Finite Group Databasebeta

← #11791859

Subgroups — #11791859

Order 1152.

390 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 65 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×C22⋊PSU(3,𝔽2)C211yesmaximal, characteristic
2C22⋊C4×#157C211yesmaximal, characteristic
3#10593568C211yesmaximal, characteristic
4#10590155C212yesmaximal
5#10590098C212yesmaximal
6C23×#157C2211yescharacteristic
7C22×PSU(3,𝔽2)C411yescharacteristic, complemented
8C2×C22⋊#157C2212yes—
9C2×C22⋊#157C2211yescharacteristic
10C2×C22⋊#157C2212yes—
11C2×C4×#157—21no—
12C22⋊PSU(3,𝔽2)—21no—
13C22⋊PSU(3,𝔽2)—21no—
14C22⋊PSU(3,𝔽2)—21no—
15C22⋊PSU(3,𝔽2)—21no—
16C2×#3777—21no—
17C2×#3777C411yescharacteristic, complemented
18C2×#3777C412yescomplemented
19C22⋊C4×C3⋊D3C2211yescharacteristic
20#63781—21no—
21#63747—21no—
22C23×C3⋊D3C2311yescharacteristic
23C22×#157—21no—
24C22×#157—22no—
25C22×#157—42no—
26C22×#157C2×C411yescharacteristic
27C22×#157C2×C412yes—
28C22×#157C2×C412yes—
29C22×#157C2×C411yescharacteristic
30C22×#157D411yescharacteristic
31C22×#157D411yescharacteristic
32C22×#157D411yescharacteristic
33C22×#157D411yescharacteristic
34C2×PSU(3,𝔽2)—21no—
35C2×PSU(3,𝔽2)—21no—
36C2×PSU(3,𝔽2)—42no—
37C2×C4×C3⋊D3—21no—
38C22⋊#157—22no—
39C22⋊#157—22no—
40C22⋊#157—21no—
41C22⋊#157—22no—
42C22⋊#157—22no—
43C22⋊#157Q812yescomplemented
44C4×#157—42no—
45#3777—22no—
46#3777—22no—
47#3777—21no—
48#3777—21no—
49#3777—42no—
50C32×C22⋊C4Q811yescharacteristic, complemented
51#3757Q811yescharacteristic, complemented
52#3752—21no—
53#1996—91nomaximal, Sylow
54D3×C22⋊C4—121no—
55C23×C32C2×Q811yescharacteristic
56C22×C3⋊D3—41no—
57C22×C3⋊D3—41no—
58C22×C3⋊D3C22×C411yescharacteristic
59C22×C3⋊D3C2×D411yescharacteristic
60C22×C3⋊D3C2×D411yescharacteristic
61C22×C3⋊D3C2×Q811yescharacteristic
62C22×C3⋊D3#4111yescharacteristic
63C22×C3⋊D3#4111yescharacteristic
64C2×#157—22no—
65C2×#157—22no—
66C2×#157—22no—
67C2×#157—22no—
68C2×#157—22no—
69C2×#157—22no—
70C2×#157—41no—
71C2×#157—41no—
72C2×#157—41no—
73C2×#157—41no—
74C2×#157—41no—
75C2×#157—42no—
76C2×#157—42no—
77C2×#157—42no—
78C2×#157—42no—
79C2×#157—42no—
80C2×#157C22⋊C411yescharacteristic
81C2×#157C22⋊C411yescharacteristic
82C2×#157C22⋊C411yescharacteristic
83C2×#157C22⋊C411yescharacteristic
84PSU(3,𝔽2)—42no—
85PSU(3,𝔽2)—42no—
86C2×C32×C4—21no—
87C2×C3⋊Dic3—21no—
88C4×C3⋊D3—41no—
89C4×C3⋊D3—41no—
90C2×C22⋊Q8—91no—
91#380—91no—
92C4×C22⋊C4—91no—
93#342—92no—
94#342—92no—
95C23×D3—121no—
96C2×C4×D3—122no—
97C3×C22⋊C4—41no—
98#217—121no—
99(C2×D3)⋊C4—122no—
100C22×C32—41no—
101C22×C32C4×Q811yescharacteristic
102C22×C32C22⋊Q811yescharacteristic
103C22×C32C4⋊Q811yescharacteristic
104C2×C3⋊D3—21no—
105C2×C3⋊D3—21no—
106C2×C3⋊D3—21no—
107C2×C3⋊D3—21no—
108C2×C3⋊D3—21no—
109C2×C3⋊D3—21no—
110C2×C3⋊D3—41no—
111C2×C3⋊D3—41no—
112C2×C3⋊D3—41no—
113C2×C3⋊D3C2×C22⋊C411yescharacteristic
114C2×C3⋊D3C4×Q811yescharacteristic
115C2×C3⋊D3C22⋊Q811yescharacteristic
116C2×C3⋊D3#12411yescharacteristic
117#157—21no—
118#157—21no—
119#157—21no—
120#157—42no—
121#157—42no—
122#157—42no—
123#157—82no—
124#157#9912yes—
125C32×C4—41no—
126C3⋊Dic3—41no—
127C22×Q8—91no—
128C23×C4—91no—
129C22⋊Q8—181no—
130C22⋊Q8—181no—
131C22⋊Q8—181no—
132C22⋊Q8—181no—
133C2×C4⋊C4—91no—
134C2×C4⋊C4—92no—
135C2×C4⋊C4—181no—
136C2×C22⋊C4—91no—
137C2×C22⋊C4—91no—
