Finite Group Databasebeta

← #30595496

Subgroups — #30595496

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—44no—
33C2×C4⋊Dic61—42no—
34C2×C4⋊Dic61C2×C412yes—
35C2×C4×Dic61—42no—
36C2×C4×Dic61C2×C411yescharacteristic
37C2×C4×Dic61D411yescharacteristic
38C2×C4×Dic61D411yescharacteristic
39C2×C4×Dic61C2311yescharacteristic
40#14519684—22no—
41#14519684—22no—
42C23×C4×C61—21no—
43C23×C4×C61C2×C411yescharacteristic
44C2×C4⋊C4×C61—21no—
45C2×C4⋊C4×C61—24no—
46C2×C4⋊C4×C61C2×C412yes—
47C24×C61C22⋊C411yescharacteristic
48C22×C4×C61—22no—
49C22×C4×C61—22no—
50C22×C4×C61—22no—
51C22×C4×C61—24no—
52C22×C4×C61—22no—
53C22×C4×C61—42no—
54C22×C4×C61—42no—
55C22×C4×C61C22⋊C411yescharacteristic
56C22×C4×C61C22×C411yescharacteristic
57C22×Dic61—22no—
58C22×Dic61—22no—
59C22×Dic61—22no—
60C22×Dic61—22no—
61C22×Dic61—22no—
62C22×Dic61—41no—
63C22×Dic61—42no—
64C22×Dic61—42no—
65C22×Dic61—42no—
66C22×Dic61C22⋊C411yescharacteristic
67C22×Dic61C22⋊C411yescharacteristic
68C22×Dic61SD1612yes—
69C22×Dic61Q1612yes—
70C22×Dic61C2×D411yescharacteristic
71C22×Dic61C2×D411yescharacteristic
72#11758217—48no—
73C2×#92420—24no—
74C2×#92420—42no—
75C2×#92420—42no—
76C2×#92420—42no—
77C2×#92420—42no—
78C2×#92420—42no—
79C4⋊C4×C61—24no—
80C4⋊C4×C61—22no—
81C4⋊C4×C61M1614yescomplemented
82C4⋊Dic61—44no—
83C4⋊Dic61—42no—
84C4⋊Dic61—44no—
85C4⋊Dic61—42no—
86C4×Dic61—41no—
87C4×Dic61—41no—
88C4×Dic61—41no—
89C4×Dic61—41no—
90C23×C61—21no—
91C23×C61—21no—
92C23×C61—21no—
93C23×C61—21no—
94C23×C61#9912yes—
95C23×C61#10312yes—
96C23×C61#10412yes—
97C23×C61C2×C22⋊C411yescharacteristic
98C2×C4×C61—28no—
99C2×C4×C61—21no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—24no—
104C2×C4×C61—48no—
105C2×C4×C61—42no—
106C2×C4×C61—42no—
107C2×C4×C61#10112yes—
108C2×C4×C61C2×M1612yes—
109C2×Dic61—24no—
110C2×Dic61—24no—
111C2×Dic61—21no—
112C2×Dic61—21no—
113C2×Dic61—21no—
114C2×Dic61—21no—
115C2×Dic61—21no—
116C2×Dic61—21no—
117C2×Dic61—21no—
118C2×Dic61—21no—
119C2×Dic61—21no—
120C2×Dic61—44no—
121C2×Dic61—42no—
122C2×Dic61—42no—
123C2×Dic61—42no—
124C2×Dic61—84no—
125C2×Dic61#10312yes—
126C2×Dic61C2×SD1611yescharacteristic
127C2×Dic61C2×Q1611yescharacteristic
128C2×Dic61AGL(1,ℤ/8)12yes—
129#92420—44no—
130#92420—82no—
131#7607—611nomaximal, Sylow
132C22×C61—22no—
133C22×C61—22no—
134C22×C61—22no—
135C22×C61—22no—
136C22×C61—22no—
137C22×C61—22no—
138C22×C61#32814yes—
139C22×C61#40711yescharacteristic
140C22×C61C2×#9911yescharacteristic
141C22×C61C2×#10111yescharacteristic
142C22×C61C2×#10311yescharacteristic
143C22×C61#41811yescharacteristic
144C22×C61C2×#10411yescharacteristic
145C22×C61#42111yescharacteristic
146C4×C61—22no—
147C4×C61—22no—
148C4×C61—44no—
149Dic61—21no—
150Dic61—22no—
151Dic61—21no—
152Dic61—22no—
153Dic61—41no—
154Dic61—82no—
155C2×#389—611no—
156#1817—611no—
157#1728—611no—
158#1271—614no—
159C2×C61—24no—
160C2×C61#146311yescharacteristic
161C2×C61#148711yescharacteristic
162C2×C61#149011yescharacteristic
163C2×C61C2×#32811yescharacteristic
164C2×C61#151011yescharacteristic
165C2×C61#152811yescharacteristic
166C2×C61#154911yescharacteristic
167C22×M16—1221no—
168C22×C4⋊C4—611no—
169C22×C4⋊C4—1221no—
