Finite Group Databasebeta

← #30611987

Subgroups — #30611987

Order 15616.

870 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462745C212yesmaximal
2#21461355C212yesmaximal, complemented
3#21461352C211yesmaximal, characteristic, complemented
4#21461345C212yesmaximal, complemented
5#21461340C212yesmaximal, complemented
6#21461337C212yesmaximal, complemented
7#21461321C212yesmaximal, complemented
8C2×#16840622C211yesmaximal, characteristic, complemented
9#21459704C211yesmaximal, characteristic
10#16840813—22no—
11#16840793—22no—
12#16840793—22no—
13D4×#92420—22no—
14#16840782C2212yes—
15#16840782C2212yes—
16#16840779C2212yes—
17#16840765C2212yes—
18C2×#14519844C2212yes—
19C2×#14519844C2212yes—
20#16840622C2211yescharacteristic
21#16840622C2212yes—
22#16840622C2211yescharacteristic
23#16840622C2212yes—
24#16840622C2211yescharacteristic
25#16840622C2211yescharacteristic
26#16840621C2212yes—
27#16840621C2212yes—
28#16840621C2212yes—
29#16840621C2212yes—
30#16840487—22no—
31#16840481—22no—
32#16840449—22no—
33#16840449—22no—
34#16840372C412yes—
35#16840372C412yes—
36#16840333C411yescharacteristic
37#16840315C2211yescharacteristic
38#16840314C2212yes—
39#16840314C2212yes—
40C2×#14519775C2211yescharacteristic
41#16840240C411yescharacteristic
42#16840239C411yescharacteristic
43#41×Dic61C2211yescharacteristic
44C61×#536C411yescharacteristic
45C2×#11758217—22no—
46C22×#92420—22no—
47C22×#92420—22no—
48#14519844—22no—
49#14519844—22no—
50#14519844C2312yes—
51#14519844C2312yes—
52#14519844C2312yes—
53#14519844C2312yes—
54C4⋊#92420—22no—
55C22⋊#92420—22no—
56C22⋊#92420—22no—
57C22⋊#92420—22no—
58#14519832—22no—
59#14519832—22no—
60C4×#92420—22no—
61C22×Dic122C2×C411yescharacteristic
62#14519776C2312yes—
63#14519775—21no—
64#14519775—22no—
65#14519775C2×C411yescharacteristic
66#14519775C2×C411yescharacteristic
67#14519775C2311yescharacteristic
68#14519775C2311yescharacteristic
69#14519770—22no—
70#14519770—22no—
71#14519770—22no—
72#14519770—22no—
73#14519768—22no—
74#14519768—22no—
75#14519768C2×C412yes—
76#14519767—22no—
77#14519767—22no—
78#14519767—22no—
79#14519767—22no—
80#14519764—21no—
81#14519758—22no—
82#14519756C2×C412yes—
83#14519756C2×C412yes—
84#14519754—22no—
85#14519754—22no—
86#14519754—22no—
87Q8×Dic61—22no—
88Q8×Dic61C2×C412yes—
89Q8×Dic61C2×C412yes—
90Q8×Dic61D412yes—
91C4×Dic122—22no—
92C4×Dic122—22no—
93#14519742—22no—
94#14519742—22no—
95#14519742C2×C412yes—
96#14519742C2×C412yes—
97D4×Dic61D412yes—
98D4×Dic61D412yes—
99D4×Dic61D412yes—
100#14519736—21no—
101#14519736—21no—
102#14519736C2×C411yescharacteristic
103#14519736C2×C411yescharacteristic
104#14519736C2×C412yes—
105#14519733—22no—
106#14519733—22no—
107#14519733—21no—
108#14519733—21no—
109C2×#11758196—22no—
