Finite Group Databasebeta

← #29665170

Subgroups — #29665170

Order 15616.

738 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456760C211yesmaximal, characteristic, complemented
2#21455616C211yesmaximal, characteristic, complemented
3#21454843C211yesmaximal, characteristic, complemented
4#21445226C214yesmaximal, complemented
5#21445158C214yesmaximal, complemented
6#21444589C211yesmaximal, characteristic, complemented
7#21444487C211yesmaximal, characteristic, complemented
8#21444477C211yesmaximal, characteristic, complemented
9C2×C61×#459C211yesmaximal, characteristic
10#16840319C2211yescharacteristic
11#16840134C2212yescomplemented
12#16840108C2212yescomplemented
13#16840062—22no—
14#16840052C2212yescomplemented
15#16840042C2212yescomplemented
16#16839959—22no—
17#16839913—22no—
18D4×#92418—22no—
19D4×#92418C2214yescomplemented
20#16839815C2211yescharacteristic
21#16839815C2211yescharacteristic
22#16839802—21no—
23#16839786C2211yescharacteristic
24#16839767—21no—
25C2×C22⋊#92418—22no—
26#16839512C2214yescomplemented
27#16839510C2212yescomplemented
28#16839507C2212yescomplemented
29#16839506C2214yescomplemented
30C22×D8×C61—21no—
31C2×#127×C61C2211yescharacteristic
32C2×C4×D4×C61C2211yescharacteristic
33C61×#459—22no—
34C61×#459C2214yes—
35C2×C4⋊C8×C61C2211yescharacteristic
36C2×#102×C61—21no—
37C2×#11758200—24no—
38C2×#11758194—21no—
39C22×#92418—24no—
40#14519777C2311yescharacteristic
41#14519777C2311yescharacteristic
42#14519769—21no—
43#14519756—21no—
44#14519756—21no—
45C4×Dic122—21no—
46C4×Dic122—21no—
47#14519736—21no—
48#14519710—22no—
49#14519708—22no—
50C4⋊#92418—21no—
51C4⋊#92418D412yes—
52C4⋊#92418C2312yes—
53C4⋊#92418C2312yes—
54#14519698—22no—
55#14519698—22no—
56#14519698—24no—
57C22⋊#92418—21no—
58C22⋊#92418—24no—
59C22⋊#92418D412yes—
60C4×#92418—21no—
61C4×#92418C2312yes—
62C22×D4×C61—21no—
63C22×D4×C61—21no—
64C22×D4×C61—21no—
65C22×D4×C61D411yescharacteristic
66C23×C4×C61—21no—
67C2×D8×C61—24no—
68C2×D8×C61—44no—
69C22×C8×C61—21no—
70#127×C61—21no—
71#127×C61—21no—
72#127×C61—21no—
73#127×C61C2312yes—
74C4×D4×C61—21no—
75C4×D4×C61—21no—
76C4×D4×C61D412yes—
77C4×D4×C61C2312yes—
78C2×C4⋊C4×C61D411yescharacteristic
79C2×C22⋊C4×C61—21no—
80C2×C42×C61C2311yescharacteristic
81C4⋊C8×C61—21no—
82C4⋊C8×C61C2312yes—
83#102×C61—24no—
84C24×C61—21no—
85C24×C61—41no—
86C2×D4×C61—21no—
87C2×D4×C61—22no—
88C2×D4×C61—22no—
89C2×D4×C61—22no—
90C2×D4×C61—22no—
91C2×D4×C61—41no—
92C2×D4×C61—41no—
93C2×D4×C61—41no—
94C2×D4×C61—42no—
95C2×D4×C61—42no—
96C2×D4×C61—82no—
97C2×D4×C61—82no—
98C2×D4×C61SD1614yescomplemented
99C2×D4×C61C2×D412yes—
100C22×C4×C61—21no—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—82no—
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61#4111yescharacteristic
107C2×Dic122—21no—
108C2×Dic122—21no—
109D8×C61—48no—
110C2×C8×C61—22no—
111C2×C8×C61—42no—
112C4⋊C4×C61—21no—
113C4⋊C4×C61C2×D412yes—
114C22⋊C4×C61—22no—
115C22⋊C4×C61—22no—
116C42×C61C2×D411yescharacteristic
117C42×C61C2×D411yescharacteristic
118C42×C61#4111yescharacteristic
119C42×C61C2411yescharacteristic
120C22⋊Dic61—41no—
121#11758200—84no—
122C4⋊Dic61—21no—
123C4⋊Dic61—21no—
124C4⋊Dic61—21no—
