Finite Group Databasebeta

← C2×Q8×((C3⋊D3)⋊C3)⋊C2 #13164766

Subgroups — C2×Q8×((C3⋊D3)⋊C3)⋊C2 #13164766

Order 1728.

3131 conjugacy classes of subgroups; only the 619 normal subgroups are shown.

normal subgroups only

This group has 3131 conjugacy classes of subgroups — too many to draw. Only its 619 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
1C2×Q8×#272C211yesmaximal, characteristic, complemented
2C2×Q8×(C3⋊D3)⋊C3C212yesmaximal, complemented
3Q8×((C3⋊D3)⋊C3)⋊C2C2116yesmaximal, direct product factor
4C2×C4×((C3⋊D3)⋊C3)⋊C2C213yesmaximal
5C2×#88651C213yesmaximal, complemented
6C2×((C3⋊D3)⋊C3)⋊Q8C216yesmaximal, complemented
7C2×D32×Q8—32nomaximal
8C22×((C3⋊D3)⋊C3)⋊C2C2211yescharacteristic
9C2×Q8×H(3,𝔽3)C2211yescharacteristic, complemented
10Q8×#272C2218yescomplemented
11C2×C4×#272C2213yes—
12C2×#6229C2213yescomplemented
13Q8×(C3⋊D3)⋊C3C22116yescomplemented
14C2×C4×(C3⋊D3)⋊C3C2216yes—
15C2×#634⋊C3C2216yescomplemented
16C2×#6199C2213yes—
17C2×((C3⋊D3)⋊C3)⋊C4C2216yes—
18C2×#6196C2216yescomplemented
19C4×((C3⋊D3)⋊C3)⋊C2C22124yes—
20#88651C22124yescomplemented
21((C3⋊D3)⋊C3)⋊Q8C22148yescomplemented
22C2×Q8×C3⋊D3D312yescomplemented
23C2×C3×D3×Q8—32no—
24C2×C3×D3×Q8—32no—
25C2×C3×D3×Q8—61no—
26D32×Q8—332no—
27C2×C4×D32—36no—
28C2×#3797—36no—
29C2×D3×Dic6—36no—
30C2×D3×Dic6—36no—
31C22×#272C2311yescharacteristic
32C22×(C3⋊D3)⋊C3C2312yes—
33C2×((C3⋊D3)⋊C3)⋊C2Q818yesdirect product factor
34C2×((C3⋊D3)⋊C3)⋊C2C2314yes—
35Q8×H(3,𝔽3)C2314yescomplemented
36C2×C4×H(3,𝔽3)C2313yes—
37C2×#1084C2313yes—
38C4×#272C23112yes—
39#6229C23112yescomplemented
40C2×(C3⋊Dic3)⋊C3C2316yes—
41C4×(C3⋊D3)⋊C3C23124yes—
42#634⋊C3C23124yescomplemented
43#6199C23112yes—
44((C3⋊D3)⋊C3)⋊C4C23124yes—
45#6196C23124yescomplemented
46C22×D3×Q8—91no—
47C22×D3×Q8—92no—
48C22×D32—32no—
49C2×C32×Q8—21no—
50C2×C32×Q8C2×D312yescomplemented
51Q8×C3⋊D3C2×D3116yescomplemented
52C2×C4×C3⋊D3C2×D316yes—
53C2×#634C2×D316yescomplemented
54C3×D3×Q8—316no—
55C3×D3×Q8—316no—
56C3×D3×Q8—68no—
57C2×C3×C4×D3—36no—
58C2×C3×C4×D3—36no—
59C2×C3×C4×D3—63no—
60C2×C3×Dic6—36no—
61C2×C3×Dic6—36no—
62C2×C3×Dic6—63no—
63C2×#627—312no—
64C2×#624—36no—
65C2×D3×Dic3—36no—
66C2×D3×Dic3—36no—
67C4×D32—348no—
68#3797—348no—
69D3×Dic6—348no—
70D3×Dic6—348no—
71C22×H(3,𝔽3)C2411yescharacteristic
72C2×#272C2×Q814yescomplemented
73C2×#272C2412yes—
74C2×(C3⋊D3)⋊C3C2×Q818yescomplemented
75C2×(C3⋊D3)⋊C3C2414yes—
76((C3⋊D3)⋊C3)⋊C2C2×Q8116yesdirect product factor
77C4×H(3,𝔽3)C2416yes—
78#1084C2416yes—
79(C3⋊Dic3)⋊C3C24112yes—
80C22×C3×Q8—92no—
81C22×C3×Q8—91no—
82C2×D3×Q8—32no—
83C2×D3×Q8—32no—
84C2×D3×Q8—61no—
85C2×D3×Q8—92no—
86C2×D3×Q8—96no—
87C2×D3×Q8—96no—
88C2×D3×Q8—96no—
89C2×D3×Q8—916no—
90C2×D3×Q8—932no—
91C22×C4×D3—93no—
92C22×C4×D3—96no—
93C22×Dic6—93no—
94C22×Dic6—96no—
95C22×C3⋊D3C22×D312yes—
96C22×C3×D3—32no—
97C22×C3×D3—32no—
98C22×C3×D3—61no—
99C2×D32—316no—
100C2×D32—38no—
101C32×Q8—24no—
