Finite Group Databasebeta

← #30615926

Subgroups — #30615926

Order 15616.

1026 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840777C211yesmaximal, characteristic, complemented
2C2×#16840760C211yesmaximal, characteristic
3C2×#16840736C211yesmaximal, characteristic
4#21462379C211yesmaximal, characteristic, complemented
5#21461901C214yesmaximal, complemented
6#21461900C214yesmaximal, complemented
7#21461499C211yesmaximal, characteristic, complemented
8#21461487C211yesmaximal, characteristic, complemented
9#21454333C211yesmaximal, characteristic
10C22×#11758213C2211yescharacteristic
11C2×D61⋊M16C2211yescharacteristic
12C22×D61⋊C8C2211yescharacteristic
13C2×#14519856C2211yescharacteristic
14#16840777—24no—
15C2×#14519848C2212yescomplemented
16C2×#14519847C2212yescomplemented
17#16840760—24no—
18C2×#14519841C2212yes—
19C2×#14519841C2212yes—
20#16840736—21no—
21#16840736—21no—
22#16840736C2214yes—
23C2×C22⋊#92420—22no—
24#16840727C2212yescomplemented
25#16840727C2212yescomplemented
26#16840726—22no—
27#16840721—21no—
28C4×D61⋊C8—21no—
29#16840632C2214yescomplemented
30(C2×#6923)⋊C8C2214yescomplemented
31C22×#11758224C2211yescharacteristic
32C2×M16×D61C2211yescharacteristic
33#117×D61C2211yescharacteristic
34#16839772C411yescharacteristic, complemented
35#16839770C411yescharacteristic, complemented
36#16839767C411yescharacteristic, complemented
37#16839760C414yescomplemented
38#16839755C2214yes—
39C61×#418C411yescharacteristic, complemented
40C2×#11758213—22no—
41C2×#11758213—22no—
42C2×#11758213—21no—
43C2×#11758213C2312yes—
44C2×#11758213C2311yescharacteristic
45C2×#11758213C2311yescharacteristic
46D61⋊M16—22no—
47D61⋊M16D414yes—
48C2×#11758217—21no—
49C2×D61⋊C8—21no—
50C2×D61⋊C8—22no—
51C2×D61⋊C8—22no—
52C2×D61⋊C8D411yescharacteristic
53C2×D61⋊C8D411yescharacteristic
54C2×D61⋊C8C2312yes—
55C22×#92420—21no—
56C22×#92420D412yes—
57#14519856—22no—
58#14519856—22no—
59C2×C4⋊#6923—21no—
60C2×C22⋊#6923—21no—
61C2×C4×#6923C2312yes—
62#14519848—24no—
63#14519847—24no—
64#14519841—24no—
65#14519841—24no—
66C22⋊#92420—24no—
67C22⋊#92420—24no—
68#14519832—22no—
69#14519832—22no—
70C4×#92420—22no—
71C4×#92420—22no—
72C2×#11758224—22no—
73C2×#11758224—21no—
74C2×#11758224—22no—
75C2×#11758224—22no—
76C2×#11758224—22no—
77C2×#11758224—21no—
78C2×#11758224C2×C412yes—
79C2×#11758224C2312yes—
80C22×Dic122C2×C411yescharacteristic
81C22×C22⋊D61—21no—
82C23×Dic61—21no—
83C22×C4×D61C2311yescharacteristic
84C2×#11758194C2×C411yescharacteristic
85C2×#11758189C2×C412yes—
86M16×D61—22no—
87M16×D61—22no—
88C2×C8×D61C2312yes—
89#14519736C2×C411yescharacteristic
90#14519735C2312yes—
91#14519734—21no—
92#14519733—21no—
93C4⋊C4×D61C2×C412yes—
94C22⋊C4×D61—21no—
95C42×D61—21no—
96#14519702C2×C414yes—
97#14519700C2×C414yes—
98#14519698C2×C414yes—
99C22×D4×C61C2×C411yescharacteristic
100C2×M16×C61C2×C411yescharacteristic
101#117×C61C2×C411yescharacteristic
102#102×C61C2×C414yes—
103C24×C61—21no—
104C23×D61#4111yescharacteristic
105C22×#6923—22no—
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61—21no—