138C2×C22⋊C4—92no—
139C2×C22⋊C4—92no—
140C2×C42—181no—
141#95—181no—
142#95—181no—
143C23×C3—41no—
144C22×D3—121no—
145C22×D3—121no—
146C22×D3—121no—
147C22×D3—121no—
148C22×D3—121no—
149C22×D3—121no—
150C22×D3—242no—
151C22×D3—242no—
152C2×C3×C4—42no—
153C2×Dic3—122no—
154C4×D3—242no—
155C4×D3—242no—
156C2×C32—21no—
157C2×C32—21no—
158C2×C32—21no—
159C2×C32#39111yescharacteristic
160C3⋊D3—21no—
161C3⋊D3—21no—
162C3⋊D3—21no—
163C3⋊D3C2×#9911yescharacteristic
164C3⋊D3#41011yescharacteristic
165C24—91no—
166C2×Q8—181no—
167C2×Q8—181no—
168C2×Q8—362no—
169C22×C4—91no—
170C22×C4—91no—
171C22×C4—91no—
172C22×C4—91no—
173C22×C4—91no—
174C22×C4—91no—
175C22×C4—92no—
176C22×C4—92no—
177C22×C4—181no—
178C22×C4—181no—
179C22×C4—182no—
180C22×C4—362no—
181C4⋊C4—181no—
182C4⋊C4—182no—
183C4⋊C4—182no—
184C4⋊C4—181no—
185C4⋊C4—362no—
186C22⋊C4—92no—
187C22⋊C4—91no—
188C22⋊C4—181no—
189C22⋊C4—182no—
190C22⋊C4—182no—
191C22⋊C4—181no—
192C22⋊C4—182no—
193C22⋊C4—182no—
194C22⋊C4PSU(3,𝔽2)11yescharacteristic, complemented
195C42—362no—
196C22×C3—41no—
197C22×C3—41no—
198C22×C3—41no—
199C22×C3—82no—
200C2×D3—121no—
201C2×D3—121no—
202C2×D3—121no—
203C2×D3—121no—
204C2×D3—122no—
205C2×D3—122no—
206C2×D3—241no—
207C2×D3—241no—
208C2×D3—241no—
209C2×D3—241no—
210C2×D3—242no—
211C2×D3—242no—
212C2×D3—242no—
213C3×C4—82no—
214Dic3—242no—
215C32#199611yescharacteristic, complemented, Sylow
216C23—91no—
217C23—91no—
218C23—91no—
219C23—91no—
220C23—91no—
221C23—91no—
222C23—361no—
223C23—361no—
224C23C2×PSU(3,𝔽2)11yescharacteristic
225Q8—362no—
226Q8—362no—
227C2×C4—21no—
228C2×C4—91no—
229C2×C4—91no—
230C2×C4—91no—
231C2×C4—91no—
232C2×C4—182no—
233C2×C4—182no—
234C2×C4—182no—
235C2×C4—182no—
236C2×C4—182no—
237C2×C4—182no—
238C2×C4—181no—
239C2×C4—361no—
240C2×C4—361no—
241C2×C4—361no—
242C2×C4—361no—
243C2×C4—361no—
244C2×C4—361no—
245C2×C4—361no—
246C2×C4—362no—
247C2×C4—362no—
248C2×C4—362no—
249C2×C4—362no—
250C2×C4—362no—
251C2×C3—42no—
252C2×C3—41no—
253C2×C3—81no—
254C2×C3—81no—
255D3—121no—
256D3—121no—
257D3—122no—
258D3—241no—
259D3—241no—
260C22—41no—
261C22—91no—
262C22—91no—
263C22—91no—
264C22—91no—
265C22—181no—
266C22—181no—
267C22—181no—
268C22—181no—
269C22—181no—
270C22—181no—
271C22—361no—
272C22—361no—
273C22—361no—
274C22C4×PSU(3,𝔽2)11yescharacteristic
275C22C4⋊PSU(3,𝔽2)11yescharacteristic
276C22C22⋊PSU(3,𝔽2)11yescharacteristic
277C4—41no—
278C4—92no—
279C4—181no—
280C4—181no—
281C4—181no—
282C4—361no—
283C4—362no—
284C4—362no—
285C4—362no—
286C4—722no—
287C3—41no—
288C2—21no—
289C2—21no—
290C2—21no—
291C2—91no—
292C2—91no—
293C2—181no—
294C2—181no—
295C2—181no—
296C2#1059355511yescharacteristic
subgroupG/H
centerC2#10593555
commutatorC22×C3⋊D3C22×C4
FrattiniC23C2×PSU(3,𝔽2)
FittingC32×C22⋊C4Q8
socleC2×C32#391
prime radical, cosocleC23×C3⋊D3C23
hypercenterC22⋊C4PSU(3,𝔽2)
hypocenterC32#1996
LCSG ▷ C22×C3⋊D3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C23 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
7. C22×PSU(3,𝔽2)C4semidirect
17. C2×#3777C4semidirect
18. C2×#3777C4semidirect
43. C22⋊#157Q8semidirect
50. C32×C22⋊C4Q8semidirect
51. #3757Q8semidirect
194. C22⋊C4PSU(3,𝔽2)semidirect
215. C32#1996semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊PSU(3,𝔽2)2
C22⋊C4×#1572
#105935682
#105901552
#105900982
#19969