Finite Group Databasebeta

← #30599262

Subgroups — #30599262

Order 15616.

588 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 110 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#21461711C211yesmaximal, characteristic
2#21460995C211yesmaximal, characteristic
3#21460692C214yesmaximal
4#21448224C211yesmaximal, characteristic
5C2×#14519834C2212yes—
6C2×(C2×#6923)⋊C4—21no—
7C2×(C2×#6923)⋊C4—21no—
8C2×(C2×#6923)⋊C4C2212yes—
9#16840628—21no—
10#16840628—21no—
11C2×#14519735—21no—
12C2×C42×D61C2211yescharacteristic
13#16839632C411yescharacteristic, complemented
14#16839627—21no—
15#16839619—21no—
16#16839619C2212yes—
17C4⋊C4×Dic61—21no—
18C4⋊C4×Dic61C412yescomplemented
19C61×#384C411yescharacteristic, complemented
20C23×#6923—22no—
21C22×#92420—22no—
22#14519834—24no—
23(C2×#6923)⋊C4—24no—
24(C2×#6923)⋊C4—24no—
25(C2×#6923)⋊C4—42no—
26C22×C4×D61D411yescharacteristic
27C22×C4×D61D411yescharacteristic
28C22×C4×D61C2311yescharacteristic
29#14519735—42no—
30#14519735—42no—
31C2×C4⋊Dic61—21no—
32C2×C4⋊Dic61—21no—
33C2×C4⋊Dic61C2×C412yes—
34C2×#11758198—22no—
35C2×C4×Dic61—22no—
36C2×C4×Dic61C2×C411yescharacteristic
37C2×C4×Dic61D411yescharacteristic
38C2×C4×Dic61D411yescharacteristic
39C42×D61—24no—
40#14519684—22no—
41#14519684—22no—
42C2×C4⋊C4×C61—21no—
43C2×C4⋊C4×C61—21no—
44C2×C4⋊C4×C61C2×C412yes—
45C2×C42×C61C2×C411yescharacteristic
46C23×D61C2×D411yescharacteristic
47C22×#6923—24no—
48C22×#6923—42no—
49C22×#6923—42no—
50C22×#6923—42no—
51C22×#6923—42no—
52C22×#6923—42no—
53C22×#6923—42no—
54C22×C4×C61—22no—
55C22×C4×C61C22⋊C411yescharacteristic
56C22×C4×C61C22⋊C411yescharacteristic
57C22×C4×C61C22×C411yescharacteristic
58C22×Dic61—22no—
59C22×Dic61C22⋊C411yescharacteristic
60C22×Dic61C22⋊C411yescharacteristic
61C22×Dic61C2×D411yescharacteristic
62C2×C4×D61—22no—
63C2×C4×D61—22no—
64C2×C4×D61—22no—
65C2×C4×D61—22no—
66C2×C4×D61—41no—
67C2×C4×D61—41no—
68C2×C4×D61—41no—
69C2×C4×D61D812yes—
70C2×C4×D61SD1612yes—
71C2×C4×D61#4114yes—
72C2×#92420—42no—
73C2×#92420—42no—
74C2×#92420—42no—
75C4⋊C4×C61—42no—
76C4⋊C4×C61—42no—
77C4⋊C4×C61—42no—
78C42×C61—22no—
79#11758198—42no—
80#11758198—42no—
81C4⋊Dic61—42no—
82C4⋊Dic61—42no—
83C4⋊Dic61—42no—
84C4×Dic61—22no—
85C4×Dic61—22no—
86C4×Dic61—22no—
87C4×Dic61—44no—
88C23×C61C2×C22⋊C411yescharacteristic
89C22×D61—21no—
90C22×D61—21no—
91C22×D61—21no—
92C22×D61—21no—
93C22×D61#12312yes—
94C22×D61C2×D811yescharacteristic
95C22×D61C2×SD1611yescharacteristic
96C22×D61#13712yes—
97C2×#6923—44no—
98C2×#6923—44no—
99C2×#6923—44no—
100C2×#6923—82no—
101C2×#6923—82no—
102C2×#6923—82no—
103C2×#6923—82no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—42no—
111C2×C4×C61—42no—
112C2×C4×C61—42no—
113C2×C4×C61#10212yes—
114C2×C4×C61#10412yes—
115C2×C4×C61#11712yes—
116C2×Dic61—21no—
117C2×Dic61—21no—
118C2×Dic61—21no—
119C2×Dic61—21no—
120C2×Dic61—21no—
121C2×Dic61—21no—
122C2×Dic61—21no—
123C2×Dic61—42no—
124C2×Dic61—42no—
125C2×Dic61—42no—
126C2×Dic61#10212yes—
127C2×Dic61#12312yes—
128C4×D61—24no—
129C4×D61—24no—
130C4×D61—44no—
131C4×D61—44no—
132#92420—82no—
133#8716—611nomaximal, Sylow
134C22×C61#38111yescharacteristic
135C22×C61#41011yescharacteristic
136C22×C61#41011yescharacteristic
137C22×C61C2×#10211yescharacteristic
138C22×C61#41911yescharacteristic
139C22×C61C2×#10411yescharacteristic
140C22×C61#42111yescharacteristic
141C2×D61—22no—
142C2×D61—22no—
143C2×D61—21no—
144C2×D61—21no—
145C2×D61—22no—
146C2×D61—22no—
147C2×D61—21no—
148C2×D61—21no—
149C2×D61#48012yes—
150C2×D61#48312yes—
151#6923—84no—
152C4×C61—22no—
153C4×C61—22no—
154C4×C61—22no—
155C4×C61—82no—
156Dic61—22no—
157Dic61—22no—
158Dic61—41no—
159Dic61—82no—
160#2285—611no—
161#1817—611no—
162#1434—611no—
163#1294—614no—
164C2×C61#150211yescharacteristic
