Finite Group Databasebeta

← Q8×(C3⋊S4)⋊C3 #13164417

Subgroups — Q8×(C3⋊S4)⋊C3 #13164417

Order 1728.

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

normal subgroups only

This group has 632 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×C22⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C4×(C3⋊S4)⋊C3C213yesmaximal
3#64259⋊C3C213yesmaximal, complemented
4Q8×C3⋊S4C311yesmaximal, characteristic, complemented
5C3×Q8×#67—31nomaximal
6C2×(C3⋊S4)⋊C3C2211yescharacteristic
7Q8×(C3⋊D3)⋊C3—41nomaximal
8C4×C22⋊H(3,𝔽3)C2213yes—
9(C3⋊SL(2,ℤ/4))⋊C3C2213yes—
10C22×C32×Q8D311yescharacteristic, complemented
11C2×C3×D3×Q8—31no—
12C3×Q8×A4—21no—
13C3×Q8×A4C2×C311yescharacteristic, complemented
14C4×C3⋊S4C2×C313yes—
15#64259C2×C313yescomplemented
16C3×#946—33no—
17C3×Q8×Dic3—31no—
18C3×#944—33no—
19C3×C4×#67—33no—
20C3×#924—33no—
21C2×C22⋊H(3,𝔽3)C2311yescharacteristic
22(C3⋊S4)⋊C3Q812yesdirect product factor
23Q8×H(3,𝔽3)—41no—
24C4×(C3⋊D3)⋊C3—43no—
25#634⋊C3—43no—
26Q8×S4—91no—
27C3×D4×Q8—91no—
28Q8×#67—31no—
29C2×C3⋊S4C22×C311yescharacteristic
30C2×C32×Q8—31no—
31C2×C32×Q8—33no—
32C22×C32×C4C2×D313yes—
33Q8×C3⋊D3—41no—
34C2×C3×#67—31no—
35C3×D3×Q8—63no—
36C3×D3×Q8—61no—
37C2×C3×C4×D3—33no—
38C2×C3×Dic6—33no—
39C3×C4×A4—23no—
40C3×C4×A4C22×C313yes—
41C3⋊SL(2,ℤ/4)C22×C313yes—
42C3×#217—33no—
43C3×(C2×D3)⋊C4—33no—
44C3×C4⋊Dic3—33no—
45C3×Dic3⋊C4—36no—
46C3×Dic3⋊C4—33no—
47C3×C4×Dic3—33no—
48C2×(C3⋊D3)⋊C3—41no—
49C22⋊H(3,𝔽3)C2×Q811yescharacteristic, complemented
50C4×H(3,𝔽3)—43no—
51(C3⋊Dic3)⋊C3—43no—
52C22×C3×Q8—31no—
53C22×C3×Q8—91no—
54C22×C3×Q8C3×D311yescharacteristic
55C2×D3×Q8—31no—
56Q8×A4—31no—
57Q8×A4—61no—
58C4×S4—93no—
59A4⋊Q8—93no—
60C3×C4⋊Q8—93no—
61C3×C22⋊Q8—93no—
62C3×C22⋊Q8—93no—
63C3×C4×Q8—91no—
64C3×C4×D4—93no—
65#946—33no—
66Q8×Dic3—31no—
67#944—33no—
68C4×#67—33no—
69#924—33no—
70C23×C32C22×D311yescharacteristic
71C22×C3×D3—31no—
72C2×C3×A4—21no—
73C2×C3×A4C23×C311yescharacteristic
74C3⋊S4C3×Q812yescomplemented
75C32×Q8—33no—
76C32×Q8—41no—
77C32×Q8—61no—
78C32×Q8—81no—
79C32×Q8S411yescharacteristic, complemented
80C2×C32×C4—33no—
81C2×C32×C4—33no—
82C4×C3⋊D3—43no—
83#634—43no—
84C3×#67—32no—
85C3×#67—32no—
86C2×C3×Dic3—31no—
87C2×C3×Dic3—33no—
88C2×C3×Dic3—33no—
89C3×C4×D3—63no—
90C3×C4×D3—63no—
91C3×Dic6—63no—
92C3×Dic6—63no—
