Finite Group Databasebeta

← #29698506

Subgroups — #29698506

Order 15616.

862 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839877C211yesmaximal, characteristic
2#21454690C212yesmaximal
3#21454419C211yesmaximal, characteristic, complemented
4#21454415C211yesmaximal, characteristic, complemented
5#21446603C212yesmaximal, complemented
6#21446048C212yesmaximal, complemented
7#21446043C212yesmaximal, complemented
8#21445977C211yesmaximal, characteristic, complemented
9#21445880C211yesmaximal, characteristic, complemented
10#21445701C211yesmaximal, characteristic, complemented
11C61×#1816C211yesmaximal, characteristic, complemented
12C2×#14519797C2211yescharacteristic
13C2×#14519783C2211yescharacteristic
14#16840245C2212yes—
15#16840243C2212yes—
16#16839877C418yescomplemented
17#16839865—22no—
18#16839859—22no—
19#16839854—22no—
20#16839842—22no—
21C2×#14519712C2211yescharacteristic
22#16839804—22no—
23#16839804C2212yes—
24#16839803—22no—
25#16839802C2212yes—
26#16839800—21no—
27#16839800C2211yescharacteristic
28#16839799—21no—
29C2×#14519711—21no—
30C2×#14519711C2211yescharacteristic
31C2×#14519710C2212yes—
32#107×D61—21no—
33C2×#14519704C2211yescharacteristic
34C2×#14519703C2211yescharacteristic
35#16839764C2212yes—
36#16839720C2212yes—
37C2×C8×Dic61C2211yescharacteristic
38#16839631C2212yes—
39#16839530C2212yes—
40#16839523C2212yes—
41C2×#131×C61C2211yescharacteristic
42C2×#108×C61C2211yescharacteristic
43C2×#107×C61C2211yescharacteristic
44C61×#417C2212yes—
45C61×#343C2212yes—
46C2×#11758222C2311yescharacteristic
47#14519797C2×C414yes—
48#14519797Q814yes—
49C2×#11758202D411yescharacteristic
50C2×#11758200D411yescharacteristic
51C2×#11758201D411yescharacteristic
52C2×#11758199D411yescharacteristic
53#14519783—22no—
54#14519783C2×C414yescomplemented
55C2×#11758194C2311yescharacteristic
56C2×#11758189—21no—
57C22×#92418C2311yescharacteristic
58C2×C8×D61—21no—
59Q8×Dic61—22no—
60#14519742—22no—
61D4×Dic61—22no—
62D4×Dic61—42no—
63#14519739—22no—
64C4×C22⋊D61—42no—
65#14519736—22no—
66#14519736C2312yes—
67#14519735—22no—
68C2×C4⋊Dic61—22no—
69C2×C4⋊Dic61C2312yes—
70C2×#11758196—42no—
71C4⋊C4×D61—22no—
72C2×C4×Dic61C2311yescharacteristic
73#14519712C2×C414yescomplemented
74#14519711—21no—
75#14519711—21no—
76#14519711—22no—
77#14519711D411yescharacteristic
78#14519711D411yescharacteristic
79#14519710—22no—
80#14519710—22no—
81#14519710—22no—
82#14519709—22no—
83#14519709D412yes—
84#14519708—22no—
85#14519704C2×C414yescomplemented
86#14519703C2×C414yescomplemented
87#14519702—22no—
88#14519699—22no—
89#14519687—22no—
90C8×Dic61—22no—
91C8×Dic61C2×C414yescomplemented
92C2×#41×C61C2311yescharacteristic
93#131×C61—22no—
94#131×C61C2×C414yescomplemented
95C2×M16×C61—21no—
96C2×M16×C61C2312yes—
97C2×M16×C61C2311yescharacteristic
98C22×C8×C61—21no—
99C22×C8×C61C2311yescharacteristic
100C2×C4⋊C4×C61C2312yes—
101#108×C61—22no—
102#107×C61—21no—
103#107×C61—21no—
104#103×C61—22no—
105#102×C61—22no—
