Finite Group Databasebeta

← #30602040

Subgroups — #30602040

Order 15616.

494 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462000C211yesmaximal, characteristic
2#21462000C211yesmaximal, characteristic
3#21462000C211yesmaximal, characteristic
4#21461946C211yesmaximal, characteristic
5#21461754C211yesmaximal, characteristic
6#21461015C211yesmaximal, characteristic
7#487×D61C211yesmaximal, characteristic
8C2×#14519845—22no—
9C2×#14519840—22no—
10C4×D61⋊C8C2211yescharacteristic
11C4×D61⋊C8C2211yescharacteristic
12#16840708C2211yescharacteristic
13#16840708C2211yescharacteristic
14#16840635—22no—
15#16840635—22no—
16#16840635—22no—
17#92420⋊C8—22no—
18C4⋊Q8×D61C2211yescharacteristic
19C4⋊Q8×D61C2211yescharacteristic
20#16840114C411yescharacteristic, complemented
21#16840113C411yescharacteristic, complemented
22#16840112C411yescharacteristic, complemented
23#16840111—22no—
24#103×D61—22no—
25C4×C8×D61C2211yescharacteristic
26C61×#487C411yescharacteristic, complemented
27C2×D61⋊C8D412yes—
28C2×D61⋊C8D412yes—
29C2×C4⋊#6923—21no—
30C2×C4⋊#6923—21no—
31C2×C4⋊#6923—21no—
32C2×C4⋊#6923—21no—
33C2×C4⋊#6923—21no—
34C2×C4⋊#6923—21no—
35#14519845—44no—
36#14519840—44no—
37C4×#92420—21no—
38C4×#92420—21no—
39C4×#92420—21no—
40C4×#92420—21no—
41C2×Q8×D61—21no—
42C2×Q8×D61—21no—
43C2×C8×D61D412yes—
44#14519777C2×C411yescharacteristic
45#14519777C2×C411yescharacteristic
46#14519776—21no—
47#14519776—21no—
48#14519776—21no—
49#14519776—21no—
50#14519775—21no—
51#14519775—21no—
52C4⋊C4×D61—21no—
53C4⋊C4×D61—21no—
54C4⋊C4×D61—21no—
55C4⋊C4×D61—21no—
56C42×D61C2311yescharacteristic
57#14519702—22no—
58#14519701—22no—
59#14519700—22no—
60C4×#92418C2×C411yescharacteristic
61C8×Dic61D412yes—
62C4⋊Q8×C61C2×C411yescharacteristic
63C4⋊Q8×C61C2×C411yescharacteristic
64#103×C61—22no—
65C4×C8×C61C2×C411yescharacteristic
66C22×#6923—41no—
67C22×#6923—41no—
68C2×Q8×C61—21no—
69C2×Q8×C61—21no—
70Q8×D61—42no—
71Q8×D61—42no—
72Q8×D61—42no—
73Q8×D61—42no—
74C2×C4×D61—41no—
75C2×C4×D61—41no—
76C2×C4×D61C2×D411yescharacteristic
77C2×C4×D61C2×D411yescharacteristic
78C2×C4×D61C2×D411yescharacteristic
79C2×Dic122—21no—
80C2×Dic122—21no—
81C2×Dic122—41no—
82C2×Dic122—41no—
83C2×#92420—22no—
84C2×#92420—22no—
85C4⋊#6923—42no—
86C4⋊#6923—42no—
87C4⋊#6923—42no—
88C4⋊#6923—42no—
89C4⋊#6923—42no—
90C4⋊#6923—42no—
91D61⋊C8—22no—
92D61⋊C8—22no—
93D61⋊C8—22no—
94D61⋊C8SD1614yes—
95C2×C8×C61C22⋊C412yes—
96C4⋊C4×C61—21no—
97C4⋊C4×C61—21no—
98C4⋊C4×C61—21no—
99C4⋊C4×C61—21no—
100C42×C61C22×C411yescharacteristic
101C4⋊Dic61—21no—
102C4⋊Dic61—21no—
103C4⋊Dic61—21no—
104C4⋊Dic61—21no—
105#11758196—41no—
106#11758196—41no—
107#11758196—41no—
108#11758196—41no—
109C4×Dic61C2×D411yescharacteristic
