Finite Group Databasebeta

← Q8×#3773 #11805578

Subgroups — Q8×#3773 #11805578

Order 1152.

866 conjugacy classes of subgroups; only the 215 normal subgroups are shown.

normal subgroups only

This group has 866 conjugacy classes of subgroups — too many to draw. Only its 215 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#10593413C213yesmaximal
4#10593412C216yesmaximal
5C4×#3773C213yesmaximal
6C2×Q8×C3⋊D3C2211yescharacteristic
7Q8×#157C418yescomplemented
8C2×C4⋊#157C2213yes—
9C2×C4×#157C2213yes—
10C2×#3773C2213yes—
11C2×#3773C2216yes—
12C2×#3773C2211yescharacteristic
13C3×Q8×Dic3—22no—
14#63900C2216yes—
15C4×#624C2216yes—
16#63844—26no—
17#63737C2216yes—
18C4×D3×Q8—62no—
19C22×#157C2313yes—
20C22×#157C2311yescharacteristic
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—26no—
31#3773—46no—
32#3773Q814yesdirect product factor
33C3×C4⋊Dic3—26no—
34C3×C4×Dic3—26no—
35#3724—26no—
36Dic32—26no—
37Q8×C4⋊C4—91nomaximal, Sylow
38C2×D3×Q8—62no—
39C3×C4×Q8—62no—
40Q8×Dic3—22no—
41D3×C4⋊C4—66no—
42#887—66no—
43C42×D3—66no—
44C4×Dic6—66no—
45C22×C3⋊D3C2411yescharacteristic
46C2×#157—23no—
47C2×#157—23no—
48C2×#157C22×C416yes—
49C2×#157C22×C412yes—
50C2×#157C2×Q812yes—
51C2×#157#4112yes—
52C32×Q8C4⋊C414yescomplemented
53C2×C32×C4C2×D413yes—
54C2×C3⋊Dic3C2×D413yes—
55C4×C3⋊D3C22×C416yes—
56C4×C3⋊D3C2×Q816yes—
57#634C4⋊C4112yescomplemented
58C2×C3×Dic3—22no—
59C2×C3×Dic3—26no—
60#624—22no—
61#624—26no—
62#624—26no—
63#624C2×Q814yes—
64C2×C4×Q8—92no—
65C2×C4×Q8—91no—
66#389—96no—
67#384—93no—
68C4×C4⋊C4—93no—
69C2×C3×Q8—22no—
70D3×Q8—68no—
71D3×Q8—68no—
72C2×C4×D3—62no—
73C2×C4×D3—66no—
74C2×C4×D3—66no—
75C2×Dic6—66no—
76C3×C4⋊C4—66no—
77C3×C42—66no—
78C4⋊Dic3—26no—
79Dic3⋊C4—612no—
80C4×Dic3—26no—
81C4×Dic3—66no—
82C4×Dic3—66no—
83C22×C32C22×D411yescharacteristic
84C2×C3⋊D3C23×C411yescharacteristic
85C2×C3⋊D3C22×Q811yescharacteristic
86C2×C3⋊D3C22×Q811yescharacteristic
87C2×C3⋊D3C2×#4111yescharacteristic
88C2×C3⋊D3H(5,𝔽2)11yescharacteristic
89C2×C3⋊D32−1+411yescharacteristic
90#157—26no—
91#157C4×Q814yescomplemented
92C32×C4C2×C4⋊C416yes—
93C3⋊Dic3C2×C4⋊C416yes—
94C3×Dic3—24no—
95C3×Dic3—46no—
96C22×Q8—91no—
97C4×Q8—62no—
98C4×Q8—98no—
99C4×Q8—186no—
100C2×C4⋊C4—96no—
101C2×C4⋊C4—93no—
102C2×C4⋊C4—91no—
103C2×C4⋊C4—93no—
104C2×C4⋊C4—96no—
105C2×C42—96no—
106C2×C42—93no—
107#95—96no—
108C22×D3—62no—
109C3×Q8—28no—
110C2×C3×C4—26no—
111C2×C3×C4—62no—
112C2×C3×C4—66no—
113C2×Dic3—22no—
114C2×Dic3—26no—
115C2×Dic3—66no—
116C4×D3—64no—
117C4×D3—612no—
118C4×D3—612no—
119C4×D3—64no—
120C4×D3—126no—
121C4×D3—126no—
122Dic6—624no—
123C2×C32C22×C4⋊C411yescharacteristic
124C2×C32D4×Q811yescharacteristic
125C2×C32#55011yescharacteristic
126C3⋊D3C2×C4×Q811yescharacteristic
127C3⋊D3#52011yescharacteristic
128C3⋊D3#55711yescharacteristic
129C3⋊D3Q8211yescharacteristic
130C2×Q8—94no—
131C2×Q8—94no—
132C2×Q8—93no—
133C2×Q8GO+(4,𝔽2)11yescharacteristic
134C22×C4—92no—
135C22×C4—96no—
136C22×C4—91no—
137C22×C4—93no—
138C22×C4—93no—
139C4⋊C4—66no—
140C4⋊C4—912no—
141C4⋊C4—94no—
142C4⋊C4—186no—
143C4⋊C4—186no—
144C4⋊C4—366no—
145C42—66no—
146C42—912no—
147C42—186no—
148C22×C3—22no—
149C2×D3—62no—
150C2×D3—62no—
151C2×D3—62no—
152C2×D3—62no—
153C2×D3—62no—
154C2×D3—62no—
155C3×C4—212no—
156C3×C4—64no—
157C3×C4—126no—
158Dic3—24no—
159Dic3—46no—
160Dic3—612no—
161C32Q8×C4⋊C411yescharacteristic, complemented, Sylow
162C23—91no—
163Q8—912no—
164Q8#377314yesdirect product factor
165C2×C4—62no—
166C2×C4—66no—
167C2×C4—94no—
168C2×C4—92no—
169C2×C4—92no—
170C2×C4—92no—
171C2×C4—93no—
172C2×C4—96no—
173C2×C4—96no—
174C2×C4—96no—
175C2×C4—182no—
176C2×C4—186no—
177C2×C4—186no—
178C2×C4—183no—
179C2×C4—183no—
180C2×C4C2×GO+(4,𝔽2)13yes—
181C2×C3—22no—
182C2×C3—22no—
183C2×C3—22no—
184D3—62no—
185D3—62no—
186D3—62no—
187D3—62no—
188C22—91no—
189C22—91no—
190C22—91no—
191C22—91no—
192C22—91no—
193C22—91no—
194C22C22×GO+(4,𝔽2)11yescharacteristic
195C4—64no—
196C4—94no—
197C4—96no—
198C4—126no—
199C4—186no—
200C4C2×#377316yes—
201C3—22no—
202C2—91no—
203C2—91no—
204C2—91no—
205C2—91no—
206C2Q8×GO+(4,𝔽2)11yescharacteristic
207C2#1059652511yescharacteristic
208C2C22×#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)
hypocenterC32Q8×C4⋊C4
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
32. #3773Q8direct (N × H)
52. C32×Q8C4⋊C4semidirect
57. #634C4⋊C4semidirect
91. #157C4×Q8semidirect
161. C32Q8×C4⋊C4semidirect
164. Q8#3773direct (N × H)

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
#105934132
#105934122
C4×#37732
Q8×C4⋊C49