Finite Group Databasebeta

← #30620788

Subgroups — #30620788

Order 15616.

778 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462330C211yesmaximal, characteristic
2#21462329C211yesmaximal, characteristic
3#21461922C212yesmaximal, complemented
4#21461883C212yesmaximal, complemented
5#21461880C212yesmaximal, complemented
6#21461724C211yesmaximal, characteristic
7#21461724C211yesmaximal, characteristic
8#21461695C211yesmaximal, characteristic
9#106×#6923C212yesmaximal, complemented
10C8×C22⋊#6923C211yesmaximal, characteristic
11#427×D61C211yesmaximal, characteristic
12C2×#14519845C2212yes—
13D61⋊(C22⋊C8)C2211yescharacteristic
14C2×C8×#6923C2212yes—
15#16840713C2212yes—
16#16840713C2212yes—
17#16840712—22no—
18#16840712—22no—
19#16840707—21no—
20#16840706—21no—
21#16840704C2211yescharacteristic
22#16840704C2211yescharacteristic
23C4⋊C4×#6923C2212yes—
24C4⋊C4×#6923C2212yes—
25C4×C4⋊#6923—21no—
26C4×C22⋊#6923C2211yescharacteristic
27#16840698—21no—
28#16840636C2212yes—
29#16840635C2212yes—
30#16840635C2212yes—
31#16840635C2212yes—
32#16840635C2212yes—
33(C2×#6923)⋊C8C2212yes—
34C22×C8×D61C2211yescharacteristic
35#117×D61C2211yescharacteristic
36#117×D61C2211yescharacteristic
37#16839800C411yescharacteristic, complemented
38#16839799C411yescharacteristic, complemented
39#16839799C411yescharacteristic, complemented
40#16839798C411yescharacteristic, complemented
41#16839798C411yescharacteristic, complemented
42#16839797C411yescharacteristic, complemented
43#16839797C411yescharacteristic, complemented
44#107×D61C2211yescharacteristic
45#107×D61C2211yescharacteristic
46#106×D61C2211yescharacteristic
47#106×D61C2211yescharacteristic
48C61×#427C411yescharacteristic, complemented
49C2×D61⋊C8C2312yes—
50C2×C4⋊#6923—22no—
51C2×C4⋊#6923—22no—
52C2×C4⋊#6923C2312yes—
53C2×C4⋊#6923C2312yes—
54C2×C22⋊#6923—21no—
55C2×C22⋊#6923—21no—
56C2×C22⋊#6923D412yes—
57C2×C4×#6923—22no—
58C2×C4×#6923—22no—
59C2×C4×#6923C2312yes—
60#14519845—24no—
61#14519834—21no—
62C22⋊#92420—21no—
63C8×#6923—24no—
64(C2×#6923)⋊C4—22no—
65(C2×#6923)⋊C4—22no—
66(C2×#6923)⋊C4—22no—
67(C2×#6923)⋊C4—22no—
68(C2×#6923)⋊C4—21no—
69(C2×#6923)⋊C4—21no—
70(C2×#6923)⋊C4D411yescharacteristic
71(C2×#6923)⋊C4D411yescharacteristic
72C22×C4×D61C2311yescharacteristic
73C22×#92418C2×C411yescharacteristic
74C2×C8×D61—24no—
75C2×C8×D61C2×C411yescharacteristic
76C2×C8×D61C2×C411yescharacteristic
77C2×C8×D61C2311yescharacteristic
78C2×C8×D61C2311yescharacteristic
79#14519736C2×C411yescharacteristic
80#14519736C2×C411yescharacteristic
81#14519735C2×C411yescharacteristic
82#14519735C2×C411yescharacteristic
83#14519735C2×C411yescharacteristic
84#14519735C2×C411yescharacteristic
85#14519734—21no—
86#14519734—21no—
87#14519733—21no—
88#14519733—21no—
89C4⋊C4×D61C2311yescharacteristic
90C4⋊C4×D61C2311yescharacteristic
91C4⋊C4×D61C2311yescharacteristic
92C4⋊C4×D61C2311yescharacteristic
93C22⋊C4×D61—21no—
94C22⋊C4×D61—21no—
95C42×D61—21no—
96C42×D61—21no—
97#14519711C2×C411yescharacteristic
98#14519711C2×C411yescharacteristic
99#14519710C2×C411yescharacteristic
100#14519710C2×C411yescharacteristic
101#14519710C2×C411yescharacteristic
102#14519710C2×C411yescharacteristic
103#14519709C2×C411yescharacteristic
