Finite Group Databasebeta

← #30606306

Subgroups — #30606306

Order 15616.

826 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 152 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×#16840741C212yesmaximal
2#21460913C214yesmaximal, complemented
3#21448407C211yesmaximal, characteristic
4C2×#14519856—22no—
5#16840741—24no—
6#16840741C418yescomplemented
7C2×#14519836C2214yes—
8C2×#14519835—22no—
9C22×#11758222C2211yescharacteristic
10C22×C4⋊Dic61C411yescharacteristic, complemented
11#16839668C412yescomplemented
12#16839664C2212yes—
13#16839639—22no—
14#16839619—22no—
15C2×#117×C61C411yescharacteristic, complemented
16#14519856—28no—
17#14519856—44no—
18C2×C4⋊#6923—42no—
19C2×C22⋊#6923—24no—
20C2×C4×#6923—42no—
21#14519836—24no—
22#14519836C2×C418yescomplemented
23#14519835—28no—
24C2×#11758222—42no—
25C2×#11758222—42no—
26C2×#11758222C2×C414yes—
27C2×#11758222D412yes—
28C2×#11758222Q812yes—
29C22×Dic122C2×C411yescharacteristic
30C22×D244C2311yescharacteristic
31C22×C22⋊D61D412yes—
32C23×Dic61—21no—
33C22×C4×D61D412yes—
34C2×C4⋊Dic61—21no—
35C2×C4⋊Dic61—42no—
36C2×C4⋊Dic61C2×C412yes—
37C2×#11758198—24no—
38#14519684—24no—
39C23×C4×C61C2×C411yescharacteristic
40#117×C61—42no—
41C2×C4⋊C4×C61—21no—
42C2×C22⋊C4×C61—21no—
43C2×C42×C61C2×C412yes—
44C24×C61C22⋊C411yescharacteristic
45C23×D61C2×D412yes—
46C22×#6923—44no—
47C22×C4×C61—22no—
48C22×C4×C61—22no—
49C22×C4×C61—22no—
50C22×C4×C61C4212yes—
51C22×C4×C61C22⋊C411yescharacteristic
52C22×C4×C61C4⋊C412yes—
53C22×C4×C61C22×C411yescharacteristic
54C2×C22⋊D61—42no—
55C2×C22⋊D61—42no—
56C2×C22⋊D61C22⋊C414yes—
57C2×C22⋊D61C4⋊C414yes—
58C22×Dic61—22no—
59C22×Dic61—42no—
60C22×Dic61—82no—
61C22×Dic61C22⋊C412yes—
62#11758222—44no—
63#11758222—44no—
64#11758222—44no—
65#11758222—44no—
66C2×D244—41no—
67C2×D244—41no—
68C2×D244C22×C412yes—
69C2×D244C2×D411yescharacteristic
70C2×D244C2×Q811yescharacteristic
71C2×C4×D61—22no—
72C2×C4×D61—22no—
73C2×C4×D61—24no—
74C2×C4×D61C22⋊C414yes—
75C2×C4×D61C4⋊C414yes—
76C2×Dic122—41no—
77C2×Dic122—41no—
78C2×Dic122C4212yes—
79C2×Dic122C4⋊C412yes—
80C22⋊#6923—28no—
81C22⋊#6923—44no—
82C4⋊#6923—48no—
83C4×#6923—48no—
84C4⋊C4×C61—42no—
85C22⋊C4×C61—42no—
86C42×C61—42no—
87#11758198—84no—
88C4⋊Dic61—42no—
89C4⋊Dic61—42no—
90C23×C61—21no—
91C23×C61—21no—
92C23×C61—41no—
93C23×C61—41no—
94C23×C61#9512yes—
95C23×C61C2×C22⋊C411yescharacteristic
96C22×D61—22no—
97C22×D61—22no—
98C22×D61—42no—
99C22×D61—42no—
100C22×D61C2×C22⋊C412yes—
101C22×D61C2×C4⋊C412yes—
102C2×#6923—48no—
103C2×#6923—84no—
104C2×#6923—84no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—24no—
108C2×C4×C61—24no—
109C2×C4×C61—42no—
110C2×C4×C61—42no—
111C2×C4×C61—42no—
112C2×C4×C61#9512yes—
113C2×C4×C61#9912yes—
114C2×C4×C61#9912yes—
115C2×C4×C61C2×C4211yescharacteristic
116C2×C4×C61C2×C4⋊C411yescharacteristic
117C22⋊D61—44no—
118C22⋊D61—44no—
119C2×Dic61—22no—
120C2×Dic61—22no—
121C2×Dic61—44no—
122C2×Dic61—42no—
123C2×Dic61—84no—
124C2×Dic61#9514yes—
125D244—42no—
126D244—41no—
127D244—41no—
128C4×D61—24no—
129C4×D61—24no—
130C4×D61—24no—
131C4×D61—24no—
132Dic122—42no—
133Dic122—42no—
134#12583—611nomaximal, Sylow
135C22×C61—21no—
136C22×C61—21no—
137C22×C61—22no—
138C22×C61—42no—
139C22×C61—42no—
140C22×C61—41no—
141C22×C61—41no—
142C22×C61C2×#9511yescharacteristic
143C22×C61C2×#9911yescharacteristic
144C22×C61C2×#9911yescharacteristic
145C2×D61—22no—
146C2×D61—22no—
147C2×D61—42no—
148C2×D61—42no—
149C2×D61—42no—
150C2×D61—42no—
151C2×D61—82no—
152#6923—88no—
153C4×C61—21no—
154C4×C61—21no—
155C4×C61—82no—
156C4×C61#34214yes—
157Dic61—24no—
158Dic61—82no—
159C2×#410—612no—
160#2311—611no—
161#1393—614no—
162C2×C61—21no—
163C2×C61—21no—
164C2×C61—42no—
165C2×C61C2×#34211yescharacteristic
