Finite Group Databasebeta

← #30599320

Subgroups — #30599320

Order 15616.

656 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 80 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#21461655C211yesmaximal, characteristic
2#21461648C211yesmaximal, characteristic
3#21460667C214yesmaximal, complemented
4#21448307C211yesmaximal, characteristic
5C2×#14519838—21no—
6C2×#14519837—21no—
7C2×#14519836—21no—
8C2×#14519835—21no—
9#16840734C2212yes—
10#16840705C2212yes—
11C22×#11758222C2211yescharacteristic
12C22×C4×Dic61C411yescharacteristic, complemented
13#16839667—21no—
14#16839663—21no—
15#16839639—21no—
16#16839639—21no—
17C4×#11758198C412yescomplemented
18C4×#11758198C2212yes—
19C2×#117×C61C411yescharacteristic, complemented
20C2×#11758217—22no—
21C2×C22⋊#6923—22no—
22#14519838—42no—
23#14519837—42no—
24#14519836—42no—
25#14519835—42no—
26#14519834—24no—
27(C2×#6923)⋊C4—24no—
28C2×#11758222—24no—
29C2×#11758222—42no—
30C2×#11758222—42no—
31C22×Dic122D411yescharacteristic
32C22×D244D411yescharacteristic
33C22×C22⋊D61D411yescharacteristic
34C22×C22⋊D61D411yescharacteristic
35C23×Dic61—21no—
36C22×C4×D61C2×C411yescharacteristic
37C22×C4×D61C2311yescharacteristic
38C2×#11758198—22no—
39C2×#11758198—22no—
40C2×C4×Dic61—21no—
41C2×C4×Dic61—42no—
42C2×C4×Dic61C2×C412yes—
43#14519684—22no—
44#14519684—22no—
45C23×C4×C61C2×C411yescharacteristic
46#117×C61—42no—
47C2×C4⋊C4×C61—21no—
48C2×C22⋊C4×C61—21no—
49C2×C42×C61C2×C412yes—
50C24×C61C22⋊C411yescharacteristic
51C23×D61C22⋊C411yescharacteristic
52C23×D61C2×D411yescharacteristic
53C22×#6923—42no—
54C22×C4×C61—22no—
55C22×C4×C61—22no—
56C22×C4×C61—22no—
57C22×C4×C61—22no—
58C22×C4×C61C22⋊C411yescharacteristic
59C22×C4×C61C22×C411yescharacteristic
60C2×C22⋊D61—22no—
61C2×C22⋊D61—22no—
62C2×C22⋊D61—22no—
63C2×C22⋊D61—22no—
64C2×C22⋊D61—41no—
65C2×C22⋊D61—41no—
66C22×Dic61—22no—
67C22×Dic61—44no—
68C22×Dic61—42no—
69C22×Dic61C22⋊C411yescharacteristic
70C22×Dic61C2×D411yescharacteristic
71#11758222—416no—
72C2×D244—22no—
73C2×D244—22no—
74C2×D244—41no—
75C2×C4×D61—22no—
76C2×C4×D61—41no—
77C2×C4×D61—41no—
78C2×C4×D61—41no—
79C2×C4×D61—41no—
80C2×C4×D61#4114yes—
81C2×Dic122—22no—
82C2×Dic122—22no—
83C2×Dic122—41no—
84C22⋊#6923—42no—
85C22⋊#6923—42no—
86#11758217—42no—
87#11758217—42no—
88C2×#92420—42no—
89C4⋊C4×C61—42no—
90C22⋊C4×C61—42no—
91C42×C61—42no—
92#11758198—44no—
93#11758198—44no—
94C4×Dic61—42no—
95C4×Dic61—42no—
96C23×C61—21no—
97C23×C61—21no—
98C23×C61—42no—
99C23×C61#9912yes—
100C23×C61C2×C22⋊C411yescharacteristic
101C22×D61—21no—
102C22×D61—21no—
103C22×D61—21no—
104C22×D61—21no—
105C22×D61—21no—
106C22×D61—42no—
107C22×D61—42no—
108C22×D61#12312yes—
109C2×#6923—82no—
110C2×#6923—84no—
111C2×C4×C61—28no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—42no—
117C2×C4×C61—42no—
118C2×C4×C61—42no—
119C2×C4×C61#9912yes—
120C2×C4×C61#11712yes—
121C22⋊D61—44no—
122C22⋊D61—44no—
123C2×Dic61—21no—
124C2×Dic61—21no—
125C2×Dic61—24no—
126C2×Dic61—21no—
127C2×Dic61—21no—
128C2×Dic61—21no—
129C2×Dic61—42no—
130C2×Dic61—48no—
131C2×Dic61—42no—
132C2×Dic61#12312yes—
133D244—44no—
134C4×D61—44no—
135C4×D61—44no—
136Dic122—44no—
137#92420—82no—
138#8738—611nomaximal, Sylow
139C22×C61—22no—
140C22×C61—22no—
141C22×C61—42no—
142C22×C61—42no—
143C22×C61—42no—
144C22×C61#38111yescharacteristic
145C22×C61C2×#9911yescharacteristic
146C22×C61C2×#9911yescharacteristic
147C2×D61—22no—
148C2×D61—24no—
149C2×D61—41no—
150C2×D61—42no—
151C2×D61—81no—
152C2×D61—81no—
153C2×D61—81no—
154C2×D61—81no—
155#6923—162no—
156C4×C61—22no—
157C4×C61—22no—
158C4×C61—82no—
159Dic61—22no—
160Dic61—44no—
161Dic61—41no—
162#2297—611no—
