Finite Group Databasebeta

← #11803000

Subgroups — #11803000

Order 1152.

718 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 718 conjugacy classes of subgroups — too many to draw. Only its 199 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
2C2×#64286C211yesmaximal, characteristic
3C22×C4⋊#157C211yesmaximal, characteristic
4#10593569C214yesmaximal
5#10593554C214yesmaximal, complemented
6#10593553C212yesmaximal, complemented
7#10593545C212yesmaximal, complemented
8C23×#157C2212yes—
9C22×C4×C3⋊D3C2211yescharacteristic
10C2×C22⋊#157C2212yes—
11C2×C22⋊#157C2212yes—
12#64286—24no—
13#64286C418yescomplemented
14C2×C4⋊#157—24no—
15C2×C4⋊#157C2211yescharacteristic
16C2×C4⋊#157C2212yes—
17C2×C4⋊#157C2211yescharacteristic
18C2×C4⋊#157C2212yes—
19C2×C4⋊#157C2212yes—
20C2×C4×#157C2212yes—
21C2×C4×#157C2212yes—
22C2×#3777C2218yes—
23#63747C2218yes—
24C23×C3⋊D3C2311yescharacteristic
25C22×#157—24no—
26C22×#157—44no—
27C22×#157C2312yes—
28C22×#157C2314yes—
29C22×#157C2312yes—
30C22×#157C2314yes—
31C22×C32×C4Q811yescharacteristic
32C22×C3⋊Dic3Q811yescharacteristic
33C2×C4×C3⋊D3C2×C414yes—
34C2×C4×C3⋊D3D414yes—
35C2×C4×C3⋊D3Q811yescharacteristic
36C2×C4×C3⋊D3Q811yescharacteristic
37C2×C4×C3⋊D3C2311yescharacteristic
38C2×C4×C3⋊D3C2311yescharacteristic
39C22⋊#157—24no—
40C22⋊#157C2×C418yes—
41C4⋊#157—24no—
42C4⋊#157—24no—
43C4⋊#157—22no—
44C4⋊#157—22no—
45C4⋊#157C2×C418yescomplemented
46C4×#157—24no—
47C4×#157C2×C418yescomplemented
48#3777—48no—
49#2317—91nomaximal, Sylow
50C22×C4×D3—121no—
51C23×C32C2×Q811yescharacteristic
52C22×C3⋊D3—22no—
53C22×C3⋊D3—22no—
54C22×C3⋊D3C22×C412yes—
55C22×C3⋊D3C2×D412yes—
56C22×C3⋊D3C2×Q811yescharacteristic
57C22×C3⋊D3C2411yescharacteristic
58C2×#157—24no—
59C2×#157—24no—
60C2×#157—24no—
61C2×#157—22no—
62C2×#157—22no—
63C2×#157—28no—
64C2×#157—24no—
65C2×#157—24no—
66C2×#157—48no—
67C2×#157C22×C418yes—
68C2×C32×C4C4⋊C414yescomplemented
69C2×C32×C4C2×Q811yescharacteristic
70C2×C32×C4C2×Q811yescharacteristic
71C2×C3⋊Dic3C4⋊C414yescomplemented
72C2×C3⋊Dic3C2×Q811yescharacteristic
73C2×C3⋊Dic3C2×Q811yescharacteristic
74C4×C3⋊D3C4⋊C414yescomplemented
75C4×C3⋊D3C4⋊C414yescomplemented
76C4×C3⋊D3C22×C412yes—
77C4×C3⋊D3C22×C412yes—
78C4×C3⋊D3C2×D412yes—
79C4×C3⋊D3C2×D412yes—
80C2×#117—91no—
81C2×#117—91no—
82C22×C4⋊C4—91no—
83#389—94no—
84#385—94no—
85#384—92no—
86#382—92no—
87C23×D3—121no—
88C22×C3×C4—41no—
89C22×Dic3—121no—
90C2×C4×D3—124no—
91C2×C4×D3—124no—
92C2×C4×D3—121no—
93C2×C4×D3—121no—
94C2×C4×D3—121no—
95C2×C4×D3—121no—
96C22×C32—22no—
97C22×C32C2×C4⋊C412yes—
98C22×C32C22×Q811yescharacteristic
99C2×C3⋊D3—22no—
100C2×C3⋊D3—24no—
101C2×C3⋊D3—22no—
102C2×C3⋊D3—22no—
103C2×C3⋊D3C2×C4⋊C412yes—
104C2×C3⋊D3C23×C411yescharacteristic
105C2×C3⋊D3C22×D411yescharacteristic
106C2×C3⋊D3H(5,𝔽2)12yes—
107C2×C3⋊D32−1+412yes—
108#157—28no—
109#157—42no—
110#157—44no—
111#157—42no—
112C32×C4C2×C4⋊C412yes—
113C32×C4C2×C4⋊C412yes—
114C3⋊Dic3C2×C4⋊C412yes—
115C3⋊Dic3C2×C4⋊C412yes—
116C23×C4—91no—
117C23×C4—92no—
118#117—98no—
119#117—184no—
120C2×C4⋊C4—92no—
121C2×C4⋊C4—92no—
122C2×C4⋊C4—92no—
123C2×C4⋊C4—91no—
