Finite Group Databasebeta

← #29665175

Subgroups — #29665175

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#21456757C211yesmaximal, characteristic, complemented
2C2×#16839957C211yesmaximal, characteristic
3#21454843C211yesmaximal, characteristic, complemented
4#21445230C214yesmaximal, complemented
5#21445162C214yesmaximal, complemented
6#21444591C211yesmaximal, characteristic, complemented
7#21444488C211yesmaximal, characteristic, complemented
8#21444476C211yesmaximal, characteristic, complemented
9C61×#3077C211yesmaximal, characteristic, complemented
10C22×#11758199—21no—
11C2×#14519774C2211yescharacteristic
12#16840136C2212yescomplemented
13#16840110C2212yescomplemented
14#16839962—22no—
15#16839957—22no—
16#16839957C2214yes—
17#16839912—22no—
18D4×#92418—22no—
19D4×#92418C2214yescomplemented
20#16839815C2211yescharacteristic
21#16839815C2211yescharacteristic
22C2×C4⋊#92418C2211yescharacteristic
23C2×#14519697—21no—
24C2×C22⋊#92418—22no—
25#16839512C2214yescomplemented
26#16839511C2212yescomplemented
27#16839508C2212yescomplemented
28#16839505C2214yescomplemented
29C61×#533C2211yescharacteristic
30C2×C4×D4×C61C2211yescharacteristic
31C61×#478—22no—
32C61×#476C2212yescomplemented
33C61×#474C2212yescomplemented
34C61×#428—21no—
35C61×#423C2211yescharacteristic
36C61×#418—21no—
37C22×D244—21no—
38C22×D244—21no—
39C22×D244—21no—
40C2×#11758201—24no—
41C2×#11758199—24no—
42C2×#11758199—44no—
43C22×#92418—21no—
44C22×#92418—24no—
45#14519774—21no—
46#14519774—21no—
47#14519774—21no—
48#14519774C2312yes—
49C4⋊#92418—21no—
50C4⋊#92418D412yes—
51C4⋊#92418C2312yes—
52C4⋊#92418C2312yes—
53#14519697—24no—
54#14519697—24no—
55C22⋊#92418—21no—
56C22⋊#92418—24no—
57C22⋊#92418D412yes—
58C4×#92418—21no—
59C4×#92418C2312yes—
60C22×D4×C61D411yescharacteristic
61C23×C4×C61—21no—
62C2×M16×C61—21no—
63C4⋊Q8×C61C2311yescharacteristic
64C4⋊Q8×C61C2311yescharacteristic
65#125×C61—21no—
66C22⋊Q8×C61—21no—
67C22⋊Q8×C61—21no—
68C4×Q8×C61—21no—
69C4×Q8×C61—21no—
70C4×D4×C61—21no—
71C4×D4×C61—21no—
72C4×D4×C61D412yes—
73C4×D4×C61C2312yes—
74#117×C61—21no—
75C2×C4⋊C4×C61D411yescharacteristic
76C2×C22⋊C4×C61—21no—
77C2×C42×C61C2311yescharacteristic
78#107×C61—22no—
79#106×C61—22no—
80C4⋊C8×C61—21no—
81C4⋊C8×C61C2312yes—
82#102×C61—22no—
83#102×C61—22no—
84C24×C61—21no—
85C23×D61—41no—
86C2×Q8×C61—21no—
87C2×Q8×C61—21no—
88C2×D4×C61—21no—
89C2×D4×C61—22no—
90C2×D4×C61SD1614yescomplemented
91C2×D4×C61C2×D412yes—
92C22×C4×C61—21no—
93C22×C4×C61—22no—
94C22×C4×C61—22no—
95C22×C4×C61—82no—
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61#4111yescharacteristic
99C2×D244—22no—
100C2×D244—22no—
101C2×D244—22no—
102C2×D244—42no—
103C2×D244—42no—
104C2×D244—41no—
105C2×D244—41no—
106C2×D244—41no—
107C2×D244—82no—
108C2×D244—82no—
109M16×C61—42no—
110C2×C8×C61—22no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—21no—
113C4⋊C4×C61—21no—
114C4⋊C4×C61—21no—
115C4⋊C4×C61—21no—
116C4⋊C4×C61—41no—
117C4⋊C4×C61—41no—
118C4⋊C4×C61C2×D412yes—
119C22⋊C4×C61—22no—
120C22⋊C4×C61—22no—
121C22⋊C4×C61—41no—