106#41×C61—21no—
107#41×C61—21no—
108#41×C61C22×C412yes—
109#41×C61C2×Q812yes—
110C2×Q8×C61C2×D411yescharacteristic
111C2×D4×C61—21no—
112C2×D4×C61C2×D411yescharacteristic
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61C2411yescharacteristic
116C2×C22⋊D61—21no—
117C22×Dic61—21no—
118C22×Dic61—22no—
119C22×Dic61—42no—
120#11758222—21no—
121#11758222—21no—
122#11758222C22×C412yes—
123#11758222C2×Q812yes—
124C2×D244C2×D411yescharacteristic
125C2×C4×D61—21no—
126C2×C4×D61—42no—
127C2×Dic122C2×D411yescharacteristic
128M16×C61—21no—
129M16×C61—24no—
130M16×C61—22no—
131M16×C61—22no—
132M16×C61C22×C412yes—
133C2×C8×C61—21no—
134C2×C8×C61—24no—
135C2×C8×C61—22no—
136C2×C8×C61C22×C412yes—
137C2×C8×C61C2×D411yescharacteristic
138C2×C8×C61C2×D411yescharacteristic
139C2×C8×C61#4112yes—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—22no—
142C22⋊Dic61—22no—
143C22⋊Dic61—42no—
144#11758202C4⋊C414yescomplemented
145#11758201C4⋊C414yescomplemented
146#11758200C4⋊C414yescomplemented
147#11758199C4⋊C414yescomplemented
148#11758198—42no—
149C4⋊Dic61—22no—
150C4⋊Dic61—22no—
151C4⋊Dic61—42no—
152C4⋊Dic61C2×D412yes—
153C4⋊Dic61C2×D412yes—
154#11758196—42no—
155#11758196—84no—
156C4×Dic61—22no—
157C4×Dic61—21no—
158C4×Dic61—42no—
159C4×Dic61—42no—
160C4×Dic61C22×C412yes—
161#11758194—21no—
162#11758194C22×C412yes—
163C2×#92418—21no—
164C2×#92418C22×C412yes—
165C2×#92418C2×D411yescharacteristic
166C2×#92418C2×D411yescharacteristic
167C2×#92418C2×Q812yes—
168#11758189—24no—
169C8×D61—24no—
170C23×C61—21no—
171C23×C61H(5,𝔽2)11yescharacteristic
172C22×D61—21no—
173Q8×C61—21no—
174Q8×C61C2×C4⋊C412yes—
175D4×C61—22no—
176D4×C61—22no—
177D4×C61—21no—
178D4×C61C2×C4⋊C412yes—
179C2×C4×C61—21no—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—42no—
183C2×C4×C61—82no—
184C2×C4×C61C23×C411yescharacteristic
185C2×C4×C61C22×D411yescharacteristic
186C2×C4×C61C22×D411yescharacteristic
187C2×C4×C61C22×Q811yescharacteristic
188C2×C4×C61C2×#4111yescharacteristic
189C2×C4×C612−1+411yescharacteristic
190C22⋊D61—22no—
191C22⋊D61—22no—
192C2×Dic61—22no—
193C2×Dic61—22no—
194C2×Dic61—22no—
195C2×Dic61—21no—
196C2×Dic61—22no—
197C2×Dic61—22no—
198C2×Dic61—42no—
199C2×Dic61—82no—
200C2×Dic61—162no—
201D244—21no—
202D244C2×C4⋊C412yes—
203C4×D61—22no—
204C4×D61—22no—
205C4×D61—162no—
206Dic122—21no—
207Dic122C2×C4⋊C412yes—
208C8×C61—22no—
209C8×C61—42no—
210C8×C61C4×D412yes—
211C8×C61C4×D412yes—
212#92418—22no—
213#92418C2×C4⋊C412yes—
214#92418C2×C4⋊C412yes—
215#22692—611nomaximal, Sylow
216C22×C61—21no—
217C22×C61—21no—
218C22×C61—22no—
219C22×C61—41no—
220C22×C61—41no—
221C22×C61#52011yescharacteristic
222C22×C61D4211yescharacteristic
223C22×C61#55411yescharacteristic
224C2×D61—22no—
225C2×D61—41no—
226C2×D61—41no—
227C4×C61—22no—
228C4×C61—82no—
229C4×C61C22×C4⋊C411yescharacteristic
230C4×C61C2×C4×D411yescharacteristic
231C4×C61#54711yescharacteristic
232C4×C61D4×Q811yescharacteristic
233Dic61—24no—
234Dic61—22no—
