Finite Group Databasebeta

← #29576069

Subgroups — #29576069

Order 15616.

516 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 111 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#21451632C211yesmaximal, characteristic
2#21448422C211yesmaximal, characteristic
3#21444392C212yesmaximal
4#21443687C212yesmaximal
5C2×C61×#324C211yesmaximal, characteristic
6C2×#14519756C2211yescharacteristic
7#16839701C412yescomplemented
8#16839695—22no—
9#16839682C412yescomplemented
10#16839648—24no—
11#16839635—21no—
12#16839625—21no—
13#16839625—21no—
14#16839622C2211yescharacteristic
15#16839622C2212yes—
16C4×C22⋊Dic61—21no—
17#16839500—24no—
18C22×C4⋊C4×C61C2211yescharacteristic
19C2×C22⋊C8×C61C2212yes—
20C61×#324—22no—
21C22×Dic122C2×C411yescharacteristic
22#14519756—22no—
23#14519756—22no—
24C2×C4⋊Dic61—21no—
25C2×C4⋊Dic61—22no—
26C2×C4⋊Dic61C2×C411yescharacteristic
27C2×#11758196—21no—
28C2×#11758196—22no—
29C2×C22⋊Dic61—21no—
30C2×C22⋊Dic61—22no—
31C2×C22⋊Dic61—21no—
32C2×C4×Dic61—22no—
33#14519684—22no—
34#14519684—24no—
35#14519684—22no—
36#14519684—22no—
37#14519684—22no—
38#14519684C2×C412yes—
39#14519684D414yes—
40C23×C4×C61D411yescharacteristic
41C23×C4×C61D411yescharacteristic
42C23×C4×C61C2311yescharacteristic
43C22×C8×C61—22no—
44C2×C4⋊C4×C61—22no—
45C2×C4⋊C4×C61—22no—
46C2×C4⋊C4×C61C2×C412yes—
47C2×C4⋊C4×C61D412yes—
48C22⋊C8×C61—22no—
49C22⋊C8×C61—22no—
50C24×C61C2×D411yescharacteristic
51C22×C4×C61—21no—
52C22×C4×C61—21no—
53C22×C4×C61—21no—
54C22×C4×C61—22no—
55C22×C4×C61—22no—
56C22×C4×C61—21no—
57C22×C4×C61—81no—
58C22×C4×C61—81no—
59C22×C4×C61C22⋊C412yes—
60C22×C4×C61C22⋊C412yes—
61C22×C4×C61C22×C411yescharacteristic
62C22×C4×C61C2×D412yes—
63C22×C4×C61C2×D411yescharacteristic
64C22×C4×C61#4112yes—
65C22×Dic61—21no—
66C22×Dic61—21no—
67C22×Dic61—22no—
68C22×Dic61—24no—
69C2×Dic122—22no—
70C2×Dic122—81no—
71C2×C8×C61—42no—
72C2×C8×C61—82no—
73C4⋊C4×C61—24no—
74C4⋊C4×C61—24no—
75C22⋊Dic61—41no—
76C22⋊Dic61—41no—
77C22⋊Dic61—42no—
78C22⋊Dic61—42no—
79C22⋊Dic61—42no—
80C4⋊Dic61—22no—
81C4⋊Dic61—82no—
82C4⋊Dic61—81no—
83#11758196—42no—
84#11758196—82no—
85C4×Dic61—82no—
86C23×C61—41no—
87C23×C61—41no—
88C23×C61C2×C22⋊C411yescharacteristic
89C23×C61#12011yescharacteristic
90C23×C61#12311yescharacteristic
91C23×C61#13512yes—
92C23×C61#13712yes—
93C2×C4×C61—22no—
94C2×C4×C61—22no—
95C2×C4×C61—22no—
96C2×C4×C61—22no—
97C2×C4×C61—41no—
98C2×C4×C61—42no—
99C2×C4×C61—42no—
100C2×C4×C61—41no—
101C2×C4×C61—42no—
102C2×C4×C61—42no—
103C2×C4×C61—42no—
104C2×C4×C61—42no—
105C2×C4×C61—81no—
106C2×C4×C61C4×D412yes—
107C2×C4×C61#12112yes—
108C2×Dic61—44no—
109C2×Dic61—42no—
110C2×Dic61—42no—
111C2×Dic61—42no—
112C2×Dic61—44no—
113C2×Dic61—42no—
114C2×Dic61—41no—
115C2×Dic61—41no—
116C2×Dic61—42no—
117C2×Dic61—44no—
118Dic122—82no—
119C8×C61—82no—
120#9202—611nomaximal, Sylow
121C22×C61—21no—
122C22×C61—21no—
123C22×C61—21no—
124C22×C61—21no—
125C22×C61—21no—
126C22×C61—21no—
127C22×C61—42no—
128C22×C61—42no—
129C22×C61#38611yescharacteristic
130C22×C61#41611yescharacteristic
131C22×C61#41911yescharacteristic
132C22×C61#45212yes—
133C22×C61#48512yes—
134C4×C61—22no—
135C4×C61—22no—
136C4×C61—42no—
137C4×C61—42no—
138Dic61—82no—
139Dic61—84no—
140Dic61—81no—
141Dic61—81no—
