Finite Group Databasebeta

← Q8×(C3⋊D3)⋊A4 #13164623

Subgroups — Q8×(C3⋊D3)⋊A4 #13164623

Order 1728.

1064 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 1064 conjugacy classes of subgroups — too many to draw. Only its 87 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⋊D3)⋊A4C213yesmaximal
3#634⋊A4C213yesmaximal, complemented
4C22×Q8×C3⋊D3C311yesmaximal, characteristic, complemented
5D3×Q8×A4—31nomaximal
6C2×(C3⋊D3)⋊A4C2211yescharacteristic
7Q8×(C3⋊D3)⋊C3—41nomaximal
8C4×C22⋊H(3,𝔽3)C2213yes—
9(C3⋊Dic3)⋊A4C2213yes—
10C22×C32×Q8C2×C311yescharacteristic, complemented
11C2×Q8×C3⋊D3—31no—
12C2×Q8×C3⋊D3—33no—
13C2×Q8×C3⋊D3—33no—
14C2×Q8×C3⋊D3—31no—
15C22×C4×C3⋊D3C2×C313yes—
16C22×#634C2×C313yescomplemented
17C3×Q8×A4—21no—
18C3×Q8×A4D311yescharacteristic, complemented
19C4×D3×A4—33no—
20A4×Dic6—33no—
21C2×C22⋊H(3,𝔽3)C2311yescharacteristic
22(C3⋊D3)⋊A4Q812yesdirect product factor
23Q8×H(3,𝔽3)—41no—
24C4×(C3⋊D3)⋊C3—43no—
25#634⋊C3—43no—
26C22×D3×Q8—31no—
27C22×D3×Q8—91no—
28C2×Q8×A4—91no—
29C23×C3⋊D3C22×C311yescharacteristic
30C2×D3×A4—31no—
31C2×C32×Q8—31no—
32C2×C32×Q8—33no—
33C22×C32×C4C22×C313yes—
34C22×C3⋊Dic3C22×C313yes—
35Q8×C3⋊D3—32no—
36Q8×C3⋊D3—36no—
37Q8×C3⋊D3—36no—
38Q8×C3⋊D3—31no—
39Q8×C3⋊D3—33no—
40Q8×C3⋊D3—33no—
41Q8×C3⋊D3A411yescharacteristic, complemented
42C2×C4×C3⋊D3—33no—
43C2×C4×C3⋊D3—33no—
44C2×C4×C3⋊D3—33no—
45C2×C4×C3⋊D3—33no—
46C2×#634—33no—
47C2×#634—33no—
48C2×#634—36no—
49C3×D3×Q8—121no—
50C3×C4×A4—23no—
51C3×C4×A4C2×D313yes—
52Dic3×A4—33no—
53C2×(C3⋊D3)⋊C3—41no—
54C22⋊H(3,𝔽3)C2×Q811yescharacteristic, complemented
55C4×H(3,𝔽3)—43no—
56(C3⋊Dic3)⋊C3—43no—
57C22×C3×Q8—31no—
58C22×C3×Q8C3×D311yescharacteristic
59C2×D3×Q8—91no—
60C2×D3×Q8—93no—
61C2×D3×Q8—99no—
62C2×D3×Q8—99no—
63C2×D3×Q8—93no—
64C2×D3×Q8—93no—
65C2×D3×Q8—93no—
66C2×D3×Q8—91no—
67C22×C4×D3—33no—
68C22×C4×D3—93no—
69C22×Dic6—33no—
70C22×Dic6—93no—
71Q8×A4—31no—
72Q8×A4—61no—
73Q8×A4—93no—
74C2×C4×A4—93no—
75C23×C32C23×C311yescharacteristic
76C22×C3⋊D3—31no—
77C22×C3⋊D3—32no—
78C22×C3⋊D3—31no—
79C22×C3⋊D3C3×Q812yescomplemented
80C2×C3×A4—21no—
81C2×C3×A4C22×D311yescharacteristic
82D3×A4—32no—
83C32×Q8—32no—
84C32×Q8—33no—
85C32×Q8—41no—
86C32×Q8—81no—
87C32×Q8C2×A411yescharacteristic, complemented
88C2×C32×C4—33no—
89C2×C32×C4—33no—
90C2×C3⋊Dic3—33no—
91C2×C3⋊Dic3—33no—
92C4×C3⋊D3—36no—
93C4×C3⋊D3—33no—
94C4×C3⋊D3—33no—
95C4×C3⋊D3—33no—
96C4×C3⋊D3C2×A413yes—
97#634—36no—
98#634—33no—
99#634—36no—
100#634C2×A413yescomplemented
101C3×C4×D3—123no—
102C3×Dic6—123no—
103C23×Q8—91noSylow
104C2×H(3,𝔽3)—41no—
