Finite Group Databasebeta

← Q8×((C22⋊C9)⋊C2)⋊C3 #13164411

Subgroups — Q8×((C22⋊C9)⋊C2)⋊C3 #13164411

Order 1728.

582 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 74 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×#1094C211yesmaximal, characteristic, complemented
2C4×((C22⋊C9)⋊C2)⋊C3C213yesmaximal
3#1094⋊Q8C213yesmaximal, complemented
4C3×Q8×#67—31nomaximal
5Q8×(C22⋊C9)⋊C2C311yesmaximal, characteristic, complemented
6C2×((C22⋊C9)⋊C2)⋊C3C2211yescharacteristic
7Q8×AGL(1,ℤ/9)—41nomaximal
8C4×#1094C2213yes—
9#1094⋊C4C2213yes—
10C22×C32×Q8D311yescharacteristic
11C2×C3×D3×Q8—31no—
12C3×#946—33no—
13C3×Q8×Dic3—31no—
14C3×#944—33no—
15C3×C4×#67—33no—
16C3×#924—33no—
17Q8×C22⋊C9—21no—
18Q8×C22⋊C9C2×C311yescharacteristic, complemented
19C4×(C22⋊C9)⋊C2C2×C313yes—
20(C22⋊C9)⋊Q8C2×C313yescomplemented
21C2×#1094C2311yescharacteristic
22((C22⋊C9)⋊C2)⋊C3Q812yesdirect product factor
23Q8×3−1+2—41no—
24C4×AGL(1,ℤ/9)—43no—
253−1+2⋊Q8—43no—
26C3×D4×Q8—91no—
27Q8×#67—31no—
28C2×C32×Q8—31no—
29C2×C32×Q8—33no—
30C22×C32×C4C2×D313yes—
31C2×C3×#67—31no—
32C3×D3×Q8—63no—
33C3×D3×Q8—61no—
34C2×C3×C4×D3—33no—
35C2×C3×Dic6—33no—
36C2×(C22⋊C9)⋊C2C22×C311yescharacteristic
37C3×#217—33no—
38C3×(C2×D3)⋊C4—33no—
39C3×C4⋊Dic3—33no—
40C3×Dic3⋊C4—36no—
41C3×Dic3⋊C4—33no—
42C3×C4×Dic3—33no—
43Q8×D9—41no—
44C4×C22⋊C9—23no—
45C4×C22⋊C9C22×C313yes—
46(C22⋊C9)⋊C4C22×C313yes—
47C2×AGL(1,ℤ/9)—41no—
48#1094C2×Q811yescharacteristic, complemented
49C4×3−1+2—43no—
503−1+2⋊C4—43no—
51C22×C3×Q8—31no—
52C22×C3×Q8—91no—
53C22×C3×Q8C3×D311yescharacteristic
54C2×D3×Q8—31no—
55C3×C4⋊Q8—93no—
56C3×C22⋊Q8—93no—
57C3×C22⋊Q8—93no—
58C3×C4×Q8—91no—
59C3×C4×D4—93no—
60#946—33no—
61Q8×Dic3—31no—
62#944—33no—
63C4×#67—33no—
64#924—33no—
65C23×C32C22×D311yescharacteristic
66C22×C3×D3—31no—
67C32×Q8—33no—
68C32×Q8—61no—
69C32×Q8S411yescharacteristic
70C2×C32×C4—33no—
71C2×C32×C4—33no—
72C3×#67—32no—
73C3×#67—32no—
74C2×C3×Dic3—31no—
75C2×C3×Dic3—33no—
76C2×C3×Dic3—33no—
77C3×C4×D3—63no—
78C3×C4×D3—63no—
79C3×Dic6—63no—
80C3×Dic6—63no—
81C2×C22⋊C9—21no—
82C2×C22⋊C9C23×C311yescharacteristic
83(C22⋊C9)⋊C2C3×Q812yescomplemented
