Finite Group Databasebeta

← #29667916

Subgroups — #29667916

Order 15616.

784 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456764C211yesmaximal, characteristic, complemented
2#21456737C211yesmaximal, characteristic
3#21456379C211yesmaximal, characteristic, complemented
4#21456213C211yesmaximal, characteristic, complemented
5C2×#16840108C211yesmaximal, characteristic, complemented
6#21445480C212yesmaximal, complemented
7#21445459C212yesmaximal, complemented
8#21445430C212yesmaximal, complemented
9#21445423C212yesmaximal, complemented
10M16×#92418C211yesmaximal, characteristic, complemented
11C61×#3138C211yesmaximal, characteristic, complemented
12#16840312C2211yescharacteristic
13C2×#14519777C2211yescharacteristic
14#16840158C2211yescharacteristic, complemented
15#16840141C2212yescomplemented
16#16840138C2212yes—
17#16840136C2211yescharacteristic, complemented
18#16840122—22no—
19#16840114C2212yescomplemented
20#16840113C2212yescomplemented
21#16840112C2211yescharacteristic, complemented
22#16840110C2211yescharacteristic, complemented
23#16840109C2212yescomplemented
24#16840108—22no—
25#16840108C2212yes—
26#16840108C2212yes—
27#16840108C2212yes—
28#16839993—22no—
29#16839972—22no—
30#16839954—22no—
31#16839917—22no—
32#16839896—22no—
33C4×#11758201—22no—
34#16839815C2211yescharacteristic
35#16839764—22no—
36C2×#14519698—22no—
37C2×C4×#92418C2211yescharacteristic
38#16839483C2212yescomplemented
39C8×#92418C2212yescomplemented
40C61×#533C2211yescharacteristic
41C2×#127×C61C2211yescharacteristic
42C61×#490—22no—
43C61×#486C2212yes—
44C61×#486C2212yes—
45C61×#418—22no—
46C4×M16×C61C2211yescharacteristic
47C2×#11758222—21no—
48C22×Dic122—21no—
49C2×#11758202—24no—
50C2×#11758201—22no—
51C2×#11758199—24no—
52C22×#92418D412yes—
53#14519777—21no—
54#14519777—21no—
55#14519777—21no—
56#14519777C2312yes—
57#14519777C2311yescharacteristic
58#14519774C2311yescharacteristic
59#14519765—21no—
60#14519751—21no—
61#14519751—21no—
62C4×D244—21no—
63C4×D244—21no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—21no—
66#14519708—22no—
67C4⋊#92418D412yes—
68#14519702—24no—
69#14519701—22no—
70#14519700—24no—
71#14519699—24no—
72#14519698—24no—
73#14519698—24no—
74#14519698—24no—
75#14519697—22no—
76C22⋊#92418D412yes—
77C4×#92418D412yes—
78C4×#92418C2311yescharacteristic
79C4×#92418C2311yescharacteristic
80C4×#92418C2312yes—
81C22×D4×C61—21no—
82C22×D4×C61—21no—
83C22×D4×C61—21no—
84C2×M16×C61D412yes—
85C4⋊Q8×C61C2311yescharacteristic
86C4⋊Q8×C61C2311yescharacteristic
87#127×C61—21no—
88#127×C61—21no—
89#127×C61—21no—
90#127×C61C2312yes—
91#125×C61—21no—
92C22⋊Q8×C61—21no—
93C22⋊Q8×C61—21no—
94C4×Q8×C61—21no—
95C4×Q8×C61—21no—
96#117×C61—21no—
97C2×C42×C61C2311yescharacteristic
98#102×C61—24no—
99#102×C61—24no—
100#97×C61D412yes—
101C4×C8×C61C2312yes—
102C24×C61—41no—
103C2×Q8×C61—21no—
104C2×Q8×C61—21no—
105C2×D4×C61—22no—
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61—44no—
109C2×D4×C61—41no—
110C2×D4×C61—41no—
111C2×D4×C61—41no—
112C2×D4×C61—44no—
113C2×D4×C61—82no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C2×C22⋊D61—41no—
118C22×Dic61—41no—
119#11758222—44no—
120C2×D244—21no—
121C2×D244—21no—
122C2×D244—21no—
123C2×C4×D61—41no—
124C2×Dic122—22no—
125C2×Dic122—21no—
126C2×Dic122—41no—
127C2×Dic122—82no—
128M16×C61—24no—
129C2×C8×C61C2×D414yes—
130C4⋊C4×C61—21no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—41no—
135C4⋊C4×C61—41no—
136C22⋊C4×C61—41no—
137C42×C61—41no—
