Finite Group Databasebeta

← #30601199

Subgroups — #30601199

Order 15616.

474 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 94 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#21462714C211yesmaximal, characteristic
2#21461686C211yesmaximal, characteristic
3#21461147C212yesmaximal
4#21460672C212yesmaximal
5#21451180C211yesmaximal, characteristic
6#16840811—24no—
7#16840779—21no—
8#16840779C2212yes—
9#16840778—21no—
10#16840765C2212yes—
11C2×#14519838—21no—
12#16840642—21no—
13Q8⋊#92420—24no—
14C2×Q8×Dic61C2211yescharacteristic
15C2×#14519742C2211yescharacteristic
16#16839702C411yescharacteristic
17#16839701C411yescharacteristic
18#16839681C411yescharacteristic
19#16839666—21no—
20#16839657—21no—
21#16839649—21no—
22#16839648—21no—
23#16839626C2211yescharacteristic
24C61×#395C411yescharacteristic
25C2×#11758217—22no—
26#14519844—22no—
27#14519844—22no—
28#14519844—22no—
29C4⋊#92420—22no—
30C4⋊#92420—22no—
31#14519838—42no—
32C22⋊#92420—22no—
33C4×#92420—22no—
34C4×#92420—22no—
35C4×#92420D414yes—
36C22×Dic122D411yescharacteristic
37Q8×Dic61—22no—
38Q8×Dic61—22no—
39#14519742—22no—
40#14519742—22no—
41C2×C4⋊Dic61—21no—
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
58C22×Q8×C61C2×C411yescharacteristic
59C2×C4⋊C4×C61—21no—
60C2×C4⋊C4×C61C2×C411yescharacteristic
61#95×C61—21no—
62#95×C61C2×C411yescharacteristic
63C2×Q8×C61—22no—
64C2×Q8×C61—42no—
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61C22⋊C411yescharacteristic
68C22×C4×C61C22⋊C411yescharacteristic
69C22×C4×C61C22×C411yescharacteristic
70C22×Dic61—21no—
71C22×Dic61—21no—
72C22×Dic61C22⋊C411yescharacteristic
73C22×Dic61C22⋊C411yescharacteristic
74C22×Dic61C2×D411yescharacteristic
75C22×Dic61C2×D411yescharacteristic
76C2×Dic122—22no—
77C2×Dic122—22no—
78C2×Dic122SD1612yes—
79C2×Dic122Q1612yes—
80#11758217—42no—
81#11758217—42no—
82C2×#92420—24no—
83C2×#92420—42no—
84C4⋊C4×C61—22no—
85C4⋊C4×C61—42no—
86C4⋊Dic61—22no—
87C4⋊Dic61—41no—
88C4⋊Dic61—41no—
89#11758196—22no—
90#11758196—22no—
91#11758196—42no—
92#11758196—42no—
93C4×Dic61—21no—
94C4×Dic61—21no—
95C4×Dic61—21no—
96C4×Dic61—21no—
97C4×Dic61—42no—
98C4×Dic61C2×D412yes—
99C4×Dic61#4112yes—
100C23×C61C2×C22⋊C411yescharacteristic
101Q8×C61—24no—
102Q8×C61—42no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—22no—
106C2×C4×C61—21no—
107C2×C4×C61—41no—
108C2×C4×C61—41no—
109C2×C4×C61—41no—
110C2×C4×C61—41no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61#10312yes—
115C2×C4×C61C4×D412yes—
116C2×Dic61—21no—
117C2×Dic61—21no—
118C2×Dic61—21no—
119C2×Dic61—41no—
120C2×Dic61—41no—
121C2×Dic61—41no—
122C2×Dic61—41no—
123C2×Dic61—42no—
124C2×Dic61—41no—
125C2×Dic61—41no—
126C2×Dic61#10312yes—
127C2×Dic61#12011yescharacteristic
128C2×Dic61#12112yes—
129C2×Dic61#12311yescharacteristic
130C2×Dic61C2×SD1611yescharacteristic
131C2×Dic61C2×Q1611yescharacteristic
132C2×Dic61#13512yes—
133Dic122—24no—
134Dic122—24no—
135#92420—44no—
136#92420—82no—
137#9484—611nomaximal, Sylow
138C22×C61—21no—
139C22×C61—21no—
140C22×C61#38611yescharacteristic
141C22×C61C2×#10311yescharacteristic
142C22×C61#41611yescharacteristic
143C4×C61—22no—
144C4×C61—22no—
145C4×C61—41no—
146C4×C61—42no—
147C4×C61—81no—
148Dic61—22no—
