Finite Group Databasebeta

← #11774714

Subgroups — #11774714

Order 1152.

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

normal subgroups only

This group has 559 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
2#10592534C211yesmaximal, characteristic
3C4×#3712C211yesmaximal, characteristic
4C3×#4484C212yesmaximal, complemented
5#10589084C212yesmaximal, complemented
6#68239—32nomaximal
7C2×#3797C411yescharacteristic
8C42×C3⋊D3C2211yescharacteristic
9C3×C4×Dic6C2212yes—
10C4×#627—21no—
11#63900C411yescharacteristic
12#63883—21no—
13C4×#624—21no—
14#63843C412yes—
15#63838—21no—
16C2×#3712—21no—
17C32×C4⋊C8C2211yescharacteristic
18#63647—21no—
19#63642—21no—
20C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C3×C4×C3⋊C8C2212yes—
22#63555—21no—
23C4×D3×Q8—32no—
24#4835—32no—
25#4835—61no—
26C4×ΓL(1,𝔽25)—32no—
27C3×Q8⋊C8—32no—
28#4535—32no—
29#4484D312yescomplemented
30#4456—32no—
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—
45C3×C4⋊Dic3C2×C412yes—
46C3×Dic3⋊C4—22no—
47C3×C4×Dic3—22no—
48C2×C3×C3⋊C8—22no—
49#3724—21no—
50Dic3⋊Dic3—22no—
51Dic32—22no—
52#3712—24no—
53#1491—91noSylow
54C2×D3×Q8—32no—
55C3×C4×Q8—32no—
56Q8×Dic3—32no—
57C2×ΓL(1,𝔽25)—62no—
58D3×C4⋊C4—32no—
59D3×C4⋊C4—62no—
60#887—32no—
61#887—62no—
62C42×D3—32no—
63C42×D3—61no—
64C42×D3—62no—
65C4×Dic6—62no—
66C4×Dic6C2×D312yes—
67C3×C4⋊C8—21no—
68C3×C4⋊C8C2×D312yes—
69C3×C4×C8—32no—
70(C2×D3)⋊C8—62no—
71(C2×D3)⋊C8—62no—
72#815—62no—
73#815—62no—
74#815—62no—
75C4⋊(C3⋊C8)—32no—
76C4⋊(C3⋊C8)—61no—
77C4×C3⋊C8C2×D312yes—
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—21no—
88C4×C3⋊D3—21no—
89C2×C3×Dic3—22no—
90C2×C3×Dic3—22no—
91C3×Dic6—22no—
92C3×Dic6M1614yescomplemented
93#627—24no—
94#624—24no—
95#624—42no—
96C32×C8—41no—
97C3⋊(C3⋊C8)—41no—
98C3×C3⋊C8—24no—
99C2×C4×Q8—91no—
100#432—91no—
101C4×M16—91no—
102Q8⋊C8—32no—
103Q8⋊C8—92no—
104C2×C3×Q8—32no—
105D3×Q8—64no—
106D3×Q8—64no—
107C2×C4×D3—32no—
108C2×C4×D3—32no—
109C2×C4×D3—32no—
110C2×C4×D3—61no—
111C2×C4×D3—61no—
112C2×C4×D3—61no—
113C2×C4×D3—62no—
114C2×C4×D3—62no—
115C2×Dic6—62no—
116C2×Dic6C4×D312yes—
117C2×C3×C8—22no—
118C2×C3×C8—22no—
119C2×C3×C8—62no—
120C3×C4⋊C4—32no—
121C3×C4⋊C4—62no—
122C3×C42—21no—
123C3×C42—62no—
124C3×C42C22×D312yes—
125C4⋊Dic3—62no—
126C4⋊Dic3C4×D312yes—
127Dic3⋊C4—22no—
128Dic3⋊C4—62no—
129Dic3⋊C4—62no—
130C4×Dic3—22no—
131C4×Dic3—32no—
132C4×Dic3—32no—
133C4×Dic3—32no—
134C4×Dic3—61no—
135C4×Dic3—61no—
136C4×Dic3—61no—
137C4×Dic3—62no—
138C4×Dic3—62no—
139C2×C3⋊C8—22no—
140C2×C3⋊C8—62no—
141C2×C3⋊C8—62no—
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#13711yescharacteristic
150C32×C4#13711yescharacteristic
151C3⋊Dic3—21no—
152C3⋊Dic3—21no—
153C3⋊Dic3—21no—
154C3×Dic3—24no—
155C3×Dic3—42no—
156C22×Q8—91no—
157C2×M16—181no—
158C4×Q8—32no—
159C4×Q8—92no—
160C4×Q8—181no—
161C4×Q8—181no—
162C2×C4⋊C4—91no—
163C2×C4⋊C4—181no—
164C2×C42—91no—
165C2×C42—181no—
166C4⋊C8—91no—
167C4⋊C8D3211yescharacteristic
168C22⋊C8—181no—