102C32×Q8C22×D318yescomplemented
103C2×C32×C4—23no—
104C2×C32×C4C22×D316yes—
105C2×C3⋊Dic3C22×D316yes—
106C4×C3⋊D3C22×D3124yes—
107#634C22×D3124yescomplemented
108C2×C3×Dic3—36no—
109C2×C3×Dic3—36no—
110C2×C3×Dic3—63no—
111C3×C4×D3—324no—
112C3×C4×D3—324no—
113C3×C4×D3—612no—
114C3×Dic6—324no—
115C3×Dic6—324no—
116C3×Dic6—612no—
117#627—348no—
118#624—324no—
119D3×Dic3—324no—
120D3×Dic3—324no—
121C23×Q8—271noSylow
122C2×H(3,𝔽3)C22×Q812yescomplemented
123C2×H(3,𝔽3)C2511yescharacteristic
124#272C22×Q814yescomplemented
125(C3⋊D3)⋊C3C22×Q818yescomplemented
126C23×D3—91no—
127C23×D3—92no—
128C2×C3×Q8—32no—
129C2×C3×Q8—61no—
130C2×C3×Q8—96no—
131C2×C3×Q8—916no—
132C2×C3×Q8—98no—
133C2×C3×Q8—93no—
134C2×C3×Q8D3211yescharacteristic
135C22×C3×C4—96no—
136C22×C3×C4—93no—
137C22×Dic3—93no—
138C22×Dic3—96no—
139D3×Q8—316no—
140D3×Q8—316no—
141D3×Q8—68no—
142D3×Q8—916no—
143D3×Q8—948no—
144D3×Q8—948no—
145D3×Q8—948no—
146C2×C4×D3—36no—
147C2×C4×D3—36no—
148C2×C4×D3—63no—
149C2×C4×D3—96no—
150C2×C4×D3—924no—
151C2×C4×D3—948no—
152C2×C4×D3—96no—
153C2×C4×D3—96no—
154C2×C4×D3—96no—
155C2×Dic6—36no—
156C2×Dic6—36no—
157C2×Dic6—63no—
158C2×Dic6—96no—
159C2×Dic6—924no—
160C2×Dic6—948no—
161C2×Dic6—96no—
162C2×Dic6—912no—
163C22×C32—21no—
164C22×C32C23×D312yes—
165C2×C3⋊D3D3×Q818yescomplemented
166C2×C3⋊D3C23×D314yes—
167C2×C3×D3—38no—
168C2×C3×D3—38no—
169C2×C3×D3—34no—
170C2×C3×D3—34no—
171C2×C3×D3—64no—
172C2×C3×D3—62no—
173D32—332no—
174C32×C4—26no—
175C32×C4C23×D3112yes—
176C3⋊Dic3C23×D3112yes—
177C3×Dic3—312no—
178C3×Dic3—312no—
179C3×Dic3—66no—
180C22×Q8—91no—
181C22×Q8—92no—
182C22×Q8—273no—
183C22×Q8—276no—
184C22×Q8—2716no—
185C23×C4—273no—
186H(3,𝔽3)C23×Q811yescharacteristic, complemented, Sylow
187C23×C3—92no—
188C23×C3—91no—
189C22×D3—32no—
190C22×D3—32no—
191C22×D3—61no—
192C22×D3—92no—
193C22×D3—94no—
194C22×D3—98no—
195C22×D3—98no—
196C22×D3—916no—
197C3×Q8—38no—
198C3×Q8—64no—
199C3×Q8—924no—
200C3×Q8—912no—
201C3×Q8C2×D3214yes—
202C2×C3×C4—36no—
203C2×C3×C4—63no—
204C2×C3×C4—924no—
205C2×C3×C4—96no—
206C2×C3×C4—93no—
207C2×C3×C4—912no—
208C2×C3×C4C2×D3213yes—
209C2×Dic3—36no—
210C2×Dic3—36no—
211C2×Dic3—63no—
212C2×Dic3—96no—
213C2×Dic3—912no—
214C2×Dic3—924no—
215C4×D3—324no—
216C4×D3—324no—
217C4×D3—612no—
218C4×D3—924no—
219C4×D3—924no—
220C4×D3—924no—
221C4×D3—924no—
222Dic6—324no—
223Dic6—324no—
224Dic6—612no—
225Dic6—924no—
226Dic6—924no—
227Dic6—948no—
228C2×C32—22no—
229C2×C32—21no—
230C2×C32C2×D3×Q814yescomplemented
231C2×C32C24×D312yes—
232C3⋊D3C2×D3×Q818yescomplemented
233C3×D3—38no—
234C3×D3—38no—
235C3×D3—64no—
236C24—271no—
237C2×Q8—98no—
238C2×Q8—916no—
239C2×Q8—93no—
240C2×Q8—96no—
241C2×Q8—276no—
242C2×Q8—2724no—
243C2×Q8—2748no—
244C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic, direct product factor
245C22×C4—93no—
246C22×C4—96no—
247C22×C4—2724no—
248C22×C4—276no—
249C22×C4—273no—
250C22×C3—32no—
