Finite Group Databasebeta

← #29650911

Subgroups — #29650911

Order 15616.

680 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455825C211yesmaximal, characteristic
2#21454767C211yesmaximal, characteristic, complemented
3D4×#11758189C211yesmaximal, characteristic
4#21444709C211yesmaximal, characteristic, complemented
5#21444708C211yesmaximal, characteristic, complemented
6#21444664C211yesmaximal, characteristic, complemented
7#21444659C211yesmaximal, characteristic, complemented
8#21444652C211yesmaximal, characteristic, complemented
9SD16×#92418C211yesmaximal, characteristic, complemented
10C8×#11758200C211yesmaximal, characteristic, complemented
11#21444588C211yesmaximal, characteristic, complemented
12#21444272C211yesmaximal, characteristic, complemented
13D4⋊C8×D61C211yesmaximal, characteristic
14#21444098C211yesmaximal, characteristic, complemented
15C61×#1582C211yesmaximal, characteristic, complemented
16C22×#11758189—21no—
17#16840316C2211yescharacteristic
18C4×D4×D61C2211yescharacteristic
19#16840094C411yescharacteristic
20#16840091C411yescharacteristic
21#16840051C411yescharacteristic
22#16840050C411yescharacteristic
23#16839966C411yescharacteristic
24#16839965C411yescharacteristic
25#16839914C411yescharacteristic
26#16839911C411yescharacteristic
27#16839890—21no—
28#16839889—21no—
29#16839889C2211yescharacteristic, complemented
30Q8×#92418C2211yescharacteristic, complemented
31C8×Dic122C2211yescharacteristic, complemented
32C4×#11758190C2211yescharacteristic
33C4×#11758201C2211yescharacteristic
34C4×#11758200C2211yescharacteristic
35#16839841C2211yescharacteristic
36#16839839C2211yescharacteristic
37#16839833—21no—
38#16839830—21no—
39D4×#92418C2211yescharacteristic
40#16839813C2211yescharacteristic
41#16839787—21no—
42#16839784C2211yescharacteristic
43C4⋊C8×D61C2211yescharacteristic
44#16839724—21no—
45#16839719—21no—
46C4×#11758189C2211yescharacteristic
47C4×C8×D61C2211yescharacteristic
48#16839513C2211yescharacteristic, complemented
49#16839512C2211yescharacteristic, complemented
50#16839512C2211yescharacteristic, complemented
51#16839494C2211yescharacteristic, complemented
52Q8⋊#92418C2211yescharacteristic, complemented
53#16839492C2211yescharacteristic, complemented
54#16839491C2211yescharacteristic
55D4⋊#92418C2211yescharacteristic
56#16839489C2211yescharacteristic
57#16839484C2211yescharacteristic, complemented
58C8×#92418C2211yescharacteristic, complemented
59C8×#92418C2211yescharacteristic, complemented
60C61×#446C2211yescharacteristic, complemented
61C4×SD16×C61C2211yescharacteristic
62C8×D4×C61C2211yescharacteristic
63C61×#334C2211yescharacteristic, complemented
64C61×Q8⋊C8C2211yescharacteristic, complemented
65C61×D4⋊C8C2211yescharacteristic
66C61×#322C2211yescharacteristic, complemented
67C2×D4×D61C2×C411yescharacteristic
68C22×C4×D61—21no—
69C2×#11758190C2×C411yescharacteristic
70C2×#11758200C2×C411yescharacteristic
71C2×#11758201C2×C411yescharacteristic
72C2×#11758189—21no—
73C2×#11758189—21no—
74C2×#11758189—21no—
75C2×#11758189—44no—
76C2×#11758189D411yescharacteristic
77C2×#11758189D411yescharacteristic
78C22×#92418—21no—
79C2×C8×D61—21no—
80C2×C8×D61—21no—
81#14519776C2×C411yescharacteristic
82#14519775C2×C411yescharacteristic
83#14519767—21no—
84#14519757C2×C411yescharacteristic
85#14519755—21no—
86#14519753C2×C411yescharacteristic
87C4×Dic122—21no—
88C4×Dic122C2311yescharacteristic
89C4×D244C2311yescharacteristic
90D4×Dic61C2×C411yescharacteristic
91#14519739C2×C411yescharacteristic
92C4×C22⋊D61—21no—