84Q8×C9—41no—
85Q8×C9—81no—
86C4×D9—43no—
87Dic18—43no—
88D4×Q8—91noSylow
89C2×3−1+2—41no—
90AGL(1,ℤ/9)—42no—
91C2×C3×Q8—31no—
92C2×C3×Q8—31no—
93C2×C3×Q8—33no—
94C2×C3×Q8—33no—
95C2×C3×Q8—61no—
96C2×C3×Q8—63no—
97C2×C3×Q8—93no—
98C2×C3×Q8—183no—
99C2×C3×Q8—181no—
100C2×C3×D4—91no—
101C22×C3×C4—33no—
102C22×C3×C4—93no—
103C22×C3×C4C2×C3×D313yes—
104C2×#67—31no—
105D3×Q8—63no—
106D3×Q8—61no—
107C2×C4×D3—33no—
108C2×Dic6—33no—
109C3×C4⋊C4—93no—
110C3×C4⋊C4—96no—
111C3×C4⋊C4—93no—
112C3×C22⋊C4—93no—
113C3×C22⋊C4—93no—
114C3×C42—93no—
115#217—33no—
116(C2×D3)⋊C4—33no—
117C4⋊Dic3—33no—
118Dic3⋊C4—36no—
119Dic3⋊C4—33no—
120C4×Dic3—33no—
121C22×C32—31no—
122C22×C32—31no—
123C22×C32D3×Q811yescharacteristic
124C2×C3×D3—32no—
125C2×C3×D3—61no—
126C2×C3×D3—61no—
127C32×C4—33no—
128C32×C4C2×S413yes—
129C3×Dic3—32no—
130C3×Dic3—63no—
131C3×Dic3—63no—
132C2×D9—41no—
133C22⋊C9—21no—
134C22⋊C9C2×C3×Q811yescharacteristic, complemented
135C4×C9—43no—
136C4×C9—83no—
137Dic9—43no—
138C22×Q8—91no—
139C22×Q8AGL(1,ℤ/9)11yescharacteristic, complemented
140C4⋊Q8—93no—
141C22⋊Q8—93no—
142C22⋊Q8—93no—
143C4×Q8—91no—
144C4×D4—93no—
1453−1+2—41noSylow
146C23×C3—31no—
147C23×C3—91no—
148C23×C3C22×C3×D311yescharacteristic
149C22×D3—31no—
150C3×Q8—33no—
151C3×Q8—33no—
152C3×Q8—31no—
153C3×Q8—63no—
154C3×Q8—61no—
155C3×Q8—63no—
156C3×Q8—183no—
157C3×Q8—183no—
158C3×Q8C3×S411yescharacteristic
159C3×D4—92no—
160C3×D4—92no—
161C2×C3×C4—33no—
162C2×C3×C4—33no—
163C2×C3×C4—33no—
164C2×C3×C4—33no—
165C2×C3×C4—63no—
166C2×C3×C4—63no—
167C2×C3×C4—91no—
168C2×C3×C4—93no—
169C2×C3×C4—93no—
170C2×C3×C4—183no—
171C2×C3×C4—183no—
172#67—32no—
173#67—32no—
174C2×Dic3—31no—
175C2×Dic3—33no—
176C2×Dic3—33no—
177C4×D3—63no—
178C4×D3—63no—
179Dic6—63no—
180Dic6—63no—
181C2×C32—31no—
182C2×C32—31no—
183C2×C32C22×S411yescharacteristic
184C3×D3—62no—
185C2×C9—41no—
186C2×C9—81no—
187D9—42no—
188C2×Q8—31no—
189C2×Q8—33no—
190C2×Q8—93no—
191C2×Q8—183no—
192C2×Q8—181no—
193C2×D4—91no—
194C22×C4—93no—
195C22×C4C2×AGL(1,ℤ/9)13yes—
196C4⋊C4—93no—
197C4⋊C4—93no—
198C4⋊C4—96no—
199C22⋊C4—93no—
200C22⋊C4—93no—
201C42—93no—
202C22×C3—31no—