110C2×C4×Dic61C2311yescharacteristic
111C2×C4×Dic61C2312yes—
112C2×#41×C61C2×C411yescharacteristic
113C22×Q8×C61C2×C411yescharacteristic
114C4⋊Q8×C61—21no—
115#124×C61—21no—
116#123×C61—22no—
117C22⋊Q8×C61C2×C412yes—
118C22⋊Q8×C61C2×C412yes—
119#117×C61C2×C411yescharacteristic
120#41×C61—42no—
121C2×Q8×C61—21no—
122C2×Q8×C61—42no—
123C2×Q8×C61C22⋊C411yescharacteristic
124C2×Q8×C61C22×C411yescharacteristic
125C2×Q8×C61C22×C411yescharacteristic
126C2×D4×C61C22⋊C412yes—
127C2×D4×C61C22⋊C411yescharacteristic
128C22×C4×C61—21no—
129C22×C4×C61C22×C411yescharacteristic
130C22×C4×C61C22×C412yes—
131C22×Dic61—21no—
132C22×Dic61C2×D412yes—
133C22×Dic61C2×D412yes—
134C22×Dic61C2×D412yes—
135C2×Dic122—21no—
136C2×Dic122—22no—
137C2×Dic122—42no—
138C2×Dic122C22×C411yescharacteristic
139C2×Dic122C22×C411yescharacteristic
140#11758217—82no—
141C2×#92420—22no—
142C2×#92420—22no—
143C2×#92420—22no—
144C2×#92420—22no—
145C2×#92420—82no—
146C2×#92420—82no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61C22×C412yes—
150C22⋊C4×C61—21no—
151C22⋊C4×C61—22no—
152C22⋊C4×C61—22no—
153C42×C61—21no—
154C22⋊Dic61—22no—
155C22⋊Dic61—22no—
156C22⋊Dic61—21no—
157C22⋊Dic61C22⋊C411yescharacteristic
158C22⋊Dic61C22⋊C411yescharacteristic
159C22⋊Dic61C22⋊C412yes—
160C22⋊Dic61C22⋊C412yes—
161C4⋊Dic61—22no—
162C4⋊Dic61—22no—
163C4⋊Dic61—22no—
164C4⋊Dic61C22⋊C412yes—
165C4⋊Dic61C22⋊C412yes—
166C4⋊Dic61C22⋊C412yes—
167C4⋊Dic61C22×C412yes—
168#11758196—22no—
169#11758196—22no—
170#11758196—22no—
171#11758196—22no—
172#11758196—22no—
173#11758196—22no—
174#11758196—22no—
175#11758196—22no—
176#11758196—22no—
177#11758196—22no—
178C4×Dic61—21no—
179C4×Dic61—22no—
180C4×Dic61—22no—
181C4×Dic61—22no—
182C4×Dic61—22no—
183C4×Dic61C22×C412yes—
184C4×Dic61C22×C411yescharacteristic
185C4×Dic61C2×D412yes—
186C4×Dic61C2×D412yes—
187C4×Dic61C2×D412yes—
188C4×Dic61C2411yescharacteristic
189C23×C61C2×C22⋊C412yes—
190C23×C61C2×C22⋊C411yescharacteristic
191Q8×C61—41no—
192Q8×C61—42no—
193Q8×C61—41no—
194Q8×C61—41no—
195D4×C61—24no—
196D4×C61—41no—
197C2×C4×C61—22no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—22no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61C2×C22⋊C411yescharacteristic
204C2×C4×C61C2×C22⋊C412yes—
205C2×C4×C61C23×C411yescharacteristic
206C2×Dic61—22no—
207C2×Dic61—22no—
208C2×Dic61—22no—
209C2×Dic61—22no—
210C2×Dic61—22no—
211C2×Dic61—22no—
212C2×Dic61—22no—
213C2×Dic61—21no—
214C2×Dic61—21no—
215C2×Dic61—42no—
216C2×Dic61C2×C22⋊C412yes—
217C2×Dic61C2×C22⋊C412yes—
218C2×Dic61C2×C22⋊C412yes—
219C2×Dic61C22×D412yes—
220Dic122—42no—
221Dic122—42no—
222Dic122—42no—
223Dic122—41no—
224Dic122—41no—
225#92420—42no—
226#92420—42no—
227#92420—42no—
228#92420—42no—
229#17780—611nomaximal, Sylow
