Finite Group Databasebeta

← #13178355

Subgroups — #13178355

Order 1728.

1430 conjugacy classes of subgroups; only the 255 normal subgroups are shown.

normal subgroups only

This group has 1430 conjugacy classes of subgroups — too many to draw. Only its 255 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×#89042C211yesmaximal, characteristic
2C3×#64101C213yesmaximal, complemented
3C4×#6296C212yesmaximal, complemented
4#11723689C213yesmaximal, complemented
5#11723685C216yesmaximal, complemented
6#10594511—33nomaximal
7C2×C3×C4×C3⋊D3C2213yes—
8C2×#6296C2211yescharacteristic
9C2×#6295C2213yes—
10C2×#6294C2216yes—
11#89042C418yescomplemented
12C33×C4⋊C4C2211yescharacteristic
13C3×#3752C2216yes—
14C3×C4⋊(C3⋊Dic3)C2213yes—
15C3×C4×C3⋊Dic3C2216yes—
16#88815C2216yes—
17C2×#3798—33no—
18#64101D313yescomplemented
19C3×#892—36no—
20C3×#892—63no—
21C4×#627—36no—
22#63902—36no—
23#63900—33no—
24D3×C4⋊Dic3—36no—
25#63880—312no—
26C22×C3×C3⋊D3C2313yes—
27C2×C33×C4C2312yes—
28C2×C33×C4C2311yescharacteristic
29C2×C3×C3⋊Dic3C2316yes—
30C2×C3×C3⋊Dic3C2313yes—
31C3×C4×C3⋊D3C2×C4112yescomplemented
32#6296C2×C414yescomplemented
33#6295C2×C4112yescomplemented
34#6294—46no—
35#5523—93no—
36C2×C4×C3⋊D3C2×D313yes—
37C2×C3×C4×D3—36no—
38C2×C3×C4×D3—63no—
39C2×#627—33no—
40C2×#626—36no—
41C2×#625—33no—
42C2×#624—36no—
43C2×D3×Dic3—312no—
44#3798—324no—
45C32×C4⋊C4—23no—
46C32×C4⋊C4—41no—
47C32×C4⋊C4C2×D313yes—
48#3752C2×D316yes—
49C4⋊(C3⋊Dic3)C2×D313yes—
50C4×C3⋊Dic3C2×D316yes—
51C3×(C2×D3)⋊C4—312no—
52C3×(C2×D3)⋊C4—66no—
53C3×C4⋊Dic3—36no—
54C3×C4⋊Dic3—63no—
55C3×C4×Dic3—312no—
56C3×C4×Dic3—66no—
57#3724—36no—
58Dic3⋊Dic3—312no—
59C22×C33C2411yescharacteristic
60C2×C3×C3⋊D3—23no—
61C2×C3×C3⋊D3—23no—
62C2×C3×C3⋊D3C22×C416yes—
63C33×C4—22no—
64C33×C4C22×C412yes—
65C3×C3⋊Dic3—26no—
66C3×C3⋊Dic3C22×C416yes—
67C2×#235—93no—
68C3×#117—93no—
69C2×C4⋊Dic3—93no—
70#892—36no—
71#892—63no—
72D3×C4⋊C4—96no—
73#67⋊C4—912no—
74C4×D12—96no—
75C4×Dic6—96no—
76C22×C3⋊D3C22×D313yes—
77C22×C3×D3—36no—
78C22×C3×D3—63no—
79C2×C32×C4—23no—
80C2×C32×C4—26no—
81C2×C32×C4—41no—
82C2×C32×C4—42no—
83C2×C32×C4C22×D316yes—
84C2×C32×C4C22×D313yes—
85C2×C3⋊Dic3C22×D316yes—
86C2×C3⋊Dic3C22×D313yes—
87C4×C3⋊D3C4×D3112yescomplemented
88C2×C3×Dic3—36no—
89C2×C3×Dic3—312no—
90C2×C3×Dic3—63no—
91C2×C3×Dic3—66no—
92C3×C4×D3—324no—
93C3×C4×D3—612no—
94#627—312no—
95#626—324no—
96#625—312no—
97#624—126no—
98D3×Dic3—1212no—
99#520—271noSylow
100C2×C33C23×C411yescharacteristic
101C2×C33H(5,𝔽2)11yescharacteristic
102C2×C332−1+411yescharacteristic
103C3×C3⋊D3—26no—
104C22×C3×C4—93no—
105C2×#67—96no—
106C22×Dic3—96no—
107#235—924no—
108C2×D12—93no—
109C2×C4×D3—36no—
110C2×C4×D3—63no—
111C2×C4×D3—912no—
112C2×Dic6—93no—
113C3×C4⋊C4—23no—
114C3×C4⋊C4—41no—
115C3×C4⋊C4—93no—
116C3×C4⋊C4D3213yes—
117C3×C22⋊C4—96no—
118C3×C42—96no—
119(C2×D3)⋊C4—312no—
120(C2×D3)⋊C4—66no—
121C4⋊Dic3—36no—
122C4⋊Dic3—63no—
123C4⋊Dic3—96no—
124Dic3⋊C4—912no—
125C4×Dic3—312no—
126C4×Dic3—66no—
127C22×C32—23no—
128C22×C32—41no—
129C22×C32C23×D313yes—
130C2×C3⋊D3—23no—
131C2×C3⋊D3—23no—
