Finite Group Databasebeta

← #30601026

Subgroups — #30601026

Order 15616.

450 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 86 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#21462718C211yesmaximal, characteristic
2#21461686C211yesmaximal, characteristic
3#21461334C212yesmaximal
4#21460717C212yesmaximal
5#21451030C211yesmaximal, characteristic
6#16840814—24no—
7#16840783—21no—
8#16840782—21no—
9#16840779C2212yes—
10#16840765C2212yes—
11C2×#14519838—21no—
12#16840642—21no—
13#16840621—24no—
14C2×#14519776C2211yescharacteristic
15C2×#14519742C2211yescharacteristic
16#16839702C411yescharacteristic
17#16839701C411yescharacteristic
18#16839700C411yescharacteristic
19#16839666—21no—
20#16839666—21no—
21#16839666—21no—
22#16839626—21no—
23#16839613C2211yescharacteristic
24C61×#400C411yescharacteristic
25C2×#11758217—22no—
26#14519844—22no—
27#14519844—22no—
28#14519844D414yes—
29C4⋊#92420—22no—
30C4⋊#92420—22no—
31#14519838—42no—
32C22⋊#92420—22no—
33#14519832—22no—
34#14519832—22no—
35C4×#92420—22no—
36C22×Dic122—21no—
37C22×Dic122D411yescharacteristic
38#14519776—22no—
39#14519776—22no—
40#14519742—22no—
41#14519742—22no—
42C2×C4⋊Dic61C2×C411yescharacteristic
43C2×#11758196—21no—
44C2×#11758196—21no—
45C2×#11758196C2×C411yescharacteristic
46C2×#11758196D411yescharacteristic
47C2×C4×Dic61—21no—
48C2×C4×Dic61D411yescharacteristic
49C2×C4×Dic61D411yescharacteristic
50C2×C4×Dic61C2311yescharacteristic
51#14519684—21no—
52#14519684—21no—
53#14519684—21no—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57#14519684C2×C411yescharacteristic
58C2×C4⋊C4×C61C2×C411yescharacteristic
59C2×C4⋊C4×C61C2×C411yescharacteristic
60#95×C61—21no—
61#95×C61—21no—
62#95×C61C2×C411yescharacteristic
63C22×C4×C61—21no—
64C22×C4×C61—21no—
65C22×C4×C61C22⋊C411yescharacteristic
66C22×C4×C61C22⋊C411yescharacteristic
67C22×C4×C61C22×C411yescharacteristic
68C22×Dic61—21no—
69C22×Dic61—21no—
70C22×Dic61C22⋊C411yescharacteristic
71C22×Dic61C22⋊C411yescharacteristic
72C22×Dic61C2×D411yescharacteristic
73C22×Dic61C2×D411yescharacteristic
74C2×Dic122—22no—
75C2×Dic122—21no—
76C2×Dic122—22no—
77C2×Dic122—21no—
78C2×Dic122—44no—
79C2×Dic122—41no—
80C2×Dic122—41no—
81#11758217—42no—
82#11758217—42no—
83C2×#92420—24no—
84C2×#92420—42no—
85C4⋊C4×C61—22no—
86C4⋊C4×C61—22no—
87C4⋊Dic61—22no—
88#11758196—22no—
89#11758196—22no—
90#11758196—42no—
91#11758196—42no—
92C4×Dic61—21no—
93C4×Dic61—21no—
94C4×Dic61—21no—
95C4×Dic61—21no—
96C4×Dic61—41no—
97C4×Dic61—41no—
98C4×Dic61C2×D412yes—
99C4×Dic61#4112yes—
100C23×C61C2×C22⋊C411yescharacteristic
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—41no—
106C2×C4×C61—41no—
107C2×C4×C61—41no—
108C2×C4×C61—42no—
109C2×C4×C61—41no—
110C2×C4×C61—41no—
111C2×C4×C61—41no—
112C2×C4×C61C4×D412yes—
113C2×Dic61—22no—
114C2×Dic61—21no—
115C2×Dic61—21no—
116C2×Dic61—21no—
117C2×Dic61—21no—
118C2×Dic61—41no—
119C2×Dic61—41no—
120C2×Dic61—42no—
121C2×Dic61—41no—
122C2×Dic61—41no—
123C2×Dic61—41no—
124C2×Dic61#12011yescharacteristic
125C2×Dic61#12112yes—
126C2×Dic61#12311yescharacteristic
127C2×Dic61AGL(1,ℤ/8)11yescharacteristic
128C2×Dic61#13712yes—
129C2×Dic61#13711yescharacteristic
130Dic122—42no—
131Dic122—42no—
132Dic122—82no—
133Dic122—82no—
134#92420—44no—
135#92420—82no—
136#9404—611nomaximal, Sylow
137C22×C61—21no—
138C22×C61—21no—
139C22×C61#38611yescharacteristic
140C22×C61#41711yescharacteristic
141C22×C61#41911yescharacteristic
142C4×C61—22no—
143C4×C61—41no—
144C4×C61—41no—
145C4×C61—81no—
146C4×C61—81no—
