Finite Group Databasebeta

← #29575005

Subgroups — #29575005

Order 15616.

756 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C2×#16839682C211yesmaximal, characteristic
2#21448422C211yesmaximal, characteristic
3#21445755C212yesmaximal
4#21443699C212yesmaximal
5C2×C61×#324C211yesmaximal, characteristic
6C23×Dic122C2211yescharacteristic
7C22×#11758196—21no—
8C2×#14519702—24no—
9#16839682—22no—
10#16839682C414yescomplemented
11#16839648—24no—
12#16839625—21no—
13#16839622C2212yes—
14C4×C22⋊Dic61—21no—
15C2×#14519684—21no—
16C2×#14519684C2211yescharacteristic
17C22×C4⋊C4×C61C2211yescharacteristic
18C2×C22⋊C8×C61C2212yes—
19C61×#324—22no—
20C22×Dic122—22no—
21C22×Dic122—22no—
22C22×Dic122—21no—
23C22×Dic122—44no—
24C22×Dic122C2×C412yes—
25C23×Dic61—21no—
26C23×Dic61—21no—
27C2×C4⋊Dic61D414yes—
28C2×#11758196—24no—
29C2×#11758196—44no—
30C2×C22⋊Dic61—22no—
31C2×C4×Dic61—22no—
32#14519702—48no—
33#14519684—22no—
34#14519684—24no—
35#14519684—24no—
36#14519684—21no—
37#14519684—22no—
38#14519684C2×C412yes—
39C23×C4×C61D411yescharacteristic
40C23×C4×C61D411yescharacteristic
41C23×C4×C61C2311yescharacteristic
42C22×C8×C61—22no—
43C2×C4⋊C4×C61—22no—
44C2×C4⋊C4×C61—22no—
45C2×C4⋊C4×C61C2×C412yes—
46C2×C4⋊C4×C61D412yes—
47C22⋊C8×C61—22no—
48C22⋊C8×C61—22no—
49C24×C61C2×D411yescharacteristic
50C22×C4×C61—21no—
51C22×C4×C61—21no—
52C22×C4×C61—21no—
53C22×C4×C61—22no—
54C22×C4×C61—22no—
55C22×C4×C61—21no—
56C22×C4×C61—42no—
57C22×C4×C61—42no—
58C22×C4×C61C22⋊C412yes—
59C22×C4×C61C22⋊C412yes—
60C22×C4×C61C22×C411yescharacteristic
61C22×C4×C61C2×D412yes—
62C22×C4×C61C2×D411yescharacteristic
63C22×C4×C61#4112yes—
64C22×Dic61—24no—
65C22×Dic61—24no—
66C22×Dic61—22no—
67C22×Dic61—22no—
68C22×Dic61—24no—
69C22×Dic61—42no—
70C22×Dic61—42no—
71C22×Dic61—41no—
72C22×Dic61—41no—
73C22×Dic61—42no—
74C2×Dic122—28no—
75C2×Dic122—24no—
76C2×Dic122—44no—
77C2×Dic122—48no—
78C2×Dic122—44no—
79C2×Dic122—44no—
80C2×Dic122—42no—
81C2×C8×C61—44no—
82C2×C8×C61—42no—
83C4⋊C4×C61—24no—
84C4⋊C4×C61—24no—
85C22⋊Dic61—42no—
86C22⋊Dic61—42no—
87C4⋊Dic61—44no—
88#11758196—48no—
89C4×Dic61—82no—
90C23×C61—22no—
91C23×C61—22no—
92C23×C61C2×C22⋊C411yescharacteristic
93C23×C61#12011yescharacteristic
94C23×C61#12311yescharacteristic
95C23×C61#13512yes—
96C23×C61#13712yes—
97C2×C4×C61—24no—
98C2×C4×C61—22no—
99C2×C4×C61—22no—
100C2×C4×C61—22no—
101C2×C4×C61—22no—
102C2×C4×C61—41no—
103C2×C4×C61—41no—
104C2×C4×C61—42no—
105C2×C4×C61—42no—
106C2×C4×C61—42no—
107C2×C4×C61—42no—
108C2×C4×C61—42no—
109C2×C4×C61—81no—
110C2×C4×C61C4×D412yes—
111C2×C4×C61#12112yes—
112C2×Dic61—24no—
113C2×Dic61—24no—
114C2×Dic61—42no—
115C2×Dic61—42no—
116C2×Dic61—44no—
117C2×Dic61—48no—
118C2×Dic61—42no—
119C2×Dic61—42no—
120C2×Dic61—44no—
121C2×Dic61—44no—
122C2×Dic61—44no—
123C2×Dic61—44no—
124Dic122—28no—
125Dic122—48no—
126Dic122—44no—
127C8×C61—82no—
128#9014—611nomaximal, Sylow
129C22×C61—24no—
130C22×C61—24no—
131C22×C61—21no—
132C22×C61—21no—
133C22×C61—21no—
134C22×C61—21no—
135C22×C61#38611yescharacteristic
136C22×C61#41611yescharacteristic
137C22×C61#41911yescharacteristic
138C22×C61#45212yes—
139C22×C61#45612yes—
140C22×C61B(4,𝔽2)12yes—
141C22×C61#48512yes—
142C4×C61—22no—
143C4×C61—22no—
144C4×C61—42no—
145C4×C61—42no—
146Dic61—44no—
147Dic61—44no—
148Dic61—84no—
149C2×#395—611no—
150#2313—611no—
151#1787—612no—
