Finite Group Databasebeta

← #11774721

Subgroups — #11774721

Order 1152.

575 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 575 conjugacy classes of subgroups — too many to draw. Only its 108 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
1C4×#3797C211yesmaximal, characteristic
2C4×#3743C211yesmaximal, characteristic
3#10590260C211yesmaximal, characteristic
4C3×#4456C212yesmaximal, complemented
5#10589083C212yesmaximal, complemented
6#68239—32nomaximal
7C2×#3797C411yescharacteristic
8C2×#3743—21no—
9C42×C3⋊D3C2211yescharacteristic
10C3×C4×Dic6C2212yes—
11C4×#627—21no—
12#63900C411yescharacteristic
13#63883—21no—
14C4×#624—21no—
15#63843C412yes—
16#63838—21no—
17C32×C4×C8C2211yescharacteristic
18#63642—21no—
19C4×C3⋊(C3⋊C8)C2211yescharacteristic
20C3×C4⋊(C3⋊C8)C2212yes—
21#63557—21no—
22#63555—21no—
23C4×D3×Q8—32no—
24#4835—32no—
25C4×ΓL(1,𝔽25)—32no—
26C4×ΓL(1,𝔽25)—61no—
27C3×Q8⋊C8—32no—
28#4535—32no—
29#4484—32no—
30#4456D312yescomplemented
31C2×C4×C3⋊D3C2×C411yescharacteristic
32C2×C4×C3⋊D3D411yescharacteristic
33C2×C4×C3⋊D3D411yescharacteristic
34C2×C3×Dic6C2×C412yes—
35C2×#627—21no—
36C2×#624—21no—
37C2×#624—21no—
38#3797—24no—
39C2×C32×C8—21no—
40C32×C42C2311yescharacteristic
41C4×C3⋊Dic3C2×C411yescharacteristic
42C4×C3⋊Dic3D411yescharacteristic
43C4×C3⋊Dic3D411yescharacteristic
44C2×C3⋊(C3⋊C8)—21no—
45#3743—24no—
46C3×C4⋊Dic3C2×C412yes—
47C3×Dic3⋊C4—22no—
48C3×C4×Dic3—22no—
49C2×C3×C3⋊C8—22no—
50#3724—21no—
51Dic3⋊Dic3—22no—
52Dic32—22no—
53#1491—91noSylow
54C2×D3×Q8—32no—
55C3×C4×Q8—32no—
56Q8×Dic3—32no—
57C2×ΓL(1,𝔽25)—62no—
58C2×ΓL(1,𝔽25)—62no—
59D3×C4⋊C4—32no—
60D3×C4⋊C4—62no—
61#887—32no—
62#887—62no—
63C42×D3—32no—
64C42×D3—61no—
65C42×D3—62no—
66C4×Dic6—62no—
67C4×Dic6C2×D312yes—
68C3×C4⋊C8—32no—
69C3×C4×C8—21no—
70C3×C4×C8C2×D312yes—
71(C2×D3)⋊C8—62no—
72#815—62no—
73#815—62no—
74#815—62no—
75C4⋊(C3⋊C8)C2×D312yes—
76C4×C3⋊C8—32no—
77C4×C3⋊C8—61no—
78C22×C3⋊D3C22⋊C411yescharacteristic
79C2×C32×C4C22×C411yescharacteristic
80C2×C32×C4C2×D411yescharacteristic
81C2×C32×C4C2×D411yescharacteristic
82C2×C3⋊Dic3C22⋊C411yescharacteristic
83C2×C3⋊Dic3C22⋊C411yescharacteristic
84C2×C3⋊Dic3C22⋊C411yescharacteristic
85C4×C3⋊D3—22no—
86C4×C3⋊D3—22no—
87C4×C3⋊D3—22no—
88C2×C3×Dic3—22no—
89C2×C3×Dic3—22no—
90C3×Dic6—22no—
91C3×Dic6M1614yescomplemented
92#627—24no—
93#624—24no—
94#624—42no—
95C32×C8—22no—
96C3⋊(C3⋊C8)—22no—
97C3×C3⋊C8—42no—
98C2×C4×Q8—91no—
99#432—91no—
100C4×M16—91no—
101Q8⋊C8—32no—
102Q8⋊C8—92no—
103C2×C3×Q8—32no—
104D3×Q8—64no—
105D3×Q8—64no—
106C2×C4×D3—32no—
107C2×C4×D3—32no—
108C2×C4×D3—32no—
109C2×C4×D3—61no—
110C2×C4×D3—61no—
111C2×C4×D3—61no—
112C2×C4×D3—62no—
113C2×C4×D3—62no—
114C2×Dic6—62no—
115C2×Dic6C4×D312yes—
116C2×C3×C8—22no—
117C2×C3×C8—22no—
118C2×C3×C8—62no—
119C3×C4⋊C4—32no—
120C3×C4⋊C4—62no—
121C3×C42—21no—
122C3×C42—62no—
123C3×C42C22×D312yes—
124C4⋊Dic3—62no—
125C4⋊Dic3C4×D312yes—
126Dic3⋊C4—22no—
127Dic3⋊C4—62no—
128Dic3⋊C4—62no—
129C4×Dic3—22no—
130C4×Dic3—32no—
131C4×Dic3—32no—
132C4×Dic3—32no—
133C4×Dic3—61no—
134C4×Dic3—61no—
135C4×Dic3—61no—
136C4×Dic3—62no—
137C4×Dic3—62no—
138C2×C3⋊C8—22no—
139C2×C3⋊C8—62no—
140C2×C3⋊C8—62no—
141ΓL(1,𝔽25)—68no—
142ΓL(1,𝔽25)—68no—
143C22×C32C2×C22⋊C411yescharacteristic
144C2×C3⋊D3—21no—
145C2×C3⋊D3—21no—
146C2×C3⋊D3#10412yes—
147C32×C4—21no—
148C32×C4C2×M1612yes—
149C32×C4#13712yes—
150C3⋊Dic3—21no—
151C3⋊Dic3—21no—
152C3⋊Dic3—21no—
