Finite Group Databasebeta

← #30595506

Subgroups — #30595506

Order 15616.

612 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 122 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#21461710C211yesmaximal, characteristic
2#21461482C211yesmaximal, characteristic
3#21460637C214yesmaximal, complemented
4C2×#16839661C211yesmaximal, characteristic
5C22×#11758217—21no—
6C2×C22⋊#92420—21no—
7#16840628—21no—
8#16840628—21no—
9#16840628C2212yes—
10#16840628C2212yes—
11C22×C4⋊Dic61—21no—
12C22×C4⋊Dic61C411yescharacteristic
13C22×C4×Dic61C2211yescharacteristic
14#16839661—21no—
15#16839661—21no—
16#16839661C412yes—
17#16839661C2212yes—
18C2×#14519684—21no—
19C22×C4⋊C4×C61C411yescharacteristic
20C2×#11758217—24no—
21C2×#11758217—42no—
22C2×#11758217—42no—
23C22×#92420—22no—
24C22×#92420—22no—
25C22⋊#92420—42no—
26C23×Dic61—21no—
27C23×Dic61D411yescharacteristic
28C23×Dic61D411yescharacteristic
29C2×C4⋊Dic61—21no—
30C2×C4⋊Dic61—22no—
31C2×C4⋊Dic61—22no—
32C2×C4⋊Dic61—42no—
33C2×C4⋊Dic61—42no—
34C2×C4⋊Dic61—42no—
35C2×C4⋊Dic61C2×C412yes—
36C2×C4×Dic61—42no—
37C2×C4×Dic61C2×C411yescharacteristic
38C2×C4×Dic61D411yescharacteristic
39C2×C4×Dic61D411yescharacteristic
40C2×C4×Dic61C2311yescharacteristic
41#14519684—22no—
42#14519684—22no—
43C23×C4×C61—21no—
44C23×C4×C61C2×C411yescharacteristic
45C2×C4⋊C4×C61—21no—
46C2×C4⋊C4×C61—24no—
47C2×C4⋊C4×C61C2×C412yes—
48C24×C61C22⋊C411yescharacteristic
49C22×C4×C61—22no—
50C22×C4×C61—22no—
51C22×C4×C61—22no—
52C22×C4×C61—24no—
53C22×C4×C61—22no—
54C22×C4×C61—42no—
55C22×C4×C61—42no—
56C22×C4×C61C22⋊C411yescharacteristic
57C22×C4×C61C22×C411yescharacteristic
58C22×Dic61—22no—
59C22×Dic61—22no—
60C22×Dic61—22no—
61C22×Dic61—22no—
62C22×Dic61—22no—
63C22×Dic61—41no—
64C22×Dic61—42no—
65C22×Dic61—42no—
66C22×Dic61—42no—
67C22×Dic61C22⋊C411yescharacteristic
68C22×Dic61C22⋊C411yescharacteristic
69C22×Dic61D812yes—
70C22×Dic61SD1612yes—
71C22×Dic61C2×D411yescharacteristic
72C22×Dic61C2×D411yescharacteristic
73#11758217—48no—
74C2×#92420—24no—
75C2×#92420—42no—
76C2×#92420—42no—
77C2×#92420—42no—
78C2×#92420—42no—
79C2×#92420—42no—
80C4⋊C4×C61—24no—
81C4⋊C4×C61—22no—
82C4⋊C4×C61M1614yescomplemented
83C4⋊Dic61—42no—
84C4⋊Dic61—44no—
85C4⋊Dic61—42no—
86C4⋊Dic61—42no—
87C4⋊Dic61—42no—
88C4×Dic61—41no—
89C4×Dic61—41no—
90C4×Dic61—41no—
91C4×Dic61—41no—
92C23×C61—21no—
93C23×C61—21no—
94C23×C61—21no—
95C23×C61—21no—
96C23×C61#9912yes—
97C23×C61#10212yes—
98C23×C61#10412yes—
99C23×C61C2×C22⋊C411yescharacteristic
100C2×C4×C61—28no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—24no—
106C2×C4×C61—48no—
107C2×C4×C61—42no—
108C2×C4×C61—42no—
109C2×C4×C61#10112yes—
110C2×C4×C61C2×M1612yes—
111C2×Dic61—24no—
112C2×Dic61—24no—
113C2×Dic61—21no—
114C2×Dic61—21no—
115C2×Dic61—21no—
116C2×Dic61—21no—
117C2×Dic61—21no—
118C2×Dic61—21no—
119C2×Dic61—21no—
120C2×Dic61—21no—
121C2×Dic61—21no—
122C2×Dic61—44no—
123C2×Dic61—42no—
124C2×Dic61—42no—
125C2×Dic61—42no—
126C2×Dic61—84no—
127C2×Dic61#10212yes—
128C2×Dic61C2×D811yescharacteristic
129C2×Dic61C2×SD1611yescharacteristic
130C2×Dic61#13712yes—
131#92420—44no—
132#92420—82no—
133#7610—611nomaximal, Sylow
134C22×C61—22no—
135C22×C61—22no—
136C22×C61—22no—
137C22×C61—22no—
138C22×C61—22no—
139C22×C61—22no—
140C22×C61#32714yes—
141C22×C61#40711yescharacteristic
142C22×C61C2×#9911yescharacteristic
143C22×C61C2×#10111yescharacteristic
144C22×C61C2×#10211yescharacteristic
145C22×C61#41911yescharacteristic
146C22×C61C2×#10411yescharacteristic
147C22×C61#42111yescharacteristic
148C4×C61—22no—
149C4×C61—22no—
150C4×C61—44no—
151Dic61—21no—
152Dic61—22no—
153Dic61—21no—
154Dic61—22no—
155Dic61—41no—
156Dic61—82no—
157C2×#389—611no—
158#1817—611no—
159#1728—611no—
160#1271—614no—