166C2×C61#212811yescharacteristic
167C2×C61#212911yescharacteristic
168D61—42no—
169D61—42no—
170C22×#41—611no—
171C2×#117—1222no—
172C2×#117#692311yescharacteristic, complemented
173C22×C4⋊C4—611no—
174#410—618no—
175#410—1224no—
176C2×#99—614no—
177C2×#99—1222no—
178#391—612no—
179#390—612no—
180#386—1222no—
181#380—1222no—
182C61#1258311yescharacteristic, complemented, Sylow
183C2×#41—614no—
184C2×#41—612no—
185C2×#41—612no—
186C2×#41—2442no—
187C2×#41—2442no—
188C22×Q8—611no—
189C22×D4—612no—
190C22×D4—611no—
191C23×C4—612no—
192C23×C4—1221no—
193C23×C4C2×#692311yescharacteristic
194#117—42no—
195#117—1228no—
196#117—2444no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—612no—
199C2×C4⋊C4—1221no—
200C2×C4⋊C4—2442no—
201C2×C4⋊C4—2442no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—1224no—
204C2×C22⋊C4—1224no—
205C2×C42—2442no—
206C2×C42C2×#692312yes—
207#99—618no—
208#99—1228no—
209#99—1224no—
210#95—1224no—
211C24—612no—
212C24C22⋊#692311yescharacteristic
213#41—2444no—
214#41—2444no—
215#41—2444no—
216#41—2444no—
217C2×Q8—612no—
218C2×Q8—612no—
219C2×Q8—2441no—
220C2×Q8—2441no—
221C2×D4—614no—
222C2×D4—614no—
223C2×D4—612no—
224C2×D4—611no—
225C2×D4—611no—
226C2×D4—2441no—
227C2×D4—2441no—
228C2×D4—2442no—
229C2×D4—2442no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—612no—
234C22×C4—614no—
235C22×C4—614no—
236C22×C4—1224no—
237C22×C4—1222no—
238C22×C4—1222no—
239C22×C4—1222no—
240C22×C4—4882no—
241C22×C4—2442no—
242C22×C4—2444no—
243C22×C4C4×#692312yes—
244C22×C4C4⋊#692312yes—
245C22×C4C22⋊#692311yescharacteristic
246C22×C4C22×#692311yescharacteristic
247C4⋊C4—42no—
248C4⋊C4—2442no—
249C4⋊C4—2448no—
250C4⋊C4—2442no—
251C22⋊C4—42no—
252C22⋊C4—1228no—
253C22⋊C4—4884no—
254C22⋊C4—2444no—
255C42—42no—
256C42—2448no—
257C23—21no—
258C23—21no—
259C23—41no—
260C23—41no—
261C23—612no—
262C23—612no—
263C23—1222no—
264C23—1222no—
265C23—2442no—
266C23—2442no—
267C23(C2×#6923)⋊C412yes—
268C23C2×C22⋊#692311yescharacteristic
269Q8—2442no—
270Q8—2442no—
271D4—2442no—
272D4—2441no—
273D4—2441no—
274D4—2444no—
275D4—2444no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—614no—
284C2×C4—1222no—
285C2×C4—1222no—
286C2×C4—1224no—
287C2×C4—1224no—
288C2×C4—1224no—
289C2×C4—1224no—
290C2×C4—4884no—
291C2×C4—4884no—
292C2×C4—4884no—
293C2×C4—2444no—
294C2×C4—2448no—
295C2×C4—2442no—
296C2×C4(C2×#6923)⋊C412yes—
297C2×C4#1451983512yes—
298C2×C4#1451983612yes—
299C2×C4C2×C4×#692311yescharacteristic
300C2×C4C2×C4⋊#692311yescharacteristic
301C22—21no—
302C22—21no—
303C22—22no—
304C22—42no—
305C22—42no—
306C22—41no—
307C22—41no—
308C22—1222no—
309C22—1222no—
310C22—4882no—
311C22—2442no—
312C22—2442no—
313C22—2442no—
314C22—2442no—
315C22C2×(C2×#6923)⋊C411yescharacteristic
316C22C2×#1451983511yescharacteristic
317C22C2×#1451983611yescharacteristic
318C4—21no—
319C4—21no—
320C4—82no—
321C4—1224no—
322C4—4888no—
323C4—4882no—
324C4#1684064014yes—
325C2—21no—
326C2—21no—
327C2—42no—
328C2—2442no—
329C2—2442no—
330C2C2×#1684064011yescharacteristic
331C2#2146210011yescharacteristic
332C2#2146210311yescharacteristic
subgroupG/H
centerC22C2×#14519836
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC2×#117×C61C4
socleC22×C61C2×#99
prime radicalC22×#11758222C22
cosocleC22×D244C23
hypercenterC2×#117#6923
hypocenterC61#12583
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #21460913C2semidirect
6. #16840741C4semidirect
10. C22×C4⋊Dic61C4semidirect
11. #16839668C4semidirect
15. C2×#117×C61C4semidirect
22. #14519836C2×C4semidirect
172. C2×#117#6923semidirect
182. C61#12583semidirect

Maximal subgroups

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

subgroupindex
C2×#168407412
#214609132
#214484072
#1258361