125—1224no—
371#125—1222no—
372#125—1222no—
373C22⋊Q8—1224no—
374C22⋊Q8—1224no—
375C22⋊Q8—1224no—
376C22⋊Q8—1224no—
377C22⋊Q8—1222no—
378C22⋊Q8—1222no—
379C22⋊Q8—1222no—
380C22⋊Q8—1222no—
381C4×Q8—1221no—
382C4×Q8—1221no—
383C4×Q8—1221no—
384C4×Q8—1221no—
385C4×D4—611no—
386C4×D4—611no—
387C4×D4—612no—
388C4×D4—612no—
389C4×D4—611no—
390C4×D4—611no—
391C4×D4—1222no—
392C4×D4—1224no—
393C4×D4—1222no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—614no—
396C2×C4⋊C4—612no—
397C2×C4⋊C4—614no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—1222no—
400C2×C4⋊C4—1222no—
401C2×C4⋊C4—1222no—
402C2×C4⋊C4—1222no—
403C2×C4⋊C4C4×D6112yes—
404C2×C4⋊C4C22×D6112yes—
405C2×C22⋊C4—612no—
406C2×C22⋊C4—614no—
407C2×C42—612no—
408C2×C42C22×D6111yescharacteristic
409C4⋊C8—21no—
410C4⋊C8—21no—
411C4⋊C8—1221no—
412C4⋊C8—1221no—
413#101—1222no—
414#101—1222no—
415#101—2442no—
416C22⋊C8—1222no—
417C22⋊C8—1222no—
418C22⋊C8—1222no—
419C22⋊C8—1222no—
420#97—21no—
421#97—1221no—
422C4×C8—21no—
423C4×C8—1221no—
424C24—612no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—82no—
428C2×Q8—1221no—
429C2×Q8—1221no—
430C2×Q8—1221no—
431C2×Q8—1221no—
432C2×Q8—1221no—
433C2×Q8—1221no—
434C2×D4—611no—
435C2×D4—612no—
436C2×D4—611no—
437C2×D4—1224no—
438C22×C4—612no—
439C22×C4—612no—
440C22×C4—612no—
441C22×C4—612no—
442C22×C4—612no—
443C22×C4—612no—
444C22×C4—612no—
445C22×C4—1224no—
446C22×C4—1224no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—4884no—
452C22×C4C2×C4×D6111yescharacteristic
453C22×C4C2×C4×D6114yes—
454C22×C4C2×C4×D6111yescharacteristic
455C22×C4C23×D6111yescharacteristic
456M16—21no—
457M16—21no—
458M16—22no—
459M16—22no—
460M16—21no—
461M16—612no—
462M16—1221no—
463M16—1221no—
464M16—1222no—
465M16—1222no—
466M16—1221no—
467M16—2444no—
468M16—2444no—
469M16#1175822612yes—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—22no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—1222no—
476C2×C8—2444no—
477C2×C8—2444no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—612no—
483C4⋊C4—611no—
484C4⋊C4—611no—
485C4⋊C4—1224no—
486C4⋊C4—1224no—
487C4⋊C4—1224no—
488C4⋊C4—1224no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1222no—
492C4⋊C4—1222no—
493C22⋊C4—612no—
494C22⋊C4—612no—
495C22⋊C4—612no—
496C22⋊C4—612no—
497C22⋊C4—1224no—
498C22⋊C4—1224no—
499C42—1222no—
500C42—1222no—
501C42D4×D6111yescharacteristic
502C42D4×D6111yescharacteristic
503C42D4×D6111yescharacteristic
504C42D4×D6111yescharacteristic
505C23—612no—
506C23—612no—
507C23—1224no—
508C23—1222no—
509C23—1222no—
510C23—2442no—
511C23C22×C4×D6111yescharacteristic
512Q8—82no—
513Q8—82no—
514Q8—2441no—
515Q8—2441no—
516Q8—2441no—
517Q8—2441no—
518D4—614no—
519D4—1224no—
520D4—1222no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—41no—
524C2×C4—44no—
525C2×C4—612no—
526C2×C4—612no—
527C2×C4—1222no—
528C2×C4—1222no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—1222no—
532C2×C4—1222no—
533C2×C4—1222no—
534C2×C4—1222no—
535C2×C4—1222no—
536C2×C4—1222no—
537C2×C4—1222no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—4884no—
542C2×C4—4884no—
543C2×C4—2442no—
544C2×C4C22⋊C4×D6111yescharacteristic
545C2×C4C22⋊C4×D6111yescharacteristic
546C2×C4C22⋊C4×D6111yescharacteristic
547C2×C4C22⋊C4×D6111yescharacteristic
548C2×C4C2×D4×D6111yescharacteristic
549C2×C4C2×D4×D6111yescharacteristic
550C2×C4#41×D6111yescharacteristic
551C2×C4C2×#1175822611yescharacteristic
552C2×C4#1451982311yescharacteristic
553C2×C4#1451982511yescharacteristic
554C8—22no—
555C8—42no—
556C8—41no—
557C8—1222no—
558C8—2442no—
559C8—2441no—
560C22—21no—
561C22—21no—
562C22—614no—
563C22—1222no—
564C22—1222no—
565C22—1222no—
566C22—1222no—
567C22—1222no—
568C22—2444no—
569C22—2444no—
570C22C2×C22⋊C4×D6111yescharacteristic
571C22C4×#1175822611yescharacteristic
572C22#1684024411yescharacteristic
573C4—21no—
574C4—21no—
575C4—21no—
576C4—44no—
577C4—1222no—
578C4—1222no—
579C4—2442no—
580C4—2442no—
581C4#101×D6112yes—
582C4#1684040311yescharacteristic
583C4#1684040511yescharacteristic
584C4#1684041411yescharacteristic
585C4#1684041511yescharacteristic
586C2—21no—
587C2—21no—
588C2—1224no—
589C2—2442no—
590C2#2144864811yescharacteristic
591C2C2×#101×D6111yescharacteristic
592C2#2145429511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1997C2
socleC22×C61C22×C22⋊C4
prime radicalC22×D244C2×C4
cosocleC22×C4×C61C24
hypercenter#1997D61
hypocenterC61#22481
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1997

Complements

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

NH ≅ G/Ntype
4. #21446809C2semidirect
5. #21446286C2semidirect
6. #21446286C2semidirect
7. #21445898C2semidirect
8. #21445812C2semidirect
9. C4×#14519695C2semidirect
10. #343×D61C2semidirect
11. C61×#1997C2semidirect
242. D244#101semidirect
294. #1997D61semidirect
346. C61#22481semidirect

Maximal subgroups

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

subgroupindex
C2×#168404732
M16×D2442
#214545542
#214468092
#214462862
#214458982
#214458122
C4×#145196952
#343×D612
C61×#19972
#2248161