230C22×C61—22no—
231C22×C61—22no—
232C22×C61—22no—
233C22×C61—22no—
234C22×C61#41312yes—
235C22×C61C22×C22⋊C411yescharacteristic
236C4×C61—21no—
237C4×C61—21no—
238C4×C61—22no—
239C4×C61—41no—
240C4×C61—41no—
241C4×C61—42no—
242Dic61—22no—
243Dic61—22no—
244Dic61—22no—
245Dic61—41no—
246Dic61—41no—
247Dic61—42no—
248Dic61#57712yes—
249Dic61#57812yes—
250#3509—611no—
251#2990—612no—
252#1559—612no—
253#1558—611no—
254#1554—612no—
255#1552—612no—
256#1550—612no—
257#1542—612no—
258C2×#333—611no—
259C2×C61—22no—
260C2×C61—42no—
261C2×C61C2×#41311yescharacteristic
262C2×C61#289812yes—
263#557—1222no—
264#556—1222no—
265#552—1222no—
266#552—1222no—
267#541—612no—
268#541—612no—
269#536—611no—
270#536#692311yescharacteristic
271#533—611no—
272#533—612no—
273#533—612no—
274C2×C4⋊Q8—611no—
275#519—611no—
276#519—611no—
277C4×#41—611no—
278#446—1222no—
279#435—1222no—
280#435—1222no—
281C8×D4—1222no—
282#432—612no—
283#432—612no—
284#431—612no—
285#423—612no—
286C2×C4⋊C8—612no—
287C2×C4⋊C8—612no—
288#333—611no—
289#333—612no—
290#333—611no—
291#333—612no—
292#333—611no—
293#333—611no—
294#332—612no—
295#332—612no—
296#332—612no—
297#332—612no—
298C61#1778011yescharacteristic, complemented, Sylow
299C2×#41C2×#692311yescharacteristic
300C22×Q8—611no—
301C22×Q8C2×#692311yescharacteristic
302C2×M16—1222no—
303C22×C8—1222no—
304C22×C8—1222no—
305C4⋊Q8—21no—
306C4⋊Q8—612no—
307C4⋊Q8—611no—
308C4⋊Q8—611no—
309C4⋊Q8—611no—
310C4⋊Q8—611no—
311C4⋊Q8—1222no—
312C4⋊Q8—1221no—
313#126—1222no—
314#126—1222no—
315#126—1222no—
316#126—1222no—
317#125—612no—
318#125—1222no—
319#125—1222no—
320#125—1222no—
321#125—1222no—
322#125—1222no—
323#125—1222no—
324#124—21no—
325#124—1221no—
326#123—22no—
327#123—1222no—
328C22⋊Q8—612no—
329C22⋊Q8—612no—
330C22⋊Q8—1222no—
331C22⋊Q8—1222no—
332C22⋊Q8—1222no—
333C22⋊Q8C2×#692312yes—
334C22⋊Q8C2×#692312yes—
335C4×Q8—612no—
336C4×Q8—612no—
337C4×Q8—612no—
338C4×Q8—612no—
339C4×Q8—612no—
340C4×Q8—1222no—
341C4×Q8—1222no—
342C4×Q8—1222no—
343C4×Q8—1222no—
344C4×Q8—1222no—
345C4×D4—612no—
346C4×D4—612no—
347C4×D4—612no—
348#117—611no—
349#117—612no—
350#117—611no—
351#117—1221no—
352#117—1221no—
353#117—1222no—
354#117—1222no—
355#117—1221no—
356#117—1221no—
357#117C2×#692311yescharacteristic
358C2×C4⋊C4—1222no—
359C2×C42—611no—
360C2×C42—612no—
361C4⋊C8—612no—
362C4⋊C8—612no—
363C4⋊C8—612no—
364C4⋊C8—612no—
365C4⋊C8—1222no—
366C4⋊C8—1222no—
367C4⋊C8—1222no—
368C22⋊C8—1222no—
369C22⋊C8—1222no—
370C22⋊C8—1222no—
371#97—1222no—
372#97—1222no—
373C4×C8—1222no—
374#41—42no—
375C2×Q8—21no—
376C2×Q8—42no—
377C2×Q8—611no—
378C2×Q8—611no—
379C2×Q8—1222no—
380C2×Q8—1221no—
381C2×Q8—2442no—
382C2×Q8C22⋊#692311yescharacteristic
383C2×Q8C22×#692311yescharacteristic
