Finite Group Databasebeta

← #30605162

Subgroups — #30605162

Order 15616.

382 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 64 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#21461717C211yesmaximal, characteristic
2#21461073C211yesmaximal, characteristic
3#21460816C212yesmaximal
4#21460815C212yesmaximal
5#21451647C211yesmaximal, characteristic
6#16840783—21no—
7#16840705—21no—
8#16840640—21no—
9#16840640C2212yes—
10#16840607—21no—
11#16840606C2212yes—
12C2×#14519776C411yescharacteristic, complemented
13C2×#14519735C2211yescharacteristic
14#16839701C411yescharacteristic, complemented
15#16839699C2211yescharacteristic
16#16839697C2211yescharacteristic
17#16839683—21no—
18#16839675—21no—
19#16839659—21no—
20#16839626C411yescharacteristic, complemented
21#16839621—21no—
22C61×#400C411yescharacteristic, complemented
23C2×C22⋊#6923—22no—
24C4⋊#92420—41no—
25C22⋊#92420—22no—
26#14519832—41no—
27(C2×#6923)⋊C4—41no—
28(C2×#6923)⋊C4—41no—
29C22×Dic122—21no—
30C22×C4×D61D411yescharacteristic
31#14519776—41no—
32#14519776—41no—
33#14519735—41no—
34#14519735—41no—
35C2×C4⋊Dic61—21no—
36C2×C4⋊Dic61C2×C411yescharacteristic
37C2×C4⋊Dic61D411yescharacteristic
38C2×#11758196—21no—
39C2×#11758196—21no—
40C2×#11758196—21no—
41C2×#11758198—21no—
42C2×#11758198—21no—
43C2×#11758198D411yescharacteristic
44C2×#11758198C2311yescharacteristic
45C2×C4×Dic61C2×C411yescharacteristic
46C2×C4×Dic61D411yescharacteristic
47#14519684—21no—
48#14519684—21no—
49#14519684—21no—
50#14519684C2×C411yescharacteristic
51C2×C4⋊C4×C61C2×C411yescharacteristic
52C2×C4⋊C4×C61C2×C411yescharacteristic
53#95×C61—21no—
54#95×C61—21no—
55#95×C61C2×C411yescharacteristic
56C23×D61C2×D411yescharacteristic
57C22×#6923—42no—
58C22×C4×C61—21no—
59C22×C4×C61—21no—
60C22×C4×C61C22⋊C411yescharacteristic
61C22×C4×C61C22⋊C411yescharacteristic
62C22×C4×C61C22×C411yescharacteristic
63C22×Dic61—21no—
64C22×Dic61—21no—
65C22×Dic61C22⋊C411yescharacteristic
66C22×Dic61C22⋊C411yescharacteristic
67C22×Dic61C2×D411yescharacteristic
68C2×C4×D61—22no—
69C2×C4×D61—21no—
70C2×C4×D61—21no—
71C2×C4×D61—41no—
72C2×Dic122—42no—
73C2×Dic122—81no—
74C2×Dic122—81no—
75C22⋊#6923—42no—
76C22⋊#6923—42no—
77C2×#92420—42no—
78C4⋊C4×C61—41no—
79C4⋊C4×C61—41no—
80#11758198—41no—
81#11758198—41no—
82#11758198—81no—
83#11758198—81no—
84C4⋊Dic61—41no—
85C4⋊Dic61—41no—
86C4⋊Dic61—81no—
87#11758196—42no—
88#11758196—41no—
89#11758196—41no—
90#11758196—81no—
91C4×Dic61—21no—
92C4×Dic61—21no—
93C4×Dic61—21no—
94C4×Dic61—21no—
95C23×C61C2×C22⋊C411yescharacteristic
96C22×D61—21no—
97C22×D61—21no—
98C22×D61—41no—
99C22×D61—41no—
100C22×D61#13712yes—
101C2×#6923—82no—
102C2×#6923—82no—
103C2×#6923—82no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—41no—
110C2×C4×C61—41no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×Dic61—21no—
118C2×Dic61—21no—
119C2×Dic61—21no—
120C2×Dic61—21no—
121C2×Dic61—41no—
122C2×Dic61—41no—
123C2×Dic61—41no—
124C2×Dic61—41no—
125C2×Dic61—41no—
126C2×Dic61—41no—
127C2×Dic61—41no—
128C2×Dic61—41no—
129C2×Dic61#9911yescharacteristic
130C2×Dic61#9911yescharacteristic
131C4×D61—42no—
132C4×D61—42no—
133Dic122—84no—
134#92420—82no—
135#11813—611nomaximal, Sylow
136C22×C61—21no—
137C22×C61—21no—
138C22×C61C2×#9911yescharacteristic
139C22×C61#41911yescharacteristic
140C22×C61#42111yescharacteristic
141C2×D61—22no—
142C2×D61—41no—
143C2×D61—41no—
144C2×D61—42no—
145C2×D61—81no—
146#6923—162no—
147C4×C61—41no—
148C4×C61—41no—
149C4×C61—41no—
150C4×C61—81no—
151C4×C61—81no—
152Dic61—21no—
153Dic61—21no—
154Dic61—41no—
155Dic61—41no—
156Dic61—81no—
157Dic61—81no—