93#14519737—21no—
94#14519734—21no—
95C4⋊C4×D61C2×C411yescharacteristic
96C22⋊C4×D61—21no—
97C42×D61C2311yescharacteristic
98#14519709C2×C411yescharacteristic
99#14519708C2×C411yescharacteristic
100#14519708C2×C411yescharacteristic
101C4⋊#92418—21no—
102C4⋊#92418C2311yescharacteristic
103C4⋊#92418C2311yescharacteristic
104#14519706—21no—
105#14519706—21no—
106#14519706—21no—
107#14519706—21no—
108#14519706C2×C411yescharacteristic
109#14519706C2×C411yescharacteristic
110#14519702C2×C411yescharacteristic
111#14519701C2×C411yescharacteristic
112#14519700C2×C411yescharacteristic
113#14519699C2×C411yescharacteristic
114#14519698C2×C411yescharacteristic
115#14519697C2×C411yescharacteristic
116C22⋊#92418—21no—
117#14519689—21no—
118#14519689—21no—
119#14519689C2×C411yescharacteristic
120#14519689C2×C411yescharacteristic
121#14519687—21no—
122#14519687D411yescharacteristic
123#14519687D411yescharacteristic
124C4×#92418—21no—
125C4×#92418C2311yescharacteristic
126C4×#92418C2311yescharacteristic
127C4×#92418C2311yescharacteristic
128C8×Dic61—21no—
129C8×Dic61—21no—
130C2×SD16×C61C2×C411yescharacteristic
131C22×C8×C61—21no—
132C4×Q8×C61C2311yescharacteristic
133C4×D4×C61C2311yescharacteristic
134#106×C61C2×C411yescharacteristic
135C4⋊C8×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#103×C61C2×C411yescharacteristic
139#102×C61C2×C411yescharacteristic
140C22⋊C8×C61—21no—
141#97×C61—21no—
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144C4×C8×C61C2311yescharacteristic
145C23×D61—21no—
146C2×Q8×C61C22×C411yescharacteristic
147C2×D4×C61C22×C411yescharacteristic
148C22×C4×C61—21no—
149C2×C22⋊D61—21no—
150C22×Dic61—21no—
151D4×D61—22no—
152D4×D61—41no—
153C2×D244C22×C411yescharacteristic
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×C4×D61—21no—
157C2×C4×D61—42no—
158C2×C4×D61—42no—
159C2×C4×D61C22×C411yescharacteristic
160C2×C4×D61C2×D411yescharacteristic
161C2×C4×D61#4111yescharacteristic
162C2×Dic122—21no—
163C2×Dic122C22×C411yescharacteristic
164SD16×C61—22no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—42no—
169C2×C8×C61C22×C411yescharacteristic
170C2×C8×C61C22×C411yescharacteristic
171C2×C8×C61C2×D411yescharacteristic
172C2×C8×C61C2×D411yescharacteristic
173C4⋊C4×C61—21no—
174C4⋊C4×C61C22×C411yescharacteristic
175C4⋊C4×C61C22×C411yescharacteristic
176C22⋊C4×C61—21no—
177C42×C61—21no—
178C42×C61C2411yescharacteristic
179C22⋊Dic61—21no—
180#11758201—41no—
181#11758200—22no—
182#11758198—21no—
183#11758198—21no—
184#11758198—21no—
185#11758198—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61C22×C411yescharacteristic
188C4⋊Dic61C22×C411yescharacteristic
189#11758196—21no—
190#11758196—21no—
191#11758196—21no—
192#11758196—21no—
193#11758196—21no—
194#11758196—21no—
195C4×Dic61—21no—
196C4×Dic61C22×C411yescharacteristic
197C4×Dic61C2×D411yescharacteristic
198C4×Dic61#4111yescharacteristic
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—42no—
203C2×#92418C22×C411yescharacteristic
204C2×#92418C22×C411yescharacteristic
205C2×#92418C2×D411yescharacteristic
206C2×#92418C2×D411yescharacteristic
207#11758190—41no—
208#11758189—22no—
209#11758189—42no—
210#11758189—42no—
211#11758189—42no—
212#11758189SD1614yescomplemented
213C8×D61—42no—
214C8×D61—42no—
215C23×C61—21no—
216C22×D61—21no—
217C22×D61—42no—
218C22×D61—42no—
219C22×D61—81no—
220C22×D61C4×D411yescharacteristic
221Q8×C61—21no—
222Q8×C61—21no—
223D4×C61—21no—