110C4×Dic61#4111yescharacteristic
111C4×Dic61#4111yescharacteristic
112C2×#92418C22⋊C412yes—
113C8×D61—22no—
114C8×D61—22no—
115C22×D61#12711yescharacteristic
116C2×#6923—81no—
117C2×#6923—81no—
118C2×#6923—81no—
119C2×#6923—81no—
120C2×#6923—81no—
121C2×#6923—81no—
122Q8×C61—42no—
123Q8×C61—42no—
124C2×C4×C61—41no—
125C2×C4×C61—41no—
126C2×C4×C61C2×C22⋊C411yescharacteristic
127C2×C4×C61C4×D411yescharacteristic
128C2×C4×C61C4×D411yescharacteristic
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61#12111yescharacteristic
132C2×Dic61#12111yescharacteristic
133C2×Dic61#12411yescharacteristic
134C4×D61—81no—
135C4×D61—81no—
136C4×D61—81no—
137C4×D61—81no—
138C4×D61C2×SD1612yes—
139C4×D61#13512yes—
140C4×D61AGL(1,ℤ/8)12yes—
141C4×D61#13712yes—
142C4×D61#13712yes—
143C4×D61#13712yes—
144Dic122—42no—
145Dic122—42no—
146Dic122—82no—
147Dic122—82no—
148#92420—24no—
149#92420—42no—
150C8×C61—22no—
151#92418—22no—
152#9826—611nomaximal, Sylow
153C22×C61#39011yescharacteristic
154C2×D61#49211yescharacteristic
155C2×D61#49511yescharacteristic
156C2×D61#49611yescharacteristic
157C2×D61#49711yescharacteristic
158C2×D61#49711yescharacteristic
159C2×D61#49711yescharacteristic
160#6923—161no—
161#6923—161no—
162C4×C61—81no—
163C4×C61—81no—
164C4×C61#41712yes—
165C4×C61C4×SD1612yes—
166C4×C61#44112yes—
167Dic61—81no—
168Dic61—81no—
169Dic61#46512yes—
170Dic61#47012yes—
171Dic61#48912yes—
172C2×#487—611no—
173#1990—611no—
174#1990—611no—
175#1990—611no—
176#1956—611no—
177#1844—611no—
178C2×#322—611no—
179C2×C61#195911yescharacteristic
180C2×C61#196811yescharacteristic
181C2×C61#197111yescharacteristic
182D61#171411yescharacteristic
183D61#171611yescharacteristic
184D61#171811yescharacteristic
185D61#171911yescharacteristic
186C2×C4⋊Q8—611no—
187C2×C4⋊Q8—611no—
188#487—611no—
189#487—611no—
190#487—611no—
191#487—1222no—
192#487#692311yescharacteristic, complemented
193C2×#106—1222no—
194C2×#103—1222no—
195C2×#103—1222no—
196C2×C4×C8—611no—
197C2×C4×C8—611no—
198C2×C4×C8—611no—
199#384—611no—
200#384—611no—
201#340—1222no—
202#340—1222no—
203#340—1222no—
204#322—1222no—
205C61#982611yescharacteristic, complemented, Sylow
206C22×Q8—1221no—
207C22×Q8—1221no—
208C22×C8—612no—
209C22×C8—612no—
210C22×C8—612no—
211C4⋊Q8—611no—
212C4⋊Q8—611no—
213C4⋊Q8—1221no—
214C4⋊Q8—1221no—
215C4⋊Q8—1221no—
216C4⋊Q8—1221no—
217C4⋊Q8—1221no—
218C4⋊Q8—1221no—
219C4⋊Q8C2×#692311yescharacteristic
220C4⋊Q8C2×#692311yescharacteristic
221C2×C4⋊C4—1221no—
222C2×C4⋊C4—1221no—
223C2×C4⋊C4—1221no—
224C2×C4⋊C4—1221no—
225C2×C4⋊C4—1221no—
226C2×C4⋊C4—1221no—
227C2×C4⋊C4—1221no—
228C2×C4⋊C4—1221no—
229C2×C4⋊C4—1221no—
230C2×C4⋊C4—1221no—
231C2×C42—611no—
232#106—2444no—
233#103—22no—
