Finite Group Databasebeta

← #30596254

Subgroups — #30596254

Order 15616.

1058 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 1058 conjugacy classes of subgroups — too many to draw. Only its 188 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
1C2×#16840732C212yesmaximal
2#21460657C214yesmaximal, complemented
3C2×C22⋊C4×Dic61C211yesmaximal, characteristic
4C22×#11758217—22no—
5#16840732—24no—
6#16840732C418yescomplemented
7C2×C22⋊#92420—22no—
8C2×C22⋊#92420C2214yes—
9C24×Dic61C2211yescharacteristic
10C22×C22⋊Dic61C411yescharacteristic
11C22×C4×Dic61C412yes—
12C22⋊C4×Dic61—22no—
13C22⋊C4×Dic61—22no—
14C2×#14519684C2212yes—
15C22×C22⋊C4×C61C411yescharacteristic
16C2×#11758217—28no—
17C2×#11758217—48no—
18C22×#92420—24no—
19C22⋊#92420—28no—
20C22⋊#92420—24no—
21C22⋊#92420C2×C418yes—
22C23×Dic61—44no—
23C23×Dic61C2×C411yescharacteristic
24C23×Dic61C2×C412yes—
25C23×Dic61C2×C414yes—
26C23×Dic61D412yes—
27C23×Dic61D412yes—
28C23×Dic61D412yes—
29C23×Dic61Q812yes—
30C23×Dic61C2311yescharacteristic
31C2×C22⋊Dic61—22no—
32C2×C22⋊Dic61—24no—
33C2×C4×Dic61—22no—
34C2×C4×Dic61—22no—
35C2×C4×Dic61—44no—
36#14519684—24no—
37C25×C61C2×C411yescharacteristic
38C23×C4×C61C2×C412yes—
39C2×C22⋊C4×C61—22no—
40C2×C22⋊C4×C61—24no—
41C24×C61—22no—
42C24×C61—22no—
43C24×C61—42no—
44C24×C61—42no—
45C24×C61C4212yes—
46C24×C61C22⋊C412yes—
47C24×C61C4⋊C412yes—
48C24×C61C22×C411yescharacteristic
49C22×C4×C61—22no—
50C22×C4×C61—22no—
51C22×C4×C61—42no—
52C22×C4×C61—42no—
53C22×C4×C61M1614yescomplemented
54C22×Dic61—24no—
55C22×Dic61—28no—
56C22×Dic61—22no—
57C22×Dic61—22no—
58C22×Dic61—48no—
59C22×Dic61—44no—
60C22×Dic61—44no—
61C22×Dic61—44no—
62C22×Dic61—42no—
63C22×Dic61—42no—
64C22×Dic61—42no—
65C22×Dic61C4212yes—
66C22×Dic61C22⋊C411yescharacteristic
67C22×Dic61C22⋊C414yes—
68C22×Dic61C22⋊C414yes—
69C22×Dic61C22⋊C411yescharacteristic
70C22×Dic61C4⋊C412yes—
71C22×Dic61C4⋊C414yes—
72C22×Dic61C4⋊C414yes—
73C22×Dic61M1614yescomplemented
74C22×Dic61C22×C412yes—
75C22×Dic61C2×D412yes—
76C22×Dic61C2×D411yescharacteristic
77C22×Dic61C2×Q811yescharacteristic
78#11758217—416no—
79C2×#92420—28no—
80C2×#92420—48no—
81C22⋊C4×C61—28no—
82C22⋊Dic61—28no—
83C4×Dic61—44no—
84C4×Dic61—44no—
85C23×C61—22no—
86C23×C61—22no—
87C23×C61—24no—
88C23×C61—22no—
89C23×C61—22no—
90C23×C61—24no—
91C23×C61—24no—
92C23×C61—24no—
93C23×C61—42no—
94C23×C61—42no—
95C23×C61—42no—
96C23×C61—42no—
97C23×C61—44no—
98C23×C61—44no—
99C23×C61—44no—
100C23×C61—44no—
101C23×C61#9514yes—
102C23×C61C2×C4211yescharacteristic
103C23×C61C2×C22⋊C411yescharacteristic
104C23×C61C2×C4⋊C411yescharacteristic
105C23×C61C2×M1612yes—
106C23×C61C2×M1612yes—
107C2×C4×C61—24no—
108C2×C4×C61—44no—
109C2×C4×C61—42no—
110C2×C4×C61—42no—
111C2×C4×C61—44no—
112C2×Dic61—22no—
113C2×Dic61—24no—
114C2×Dic61—28no—
115C2×Dic61—24no—
116C2×Dic61—24no—
117C2×Dic61—28no—
118C2×Dic61—24no—
119C2×Dic61—44no—
120C2×Dic61—44no—
121C2×Dic61—42no—
122C2×Dic61—41no—
123C2×Dic61—41no—
124C2×Dic61—44no—
125C2×Dic61—42no—
126C2×Dic61—42no—
127C2×Dic61—44no—
128C2×Dic61#9512yes—
129C2×Dic61#9512yes—
130C2×Dic61#9914yes—
131C2×Dic61C2×C22⋊C412yes—
132C2×Dic61C2×C4⋊C412yes—
133#92420—48no—
134#7911—611nomaximal, Sylow
135C22×C61—22no—
136C22×C61—22no—
137C22×C61—22no—
138C22×C61—22no—
139C22×C61—21no—
140C22×C61—21no—
141C22×C61—21no—
142C22×C61—21no—
143C22×C61—24no—
144C22×C61—24no—
145C22×C61—24no—
146C22×C61—48no—
147C22×C61—48no—
148C22×C61—44no—
149C22×C61—44no—
150C22×C61C2×#9511yescharacteristic
151C22×C61C4×M1612yes—
152C22×C61#40711yescharacteristic
153C22×C61#40711yescharacteristic
154C22×C61C2×#9912yes—
155C22×C61#41312yes—
156C22×C61#42312yes—
157C4×C61—44no—
158Dic61—22no—
159Dic61—22no—
160Dic61—22no—
161Dic61—44no—
162Dic61#34214yes—
163C2×#407—612no—
164C2×C4×C22⋊C4—611no—
165#1280—614no—