125C4⋊Dic61—21no—
126#11758196—41no—
127#11758196—41no—
128C4×Dic61—41no—
129#11758194—42no—
130C2×#92418—22no—
131C2×#92418—22no—
132C2×#92418—84no—
133C2×#92418C2×D414yes—
134C23×C61—22no—
135C23×C61—24no—
136C23×C61—42no—
137C23×C61—81no—
138C23×C61—81no—
139C23×C61—82no—
140C23×C61—82no—
141C23×C61—82no—
142C23×C61#12111yescharacteristic
143D4×C61—22no—
144D4×C61—24no—
145D4×C61—42no—
146D4×C61—42no—
147D4×C61—44no—
148D4×C61—82no—
149D4×C61—82no—
150D4×C61—82no—
151D4×C61—82no—
152D4×C61C2×SD1614yes—
153C2×C4×C61—22no—
154C2×C4×C61—22no—
155C2×C4×C61—41no—
156C2×C4×C61—44no—
157C2×C4×C61—82no—
158C2×C4×C61—81no—
159C2×C4×C61#12111yescharacteristic
160C2×C4×C61#12111yescharacteristic
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61C2×SD1612yes—
163C2×C4×C61#13512yes—
164C2×C4×C61C22×D411yescharacteristic
165C2×C4×C61C22×D411yescharacteristic
166C2×C4×C61C2×#4111yescharacteristic
167C2×Dic61—41no—
168C2×Dic61—41no—
169Dic122—42no—
170Dic122—81no—
171C8×C61—42no—
172#92418—24no—
173#92418—42no—
174#92418—42no—
175#19384—611nomaximal, Sylow
176C22×C61—21no—
177C22×C61—21no—
178C22×C61—24no—
179C22×C61—42no—
180C22×C61—42no—
181C22×C61—84no—
182C22×C61—82no—
183C22×C61—82no—
184C22×C61—82no—
185C22×C61—164no—
186C22×C61#46512yes—
187C22×C61C2×#12111yescharacteristic
188C4×C61—21no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—42no—
192C4×C61#46512yes—
193C4×C61C22×SD1611yescharacteristic
194C4×C61C2×#13511yescharacteristic
195C4×C61#57612yes—
196Dic61—81no—
197Dic61—81no—
198#3208—611no—
199#3077—611no—
200C2×#459D6111yescharacteristic
201#2972—611no—
202#1677—614no—
203#1665—614no—
204#1570—611no—
205#1545—611no—
206#1542—611no—
207C2×C61—22no—
208C2×C61—44no—
209C2×C61—162no—
210C2×C61#304111yescharacteristic
211C2×C61C2×#46511yescharacteristic
212C2×C61#306811yescharacteristic
213C22×D8—21no—
214#533—611no—
215C2×#127C2×D6111yescharacteristic
216C2×C4×D4C2×D6111yescharacteristic
217#492—612no—
218#486—612no—
219#478—1222no—
220#476—612no—
221#474—612no—
222#460—1222no—
223#459—22no—
224#459C2×D6114yes—
225#450—1222no—
226C8×D4—614no—
227C8×D4—1222no—
228#431—611no—
229#431—611no—
230#428—1221no—
231#423—611no—
232C2×C4⋊C8C2×D6111yescharacteristic
233#418—1221no—
234C2×#102—21no—
235C2×C22⋊C8—1222no—
236#334—614no—
237#333—612no—
238#332—612no—
239#331—614no—
240C61#1938411yescharacteristic, complemented, Sylow
241C22×D4—21no—
242C22×D4—21no—
243C22×D4—21no—
244C22×D4C22⋊D6111yescharacteristic
245C23×C4—21no—
246C2×SD16—1224no—
247C2×D8—24no—
248C2×D8—44no—
249C2×M16—1221no—
250C22×C8—21no—
251C22×C8—1224no—
252C4⋊Q8—611no—
253C4⋊Q8—611no—
254#127—21no—
255#127—21no—
256#127—21no—
257#127C22×D6112yes—
258#125—1221no—
259C22⋊Q8—1221no—
260C22⋊Q8—1221no—
261C4×Q8—1221no—
262C4×Q8—1221no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4C22⋊D6112yes—
266C4×D4C22×D6112yes—
267#117—1221no—
268C2×C4⋊C4C22⋊D6111yescharacteristic
269C2×C22⋊C4—21no—
270C2×C42C22×D6111yescharacteristic
271#107—1222no—
272#106—1222no—
273C4⋊C8—21no—
274C4⋊C8—612no—
275C4⋊C8—612no—
276C4⋊C8—612no—
277C4⋊C8—1221no—
278C4⋊C8C22×D6112yes—
279#102—24no—
280#102—1224no—
281#102—1222no—
282#102—1222no—
283C22⋊C8—612no—