109C2×D4×C61C22×C412yes—
110C22×C4×C61C22×C411yescharacteristic
111C2×C22⋊D61—22no—
112C2×C22⋊D61—22no—
113C2×C22⋊D61—42no—
114C2×C22⋊D61—42no—
115C22×Dic61—21no—
116C22×Dic61—22no—
117C22×Dic61—22no—
118C22×Dic61—42no—
119C22×Dic61—42no—
120C22×Dic61C2×D411yescharacteristic
121#11758224—24no—
122#11758224—24no—
123#11758224—22no—
124#11758224—22no—
125#11758224—22no—
126#11758224—22no—
127#11758224—42no—
128#11758224—42no—
129#11758224—42no—
130#11758224—42no—
131C2×C4×D61—22no—
132C2×C4×D61C22×C411yescharacteristic
133C2×C4×D61C2×D412yes—
134C2×C4×D61C2×D411yescharacteristic
135C2×C4×D61#4112yes—
136C2×C4×D61#4112yes—
137C2×C4×D61#4112yes—
138C2×C4×D61#4111yescharacteristic
139C2×C4×D61C2411yescharacteristic
140C2×Dic122—22no—
141C2×Dic122—22no—
142C2×Dic122—21no—
143C2×Dic122C22×C412yes—
144C22⋊#6923—42no—
145#11758217—24no—
146C2×#92420—24no—
147C2×#92420—21no—
148C2×#92420—21no—
149C2×#92420—22no—
150C2×#92420—42no—
151C2×#92420C2×D412yes—
152C2×#92420C2×D412yes—
153C4⋊#6923—42no—
154C4×#6923—22no—
155C4×#6923—22no—
156#11758213—21no—
157#11758213—21no—
158#11758213—22no—
159#11758213—22no—
160#11758213—42no—
161#11758213C2×D412yes—
162#11758213C2×D412yes—
163D61⋊C8—24no—
164D61⋊C8—22no—
165D61⋊C8—22no—
166D61⋊C8—22no—
167D61⋊C8—22no—
168M16×C61—22no—
169C2×C8×C61C22×C412yes—
170C4⋊C4×C61C22×C412yes—
171C22⋊C4×C61—21no—
172C42×C61—21no—
173C22⋊Dic61—21no—
174#11758198—22no—
175C4⋊Dic61C22×C412yes—
176#11758196—22no—
177C4×Dic61—21no—
178C4×Dic61—22no—
179#11758194—22no—
180C2×#92418C22×C412yes—
181#11758189—24no—
182C8×D61—22no—
183C8×D61—22no—
184C23×C61—22no—
185C23×C61—44no—
186C23×C61—42no—
187C23×C61C4×D411yescharacteristic
188C22×D61—42no—
189C22×D61C4×D411yescharacteristic
190C22×D61#12412yes—
191C22×D61#12612yes—
192C22×D61C2×#4111yescharacteristic
193C2×#6923—44no—
194C2×#6923—42no—
195D4×C61—22no—
196D4×C61—22no—
197D4×C61—42no—
198C2×C4×C61—22no—
199C2×C4×C61#11712yes—
200C2×C4×C61C4×D412yes—
201C2×C4×C61C4×D411yescharacteristic
202C2×C4×C61C23×C411yescharacteristic
203C22⋊D61—42no—
204C22⋊D61—42no—
205C22⋊D61—44no—
206C2×Dic61—22no—
207C2×Dic61—22no—
208C2×Dic61—42no—
209C2×Dic61—42no—
210C2×Dic61—42no—
211C2×Dic61—44no—
212C2×Dic61C4×D411yescharacteristic
213C2×Dic61#12112yes—
214C2×Dic61#12312yes—
215C2×Dic61C22×D411yescharacteristic
216C4×D61—44no—
217C4×D61#11712yes—
218C4×D61C4×D412yes—
219C4×D61#12112yes—
220C4×D61#12312yes—
221C4×D61#12412yes—
222C4×D61#12612yes—
223C4×D61C22×D411yescharacteristic
224C4×D61C2×#4111yescharacteristic
225C4×D61C2×#4111yescharacteristic
226C4×D61C2×#4111yescharacteristic
227Dic122—22no—
228Dic122—22no—
229Dic122—42no—
230#92420—22no—
231#92420—22no—
232#92420—22no—
233#92420—22no—
234#92420—42no—
235C8×C61—22no—
236#92418—22no—
237#21070—611nomaximal, Sylow
238C22×C61—22no—
239C22×C61—44no—
240C22×C61—42no—
241C22×C61—84no—
242C22×C61#38812yes—
243C22×C61C2×C4×D411yescharacteristic
244C2×D61—22no—
245C2×D61—22no—
246C2×D61—42no—
247C2×D61—42no—
248C2×D61#38812yes—
