Finite Group Databasebeta

← #11783282

Subgroups — #11783282

Order 1152.

746 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 746 conjugacy classes of subgroups — too many to draw. Only its 138 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×#64286C211yesmaximal, characteristic
2C22×C4⋊#157C211yesmaximal, characteristic
3C2×M16×C3⋊D3C211yesmaximal, characteristic
4#10590182C214yesmaximal, complemented
5C23×#157—21no—
6C22×C4×C3⋊D3C2211yescharacteristic
7C2×C22⋊#157—21no—
8#64286—22no—
9#64286C414yescomplemented
10C2×C4⋊#157—22no—
11C2×C4⋊#157—21no—
12C2×C4⋊#157C414yescomplemented
13C2×C4⋊#157C2212yes—
14C2×C4⋊#157C2212yes—
15C2×C4×#157—21no—
16C2×C32×M16C411yescharacteristic, complemented
17C2×#3748C411yescharacteristic, complemented
18M16×C3⋊D3—22no—
19M16×C3⋊D3—22no—
20C2×#3743C412yescomplemented
21C2×C8×C3⋊D3C2212yes—
22C2×D3×M16—62no—
23C23×C3⋊D3D411yescharacteristic
24C22×#157—24no—
25C22×#157—22no—
26C22×#157—22no—
27C22×#157—44no—
28C22×C32×C4C2×C411yescharacteristic
29C22×C3⋊Dic3Q811yescharacteristic
30C2×C4×C3⋊D3C2×C411yescharacteristic
31C2×C4×C3⋊D3C2×C412yes—
32C2×C4×C3⋊D3C2×C412yes—
33C2×C4×C3⋊D3D411yescharacteristic
34C2×C4×C3⋊D3D412yes—
35C2×C4×C3⋊D3D412yes—
36C2×C4×C3⋊D3Q811yescharacteristic
37C2×C4×C3⋊D3C2311yescharacteristic
38C22⋊#157—24no—
39C4⋊#157—24no—
40C4⋊#157—22no—
41C4⋊#157—22no—
42C4⋊#157C2×C414yes—
43C4⋊#157C2×C414yescomplemented
44C4×#157—24no—
45C32×M16—22no—
46C2×C32×C8C2×C412yescomplemented
47#3748—22no—
48C2×C3⋊(C3⋊C8)C2×C412yescomplemented
49#3743—24no—
50C8×C3⋊D3—22no—
51C8×C3⋊D3—22no—
52#1736—91nomaximal, Sylow
53C22×C4×D3—62no—
54C2×C3×M16—22no—
55C2×#208—62no—
56D3×M16—68no—
57D3×M16—68no—
58C2×ΓL(1,𝔽25)—64no—
59C2×D3×C8—64no—
60C23×C32C4⋊C411yescharacteristic
61C22×C3⋊D3—22no—
62C22×C3⋊D3—22no—
63C22×C3⋊D3C22⋊C412yes—
64C22×C3⋊D3C22⋊C412yes—
65C22×C3⋊D3C4⋊C411yescharacteristic
66C22×C3⋊D3C2×D411yescharacteristic
67C2×#157—24no—
68C2×#157—24no—
69C2×#157—48no—
70C2×#157—42no—
71C2×#157—44no—
72C2×#157—42no—
73C2×C32×C4C4212yes—
74C2×C32×C4C22⋊C412yes—
75C2×C32×C4C4⋊C411yescharacteristic
76C2×C32×C4C22×C411yescharacteristic
77C2×C3⋊Dic3C4⋊C411yescharacteristic
78C2×C3⋊Dic3C4⋊C412yes—
79C2×C3⋊Dic3C4⋊C412yes—
80C2×C3⋊Dic3C2×Q811yescharacteristic
81C4×C3⋊D3C4212yes—
82C4×C3⋊D3C22⋊C412yes—
83C4×C3⋊D3C22⋊C412yes—
84C4×C3⋊D3C22⋊C412yes—
85C4×C3⋊D3C4⋊C412yes—
