Finite Group Databasebeta

← C2×#11722579 #13164861

Subgroups — C2×#11722579 #13164861

Order 1728.

1052 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 1052 conjugacy classes of subgroups — too many to draw. Only its 169 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
1#11722579C214yesmaximal, direct product factor
2C2×#88691C211yesmaximal, characteristic, complemented
3C22×#6204C212yesmaximal, complemented
4C2×C32×2−1+4C311yesmaximal, characteristic
5C2×2−1+4⋊C9C313yesmaximal, complemented
6C2×D4×3−1+2—41nomaximal
7#88691C2212yescomplemented
8C2×#6204C2211yescharacteristic
9C2×#6204C2214yescomplemented
10C32×2−1+4—34no—
11C32×2−1+4C2×C314yes—
12C2×C32×#41—31no—
13C2×C32×#41—32no—
14C2×C32×#41C2×C311yescharacteristic
15C22×C32×Q8—31no—
16C22×C32×Q8C2×C312yes—
172−1+4⋊C9C2×C3112yescomplemented
18C2×#41⋊C9C2×C313yescomplemented
19C22×Q8⋊C9C2×C316yescomplemented
20C23×3−1+2—42no—
21D4×3−1+2—44no—
22C2×C4×3−1+2—41no—
23#6204C2311yescharacteristic
24C2×C3×2−1+4—31no—
25C2×C3×2−1+4C3211yescharacteristic
26C32×#41—38no—
27C32×#41—34no—
28C32×#41—34no—
29C32×#41—38no—
30C32×#41—32no—
31C32×#41C22×C312yes—
32C2×C32×Q8—34no—
33C2×C32×Q8—34no—
34C2×C32×Q8—31no—
35C2×C32×Q8—32no—
36C2×C32×Q8—34no—
37C2×C32×Q8C22×C311yescharacteristic
38C2×C32×Q8C22×C314yes—
39C2×C32×D4—32no—
40C2×C32×D4—31no—
41C2×C32×D4A411yescharacteristic
42C22×C32×C4—32no—
43C22×C32×C4—32no—
44C22×C32×C4—31no—
45C2×D4×C9—43no—
46#41⋊C9C22×C316yescomplemented
47C2×Q8⋊C9C22×C313yes—
48C2×Q8⋊C9C22×C3112yescomplemented
49C22×3−1+2—41no—
50C22×3−1+2—44no—
51C22×3−1+2—84no—
52C4×3−1+2—42no—
53C3×2−1+4—34no—
54C3×2−1+4—312no—
55C3×2−1+4—34no—
56C3×2−1+4C2×C3214yes—
57C2×C3×#41—33no—
58C2×C3×#41—36no—
59C2×C3×#41—31no—
60C2×C3×#41—32no—
61C2×C3×#41—31no—
62C2×C3×#41C2×C3211yescharacteristic
63C22×C3×Q8—31no—
64C22×C3×Q8—33no—
65C22×C3×Q8—32no—
66C22×C3×Q8C2×C3212yes—
67C23×C32—31no—
68C23×C32C2×A412yes—
69C32×Q8—34no—
70C32×Q8—31no—
71C32×Q8—34no—
72C32×Q8—34no—
73C32×Q8C23×C311yescharacteristic
74C32×D4—32no—
75C32×D4—32no—
76C32×D4—38no—
77C32×D4C2×A414yes—
78C2×C32×C4—38no—
79C2×C32×C4—32no—
80C2×C32×C4—31no—
81C2×C32×C4—34no—
82C2×C32×C4—34no—
83C2×C32×C4—32no—
84C2×C32×C4—32no—
85C2×C32×C4C2×A411yescharacteristic
86C23×C9—46no—
87D4×C9—412no—
88C2×C4×C9—43no—
89Q8⋊C9C23×C313yes—
90C2×2−1+43−1+211yescharacteristic, complemented, Sylow
91C2×3−1+2—42no—
92C2×3−1+2—41no—
93C2×3−1+2—84no—
94C3×#41—324no—
95C3×#41—38no—
96C3×#41—312no—
97C3×#41—312no—
98C3×#41—324no—
99C3×#41—36no—
100C3×#41—34no—
101C3×#41—34no—
102C3×#41—38no—
103C3×#41—32no—
104C3×#41—32no—
105C3×#41C22×C3212yes—
106C2×C3×Q8—34no—
107C2×C3×Q8—33no—
108C2×C3×Q8—36no—
109C2×C3×Q8—312no—
110C2×C3×Q8—312no—
111C2×C3×Q8—34no—
112C2×C3×Q8—312no—
113C2×C3×Q8—31no—
114C2×C3×Q8—32no—
115C2×C3×Q8—34no—
116C2×C3×Q8—31no—
117C2×C3×Q8—34no—
118C2×C3×Q8C22×C3211yescharacteristic
119C2×C3×Q8C22×C3214yes—
120C2×C3×D4—31no—
121C2×C3×D4—33no—
122C2×C3×D4—36no—
123C2×C3×D4—32no—
124C2×C3×D4—31no—
125C2×C3×D4C3×A411yescharacteristic
126C22×C3×C4—36no—
127C22×C3×C4—32no—
128C22×C3×C4—36no—
129C22×C3×C4—33no—
130C22×C3×C4—32no—
131C22×C3×C4—31no—
132C22×C32—24no—
133C22×C32—32no—
134C22×C32—62no—
135C22×C32C22×A414yes—
136C22×C32C22×A411yescharacteristic
137C32×C4—34no—
138C32×C4—31no—
139C32×C4—31no—
140C32×C4C22×A412yes—
141C22×C9—412no—
142C22×C9—43no—
143C22×C9—812no—
144C4×C9—46no—
1452−1+4—34no—
1462−1+4C2×3−1+214yescomplemented
147C2×#41—31no—