93D4×Q8—91noSylow
94C2×H(3,𝔽3)—41no—
95(C3⋊D3)⋊C3—42no—
96C2×S4—91no—
97C2×C3×Q8—31no—
98C2×C3×Q8—31no—
99C2×C3×Q8—33no—
100C2×C3×Q8—33no—
101C2×C3×Q8—61no—
102C2×C3×Q8—63no—
103C2×C3×Q8—93no—
104C2×C3×Q8—183no—
105C2×C3×Q8—181no—
106C2×C3×D4—91no—
107C22×C3×C4—33no—
108C22×C3×C4—93no—
109C22×C3×C4C2×C3×D313yes—
110C2×#67—31no—
111D3×Q8—63no—
112D3×Q8—61no—
113D3×Q8—361no—
114C2×C4×D3—33no—
115C2×Dic6—33no—
116C4×A4—33no—
117C4×A4—63no—
118SL(2,ℤ/4)—93no—
119C3×C4⋊C4—93no—
120C3×C4⋊C4—96no—
121C3×C4⋊C4—93no—
122C3×C22⋊C4—93no—
123C3×C22⋊C4—93no—
124C3×C42—93no—
125#217—33no—
126(C2×D3)⋊C4—33no—
127C4⋊Dic3—33no—
128Dic3⋊C4—36no—
129Dic3⋊C4—33no—
130C4×Dic3—33no—
131C22×C32—31no—
132C22×C32—31no—
133C22×C32D3×Q811yescharacteristic, complemented
134C2×C3⋊D3—41no—
135C2×C3×D3—32no—
136C2×C3×D3—61no—
137C2×C3×D3—61no—
138C3×A4—21no—
139C3×A4C2×C3×Q811yescharacteristic, complemented
140C32×C4—33no—
141C32×C4—43no—
142C32×C4—83no—
143C32×C4C2×S413yes—
144C3⋊Dic3—43no—
145C3×Dic3—32no—
146C3×Dic3—63no—
147C3×Dic3—63no—
148C22×Q8—91no—
149C22×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
150C4⋊Q8—93no—
151C22⋊Q8—93no—
152C22⋊Q8—93no—
153C4×Q8—91no—
154C4×D4—93no—
155H(3,𝔽3)—41noSylow
156C23×C3—31no—
157C23×C3—91no—
158C23×C3C22×C3×D311yescharacteristic
159C22×D3—31no—
160C2×A4—31no—
161C2×A4—61no—
162S4—92no—
163C3×Q8—33no—
164C3×Q8—33no—
165C3×Q8—31no—
166C3×Q8—63no—
167C3×Q8—61no—
168C3×Q8—63no—
169C3×Q8—121no—
170C3×Q8—183no—
171C3×Q8—183no—
172C3×Q8—241no—
173C3×Q8C3×S411yescharacteristic
174C3×D4—92no—
175C3×D4—92no—
176C2×C3×C4—33no—
177C2×C3×C4—33no—
178C2×C3×C4—33no—
179C2×C3×C4—33no—
180C2×C3×C4—63no—
181C2×C3×C4—63no—
182C2×C3×C4—91no—
183C2×C3×C4—93no—
184C2×C3×C4—93no—
185C2×C3×C4—183no—
186C2×C3×C4—183no—
187#67—32no—
188#67—32no—
189C2×Dic3—31no—
190C2×Dic3—33no—
191C2×Dic3—33no—
192C4×D3—63no—
193C4×D3—63no—
194C4×D3—363no—
195Dic6—63no—
196Dic6—63no—
197Dic6—363no—
198C2×C32—31no—
199C2×C32—31no—
200C2×C32—41no—
201C2×C32—81no—
202C2×C32C22×S411yescharacteristic
203C3⋊D3—42no—
204C3×D3—62no—
205C2×Q8—31no—
206C2×Q8—33no—
207C2×Q8—93no—
208C2×Q8—183no—
209C2×Q8—181no—
210C2×D4—91no—
211C22×C4—93no—
212C22×C4C2×(C3⋊D3)⋊C313yes—
213C4⋊C4—93no—
214C4⋊C4—93no—
215C4⋊C4—96no—
216C22⋊C4—93no—
217C22⋊C4—93no—
218C42—93no—
219C22×C3—31no—
