Finite Group Databasebeta

← #11868774

Subgroups — #11868774

Order 1152.

454 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 111 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×Q8×#157C211yesmaximal, characteristic
2#10596542C211yesmaximal, characteristic
3#10593829C212yesmaximal
4#10593562C212yesmaximal, complemented
5#10593561C212yesmaximal, complemented
6#10593560C214yesmaximal, complemented
7#10593544C212yesmaximal
8#10593539C211yesmaximal, characteristic
9C2×Q8×C3⋊D3C2211yescharacteristic
10Q8×#157—21no—
11Q8×#157—21no—
12Q8×#157C2212yes—
13Q8×#157C2212yes—
14#64286C2212yes—
15C2×C4⋊#157—21no—
16C2×C4⋊#157C2211yescharacteristic
17C2×C4×#157—21no—
18C2×C4×#157C2211yescharacteristic
19#64283C2212yes—
20C22⋊PSU(3,𝔽2)—21no—
21C22⋊PSU(3,𝔽2)—21no—
22C4⋊PSU(3,𝔽2)C2211yescharacteristic
23C4⋊PSU(3,𝔽2)C2211yescharacteristic
24C4×PSU(3,𝔽2)—21no—
25C4×PSU(3,𝔽2)—21no—
26#64243C2212yes—
27#63784C2214yes—
28#63784C2214yes—
29#63746C2214yes—
30#63745C2214yes—
31#63744C2212yes—
32#63744C2212yes—
33C22×#157—21no—
34C22×#157D411yescharacteristic
35C22×#157D411yescharacteristic
36C2×PSU(3,𝔽2)—21no—
37C2×PSU(3,𝔽2)—21no—
38C2×C32×Q8Q811yescharacteristic, complemented
39Q8×C3⋊D3—21no—
40Q8×C3⋊D3—21no—
41Q8×C3⋊D3Q812yescomplemented
42Q8×C3⋊D3C2312yes—
43C2×C4×C3⋊D3—21no—
44C2×C4×C3⋊D3C2311yescharacteristic
45C2×#634—21no—
46C2×#634Q811yescharacteristic, complemented
47C22⋊#157—22no—
48#3792—22no—
49C4⋊#157—22no—
50C4⋊#157—22no—
51C4⋊#157—21no—
52C4⋊#157C2312yes—
53C4⋊#157C2312yes—
54C4⋊#157C2312yes—
55C4×#157—22no—
56C4×#157—22no—
57C4×#157—22no—
58C4×#157D411yescharacteristic
59C4×#157D411yescharacteristic
60C4×#157C2311yescharacteristic
61C4×#157C2311yescharacteristic
62#3788—22no—
63#3787C2314yes—
64#3777—22no—
65#3777—22no—
66#3777—21no—
67#3777—21no—
68#3078—91nomaximal, Sylow
69C2×D3×Q8—121no—
70C22×C3⋊D3C2×D411yescharacteristic
71C2×#157—22no—
72C2×#157—22no—
73C2×#157—22no—
74C2×#157—22no—
75C2×#157—41no—
76C2×#157—42no—
77C2×#157C2×D411yescharacteristic
78C2×#157C2×D411yescharacteristic
79C2×#157C2×D411yescharacteristic
80C2×#157C2×D411yescharacteristic
81PSU(3,𝔽2)—42no—
82PSU(3,𝔽2)—42no—
83C32×Q8—21no—
84C32×Q8C2×Q812yes—
85C2×C32×C4—21no—
86C2×C32×C4C2×Q811yescharacteristic
87C2×C3⋊Dic3—21no—
88C2×C3⋊Dic3#4111yescharacteristic
89C4×C3⋊D3—22no—
90C4×C3⋊D3—22no—
91C4×C3⋊D3C2×D411yescharacteristic
92C4×C3⋊D3C2×Q811yescharacteristic
93C4×C3⋊D3#4111yescharacteristic
94C4×C3⋊D3C2411yescharacteristic
95#634—21no—
96#634—22no—
97#634—22no—
98#634C2×Q812yes—
99#622—24no—
100#533—91no—
101C2×C4×Q8—91no—
102#479—94no—
103#477—92no—
104#475—92no—
105#428—92no—
106#423—91no—
107#419—92no—
108C2×C3×Q8—41no—
109D3×Q8—122no—
110D3×Q8—122no—
111D3×Q8—122no—
112D3×Q8—122no—
113C2×C4×D3—121no—
114C2×C4×D3—122no—
115C2×Dic6—121no—
116C2×Dic6—122no—
117C22×C32C22⋊Q811yescharacteristic
118C2×C3⋊D3—21no—
119C2×C3⋊D3—21no—
120C2×C3⋊D3C22⋊Q811yescharacteristic
121C2×C3⋊D3C22×D411yescharacteristic
122#157—21no—
123#157—21no—
124#157—21no—
125#157—42no—
126#157—42no—
127#157—42no—
128#157#13712yes—
129C32×C4—22no—
130C32×C4C22⋊Q811yescharacteristic
131C32×C4C22×Q811yescharacteristic
132C3⋊Dic3—22no—
133C3⋊Dic3C22⋊Q811yescharacteristic
