Finite Group Databasebeta

← #29571432

Subgroups — #29571432

Order 15616.

1146 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1146 conjugacy classes of subgroups — too many to draw. Only its 203 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#21444210C214yesmaximal, complemented
2#21444133C212yesmaximal, complemented
3C2×C61×#518C211yesmaximal, characteristic
4#16839815C2214yescomplemented
5D4⋊#92418—28no—
6#16839488—22no—
7#16839486—22no—
8C23×D4×C61C411yescharacteristic
9C61×#518—28no—
10C2×C4×D4×C61—22no—
11C2×C4×D4×C61C412yes—
12C2×C4×D4×C61C2212yes—
13C2×#117×C61C411yescharacteristic
14C2×#117×C61C2211yescharacteristic
15C22×C22⋊C4×C61—22no—
16C4⋊#92418—24no—
17C22⋊#92418—24no—
18C4×#92418—24no—
19C25×C61—21no—
20C22×D4×C61—24no—
21C22×D4×C61—24no—
22C22×D4×C61—21no—
23C22×D4×C61—22no—
24C22×D4×C61C814yescomplemented
25C22×D4×C61C2×C412yes—
26C23×C4×C61—22no—
27C23×C4×C61—22no—
28C23×C4×C61C2×C411yescharacteristic
29C4×D4×C61—216no—
30C4×D4×C61—24no—
31C4×D4×C61—24no—
32C4×D4×C61—24no—
33C4×D4×C61D418yes—
34#117×C61—24no—
35#117×C61—24no—
36C2×C4⋊C4×C61C2×C412yes—
37C2×C4⋊C4×C61D412yes—
38C2×C22⋊C4×C61—216no—
39C2×C22⋊C4×C61—21no—
40C2×C22⋊C4×C61—22no—
41C2×C22⋊C4×C61—22no—
42C2×C22⋊C4×C61D412yes—
43C2×C42×C61—21no—
44C2×C42×C61C2×C411yescharacteristic
45C2×C42×C61C2311yescharacteristic
46C24×C61—24no—
47C24×C61—24no—
48C24×C61—22no—
49C24×C61—24no—
50C24×C61—84no—
51C24×C61C22⋊C411yescharacteristic
52C2×D4×C61—24no—
53C2×D4×C61—24no—
54C2×D4×C61—216no—
55C2×D4×C61—216no—
56C2×D4×C61—24no—
57C2×D4×C61—22no—
58C2×D4×C61—22no—
59C2×D4×C61—22no—
60C2×D4×C61C22⋊C414yes—
61C2×D4×C61C2×C814yes—
62C2×D4×C61M1614yes—
63C22×C4×C61—22no—
64C22×C4×C61—22no—
65C22×C4×C61—22no—
66C22×C4×C61—28no—
67C22×C4×C61—28no—
68C22×C4×C61—84no—
69C22×C4×C61—84no—
70C22×C4×C61C22⋊C411yescharacteristic
71C22×C4×C61C22⋊C412yes—
72C22×C4×C61C2×C812yes—
73C22×C4×C61M1612yes—
74C22×C4×C61C22×C411yescharacteristic
75C22×C4×C61C2×D412yes—
76C4⋊C4×C61—22no—
77C4⋊C4×C61—24no—
78C4⋊C4×C61C22⋊C414yes—
79C22⋊C4×C61—24no—
80C22⋊C4×C61—28no—
81C22⋊C4×C61—24no—
82C22⋊C4×C61—28no—
83C42×C61—24no—
84C42×C61—22no—
85C42×C61C2×D414yes—
86C2×#92418—44no—
87C23×C61—216no—
88C23×C61—21no—
89C23×C61—24no—
90C23×C61—24no—
91C23×C61—21no—
92C23×C61—24no—
93C23×C61—44no—
94C23×C61—44no—
95C23×C61—42no—
96C23×C61—88no—
97C23×C61—88no—
98C23×C61—88no—
99C23×C61C22⋊C812yes—
100C23×C61C2×C22⋊C411yescharacteristic
101D4×C61—216no—
102D4×C61—24no—
103D4×C61—24no—
104D4×C61—24no—
105D4×C61C22⋊C818yes—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—24no—
109C2×C4×C61—24no—
110C2×C4×C61—22no—
111C2×C4×C61—24no—
112C2×C4×C61—42no—
113C2×C4×C61—44no—
114C2×C4×C61—42no—
115C2×C4×C61—88no—
116C2×C4×C61—88no—
117C2×C4×C61C22⋊C812yes—
118C2×C4×C61C2×C22⋊C412yes—
119C2×C4×C61#12014yes—
120C2×C4×C61C22×C811yescharacteristic
121C2×C4×C61C2×M1612yes—
122C2×C4×C61C2×M1611yescharacteristic
123#92418—84no—
124#8322—611nomaximal, Sylow
125C22×C61—22no—
126C22×C61—22no—
127C22×C61—28no—
128C22×C61—42no—
129C22×C61—48no—
130C22×C61—48no—
131C22×C61—42no—
132C22×C61—44no—
133C22×C61—44no—
134C22×C61—42no—
135C22×C61—816no—
136C22×C61#37911yescharacteristic
137C22×C61C2×C22⋊C811yescharacteristic
138C22×C61#40711yescharacteristic
139C4×C61—24no—
140C4×C61—22no—
141C4×C61—44no—
142C4×C61—42no—
143C4×C61C2×C22⋊C812yes—
144C4×C61#40712yes—
145C2×#518D6111yescharacteristic
146#1478—614no—
147#1458—612no—
148C2×C61—21no—
149C2×C61—21no—