104#14519709C2×C411yescharacteristic
105#14519708C2×C411yescharacteristic
106#14519708C2×C411yescharacteristic
107#14519708C2×C411yescharacteristic
108#14519708C2×C411yescharacteristic
109C22×C8×C61C2×C411yescharacteristic
110#117×C61C2×C411yescharacteristic
111#117×C61C2×C411yescharacteristic
112#107×C61C2×C411yescharacteristic
113#107×C61C2×C411yescharacteristic
114#106×C61C2×C411yescharacteristic
115#106×C61C2×C411yescharacteristic
116C23×D61C2×D411yescharacteristic
117C22×#6923—22no—
118C22×#6923—22no—
119C22×#6923C2×D412yes—
120C22×#6923C2×D412yes—
121C22×C4×C61C22×C411yescharacteristic
122C22×Dic61#4111yescharacteristic
123C2×C4×D61—22no—
124C2×C4×D61—22no—
125C2×C4×D61—21no—
126C2×C4×D61—21no—
127C2×C4×D61—21no—
128C2×C4×D61—21no—
129C2×C4×D61C22×C411yescharacteristic
130C2×C4×D61C2×D411yescharacteristic
131C2×C4×D61#4111yescharacteristic
132C2×C4×D61C2411yescharacteristic
133C22⋊#6923—24no—
134C22⋊#6923—81no—
135C22⋊#6923—81no—
136C2×#92420—22no—
137C4⋊#6923—22no—
138C4⋊#6923—22no—
139C4⋊#6923—22no—
140C4⋊#6923—22no—
141C4⋊#6923—44no—
142C4⋊#6923—82no—
143C4×#6923—22no—
144C4×#6923—22no—
145C4×#6923—44no—
146C4×#6923—82no—
147D61⋊C8—22no—
148D61⋊C8—22no—
149C2×C8×C61—22no—
150C2×C8×C61C22×C411yescharacteristic
151C2×C8×C61C22×C411yescharacteristic
152C4⋊C4×C61C22×C411yescharacteristic
153C4⋊C4×C61C22×C411yescharacteristic
154C4⋊C4×C61C22×C411yescharacteristic
155C4⋊C4×C61C22×C411yescharacteristic
156C22⋊C4×C61—21no—
157C22⋊C4×C61—21no—
158C42×C61—21no—
159C42×C61—21no—
160C22⋊Dic61—21no—
161C22⋊Dic61—21no—
162#11758198—21no—
163#11758198—21no—
164#11758198—21no—
165#11758198—21no—
166C4⋊Dic61C22×C411yescharacteristic
167C4⋊Dic61C22×C411yescharacteristic
168C4⋊Dic61C22×C411yescharacteristic
169C4⋊Dic61C22×C411yescharacteristic
170#11758196—21no—
171#11758196—21no—
172#11758196—21no—
173#11758196—21no—
174C4×Dic61—21no—
175C4×Dic61—21no—
176C4×Dic61—21no—
177C4×Dic61—21no—
178C4×Dic61—21no—
179C4×Dic61—21no—
180C2×#92418—22no—
181C2×#92418C22×C411yescharacteristic
182C2×#92418C22×C411yescharacteristic
183C8×D61—22no—
184C8×D61—22no—
185C8×D61—22no—
186C8×D61#4114yes—
187C23×C61C4×D411yescharacteristic
188C22×D61—22no—
189C22×D61—22no—
190C22×D61—21no—
191C22×D61—21no—
192C22×D61C4×D411yescharacteristic
193C22×D61C22×D411yescharacteristic
194C2×#6923—22no—
195C2×#6923—22no—
196C2×#6923—22no—
197C2×#6923—22no—
198C2×#6923—22no—
199C2×#6923—42no—
200C2×#6923—42no—
201C2×#6923—42no—
202C2×#6923—42no—
203C2×#6923—42no—
204C2×#6923—42no—
205C2×#6923#13514yes—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61C4×D411yescharacteristic
213C2×C4×C61C23×C411yescharacteristic
214C2×Dic61—22no—
215C2×Dic61—21no—
216C2×Dic61—21no—
217C2×Dic61—21no—
218C2×Dic61—21no—
219C2×Dic61C4×D411yescharacteristic
220C2×Dic61C2×#4111yescharacteristic
221C4×D61—22no—
222C4×D61—21no—
223C4×D61—21no—
224C4×D61—22no—
225C4×D61—42no—
226C4×D61—42no—
227C4×D61—41no—
228C4×D61—41no—
229C4×D61—41no—
230C4×D61—41no—
231C4×D61C2×#4112yes—
232C4×D612−1+412yes—
233#92420—42no—
234C8×C61—21no—
235C8×C61#11712yes—
236#92418—21no—
237#92418#11712yes—
238#22610—611nomaximal, Sylow
239C22×C61—21no—
240C22×C61—21no—
241C22×C61—21no—
242C22×C61C2×C4×D411yescharacteristic
