Finite Group Databasebeta

← #11772551

Subgroups — #11772551

Order 1152.

702 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C32×#386C211yesmaximal, characteristic
2C2×#63661C211yesmaximal, characteristic
3C2×#63660C211yesmaximal, characteristic
4#67637—34nomaximal
5C22×C32×D4C2211yescharacteristic
6C23×C32×C4C411yescharacteristic, complemented
7C2×C32×C22⋊C4—21no—
8C2×C32×C22⋊C4C411yescharacteristic, complemented
9C2×#3757—21no—
10C2×#3748—21no—
11C32×#95—21no—
12#63661—21no—
13#63661C412yescomplemented
14#63660—21no—
15#63660C412yescomplemented
16C3×#386D314yes—
17C2×#834—34no—
18C2×#833—34no—
19C24×C32C2×C411yescharacteristic
20C24×C32D411yescharacteristic
21C2×C32×D4—41no—
22C2×C32×D4—41no—
23C2×C32×D4C2×C411yescharacteristic
24C2×C32×D4C2×C411yescharacteristic
25C2×C32×D4D411yescharacteristic
26C2×C32×D4Q811yescharacteristic
27C22×C32×C4—21no—
28C22×C32×C4—21no—
29C22×C32×C4—42no—
30C22×C32×C4—41no—
31C22×C32×C4D411yescharacteristic
32C22×C3⋊Dic3—41no—
33C32×C22⋊C4—42no—
34C32×C22⋊C4—41no—
35#3757—22no—
36#3757—41no—
37#3748—22no—
38#3748—41no—
39C2×C3⋊(C3⋊C8)—41no—
40#1390—91noSylow
41C22×C3×D4C2×D314yes—
42C23×C3×C4Dic314yescomplemented
43C2×C3×C22⋊C4—24no—
44C2×C3×C22⋊C4Dic314yescomplemented
45C2×#217—64no—
46C2×#208—64no—
47C3×#95—24no—
48#834—38no—
49#834—64no—
50#833—38no—
51#833—64no—
52C23×C32—21no—
53C23×C32—21no—
54C23×C32—21no—
55C23×C32—21no—
56C23×C32—41no—
57C23×C32—41no—
58C23×C32—41no—
59C23×C32—41no—
60C23×C32C4211yescharacteristic
61C23×C32C22⋊C411yescharacteristic
62C23×C32C22⋊C411yescharacteristic
63C23×C32C4⋊C411yescharacteristic
64C23×C32C4⋊C411yescharacteristic
65C32×D4—41no—
66C32×D4—41no—
67C32×D4—41no—
68C32×D4—41no—
69C2×C32×C4—21no—
70C2×C32×C4—21no—
71C2×C32×C4—22no—
72C2×C32×C4—41no—
73C2×C32×C4—42no—
74C2×C32×C4—41no—
75C2×C32×C4—42no—
76C2×C32×C4—41no—
77C2×C32×C4—41no—
78C2×C32×C4—41no—
79C2×C32×C4C22⋊C411yescharacteristic
80C2×C32×C4C4⋊C411yescharacteristic
81C2×C3⋊Dic3—42no—
82C2×C3⋊Dic3—81no—
83C2×C3⋊Dic3—81no—
84C3⋊(C3⋊C8)—42no—
85C2×#100—91no—
86C2×#99—91no—
87#386C3⋊D311yescharacteristic
88C24×C3C2×Dic314yes—
89C24×C3#6714yes—
90C2×C3×D4—44no—
91C2×C3×D4—44no—
92C2×C3×D4Dic614yes—
93C2×C3×D4C4×D314yes—
94C2×C3×D4C4×D314yes—
95C2×C3×D4D1214yes—
96C22×C3×C4—24no—
97C22×C3×C4—24no—
98C22×C3×C4—48no—
99C22×C3×C4—44no—
100C22×C3×C4#6714yes—
101C22×Dic3—124no—
102C3×C22⋊C4—48no—
103C3×C22⋊C4—44no—
104#217—68no—
105#217—124no—
106#208—68no—
107#208—124no—
108C2×C3⋊C8—124no—
109C22×C32—21no—
110C22×C32—21no—
111C22×C32—21no—
112C22×C32—21no—
113C22×C32—21no—
114C22×C32—21no—
115C22×C32—41no—
116C22×C32—41no—
117C22×C32—41no—
118C22×C32—41no—
119C22×C32—42no—
120C22×C32—41no—
121C22×C32—41no—
122C22×C32—41no—
123C22×C32—41no—
124C22×C32—81no—
125C22×C32#9511yescharacteristic
126C22×C32#9911yescharacteristic
127C22×C32#9911yescharacteristic
128C32×C4—21no—
129C32×C4—21no—
130C32×C4—42no—
131C32×C4—81no—
132C3⋊Dic3—82no—
133C22×D4C2×C3⋊D311yescharacteristic
134C23×C4C3⋊Dic311yescharacteristic, complemented
135C2×M16—181no—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—181no—
138C2×C22⋊C4C3⋊Dic311yescharacteristic, complemented
139#100—92no—
140#100—181no—
141#99—92no—
142#99—181no—
143#95—21no—
144C23×C3—24no—
145C23×C3—24no—
146C23×C3—24no—
147C23×C3—24no—
148C23×C3—44no—
149C23×C3—44no—
150C23×C3—44no—
151C23×C3—44no—
152C23×C3C4×Dic314yes—
