Finite Group Databasebeta

← #11771250

Subgroups — #11771250

Order 1152.

479 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 479 conjugacy classes of subgroups — too many to draw. Only its 104 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#10592906C211yesmaximal, characteristic
2C3×#5211C211yesmaximal, characteristic
3C3×#5207C211yesmaximal, characteristic
4#67686—31nomaximal
5#67629—31nomaximal
6C23×C32×C4C2211yescharacteristic
7C2×#3748—21no—
8C2×C3×#217C412yes—
9C2×C3×#208—21no—
10#63657—21no—
11#63657C412yescomplemented
12C3×C3⋊#95—21no—
13C3×C3⋊#95—21no—
14C3×#830—21no—
15C3×#830C412yescomplemented
16C3×#407—31no—
17C3×#381—31no—
18#5211D311yescharacteristic
19#5207—31no—
20#5207—31no—
21#5207—61no—
22#5207D311yescharacteristic
23C24×C32C2×C411yescharacteristic
24C22×C32×C4—42no—
25C22×C32×C4C2×C411yescharacteristic
26C22×C32×C4C2×C411yescharacteristic
27C22×C32×C4D411yescharacteristic
28C22×C32×C4D411yescharacteristic
29C22×C32×C4D411yescharacteristic
30C22×C32×C4Q811yescharacteristic
31C22×C3×Dic3—22no—
32#3748—42no—
33C2×C3⋊(C3⋊C8)—22no—
34C3×#217—42no—
35C3×#208—42no—
36C2×C3×C3⋊C8—22no—
37#1306—91noSylow
38C23×C3×C4—21no—
39C23×C3×C4C2×D311yescharacteristic
40C23×C3×C4C2×D311yescharacteristic
41C2×C3×M16—61no—
42C2×C3×C22⋊C4—32no—
43C2×#217Dic312yes—
44C2×#208—21no—
45C2×#208—62no—
46C2×#208—61no—
47C2×#208—61no—
48C3×C22⋊C8—32no—
49C3×C22⋊C8—61no—
50C3×#95—61no—
51C3×#95—61no—
52C3⋊#95—21no—
53C3⋊#95—21no—
54#830—21no—
55#830—32no—
56#830—32no—
57#830—62no—
58#830—62no—
59#830—61no—
60#830—61no—
61#830Dic312yescomplemented
62C23×C32—22no—
63C23×C32—41no—
64C23×C32—41no—
65C23×C32C4211yescharacteristic
66C23×C32C22⋊C411yescharacteristic
67C23×C32C4⋊C411yescharacteristic
68C2×C32×C4—22no—
69C2×C32×C4—22no—
70C2×C32×C4—41no—
71C2×C32×C4—41no—
72C2×C32×C4—41no—
73C2×C32×C4—41no—
74C2×C32×C4C22⋊C411yescharacteristic
75C2×C32×C4C22⋊C411yescharacteristic
76C2×C32×C4C4⋊C411yescharacteristic
77C2×C32×C4C4⋊C411yescharacteristic
78C2×C3×Dic3—42no—
79C2×C3×Dic3—42no—
80C2×C3×Dic3—42no—
81C3⋊(C3⋊C8)—42no—
82C3×C3⋊C8—42no—
83#407—31no—
84#407—91no—
85#381—31no—
86C24×C3—21no—
87C24×C3C4×D311yescharacteristic
88C24×C3C2×Dic311yescharacteristic
89C22×C3×C4—21no—
90C22×C3×C4—21no—
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4—42no—
96C22×C3×C4—42no—
97C22×C3×C4—44no—
98C22×C3×C4—62no—
99C22×C3×C4Dic611yescharacteristic
100C22×C3×C4Dic611yescharacteristic
101C22×C3×C4C4×D311yescharacteristic
102C22×C3×C4C4×D311yescharacteristic
103C22×C3×C4C4×D311yescharacteristic
104C22×C3×C4D1211yescharacteristic
105C22×C3×C4D1211yescharacteristic
106C22×C3×C4C2×Dic311yescharacteristic
107C22×C3×C4#6711yescharacteristic
108C22×C3×C4#6711yescharacteristic
109C22×C3×C4#6711yescharacteristic
110C22×C3×C4#6711yescharacteristic
111C22×Dic3—22no—
112C3×M16—122no—
113C2×C3×C8—62no—
114C3×C22⋊C4—122no—
115#217—42no—
116#208—42no—
117#208—124no—
118#208—122no—
119#208—122no—
120C2×C3⋊C8—22no—
121C2×C3⋊C8—64no—
122C2×C3⋊C8—62no—
123C2×C3⋊C8—62no—
124C22×C32—21no—
125C22×C32—21no—
126C22×C32—21no—
127C22×C32—21no—
128C22×C32—21no—
129C22×C32—21no—
130C22×C32—21no—
131C22×C32—21no—
132C22×C32—44no—
133C22×C32#9511yescharacteristic
134C22×C32#9912yes—
135C32×C4—21no—
136C32×C4—21no—
137C32×C4—21no—
138C32×C4—21no—
139C3×Dic3—82no—
140C23×C4D3211yescharacteristic
141C2×M16—61no—
142C2×M16—181no—
143C2×C22⋊C4—32no—
144C22⋊C8—32no—
145C22⋊C8—61no—
146C22⋊C8—92no—
147C22⋊C8—181no—
148#95—61no—
149#95—61no—
150C23×C3—22no—
151C23×C3—22no—
152C23×C3—21no—
153C23×C3—21no—
154C23×C3—21no—
155C23×C3—42no—
156C23×C3—41no—
157C23×C3—41no—
158C23×C3—41no—
159C23×C3—41no—
160C23×C3—42no—
161C23×C3—42no—
162C23×C3C4×Dic311yescharacteristic
163C23×C3C4×Dic311yescharacteristic
164C23×C3Dic3⋊C411yescharacteristic