138C42×C61C2×D411yescharacteristic
139C42×C61C2×D411yescharacteristic
140C42×C61C2×D411yescharacteristic
141C42×C61C2411yescharacteristic
142#11758202—84no—
143#11758201—44no—
144#11758199—84no—
145#11758198—41no—
146#11758198—41no—
147C4⋊Dic61—22no—
148C4⋊Dic61—21no—
149C4⋊Dic61—22no—
150C4⋊Dic61—21no—
151C4⋊Dic61—41no—
152C4⋊Dic61—41no—
153C2×#92418—42no—
154C2×#92418C2×D412yes—
155C2×#92418C2×D412yes—
156C2×#92418C2×D414yes—
157C23×C61—42no—
158C23×C61—82no—
159C23×C61—82no—
160C23×C61—82no—
161C23×C61#12711yescharacteristic
162C22×D61—41no—
163Q8×C61—42no—
164Q8×C61—42no—
165D4×C61—44no—
166D4×C61—44no—
167D4×C61—44no—
168D4×C61—82no—
169D4×C61—82no—
170D4×C61—82no—
171C2×C4×C61—22no—
172C2×C4×C61—22no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61#12711yescharacteristic
176C2×C4×C61#12711yescharacteristic
177C2×C4×C61#12711yescharacteristic
178C2×C4×C61#13514yes—
179C2×C4×C61C22×D411yescharacteristic
180C2×C4×C61C22×D411yescharacteristic
181C2×C4×C61C22×D411yescharacteristic
182C22⋊D61—82no—
183C2×Dic61—42no—
184C2×Dic61—41no—
185C2×Dic61—82no—
186D244—42no—
187D244—42no—
188D244—42no—
189C4×D61—82no—
190Dic122—42no—
191Dic122—44no—
192Dic122—82no—
193C8×C61—24no—
194#92418—22no—
195#92418—22no—
196#92418—24no—
197#19627—611nomaximal, Sylow
198C22×C61—21no—
199C22×C61—21no—
200C22×C61—82no—
201C22×C61—82no—
202C22×C61—82no—
203C22×C61—164no—
204C22×C61#49512yes—
205C22×C61C2×#12711yescharacteristic
206C2×D61—81no—
207C2×D61—81no—
208C2×D61—81no—
209C4×C61—21no—
210C4×C61—21no—
211C4×C61—81no—
212C4×C61—81no—
213C4×C61#49512yes—
214C4×C61C2×#13512yes—
215C4×C61#57512yes—
216C4×C61#57612yes—
217Dic61—81no—
218Dic61—82no—
219#3209—611no—
220#3205—611no—
221#3167—611no—
222#3141—611no—
223#3138D6111yescharacteristic, complemented
224C2×#486—611no—
225#1720—612no—
226#1717—612no—
227#1711—612no—
228#1709—612no—
229C8×M16—611no—
230C2×C61—22no—
231C2×C61—162no—
232C2×C61C2×#49511yescharacteristic
233C2×C61#315311yescharacteristic
234C2×C61#315511yescharacteristic
235D61—161no—
236#533C2×D6111yescharacteristic
237#532—611no—
238C2×C4⋊Q8—611no—
239C2×#127C2×D6111yescharacteristic
240#498—611no—
241#494—612no—
242#493—612no—
243#492—611no—
244#490—22no—
245#489—1222no—
246#487—611no—
247#487—612no—
248#487—612no—
249#486—611no—
250#486—612no—
251#486—612no—
252#486—612no—
253#486—612no—
254#486—1222no—
255#486C2×D6112yes—
256#486C2×D6112yes—
257#465—1222no—
258#462—1222no—
259#459—1222no—
260#451—1222no—
261#447—1222no—
262C4×SD16—1222no—
263#431—611no—
264#418—22no—
265#417—1222no—
266C2×#102—1222no—
267C4×M16C2×D6111yescharacteristic
268C2×C4×C8—611no—
269#322—612no—
270C82—612no—
271C61#1962711yescharacteristic, complemented, Sylow
272C2×#41—1221no—
273C22×Q8—1221no—
274C22×D4—21no—
275C22×D4—21no—
276C22×D4—21no—
277C2×Q16—1224no—
278C2×SD16—1222no—
279C2×D8—1224no—
280C2×M16C22⋊D6112yes—
281C22×C8—612no—
282C4⋊Q8—612no—
283C4⋊Q8—611no—
284C4⋊Q8—1221no—
285C4⋊Q8—1221no—
286C4⋊Q8—1221no—
287C4⋊Q8C22×D6111yescharacteristic
288C4⋊Q8C22×D6111yescharacteristic
289#127—21no—
290#127—21no—
291#127—21no—
292#127—611no—
293#127C22×D6112yes—
294#125—21no—
295#124—1221no—
296C22⋊Q8—21no—
297C22⋊Q8—21no—
298#121—1221no—
299#121—1221no—
300C4×Q8—21no—
301C4×Q8—21no—
302C4×D4—1221no—
303C4×D4—1221no—
304#117—21no—
305C2×C4⋊C4—1221no—
306C2×C4⋊C4—1221no—
307C2×C42C22×D6111yescharacteristic
308#106—1222no—
309C4⋊C8—612no—
310#103—1224no—
311#103—1222no—
312#103—1224no—
313#102—24no—
314#102—24no—
315#102—1222no—
316#102—1224no—