122C42×C61—41no—
123C42×C61C2×D411yescharacteristic
124C42×C61C2×D411yescharacteristic
125C42×C61#4111yescharacteristic
126C42×C61C2411yescharacteristic
127#11758201—84no—
128#11758199—48no—
129C2×#92418—22no—
130C2×#92418—22no—
131C2×#92418—42no—
132C2×#92418—84no—
133C2×#92418C2×D414yes—
134C23×C61—22no—
135C23×C61—24no—
136C23×C61—81no—
137C23×C61—81no—
138C23×C61#12111yescharacteristic
139C22×D61—42no—
140C22×D61—82no—
141C22×D61—82no—
142C22×D61—82no—
143Q8×C61—42no—
144Q8×C61—81no—
145D4×C61—22no—
146D4×C61—24no—
147D4×C61C2×SD1614yes—
148C2×C4×C61—22no—
149C2×C4×C61—22no—
150C2×C4×C61—41no—
151C2×C4×C61—44no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—82no—
155C2×C4×C61—81no—
156C2×C4×C61#12111yescharacteristic
157C2×C4×C61#12111yescharacteristic
158C2×C4×C61#12111yescharacteristic
159C2×C4×C61C2×SD1612yes—
160C2×C4×C61#13512yes—
161C2×C4×C61C22×D411yescharacteristic
162C2×C4×C61C22×D411yescharacteristic
163C2×C4×C61C2×#4111yescharacteristic
164D244—42no—
165D244—42no—
166D244—44no—
167D244—82no—
168D244—82no—
169D244—82no—
170D244—82no—
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#46512yes—
183C22×C61C2×#12111yescharacteristic
184C2×D61—82no—
185C2×D61—82no—
186C2×D61—82no—
187C2×D61—164no—
188C4×C61—21no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—42no—
192C4×C61—81no—
193C4×C61—81no—
194C4×C61#46512yes—
195C4×C61C22×SD1611yescharacteristic
196C4×C61C2×#13511yescharacteristic
197C4×C61#57612yes—
198#3208—611no—
199#3077D6111yescharacteristic, complemented
200C2×#459—611no—
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#304111yescharacteristic
210C2×C61C2×#46511yescharacteristic
211C2×C61#306811yescharacteristic
212D61—162no—
213C22×D8—1221no—
214#533C2×D6111yescharacteristic
215C2×#127—611no—
216C2×C4×D4C2×D6111yescharacteristic
217#492—612no—
218#486—612no—
219#478—22no—
220#476C2×D6112yescomplemented
221#474C2×D6112yescomplemented
222#460—1222no—
223#459—614no—
224#459—1222no—
225#450—1222no—
226C8×D4—614no—
227C8×D4—1222no—
228#431—611no—
229#431—611no—
230#428—21no—
231#423C2×D6111yescharacteristic
232C2×C4⋊C8—611no—
233#418—21no—
234C2×#102—1221no—
235C2×C22⋊C8—1222no—
236#334—614no—
237#333—612no—
238#332—612no—
239#331—614no—
240C61#1938411yescharacteristic, complemented, Sylow
241C22×D4—1221no—
242C22×D4—1221no—
243C22×D4—1221no—
244C22×D4C22⋊D6111yescharacteristic
245C23×C4—21no—
246C2×SD16—1224no—
247C2×D8—1224no—
248C2×D8—2444no—
249C2×M16—21no—
250C22×C8—1224no—
251C22×C8—1221no—
252C4⋊Q8C22×D6111yescharacteristic
253C4⋊Q8C22×D6111yescharacteristic
254#127—612no—
255#127—1221no—
256#127—1221no—
257#127—1221no—
258#125—21no—
259C22⋊Q8—21no—
260C22⋊Q8—21no—
261C4×Q8—21no—
262C4×Q8—21no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4C22⋊D6112yes—
266C4×D4C22×D6112yes—
267#117—21no—
268C2×C4⋊C4C22⋊D6111yescharacteristic
269C2×C22⋊C4—21no—
270C2×C42C22×D6111yescharacteristic
271#107—22no—
272#106—22no—
273C4⋊C8—21no—
274C4⋊C8—612no—
275C4⋊C8—612no—
276C4⋊C8—612no—
277C4⋊C8—1221no—
278C4⋊C8C22×D6112yes—
279#102—22no—