105(C3⋊D3)⋊C3—42no—
106C23×D3—31no—
107C23×D3—91no—
108C22×A4—91no—
109C2×C3×Q8—33no—
110C2×C3×Q8—39no—
111C2×C3×Q8—33no—
112C2×C3×Q8—31no—
113C2×C3×Q8—361no—
114C22×C3×C4—33no—
115C22×C3×C4C2×C3×D313yes—
116C22×Dic3—33no—
117C22×Dic3—93no—
118D3×Q8—31no—
119D3×Q8—91no—
120D3×Q8—93no—
121D3×Q8—93no—
122D3×Q8—99no—
123D3×Q8—99no—
124D3×Q8—96no—
125D3×Q8—92no—
126D3×Q8—918no—
127D3×Q8—96no—
128D3×Q8—918no—
129D3×Q8—96no—
130D3×Q8—93no—
131D3×Q8—91no—
132C2×C4×D3—93no—
133C2×C4×D3—99no—
134C2×C4×D3—99no—
135C2×C4×D3—99no—
136C2×C4×D3—99no—
137C2×C4×D3—93no—
138C2×C4×D3—93no—
139C2×C4×D3—93no—
140C2×Dic6—93no—
141C2×Dic6—99no—
142C2×Dic6—99no—
143C2×Dic6—918no—
144C2×Dic6—96no—
145C2×Dic6—93no—
146C4×A4—33no—
147C4×A4—63no—
148C4×A4—93no—
149C22×C32—31no—
150C22×C32—31no—
151C22×C32C2×C3×Q811yescharacteristic, complemented
152C2×C3⋊D3—32no—
153C2×C3⋊D3—31no—
154C2×C3⋊D3—32no—
155C2×C3⋊D3—32no—
156C2×C3⋊D3—32no—
157C2×C3⋊D3C22×A411yescharacteristic
158C2×C3×D3—121no—
159C3×A4—21no—
160C3×A4D3×Q811yescharacteristic, complemented
161C32×C4—33no—
162C32×C4—43no—
163C32×C4—83no—
164C32×C4C22×A413yes—
165C3⋊Dic3—33no—
166C3⋊Dic3C22×A413yes—
167C3×Dic3—123no—
168C22×Q8—93no—
169C22×Q8—271no—
170C22×Q8—273no—
171C22×Q8—273no—
172C22×Q8—271no—
173C22×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
174C23×C4—93no—
175H(3,𝔽3)—41noSylow
176C23×C3—31no—
177C23×C3C22×C3×D311yescharacteristic
178C22×D3—32no—
179C22×D3—92no—
180C22×D3—91no—
181C22×D3—93no—
182C22×D3—93no—
183C22×D3—96no—
184C22×D3—91no—
185C22×D3—92no—
186C2×A4—31no—
187C2×A4—61no—
188C2×A4—92no—
189C3×Q8—36no—
190C3×Q8—32no—
191C3×Q8—39no—
192C3×Q8—33no—
193C3×Q8—31no—
194C3×Q8—121no—
195C3×Q8—241no—
196C3×Q8—363no—
197C3×Q8D3×A411yescharacteristic
198C2×C3×C4—39no—
199C2×C3×C4—39no—
200C2×C3×C4—33no—
201C2×C3×C4—33no—
202C2×C3×C4—363no—
203C2×Dic3—93no—
204C2×Dic3—99no—
205C2×Dic3—99no—
206C2×Dic3—93no—
207C4×D3—33no—
208C4×D3—93no—
209C4×D3—93no—
210C4×D3—93no—
211C4×D3—99no—
212C4×D3—99no—
213C4×D3—918no—
214C4×D3—96no—
215C4×D3—99no—
216C4×D3—93no—
217Dic6—33no—
218Dic6—93no—
219Dic6—96no—
220Dic6—918no—
221Dic6—918no—
222Dic6—96no—
223Dic6—99no—
224Dic6—93no—
225C2×C32—31no—
226C2×C32—31no—
227C2×C32—41no—
228C2×C32—81no—
229C2×C32C23×A411yescharacteristic
230C3⋊D3—32no—
231C3⋊D3Q8×A412yescomplemented
232C3×D3—122no—
233C24—91no—
234C2×Q8—33no—
235C2×Q8—31no—
236C2×Q8—91no—
237C2×Q8—273no—
238C2×Q8—273no—
239C2×Q8—276no—
240C2×Q8—273no—
241C2×Q8—273no—
242C2×Q8—276no—
243C2×Q8—272no—
244C2×Q8—276no—
245C2×Q8—271no—