153C3×Dic3—24no—
154C3×Dic3—42no—
155C22×Q8—91no—
156C2×M16—181no—
157C4×Q8—32no—
158C4×Q8—92no—
159C4×Q8—181no—
160C4×Q8—181no—
161C2×C4⋊C4—91no—
162C2×C4⋊C4—181no—
163C2×C42—91no—
164C2×C42—181no—
165C4⋊C8—32no—
166C22⋊C8—181no—
167#97—181no—
168#97—181no—
169C4×C8—91no—
170C4×C8D3211yescharacteristic
171C22×D3—32no—
172C22×D3—61no—
173C3×Q8—34no—
174C3×Q8—62no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—62no—
179C2×C3×C4—62no—
180C2×C3×C4C2×C4×D312yes—
181C2×C3×C4D3×D412yes—
182C2×C3×C4D3×D412yes—
183C2×Dic3—22no—
184C2×Dic3—22no—
185C2×Dic3—32no—
186C2×Dic3—32no—
187C2×Dic3—32no—
188C2×Dic3—61no—
189C2×Dic3—61no—
190C2×Dic3—61no—
191C4×D3—68no—
192C4×D3—64no—
193C4×D3—64no—
194C4×D3—62no—
195C4×D3—62no—
196C4×D3—62no—
197C4×D3—62no—
198C4×D3—122no—
199C4×D3—122no—
200C4×D3—124no—
201Dic6—22no—
202Dic6—68no—
203Dic6ΓL(1,𝔽25)14yescomplemented
204C3×C8—24no—
205C3×C8—24no—
206C3×C8—122no—
207C3⋊C8—42no—
208C3⋊C8—64no—
209C3⋊C8—64no—
210C2×C32#40711yescharacteristic
211C2×C32#41911yescharacteristic
212C2×C32C2×#10411yescharacteristic
213C3⋊D3—22no—
214C2×Q8—32no—
215C2×Q8—181no—
216C2×Q8—184no—
217C22×C4—91no—
218C22×C4—91no—
219C22×C4—91no—
220C22×C4—181no—
221C22×C4—181no—
222M16—184no—
223C2×C8—21no—
224C2×C8—62no—
225C2×C8—181no—
226C4⋊C4—32no—
227C4⋊C4—62no—
228C4⋊C4—181no—
229C4⋊C4—182no—
230C42—62no—
231C42—91no—
232C42—91no—
233C42—91no—
234C42—182no—
235C42C2×D3211yescharacteristic
236C22×C3—21no—
237C22×C3D3×C22⋊C412yes—
238C2×D3—34no—
239C2×D3—62no—
240C2×D3—62no—
241C2×D3—62no—
242C2×D3—121no—
243C2×D3—121no—
244C3×C4—22no—
245C3×C4—22no—
246C3×C4—22no—
247C3×C4—22no—
248C3×C4—64no—
249C3×C4—122no—
250C3×C4C2×ΓL(1,𝔽25)12yes—
251C3×C4D3×M1612yes—
252C3×C4#91512yes—
253C3×C4#91812yes—
254Dic3—24no—
255Dic3—42no—
256Dic3—62no—
257Dic3—62no—
258Dic3—62no—
259Dic3—62no—
260Dic3—121no—
261Dic3—121no—
262C32#149111yescharacteristic, complemented, Sylow
263C23—91no—
264Q8—34no—
265Q8—62no—
266Q8—184no—
267C2×C4—62no—
268C2×C4—62no—
269C2×C4—91no—
270C2×C4—91no—
271C2×C4—91no—
272C2×C4—182no—
273C2×C4—184no—
274C2×C4—182no—
275C2×C4—182no—
276C2×C4—362no—
277C2×C4C4×D3211yescharacteristic
278C2×C4#380211yescharacteristic
279C2×C4#381111yescharacteristic
280C8—22no—
281C8—122no—
282C8—182no—
283C2×C3—21no—
284C2×C3—21no—
285C2×C3—21no—
286C2×C3#472612yes—
287C2×C3#480212yes—
288C2×C3D3×#10412yes—
289D3—64no—
290D3—122no—
291C22—92no—
292C22—181no—
293C22—181no—
294C22#6392111yescharacteristic
295C4—21no—
296C4—64no—
297C4—122no—
298C4—181no—
299C4—181no—
300C4—181no—
301C4D3×ΓL(1,𝔽25)12yes—
302C4#6381112yes—
303C3—21no—
304C3#6823912yescomplemented
305C2—182no—
306C2#1059039311yescharacteristic
307C2#1059069111yescharacteristic
308C2#1059093711yescharacteristic
subgroupG/H
centerC2×C4#3802
commutatorC2×C32×C4C22×C4
FrattiniC42C2×D32
FittingC32×C4×C8C22
socleC22×C32C2×C22⋊C4
prime radicalC22×C3⋊D3C22⋊C4
cosocleC32×C42C23
hypercenterC4×C8D32
hypocenterC32#1491
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×C8

Complements

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

NH ≅ G/Ntype
4. C3×#4456C2semidirect
5. #10589083C2semidirect
30. #4456D3semidirect
91. C3×Dic6M16semidirect
203. Dic6ΓL(1,𝔽25)semidirect
262. C32#1491semidirect
304. C3#68239semidirect

Maximal subgroups

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

subgroupindex
C4×#37972
C4×#37432
#105902602
C3×#44562
#105890832
#682393