284C22⋊C8—1224no—
285C22⋊C8—1221no—
286C4×C8—612no—
287C4×C8—1221no—
288C24—21no—
289C24—41no—
290C2×Q8—1221no—
291C2×Q8—1221no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—21no—
296C2×D4—22no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—82no—
303C2×D4—82no—
304C2×D4#1175820014yescomplemented
305C2×D4C2×C22⋊D6112yes—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—82no—
310C22×C4D4×D6111yescharacteristic
311C22×C4#1175822411yescharacteristic
312C22×C4C2×C22⋊D6111yescharacteristic
313SD16—4884no—
314D8—48no—
315M16—2442no—
316C2×C8—22no—
317C2×C8—42no—
318C2×C8—614no—
319C2×C8—1222no—
320C2×C8—1222no—
321C2×C8—4884no—
322C4⋊C4—21no—
323C4⋊C4—1221no—
324C4⋊C4—1221no—
325C4⋊C4—1221no—
326C4⋊C4—1221no—
327C4⋊C4—2441no—
328C4⋊C4—2441no—
329C4⋊C4C2×C22⋊D6112yes—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—2441no—
333C42—2441no—
334C42D4×D6111yescharacteristic
335C42#1175822411yescharacteristic
336C42C2×C22⋊D6111yescharacteristic
337C42C23×D6111yescharacteristic
338C23—22no—
339C23—24no—
340C23—42no—
341C23—81no—
342C23—81no—
343C23—82no—
344C23—82no—
345C23—82no—
346C23#1451974911yescharacteristic
347Q8—4881no—
348Q8—2442no—
349D4—24no—
350D4—22no—
351D4—42no—
352D4—42no—
353D4—44no—
354D4—82no—
355D4—82no—
356D4—82no—
357D4—82no—
358D4C2×#1175820014yes—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—44no—
363C2×C4—82no—
364C2×C4—81no—
365C2×C4—2441no—
366C2×C4—2441no—
367C2×C4#1451974911yescharacteristic
368C2×C4#1451974911yescharacteristic
369C2×C4#1451974911yescharacteristic
370C2×C4C2×#1175820012yes—
371C2×C4#1451979712yes—
372C2×C4C2×D4×D6111yescharacteristic
373C2×C4C22×C22⋊D6111yescharacteristic
374C2×C4C2×#1175822411yescharacteristic
375C8—42no—
376C8—1224no—
377C8—2442no—
378C8—2442no—
379C22—21no—
380C22—21no—
381C22—24no—
382C22—42no—
383C22—42no—
384C22—84no—
385C22—82no—
386C22—82no—
387C22—82no—
388C22—164no—
389C22#1683998912yes—
390C22C2×#1451974911yescharacteristic
391C4—21no—
392C4—21no—
393C4—21no—
394C4—42no—
395C4—4881no—
396C4—4881no—
397C4#1683998912yes—
398C4C22×#1175820011yescharacteristic
399C4C2×#1451979711yescharacteristic
400C4#1684056312yes—
401C2—22no—
402C2—44no—
403C2—162no—
404C2#2145531911yescharacteristic
405C2C2×#1683998911yescharacteristic
406C2#2145554411yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC2×C61×#459C2
socleC22×C61C2×#121
hypercenterC2×#459D61
hypocenterC61#19384
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
1. #21456760C2semidirect
2. #21455616C2semidirect
3. #21454843C2semidirect
4. #21445226C2semidirect
5. #21445158C2semidirect
6. #21444589C2semidirect
7. #21444487C2semidirect
8. #21444477C2semidirect
11. #16840134C22semidirect
12. #16840108C22semidirect
14. #16840052C22semidirect
15. #16840042C22semidirect
19. D4×#92418C22semidirect
26. #16839512C22semidirect
27. #16839510C22semidirect
28. #16839507C22semidirect
29. #16839506C22semidirect
98. C2×D4×C61SD16semidirect
240. C61#19384semidirect
304. C2×D4#11758200semidirect

Maximal subgroups

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

subgroupindex
#214567602
#214556162
#214548432
#214452262
#214451582
#214445892
#214444872
#214444772
C2×C61×#4592
#1938461