251C22×C3—61no—
252C22×C3—94no—
253C22×C3—98no—
254C22×C3—94no—
255C22×C3—92no—
256C22×C3C22×D3211yescharacteristic
257C2×D3—34no—
258C2×D3—38no—
259C2×D3—38no—
260C2×D3—34no—
261C2×D3—62no—
262C2×D3—64no—
263C2×D3—94no—
264C2×D3—98no—
265C2×D3—916no—
266C2×D3—932no—
267C3×C4—312no—
268C3×C4—66no—
269C3×C4—912no—
270C3×C4—96no—
271C3×C4C22×D3216yes—
272Dic3—312no—
273Dic3—312no—
274Dic3—66no—
275Dic3—912no—
276C32—21no—
277C32C22×D3×Q812yescomplemented
278C23—91no—
279C23—92no—
280C23—274no—
281C23—278no—
282Q8—912no—
283Q8—924no—
284Q8—2724no—
285Q8C2×((C3⋊D3)⋊C3)⋊C214yesdirect product factor
286C2×C4—93no—
287C2×C4—96no—
288C2×C4—912no—
289C2×C4—924no—
290C2×C4—2724no—
291C2×C4—2712no—
292C2×C4C2×((C3⋊D3)⋊C3)⋊C213yes—
293C2×C3—34no—
294C2×C3—32no—
295C2×C3—62no—
296C2×C3—61no—
297C2×C3—98no—
298C2×C3—94no—
299C2×C3D32×Q812yes—
300C2×C3C23×D3211yescharacteristic
301D3—38no—
302D3—38no—
303D3—64no—
304D3—98no—
305C22—94no—
306C22—98no—
307C22—92no—
308C22—94no—
309C22—2716no—
310C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
311C4—96no—
312C4—912no—
313C4C22×((C3⋊D3)⋊C3)⋊C216yes—
314C3—32no—
315C3—61no—
316C3C2×D32×Q811yescharacteristic
317C2—94no—
318C2—98no—
319C2Q8×((C3⋊D3)⋊C3)⋊C212yesdirect product factor
320C2C23×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC2×H(3,𝔽3)C25
FrattiniC2×C3C23×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3C22×D32
prime radicalC22×((C3⋊D3)⋊C3)⋊C2C22
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C23×Q8
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×Q8×#272C2semidirect
2. C2×Q8×(C3⋊D3)⋊C3C2semidirect
3. Q8×((C3⋊D3)⋊C3)⋊C2C2direct (N × H)
5. C2×#88651C2semidirect
6. C2×((C3⋊D3)⋊C3)⋊Q8C2semidirect
9. C2×Q8×H(3,𝔽3)C22semidirect
10. Q8×#272C22semidirect
12. C2×#6229C22semidirect
13. Q8×(C3⋊D3)⋊C3C22semidirect
15. C2×#634⋊C3C22semidirect
18. C2×#6196C22semidirect
20. #88651C22semidirect
21. ((C3⋊D3)⋊C3)⋊Q8C22semidirect
22. C2×Q8×C3⋊D3D3semidirect
33. C2×((C3⋊D3)⋊C3)⋊C2Q8direct (N × H)
35. Q8×H(3,𝔽3)C23semidirect
39. #6229C23semidirect
42. #634⋊C3C23semidirect
45. #6196C23semidirect
50. C2×C32×Q8C2×D3semidirect
51. Q8×C3⋊D3C2×D3semidirect
53. C2×#634C2×D3semidirect
72. C2×#272C2×Q8semidirect
74. C2×(C3⋊D3)⋊C3C2×Q8semidirect
76. ((C3⋊D3)⋊C3)⋊C2C2×Q8direct (N × H)
102. C32×Q8C22×D3semidirect
107. #634C22×D3semidirect
122. C2×H(3,𝔽3)C22×Q8semidirect
124. #272C22×Q8semidirect
125. (C3⋊D3)⋊C3C22×Q8semidirect
165. C2×C3⋊D3D3×Q8semidirect
186. H(3,𝔽3)C23×Q8semidirect
230. C2×C32C2×D3×Q8semidirect
232. C3⋊D3C2×D3×Q8semidirect
244. C2×Q8((C3⋊D3)⋊C3)⋊C2direct (N × H)
277. C32C22×D3×Q8semidirect
285. Q8C2×((C3⋊D3)⋊C3)⋊C2direct (N × H)
319. C2Q8×((C3⋊D3)⋊C3)⋊C2direct (N × H)

Maximal subgroups

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

subgroupindex
C2×Q8×#2722
C2×Q8×(C3⋊D3)⋊C32
Q8×((C3⋊D3)⋊C3)⋊C22
C2×C4×((C3⋊D3)⋊C3)⋊C22
C2×#886512
C2×((C3⋊D3)⋊C3)⋊Q82
C2×D32×Q83