169#97—181no—
170#97—181no—
171C4×C8—32no—
172C22×D3—32no—
173C22×D3—61no—
174C3×Q8—34no—
175C3×Q8—62no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—62no—
180C2×C3×C4—62no—
181C2×C3×C4C2×C4×D312yes—
182C2×C3×C4D3×D412yes—
183C2×C3×C4D3×D412yes—
184C2×Dic3—22no—
185C2×Dic3—22no—
186C2×Dic3—32no—
187C2×Dic3—32no—
188C2×Dic3—32no—
189C2×Dic3—61no—
190C2×Dic3—61no—
191C2×Dic3—61no—
192C4×D3—68no—
193C4×D3—64no—
194C4×D3—62no—
195C4×D3—62no—
196C4×D3—62no—
197C4×D3—62no—
198C4×D3—122no—
199C4×D3—121no—
200C4×D3—121no—
201C4×D3—122no—
202C4×D3—124no—
203Dic6—22no—
204Dic6—68no—
205Dic6ΓL(1,𝔽25)14yescomplemented
206C3×C8—42no—
207C3×C8—42no—
208C3×C8—64no—
209C3⋊C8—24no—
210C3⋊C8—122no—
211C3⋊C8—122no—
212C2×C32#40711yescharacteristic
213C2×C32#41911yescharacteristic
214C2×C32C2×#10411yescharacteristic
215C3⋊D3—22no—
216C2×Q8—32no—
217C2×Q8—181no—
218C2×Q8—184no—
219C22×C4—91no—
220C22×C4—91no—
221C22×C4—91no—
222C22×C4—181no—
223C22×C4—181no—
224M16—184no—
225C2×C8—21no—
226C2×C8—62no—
227C2×C8—181no—
228C4⋊C4—32no—
229C4⋊C4—62no—
230C4⋊C4—181no—
231C4⋊C4—182no—
232C42—62no—
233C42—91no—
234C42—91no—
235C42—91no—
236C42—182no—
237C42C2×D3211yescharacteristic
238C22×C3—21no—
239C22×C3D3×C22⋊C412yes—
240C2×D3—34no—
241C2×D3—62no—
242C2×D3—62no—
243C2×D3—62no—
244C2×D3—121no—
245C2×D3—121no—
246C3×C4—22no—
247C3×C4—22no—
248C3×C4—21no—
249C3×C4—21no—
250C3×C4—41no—
251C3×C4—64no—
252C3×C4—122no—
253C3×C4C2×ΓL(1,𝔽25)12yes—
254C3×C4D3×M1612yes—
255C3×C4#91512yes—
256C3×C4#91812yes—
257Dic3—24no—
258Dic3—42no—
259Dic3—62no—
260Dic3—62no—
261Dic3—62no—
262Dic3—121no—
263Dic3—121no—
264Dic3—121no—
265C32#149111yescharacteristic, complemented, Sylow
266C23—91no—
267Q8—34no—
268Q8—62no—
269Q8—184no—
270C2×C4—62no—
271C2×C4—62no—
272C2×C4—91no—
273C2×C4—91no—
274C2×C4—91no—
275C2×C4—182no—
276C2×C4—184no—
277C2×C4—182no—
278C2×C4—181no—
279C2×C4—181no—
280C2×C4—362no—
281C2×C4C4×D3211yescharacteristic
282C2×C4#380211yescharacteristic
283C2×C4#381111yescharacteristic
284C8—41no—
285C8—64no—
286C8—361no—
287C2×C3—21no—
288C2×C3—21no—
289C2×C3—21no—
290C2×C3#472612yes—
291C2×C3#480212yes—
292C2×C3D3×#10412yes—
293D3—64no—
294D3—122no—
295C22—92no—
296C22—181no—
297C22—181no—
298C22#6392111yescharacteristic
299C4—21no—
300C4—64no—
301C4—122no—
302C4—181no—
303C4—181no—
304C4—181no—
305C4D3×ΓL(1,𝔽25)12yes—
306C4#6393111yescharacteristic
307C4#6394511yescharacteristic
308C3—21no—
309C3#6823912yescomplemented
310C2—182no—
311C2#1059060211yescharacteristic
312C2#1059093711yescharacteristic
313C2#1059100811yescharacteristic
subgroupG/H
centerC2×C4#3811
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×#4484C2semidirect
5. #10589084C2semidirect
29. #4484D3semidirect
92. C3×Dic6M16semidirect
205. Dic6ΓL(1,𝔽25)semidirect
265. C32#1491semidirect
309. C3#68239semidirect

Maximal subgroups

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

subgroupindex
C4×#37972
#105925342
C4×#37122
C3×#44842
#105890842
#682393