234#103—1222no—
235#103—1222no—
236#103—1222no—
237#103—2444no—
238C4×C8—612no—
239C4×C8—611no—
240C4×C8—1221no—
241C4×C8—1221no—
242C4×C8—1221no—
243C4×C8—1221no—
244C4×C8C2×#692311yescharacteristic
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—1221no—
248C2×Q8—1221no—
249C2×Q8—2441no—
250C2×Q8—2441no—
251C2×Q8—2442no—
252C2×Q8—2442no—
253C2×Q8—2442no—
254C2×Q8—2442no—
255C22×C4—611no—
256C22×C4—611no—
257C22×C4—611no—
258C22×C4—2441no—
259C22×C4—2441no—
260C22×C4—2441no—
261C22×C4—2441no—
262C2×C8—614no—
263C2×C8—612no—
264C2×C8—1222no—
265C2×C8—1222no—
266C2×C8—1222no—
267C2×C8—1222no—
268C2×C8—1222no—
269C2×C8—1222no—
270C2×C8—1222no—
271C2×C8C22⋊#692312yes—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—1221no—
277C4⋊C4—1221no—
278C4⋊C4—1221no—
279C4⋊C4—1221no—
280C4⋊C4—2441no—
281C4⋊C4—2441no—
282C4⋊C4—2441no—
283C4⋊C4—2441no—
284C4⋊C4—2442no—
285C4⋊C4—2442no—
286C4⋊C4—2442no—
287C4⋊C4—2442no—
288C4⋊C4—2442no—
289C4⋊C4—2442no—
290C42—611no—
291C42—611no—
292C42—611no—
293C42C22×#692311yescharacteristic
294C23—611no—
295Q8—42no—
296Q8—42no—
297Q8—4882no—
298Q8—4882no—
299Q8—2442no—
300Q8—2442no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—611no—
304C2×C4—611no—
305C2×C4—611no—
306C2×C4—612no—
307C2×C4—612no—
308C2×C4—612no—
309C2×C4—612no—
310C2×C4—612no—
311C2×C4—612no—
312C2×C4—4881no—
313C2×C4—4881no—
314C2×C4—4881no—
315C2×C4—4881no—
316C2×C4—4881no—
317C2×C4—4881no—
318C2×C4—4881no—
319C2×C4—4881no—
320C2×C4—4881no—
321C2×C4—4881no—
322C2×C4—2441no—
323C2×C4—2441no—
324C2×C4C2×C22⋊#692311yescharacteristic
325C2×C4D4×#692311yescharacteristic
326C2×C4D4×#692311yescharacteristic
327C8—22no—
328C8—1224no—
329C8—1222no—
330C8—2442no—
331C22—611no—
332C22—611no—
333C22—611no—
334C22—611no—
335C22—611no—
336C22—611no—
337C22#1684071411yescharacteristic
338C4—81no—
339C4—81no—
340C4—612no—
341C4—612no—
342C4—612no—
343C4—9761no—
344C4—9761no—
345C4—4881no—
346C4—4881no—
347C4#1684075312yes—
348C4SD16×#692312yes—
349C4#1684080212yes—
350C2—611no—
351C2—611no—
352C2—611no—
353C2—611no—
354C2#2146194911yescharacteristic
355C2#2146196411yescharacteristic
356C2#2146196711yescharacteristic
subgroupG/H
centerC22#16840714
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#487C4
socleC22×C61#390
prime radicalC22×D61#127
cosocleC42×D61C23
hypercenter#487#6923
hypocenterC61#9826
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
20. #16840114C4semidirect
21. #16840113C4semidirect
22. #16840112C4semidirect
26. C61×#487C4semidirect
192. #487#6923semidirect
205. C61#9826semidirect

Maximal subgroups

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

subgroupindex
#214620002
#214619462
#214617542
#214610152
#487×D612
#982661