Finite Group Databasebeta

← Q8×#157⋊D3 #13181158

Subgroups — Q8×#157⋊D3 #13181158

Order 1728.

948 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 948 conjugacy classes of subgroups — too many to draw. Only its 135 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
1Q8×#1113C212yesmaximal, complemented
2C3×Q8×#157C211yesmaximal, characteristic, complemented
3C4×#157⋊D3C213yesmaximal
4#11724335C213yesmaximal, complemented
5#11724333C216yesmaximal, complemented
6Q8×GO+(4,𝔽2)—31nomaximal
7C2×#157⋊D3C2211yescharacteristic
8C3×Q8×C3⋊D3—22no—
9C3×Q8×C3⋊D3C2211yescharacteristic
10C4×#1113C2216yes—
11#89040—26no—
12#89040C2216yes—
13C3×C4⋊#157C2213yes—
14C3×C4×#157C2213yes—
15#88940C2216yes—
16#88940C2213yes—
17#88939C2216yes—
18D32×Q8—32no—
19D32×Q8—62no—
20Q8×#157D311yescharacteristic, complemented
21C4×GO+(4,𝔽2)—33no—
22#64222—33no—
23D32⋊Q8—36no—
24C2×#1113C2312yes—
25C2×C3×#157C2313yes—
26C2×C3×#157C2311yescharacteristic
27#157⋊D3Q814yesdirect product factor
28C33×Q8D411yescharacteristic, complemented
29C3×C4×C3⋊D3—26no—
30C3×C4×C3⋊D3C2313yes—
31C3×#634—26no—
32C3×#634D413yescomplemented
33#6296—26no—
34#6294—26no—
35#6294C2316yes—
36Q8×D12—91nomaximal
37C2×GO+(4,𝔽2)—31no—
38Q8×C3⋊D3—22no—
39Q8×C3⋊D3C2×D311yescharacteristic
40C3×D3×Q8—62no—
41C3×D3×Q8—62no—
42C3×D3×Q8—62no—
43C3×D3×Q8—122no—
44C4×D32—36no—
45C4×D32—66no—
46#3797—36no—
47#3797—66no—
48D3×Dic6—66no—
49D3×Dic6—66no—
50D3×Dic6—66no—
51C4⋊#157C2×D313yes—
52C4×#157C2×D313yes—
53#3773—36no—
54#3773—33no—
55D32⋊C4—36no—
56C2×C3×C3⋊D3—22no—
57C2×C3×C3⋊D3C2411yescharacteristic
58#1113—22no—
59#1113C2×Q814yescomplemented
60C3×#157—23no—
61C3×#157C2×Q812yescomplemented
62C33×C4C2×D413yes—
63C3×C3⋊Dic3—26no—
64C3×C3⋊Dic3C2×D413yes—
65C2×D3×Q8—92no—
66C2×D3×Q8—182no—
67C3×C4×Q8—91no—
68#897—96no—
69C4×D12—93no—
70#869—93no—
71C2×D32—32no—
72C2×D32—62no—
73C2×#157C22×D313yes—
74C2×#157C22×D311yescharacteristic
75GO+(4,𝔽2)—34no—
76C32×Q8—22no—
77C32×Q8—42no—
78C32×Q8D1211yescharacteristic, complemented
79C4×C3⋊D3—26no—
80C4×C3⋊D3C22×D313yes—
81#634—26no—
82#634D1213yescomplemented
83C3×C4×D3—66no—
84C3×C4×D3—66no—
85C3×C4×D3—66no—
86C3×C4×D3—126no—
87C3×Dic6—66no—
88C3×Dic6—66no—
89C3×Dic6—66no—
90C3×Dic6—126no—
91#627—612no—
92#627—66no—
93#624—36no—
94#624—66no—
95D3×Dic3—66no—
96D3×Dic3—66no—
97D3×Dic3—66no—
98D4×Q8—271noSylow
99C2×C33C22×D411yescharacteristic
100C3×C3⋊D3—24no—
101C3×C3⋊D3C22×Q811yescharacteristic
102C3×C3⋊D32−1+411yescharacteristic
103C2×C3×Q8—91no—
104C2×C3×Q8—182no—
105D3×Q8—62no—
106D3×Q8—62no—
107D3×Q8—62no—
108D3×Q8—122no—
109D3×Q8—186no—
110D3×Q8—186no—
111D3×Q8—186no—
112C2×D12—91no—
113C2×C4×D3—96no—
114C2×C4×D3—186no—
115C2×Dic6—96no—
116C2×Dic6—186no—
117C3×C4⋊C4—93no—
118C3×C42—93no—
119(C2×D3)⋊C4—96no—
120C4⋊Dic3—93no—
121C4⋊Dic3—96no—
122C2×C3⋊D3—22no—
123C2×C3⋊D3C23×D311yescharacteristic
124C2×C3×D3—62no—
125C2×C3×D3—62no—
126C2×C3×D3—62no—
127C2×C3×D3—122no—
128D32—34no—
129D32—64no—
130D32—64no—
131D32—62no—
132#157—23no—
133#157D3×Q812yescomplemented
134C32×C4—26no—
135C32×C4—46no—