124C2×C4⋊C4—91no—
125C2×C4⋊C4—98no—
126C2×C4⋊C4—184no—
127C2×C22⋊C4—92no—
128C2×C22⋊C4—92no—
129C2×C42—92no—
130C2×C42—92no—
131#95—98no—
132C23×C3—41no—
133C22×D3—122no—
134C22×D3—122no—
135C22×D3—121no—
136C22×D3—121no—
137C22×D3—124no—
138C22×D3—124no—
139C2×C3×C4—41no—
140C2×C3×C4—41no—
141C2×C3×C4—44no—
142C2×Dic3—124no—
143C2×Dic3—121no—
144C2×Dic3—121no—
145C4×D3—124no—
146C4×D3—124no—
147C4×D3—122no—
148C4×D3—122no—
149C4×D3—122no—
150C4×D3—122no—
151C2×C32—22no—
152C2×C32C22×C4⋊C411yescharacteristic
153C2×C32#54412yes—
154C3⋊D3—22no—
155C3⋊D3#52012yes—
156C3⋊D3#53511yescharacteristic
157C3⋊D3#53611yescharacteristic
158C24—91no—
159C22×C4—91no—
160C22×C4—91no—
161C22×C4—91no—
162C22×C4—91no—
163C22×C4—91no—
164C22×C4—94no—
165C22×C4—94no—
166C22×C4—94no—
167C22×C4—92no—
168C22×C4—94no—
169C22×C4—92no—
170C22×C4—184no—
171C22×C4—364no—
172C22×C4PSU(3,𝔽2)11yescharacteristic
173C4⋊C4—98no—
174C4⋊C4—184no—
175C4⋊C4—184no—
176C4⋊C4—182no—
177C4⋊C4—182no—
178C4⋊C4—368no—
179C22⋊C4—98no—
180C22⋊C4—184no—
181C42—98no—
182C42—184no—
183C22×C3—44no—
184C22×C3—41no—
185C22×C3—42no—
186C2×D3—122no—
187C2×D3—121no—
188C2×D3—121no—
189C2×D3—122no—
190C2×D3—122no—
191C2×D3—124no—
192C2×D3—128no—
193C2×D3—124no—
194C2×D3—124no—
195C3×C4—42no—
196C3×C4—42no—
197Dic3—122no—
198Dic3—122no—
199C32#231711yescharacteristic, complemented, Sylow
200C23—91no—
201C23—91no—
202C23—92no—
203C23—92no—
204C23—182no—
205C23—182no—
206C23C2×PSU(3,𝔽2)11yescharacteristic
207C2×C4—91no—
208C2×C4—91no—
209C2×C4—94no—
210C2×C4—92no—
211C2×C4—92no—
212C2×C4—92no—
213C2×C4—92no—
214C2×C4—94no—
215C2×C4—94no—
216C2×C4—98no—
217C2×C4—184no—
218C2×C4—184no—
219C2×C4—182no—
220C2×C4—182no—
221C2×C4—184no—
222C2×C4—184no—
223C2×C4—188no—
224C2×C4—184no—
225C2×C4—368no—
226C2×C4#377714yescomplemented
227C2×C4C2×PSU(3,𝔽2)11yescharacteristic
228C2×C4C2×PSU(3,𝔽2)11yescharacteristic
229C2×C3—44no—
230C2×C3—42no—
231C2×C3—41no—
232D3—122no—
233D3—121no—
234D3—121no—
235D3—124no—
236C22—22no—
237C22—92no—
238C22—92no—
239C22—91no—
240C22—91no—
241C22—92no—
242C22—182no—
243C22—184no—
244C22—182no—
245C22—182no—
246C22C2×#377712yes—
247C22C22×PSU(3,𝔽2)11yescharacteristic
248C4—92no—
249C4—92no—
250C4—188no—
251C4—364no—
252C4—362no—
253C4—362no—
254C4C2×#377712yes—
255C4C2×#377712yes—
256C3—41no—
257C2—22no—
258C2—92no—
259C2—91no—
260C2—91no—
261C2—182no—
262C2#1059654312yes—
263C2C22×#377711yescharacteristic
subgroupG/H
center, FrattiniC22C22×PSU(3,𝔽2)
commutatorC2×C3⋊D3C23×C4
FittingC22×C32×C4Q8
socleC22×C32C22×Q8
prime radicalC23×C3⋊D3C23
cosocleC22×C3⋊D3C24
hypercenterC22×C4PSU(3,𝔽2)
hypocenterC32#2317
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
5. #10593554C2semidirect
6. #10593553C2semidirect
7. #10593545C2semidirect
13. #64286C4semidirect
45. C4⋊#157C2×C4semidirect
47. C4×#157C2×C4semidirect
68. C2×C32×C4C4⋊C4semidirect
71. C2×C3⋊Dic3C4⋊C4semidirect
74. C4×C3⋊D3C4⋊C4semidirect
75. C4×C3⋊D3C4⋊C4semidirect
199. C32#2317semidirect
226. C2×C4#3777semidirect

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
#105935692
#105935542
#105935532
#105935452
#23179