220C22×C3—31no—
221C22×C3—31no—
222C22×C3—31no—
223C22×C3—31no—
224C22×C3—61no—
225C22×C3—61no—
226C22×C3—92no—
227C22×C3—181no—
228C22×C3—181no—
229C22×C3C3×D3×Q811yescharacteristic
230C2×D3—32no—
231C2×D3—61no—
232C2×D3—61no—
233C2×D3—361no—
234A4—31no—
235A4—61no—
236C3×C4—33no—
237C3×C4—33no—
238C3×C4—33no—
239C3×C4—63no—
240C3×C4—92no—
241C3×C4—123no—
242C3×C4—183no—
243C3×C4—183no—
244C3×C4—243no—
245C3×C4C2×C3×S413yes—
246Dic3—32no—
247Dic3—63no—
248Dic3—63no—
249Dic3—363no—
250C32—41no—
251C32—81no—
252C32Q8×S411yescharacteristic, complemented
253C23—91no—
254C23C22×(C3⋊D3)⋊C311yescharacteristic
255Q8—33no—
256Q8—61no—
257Q8—183no—
258Q8—183no—
259Q8(C3⋊S4)⋊C311yescharacteristic, direct product factor
260D4—92no—
261D4—92no—
262C2×C4—33no—
263C2×C4—33no—
264C2×C4—91no—
265C2×C4—93no—
266C2×C4—93no—
267C2×C4—183no—
268C2×C4—183no—
269C2×C3—31no—
270C2×C3—31no—
271C2×C3—31no—
272C2×C3—31no—
273C2×C3—31no—
274C2×C3—61no—
275C2×C3—61no—
276C2×C3—121no—
277C2×C3—182no—
278C2×C3—241no—
279C2×C3C22×C3×S411yescharacteristic
280D3—62no—
281D3—362no—
282C22—31no—
283C22—31no—
284C22—92no—
285C22—181no—
286C22—181no—
287C22Q8×(C3⋊D3)⋊C311yescharacteristic, complemented
288C4—33no—
289C4—92no—
290C4—183no—
291C4—183no—
292C4C2×(C3⋊S4)⋊C313yes—
293C3—31no—
294C3—121no—
295C3—241no—
296C3C3×Q8×S411yescharacteristic
297C2—31no—
298C2—31no—
299C2—182no—
300C2C22×(C3⋊S4)⋊C311yescharacteristic
subgroupG/H
centerC2C22×(C3⋊S4)⋊C3
commutator, cosocleC2×C3×A4C23×C3
FrattiniC2×C3C22×C3×S4
FittingC22×C32×Q8D3
socleC23×C3C22×C3×D3
prime radicalC2×(C3⋊S4)⋊C3C22
hypercenterQ8(C3⋊S4)⋊C3
hypocenterC3×A4C2×C3×Q8
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C2 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. Q8×C22⋊H(3,𝔽3)C2semidirect
3. #64259⋊C3C2semidirect
4. Q8×C3⋊S4C3semidirect
10. C22×C32×Q8D3semidirect
13. C3×Q8×A4C2×C3semidirect
15. #64259C2×C3semidirect
22. (C3⋊S4)⋊C3Q8direct (N × H)
49. C22⋊H(3,𝔽3)C2×Q8semidirect
74. C3⋊S4C3×Q8semidirect
79. C32×Q8S4semidirect
133. C22×C32D3×Q8semidirect
139. C3×A4C2×C3×Q8semidirect
149. C22×Q8(C3⋊D3)⋊C3semidirect
252. C32Q8×S4semidirect
259. Q8(C3⋊S4)⋊C3direct (N × H)
287. C22Q8×(C3⋊D3)⋊C3semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊H(3,𝔽3)2
C4×(C3⋊S4)⋊C32
#64259⋊C32
Q8×C3⋊S43
C3×Q8×#673
Q8×(C3⋊D3)⋊C34