Finite Group Databasebeta

← #30600738

Subgroups — #30600738

Order 15616.

416 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 76 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#21462727C211yesmaximal, characteristic, complemented
2#21461213C211yesmaximal, characteristic, complemented
3#21461210C211yesmaximal, characteristic
4#21461180C211yesmaximal, characteristic
5#21460676C211yesmaximal, characteristic, complemented
6#21460642C211yesmaximal, characteristic, complemented
7#21456772C211yesmaximal, characteristic
8#16840813—22no—
9#16840812—21no—
10#16840811—21no—
11#16840787—22no—
12#16840784C2211yescharacteristic
13#16840784C2211yescharacteristic
14#16840780C2211yescharacteristic
15#16840698C2211yescharacteristic
16#16840616—22no—
17#16840614—22no—
18#16840610—21no—
19#16840610—21no—
20#16840317C2211yescharacteristic
21#16840313C2211yescharacteristic
22#16840136C411yescharacteristic, complemented
23#16840112C411yescharacteristic, complemented
24#16840109C411yescharacteristic, complemented
25#16839986—22no—
26#16839923—22no—
27#16839813C2211yescharacteristic
28C61×#494C411yescharacteristic, complemented
29C2×C4×#6923—21no—
30#14519844—22no—
31#14519844—21no—
32C4⋊#92420—22no—
33C4⋊#92420—21no—
34C4⋊#92420—21no—
35#14519834—21no—
36#14519834—21no—
37#14519834D412yes—
38#14519832—22no—
39#14519832—21no—
40#14519832—21no—
41C4×#92420—22no—
42C4×#92420D411yescharacteristic
43C4×#92420D411yescharacteristic
44(C2×#6923)⋊C4—21no—
45(C2×#6923)⋊C4—21no—
46C2×#11758226—21no—
47C2×#11758201—22no—
48#14519777C2×C411yescharacteristic
49#14519774C2×C411yescharacteristic
50#14519768—21no—
51#14519766—21no—
52#14519757—21no—
53#14519753D412yes—
54#14519752—21no—
55#14519752—21no—
56Q8×Dic61D412yes—
57#14519742—21no—
58#14519739—21no—
59#14519739—21no—
60#14519735—21no—
61C42×D61C2311yescharacteristic
62#14519706—21no—
63#14519701—22no—
64#14519699—22no—
65#14519689—21no—
66C4×#92418C2×C411yescharacteristic
67C2×Q16×C61—22no—
68C4⋊Q8×C61C2×C411yescharacteristic
69C4⋊Q8×C61C2×C411yescharacteristic
70C4×C8×C61C2×C411yescharacteristic
71C22×#6923—41no—
72C2×Q8×C61—21no—
73C2×Q8×C61C22⋊C412yes—
74#11758226—82no—
75C2×D244—21no—
76C2×D244—21no—
77C2×D244—21no—
78C2×D244—41no—
79C2×C4×D61—41no—
80C2×C4×D61C2×D411yescharacteristic
81C2×C4×D61C2×D411yescharacteristic
82C2×C4×D61#4111yescharacteristic
83C2×Dic122—21no—
84C2×#92420—21no—
85C2×#92420—22no—
86C2×#92420—21no—
87C2×#92420—41no—
88C4×#6923—22no—
89C4×#6923—41no—
90Q16×C61—82no—
91C2×C8×C61—21no—
92C4⋊C4×C61—21no—
93C4⋊C4×C61—21no—
94C4⋊C4×C61—21no—
95C4⋊C4×C61—21no—
96C42×C61C22×C411yescharacteristic
97#11758201—82no—
98#11758198—22no—
99#11758198—41no—
100#11758198—41no—
101C4⋊Dic61—21no—
102C4⋊Dic61—22no—
103C4⋊Dic61C22⋊C412yes—
104#11758196—22no—
105C4×Dic61—22no—
106C4×Dic61C2×D411yescharacteristic
107C4×Dic61C2×D411yescharacteristic
108C4×Dic61#4111yescharacteristic
109C2×#92418—21no—
110C22×D61—41no—
111C22×D61#12111yescharacteristic
112C2×#6923—42no—
113C2×#6923—81no—
114C2×#6923—81no—
115Q8×C61—42no—
116Q8×C61—42no—
117C2×C4×C61—22no—
118C2×C4×C61—41no—
119C2×C4×C61C2×C22⋊C411yescharacteristic
120C2×C4×C61C4×D411yescharacteristic
121C2×C4×C61C4×D411yescharacteristic
122C2×Dic61—22no—
123C2×Dic61#12011yescharacteristic
124C2×Dic61#12111yescharacteristic
125C2×Dic61#12311yescharacteristic
126D244—42no—
127D244—41no—
128D244—41no—
129D244—81no—
130D244—82no—
131C4×D61—21no—
132C4×D61—21no—
133C4×D61—41no—
134C4×D61—41no—
135C4×D61—82no—
136Dic122—41no—
137Dic122—41no—
138#92420—41no—
139#92420—42no—
140#92420—41no—
141#92420—81no—
142C8×C61—41no—
143#92418—41no—
144#9282—611nomaximal, Sylow
145C22×C61#38611yescharacteristic
146C2×D61—21no—
147C2×D61—21no—
148C2×D61—21no—
149C2×D61—81no—
150C2×D61—81no—
151C2×D61—81no—
152#6923—82no—
153C4×C61—21no—
154C4×C61—21no—
155C4×C61—42no—
156C4×C61—81no—