147Dic61—21no—
148Dic61—41no—
149Dic61—41no—
150Dic61—81no—
151Dic61—81no—
152Dic61#45412yes—
153Dic61#46812yes—
154Dic61#47012yes—
155#2983—611no—
156#2568—611no—
157#1801—611no—
158#1549—612no—
159#1303—612no—
160C2×C61—21no—
161C2×C61—21no—
162C2×C61#188811yescharacteristic
163C2×C61#189411yescharacteristic
164C2×C61#191511yescharacteristic
165C2×C4⋊Q8—611no—
166C2×C4×Q8—611no—
167#446—1224no—
168#432—1221no—
169#432—1221no—
170#431—612no—
171#423—612no—
172C2×#101—1221no—
173#400—611no—
174#400—611no—
175#400—611no—
176#400#692311yescharacteristic
177#391—1221no—
178#391—1221no—
179#391—1221no—
180#382—1221no—
181C4×C4⋊C4—611no—
182#343—1221no—
183#332—1224no—
184C61#940411yescharacteristic, complemented, Sylow
185C22×Q8—611no—
186C22×Q8—1221no—
187C2×M16—1222no—
188C4⋊Q8—1222no—
189C4⋊Q8—1222no—
190C4×Q8—1222no—
191C4×Q8—1222no—
192C2×C4⋊C4—611no—
193C2×C4⋊C4—611no—
194C2×C4⋊C4—611no—
195C2×C4⋊C4—1221no—
196C2×C4⋊C4—1221no—
197C2×C4⋊C4C2×#692311yescharacteristic
198C2×C4⋊C4C2×#692311yescharacteristic
199C2×C42—611no—
200C2×C42—611no—
201C2×C42—611no—
202C2×C42—1221no—
203C4⋊C8—614no—
204C4⋊C8—1222no—
205C4⋊C8—1222no—
206C4⋊C8—1222no—
207C4⋊C8—1222no—
208#101—2442no—
209C22⋊C8—1222no—
210#97—1222no—
211#97—1222no—
212C4×C8—1222no—
213#95—21no—
214#95—21no—
215#95—611no—
216#95—1221no—
217#95—1221no—
218#95—1221no—
219#95—1221no—
220#95—1221no—
221#95—1221no—
222#95C2×#692311yescharacteristic
223C2×Q8—1221no—
224C2×Q8—1222no—
225C2×Q8—1221no—
226C2×Q8—1222no—
227C2×Q8—2441no—
228C2×Q8—2441no—
229C2×Q8—2444no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—611no—
233C22×C4—611no—
234C22×C4—611no—
235C22×C4—611no—
236C22×C4—1221no—
237C22×C4—1221no—
238C22×C4C22⋊#692311yescharacteristic
239C22×C4C22⋊#692311yescharacteristic
240C22×C4C22×#692311yescharacteristic
241M16—2442no—
242M16—2442no—
243C2×C8—1224no—
244C2×C8—2442no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—1222no—
248C4⋊C4—1222no—
249C4⋊C4—1222no—
250C4⋊C4—2442no—
251C4⋊C4—2442no—
252C42—612no—
253C42—612no—
254C42—1221no—
255C42—1221no—
256C42—1221no—
257C42—1221no—
258C42—2441no—
259C42—2441no—
260C23C2×C22⋊#692311yescharacteristic
261Q8—4882no—
262Q8—4882no—
263Q8—2442no—
264Q8—2442no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—611no—
277C2×C4—612no—
278C2×C4—611no—
279C2×C4—611no—
280C2×C4—612no—
281C2×C4—611no—
282C2×C4—1222no—
283C2×C4—1221no—
284C2×C4—1221no—
285C2×C4—1221no—
286C2×C4—1221no—
287C2×C4—2441no—
288C2×C4—2441no—
289C2×C4—2442no—
290C2×C4—2441no—
291C2×C4—2441no—
292C2×C4—2441no—
293C2×C4D4×#692312yes—
294C8—4882no—
295C8—2444no—
296C22—21no—
297C22—21no—
298C22#1684071011yescharacteristic
299C22#1684075311yescharacteristic
300C22#1684075511yescharacteristic
301C4—22no—
302C4—41no—
303C4—41no—
304C4—81no—
305C4—81no—
306C4—612no—
307C4—612no—
308C4—612no—
309C4—1221no—
310C4—4881no—
311C4—4881no—
312C4—2441no—
313C4—2441no—
314C2—21no—
315C2—21no—
316C2#2146182711yescharacteristic
317C2#2146183411yescharacteristic
318C2#2146187111yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#400C4
socleC22×C61#386
prime radicalC23×C61C2×C22⋊C4
cosocleC2×C4×Dic61C23
hypercenter#400#6923
hypocenterC61#9404
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #400

Complements

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

NH ≅ G/Ntype
184. C61#9404semidirect

Maximal subgroups

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

subgroupindex
#214627182
#214616862
#214613342
#214607172
#214510302
#940461