134C3⋊Dic3C2×#4111yescharacteristic
135C22×Q8—91no—
136C2×M16—92no—
137C4⋊Q8—91no—
138C4⋊Q8—91no—
139#125—92no—
140C22⋊Q8—181no—
141C22⋊Q8—181no—
142C4×Q8—92no—
143C4×Q8—92no—
144C4×Q8—181no—
145C4×Q8—181no—
146C4×Q8—181no—
147C4×Q8—181no—
148#117—92no—
149C2×C4⋊C4—91no—
150C2×C4⋊C4—181no—
151C2×C42—91no—
152C2×C42—181no—
153#107—94no—
154#106—94no—
155C4⋊C8—92no—
156C4⋊C8—92no—
157#103—94no—
158#103—94no—
159C22×D3—121no—
160C3×Q8—42no—
161C3×Q8—42no—
162C2×C3×C4—42no—
163C2×C3×C4—41no—
164C2×Dic3—121no—
165C2×Dic3—122no—
166C4×D3—121no—
167C4×D3—121no—
168C4×D3—121no—
169C4×D3—121no—
170C4×D3—124no—
171C4×D3—124no—
172Dic6—122no—
173Dic6—124no—
174Dic6—124no—
175Dic6—122no—
176C2×C32—21no—
177C2×C32C2×C22⋊Q811yescharacteristic
178C3⋊D3—21no—
179C3⋊D3C2×#13711yescharacteristic
180C3⋊D3#57511yescharacteristic
181C2×Q8—92no—
182C2×Q8—92no—
183C2×Q8—91no—
184C2×Q8—181no—
185C2×Q8—181no—
186C2×Q8—181no—
187C2×Q8—181no—
188C2×Q8—181no—
189C2×Q8PSU(3,𝔽2)11yescharacteristic, complemented
190C22×C4—91no—
191C22×C4—91no—
192C22×C4—91no—
193C22×C4—181no—
194C22×C4—181no—
195M16—182no—
196M16—182no—
197C2×C8—94no—
198C4⋊C4—92no—
199C4⋊C4—92no—
200C4⋊C4—92no—
201C4⋊C4—181no—
202C4⋊C4—182no—
203C4⋊C4—182no—
204C4⋊C4—182no—
205C4⋊C4—182no—
206C4⋊C4—181no—
207C4⋊C4—181no—
208C22⋊C4—182no—
209C42—91no—
210C42—91no—
211C42—91no—
212C42—91no—
213C42—182no—
214C42—182no—
215C42—182no—
216C22×C3—41no—
217C2×D3—121no—
218C2×D3—121no—
219C2×D3—122no—
220C2×D3—122no—
221C3×C4—44no—
222C3×C4—41no—
223C3×C4—41no—
224Dic3—121no—
225Dic3—121no—
226Dic3—124no—
227C32#307811yescharacteristic, complemented, Sylow
228C23—91no—
229Q8—21no—
230Q8—92no—
231Q8—182no—
232Q8—182no—
233Q8—181no—
234Q8—362no—
235Q8—362no—
236Q8C2×PSU(3,𝔽2)12yes—
237C2×C4—21no—
238C2×C4—91no—
239C2×C4—91no—
240C2×C4—91no—
241C2×C4—91no—
242C2×C4—91no—
243C2×C4—91no—
244C2×C4—91no—
245C2×C4—91no—
246C2×C4—91no—
247C2×C4—182no—
248C2×C4—182no—
249C2×C4—182no—
250C2×C4—181no—
251C2×C4—182no—
252C2×C4—182no—
253C2×C4—182no—
254C2×C4—361no—
255C2×C4—362no—
256C2×C4C2×PSU(3,𝔽2)11yescharacteristic
257C8—184no—
258C2×C3—42no—
259C2×C3—41no—
260D3—121no—
261D3—121no—
262D3—122no—
263C22—91no—
264C22—91no—
265C22—181no—
266C22—181no—
267C22C22⋊PSU(3,𝔽2)11yescharacteristic
268C4—22no—
269C4—92no—
270C4—91no—
271C4—91no—
272C4—181no—
273C4—181no—
274C4—181no—
275C4—182no—
276C4—362no—
277C4—362no—
278C4—362no—
279C4C22⋊PSU(3,𝔽2)11yescharacteristic
280C4C22×PSU(3,𝔽2)11yescharacteristic
281C3—41no—
282C2—21no—
283C2—91no—
284C2—91no—
285C2—181no—
286C2C2×C22⋊PSU(3,𝔽2)11yescharacteristic
subgroupG/H
centerC2C2×C22⋊PSU(3,𝔽2)
commutator, cosocleC4×C3⋊D3C24
FrattiniC4C22×PSU(3,𝔽2)
FittingC2×C32×Q8Q8
socleC2×C32C2×C22⋊Q8
prime radicalC22×C3⋊D3C2×D4
hypercenterC2×Q8PSU(3,𝔽2)
hypocenterC32#3078
LCSG ▷ C4×C3⋊D3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
4. #10593562C2semidirect
5. #10593561C2semidirect
6. #10593560C2semidirect
38. C2×C32×Q8Q8semidirect
41. Q8×C3⋊D3Q8semidirect
46. C2×#634Q8semidirect
189. C2×Q8PSU(3,𝔽2)semidirect
227. C32#3078semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×#1572
#105965422
#105938292
#105935622
#105935612
#105935602
#105935442
#105935392
#30789