158#2628—611no—
159#1820—611no—
160#1464—611no—
161#1352—612no—
162#1352—612no—
163C2×C61—21no—
164C2×C61—21no—
165C2×C61#151111yescharacteristic
166C2×C61#212311yescharacteristic
167C2×C61#212811yescharacteristic
168D61—42no—
169C2×C4⋊Q8—611no—
170C2×#117—611no—
171#432—1221no—
172#400—611no—
173#400#692311yescharacteristic, complemented
174#399—611no—
175#399—611no—
176#395—1221no—
177#393—1221no—
178#389—1221no—
179#382—611no—
180#381—1221no—
181#380—1221no—
182#342—612no—
183#342—1221no—
184#324—1221no—
185#323—612no—
186C61#1181311yescharacteristic, complemented, Sylow
187C22×Q8—1221no—
188C23×C4—611no—
189C4⋊Q8—2441no—
190C4⋊Q8—2441no—
191#117—2441no—
192#117—2441no—
193C2×C4⋊C4—611no—
194C2×C4⋊C4—611no—
195C2×C4⋊C4—1221no—
196C2×C4⋊C4—1221no—
197C2×C4⋊C4—1221no—
198C2×C4⋊C4—1221no—
199C2×C4⋊C4C2×#692311yescharacteristic
200C2×C4⋊C4C2×#692311yescharacteristic
201C2×C22⋊C4—611no—
202C2×C22⋊C4—611no—
203C2×C22⋊C4—1221no—
204C2×C22⋊C4—1221no—
205C2×C22⋊C4—1222no—
206C2×C42—611no—
207C2×C42—611no—
208C4⋊C8—2441no—
209C22⋊C8—1222no—
210#97—2441no—
211#95—21no—
212#95—21no—
213#95—611no—
214#95—1221no—
215#95—1221no—
216#95—1221no—
217#95—2441no—
218#95—2441no—
219#95C2×#692311yescharacteristic
220C24—611no—
221C2×Q8—4881no—
222C2×Q8—4881no—
223C2×Q8—2442no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—611no—
227C22×C4—611no—
228C22×C4—611no—
229C22×C4—1221no—
230C22×C4—1221no—
231C22×C4—1222no—
232C22×C4—1221no—
233C22×C4—1221no—
234C22×C4—2441no—
235C22×C4—2442no—
236C22×C4C22⋊#692311yescharacteristic
237C22×C4C22⋊#692311yescharacteristic
238C22×C4C22×#692311yescharacteristic
239C2×C8—2442no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—4881no—
243C4⋊C4—4881no—
244C4⋊C4—2441no—
245C4⋊C4—2441no—
246C4⋊C4—2441no—
247C4⋊C4—2441no—
248C4⋊C4—2442no—
249C22⋊C4—4881no—
250C22⋊C4—4881no—
251C22⋊C4—2441no—
252C22⋊C4—2441no—
253C22⋊C4—2442no—
254C22⋊C4—2442no—
255C42—1221no—
256C42—1221no—
257C42—1221no—
258C42—1221no—
259C23—612no—
260C23—1221no—
261C23—1221no—
262C23—2441no—
263C23—2441no—
264C23C2×C22⋊#692311yescharacteristic
265Q8—4884no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—611no—
280C2×C4—611no—
281C2×C4—1221no—
282C2×C4—1221no—
283C2×C4—1221no—
284C2×C4—1221no—
285C2×C4—4882no—
286C2×C4—4882no—
287C2×C4—4882no—
288C2×C4—2442no—
289C2×C4—2442no—
290C2×C4—2441no—
291C2×C4—2441no—
292C2×C4—2441no—
293C2×C4—2441no—
294C2×C4—2441no—
295C2×C4—2441no—
296C2×C4—2441no—
297C2×C4—2441no—
298C8—4882no—
299C22—21no—
300C22—21no—
301C22—1222no—
302C22—4881no—
303C22—2442no—
304C22—2441no—
305C22—2441no—
306C22D61⋊#9911yescharacteristic
307C22#1684075511yescharacteristic
308C22#1684075911yescharacteristic
309C4—41no—
310C4—41no—
311C4—41no—
312C4—81no—
313C4—81no—
314C4—1221no—
315C4—1221no—
316C4—9762no—
317C4—4881no—
318C4—4881no—
319C4—2441no—
320C4—2441no—
321C2—21no—
322C2—21no—
323C2—2442no—
324C2#2146122811yescharacteristic
325C2#2146208211yescharacteristic
326C2#2146209911yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#400C4
socleC22×C61C2×#99
prime radicalC23×D61C2×D4
cosocleC2×#11758198C23
hypercenter#400#6923
hypocenterC61#11813
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #400

Complements

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

NH ≅ G/Ntype
12. C2×#14519776C4semidirect
14. #16839701C4semidirect
20. #16839626C4semidirect
22. C61×#400C4semidirect
173. #400#6923semidirect
186. C61#11813semidirect

Maximal subgroups

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

subgroupindex
#214617172
#214610732
#214608162
#214608152
#214516472
#1181361