165C2×C61#150611yescharacteristic
166C2×C61#151511yescharacteristic
167C2×C61#151711yescharacteristic
168C2×C61#182011yescharacteristic
169C2×C61#182311yescharacteristic
170C2×C61#182611yescharacteristic
171D61—22no—
172D61—22no—
173C2×#117—1221no—
174C22×C42—611no—
175C2×C22⋊C8—612no—
176#384—611no—
177#384#692311yescharacteristic, complemented
178#382—1221no—
179#380—612no—
180#380—1221no—
181C4×C4⋊C4—612no—
182C4×C4⋊C4—1221no—
183C2×#95—612no—
184C2×#95—1221no—
185C2×#95—1221no—
186#336—1221no—
187#336—1221no—
188C61#871611yescharacteristic, complemented, Sylow
189C23×C4—611no—
190C23×C4—611no—
191C23×C4—611no—
192C23×C4—1222no—
193C22×C8—1222no—
194#117—2442no—
195#117—2442no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—612no—
199C2×C4⋊C4—1221no—
200C2×C4⋊C4—1221no—
201C2×C4⋊C4C2×#692312yes—
202C2×C22⋊C4—1222no—
203C2×C42—611no—
204C2×C42—611no—
205C2×C42—611no—
206C2×C42—1224no—
207C2×C42—1222no—
208C2×C42C2×#692311yescharacteristic
209C22⋊C8—1224no—
210#95—1222no—
211#95—1222no—
212#95—1224no—
213#95—1224no—
214#95—2442no—
215C24—611no—
216C22×C4—22no—
217C22×C4—611no—
218C22×C4—611no—
219C22×C4—612no—
220C22×C4—612no—
221C22×C4—614no—
222C22×C4—611no—
223C22×C4—1222no—
224C22×C4—1222no—
225C22×C4—1222no—
226C22×C4—1222no—
227C22×C4—1224no—
228C22×C4—1222no—
229C22×C4—2441no—
230C22×C4—2441no—
231C22×C4—2441no—
232C22×C4—2442no—
233C22×C4—2442no—
234C22×C4—2442no—
235C22×C4—2442no—
236C22×C4—2442no—
237C22×C4—2442no—
238C22×C4C22⋊#692311yescharacteristic
239C22×C4C22⋊#692311yescharacteristic
240C22×C4C22×#692311yescharacteristic
241C2×C8—2442no—
242C2×C8—2442no—
243C2×C8—2442no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—2442no—
248C4⋊C4—2442no—
249C4⋊C4—2442no—
250C22⋊C4—2442no—
251C22⋊C4—2442no—
252C42—22no—
253C42—1222no—
254C42—1222no—
255C42—1222no—
256C42—2444no—
257C23—612no—
258C23—611no—
259C23—611no—
260C23—612no—
261C23—1221no—
262C23—1221no—
263C23—1221no—
264C23—1221no—
265C23C2×C22⋊#692311yescharacteristic
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—612no—
276C2×C4—612no—
277C2×C4—1224no—
278C2×C4—1224no—
279C2×C4—1221no—
280C2×C4—1221no—
281C2×C4—1221no—
282C2×C4—1221no—
283C2×C4—1221no—
284C2×C4—1221no—
285C2×C4—1221no—
286C2×C4—4882no—
287C2×C4—4882no—
288C2×C4—4882no—
289C2×C4—4882no—
290C2×C4—2444no—
291C2×C4—2444no—
292C2×C4—2444no—
293C2×C4—2444no—
294C2×C4—2444no—
295C2×C4—2442no—
296C2×C4—2442no—
297C2×C4—2442no—
298C2×C4#1451983912yes—
299C2×C4#1451984112yes—
300C2×C4#1451985612yes—
301C8—4882no—
302C22—612no—
303C22—612no—
304C22—1221no—
305C22—1221no—
306C22—1221no—
307C22—1221no—
308C22—1222no—
309C22—1222no—
310C22—1222no—
311C22—1222no—
312C22#1684070511yescharacteristic
313C22#1684074211yescharacteristic
314C22#1684074311yescharacteristic
315C22C2×#1451983911yescharacteristic
316C22#1684075511yescharacteristic
317C22C2×#1451984111yescharacteristic
318C22#1684076111yescharacteristic
319C4—22no—
320C4—22no—
321C4—22no—
322C4—82no—
323C4—1222no—
324C4—1222no—
325C4—4884no—
326C4—4882no—
327C4—2441no—
328C2—1222no—
329C2—1222no—
330C2#2146119911yescharacteristic
331C2#2146121111yescharacteristic
332C2#2146124311yescharacteristic
333C2#2146125011yescharacteristic
334C2#2146171711yescharacteristic
335C2#2146172211yescharacteristic
336C2#2146172511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#384C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenter#384#6923
hypocenterC61#8716
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
13. #16839632C4semidirect
18. C4⋊C4×Dic61C4semidirect
19. C61×#384C4semidirect
177. #384#6923semidirect
188. C61#8716semidirect

Maximal subgroups

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

subgroupindex
#214617112
#214609952
#214606922
#214482242
#871661