Finite Group Databasebeta

← #11805617

Subgroups — #11805617

Order 1152.

850 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 850 conjugacy classes of subgroups — too many to draw. Only its 203 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
2Q8×#624C212yesmaximal
3#10593412C216yesmaximal
4C4×#3773C213yesmaximal
5#10593344C213yesmaximal
6C2×Q8×C3⋊D3C2211yescharacteristic
7Q8×#157C418yescomplemented
8C2×C4⋊#157C2213yes—
9C2×C4×#157C2213yes—
10C2×#3773C2216yes—
11C3×Q8×Dic3—22no—
12#63900C2216yes—
13C4×#624C2216yes—
14#63844—26no—
15#63737C2213yes—
16#63737C2216yes—
17#63737C2211yescharacteristic
18C4×D3×Q8—62no—
19C22×#157C2311yescharacteristic
20C22×#157C2313yes—
21C2×C32×Q8D411yescharacteristic
22Q8×C3⋊D3C2×C414yes—
23Q8×C3⋊D3Q814yes—
24C2×C4×C3⋊D3C2313yes—
25C2×#634D413yes—
26C2×#624C2316yes—
27C2×#624C2312yes—
28C4⋊#157C2×C4112yes—
29C4×#157C2×C4112yes—
30#3773—212no—
31C3×C4⋊Dic3—26no—
32C3×C4×Dic3—26no—
33#3724—26no—
34Dic32—26no—
35#2351—91nomaximal, Sylow
36C2×D3×Q8—62no—
37C3×C4×Q8—62no—
38Q8×Dic3—22no—
39D3×C4⋊C4—66no—
40#887—66no—
41C42×D3—66no—
42C4×Dic6—66no—
43C22×C3⋊D3C2411yescharacteristic
44C2×#157—26no—
45C2×#157—22no—
46C2×#157C22×C412yes—
47C2×#157C22×C416yes—
48C32×Q8C4⋊C414yescomplemented
49C2×C32×C4C2×D413yes—
50C2×C3⋊Dic3C2×D413yes—
51C4×C3⋊D3C22×C416yes—
52C4×C3⋊D3C2×Q816yes—
53#634C4⋊C4112yescomplemented
54C2×C3×Dic3—22no—
55C2×C3×Dic3—26no—
56#624—22no—
57#624—26no—
58#624—26no—
59#624#4114yes—
60C2×C4×Q8—92no—
61C2×C4×Q8—91no—
62#389—96no—
63#382—93no—
64C4×C4⋊C4—93no—
65C2×C3×Q8—22no—
66D3×Q8—68no—
67D3×Q8—68no—
68C2×C4×D3—62no—
69C2×C4×D3—66no—
70C2×C4×D3—66no—
71C2×Dic6—66no—
72C3×C4⋊C4—66no—
73C3×C42—66no—
74C4⋊Dic3—26no—
75Dic3⋊C4—612no—
76C4×Dic3—26no—
77C4×Dic3—66no—
78C4×Dic3—66no—
79C22×C32C22×D411yescharacteristic
80C2×C3⋊D3C23×C411yescharacteristic
81C2×C3⋊D3C22×Q811yescharacteristic
82C2×C3⋊D3C2×#4112yes—
83C2×C3⋊D32−1+412yes—
84#157—26no—
85#157—22no—
86C32×C4C2×C4⋊C416yes—
87C3⋊Dic3C2×C4⋊C416yes—
88C3×Dic3—24no—
89C3×Dic3—46no—
90C22×Q8—91no—
91C4×Q8—62no—
92C4×Q8—98no—
93C4×Q8—186no—
94C2×C4⋊C4—96no—
95C2×C4⋊C4—93no—
96C2×C4⋊C4—96no—
97C2×C42—96no—
98C2×C42—93no—
99#95—91no—
100#95—93no—
101#95—96no—
102C22×D3—62no—
103C3×Q8—28no—
104C2×C3×C4—26no—
105C2×C3×C4—62no—
106C2×C3×C4—66no—
107C2×Dic3—22no—
108C2×Dic3—26no—
109C2×Dic3—66no—
110C4×D3—64no—
111C4×D3—612no—
112C4×D3—612no—
113C4×D3—64no—
114C4×D3—126no—
115C4×D3—126no—
116Dic6—624no—
117C2×C32C22×C4⋊C411yescharacteristic
118C2×C32#54812yes—
119C3⋊D3C4×#4111yescharacteristic
120C3⋊D3#51911yescharacteristic
121C3⋊D3#55712yes—
122C2×Q8—94no—
123C2×Q8—94no—
124C2×Q8—93no—
125C2×Q8GO+(4,𝔽2)11yescharacteristic
126C22×C4—92no—
127C22×C4—96no—
128C22×C4—93no—
129C22×C4—91no—
130C22×C4—93no—
131C4⋊C4—66no—
132C4⋊C4—912no—
133C4⋊C4—186no—
134C4⋊C4—1812no—
135C42—66no—
136C42—912no—
137C42—186no—
138C22×C3—22no—
139C2×D3—62no—
140C2×D3—62no—
141C2×D3—62no—
142C2×D3—62no—
143C2×D3—62no—
144C2×D3—62no—
145C3×C4—212no—
146C3×C4—64no—
147C3×C4—126no—
148Dic3—24no—
149Dic3—46no—
150Dic3—612no—
151C32#235111yescharacteristic, complemented, Sylow
152C23—91no—
153Q8—912no—
154Q8#377314yescomplemented
155C2×C4—62no—
156C2×C4—66no—
157C2×C4—94no—
158C2×C4—93no—
159C2×C4—96no—
160C2×C4—96no—
161C2×C4—92no—
162C2×C4—96no—
163C2×C4—182no—
164C2×C4—186no—
165C2×C4—186no—
166C2×C4—186no—
167C2×C4—182no—
168C2×C4C2×GO+(4,𝔽2)13yes—
169C2×C3—22no—
170C2×C3—22no—
171C2×C3—22no—
172D3—62no—
173D3—62no—
174D3—62no—
175D3—62no—
176C22—92no—
177C22—92no—
178C22—91no—
179C22—91no—
180C22C22×GO+(4,𝔽2)11yescharacteristic
181C4—64no—
182C4—96no—
183C4—126no—
184C4—186no—
185C4—182no—
186C4C2×#377316yes—
187C3—22no—
188C2—91no—
189C2—91no—
190C2—92no—
191C2#1059652412yes—
192C2C22×#377311yescharacteristic
subgroupG/H
center, FrattiniC22C22×GO+(4,𝔽2)
commutatorC2×C3⋊D3C23×C4
FittingC2×C32×Q8D4
socleC22×C32C22×D4
prime radical, cosocleC22×C3⋊D3C24
hypercenterC2×Q8GO+(4,𝔽2)
hypocenterC32#2351
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
7. Q8×#157C4semidirect
48. C32×Q8C4⋊C4semidirect
53. #634C4⋊C4semidirect
151. C32#2351semidirect
154. Q8#3773semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×#1572
Q8×#6242
#105934122
C4×#37732
#105933442
#23519