136C32×C4C2×D1213yes—
137C3⋊Dic3—26no—
138C3⋊Dic3C2×D1213yes—
139C3×Dic3—66no—
140C3×Dic3—66no—
141C3×Dic3—66no—
142C3×Dic3—126no—
143C22×Q8—272no—
144C4⋊Q8—273no—
145C22⋊Q8—276no—
146C4×Q8—91no—
147C4×D4—273no—
148C33D4×Q811yescharacteristic, complemented, Sylow
149C22×D3—92no—
150C22×D3—182no—
151C3×Q8—22no—
152C3×Q8—42no—
153C3×Q8—93no—
154C3×Q8—186no—
155C3×Q8GO+(4,𝔽2)11yescharacteristic, complemented
156C2×C3×C4—91no—
157C2×C3×C4—93no—
158C2×C3×C4—93no—
159C2×C3×C4—186no—
160C2×Dic3—96no—
161C2×Dic3—186no—
162D12—94no—
163C4×D3—66no—
164C4×D3—66no—
165C4×D3—66no—
166C4×D3—126no—
167C4×D3—186no—
168C4×D3—186no—
169C4×D3—186no—
170Dic6—66no—
171Dic6—66no—
172Dic6—66no—
173Dic6—126no—
174Dic6—186no—
175Dic6—1812no—
176C2×C32—22no—
177C2×C32—42no—
178C2×C32C22×D1211yescharacteristic
179C3⋊D3—24no—
180C3⋊D3C2×D3×Q811yescharacteristic
181C3⋊D3#101011yescharacteristic
182C3×D3—62no—
183C3×D3—62no—
184C3×D3—64no—
185C3×D3—64no—
186C3×D3—124no—
187C2×Q8—91no—
188C2×Q8—182no—
189C2×Q8—276no—
190C2×Q8—546no—
191C2×D4—271no—
192C22×C4—276no—
193C4⋊C4—93no—
194C4⋊C4—273no—
195C4⋊C4—276no—
196C22⋊C4—276no—
197C42—93no—
198C22×C3—91no—
199C22×C3—182no—
200C2×D3—62no—
201C2×D3—62no—
202C2×D3—62no—
203C2×D3—94no—
204C2×D3—122no—
205C2×D3—184no—
206C2×D3—182no—
207C2×D3—184no—
208C3×C4—26no—
209C3×C4—46no—
210C3×C4—92no—
211C3×C4—93no—
212C3×C4—186no—
213C3×C4—183no—
214C3×C4C2×GO+(4,𝔽2)13yes—
215Dic3—66no—
216Dic3—66no—
217Dic3—66no—
218Dic3—126no—
219C32—22no—
220C32—42no—
221C32Q8×D1211yescharacteristic, complemented
222C23—272no—
223Q8—93no—
224Q8—186no—
225Q8—546no—
226Q8#157⋊D311yescharacteristic, direct product factor
227D4—274no—
228C2×C4—91no—
229C2×C4—93no—
230C2×C4—93no—
231C2×C4—186no—
232C2×C4—276no—
233C2×C4—546no—
234C2×C3—22no—
235C2×C3—42no—
236C2×C3—91no—
237C2×C3—91no—
238C2×C3—184no—
239C2×C3C22×GO+(4,𝔽2)11yescharacteristic
240D3—62no—
241D3—62no—
242D3—64no—
243D3—64no—
244D3—124no—
245C22—91no—
246C22—182no—
247C22—274no—
248C22—542no—
249C4—92no—
250C4—93no—
251C4—183no—
252C4—186no—
253C4C2×#157⋊D313yes—
254C3—22no—
255C3—42no—
256C3Q8×GO+(4,𝔽2)11yescharacteristic, complemented
257C2—91no—
258C2—91no—
259C2—184no—
260C2C22×#157⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C22×#157⋊D3
commutator, cosocleC2×C3×C3⋊D3C24
FittingC33×Q8D4
socleC2×C33C22×D4
prime radicalC2×#157⋊D3C22
hypercenterQ8#157⋊D3
hypocenterC33D4×Q8
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. Q8×#1113C2semidirect
2. C3×Q8×#157C2semidirect
4. #11724335C2semidirect
5. #11724333C2semidirect
20. Q8×#157D3semidirect
27. #157⋊D3Q8direct (N × H)
28. C33×Q8D4semidirect
32. C3×#634D4semidirect
59. #1113C2×Q8semidirect
61. C3×#157C2×Q8semidirect
78. C32×Q8D12semidirect
82. #634D12semidirect
133. #157D3×Q8semidirect
148. C33D4×Q8semidirect
155. C3×Q8GO+(4,𝔽2)semidirect
221. C32Q8×D12semidirect
226. Q8#157⋊D3direct (N × H)
256. C3Q8×GO+(4,𝔽2)semidirect

Maximal subgroups

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

subgroupindex
Q8×#11132
C3×Q8×#1572
C4×#157⋊D32
#117243352
#117243332
Q8×GO+(4,𝔽2)3
Q8×D129