224D4×C61#13112yes—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61C23×C411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C22⋊D61—42no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—42no—
238C2×Dic61C4×D411yescharacteristic
239C2×Dic61C4×D411yescharacteristic
240C2×Dic61C4×D411yescharacteristic
241D244—21no—
242D244—21no—
243C4×D61—21no—
244C4×D61—21no—
245C4×D61—41no—
246C4×D61—42no—
247C4×D61—41no—
248C4×D61—41no—
249C4×D61—42no—
250C4×D61C2×SD1612yes—
251C4×D61#13512yes—
252Dic122—21no—
253Dic122—21no—
254Dic122—81no—
255C8×C61—21no—
256C8×C61—21no—
257C8×C61—21no—
258C8×C61—41no—
259C8×C61—41no—
260C8×C61—41no—
261C8×C61C2×SD1612yes—
262#92418—22no—
263#92418—21no—
264#92418—21no—
265#92418—21no—
266#92418—41no—
267#92418—41no—
268#92418C2×SD1612yes—
269#17995—611nomaximal, Sylow
270C22×C61—22no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61C2×C4×D411yescharacteristic
274C2×D61—21no—
275C2×D61—21no—
276C2×D61—41no—
277C2×D61—41no—
278C2×D61—41no—
279C2×D61—84no—
280C2×D61C4×SD1612yes—
281C4×C61—21no—
282C4×C61—41no—
283C4×C61—41no—
284C4×C61—41no—
285C4×C61C2×#13111yescharacteristic
286C4×C61#56811yescharacteristic
287C4×C61C22×SD1611yescharacteristic
288C4×C61C2×#13511yescharacteristic
289Dic61—21no—
290Dic61—21no—
291Dic61—41no—
292Dic61—41no—
293Dic61—41no—
294Dic61C4×SD1612yes—
295#3101—611no—
296#2962—611no—
297D4×M16—611no—
298#1590—611no—
299#1590—611no—
300#1583—611no—
301#1583—611no—
302#1582—611no—
303#1582D6111yescharacteristic, complemented
304C8×SD16—611no—
305C8×SD16—611no—
306#1570—611no—
307#1493—611no—
308C2×D4⋊C8—611no—
309C8×M16—611no—
310C2×C61—42no—
311C2×C61#293311yescharacteristic
312C2×C61C2×C4×SD1611yescharacteristic
313C2×C61#295611yescharacteristic
314D61—22no—
315D61—81no—
316C22×M16—1221no—
317#533—611no—
318C2×C4×D4—611no—
319#483—611no—
320#483—611no—
321#476—611no—
322#476—611no—
323#461—611no—
324#461—611no—
325#450—611no—
326#450—611no—
327#446—611no—
328#446—1221no—
329#446—1221no—
330#446C2×D6111yescharacteristic, complemented
331C8×Q8—611no—
332C8×Q8—611no—
333C4×SD16—611no—
334C4×SD16—611no—
335C4×SD16—611no—
336C4×SD16C2×D6111yescharacteristic
337#436—611no—
338#436—611no—
339#435—1221no—
340#435—1221no—
341C8×D4—611no—
342C8×D4C2×D6111yescharacteristic
343#431—611no—
344#424—1221no—
345#423—611no—
346C2×C4⋊C8—611no—
347#407—1221no—
348#405—1221no—
349C4×M16—611no—
350C2×C4×C8—611no—
351#334—611no—
352#334—611no—
353#334—611no—
354#334C2×D6111yescharacteristic, complemented
355Q8⋊C8—611no—
356Q8⋊C8—611no—
357Q8⋊C8—611no—
358Q8⋊C8C2×D6111yescharacteristic, complemented
359D4⋊C8—611no—
360D4⋊C8—611no—
361D4⋊C8—611no—
362D4⋊C8C2×D6111yescharacteristic
363#322—611no—
364#322C2×D6111yescharacteristic, complemented
365C82—611no—
366C82—611no—
367C61#1799511yescharacteristic, complemented, Sylow
368C22×D4—611no—
369C23×C4—1221no—
370C2×SD16—611no—
371C2×SD16—611no—
372C2×SD16—611no—
373C2×SD16C4×D6111yescharacteristic
374C2×M16—611no—
375C2×M16—611no—
376C2×M16—1221no—
377C2×M16—1221no—
378C2×M16—1221no—
379C2×M16—2444no—
380C22×C8—21no—
381C22×C8—1221no—
382C22×C8—1221no—
383C22×C8—1221no—
384C4⋊Q8—611no—
385C4⋊Q8—611no—
386#125—1221no—
387C22⋊Q8—611no—
388C22⋊Q8—611no—
389C22⋊Q8—1221no—
390C4×Q8—611no—
391C4×Q8—1221no—
392C4×Q8C22×D6111yescharacteristic
393C4×D4—611no—
394C4×D4—611no—
395C4×D4—611no—
396C4×D4—1221no—
397C4×D4—1221no—
398C4×D4C22×D6111yescharacteristic
399#117—1221no—
400C2×C4⋊C4—611no—
401C2×C22⋊C4—1221no—
402C2×C42—611no—
403#106—611no—
404#106—611no—
405#106—611no—
406#106C4×D6111yescharacteristic
407C4⋊C8—21no—
408C4⋊C8—611no—