246C22×C4—93no—
247C22×C4—273no—
248C22×C4—273no—
249C22×C4—273no—
250C22×C4—273no—
251C22×C4C2×(C3⋊D3)⋊C313yes—
252C22×C3—31no—
253C22×C3—33no—
254C22×C3—33no—
255C22×C3—31no—
256C22×C3—31no—
257C22×C3—361no—
258C22×C3C3×D3×Q811yescharacteristic
259C2×D3—31no—
260C2×D3—91no—
261C2×D3—92no—
262C2×D3—92no—
263C2×D3—93no—
264C2×D3—96no—
265C2×D3—96no—
266C2×D3—96no—
267C2×D3—96no—
268C2×D3—92no—
269C2×D3—91no—
270C2×D3—92no—
271A4—31no—
272A4—61no—
273C3×C4—39no—
274C3×C4—33no—
275C3×C4—33no—
276C3×C4—123no—
277C3×C4—243no—
278C3×C4—363no—
279C3×C4C2×D3×A413yes—
280Dic3—33no—
281Dic3—93no—
282Dic3—99no—
283Dic3—93no—
284C32—41no—
285C32—81no—
286C32C2×Q8×A411yescharacteristic, complemented
287C23—92no—
288C23—271no—
289C23—271no—
290C23—272no—
291C23C22×(C3⋊D3)⋊C311yescharacteristic
292Q8—32no—
293Q8—33no—
294Q8—93no—
295Q8—276no—
296Q8—276no—
297Q8—273no—
298Q8(C3⋊D3)⋊A411yescharacteristic, direct product factor
299C2×C4—33no—
300C2×C4—33no—
301C2×C4—93no—
302C2×C4—273no—
303C2×C4—273no—
304C2×C4—276no—
305C2×C4—273no—
306C2×C4—273no—
307C2×C4—273no—
308C2×C3—33no—
309C2×C3—33no—
310C2×C3—31no—
311C2×C3—31no—
312C2×C3—31no—
313C2×C3—121no—
314C2×C3—241no—
315C2×C3—362no—
316C2×C3C22×D3×A411yescharacteristic
317D3—32no—
318D3—92no—
319D3—96no—
320D3—92no—
321C22—31no—
322C22—31no—
323C22—91no—
324C22—272no—
325C22—272no—
326C22—272no—
327C22—272no—
328C22—271no—
329C22Q8×(C3⋊D3)⋊C311yescharacteristic, complemented
330C4—33no—
331C4—93no—
332C4—273no—
333C4C2×(C3⋊D3)⋊A413yes—
334C3—31no—
335C3—121no—
336C3—241no—
337C3D3×Q8×A411yescharacteristic
338C2—31no—
339C2—31no—
340C2—92no—
341C2—272no—
342C2C22×(C3⋊D3)⋊A411yescharacteristic
subgroupG/H
centerC2C22×(C3⋊D3)⋊A4
commutator, cosocleC23×C32C23×C3
FrattiniC2×C3C22×D3×A4
FittingC22×C32×Q8C2×C3
socleC23×C3C22×C3×D3
prime radicalC2×(C3⋊D3)⋊A4C22
hypercenterQ8(C3⋊D3)⋊A4
hypocenterC22×C32C2×C3×Q8
LCSG ▷ C23×C32 ▷ C22×C32
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. #634⋊A4C2semidirect
4. C22×Q8×C3⋊D3C3semidirect
10. C22×C32×Q8C2×C3semidirect
16. C22×#634C2×C3semidirect
18. C3×Q8×A4D3semidirect
22. (C3⋊D3)⋊A4Q8direct (N × H)
41. Q8×C3⋊D3A4semidirect
54. C22⋊H(3,𝔽3)C2×Q8semidirect
79. C22×C3⋊D3C3×Q8semidirect
87. C32×Q8C2×A4semidirect
100. #634C2×A4semidirect
151. C22×C32C2×C3×Q8semidirect
160. C3×A4D3×Q8semidirect
173. C22×Q8(C3⋊D3)⋊C3semidirect
231. C3⋊D3Q8×A4semidirect
286. C32C2×Q8×A4semidirect
298. Q8(C3⋊D3)⋊A4direct (N × H)
329. 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⋊D3)⋊A42
#634⋊A42
C22×Q8×C3⋊D33
D3×Q8×A43
Q8×(C3⋊D3)⋊C34