86C4×C3⋊D3C4⋊C412yes—
87C4×C3⋊D3C22×C411yescharacteristic
88C4×C3⋊D3C22×C411yescharacteristic
89C4×C3⋊D3C2×D411yescharacteristic
90C4×C3⋊D3C2×D411yescharacteristic
91C32×C8—22no—
92C3⋊(C3⋊C8)—22no—
93C22×M16—91no—
94C2×#117—91no—
95C22×C4⋊C4—91no—
96#340—94no—
97C23×D3—62no—
98C22×C3×C4—22no—
99C22×Dic3—62no—
100C2×C4×D3—62no—
101C2×C4×D3—62no—
102C2×C4×D3—64no—
103C2×C4×D3—64no—
104C2×C4×D3—64no—
105C2×C4×D3—64no—
106C2×C4×D3—62no—
107C2×C4×D3—62no—
108C3×M16—28no—
109C2×C3×C8—24no—
110#208—68no—
111C2×C3⋊C8—64no—
112ΓL(1,𝔽25)—616no—
113D3×C8—68no—
114D3×C8—68no—
115C22×C32—22no—
116C22×C32#9512yes—
117C22×C32C2×C4⋊C411yescharacteristic
118C2×C3⋊D3—22no—
119C2×C3⋊D3—24no—
120C2×C3⋊D3—22no—
121C2×C3⋊D3—22no—
122C2×C3⋊D3#9512yes—
123C2×C3⋊D3C2×C22⋊C411yescharacteristic
124C2×C3⋊D3C2×C22⋊C411yescharacteristic
125C2×C3⋊D3AGL(1,ℤ/8)12yes—
126C2×C3⋊D3#13712yes—
127#157—44no—
128#157—44no—
129C32×C4#9512yes—
130C32×C4C2×C4211yescharacteristic
131C32×C4C2×C22⋊C411yescharacteristic
132C3⋊Dic3#9512yes—
133C3⋊Dic3C2×C4⋊C411yescharacteristic
134C3⋊Dic3C2×C4⋊C411yescharacteristic
135C23×C4—91no—
136C23×C4—181no—
137C2×M16—91no—
138C2×M16—92no—
139C2×M16—182no—
140C2×M16—182no—
141C2×M16#15711yescharacteristic, complemented
142C22×C8—92no—
143#117—94no—
144#117—182no—
145C2×C4⋊C4—94no—
146C2×C4⋊C4—92no—
147C2×C4⋊C4—92no—
148C2×C4⋊C4—181no—
149C2×C4⋊C4—182no—
150C2×C22⋊C4—181no—
151C2×C42—181no—
152C23×C3—22no—
153C22×D3—62no—
154C22×D3—64no—
155C22×D3—64no—
156C22×D3—62no—
157C22×D3—124no—
158C22×D3—124no—
159C2×C3×C4—22no—
160C2×C3×C4—24no—
161C2×C3×C4—22no—
162C2×C3×C4—24no—
163C2×Dic3—62no—
164C2×Dic3—64no—
165C2×Dic3—64no—
166C2×Dic3—62no—
167C4×D3—64no—
168C4×D3—64no—
169C4×D3—64no—
170C4×D3—64no—
171C4×D3—64no—
172C4×D3—64no—
173C4×D3—62no—
174C4×D3—62no—
175C4×D3—62no—
176C4×D3—62no—
177C3×C8—28no—
178C3⋊C8—68no—
179C2×C32—22no—
180C2×C32C2×#9511yescharacteristic
181C2×C32#42812yes—
182C3⋊D3—22no—
183C3⋊D3#41712yes—
184C3⋊D3#41811yescharacteristic
185C3⋊D3#41911yescharacteristic
186C24—91no—
187C22×C4—91no—
188C22×C4—91no—
189C22×C4—91no—
190C22×C4—92no—
191C22×C4—92no—
192C22×C4—91no—
193C22×C4—91no—
194C22×C4—92no—
195C22×C4—92no—
196C22×C4—182no—
197C22×C4—182no—
198C22×C4—184no—
199C22×C4—364no—
200C22×C4C2×#15711yescharacteristic
201M16—22no—
202M16—184no—
203M16—182no—
204C2×C8—92no—