153C23×C3Dic3⋊C414yes—
154C23×C3C4⋊Dic314yes—
155C23×C3(C2×D3)⋊C414yes—
156C23×C3#21714yes—
157C3×D4—44no—
158C3×D4—44no—
159C3×D4—44no—
160C3×D4—44no—
161C2×C3×C4—24no—
162C2×C3×C4—24no—
163C2×C3×C4—28no—
164C2×C3×C4—44no—
165C2×C3×C4—48no—
166C2×C3×C4—44no—
167C2×C3×C4—48no—
168C2×C3×C4—44no—
169C2×C3×C4—44no—
170C2×C3×C4—44no—
171C2×C3×C4Dic3⋊C414yes—
172C2×C3×C4(C2×D3)⋊C414yes—
173C2×Dic3—128no—
174C2×Dic3—244no—
175C2×Dic3—244no—
176C3⋊C8—128no—
177C2×C32—21no—
178C2×C32—21no—
179C2×C32—41no—
180C2×C32—41no—
181C2×C32—41no—
182C2×C32—41no—
183C2×C32#34211yescharacteristic
184C2×C32AΣL(1,𝔽16)11yescharacteristic
185C2×C32#35211yescharacteristic
186C24C2×C3⋊Dic311yescharacteristic
187C24#63811yescharacteristic
188C2×D4—41no—
189C2×D4—41no—
190C2×D4#63411yescharacteristic
191C2×D4C4×C3⋊D311yescharacteristic
192C2×D4C4×C3⋊D311yescharacteristic
193C2×D4C4⋊(C3⋊D3)11yescharacteristic
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—42no—
197C22×C4—41no—
198C22×C4—361no—
199C22×C4#63811yescharacteristic
200M16—182no—
201M16—361no—
202C2×C8—361no—
203C22⋊C4—42no—
204C22⋊C4—41no—
205C22⋊C4—182no—
206C22⋊C4—361no—
207C22×C3—24no—
208C22×C3—24no—
209C22×C3—24no—
210C22×C3—24no—
211C22×C3—24no—
212C22×C3—24no—
213C22×C3—44no—
214C22×C3—44no—
215C22×C3—44no—
216C22×C3—44no—
217C22×C3—48no—
218C22×C3—44no—
219C22×C3—44no—
220C22×C3—44no—
221C22×C3—44no—
222C22×C3—84no—
223C22×C3C3⋊#9514yes—
224C22×C3#83414yes—
225C22×C3#83414yes—
226C3×C4—24no—
227C3×C4—24no—
228C3×C4—48no—
229C3×C4—84no—
230Dic3—248no—
231C32#139011yescharacteristic, complemented, Sylow
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23C4×C3⋊Dic311yescharacteristic
241C23#375011yescharacteristic
242C23C4⋊(C3⋊Dic3)11yescharacteristic
243C23#375211yescharacteristic
244C23#375711yescharacteristic
245D4—41no—
246D4—41no—
247D4—41no—
248D4—41no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—362no—
260C2×C4—721no—
261C2×C4—721no—
262C2×C4#375011yescharacteristic
263C2×C4#375211yescharacteristic
264C8—362no—
265C2×C3—24no—
266C2×C3—24no—
267C2×C3—44no—
268C2×C3—44no—
269C2×C3—44no—
270C2×C3—44no—
271C2×C3#452614yes—
272C2×C3#453614yes—
273C2×C3#453914yes—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—42no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—81no—
290C22#6365811yescharacteristic
291C22#6366111yescharacteristic
292C22#6366111yescharacteristic
293C4—21no—
294C4—21no—
295C4—42no—
296C4—81no—
297C4—722no—
298C3#6763714yescomplemented
299C2—21no—
300C2—21no—
301C2—41no—
302C2—41no—
303C2—41no—
304C2—41no—
305C2#1058939211yescharacteristic
306C2#1058940211yescharacteristic
307C2#1058940511yescharacteristic
subgroupG/H
centerC22#63661
commutatorC23×C32C42
FrattiniC22×D4C2×C3⋊D3
FittingC32×#386C2
socleC22×C32#99
prime radical, cosocleC22×C32×D4C22
hypercenter#386C3⋊D3
hypocenterC32#1390
LCSG ▷ C23×C32 ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #386

Complements

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

NH ≅ G/Ntype
6. C23×C32×C4C4semidirect
8. C2×C32×C22⋊C4C4semidirect
13. #63661C4semidirect
15. #63660C4semidirect
42. C23×C3×C4Dic3semidirect
44. C2×C3×C22⋊C4Dic3semidirect
134. C23×C4C3⋊Dic3semidirect
138. C2×C22⋊C4C3⋊Dic3semidirect
231. C32#1390semidirect
298. C3#67637semidirect

Maximal subgroups

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

subgroupindex
C32×#3862
C2×#636612
C2×#636602
#676373