157C4×C61#44311yescharacteristic
158C4×C61#44411yescharacteristic
159Dic61—21no—
160Dic61—21no—
161Dic61—21no—
162Dic61—42no—
163#3210—611no—
164#2985—611no—
165#1506—611no—
166#1505—611no—
167#1496—611no—
168#1287—611no—
169#1274—611no—
170C2×C61#187611yescharacteristic
171C2×C61#190511yescharacteristic
172C2×C61#191011yescharacteristic
173D61—41no—
174D61—161no—
175#533—611no—
176#532—611no—
177#494#692311yescharacteristic, complemented
178#492—611no—
179#487—611no—
180#486—611no—
181#464—1222no—
182#452—1222no—
183#446—1221no—
184#446—1222no—
185C8×Q8—1221no—
186#433—1222no—
187#432—611no—
188#432—611no—
189#431—611no—
190#431—611no—
191#376—611no—
192#328—1222no—
193#327—1222no—
194D4⋊C8—1221no—
195D4⋊C8—1221no—
196C61#928211yescharacteristic, complemented, Sylow
197C2×#41—1221no—
198C2×Q16—22no—
199C2×SD16—1222no—
200C4⋊Q8—611no—
201C4⋊Q8C2×#692311yescharacteristic
202C4⋊Q8C2×#692311yescharacteristic
203#127—611no—
204#125—1221no—
205#124—1221no—
206C22⋊Q8—612no—
207C22⋊Q8—1221no—
208#121—1221no—
209#121—1221no—
210C4×Q8—612no—
211C4×Q8—1221no—
212C4×D4—1221no—
213C4×D4—1221no—
214#117—1221no—
215C2×C42—611no—
216C2×C42—1221no—
217C4⋊C8—1221no—
218C4⋊C8—1221no—
219C4⋊C8—1221no—
220C4⋊C8—1221no—
221C4⋊C8—1222no—
222C4⋊C8—1222no—
223#103—1222no—
224#102—1222no—
225C22⋊C8—612no—
226C22⋊C8—1221no—
227C22⋊C8—1221no—
228C22⋊C8—1221no—
229#97—1221no—
230#97—1221no—
231#97—1222no—
232C4×C8—611no—
233C4×C8—611no—
234C4×C8—611no—
235C4×C8—1222no—
236C4×C8C2×#692311yescharacteristic
237#95—1221no—
238#95—1221no—
239#41—4882no—
240C2×Q8—21no—
241C2×Q8—1221no—
242C2×Q8C22⋊#692312yes—
243C2×D4—1221no—
244C2×D4—1221no—
245C2×D4—1221no—
246C2×D4—2441no—
247C22×C4—611no—
248C22×C4—611no—
249C22×C4—611no—
250C22×C4—2441no—
251C22×C4—2441no—
252Q16—82no—
253SD16—4882no—
254C2×C8—21no—
255C2×C8—1221no—
256C2×C8—1221no—
257C2×C8—1222no—
258C2×C8—1221no—
259C2×C8—2441no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—612no—
265C4⋊C4—1222no—
266C4⋊C4—1222no—
267C4⋊C4—1221no—
268C22⋊C4—1222no—
269C22⋊C4—2441no—
270C22⋊C4—2441no—
271C42—611no—
272C42—611no—
273C42—611no—
274C42—1222no—
275C42—1222no—
276C42—2441no—
277C42C22×#692311yescharacteristic
278C23—611no—
279C23—2441no—
280Q8—42no—
281Q8—42no—
282Q8—2441no—
283Q8—2441no—
284D4—4881no—
285D4—4882no—
286D4—2442no—
287D4—2441no—
288D4—2441no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—611no—
292C2×C4—611no—
293C2×C4—611no—
294C2×C4—1221no—
295C2×C4—1221no—
296C2×C4—1222no—
297C2×C4—4882no—
298C2×C4—4881no—
299C2×C4—4881no—
300C2×C4—2441no—
301C2×C4—2441no—
302C2×C4—2442no—
303C2×C4C2×C22⋊#692311yescharacteristic
304C2×C4D4×#692311yescharacteristic
305C2×C4D4×#692311yescharacteristic
306C8—41no—
307C8—4881no—
308C8—2441no—
309C8—2441no—
310C8—2441no—
311C8—2442no—
312C22—1221no—
313C22—1221no—
314C22—1221no—
315C22—4881no—
316C22—4881no—
317C22—4881no—
318C22#1684071011yescharacteristic
319C4—21no—
320C4—21no—
321C4—42no—
322C4—81no—
323C4—1221no—
324C4—1221no—
325C4—1221no—
326C4—4882no—
327C4—2442no—
328C4#1684080611yescharacteristic
329C4#1684080811yescharacteristic
330C2—9761no—
331C2—2441no—
332C2#2146180911yescharacteristic
333C2#2146185411yescharacteristic
334C2#2146186211yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#494C4
socleC22×C61#386
prime radical#16840313C22
cosocleC42×D61C23
hypercenter#494#6923
hypocenterC61#9282
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
1. #21462727C2semidirect
2. #21461213C2semidirect
5. #21460676C2semidirect
6. #21460642C2semidirect
22. #16840136C4semidirect
23. #16840112C4semidirect
24. #16840109C4semidirect
28. C61×#494C4semidirect
177. #494#6923semidirect
196. C61#9282semidirect

Maximal subgroups

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

subgroupindex
#214627272
#214612132
#214612102
#214611802
#214606762
#214606422
#214567722
#928261