249C2×D61C2×#12411yescharacteristic
250C2×D61C2×#12611yescharacteristic
251#6923—82no—
252C4×C61—42no—
253C4×C61#38812yes—
254C4×C61C2×#11711yescharacteristic
255C4×C61C2×C4×D411yescharacteristic
256Dic61—42no—
257Dic61—42no—
258Dic61#38812yes—
259Dic61C2×#12111yescharacteristic
260Dic61C2×#12311yescharacteristic
261C2×#430—611no—
262C2×#421—611no—
263#2896—611no—
264C2×#408—611no—
265C2×#405—611no—
266#1929—614no—
267#1929—614no—
268#1733—611no—
269#1729—611no—
270C2×C61—22no—
271C2×C61—82no—
272C2×C61C2×#38811yescharacteristic
273C2×C61#295512yes—
274D61—21no—
275D61—21no—
276D61—22no—
277C22×#41—611no—
278C22×M16—611no—
279C22×M16—611no—
280C22×M16—611no—
281C23×C8—611no—
282C2×#117—611no—
283C2×#117—611no—
284#430—1224no—
285C2×#108—612no—
286C2×#108—612no—
287#421—1224no—
288C2×#104—612no—
289C2×#104—612no—
290#419—611no—
291#419—611no—
292#418—611no—
293#418#692311yescharacteristic, complemented
294#417—614no—
295#416—614no—
296#408—614no—
297#408—1221no—
298#408—1221no—
299C2×C22⋊C8—1222no—
300#405—612no—
301#405—612no—
302#405—1222no—
303C2×#97—1221no—
304C2×C4×C8—1221no—
305#339—614no—
306#336—614no—
307C61#2107011yescharacteristic, complemented, Sylow
308C2×#41—612no—
309C2×#41—612no—
310C2×#41—1221no—
311C2×#41—1221no—
312C2×#41—1222no—
313C2×#41—1222no—
314C2×#41—1222no—
315C2×#41—1222no—
316C22×Q8—611no—
317C22×D4—1221no—
318C22×D4C2×#692311yescharacteristic
319C23×C4—611no—
320C23×C4—1221no—
321C2×M16—611no—
322C2×M16—612no—
323C2×M16—614no—
324C2×M16—611no—
325C2×M16—611no—
326C2×M16—612no—
327C2×M16—1222no—
328C2×M16—1222no—
329C2×M16—1221no—
330C2×M16—1222no—
331C2×M16—1221no—
332C2×M16—1222no—
333C2×M16—1222no—
334C2×M16C2×#692311yescharacteristic
335C22×C8—612no—
336C22×C8—611no—
337C22×C8—611no—
338C22×C8—612no—
339C22×C8—612no—
340C22×C8—1222no—
341C22×C8—1222no—
342C22×C8—1221no—
343C22×C8—1221no—
344#117—611no—
345#117—612no—
346#117—1221no—
347#117—1221no—
348#117—1222no—
349#117—1222no—
350#117C2×#692311yescharacteristic
351C2×C4⋊C4—612no—
352C2×C4⋊C4—1221no—
353C2×C22⋊C4—1221no—
354C2×C22⋊C4—1221no—
355C2×C42—612no—
356C2×C42—1221no—
357#108—1224no—
358#108—1224no—
359#104—1224no—
360#104—1224no—
361#103—614no—
362#103—614no—
363#102—614no—
364#102C2×#692314yes—
365C22⋊C8—1224no—
366C22⋊C8—1224no—
367#97—1222no—
368#97—1222no—
369C4×C8—1222no—
370C4×C8—1222no—
371C24—21no—
372C24—611no—
373#41—1222no—
374#41—1224no—
375#41—1222no—
376#41—1222no—
377#41—1224no—
378#41—1222no—
379#41—2442no—
380#41—2442no—
381#41—2442no—
382#41—2442no—
383C2×Q8—612no—
384C2×Q8—1221no—
385C2×Q8—1222no—
386C2×Q8—1222no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—21no—
390C2×D4—1222no—
391C2×D4—1222no—
392C2×D4—2442no—
393C2×D4—2442no—
394C2×D4C22×#692312yes—
395C22×C4—611no—
396C22×C4—612no—
397C22×C4—612no—
398C22×C4—611no—
399C22×C4—611no—
400C22×C4—612no—
401C22×C4—612no—
402C22×C4—611no—
403C22×C4—611no—
404C22×C4—1222no—
405C22×C4—1222no—
406C22×C4—1222no—
407C22×C4—1221no—
408C22×C4—1222no—
409C22×C4—2442no—
410C22×C4—2442no—