409C4⋊C8—611no—
410C4⋊C8—611no—
411C4⋊C8—611no—
412C4⋊C8—1221no—
413C4⋊C8—1221no—
414C4⋊C8—1221no—
415C4⋊C8—1221no—
416C4⋊C8—1221no—
417C4⋊C8C22×D6111yescharacteristic
418C4⋊C8C22×D6111yescharacteristic
419#103—611no—
420#103—611no—
421#103—611no—
422#103C4×D6111yescharacteristic
423#102—611no—
424#102—611no—
425#102—611no—
426#102C4×D6111yescharacteristic
427C22⋊C8—21no—
428C22⋊C8—611no—
429C22⋊C8—611no—
430C22⋊C8—1221no—
431C22⋊C8—1221no—
432C22⋊C8—1221no—
433#97—21no—
434#97—611no—
435#97—611no—
436#97—1221no—
437C4×C8—611no—
438C4×C8—611no—
439C4×C8—611no—
440C4×C8—1221no—
441C4×C8—1221no—
442C4×C8—1221no—
443C4×C8C22×D6111yescharacteristic
444C4×C8C22×D6111yescharacteristic
445C4×C8C22×D6111yescharacteristic
446C24—1221no—
447C2×Q8—611no—
448C2×Q8—1221no—
449C2×Q8C2×C4×D6111yescharacteristic
450C2×D4—611no—
451C2×D4—1221no—
452C2×D4—1222no—
453C2×D4—2441no—
454C2×D4C2×C4×D6111yescharacteristic
455C22×C4—21no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—2442no—
464C22×C4—2442no—
465SD16—22no—
466SD16—1222no—
467SD16—2441no—
468SD16—2441no—
469M16—614no—
470M16—1222no—
471M16—2442no—
472M16—2442no—
473M16—2442no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—42no—
478C2×C8—611no—
479C2×C8—611no—
480C2×C8—611no—
481C2×C8—611no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—2442no—
486C2×C8—2442no—
487C2×C8—2442no—
488C2×C8C2×C4×D6111yescharacteristic
489C2×C8C2×C4×D6111yescharacteristic
490C2×C8D4×D6111yescharacteristic
491C2×C8D4×D6111yescharacteristic
492C4⋊C4—21no—
493C4⋊C4—611no—
494C4⋊C4—611no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4C2×C4×D6111yescharacteristic
503C4⋊C4C2×C4×D6111yescharacteristic
504C22⋊C4—21no—
505C22⋊C4—1221no—
506C22⋊C4—1221no—
507C22⋊C4—1221no—
508C22⋊C4—1221no—
509C22⋊C4—1221no—
510C42—21no—
511C42—611no—
512C42—611no—
513C42—611no—
514C42—1221no—
515C42C23×D6111yescharacteristic
516C23—21no—
517C23—611no—
518C23—1221no—
519C23—4881no—
520C23—2442no—
521C23—2442no—
522Q8—21no—
523Q8—21no—
524Q8—1221no—
525Q8—1221no—
526Q8—4881no—
527D4—21no—
528D4—1221no—
529D4—1221no—
530D4—2442no—
531D4#1451978512yes—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—611no—
538C2×C4—612no—
539C2×C4—612no—
540C2×C4—611no—
541C2×C4—611no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—2441no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4—2442no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4C22×C4×D6111yescharacteristic
554C2×C4C2×D4×D6111yescharacteristic
555C2×C4#41×D6111yescharacteristic
556C8—21no—
557C8—21no—
558C8—21no—
559C8—41no—
560C8—41no—
561C8—41no—
562C8—612no—
563C8—1222no—
564C8—1221no—
565C8—1221no—
566C8—1221no—
567C8—2441no—
568C8—2441no—
569C8SD16×D6112yes—
570C22—22no—
571C22—41no—
572C22—41no—
573C22—612no—
574C22—1221no—
575C22—1221no—
576C22—4884no—
577C22—2441no—
578C22—2441no—
579C22—2441no—
580C22C4×D4×D6111yescharacteristic
581C4—21no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—612no—
586C4—1221no—
587C4—1221no—
588C4—2441no—
589C4—2441no—
590C4—2441no—
591C4C2×#1451978511yescharacteristic
592C4#1684052711yescharacteristic
593C4C2×SD16×D6111yescharacteristic
594C4#135×D6111yescharacteristic
595C2—42no—
596C2—1222no—
597C2—4881no—
598C2#2145454511yescharacteristic
599C2C4×SD16×D6111yescharacteristic
600C2#2145472711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1582C2
socleC22×C61C2×C4×D4
prime radicalC2×D4×D61C2×C4
cosocleC42×C61C24
hypercenter#1582D61
hypocenterC61#17995
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1582

