Finite Group Databasebeta

← #29572269

Subgroups — #29572269

Order 15616.

678 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 678 conjugacy classes of subgroups — too many to draw. Only its 131 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#21444481C211yesmaximal, characteristic, complemented
2#21444454C211yesmaximal, characteristic, complemented
3#21444253C212yesmaximal, complemented
4#21444175C212yesmaximal, complemented
5C61×#3512C211yesmaximal, characteristic
6#16839819C2212yes—
7#16839815C2212yes—
8#16839511—21no—
9#16839506—21no—
10#16839502—22no—
11Q8⋊#92418—22no—
12D4⋊#92418—22no—
13#16839488—24no—
14C61×#553—22no—
15C61×#553—24no—
16C61×#552—22no—
17C61×#541—21no—
18C61×#540—21no—
19C61×#537C412yes—
20C61×#532C2211yescharacteristic
21C61×#529—22no—
22C2×#125×C61C2211yescharacteristic
23C61×#520C412yes—
24C4×#41×C61C2211yescharacteristic
25C4⋊#92418—22no—
26C4⋊#92418—22no—
27C22⋊#92418—22no—
28C22⋊#92418—22no—
29#14519688—22no—
30C4×#92418—22no—
31C2×#41×C61C2×C412yes—
32C2×#41×C61C2×C411yescharacteristic
33C4⋊Q8×C61D411yescharacteristic
34#127×C61D411yescharacteristic
35#126×C61—24no—
36#126×C61—24no—
37#125×C61—24no—
38#125×C61—21no—
39#125×C61—22no—
40#125×C61—21no—
41#124×C61D412yes—
42#123×C61—24no—
43#123×C61—24no—
44C22⋊Q8×C61—22no—
45C22⋊Q8×C61—22no—
46#121×C61—24no—
47#121×C61—22no—
48#121×C61—22no—
49#121×C61—24no—
50#121×C61—22no—
51C4×Q8×C61—22no—
52C4×Q8×C61D412yes—
53C4×D4×C61—24no—
54C4×D4×C61—22no—
55C4×D4×C61—24no—
56C4×D4×C61—22no—
57C4×D4×C61—22no—
58C4×D4×C61D412yes—
59#117×C61—24no—
60#117×C61—22no—
61#117×C61—21no—
62#117×C61C2×C411yescharacteristic
63C2×C4⋊C4×C61—22no—
64C2×C4⋊C4×C61C2×C412yes—
65C2×C4⋊C4×C61D414yes—
66C2×C42×C61—21no—
67C2×C42×C61C2311yescharacteristic
68#41×C61—42no—
69#41×C61—44no—
70C2×Q8×C61C22⋊C411yescharacteristic
71C2×Q8×C61C22⋊C412yes—
72C2×D4×C61—22no—
73C2×D4×C61—21no—
74C2×D4×C61—24no—
75C2×D4×C61—22no—
76C2×D4×C61C22⋊C411yescharacteristic
77C2×D4×C61C22⋊C412yes—
78C22×C4×C61—24no—
79C22×C4×C61—21no—
80C22×C4×C61—22no—
81C22×C4×C61—22no—
82C22×C4×C61C22⋊C414yes—
83C22×C4×C61C22×C411yescharacteristic
84C22×C4×C61C2×D412yes—
85C4⋊C4×C61—24no—
86C4⋊C4×C61—24no—
87C4⋊C4×C61—22no—
88C4⋊C4×C61—22no—
89C4⋊C4×C61—24no—
90C4⋊C4×C61—24no—
91C4⋊C4×C61—22no—
92C4⋊C4×C61—22no—
93C4⋊C4×C61C22⋊C412yes—
94C22⋊C4×C61—24no—
95C22⋊C4×C61—24no—
96C22⋊C4×C61—24no—
97C22⋊C4×C61—21no—
98C22⋊C4×C61—22no—
99C22⋊C4×C61—22no—
100C22⋊C4×C61—22no—
101C42×C61—24no—
102C42×C61—21no—
103C42×C61—22no—
104C42×C61C2×D411yescharacteristic
105C42×C61C2×D411yescharacteristic
106C42×C61C2×D412yes—
107C2×#92418—42no—
108C2×#92418—42no—
109C23×C61—22no—
110C23×C61—21no—
111C23×C61C2×C22⋊C411yescharacteristic
112Q8×C61—22no—
113Q8×C61—42no—
114D4×C61—22no—
115D4×C61—42no—
116D4×C61—42no—
117D4×C61—44no—
118D4×C61—82no—
119D4×C61—84no—
120C2×C4×C61—22no—
121C2×C4×C61—22no—
122C2×C4×C61—21no—
123C2×C4×C61—22no—
124C2×C4×C61—24no—
125C2×C4×C61—42no—
126C2×C4×C61—42no—
127C2×C4×C61—44no—
128C2×C4×C61—44no—
129C2×C4×C61—82no—
130C2×C4×C61—84no—
131C2×C4×C61C2×C22⋊C411yescharacteristic
132C2×C4×C61C2×C22⋊C411yescharacteristic
133C2×C4×C61#12012yes—
134C2×C4×C61#12012yes—
135#92418—82no—
136#92418—82no—
137#8476—611nomaximal, Sylow
138C22×C61—22no—
139C22×C61—21no—
140C22×C61—42no—
141C22×C61—42no—
142C22×C61—42no—
143C22×C61—41no—
144C22×C61—42no—
145C22×C61#37911yescharacteristic
146C4×C61—22no—
147C4×C61—22no—
148C4×C61—21no—
149C4×C61—44no—
150C4×C61—42no—
151C4×C61—41no—
152C4×C61—42no—
