Finite Group Databasebeta

← Q8×C9⋊S4 #13164458

Subgroups — Q8×C9⋊S4 #13164458

Order 1728.

620 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 91 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×C9×A4C211yesmaximal, characteristic, complemented
2C4×C9⋊S4C213yesmaximal
3#11722366C213yesmaximal, complemented
4Q8×C3⋊S4—31nomaximal
5Q8×(C22⋊C9)⋊C2—32nomaximal
6Q8×#611—31nomaximal
7C2×C9⋊S4C2211yescharacteristic
8Q8×C3⋊D9—41nomaximal
9C4×C9×A4C2213yes—
10C9⋊SL(2,ℤ/4)C2213yes—
11C3×Q8×A4D311yescharacteristic
12C4×C3⋊S4—33no—
13#64259—33no—
14C22×Q8×C9D311yescharacteristic, complemented
15C2×Q8×D9—31no—
16Q8×C22⋊C9D312yescomplemented
17C4×(C22⋊C9)⋊C2—36no—
18(C22⋊C9)⋊Q8—36no—
19#63508—33no—
20Q8×Dic9—31no—
21#63506—33no—
22C4×#611—33no—
23#63486—33no—
24C2×C9×A4C2311yescharacteristic
25C9⋊S4Q812yesdirect product factor
26C3×Q8×C9—41no—
27C4×C3⋊D9—43no—
28#6226—43no—
29Q8×S4—93no—
30Q8×#67—91no—
31C2×C3⋊S4—31no—
32Q8×C3⋊D3—121no—
33C3×C4×A4C2×D313yes—
34C3⋊SL(2,ℤ/4)—33no—
35C2×(C22⋊C9)⋊C2—32no—
36C2×Q8×C9—31no—
37C2×Q8×C9—33no—
38C22×C4×C9C2×D313yes—
39C2×#611—31no—
40Q8×D9—63no—
41Q8×D9—61no—
42Q8×D9—122no—
43C2×C4×D9—33no—
44C2×Dic18—33no—
45C4×C22⋊C9C2×D316yes—
46(C22⋊C9)⋊C4—36no—
47#3676—33no—
48(C2×D9)⋊C4—33no—
49C4⋊Dic9—33no—
50Dic9⋊C4—33no—
51Dic9⋊C4—36no—
52C4×Dic9—33no—
53C2×C3⋊D9—41no—
54C9×A4C2×Q811yescharacteristic, complemented
55C3×C4×C9—43no—
56C3⋊Dic9—43no—
57C22×C3×Q8C3⋊D311yescharacteristic
58C2×D3×Q8—91no—
59Q8×A4D913yescomplemented
60C4×S4—99no—
61A4⋊Q8—99no—
62#946—93no—
63Q8×Dic3—91no—
64#944—93no—
65C4×#67—93no—
66#924—93no—
67C2×C3×A4C22×D311yescharacteristic
68C3⋊S4—32no—
69C32×Q8—41no—
70C4×C3⋊D3—123no—
71#634—123no—
72C23×C9C22×D311yescharacteristic
73C22×D9—31no—
74C2×C22⋊C9C22×D312yes—
75(C22⋊C9)⋊C2—34no—
76Q8×C9—33no—
77Q8×C9—42no—
78Q8×C9—61no—
79Q8×C9S411yescharacteristic, complemented
80C2×C4×C9—33no—
81C2×C4×C9—33no—
82#611—32no—
83#611—32no—
84C2×Dic9—31no—
85C2×Dic9—33no—
86C2×Dic9—33no—
87C4×D9—63no—
88C4×D9—63no—
89C4×D9—126no—
90Dic18—63no—
91Dic18—63no—
92Dic18—126no—
93D4×Q8—271noSylow
94C2×C3×C9—41no—
95C3⋊D9—42no—
96C2×S4—93no—
97C2×C3×Q8—31no—
98C2×C3×Q8—33no—
99C22×C3×C4C2×C3⋊D313yes—
100C2×#67—91no—
101D3×Q8—183no—
102D3×Q8—181no—
103D3×Q8—363no—
104C2×C4×D3—93no—
105C2×Dic6—93no—
106C4×A4C2×D919yes—
107SL(2,ℤ/4)—99no—
108#217—93no—
109(C2×D3)⋊C4—93no—
110C4⋊Dic3—93no—
111Dic3⋊C4—96no—
112Dic3⋊C4—93no—
113C4×Dic3—93no—
114C2×C3⋊D3—121no—
115C3×A4D3×Q811yescharacteristic
116C32×C4—43no—
117C3⋊Dic3—123no—
118C22×C9—31no—
119C22×C9—31no—
120C22×C9D3×Q811yescharacteristic, complemented
121C2×D9—32no—
122C2×D9—61no—
123C2×D9—61no—
124C2×D9—122no—
125C22⋊C9D3×Q812yescomplemented
126C4×C9—33no—
127C4×C9—46no—
128C4×C9C2×S413yes—
129Dic9—32no—
130Dic9—63no—
131Dic9—63no—