235Dic61—42no—
236Dic61—42no—
237Dic61—82no—
238C2×#443—611no—
239#2951—612no—
240#2913—611no—
241#2913—611no—
242#1952—612no—
243#1859—612no—
244#1859—612no—
245#1845—611no—
246#1816—611no—
247#1816D6111yescharacteristic, complemented
248#1772—611no—
249C2×C61—21no—
250C2×C61—21no—
251C2×C61—42no—
252C2×C61D4×C4⋊C411yescharacteristic
253C2×C61#329211yescharacteristic
254C2×C61#329311yescharacteristic
255D61—42no—
256C2×#135—611no—
257C2×#131—611no—
258C2×#131C2×D6111yescharacteristic
259#520—612no—
260#520—612no—
261#443—618no—
262#440—1222no—
263#440—1222no—
264C4×Q16—1222no—
265C4×D8—1222no—
266C2×#108—611no—
267C2×#108C2×D6111yescharacteristic
268#428—612no—
269#428—612no—
270#428—1222no—
271#428—1222no—
272#427—611no—
273#427—1221no—
274#427—1221no—
275C2×#107—611no—
276C2×#107—612no—
277C2×#107—1221no—
278C2×#107—1221no—
279C2×#107C2×D6111yescharacteristic
280C2×#104—611no—
281C2×#104—611no—
282#417—612no—
283#417C2×D6112yes—
284#405—612no—
285C2×C4×C8—611no—
286#384—612no—
287#343—612no—
288#343C2×D6112yes—
289#339—612no—
290C61#2269211yescharacteristic, complemented, Sylow
291C2×#41—611no—
292C2×#41C22×D6111yescharacteristic
293#135—614no—
294#135—614no—
295C2×Q16—611no—
296C2×SD16—611no—
297C2×SD16—611no—
298C2×D8—611no—
299#131—22no—
300#131—614no—
301#131—1222no—
302#131C4×D6114yescomplemented
303C2×M16—21no—
304C2×M16—611no—
305C2×M16—1221no—
306C2×M16C22×D6112yes—
307C2×M16C22×D6111yescharacteristic
308C22×C8—21no—
309C22×C8—611no—
310C22×C8—1221no—
311C22×C8C22×D6111yescharacteristic
312C4×Q8—1222no—
313C4×Q8—1222no—
314C4×D4—1222no—
315C4×D4—1222no—
316C4×D4—2442no—
317C4×D4—2442no—
318#117—612no—
319#117—1222no—
320#117—1222no—
321C2×C4⋊C4—612no—
322C2×C4⋊C4—1222no—
323C2×C4⋊C4—1222no—
324C2×C4⋊C4—2442no—
325C2×C4⋊C4C22×D6112yes—
326C2×C42—611no—
327#108—22no—
328#108—614no—
329#107—21no—
330#107—21no—
331#107—611no—
332#107—611no—
333#107—1222no—
334#107—1222no—
335#107—1222no—
336#107—1222no—
337#107—1221no—
338#107—1221no—
339#106—612no—
340#106—1222no—
341#106—1222no—
342#104—614no—
343#104—614no—
344#103—22no—
345#103—1222no—
346#102—22no—
347#102—1222no—
348#97—1222no—
349C4×C8—614no—
350C4×C8—1222no—
351#41—21no—
352#41—21no—
353#41—612no—
354#41—612no—
355#41—1221no—
356#41—1221no—
357#41C2×C4×D6112yes—
358#41Q8×D6112yes—
359C2×Q8—611no—
360C2×Q8D4×D6111yescharacteristic
361C2×D4—21no—
362C2×D4—611no—
363C2×D4—1221no—
364C2×D4D4×D6111yescharacteristic
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—1221no—
368C22×C4—1221no—
369C22×C4—1222no—
370C22×C4—2442no—
371C22×C4—2442no—
372C22×C4C23×D6111yescharacteristic
373Q16—614no—
374SD16—614no—
375SD16—614no—
376D8—614no—
377M16—21no—
378M16—24no—
379M16—22no—
380M16—22no—
381M16—612no—
382M16—1221no—
383M16—1224no—
384M16C2×C4×D6112yes—
385C2×C8—21no—
386C2×C8—24no—
387C2×C8—22no—
388C2×C8—612no—
389C2×C8—611no—
390C2×C8—611no—
391C2×C8—612no—
392C2×C8—1221no—
393C2×C8—1224no—
394C2×C8C2×C4×D6112yes—