161C2×C61—24no—
162C2×C61#146311yescharacteristic
163C2×C61#148611yescharacteristic
164C2×C61#149111yescharacteristic
165C2×C61C2×#32711yescharacteristic
166C2×C61#151111yescharacteristic
167C2×C61#152811yescharacteristic
168C2×C61#154611yescharacteristic
169C22×M16—1221no—
170C22×C4⋊C4—611no—
171C22×C4⋊C4—1221no—
172C22×C4⋊C4#692311yescharacteristic
173C22×C42—611no—
174C2×C22⋊C8—1221no—
175#389—612no—
176#389—612no—
177#389—1221no—
178#389—1221no—
179C2×#95—1221no—
180#336—612no—
181#336—612no—
182#336—1221no—
183#336—1221no—
184C61#761011yescharacteristic, complemented, Sylow
185C23×C4—21no—
186C23×C4—611no—
187C23×C4—611no—
188C23×C4—1221no—
189C23×C4C2×#692311yescharacteristic
190C2×M16—1224no—
191C2×M16—2442no—
192C2×M16—2442no—
193C22×C8—1222no—
194C22×C8—1222no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—612no—
198C2×C4⋊C4—1222no—
199C2×C4⋊C4—1222no—
200C2×C4⋊C4—1221no—
201C2×C4⋊C4—2442no—
202C2×C4⋊C4—2442no—
203C2×C4⋊C4—2442no—
204C2×C4⋊C4C2×#692312yes—
205C2×C42—611no—
206C2×C42—611no—
207C2×C42—611no—
208C2×C42—611no—
209C2×C42—2442no—
210C22⋊C8—2442no—
211#95—1222no—
212#95—1222no—
213C24C22⋊#692311yescharacteristic
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—612no—
222C22×C4—612no—
223C22×C4—611no—
224C22×C4—611no—
225C22×C4—611no—
226C22×C4—611no—
227C22×C4—1222no—
228C22×C4—1222no—
229C22×C4—1222no—
230C22×C4—1222no—
231C22×C4—1222no—
232C22×C4—2441no—
233C22×C4—2442no—
234C22×C4—2442no—
235C22×C4—2442no—
236C22×C4C22⋊#692311yescharacteristic
237C22×C4C22×#692311yescharacteristic
238M16—2448no—
239C2×C8—1224no—
240C2×C8—2442no—
241C2×C8—2442no—
242C2×C8—2442no—
243C2×C8—2442no—
244C2×C8—2442no—
245C4⋊C4—24no—
246C4⋊C4—22no—
247C4⋊C4—2442no—
248C4⋊C4—2442no—
249C4⋊C4—2444no—
250C4⋊C4—2442no—
251C4⋊C4—2442no—
252C4⋊C4#1175821714yescomplemented
253C42—2441no—
254C42—2441no—
255C42—2441no—
256C42—2441no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23#1451983612yes—
262C23#1451983912yes—
263C23#1451984212yes—
264C23C2×C22⋊#692311yescharacteristic
265C2×C4—28no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—24no—
271C2×C4—48no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—612no—
275C2×C4—611no—
276C2×C4—611no—
277C2×C4—612no—
278C2×C4—1221no—
279C2×C4—1221no—
280C2×C4—1221no—
281C2×C4—1221no—
282C2×C4—1221no—
283C2×C4—1221no—
284C2×C4—1224no—
285C2×C4—1224no—
286C2×C4—1221no—
287C2×C4—1221no—
288C2×C4—1221no—
289C2×C4—4884no—
290C2×C4—2444no—
291C2×C4—2442no—
292C2×C4—2442no—
293C2×C4—2442no—
294C2×C4#1451983812yes—
295C2×C4C2×#1175821712yes—
296C8—4882no—
297C8—2444no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22#1684061414yes—
305C22#1684073211yescharacteristic
306C22C2×#1451983611yescharacteristic
307C22C2×#1451983811yescharacteristic
308C22C2×#1451983911yescharacteristic
309C22#1684075511yescharacteristic
310C22C2×#1451984211yescharacteristic
311C22#1684076011yescharacteristic
312C4—22no—
313C4—22no—
314C4—44no—
315C4—1221no—
316C4—1222no—
317C4—1221no—
318C4—1222no—
319C4—4882no—
320C4—2441no—
321C2—24no—
322C2#2146106711yescharacteristic
323C2#2146114411yescharacteristic
324C2#2146116111yescharacteristic
325C2C2×#1684061411yescharacteristic
326C2#2146122711yescharacteristic
327C2#2146128311yescharacteristic
328C2#2146132811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC22×C4⋊C4×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenterC22×C4⋊C4#6923
hypocenterC61#7610
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #21460637C2semidirect
82. C4⋊C4×C61M16semidirect
184. C61#7610semidirect
252. C4⋊C4#11758217semidirect

Maximal subgroups

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

subgroupindex
#214617102
#214614822
#214606372
C2×#168396612
#761061