170C22×C4⋊C4#692311yescharacteristic
171C22×C42—611no—
172C2×C22⋊C8—1221no—
173#389—612no—
174#389—612no—
175#389—1221no—
176#389—1221no—
177C2×#95—1221no—
178#336—612no—
179#336—612no—
180#336—1221no—
181#336—1221no—
182C61#760711yescharacteristic, complemented, Sylow
183C23×C4—21no—
184C23×C4—611no—
185C23×C4—611no—
186C23×C4—1221no—
187C23×C4C2×#692311yescharacteristic
188C2×M16—1224no—
189C2×M16—2442no—
190C2×M16—2442no—
191C22×C8—1222no—
192C22×C8—1222no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—612no—
196C2×C4⋊C4—1222no—
197C2×C4⋊C4—1222no—
198C2×C4⋊C4—1221no—
199C2×C4⋊C4—2442no—
200C2×C4⋊C4—2444no—
201C2×C4⋊C4C2×#692312yes—
202C2×C42—611no—
203C2×C42—611no—
204C2×C42—611no—
205C2×C42—611no—
206C2×C42—2442no—
207C22⋊C8—2442no—
208#95—1222no—
209#95—1222no—
210C24C22⋊#692311yescharacteristic
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—612no—
219C22×C4—612no—
220C22×C4—611no—
221C22×C4—611no—
222C22×C4—611no—
223C22×C4—611no—
224C22×C4—1222no—
225C22×C4—1222no—
226C22×C4—1222no—
227C22×C4—1222no—
228C22×C4—1222no—
229C22×C4—2441no—
230C22×C4—2442no—
231C22×C4—2442no—
232C22×C4—2442no—
233C22×C4C22⋊#692311yescharacteristic
234C22×C4C22×#692311yescharacteristic
235M16—2448no—
236C2×C8—1224no—
237C2×C8—2442no—
238C2×C8—2442no—
239C2×C8—2442no—
240C2×C8—2442no—
241C2×C8—2442no—
242C4⋊C4—24no—
243C4⋊C4—22no—
244C4⋊C4—2442no—
245C4⋊C4—2442no—
246C4⋊C4—2444no—
247C4⋊C4—2444no—
248C4⋊C4#1175821714yescomplemented
249C42—2441no—
250C42—2441no—
251C42—2441no—
252C42—2441no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23#1451983612yes—
258C23#1451984012yes—
259C23#1451984112yes—
260C23C2×C22⋊#692311yescharacteristic
261C2×C4—28no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—24no—
267C2×C4—48no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—612no—
271C2×C4—611no—
272C2×C4—611no—
273C2×C4—612no—
274C2×C4—1221no—
275C2×C4—1221no—
276C2×C4—1221no—
277C2×C4—1221no—
278C2×C4—1221no—
279C2×C4—1221no—
280C2×C4—1224no—
281C2×C4—1224no—
282C2×C4—1221no—
283C2×C4—1221no—
284C2×C4—1221no—
285C2×C4—4884no—
286C2×C4—2444no—
287C2×C4—2442no—
288C2×C4—2442no—
289C2×C4—2442no—
290C2×C4#1451983812yes—
291C2×C4C2×#1175821712yes—
292C8—4882no—
293C8—2444no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22#1684061514yes—
301C22#1684073211yescharacteristic
302C22C2×#1451983611yescharacteristic
303C22C2×#1451983811yescharacteristic
304C22C2×#1451984011yescharacteristic
305C22#1684075411yescharacteristic
306C22C2×#1451984111yescharacteristic
307C22#1684076111yescharacteristic
308C4—22no—
309C4—22no—
310C4—44no—
311C4—1221no—
312C4—1222no—
313C4—1221no—
314C4—1222no—
315C4—4882no—
316C4—2441no—
317C2—24no—
318C2#2146106711yescharacteristic
319C2#2146114711yescharacteristic
320C2#2146115811yescharacteristic
321C2C2×#1684061511yescharacteristic
322C2#2146122611yescharacteristic
323C2#2146128111yescharacteristic
324C2#2146133411yescharacteristic
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#7607
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
81. C4⋊C4×C61M16semidirect
182. C61#7607semidirect
248. 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
#760761