150C2×C61—48no—
151C2×C61—42no—
152C2×C61#177911yescharacteristic
153C2×C61#179512yes—
154C23×D4Dic6111yescharacteristic
155#518—28no—
156C2×C4×D4—22no—
157C2×C4×D4Dic6112yes—
158C2×C4×D4C2×D6112yes—
159C2×#117Dic6111yescharacteristic
160C2×#117C2×D6111yescharacteristic
161C22×C22⋊C4—22no—
162#431—614no—
163D4⋊C8—1228no—
164#324—1222no—
165#323—1222no—
166C61#832211yescharacteristic, complemented, Sylow
167C25—21no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—22no—
171C22×D4—21no—
172C22×D4#9241814yescomplemented
173C22×D4C2×Dic6112yes—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4C2×Dic6111yescharacteristic
177C4×D4—216no—
178C4×D4—24no—
179C4×D4—24no—
180C4×D4—24no—
181C4×D4C22⋊D6118yes—
182#117—24no—
183#117—24no—
184C2×C4⋊C4C2×Dic6112yes—
185C2×C4⋊C4C22⋊D6112yes—
186C2×C22⋊C4—216no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4C22⋊D6112yes—
191C2×C42—21no—
192C2×C42C2×Dic6111yescharacteristic
193C2×C42C22×D6111yescharacteristic
194C4⋊C8—1224no—
195C22⋊C8—1224no—
196C4×C8—1224no—
197C24—24no—
198C24—24no—
199C24—22no—
200C24—24no—
201C24—84no—
202C24C22⋊Dic6111yescharacteristic
203C2×D4—24no—
204C2×D4—24no—
205C2×D4—216no—
206C2×D4—216no—
207C2×D4—24no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4C2×#9241814yes—
212C2×D4#1175819414yes—
213C2×D4C22⋊Dic6114yes—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—28no—
218C22×C4—28no—
219C22×C4—84no—
220C22×C4—84no—
221C22×C4C2×#9241812yes—
222C22×C4#1175819412yes—
223C22×C4C22⋊Dic6111yescharacteristic
224C22×C4C22⋊Dic6112yes—
225C22×C4C22×Dic6111yescharacteristic
226C22×C4C2×C22⋊D6112yes—
227C2×C8—2444no—
228C4⋊C4—22no—
229C4⋊C4—24no—
230C4⋊C4C22⋊Dic6114yes—
231C22⋊C4—24no—
232C22⋊C4—28no—
233C22⋊C4—24no—
234C22⋊C4—28no—
235C42—24no—
236C42—22no—
237C42C2×C22⋊D6114yes—
238C23—216no—
239C23—24no—
240C23—21no—
241C23—24no—
242C23—21no—
243C23—24no—
244C23—42no—
245C23—44no—
246C23—44no—
247C23—88no—
248C23—88no—
249C23—88no—
250C23C22⋊#9241812yes—
251C23C2×C22⋊Dic6111yescharacteristic
252D4—216no—
253D4—24no—
254D4—24no—
255D4—24no—
256D4C22⋊#9241818yes—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—42no—
266C2×C4—88no—
267C2×C4—88no—
268C2×C4C22⋊#9241812yes—
269C2×C4C2×C22⋊Dic6112yes—
270C2×C4#1451974614yes—
271C2×C4C22×#9241811yescharacteristic
272C2×C4C2×#1175819412yes—
273C2×C4C2×#1175819411yescharacteristic
274C8—4884no—
275C22—22no—
276C22—22no—
277C22—28no—
278C22—42no—
279C22—48no—
280C22—48no—
281C22—42no—
282C22—44no—
283C22—44no—
284C22—42no—
285C22—816no—
286C22#1683961811yescharacteristic
287C22C2×C22⋊#9241811yescharacteristic
288C22#1683972611yescharacteristic
289C4—22no—
290C4—24no—
291C4—44no—
292C4—42no—
293C4C2×C22⋊#9241812yes—
294C4#1683972612yes—
295C2—21no—
296C2—21no—
297C2—48no—
298C2—42no—
299C2#2144572911yescharacteristic
300C2#2144578912yes—
subgroupG/H
centerC22#16839618
commutatorC2×C4×C61C22×C8
FrattiniC2×C42C22×D61
FittingC2×C61×#518C2
socleC22×C61#379
prime radicalC23×D4×C61C4
cosocleC2×C42×C61C23
hypercenterC2×#518D61
hypocenterC61#8322
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#518

Complements

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

NH ≅ G/Ntype
1. #21444210C2semidirect
2. #21444133C2semidirect
4. #16839815C22semidirect
24. C22×D4×C61C8semidirect
166. C61#8322semidirect
172. C22×D4#92418semidirect

Maximal subgroups

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

subgroupindex
#214442102
#214441332
C2×C61×#5182
#832261