132C2×C3⋊D3C2×C4×D316yes—
133C2×C3×D3—312no—
134C2×C3×D3—66no—
135C2×C3×D3—66no—
136C2×C3×D3—66no—
137C2×C3×D3—123no—
138C2×C3×D3—123no—
139C32×C4—26no—
140C32×C4—26no—
141C32×C4—42no—
142C32×C4—46no—
143C32×C4—82no—
144C32×C4C2×C4×D316yes—
145C3⋊Dic3—26no—
146C3⋊Dic3C2×C4×D316yes—
147C3×Dic3—312no—
148C3×Dic3—66no—
149C3×Dic3—612no—
150C3×Dic3—126no—
151C2×#41—271no—
152C4×Q8—272no—
153C4×D4—276no—
154#117—93no—
155C2×C4⋊C4—273no—
156C33#52011yescharacteristic, complemented, Sylow
157C23×C3—93no—
158C22×D3—36no—
159C22×D3—63no—
160C2×C3×C4—23no—
161C2×C3×C4—26no—
162C2×C3×C4—41no—
163C2×C3×C4—42no—
164C2×C3×C4—93no—
165C2×C3×C4—912no—
166C2×C3×C4—96no—
167C2×C3×C4C2×D3216yes—
168C2×C3×C4C2×D3213yes—
169#67—924no—
170C2×Dic3—36no—
171C2×Dic3—312no—
172C2×Dic3—63no—
173C2×Dic3—66no—
174C2×Dic3—366no—
175D12—912no—
176C4×D3—324no—
177C4×D3—612no—
178C4×D3—3612no—
179Dic6—912no—
180C2×C32—23no—
181C2×C32—23no—
182C2×C32—23no—
183C2×C32—41no—
184C2×C32—41no—
185C2×C32—41no—
186C2×C32C22×C4×D313yes—
187C2×C32#100913yes—
188C2×C32#101013yes—
189C3⋊D3—26no—
190C3×D3—612no—
191C3×D3—126no—
192#41—278no—
193C2×Q8—271no—
194C2×D4—273no—
195C22×C4—93no—
196C22×C4—276no—
197C4⋊C4—93no—
198C4⋊C4—276no—
199C4⋊C4#111311yescharacteristic
200C22⋊C4—96no—
201C42—96no—
202C22×C3—23no—
203C22×C3—41no—
204C22×C3—96no—
205C22×C3—183no—
206C22×C3—183no—
207C22×C3C22×D3213yes—
208C2×D3—312no—
209C2×D3—66no—
210C2×D3—66no—
211C2×D3—66no—
212C2×D3—123no—
213C2×D3—123no—
214C3×C4—26no—
215C3×C4—26no—
216C3×C4—46no—
217C3×C4—42no—
218C3×C4—82no—
219C3×C4—96no—
220C3×C4—186no—
221C3×C4C4×D3216yes—
222Dic3—312no—
223Dic3—66no—
224Dic3—612no—
225Dic3—126no—
226C32—23no—
227C32—41no—
228C32#552313yescomplemented
229C23—93no—
230Q8—274no—
231D4—2712no—
232C2×C4—93no—
233C2×C4—912no—
234C2×C4—96no—
235C2×C4—1086no—
236C2×C4C2×#111312yes—
237C2×C4C2×#111311yescharacteristic
238C2×C3—23no—
239C2×C3—23no—
240C2×C3—23no—
241C2×C3—41no—
242C2×C3—41no—
243C2×C3—41no—
244C2×C3—186no—
245C2×C3C2×C4×D3213yes—
246C2×C3#6430913yes—
247C2×C3#6432013yes—
248D3—612no—
249D3—126no—
250C22—96no—
251C22—183no—
252C22—183no—
253C22C22×#111311yescharacteristic
254C4—22no—
255C4—96no—
256C4—186no—
257C4C4×#111312yes—
258C3—23no—
259C3—41no—
260C3#1059451113yescomplemented
261C2—186no—
262C2C2×C4×#111311yescharacteristic
263C2#1172462011yescharacteristic
264C2#1172462711yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutatorC2×C33C23×C4
FittingC33×C4⋊C4C22
socle, cosocleC22×C33C24
prime radicalC2×#89042C2
hypercenterC4⋊C4#1113
hypocenterC33#520
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#64101C2semidirect
3. C4×#6296C2semidirect
4. #11723689C2semidirect
5. #11723685C2semidirect
11. #89042C4semidirect
18. #64101D3semidirect
31. C3×C4×C3⋊D3C2×C4semidirect
32. #6296C2×C4semidirect
33. #6295C2×C4semidirect
87. C4×C3⋊D3C4×D3semidirect
156. C33#520semidirect
228. C32#5523semidirect
260. C3#10594511semidirect

Maximal subgroups

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

subgroupindex
C2×#890422
C3×#641012
C4×#62962
#117236892
#117236852
#105945113