203C22×C3—31no—
204C22×C3—31no—
205C22×C3—31no—
206C22×C3—31no—
207C22×C3—61no—
208C22×C3—61no—
209C22×C3—92no—
210C22×C3—181no—
211C22×C3—181no—
212C22×C3C3×D3×Q811yescharacteristic
213C2×D3—32no—
214C2×D3—61no—
215C2×D3—61no—
216C3×C4—33no—
217C3×C4—33no—
218C3×C4—33no—
219C3×C4—63no—
220C3×C4—92no—
221C3×C4—183no—
222C3×C4—183no—
223C3×C4C2×C3×S413yes—
224Dic3—32no—
225Dic3—63no—
226Dic3—63no—
227C32Q8×S411yescharacteristic
228C9—41no—
229C9—81no—
230C23—91no—
231C23C22×AGL(1,ℤ/9)11yescharacteristic
232Q8—33no—
233Q8—61no—
234Q8—183no—
235Q8—183no—
236Q8((C22⋊C9)⋊C2)⋊C311yescharacteristic, direct product factor
237D4—92no—
238D4—92no—
239C2×C4—33no—
240C2×C4—33no—
241C2×C4—91no—
242C2×C4—93no—
243C2×C4—93no—
244C2×C4—183no—
245C2×C4—183no—
246C2×C3—31no—
247C2×C3—31no—
248C2×C3—31no—
249C2×C3—31no—
250C2×C3—31no—
251C2×C3—61no—
252C2×C3—61no—
253C2×C3—182no—
254C2×C3C22×C3×S411yescharacteristic
255D3—62no—
256C22—31no—
257C22—31no—
258C22—92no—
259C22—181no—
260C22—181no—
261C22Q8×AGL(1,ℤ/9)11yescharacteristic, complemented
262C4—33no—
263C4—92no—
264C4—183no—
265C4—183no—
266C4C2×((C22⋊C9)⋊C2)⋊C313yes—
267C3—31no—
268C3C3×Q8×S411yescharacteristic
269C2—31no—
270C2—31no—
271C2—182no—
272C2C22×((C22⋊C9)⋊C2)⋊C311yescharacteristic
subgroupG/H
centerC2C22×((C22⋊C9)⋊C2)⋊C3
commutator, cosocleC2×C22⋊C9C23×C3
FrattiniC2×C3C22×C3×S4
FittingC22×C32×Q8D3
socleC23×C3C22×C3×D3
prime radicalC2×((C22⋊C9)⋊C2)⋊C3C22
hypercenterQ8((C22⋊C9)⋊C2)⋊C3
hypocenterC22⋊C9C2×C3×Q8
LCSG ▷ C2×C22⋊C9 ▷ C22⋊C9
UCSC1 ◁ C2 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. Q8×#1094C2semidirect
3. #1094⋊Q8C2semidirect
5. Q8×(C22⋊C9)⋊C2C3semidirect
18. Q8×C22⋊C9C2×C3semidirect
20. (C22⋊C9)⋊Q8C2×C3semidirect
22. ((C22⋊C9)⋊C2)⋊C3Q8direct (N × H)
48. #1094C2×Q8semidirect
83. (C22⋊C9)⋊C2C3×Q8semidirect
134. C22⋊C9C2×C3×Q8semidirect
139. C22×Q8AGL(1,ℤ/9)semidirect
236. Q8((C22⋊C9)⋊C2)⋊C3direct (N × H)
261. C22Q8×AGL(1,ℤ/9)semidirect

Maximal subgroups

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

subgroupindex
Q8×#10942
C4×((C22⋊C9)⋊C2)⋊C32
#1094⋊Q82
C3×Q8×#673
Q8×(C22⋊C9)⋊C23
Q8×AGL(1,ℤ/9)4