Finite Group Databasebeta

← #30599284

Subgroups — #30599284

Order 15616.

630 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 132 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#21461676C211yesmaximal, characteristic
2#21461676C211yesmaximal, characteristic
3#21460697C214yesmaximal, complemented
4#384×D61C211yesmaximal, characteristic
5C2×#14519844C2212yes—
6C2×#14519844C2212yes—
7C2×#14519834—21no—
8C2×#14519834—21no—
9#16840628—21no—
10#16840628—21no—
11C2×C4⋊C4×D61—21no—
12C2×C4⋊C4×D61—21no—
13C2×C4⋊C4×D61C412yes—
14C2×C42×D61C2211yescharacteristic
15#16839632C411yescharacteristic
16#16839631—21no—
17#16839631—21no—
18#16839631C2212yes—
19C61×#384C411yescharacteristic
20C22×#92420—22no—
21C22×#92420—22no—
22#14519844—24no—
23#14519844—24no—
24#14519834—42no—
25#14519834—42no—
26C22×C4×D61—22no—
27C22×C4×D61C2×C411yescharacteristic
28C22×C4×D61D411yescharacteristic
29C22×C4×D61D411yescharacteristic
30C2×C4⋊Dic61—21no—
31C2×C4⋊Dic61—21no—
32C2×C4⋊Dic61C2×C412yes—
33C2×#11758196—22no—
34C2×#11758196—22no—
35C2×#11758196—22no—
36C4⋊C4×D61—44no—
37C4⋊C4×D61—42no—
38C4⋊C4×D61—42no—
39C4⋊C4×D61—44no—
40C2×C4×Dic61D411yescharacteristic
41C2×C4×Dic61D411yescharacteristic
42C2×C4×Dic61C2311yescharacteristic
43C42×D61—24no—
44C2×C4⋊C4×C61—21no—
45C2×C4⋊C4×C61—21no—
46C2×C4⋊C4×C61C2×C412yes—
47C2×C42×C61C2×C411yescharacteristic
48C23×D61C22⋊C411yescharacteristic
49C22×C4×C61—22no—
50C22×C4×C61C22⋊C411yescharacteristic
51C22×C4×C61C22⋊C411yescharacteristic
52C22×C4×C61C22×C411yescharacteristic
53C22×Dic61—22no—
54C22×Dic61C22⋊C411yescharacteristic
55C22×Dic61C2×D411yescharacteristic
56C22×Dic61C2×D411yescharacteristic
57C2×C4×D61—22no—
58C2×C4×D61—22no—
59C2×C4×D61—22no—
60C2×C4×D61—22no—
61C2×C4×D61—22no—
62C2×C4×D61—22no—
63C2×C4×D61—22no—
64C2×C4×D61—44no—
65C2×C4×D61—44no—
66C2×C4×D61—44no—
67C2×C4×D61D812yes—
68C2×C4×D61SD1612yes—
69C2×C4×D61SD1612yes—
70C2×C4×D61Q1612yes—
71C2×#92420—42no—
72C2×#92420—42no—
73C2×#92420—42no—
74C2×#92420—42no—
75C2×#92420—42no—
76C2×#92420—42no—
77C4⋊C4×C61—42no—
78C4⋊C4×C61—42no—
79C4⋊C4×C61—42no—
80C42×C61—22no—
81C4⋊Dic61—42no—
82C4⋊Dic61—42no—
83C4⋊Dic61—42no—
84#11758196—44no—
85#11758196—42no—
86#11758196—42no—
87#11758196—44no—
88C4×Dic61—22no—
89C4×Dic61—22no—
90C4×Dic61#4114yes—
91C23×C61C2×C22⋊C411yescharacteristic
92C22×D61—21no—
93C22×D61—21no—
94C22×D61—21no—
95C22×D61—21no—
96C22×D61#10112yes—
97C22×D61#10212yes—
98C22×D61#10312yes—
99C2×C4×C61—21no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—42no—
106C2×C4×C61—42no—
107C2×C4×C61—42no—
108C2×C4×C61#10212yes—
109C2×C4×C61#10312yes—
110C2×C4×C61#11712yes—
111C2×Dic61—21no—
112C2×Dic61—21no—
113C2×Dic61—42no—
114C2×Dic61—42no—
115C2×Dic61—42no—
116C2×Dic61#10012yes—
117C2×Dic61#12312yes—
118C2×Dic61#12312yes—
119C2×Dic61C2×D811yescharacteristic
120C2×Dic61C2×SD1611yescharacteristic
121C2×Dic61C2×SD1611yescharacteristic
122C2×Dic61C2×Q1611yescharacteristic
123C2×Dic61AGL(1,ℤ/8)12yes—
124C2×Dic61#13712yes—
125C4×D61—28no—
126C4×D61—24no—
127C4×D61—24no—
128C4×D61—24no—
129C4×D61—24no—
130C4×D61—88no—
131#92420—82no—
132#92420—82no—
133#8724—611nomaximal, Sylow
134C22×C61#38111yescharacteristic
135C22×C61C2×#10011yescharacteristic
136C22×C61C2×#10111yescharacteristic
137C22×C61C2×#10211yescharacteristic
138C22×C61C2×#10311yescharacteristic
139C22×C61#41811yescharacteristic
140C22×C61#41911yescharacteristic
141C2×D61—22no—
142C2×D61—22no—
143C2×D61—22no—
144C2×D61—22no—
145C2×D61—22no—
146C2×D61—22no—
147C2×D61#33214yes—
148C4×C61—22no—
149C4×C61—22no—
150C4×C61—22no—
151C4×C61—82no—
152Dic61—22no—
153Dic61—82no—
154Dic61#48012yes—
155Dic61#48112yes—
156Dic61#48312yes—
157Dic61#48412yes—
158C2×#384—611no—