149Dic61—21no—
150Dic61—21no—
151Dic61—41no—
152Dic61—81no—
153Dic61—81no—
154Dic61#46512yes—
155Dic61#46712yes—
156#2982—611no—
157#2581—611no—
158#1801—611no—
159#1487—612no—
160#1286—612no—
161C2×C61—21no—
162C2×C61—21no—
163C2×C61#189111yescharacteristic
164C2×C61#189211yescharacteristic
165C2×C61#191211yescharacteristic
166C2×C4×Q8—611no—
167C2×C4×Q8—611no—
168C8×Q8—1224no—
169#431—612no—
170#431—1221no—
171#431—1221no—
172#423—612no—
173C2×#101—1221no—
174#400—611no—
175#400—611no—
176#395—611no—
177#395#692311yescharacteristic
178#391—1221no—
179#389—1221no—
180#387—1221no—
181#387—1221no—
182#382—611no—
183#343—1221no—
184Q8⋊C8—1224no—
185C61#948411yescharacteristic, complemented, Sylow
186C22×Q8—611no—
187C22×Q8C2×#692311yescharacteristic
188C2×M16—1222no—
189C4×Q8—1222no—
190C4×Q8—1222no—
191C4×Q8—1222no—
192C4×Q8—1222no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—611no—
195C2×C4⋊C4—611no—
196C2×C4⋊C4—611no—
197C2×C4⋊C4—1221no—
198C2×C4⋊C4—1221no—
199C2×C4⋊C4—1221no—
200C2×C4⋊C4C2×#692311yescharacteristic
201C2×C42—611no—
202C2×C42—611no—
203C2×C42—611no—
204C2×C42—1221no—
205C4⋊C8—1222no—
206C4⋊C8—1222no—
207C4⋊C8—1222no—
208C4⋊C8—1222no—
209C4⋊C8—1222no—
210#101—2442no—
211C22⋊C8—1222no—
212C4×C8—614no—
213C4×C8—1222no—
214C4×C8—1222no—
215#95—21no—
216#95—611no—
217#95—1221no—
218#95—1221no—
219#95—1221no—
220#95—1221no—
221#95—1221no—
222#95—1221no—
223#95C2×#692311yescharacteristic
224C2×Q8—22no—
225C2×Q8—42no—
226C2×Q8—612no—
227C2×Q8—612no—
228C2×Q8—1222no—
229C2×Q8—1222no—
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—42no—
247C4⋊C4—1222no—
248C4⋊C4—1222no—
249C4⋊C4—1222no—
250C4⋊C4—2442no—
251C4⋊C4—2442no—
252C4⋊C4—2441no—
253C4⋊C4—2441no—
254C42—612no—
255C42—612no—
256C42—1221no—
257C42—1221no—
258C42—1221no—
259C42—1221no—
260C42—2442no—
261C23C2×C22⋊#692311yescharacteristic
262Q8—24no—
263Q8—42no—
264Q8—1224no—
265Q8—1224no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—612no—
278C2×C4—611no—
279C2×C4—611no—
280C2×C4—612no—
281C2×C4—611no—
282C2×C4—611no—
283C2×C4—612no—
284C2×C4—1221no—
285C2×C4—1221no—
286C2×C4—1221no—
287C2×C4—2441no—
288C2×C4—2441no—
289C2×C4—2441no—
290C2×C4—2442no—
291C2×C4—2441no—
292C2×C4—2441no—
293C2×C4—2441no—
294C2×C4#1451984012yes—
295C2×C4D4×#692312yes—
296C8—4882no—
297C8—2444no—
298C22—21no—
299C22—21no—
300C22#1684071011yescharacteristic
301C22C2×#1451984011yescharacteristic
302C22#1684075211yescharacteristic
303C4—22no—
304C4—22no—
305C4—41no—
306C4—42no—
307C4—81no—
308C4—612no—
309C4—612no—
310C4—1222no—
311C4—1221no—
312C4—1221no—
313C4—4881no—
314C4—4881no—
315C4—2441no—
316C2—21no—
317C2—21no—
318C2#2146183011yescharacteristic
319C2#2146183211yescharacteristic
320C2#2146186611yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#395C4
socleC22×C61#386
prime radicalC23×C61C2×C22⋊C4
cosocleC2×C4×Dic61C23
hypercenter#395#6923
hypocenterC61#9484
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #395

Complements

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

NH ≅ G/Ntype
185. C61#9484semidirect

Maximal subgroups

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

subgroupindex
#214627142
#214616862
#214611472
#214606722
#214511802
#948461