165C23×C3C4⋊Dic311yescharacteristic
166C23×C3(C2×D3)⋊C411yescharacteristic
167C23×C3#21711yescharacteristic
168C2×C3×C4—24no—
169C2×C3×C4—22no—
170C2×C3×C4—22no—
171C2×C3×C4—22no—
172C2×C3×C4—22no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—42no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—42no—
187C2×C3×C4—42no—
188C2×C3×C4—42no—
189C2×C3×C4—42no—
190C2×C3×C4—122no—
191C2×C3×C4—122no—
192C2×C3×C4—122no—
193C2×C3×C4Dic3⋊C411yescharacteristic
194C2×C3×C4Dic3⋊C411yescharacteristic
195C2×C3×C4Dic3⋊C411yescharacteristic
196C2×C3×C4C4⋊Dic311yescharacteristic
197C2×C3×C4(C2×D3)⋊C411yescharacteristic
198C2×C3×C4(C2×D3)⋊C411yescharacteristic
199C2×C3×C4(C2×D3)⋊C411yescharacteristic
200C2×C3×C4#21711yescharacteristic
201C2×Dic3—42no—
202C2×Dic3—42no—
203C2×Dic3—42no—
204C3×C8—122no—
205C3⋊C8—42no—
206C3⋊C8—124no—
207C3⋊C8—122no—
208C3⋊C8—122no—
209C2×C32—22no—
210C2×C32—41no—
211C2×C32—41no—
212C2×C32#34211yescharacteristic
213C2×C32#34411yescharacteristic
214C2×C32#34411yescharacteristic
215C24D3×Dic311yescharacteristic
216C22×C4—42no—
217C22×C4—62no—
218C22×C4D3×Dic311yescharacteristic
219C22×C4#62411yescharacteristic
220C22×C4#62511yescharacteristic
221C22×C4#62611yescharacteristic
222C22×C4#62611yescharacteristic
223C22×C4#62711yescharacteristic
224M16—122no—
225M16—362no—
226C2×C8—62no—
227C2×C8—182no—
228C22⋊C4—122no—
229C22×C3—21no—
230C22×C3—21no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3—22no—
234C22×C3—22no—
235C22×C3—21no—
236C22×C3—21no—
237C22×C3—21no—
238C22×C3—21no—
239C22×C3—21no—
240C22×C3—21no—
241C22×C3—21no—
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—22no—
249C22×C3—41no—
250C22×C3—41no—
251C22×C3—41no—
252C22×C3—41no—
253C22×C3—41no—
254C22×C3—41no—
255C22×C3—44no—
256C22×C3—44no—
257C22×C3—48no—
258C22×C3#80612yes—
259C22×C3C3⋊#9511yescharacteristic
260C22×C3C3⋊#9511yescharacteristic
261C22×C3#83412yes—
262C3×C4—22no—
263C3×C4—22no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4—242no—
275Dic3—82no—
276C32#130611yescharacteristic, complemented, Sylow
277C23—22no—
278C23—41no—
279C23—41no—
280C23Dic3211yescharacteristic
281C23(C2×D3)⋊Dic311yescharacteristic
282C23Dic3⋊Dic311yescharacteristic
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—122no—
290C2×C4—122no—
291C2×C4—122no—
292C2×C4(C2×D3)⋊Dic311yescharacteristic
293C2×C4#372211yescharacteristic
294C2×C4Dic3⋊Dic311yescharacteristic
295C2×C4#372411yescharacteristic
296C8—122no—
297C8—362no—
298C2×C3—22no—
299C2×C3—22no—
300C2×C3—21no—
301C2×C3—21no—
302C2×C3—21no—
303C2×C3—42no—
304C2×C3—42no—
305C2×C3—42no—
306C2×C3—41no—
307C2×C3—41no—
308C2×C3—41no—
309C2×C3—41no—
310C2×C3#452611yescharacteristic
311C2×C3#452711yescharacteristic
312C2×C3#453311yescharacteristic
313C2×C3#453311yescharacteristic
314C2×C3#455611yescharacteristic
315C2×C3#455911yescharacteristic
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—44no—
325C22#6357911yescharacteristic
326C22#6358012yes—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—242no—
332C3—21no—
333C3#6762911yescharacteristic, complemented
334C3#6768611yescharacteristic, complemented
335C2—22no—
336C2—41no—
337C2—41no—
338C2#1058916911yescharacteristic
339C2#1058917011yescharacteristic
340C2#1058918611yescharacteristic
subgroupG/H
centerC22#63579
commutatorC23×C32C42
Frattini, hypercenterC23×C4D32
Fitting, cosocleC23×C32×C4C22
socleC22×C32#95
prime radicalC24×C32C2×C4
hypocenterC32#1306
LCSG ▷ C23×C32 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
11. #63657C4semidirect
15. C3×#830C4semidirect
61. #830Dic3semidirect
276. C32#1306semidirect
333. C3#67629semidirect
334. C3#67686semidirect

Maximal subgroups

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

subgroupindex
#105929062
C3×#52112
C3×#52072
#676863
#676293