395C2×C8D4×D6111yescharacteristic
396C2×C8D4×D6111yescharacteristic
397C2×C8#1175822412yes—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—612no—
401C4⋊C4—612no—
402C4⋊C4—1222no—
403C4⋊C4—1222no—
404C4⋊C4—4884no—
405C4⋊C4—2442no—
406C4⋊C4—2442no—
407C22⋊C4—1222no—
408C22⋊C4—2442no—
409C22⋊C4—2442no—
410C42—612no—
411C42—1222no—
412C42—1221no—
413C42—2442no—
414C42—2442no—
415C23—21no—
416C23—1221no—
417C23#1451982311yescharacteristic
418Q8—21no—
419Q8—612no—
420Q8—1221no—
421Q8C4⋊C4×D6112yes—
422D4—22no—
423D4—22no—
424D4—21no—
425D4—612no—
426D4—1222no—
427D4—1222no—
428D4—1221no—
429D4C4⋊C4×D6112yes—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—42no—
434C2×C4—82no—
435C2×C4—1222no—
436C2×C4—1222no—
437C2×C4—1222no—
438C2×C4—1221no—
439C2×C4—1222no—
440C2×C4—1222no—
441C2×C4—1222no—
442C2×C4—1222no—
443C2×C4—9762no—
444C2×C4—9762no—
445C2×C4—4882no—
446C2×C4—2442no—
447C2×C4C22×C4×D6111yescharacteristic
448C2×C4C2×D4×D6111yescharacteristic
449C2×C4C2×D4×D6111yescharacteristic
450C2×C4C2×Q8×D6111yescharacteristic
451C2×C4C2×#1175822411yescharacteristic
452C2×C4#1451982411yescharacteristic
453C8—22no—
454C8—42no—
455C8—612no—
456C8—612no—
457C8—1222no—
458C8#1451973712yes—
459C8#1451973712yes—
460C22—21no—
461C22—21no—
462C22—22no—
463C22—41no—
464C22—41no—
465C22—1222no—
466C22—2441no—
467C22—2441no—
468C22#1684024211yescharacteristic
469C22#1684039611yescharacteristic
470C22#1684047011yescharacteristic
471C4—22no—
472C4—82no—
473C4—1224no—
474C4—1222no—
475C4—4882no—
476C4—2442no—
477C4—2442no—
478C4C2×C4⋊C4×D6111yescharacteristic
479C4C2×#1451973711yescharacteristic
480C4#1684041211yescharacteristic
481C4#1684042511yescharacteristic
482C2—21no—
483C2—21no—
484C2—42no—
485C2—2442no—
486C2#2144871511yescharacteristic
487C2#2145761711yescharacteristic
488C2#2145762211yescharacteristic
subgroupG/H
centerC22#16840396
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1816C2
socleC22×C61D42
prime radicalC2×#14519797C22
cosocleC22×C4×C61C24
hypercenter#1816D61
hypocenterC61#22692
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1816

Complements

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

NH ≅ G/Ntype
3. #21454419C2semidirect
4. #21454415C2semidirect
5. #21446603C2semidirect
6. #21446048C2semidirect
7. #21446043C2semidirect
8. #21445977C2semidirect
9. #21445880C2semidirect
10. #21445701C2semidirect
11. C61×#1816C2semidirect
16. #16839877C4semidirect
54. #14519783C2×C4semidirect
73. #14519712C2×C4semidirect
85. #14519704C2×C4semidirect
86. #14519703C2×C4semidirect
91. C8×Dic61C2×C4semidirect
94. #131×C61C2×C4semidirect
144. #11758202C4⋊C4semidirect
145. #11758201C4⋊C4semidirect
146. #11758200C4⋊C4semidirect
147. #11758199C4⋊C4semidirect
247. #1816D61semidirect
290. C61#22692semidirect
302. #131C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398772
#214546902
#214544192
#214544152
#214466032
#214460482
#214460432
#214459772
#214458802
#214457012
C61×#18162
#2269261