280#102—22no—
281#102—1224no—
282#102—1224no—
283C22⋊C8—612no—
284C22⋊C8—1224no—
285C22⋊C8—1221no—
286C4×C8—612no—
287C4×C8—1221no—
288C24—21no—
289C24—2441no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×D4—21no—
293C2×D4—22no—
294C2×D4—1222no—
295C2×D4—1222no—
296C2×D4—1222no—
297C2×D4—4882no—
298C2×D4—4882no—
299C2×D4—2441no—
300C2×D4—2441no—
301C2×D4—2441no—
302C2×D4—2442no—
303C2×D4—2442no—
304C2×D4#1175820114yescomplemented
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—2448no—
315M16—42no—
316C2×C8—22no—
317C2×C8—614no—
318C2×C8—1222no—
319C2×C8—1222no—
320C2×C8—4884no—
321C2×C8—2442no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4C2×C22⋊D6112yes—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—41no—
333C42—41no—
334C42D4×D6111yescharacteristic
335C42#1175822411yescharacteristic
336C42C2×C22⋊D6111yescharacteristic
337C42C23×D6111yescharacteristic
338C23—22no—
339C23—24no—
340C23—81no—
341C23—81no—
342C23—4882no—
343C23—4882no—
344C23—4882no—
345C23—2442no—
346C23#1451974911yescharacteristic
347Q8—42no—
348Q8—81no—
349D4—24no—
350D4—22no—
351D4—4882no—
352D4—4882no—
353D4—4882no—
354D4—4882no—
355D4—2442no—
356D4—2442no—
357D4—2444no—
358D4C2×#1175820114yes—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—44no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4#1451974911yescharacteristic
368C2×C4#1451974911yescharacteristic
369C2×C4#1451974911yescharacteristic
370C2×C4C2×#1175820112yes—
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—9764no—
386C22—4882no—
387C22—4882no—
388C22—4882no—
389C22#1683999312yes—
390C22C2×#1451974911yescharacteristic
391C4—21no—
392C4—21no—
393C4—21no—
394C4—42no—
395C4—81no—
396C4—81no—
397C4#1683999312yes—
398C4C22×#1175820111yescharacteristic
399C4C2×#1451979711yescharacteristic
400C4#1684056612yes—
401C2—22no—
402C2—44no—
403C2—9762no—
404C2#2145532411yescharacteristic
405C2C2×#1683999311yescharacteristic
406C2#2145554911yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3077C2
socleC22×C61C2×#121
prime radicalC2×#16839957C2
hypercenter#3077D61
hypocenterC61#19384
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3077

Complements

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

NH ≅ G/Ntype
1. #21456757C2semidirect
3. #21454843C2semidirect
4. #21445230C2semidirect
5. #21445162C2semidirect
6. #21444591C2semidirect
7. #21444488C2semidirect
8. #21444476C2semidirect
9. C61×#3077C2semidirect
12. #16840136C22semidirect
13. #16840110C22semidirect
19. D4×#92418C22semidirect
25. #16839512C22semidirect
26. #16839511C22semidirect
27. #16839508C22semidirect
28. #16839505C22semidirect
32. C61×#476C22semidirect
33. C61×#474C22semidirect
90. C2×D4×C61SD16semidirect
199. #3077D61semidirect
220. #476C2×D61semidirect
221. #474C2×D61semidirect
240. C61#19384semidirect
304. C2×D4#11758201semidirect

Maximal subgroups

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

subgroupindex
#214567572
C2×#168399572
#214548432
#214452302
#214451622
#214445912
#214444882
#214444762
C61×#30772
#1938461