152C2×#324D6111yescharacteristic
153#1294—612no—
154C2×C61—21no—
155C2×C61—21no—
156C2×C61—22no—
157C2×C61#160314yes—
158C2×C61#187311yescharacteristic
159C2×C61#188611yescharacteristic
160C2×C61#188911yescharacteristic
161C23×Q8—611no—
162C22×C4⋊C4—1221no—
163C22×C4⋊C4C2×D6111yescharacteristic
164C2×#103—1224no—
165C2×C22⋊C8C2×D6112yes—
166#395—614no—
167#395—1222no—
168#387—1224no—
169#381—1221no—
170#380—612no—
171C4×C22⋊C4—1221no—
172C2×#95—611no—
173C2×#95—1221no—
174#324—22no—
175C61#901411yescharacteristic, complemented, Sylow
176C22×Q8—612no—
177C22×Q8—1221no—
178C22×Q8—1222no—
179C22×Q8—1222no—
180C22×Q8—2444no—
181C23×C4—1221no—
182C23×C4—1221no—
183C23×C4D24411yescharacteristic
184C23×C4D24411yescharacteristic
185C23×C4C22×D6111yescharacteristic
186C22×C8—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—614no—
190C2×C4⋊C4—1224no—
191C2×C4⋊C4—2444no—
192C2×C4⋊C4C4×D6112yes—
193C2×C4⋊C4C22⋊D6112yes—
194C2×C22⋊C4—1222no—
195C2×C42—1222no—
196#103—2448no—
197C22⋊C8—22no—
198C22⋊C8—22no—
199#95—612no—
200#95—1222no—
201#95—1224no—
202#95—1224no—
203#95—1222no—
204#95—1221no—
205C24C2×D24411yescharacteristic
206C2×Q8—1228no—
207C2×Q8—1224no—
208C2×Q8—2444no—
209C2×Q8—2444no—
210C2×Q8—2444no—
211C2×Q8—2448no—
212C2×Q8—2442no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—1224no—
222C22×C4—1224no—
223C22×C4—1222no—
224C22×C4—1222no—
225C22×C4—1224no—
226C22×C4—2442no—
227C22×C4—2442no—
228C22×C4—2441no—
229C22×C4—2441no—
230C22×C4—2442no—
231C22×C4#1175819812yes—
232C22×C4#1175819812yes—
233C22×C4C2×C4×D6111yescharacteristic
234C22×C4D4×D6112yes—
235C22×C4#1175822412yes—
236C22×C4C2×C22⋊D6111yescharacteristic
237C2×C8—44no—
238C2×C8—42no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—2444no—
242C4⋊C4—2448no—
243C22⋊C4—2442no—
244C22⋊C4—2442no—
245C42—4882no—
246C23—22no—
247C23—22no—
248C23C2×#1175819811yescharacteristic
249C23#1451974711yescharacteristic
250C23#1451976111yescharacteristic
251C23#1451979912yes—
252C23#1451981012yes—
253Q8—1228no—
254Q8—2448no—
255Q8—2444no—
256C2×C4—24no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—81no—
269C2×C4—1224no—
270C2×C4—1224no—
271C2×C4—2444no—
272C2×C4—2444no—
273C2×C4—2444no—
274C2×C4—2442no—
275C2×C4—2448no—
276C2×C4—2442no—
277C2×C4—2444no—
278C2×C4—2444no—
279C2×C4—2442no—
280C2×C4—2442no—
281C2×C4#1451973712yes—
282C2×C4#1451974912yes—
283C8—82no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—24no—
289C22—24no—
290C22#1683963911yescharacteristic
291C22#1683975911yescharacteristic
292C22#1683977211yescharacteristic
293C22#1683992612yes—
294C22#1683994712yes—
295C22#1683995012yes—
296C22#1684010412yes—
297C4—22no—
298C4—22no—
299C4—42no—
300C4—42no—
301C4—4884no—
302C4—2444no—
303C4—2444no—
304C2—21no—
305C2—21no—
306C2—22no—
307C2#2144478414yes—
308C2#2144612911yescharacteristic
309C2#2144621011yescharacteristic
310C2#2144622511yescharacteristic
subgroupG/H
centerC23#14519747
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC2×C61×#324C2
socleC23×C61#120
prime radicalC24×C61C2×D4
cosocleC23×C4×C61C23
hypercenterC2×#324D61
hypocenterC61#9014
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
10. #16839682C4semidirect
175. C61#9014semidirect

Maximal subgroups

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

subgroupindex
C2×#168396822
#214484222
#214457552
#214436992
C2×C61×#3242
#901461