317#102—1224no—
318#102—1224no—
319#102—1224no—
320C22⋊C8—612no—
321#97C22⋊D6112yes—
322C4×C8—612no—
323C4×C8—611no—
324C4×C8—612no—
325C4×C8—611no—
326C4×C8C22×D6112yes—
327C24—41no—
328#41—2444no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—1222no—
332C2×Q8—1221no—
333C2×Q8—4882no—
334C2×Q8—2441no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—44no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—44no—
343C2×D4—82no—
344C2×D4—1221no—
345C2×D4—1221no—
346C2×D4—1221no—
347C2×D4—2441no—
348C22×C4—2441no—
349C22×C4—2441no—
350C22×C4D4×D6111yescharacteristic
351C22×C4D4×D6111yescharacteristic
352C22×C4C2×C22⋊D6111yescharacteristic
353Q16—4884no—
354SD16—2444no—
355D8—4884no—
356M16—24no—
357C2×C8—612no—
358C2×C8—612no—
359C2×C8—614no—
360C2×C8—2442no—
361C2×C8C2×C22⋊D6114yes—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—1222no—
369C4⋊C4—1221no—
370C4⋊C4—1221no—
371C4⋊C4—1222no—
372C4⋊C4—2441no—
373C4⋊C4—2441no—
374C22⋊C4—41no—
375C22⋊C4—2441no—
376C22⋊C4—2441no—
377C42—41no—
378C42D4×D6111yescharacteristic
379C42D4×D6111yescharacteristic
380C42C2×C22⋊D6111yescharacteristic
381C42C23×D6111yescharacteristic
382C23—42no—
383C23—82no—
384C23—82no—
385C23—82no—
386C23—2441no—
387C23#1451977311yescharacteristic
388Q8—42no—
389Q8—42no—
390Q8—4882no—
391Q8—2442no—
392Q8—2444no—
393D4—44no—
394D4—44no—
395D4—44no—
396D4—82no—
397D4—82no—
398D4—82no—
399D4—4882no—
400D4—2442no—
401D4—2442no—
402D4—2442no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—4882no—
408C2×C4—4882no—
409C2×C4—2441no—
410C2×C4—2442no—
411C2×C4#1451977311yescharacteristic
412C2×C4#1451977311yescharacteristic
413C2×C4#1451977311yescharacteristic
414C2×C4#1451979714yes—
415C2×C4C2×D4×D6111yescharacteristic
416C2×C4C2×D4×D6111yescharacteristic
417C2×C4C22×C22⋊D6111yescharacteristic
418C8—24no—
419C8—1222no—
420C8—1222no—
421C8—1224no—
422C22—21no—
423C22—21no—
424C22—82no—
425C22—82no—
426C22—82no—
427C22—164no—
428C22—4881no—
429C22—4881no—
430C22—4881no—
431C22#1684014412yes—
432C22C2×#1451977311yescharacteristic
433C4—21no—
434C4—21no—
435C4—81no—
436C4—81no—
437C4—4881no—
438C4—4882no—
439C4#1684014412yes—
440C4C2×#1451979712yes—
441C4#1684055812yes—
442C4#1684056612yes—
443C2—22no—
444C2—162no—
445C2—9761no—
446C2C2×#1684014411yescharacteristic
447C2#2145628111yescharacteristic
448C2#2145630411yescharacteristic
subgroupG/H
centerC22C2×#14519773
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3138C2
socleC22×C61C2×#127
prime radical#21456737C2
hypercenter#3138D61
hypocenterC61#19627
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3138

Complements

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

NH ≅ G/Ntype
1. #21456764C2semidirect
3. #21456379C2semidirect
4. #21456213C2semidirect
5. C2×#16840108C2semidirect
6. #21445480C2semidirect
7. #21445459C2semidirect
8. #21445430C2semidirect
9. #21445423C2semidirect
10. M16×#92418C2semidirect
11. C61×#3138C2semidirect
14. #16840158C22semidirect
15. #16840141C22semidirect
17. #16840136C22semidirect
19. #16840114C22semidirect
20. #16840113C22semidirect
21. #16840112C22semidirect
22. #16840110C22semidirect
23. #16840109C22semidirect
38. #16839483C22semidirect
39. C8×#92418C22semidirect
223. #3138D61semidirect
271. C61#19627semidirect

Maximal subgroups

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

subgroupindex
#214567642
#214567372
#214563792
#214562132
C2×#168401082
#214454802
#214454592
#214454302
#214454232
M16×#924182
C61×#31382
#1962761