243C2×D61—21no—
244C2×D61—21no—
245C2×D61—21no—
246C2×D61—42no—
247C2×D61—42no—
248C2×D61#54712yes—
249C2×D61C2×#13512yes—
250C2×D61#57711yescharacteristic
251C2×D61#57711yescharacteristic
252#6923—24no—
253#6923—42no—
254#6923—82no—
255#6923—82no—
256C4×C61—21no—
257C4×C61—41no—
258C4×C61—41no—
259C4×C61—41no—
260C4×C61—41no—
261C4×C61C2×#11711yescharacteristic
262C4×C61#51911yescharacteristic
263Dic61—21no—
264Dic61—41no—
265Dic61—41no—
266Dic61—41no—
267Dic61—41no—
268Dic61#55212yes—
269C2×#427—611no—
270#2347—611no—
271#2346—611no—
272#1941—612no—
273#1921—612no—
274#1918—612no—
275#1825—611no—
276#1825—611no—
277#1807—611no—
278C4×#106—612no—
279C8×C22⋊C4—611no—
280C2×C61—41no—
281C2×C61#234411yescharacteristic
282C2×C61C4×#13511yescharacteristic
283C2×C61#295211yescharacteristic
284D61—41no—
285D61#333712yes—
286D61#334012yes—
287C23×C8—611no—
288C2×#117—611no—
289C2×#117—611no—
290#427—611no—
291#427—611no—
292#427—611no—
293#427—611no—
294#427—611no—
295#427—611no—
296#427—611no—
297#427#692311yescharacteristic, complemented
298C2×#107—611no—
299C2×#107—611no—
300C2×#106—611no—
301C2×#106—611no—
302C2×#106—612no—
303C2×C22⋊C8—611no—
304C2×C4×C8—612no—
305#389—612no—
306#389—612no—
307#388—1222no—
308#388—1222no—
309#383—1221no—
310#382—1221no—
311#380—611no—
312#380—611no—
313C4×C4⋊C4—612no—
314C4×C4⋊C4—612no—
315C4×C4⋊C4—1221no—
316C4×C22⋊C4—611no—
317#376—1221no—
318#340—612no—
319#340—612no—
320#340—612no—
321#340—612no—
322#340—612no—
323#336—612no—
324C61#2261011yescharacteristic, complemented, Sylow
325C23×C4—611no—
326C22×C8—611no—
327C22×C8—611no—
328C22×C8—611no—
329C22×C8—611no—
330C22×C8—611no—
331C22×C8—612no—
332C22×C8—1224no—
333C22×C8C2×#692311yescharacteristic
334#117—611no—
335#117—611no—
336#117—611no—
337#117—611no—
338#117—611no—
339#117—611no—
340#117—1221no—
341#117—1221no—
342#117—1221no—
343#117—1221no—
344#117C2×#692311yescharacteristic
345#117C2×#692311yescharacteristic
346C2×C4⋊C4—611no—
347C2×C4⋊C4—611no—
348C2×C4⋊C4—611no—
349C2×C4⋊C4—611no—
350C2×C4⋊C4—612no—
351C2×C4⋊C4—612no—
352C2×C4⋊C4—1222no—
353C2×C4⋊C4—1222no—
354C2×C22⋊C4—612no—
355C2×C22⋊C4—1221no—
356C2×C22⋊C4—1221no—
357C2×C22⋊C4—1221no—
358C2×C22⋊C4—1221no—
359C2×C42—612no—
360C2×C42—1221no—
361C2×C42—1221no—
362C2×C42—1222no—
363C2×C42—1222no—
364#107—611no—
365#107—611no—
366#107—611no—
367#107—611no—
368#107—611no—
369#107—611no—
370#107C2×#692311yescharacteristic
371#107C2×#692311yescharacteristic
372#106—611no—
373#106—611no—
374#106—611no—
375#106—611no—
376#106—611no—
377#106—611no—
378#106—1224no—
379#106C2×#692311yescharacteristic
380#106C2×#692311yescharacteristic
381C22⋊C8—1221no—
382C22⋊C8—1221no—
383C4×C8—1224no—
384#95—611no—
385#95—611no—
386#95—1221no—
387#95—1222no—
388#95—1222no—
389#95—1222no—
390#95—1221no—
391#95—1222no—
392C24—611no—
393C22×C4—611no—
394C22×C4—612no—
395C22×C4—611no—
396C22×C4—611no—
397C22×C4—611no—
398C22×C4—611no—
399C22×C4—612no—
400C22×C4—1222no—
401C22×C4—1222no—
402C22×C4—1221no—
403C22×C4—1221no—
404C22×C4—1221no—
405C22×C4—1221no—
406C22×C4—1222no—
407C22×C4—1222no—
408C22×C4C22×#692311yescharacteristic
409C2×C8—22no—
410C2×C8—611no—
411C2×C8—614no—
412C2×C8—611no—