159#1797—611no—
160#1797—611no—
161#1296—614no—
162C2×C61C2×#33211yescharacteristic
163C2×C61#154511yescharacteristic
164C2×C61#154811yescharacteristic
165C2×C61#155611yescharacteristic
166C2×C61#182111yescharacteristic
167C2×C61#182311yescharacteristic
168C2×C61#182411yescharacteristic
169D61—24no—
170C22×C4⋊C4—612no—
171C22×C4⋊C4—1221no—
172C22×C4⋊C4—1221no—
173C22×C42—611no—
174C2×C4⋊C8—612no—
175C2×C4⋊C8—612no—
176C2×C22⋊C8—1221no—
177C2×C22⋊C8—1221no—
178#384—612no—
179#384—611no—
180#384—1221no—
181#384—1221no—
182#384#692311yescharacteristic
183#336—1221no—
184#336—1221no—
185C61#872411yescharacteristic, complemented, Sylow
186C23×C4—611no—
187C23×C4—611no—
188C23×C4—611no—
189C23×C4—1222no—
190C22×C8—1222no—
191C22×C8—1222no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—612no—
195C2×C4⋊C4—1222no—
196C2×C4⋊C4—1222no—
197C2×C4⋊C4—1222no—
198C2×C4⋊C4—1221no—
199C2×C4⋊C4—1221no—
200C2×C4⋊C4—2442no—
201C2×C4⋊C4—2442no—
202C2×C4⋊C4—2444no—
203C2×C4⋊C4—2444no—
204C2×C4⋊C4C2×#692312yes—
205C2×C42—611no—
206C2×C42—611no—
207C2×C42—611no—
208C2×C42—1224no—
209C2×C42C2×#692311yescharacteristic
210C4⋊C8—1224no—
211C4⋊C8—1224no—
212C22⋊C8—2442no—
213C22⋊C8—2442no—
214C24—611no—
215C22×C4—22no—
216C22×C4—611no—
217C22×C4—611no—
218C22×C4—611no—
219C22×C4—612no—
220C22×C4—612no—
221C22×C4—612no—
222C22×C4—612no—
223C22×C4—1222no—
224C22×C4—1222no—
225C22×C4—1222no—
226C22×C4—1222no—
227C22×C4—1222no—
228C22×C4—1222no—
229C22×C4—1222no—
230C22×C4—1222no—
231C22×C4—2444no—
232C22×C4—2444no—
233C22×C4—2444no—
234C22×C4C22⋊#692311yescharacteristic
235C22×C4C22⋊#692311yescharacteristic
236C22×C4C22×#692311yescharacteristic
237C2×C8—2442no—
238C2×C8—2442no—
239C2×C8—2442no—
240C2×C8—2442no—
241C2×C8—2442no—
242C2×C8—2442no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—2442no—
247C4⋊C4—2442no—
248C4⋊C4—2442no—
249C4⋊C4—2442no—
250C4⋊C4—2444no—
251C4⋊C4—2444no—
252C4⋊C4—2442no—
253C42—22no—
254C42—614no—
255C42—1222no—
256C42—1222no—
257C23—612no—
258C23—612no—
259C23—612no—
260C23—1221no—
261C23—1221no—
262C23—1221no—
263C23—1221no—
264C23C2×C22⋊#692311yescharacteristic
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—612no—
275C2×C4—612no—
276C2×C4—611no—
277C2×C4—611no—
278C2×C4—611no—
279C2×C4—611no—
280C2×C4—612no—
281C2×C4—612no—
282C2×C4—612no—
283C2×C4—1224no—
284C2×C4—1224no—
285C2×C4—1224no—
286C2×C4—1224no—
287C2×C4—1228no—
288C2×C4—1221no—
289C2×C4—1221no—
290C2×C4—4888no—
291C2×C4—2442no—
292C2×C4—2442no—
293C2×C4—2442no—
294C2×C4#1451983912yes—
295C2×C4#1451984012yes—
296C2×C4#1451985612yes—
297C8—4882no—
298C8—4882no—
299C22—614no—
300C22—1222no—
301C22—1222no—
302C22—1222no—
303C22—1222no—
304C22—1222no—
305C22—1222no—
306C22#1684070511yescharacteristic
307C22D61⋊#10011yescharacteristic
308C22D61⋊#10111yescharacteristic
309C22C2×#1451983911yescharacteristic
310C22C2×#1451984011yescharacteristic
311C22#1684075411yescharacteristic
312C22#1684075511yescharacteristic
313C4—22no—
314C4—22no—
315C4—22no—
316C4—82no—
317C4—612no—
318C4—612no—
319C4—612no—
320C4—612no—
321C4—1222no—
322C4—4882no—
323C2—1224no—
324C2D61⋊#33211yescharacteristic
325C2#2146132711yescharacteristic
326C2#2146133311yescharacteristic
327C2#2146134911yescharacteristic
328C2#2146171911yescharacteristic
329C2#2146172211yescharacteristic
330C2#2146172311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#384C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenter#384#6923
hypocenterC61#8724
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
3. #21460697C2semidirect
185. C61#8724semidirect

Maximal subgroups

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

subgroupindex
#214616762
#214606972
#384×D612
#872461