Complements

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

NH ≅ G/Ntype
2. #21454767C2semidirect
4. #21444709C2semidirect
5. #21444708C2semidirect
6. #21444664C2semidirect
7. #21444659C2semidirect
8. #21444652C2semidirect
9. SD16×#92418C2semidirect
10. C8×#11758200C2semidirect
11. #21444588C2semidirect
12. #21444272C2semidirect
14. #21444098C2semidirect
15. C61×#1582C2semidirect
29. #16839889C22semidirect
30. Q8×#92418C22semidirect
31. C8×Dic122C22semidirect
48. #16839513C22semidirect
49. #16839512C22semidirect
50. #16839512C22semidirect
51. #16839494C22semidirect
52. Q8⋊#92418C22semidirect
53. #16839492C22semidirect
57. #16839484C22semidirect
58. C8×#92418C22semidirect
59. C8×#92418C22semidirect
60. C61×#446C22semidirect
63. C61×#334C22semidirect
64. C61×Q8⋊C8C22semidirect
66. C61×#322C22semidirect
212. #11758189SD16semidirect
303. #1582D61semidirect
330. #446C2×D61semidirect
354. #334C2×D61semidirect
358. Q8⋊C8C2×D61semidirect
364. #322C2×D61semidirect
367. C61#17995semidirect

Maximal subgroups

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

subgroupindex
#214558252
#214547672
D4×#117581892
#214447092
#214447082
#214446642
#214446592
#214446522
SD16×#924182
C8×#117582002
#214445882
#214442722
D4⋊C8×D612
#214440982
C61×#15822
#1799561