205C2×C8—182no—
206C2×C8—182no—
207C2×C8C2×#15712yescomplemented
208C4⋊C4—94no—
209C4⋊C4—94no—
210C4⋊C4—184no—
211C4⋊C4—182no—
212C4⋊C4—182no—
213C22⋊C4—184no—
214C42—184no—
215C22×C3—22no—
216C22×C3—24no—
217C22×C3—44no—
218C2×D3—62no—
219C2×D3—62no—
220C2×D3—64no—
221C2×D3—64no—
222C2×D3—64no—
223C2×D3—124no—
224C2×D3—128no—
225C2×D3—124no—
226C2×D3—124no—
227C3×C4—22no—
228C3×C4—24no—
229C3×C4—22no—
230Dic3—64no—
231Dic3—62no—
232Dic3—62no—
233C32#173611yescharacteristic, complemented, Sylow
234C23—91no—
235C23—91no—
236C23—92no—
237C23—92no—
238C23—182no—
239C23—182no—
240C23C4⋊#15711yescharacteristic
241C2×C4—91no—
242C2×C4—91no—
243C2×C4—92no—
244C2×C4—92no—
245C2×C4—91no—
246C2×C4—91no—
247C2×C4—92no—
248C2×C4—92no—
249C2×C4—92no—
250C2×C4—92no—
251C2×C4—92no—
252C2×C4—91no—
253C2×C4—91no—
254C2×C4—92no—
255C2×C4—184no—
256C2×C4—184no—
257C2×C4—364no—
258C2×C4—362no—
259C2×C4—362no—
260C2×C4—368no—
261C2×C4C4×#15712yes—
262C2×C4C4⋊#15711yescharacteristic
263C2×C4C22⋊#15712yes—
264C2×C4C22×#15711yescharacteristic
265C8—22no—
266C8—182no—
267C2×C3—22no—
268C2×C3—24no—
269C2×C3—44no—
270D3—64no—
271D3—62no—
272D3—62no—
273D3—124no—
274C22—22no—
275C22—91no—
276C22—91no—
277C22—92no—
278C22—92no—
279C22—92no—
280C22—182no—
281C22—184no—
282C22—182no—
283C22—182no—
284C22#6378112yes—
285C22C2×C4⋊#15711yescharacteristic
286C4—91no—
287C4—91no—
288C4—92no—
289C4—364no—
290C4—364no—
291C4#6378112yes—
292C4C2×C4×#15711yescharacteristic
293C4C2×C22⋊#15711yescharacteristic
294C3—22no—
295C2—22no—
296C2—91no—
297C2—92no—
298C2—91no—
299C2—182no—
300C2#1059377412yes—
301C2C2×#6378111yescharacteristic
subgroupG/H
centerC22C2×C4⋊#157
commutatorC32×C4C2×C42
FrattiniC2×C4C22×#157
FittingC2×C32×M16C4
socleC22×C32C2×C4⋊C4
prime radicalC23×C3⋊D3D4
cosocleC2×C4×C3⋊D3C23
hypercenterC2×M16#157
hypocenterC32#1736
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
4. #10590182C2semidirect
9. #64286C4semidirect
12. C2×C4⋊#157C4semidirect
16. C2×C32×M16C4semidirect
17. C2×#3748C4semidirect
20. C2×#3743C4semidirect
43. C4⋊#157C2×C4semidirect
46. C2×C32×C8C2×C4semidirect
48. C2×C3⋊(C3⋊C8)C2×C4semidirect
141. C2×M16#157semidirect
207. C2×C8C2×#157semidirect
233. C32#1736semidirect

Maximal subgroups

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

subgroupindex
C2×#642862
C22×C4⋊#1572
C2×M16×C3⋊D32
#105901822
#17369