384C2×Q8C22×#692311yescharacteristic
385C2×D4C22⋊#692312yes—
386C2×D4C22⋊#692311yescharacteristic
387C22×C4—21no—
388C22×C4—612no—
389C22×C4—612no—
390C22×C4—612no—
391C22×C4—1221no—
392C22×C4C22×#692311yescharacteristic
393C22×C4C22×#692312yes—
394M16—4882no—
395C2×C8—1222no—
396C2×C8—1222no—
397C2×C8—1222no—
398C2×C8—1222no—
399C2×C8—4882no—
400C2×C8—4882no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—612no—
404C4⋊C4—612no—
405C4⋊C4—612no—
406C4⋊C4—612no—
407C4⋊C4—1222no—
408C4⋊C4—1222no—
409C4⋊C4—1222no—
410C4⋊C4—1222no—
411C4⋊C4—1222no—
412C4⋊C4—1222no—
413C4⋊C4—1222no—
414C4⋊C4—1222no—
415C4⋊C4—1222no—
416C4⋊C4—1222no—
417C4⋊C4—1222no—
418C4⋊C4—1222no—
419C4⋊C4—1222no—
420C4⋊C4C22×#692312yes—
421C22⋊C4—21no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—611no—
425C22⋊C4—611no—
426C22⋊C4—612no—
427C22⋊C4—612no—
428C22⋊C4—1221no—
429C22⋊C4—1222no—
430C22⋊C4—1222no—
431C42—21no—
432C42—612no—
433C42—612no—
434C42—612no—
435C42—611no—
436C42—612no—
437C42—611no—
438C42—1222no—
439C42—1222no—
440C42—1222no—
441C42—1222no—
442C42—1221no—
443C23C2×C22⋊#692312yes—
444C23C2×C22⋊#692311yescharacteristic
445Q8—41no—
446Q8—41no—
447Q8—41no—
448Q8—42no—
449Q8—2441no—
450Q8—2441no—
451Q8—2442no—
452Q8—2442no—
453Q8—2442no—
454D4—24no—
455D4—41no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—612no—
463C2×C4—612no—
464C2×C4—612no—
465C2×C4—612no—
466C2×C4—1222no—
467C2×C4—1222no—
468C2×C4—1222no—
469C2×C4—1222no—
470C2×C4—1222no—
471C2×C4—1222no—
472C2×C4—1222no—
473C2×C4—1221no—
474C2×C4—1221no—
475C2×C4—2442no—
476C2×C4C2×C22⋊#692311yescharacteristic
477C2×C4C2×C22⋊#692312yes—
478C2×C4C23×#692311yescharacteristic
479C8—2442no—
480C8—2442no—
481C8—2442no—
482C8—2442no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22#1684074912yes—
488C22C22×C22⋊#692311yescharacteristic
489C4—21no—
490C4—21no—
491C4—22no—
492C4—41no—
493C4—41no—
494C4—42no—
495C4—612no—
496C4—612no—
497C4—1222no—
498C4—1222no—
499C4—1222no—
500C4—2441no—
501C4—2441no—
502C4—2442no—
503C2—22no—
504C2—42no—
505C2C2×#1684074911yescharacteristic
506C2#2146244112yes—
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#536C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#41×C61C2×C4
cosocleC4×Dic61C24
hypercenter#536#6923
hypocenterC61#17780
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
2. #21461355C2semidirect
3. #21461352C2semidirect
4. #21461345C2semidirect
5. #21461340C2semidirect
6. #21461337C2semidirect
7. #21461321C2semidirect
8. C2×#16840622C2semidirect
298. C61#17780semidirect

Maximal subgroups

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

subgroupindex
#214627452
#214613552
#214613522
#214613452
#214613402
#214613372
#214613212
C2×#168406222
#214597042
#1778061