411C22×C4C22×#692311yescharacteristic
412M16—22no—
413M16—612no—
414M16—612no—
415M16—1224no—
416M16—1221no—
417M16—1221no—
418M16—1224no—
419M16—1222no—
420M16—1222no—
421M16—1222no—
422M16—2442no—
423C2×C8—612no—
424C2×C8—612no—
425C2×C8—612no—
426C2×C8—1222no—
427C2×C8—1222no—
428C2×C8—1221no—
429C2×C8—1221no—
430C2×C8—1222no—
431C2×C8—1222no—
432C2×C8—1222no—
433C2×C8—1222no—
434C2×C8—1224no—
435C2×C8—1224no—
436C2×C8—1222no—
437C2×C8—2442no—
438C2×C8C22×#692312yes—
439C4⋊C4—612no—
440C4⋊C4—1222no—
441C4⋊C4—2442no—
442C4⋊C4C22×#692312yes—
443C22⋊C4—21no—
444C22⋊C4—1222no—
445C22⋊C4—1221no—
446C22⋊C4—2442no—
447C42—21no—
448C42—1222no—
449C42—1222no—
450C42—1222no—
451C42—1221no—
452C23—22no—
453C23—44no—
454C23—42no—
455C23—611no—
456C23—611no—
457C23—612no—
458C23—612no—
459C23—2442no—
460C23D4×#692311yescharacteristic
461Q8—1222no—
462Q8—1222no—
463Q8—2442no—
464D4—22no—
465D4—22no—
466D4—42no—
467D4—2442no—
468D4—2442no—
469D4—2444no—
470C2×C4—22no—
471C2×C4—612no—
472C2×C4—612no—
473C2×C4—611no—
474C2×C4—611no—
475C2×C4—612no—
476C2×C4—612no—
477C2×C4—612no—
478C2×C4—612no—
479C2×C4—612no—
480C2×C4—611no—
481C2×C4—611no—
482C2×C4—612no—
483C2×C4—611no—
484C2×C4—611no—
485C2×C4—1222no—
486C2×C4—1222no—
487C2×C4—2444no—
488C2×C4—2442no—
489C2×C4—2442no—
490C2×C4—2444no—
491C2×C4—2442no—
492C2×C4—2444no—
493C2×C4—2442no—
494C2×C4#1451985612yes—
495C2×C4D4×#692311yescharacteristic
496C2×C4D4×#692312yes—
497C2×C4C23×#692311yescharacteristic
498C8—22no—
499C8—1222no—
500C8—1222no—
501C8—1222no—
502C8—1222no—
503C8—1222no—
504C8—2442no—
505C22—22no—
506C22—42no—
507C22—44no—
508C22—84no—
509C22—611no—
510C22—611no—
511C22—612no—
512C22—1222no—
513C22—1222no—
514C22—2442no—
515C22—2442no—
516C22#1684071212yes—
517C22C2×D4×#692311yescharacteristic
518C4—42no—
519C4—611no—
520C4—611no—
521C4—612no—
522C4—4882no—
523C4—2442no—
524C4—2442no—
525C4#1684071212yes—
526C4C2×#1451985611yescharacteristic
527C4C2×D4×#692311yescharacteristic
528C2—22no—
529C2—82no—
530C2—1221no—
531C2—1221no—
532C2—1222no—
533C2C2×#1684071211yescharacteristic
534C2#2146259112yes—
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#418C4
socleC22×C61C2×C4×D4
prime radicalC22×#11758224C22
cosocleC2×C4×D61C24
hypercenter#418#6923
hypocenterC61#21070
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#16840777C2semidirect
4. #21462379C2semidirect
5. #21461901C2semidirect
6. #21461900C2semidirect
7. #21461499C2semidirect
8. #21461487C2semidirect
15. C2×#14519848C22semidirect
16. C2×#14519847C22semidirect
24. #16840727C22semidirect
25. #16840727C22semidirect
29. #16840632C22semidirect
30. (C2×#6923)⋊C8C22semidirect
34. #16839772C4semidirect
35. #16839770C4semidirect
36. #16839767C4semidirect
37. #16839760C4semidirect
39. C61×#418C4semidirect
293. #418#6923semidirect
307. C61#21070semidirect

Maximal subgroups

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

subgroupindex
C2×#168407772
C2×#168407602
C2×#168407362
#214623792
#214619012
#214619002
#214614992
#214614872
#214543332
#2107061