153C4×C61#41312yes—
154#3512D6111yescharacteristic
155#1543—611no—
156#1536—611no—
157#1488—612no—
158#1470—612no—
159C2×C61—41no—
160C2×C61—82no—
161C2×C61—81no—
162C2×C61#178511yescharacteristic
163C2×C61#179412yes—
164#553—22no—
165#553—24no—
166#552—22no—
167#541—21no—
168#540—21no—
169#537Dic6112yes—
170#532C2×D6111yescharacteristic
171#529—22no—
172C2×#125C2×D6111yescharacteristic
173#520Dic6112yes—
174C4×#41C2×D6111yescharacteristic
175#432—612no—
176#431—612no—
177#333—1221no—
178#331—1221no—
179#329—1222no—
180Q8⋊C8—1222no—
181D4⋊C8—1222no—
182#324—1224no—
183C61#847611yescharacteristic, complemented, Sylow
184C2×#41C2×Dic6111yescharacteristic
185C2×#41C2×Dic6112yes—
186C4⋊Q8C22⋊D6111yescharacteristic
187#127C22⋊D6111yescharacteristic
188#126—24no—
189#126—24no—
190#125—24no—
191#125—21no—
192#125—21no—
193#125—22no—
194#124C22⋊D6112yes—
195#123—24no—
196#123—24no—
197C22⋊Q8—22no—
198C22⋊Q8—22no—
199#121—24no—
200#121—22no—
201#121—22no—
202#121—24no—
203#121—22no—
204C4×Q8—22no—
205C4×Q8C22⋊D6112yes—
206C4×D4—24no—
207C4×D4—22no—
208C4×D4—24no—
209C4×D4—22no—
210C4×D4—22no—
211C4×D4C22⋊D6112yes—
212#117—24no—
213#117—22no—
214#117—21no—
215#117C2×Dic6111yescharacteristic
216C2×C4⋊C4—22no—
217C2×C4⋊C4C2×Dic6112yes—
218C2×C4⋊C4C22⋊D6114yes—
219C2×C42—21no—
220C2×C42C22×D6111yescharacteristic
221C4⋊C8—1222no—
222C4⋊C8—1222no—
223C22⋊C8—1222no—
224C22⋊C8—1222no—
225#97—1222no—
226C4×C8—1222no—
227#41—42no—
228#41—44no—
229C2×Q8C22⋊Dic6111yescharacteristic
230C2×Q8C22⋊Dic6112yes—
231C2×D4—22no—
232C2×D4—21no—
233C2×D4—24no—
234C2×D4—22no—
235C2×D4C22⋊Dic6111yescharacteristic
236C2×D4C22⋊Dic6112yes—
237C22×C4—24no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4C22⋊Dic6114yes—
242C22×C4C22×Dic6111yescharacteristic
243C22×C4C2×C22⋊D6112yes—
244C2×C8—2442no—
245C2×C8—2442no—
246C4⋊C4—24no—
247C4⋊C4—24no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4C22⋊Dic6112yes—
255C22⋊C4—24no—
256C22⋊C4—24no—
257C22⋊C4—24no—
258C22⋊C4—21no—
259C22⋊C4—22no—
260C22⋊C4—22no—
261C22⋊C4—22no—
262C42—24no—
263C42—21no—
264C42—22no—
265C42C2×C22⋊D6112yes—
266C42C2×C22⋊D6111yescharacteristic
267C42C2×C22⋊D6111yescharacteristic
268C23—21no—
269C23—22no—
270C23C2×C22⋊Dic6111yescharacteristic
271Q8—22no—
272Q8—42no—
273D4—22no—
274D4—42no—
275D4—42no—
276D4—44no—
277D4—82no—
278D4—84no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—82no—
289C2×C4—84no—
290C2×C4C2×C22⋊Dic6111yescharacteristic
291C2×C4C2×C22⋊Dic6111yescharacteristic
292C2×C4#1451974612yes—
293C2×C4#1451974612yes—
294C8—4882no—
295C8—4882no—
296C22—21no—
297C22—22no—
298C22—41no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22#1683961811yescharacteristic
304C4—21no—
305C4—22no—
306C4—22no—
307C4—41no—
308C4—42no—
309C4—44no—
310C4—42no—
311C4#1683974612yes—
312C2—41no—
313C2—81no—
314C2—82no—
315C2#2144574711yescharacteristic
316C2#2144578312yes—
subgroupG/H
centerC22#16839618
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
Fitting, prime radicalC61×#3512C2
socleC22×C61#379
cosocleC2×C42×C61C23
hypercenter#3512D61
hypocenterC61#8476
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3512

Complements

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

NH ≅ G/Ntype
1. #21444481C2semidirect
2. #21444454C2semidirect
3. #21444253C2semidirect
4. #21444175C2semidirect
183. C61#8476semidirect

Maximal subgroups

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

subgroupindex
#214444812
#214444542
#214442532
#214441752
C61×#35122
#847661