132Dic9—126no—
133C22×Q8—271no—
134C22×Q8C3⋊D911yescharacteristic, complemented
135C4⋊Q8—273no—
136C22⋊Q8—273no—
137C22⋊Q8—273no—
138C4×Q8—271no—
139C4×D4—273no—
140C3×C9—41noSylow
141C23×C3C22×C3⋊D311yescharacteristic
142C22×D3—91no—
143C2×A4C22×D913yes—
144S4—96no—
145C3×Q8—33no—
146C3×Q8—43no—
147C3×Q8—61no—
148C3×Q8C3⋊S411yescharacteristic
149C2×C3×C4—33no—
150C2×C3×C4—33no—
151#67—92no—
152#67—92no—
153C2×Dic3—91no—
154C2×Dic3—93no—
155C2×Dic3—93no—
156C4×D3—183no—
157C4×D3—183no—
158C4×D3—369no—
159Dic6—183no—
160Dic6—183no—
161Dic6—369no—
162C2×C32—41no—
163C3⋊D3—122no—
164C2×C9—31no—
165C2×C9—31no—
166C2×C9—42no—
167C2×C9C22×S411yescharacteristic
168D9—62no—
169D9—124no—
170C2×Q8—31no—
171C2×Q8—33no—
172C2×Q8—273no—
173C2×Q8—543no—
174C2×Q8—541no—
175C2×D4—271no—
176C22×C4—273no—
177C22×C4C2×C3⋊D913yes—
178C4⋊C4—273no—
179C4⋊C4—273no—
180C4⋊C4—276no—
181C22⋊C4—273no—
182C22⋊C4—273no—
183C42—273no—
184C22×C3—31no—
185C22×C3—31no—
186C22×C3Q8×C3⋊D311yescharacteristic
187C2×D3—92no—
188C2×D3—181no—
189C2×D3—181no—
190C2×D3—363no—
191A4Q8×D913yescomplemented
192C3×C4—33no—
193C3×C4—49no—
194C3×C4C2×C3⋊S413yes—
195Dic3—92no—
196Dic3—183no—
197Dic3—183no—
198Dic3—369no—
199C32—41no—
200C9—42no—
201C9Q8×S411yescharacteristic, complemented
202C23—271no—
203C23C22×C3⋊D911yescharacteristic
204Q8—33no—
205Q8—61no—
206Q8—543no—
207Q8—543no—
208Q8C9⋊S411yescharacteristic, direct product factor
209D4—272no—
210D4—272no—
211C2×C4—33no—
212C2×C4—33no—
213C2×C4—271no—
214C2×C4—273no—
215C2×C4—273no—
216C2×C4—543no—
217C2×C4—543no—
218C2×C3—31no—
219C2×C3—31no—
220C2×C3—43no—
221C2×C3C22×C3⋊S411yescharacteristic
222D3—182no—
223D3—366no—
224C22—31no—
225C22—31no—
226C22—272no—
227C22—541no—
228C22—541no—
229C22Q8×C3⋊D911yescharacteristic, complemented
230C4—33no—
231C4—272no—
232C4—543no—
233C4—543no—
234C4C2×C9⋊S413yes—
235C3—43no—
236C3Q8×C3⋊S411yescharacteristic
237C2—31no—
238C2—31no—
239C2—542no—
240C2C22×C9⋊S411yescharacteristic
subgroupG/H
centerC2C22×C9⋊S4
commutator, cosocleC2×C9×A4C23
FrattiniC2×C3C22×C3⋊S4
FittingC22×Q8×C9D3
socleC23×C3C22×C3⋊D3
prime radicalC2×C9⋊S4C22
hypercenterQ8C9⋊S4
hypocenterC9×A4C2×Q8
LCSG ▷ C2×C9×A4 ▷ C9×A4
UCSC1 ◁ C2 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. Q8×C9×A4C2semidirect
3. #11722366C2semidirect
14. C22×Q8×C9D3semidirect
16. Q8×C22⋊C9D3semidirect
25. C9⋊S4Q8direct (N × H)
54. C9×A4C2×Q8semidirect
59. Q8×A4D9semidirect
79. Q8×C9S4semidirect
120. C22×C9D3×Q8semidirect
125. C22⋊C9D3×Q8semidirect
134. C22×Q8C3⋊D9semidirect
191. A4Q8×D9semidirect
201. C9Q8×S4semidirect
208. Q8C9⋊S4direct (N × H)
229. C22Q8×C3⋊D9semidirect

Maximal subgroups

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

subgroupindex
Q8×C9×A42
C4×C9⋊S42
#117223662
Q8×C3⋊S43
Q8×(C22⋊C9)⋊C23
Q8×#6113
Q8×C3⋊D94