413C2×C8—1222no—
414C2×C8—1222no—
415C2×C8—1222no—
416C2×C8—1222no—
417C2×C8—1222no—
418C2×C8—1222no—
419C2×C8—1222no—
420C2×C8C22×#692311yescharacteristic
421C2×C8C22×#692311yescharacteristic
422C4⋊C4—611no—
423C4⋊C4—611no—
424C4⋊C4—611no—
425C4⋊C4—611no—
426C4⋊C4—1221no—
427C4⋊C4—1221no—
428C4⋊C4—1221no—
429C4⋊C4—1221no—
430C4⋊C4—1222no—
431C4⋊C4—1222no—
432C4⋊C4—1222no—
433C4⋊C4—1222no—
434C4⋊C4—4882no—
435C4⋊C4—2444no—
436C4⋊C4C22×#692311yescharacteristic
437C4⋊C4C22×#692311yescharacteristic
438C4⋊C4C22×#692311yescharacteristic
439C4⋊C4C22×#692311yescharacteristic
440C22⋊C4—21no—
441C22⋊C4—21no—
442C22⋊C4—1221no—
443C22⋊C4—1221no—
444C22⋊C4—1221no—
445C22⋊C4—1221no—
446C22⋊C4—1224no—
447C22⋊C4—1221no—
448C22⋊C4—1221no—
449C22⋊C4—4881no—
450C22⋊C4—4881no—
451C42—21no—
452C42—21no—
453C42—1221no—
454C42—1221no—
455C42—1221no—
456C42—1221no—
457C42—1222no—
458C42—1222no—
459C42—1221no—
460C42—1221no—
461C42—4882no—
462C42—2444no—
463C23—611no—
464C23—611no—
465C23—1222no—
466C23—1221no—
467C23—1221no—
468C23—1222no—
469C23D4×#692311yescharacteristic
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—611no—
477C2×C4—612no—
478C2×C4—612no—
479C2×C4—614no—
480C2×C4—611no—
481C2×C4—1222no—
482C2×C4—1222no—
483C2×C4—1222no—
484C2×C4—1222no—
485C2×C4—1222no—
486C2×C4—1221no—
487C2×C4—1221no—
488C2×C4—1222no—
489C2×C4—1222no—
490C2×C4—1222no—
491C2×C4—1221no—
492C2×C4—1221no—
493C2×C4—1221no—
494C2×C4—1221no—
495C2×C4—2441no—
496C2×C4—2442no—
497C2×C4—2441no—
498C2×C4—2442no—
499C2×C4—2441no—
500C2×C4—2441no—
501C2×C4—2442no—
502C2×C4—2442no—
503C2×C4—2442no—
504C2×C4—2442no—
505C2×C4—2442no—
506C2×C4—2442no—
507C2×C4D4×#692311yescharacteristic
508C2×C4C23×#692311yescharacteristic
509C8—21no—
510C8—612no—
511C8—1221no—
512C8—2442no—
513C8#1451985612yes—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—612no—
518C22—611no—
519C22—611no—
520C22—612no—
521C22—1221no—
522C22—1221no—
523C22—1221no—
524C22—2442no—
525C22—2442no—
526C22C2×D4×#692311yescharacteristic
527C4—21no—
528C4—41no—
529C4—41no—
530C4—41no—
531C4—41no—
532C4—612no—
533C4—1224no—
534C4—1221no—
535C4—4882no—
536C4—4882no—
537C4—2442no—
538C4—2441no—
539C4—2441no—
540C4—2441no—
541C4—2441no—
542C4C2×#1451985611yescharacteristic
543C4#1684083011yescharacteristic
544C2—41no—
545C2—612no—
546C2—612no—
547C2—2441no—
548C2#2146232711yescharacteristic
549C2#135×#692311yescharacteristic
550C2#2146258211yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#427C4
socleC22×C61C2×C4×D4
prime radicalC23×D61C2×D4
cosocleC2×C4×D61C24
hypercenter#427#6923
hypocenterC61#22610
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
3. #21461922C2semidirect
4. #21461883C2semidirect
5. #21461880C2semidirect
9. #106×#6923C2semidirect
37. #16839800C4semidirect
38. #16839799C4semidirect
39. #16839799C4semidirect
40. #16839798C4semidirect
41. #16839798C4semidirect
42. #16839797C4semidirect
43. #16839797C4semidirect
48. C61×#427C4semidirect
297. #427#6923semidirect
324. C61#22610semidirect

Maximal subgroups

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

subgroupindex
#214623302
#214623292
#214619222
#214618832
#214618802
#214617242
#214616952
#106×#69232
C8×C22⋊#69232
#427×D612
#2261061