166C2×C61—22no—
167C2×C61—22no—
168C2×C61—44no—
169C2×C61#146512yes—
170C2×C61#146512yes—
171C2×C61#172811yescharacteristic
172C2×C61C2×#34211yescharacteristic
173C2×C61#174211yescharacteristic
174C24×C4—611no—
175C22×M16—1222no—
176C22×C22⋊C4—611no—
177C22×C22⋊C4#692311yescharacteristic
178C22×C42—612no—
179#407—618no—
180#407—1224no—
181C2×C22⋊C8—614no—
182C2×C22⋊C8—1222no—
183C4×C22⋊C4—1222no—
184C4×C22⋊C4—1222no—
185C2×#95—612no—
186C61#791111yescharacteristic, complemented, Sylow
187C25C2×#692311yescharacteristic
188C23×C4—612no—
189C23×C4—611no—
190C23×C4—614no—
191C23×C4—612no—
192C23×C4—612no—
193C23×C4—612no—
194C23×C4—611no—
195C23×C4—612no—
196C23×C4—2444no—
197C23×C4C2×#692312yes—
198C2×M16—1228no—
199C2×M16—2448no—
200C22×C8—1224no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—1224no—
204C2×C22⋊C4—1222no—
205C2×C42—1222no—
206C2×C42—1222no—
207C2×C42—2444no—
208C22⋊C8—618no—
209C22⋊C8—1224no—
210C22⋊C8—1228no—
211#95—1224no—
212C24—22no—
213C24—22no—
214C24—42no—
215C24—42no—
216C24C4×#692312yes—
217C24C4⋊#692312yes—
218C24C22⋊#692312yes—
219C24C22×#692311yescharacteristic
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—611no—
225C22×C4—614no—
226C22×C4—614no—
227C22×C4—612no—
228C22×C4—612no—
229C22×C4—611no—
230C22×C4—612no—
231C22×C4—614no—
232C22×C4—614no—
233C22×C4—612no—
234C22×C4—611no—
235C22×C4—611no—
236C22×C4—614no—
237C22×C4—1224no—
238C22×C4—1228no—
239C22×C4—1222no—
240C22×C4—1222no—
241C22×C4—2444no—
242C22×C4—2448no—
243C22×C4—2444no—
244C22×C4—2444no—
245C22×C4—2442no—
246C22×C4—2442no—
247C22×C4—2442no—
248C22×C4#1175821714yescomplemented
249M16—24416no—
250C2×C8—1228no—
251C2×C8—2448no—
252C22⋊C4—28no—
253C22⋊C4—1228no—
254C42—2444no—
255C42—2444no—
256C23—22no—
257C23—22no—
258C23—24no—
259C23—22no—
260C23—22no—
261C23—24no—
262C23—24no—
263C23—24no—
264C23—42no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23—44no—
269C23—44no—
270C23—44no—
271C23—44no—
272C23(C2×#6923)⋊C414yes—
273C23C2×C4×#692311yescharacteristic
274C23C2×C22⋊#692311yescharacteristic
275C23C2×C4⋊#692311yescharacteristic
276C23C2×#1175821712yes—
277C23D61⋊M1612yes—
278C2×C4—24no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—614no—
284C2×C4—612no—
285C2×C4—612no—
286C2×C4—612no—
287C2×C4—612no—
288C2×C4—1228no—
289C2×C4—1224no—
290C2×C4—1224no—
291C2×C4—1228no—
292C2×C4—1222no—
293C2×C4—1224no—
294C2×C4—1224no—
295C2×C4—2441no—
296C2×C4—2442no—
297C2×C4—2444no—
298C2×C4—2441no—
299C2×C4—2444no—
300C2×C4—2444no—
301C2×C4—2442no—
302C2×C4—2442no—
303C2×C4—2444no—
304C8—2448no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—24no—
314C22—24no—
315C22—24no—
316C22—48no—
317C22—48no—
318C22—44no—
319C22—44no—
320C22C2×(C2×#6923)⋊C411yescharacteristic
321C22C4×#1175821712yes—
322C22#1684073211yescharacteristic
323C22#1684073411yescharacteristic
324C22D61⋊#9912yes—
325C22#1684074912yes—
326C22#1684076512yes—
327C4—44no—
328C4—614no—
329C4—1222no—
330C4—1222no—
331C4—1222no—
332C4—2444no—
333C2—22no—
334C2—22no—
335C2—44no—
336C2#2146107512yes—
337C2#2146107712yes—
338C2#2146148211yescharacteristic
339C2#2146151611yescharacteristic
340C2#2146152511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC23×C61C2×C42
FrattiniC24C22×#6923
FittingC22×C22⋊C4×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC25×C61C2×C4
cosocleC23×Dic61C23
hypercenterC22×C22⋊C4#6923
hypocenterC61#7911
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C25 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #21460657C2semidirect
6. #16840732C4semidirect
53. C22×C4×C61M16semidirect
73. C22×Dic61M16semidirect
186. C61#7911semidirect
248. C22×C4#11758217semidirect

Maximal subgroups

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

subgroupindex
C2×#168407322
#214606572
C2×C22⋊C4×Dic612
#791161