163#1785—611no—
164#1782—611no—
165#1284—614no—
166C2×C61—21no—
167C2×C61—21no—
168C2×C61—42no—
169C2×C61#181911yescharacteristic
170C2×C61#182712yes—
171D61—42no—
172D61—81no—
173C22×#41—611no—
174C2×#117#692311yescharacteristic, complemented
175C22×C42—611no—
176C2×#101—1221no—
177C2×#100—1221no—
178C2×#99—1221no—
179C2×#99—1221no—
180#407—612no—
181#391—1221no—
182#390—1221no—
183#386—1221no—
184#386—1221no—
185#381—612no—
186C4×C22⋊C4—612no—
187C4×C22⋊C4—612no—
188C61#873811yescharacteristic, complemented, Sylow
189C2×#41—1224no—
190C2×#41—2442no—
191C2×#41—2442no—
192C22×Q8—611no—
193C22×D4—611no—
194C22×D4—611no—
195C22×D4—611no—
196C23×C4—611no—
197C23×C4—611no—
198C23×C4—1221no—
199C23×C4C2×#692311yescharacteristic
200C2×M16—1222no—
201#117—42no—
202C2×C4⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—1222no—
205C2×C22⋊C4—1222no—
206C2×C22⋊C4—1222no—
207C2×C42—612no—
208C2×C42—1221no—
209C2×C42—2442no—
210C2×C42C2×#692312yes—
211#101—2442no—
212#100—2442no—
213#99—2442no—
214#99—2442no—
215C22⋊C8—1224no—
216#95—1222no—
217#95—1222no—
218#95—1224no—
219C24—611no—
220C24—611no—
221C24C22⋊#692311yescharacteristic
222#41—24416no—
223C2×Q8—1222no—
224C2×Q8—1222no—
225C2×Q8—2441no—
226C2×D4—1222no—
227C2×D4—1222no—
228C2×D4—1222no—
229C2×D4—1222no—
230C2×D4—1222no—
231C2×D4—1222no—
232C2×D4—2441no—
233C2×D4—2441no—
234C2×D4—2441no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—611no—
240C22×C4—614no—
241C22×C4—611no—
242C22×C4—1222no—
243C22×C4—1222no—
244C22×C4—2441no—
245C22×C4—2441no—
246C22×C4—2441no—
247C22×C4—2441no—
248C22×C4—2444no—
249C22×C4—2442no—
250C22×C4—2442no—
251C22×C4C22⋊#692311yescharacteristic
252C22×C4C22×#692311yescharacteristic
253M16—2442no—
254M16—2442no—
255C2×C8—2442no—
256C4⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—2444no—
259C22⋊C4—2444no—
260C22⋊C4—2442no—
261C22⋊C4—2442no—
262C42—42no—
263C42—2442no—
264C42—2442no—
265C23—21no—
266C23—21no—
267C23—42no—
268C23—612no—
269C23—1221no—
270C23—1221no—
271C23—1221no—
272C23—1221no—
273C23—1221no—
274C23—2442no—
275C23—2442no—
276C23#1451983612yes—
277C23C2×C22⋊#692311yescharacteristic
278Q8—2444no—
279D4—2444no—
280D4—2444no—
281D4—2444no—
282C2×C4—28no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—612no—
291C2×C4—1221no—
292C2×C4—1221no—
293C2×C4—1224no—
294C2×C4—1221no—
295C2×C4—1221no—
296C2×C4—1221no—
297C2×C4—4882no—
298C2×C4—4884no—
299C2×C4—2444no—
300C2×C4—2444no—
301C2×C4—2442no—
302C2×C4—2448no—
303C2×C4—2442no—
304C2×C4#1451983512yes—
305C2×C4#1451985612yes—
306C8—4882no—
307C22—22no—
308C22—22no—
309C22—42no—
310C22—42no—
311C22—42no—
312C22—1222no—
313C22—1224no—
314C22—4881no—
315C22—4881no—
316C22—4881no—
317C22—4881no—
318C22—2441no—
319C22—2442no—
320C22#1684070511yescharacteristic
321C22C2×#1451983511yescharacteristic
322C22C2×#1451983611yescharacteristic
323C4—22no—
324C4—22no—
325C4—82no—
326C4—1222no—
327C4—9762no—
328C4—2444no—
329C4—2441no—
330C2—21no—
331C2—21no—
332C2—42no—
333C2—4881no—
334C2—2442no—
335C2#2146171411yescharacteristic
336C2#2146172912yes—
subgroupG/H
centerC22#16840705
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#117×C61C4
socleC22×C61#381
prime radicalC22×#11758222C22
cosocleC22×C4×D61C23
hypercenterC2×#117#6923
hypocenterC61#8738
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
3. #21460667C2semidirect
12. C22×C4×Dic61C4semidirect
17. C4×#11758198C4semidirect
19. C2×#117×C61C4semidirect
174. C2×#117#6923semidirect
188. C61#8738semidirect

Maximal subgroups

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

subgroupindex
#214616552
#214616482
#214606672
#214483072
#873861