142#2627—611no—
143#2313—611no—
144#1520—612no—
145C2×#324D6111yescharacteristic
146#1289—612no—
147C2×C61—21no—
148C2×C61—22no—
149C2×C61—21no—
150C2×C61—22no—
151C2×C61#187311yescharacteristic
152C2×C61#191211yescharacteristic
153C2×C61#191311yescharacteristic
154C2×C22⋊Q8—611no—
155C22×C4⋊C4C2×D6111yescharacteristic
156C2×C22⋊C8C2×D6112yes—
157#400—612no—
158#398—1222no—
159#395—612no—
160#387—1224no—
161#385—1221no—
162#381—1221no—
163#381—1221no—
164#380—611no—
165#380—612no—
166C4×C22⋊C4—1221no—
167#328—1224no—
168#324—22no—
169C61#920211yescharacteristic, complemented, Sylow
170C22×Q8—611no—
171C23×C4D24411yescharacteristic
172C23×C4D24411yescharacteristic
173C23×C4C22×D6111yescharacteristic
174C22×C8—22no—
175C22⋊Q8—1222no—
176C22⋊Q8—1222no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—611no—
180C2×C4⋊C4—1222no—
181C2×C4⋊C4—1222no—
182C2×C4⋊C4—1221no—
183C2×C4⋊C4—1221no—
184C2×C4⋊C4C4×D6112yes—
185C2×C4⋊C4C22⋊D6112yes—
186C2×C22⋊C4—1221no—
187C2×C22⋊C4—1222no—
188C2×C22⋊C4—1221no—
189C2×C42—1222no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192#95—612no—
193#95—614no—
194#95—1222no—
195#95—1224no—
196#95—1222no—
197#95—1222no—
198#95—1222no—
199C24C2×D24411yescharacteristic
200C2×Q8—1222no—
201C2×Q8—4881no—
202C22×C4—22no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—81no—
209C22×C4—81no—
210C22×C4—1222no—
211C22×C4—1221no—
212C22×C4—1221no—
213C22×C4—1224no—
214C22×C4#1175819812yes—
215C22×C4#1175819812yes—
216C22×C4C2×C4×D6111yescharacteristic
217C22×C4D4×D6112yes—
218C22×C4#1175822412yes—
219C22×C4C2×C22⋊D6111yescharacteristic
220C2×C8—42no—
221C2×C8—82no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—1222no—
225C4⋊C4—4882no—
226C4⋊C4—4882no—
227C4⋊C4—4881no—
228C4⋊C4—2442no—
229C22⋊C4—2442no—
230C22⋊C4—2441no—
231C22⋊C4—2441no—
232C22⋊C4—2442no—
233C22⋊C4—2442no—
234C42—4882no—
235C23—41no—
236C23—41no—
237C23C2×#1175819811yescharacteristic
238C23#1451974711yescharacteristic
239C23#1451976111yescharacteristic
240C23#1451979912yes—
241C23#1451981012yes—
242Q8—4882no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—81no—
256C2×C4—2444no—
257C2×C4—2442no—
258C2×C4—2442no—
259C2×C4—2442no—
260C2×C4—2444no—
261C2×C4—2442no—
262C2×C4—2442no—
263C2×C4—2441no—
264C2×C4—2441no—
265C2×C4—2444no—
266C2×C4#1451973712yes—
267C2×C4#1451974912yes—
268C8—82no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—42no—
276C22—42no—
277C22#1683963911yescharacteristic
278C22#1683975911yescharacteristic
279C22#1683977211yescharacteristic
280C22#1683992612yes—
281C22#1684010412yes—
282C4—22no—
283C4—22no—
284C4—42no—
285C4—42no—
286C4—4882no—
287C4—4884no—
288C4—4881no—
289C4—4881no—
290C2—21no—
291C2—22no—
292C2—21no—
293C2—22no—
294C2#2144612911yescharacteristic
295C2#2144635411yescharacteristic
296C2#2144635911yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC2×C61×#324C2
socleC22×C61#386
prime radicalC24×C61C2×D4
cosocleC23×C4×C61C23
hypercenterC2×#324D61
hypocenterC61#9202
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
7. #16839701C4semidirect
9. #16839682C4semidirect
169. C61#9202semidirect

Maximal subgroups

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

subgroupindex
#214516322
#214484222
#214443922
#214436872
C2×C61×#3242
#920261