148C2×#41—32no—
149C2×#41C2×3−1+211yescharacteristic, complemented
150C22×Q8—31no—
151C22×Q8C2×3−1+212yescomplemented
1523−1+2—41noSylow
153C23×C3—33no—
154C23×C3—31no—
155C23×C3—32no—
156C23×C3C2×C3×A412yes—
157C3×Q8—31no—
158C3×Q8—34no—
159C3×Q8—33no—
160C3×Q8—312no—
161C3×Q8—312no—
162C3×Q8—34no—
163C3×Q8—312no—
164C3×Q8—34no—
165C3×Q8—31no—
166C3×Q8C23×C3211yescharacteristic
167C3×D4—32no—
168C3×D4—36no—
169C3×D4—36no—
170C3×D4—32no—
171C3×D4—324no—
172C3×D4—38no—
173C3×D4—34no—
174C3×D4C2×C3×A414yes—
175C2×C3×C4—33no—
176C2×C3×C4—36no—
177C2×C3×C4—324no—
178C2×C3×C4—38no—
179C2×C3×C4—312no—
180C2×C3×C4—312no—
181C2×C3×C4—36no—
182C2×C3×C4—36no—
183C2×C3×C4—34no—
184C2×C3×C4—34no—
185C2×C3×C4—32no—
186C2×C3×C4—31no—
187C2×C3×C4—32no—
188C2×C3×C4—32no—
189C2×C3×C4—31no—
190C2×C3×C4C2×C3×A411yescharacteristic
191C2×C32—24no—
192C2×C32—62no—
193C2×C322−1+4⋊C312yes—
194C2×C32C23×A411yescharacteristic
195C2×C9—46no—
196C2×C9—43no—
197C2×C9—812no—
198#41—38no—
199#41—34no—
200#41—34no—
201#41—38no—
202#41—32no—
203#41C22×3−1+212yescomplemented
204C2×Q8—34no—
205C2×Q8—34no—
206C2×Q8—31no—
207C2×Q8—32no—
208C2×Q8—34no—
209C2×Q8C22×3−1+211yescharacteristic
210C2×Q8C22×3−1+214yescomplemented
211C2×D4—31no—
212C2×D4—32no—
213C2×D4#109411yescharacteristic
214C22×C4—32no—
215C22×C4—32no—
216C22×C4—31no—
217C22×C3—24no—
218C22×C3—36no—
219C22×C3—32no—
220C22×C3—31no—
221C22×C3—34no—
222C22×C3—64no—
223C22×C3—66no—
224C22×C3—62no—
225C22×C3C22×C3×A414yes—
226C22×C3C22×C3×A411yescharacteristic
227C3×C4—33no—
228C3×C4—312no—
229C3×C4—33no—
230C3×C4—31no—
231C3×C4—34no—
232C3×C4—31no—
233C3×C4—32no—
234C3×C4C22×C3×A412yes—
235C32C2×2−1+4⋊C311yescharacteristic
236C9—43no—
237C23—31no—
238C23C2×#109412yes—
239Q8—31no—
240Q8—34no—
241Q8—34no—
242Q8—34no—
243Q8C23×3−1+211yescharacteristic
244D4—32no—
245D4—32no—
246D4—38no—
247D4C2×#109414yes—
248C2×C4—38no—
249C2×C4—31no—
250C2×C4—32no—
251C2×C4—34no—
252C2×C4—34no—
253C2×C4—32no—
254C2×C4—32no—
255C2×C4C2×#109411yescharacteristic
256C2×C3—24no—
257C2×C3—31no—
258C2×C3—32no—
259C2×C3—64no—
260C2×C3—66no—
261C2×C3—62no—
262C2×C3C3×2−1+4⋊C312yes—
263C2×C3C23×C3×A411yescharacteristic
264C22—24no—
265C22—32no—
266C22—62no—
267C22C22×#109414yes—
268C22C22×#109411yescharacteristic
269C4—31no—
270C4—34no—
271C4—31no—
272C4C22×#109412yes—
273C3—31no—
274C3C2×C3×2−1+4⋊C311yescharacteristic
275C2—24no—
276C2—62no—
277C2#1172257912yesdirect product factor
278C2C23×#109411yescharacteristic
subgroupG/H
center, socleC22×C3C22×C3×A4
commutator, cosocleC3×Q8C23×C32
FrattiniC2×C3C23×C3×A4
Fitting, prime radicalC2×C32×2−1+4C3
hypercenterC2×C32×D4A4
hypocenterQ8C23×3−1+2
LCSG ▷ C3×Q8 ▷ Q8
UCSC1 ◁ C22×C3 ◁ C2×C32×D4

Complements

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

NH ≅ G/Ntype
1. #11722579C2direct (N × H)
2. C2×#88691C2semidirect
3. C22×#6204C2semidirect
5. C2×2−1+4⋊C9C3semidirect
7. #88691C22semidirect
9. C2×#6204C22semidirect
17. 2−1+4⋊C9C2×C3semidirect
18. C2×#41⋊C9C2×C3semidirect
19. C22×Q8⋊C9C2×C3semidirect
46. #41⋊C9C22×C3semidirect
48. C2×Q8⋊C9C22×C3semidirect
90. C2×2−1+43−1+2semidirect
146. 2−1+4C2×3−1+2semidirect
149. C2×#41C2×3−1+2semidirect
151. C22×Q8C2×3−1+2semidirect
203. #41C22×3−1+2semidirect
210. C2×Q8C22×3−1+2semidirect
277. C2#11722579direct (N × H)

Maximal subgroups

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

subgroupindex
#117225792
C2×#886912
C22×#62042
C2×C32×2−1+43
C2×2−1+4⋊C93
C2×D4×3−1+24