Finite Group Databasebeta

← #13169192

Subgroups — #13169192

Order 1728.

1430 conjugacy classes of subgroups; only the 252 normal subgroups are shown.

normal subgroups only

This group has 1430 conjugacy classes of subgroups — too many to draw. Only its 252 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
2#11723404C213yesmaximal, complemented
3C3×#63900C213yesmaximal, complemented
4#11722945C212yesmaximal, complemented
5#11722931C216yesmaximal, complemented
6#10594509—33nomaximal
7C2×C3×C4×C3⋊D3C2213yes—
8C2×#6296C2211yescharacteristic
9C2×#6295C2213yes—
10#89042C418yescomplemented
11C2×C3×#624C2216yes—
12C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
13C32×C4⋊Dic3C2213yes—
14#88803C2216yes—
15Dic3×C3⋊Dic3C2216yes—
16C3×#3724C2216yes—
17C2×#3798—33no—
18#64101—33no—
19C3×D3×C4⋊C4—36no—
20#63904—312no—
21C4×#625—36no—
22#63900D313yescomplemented
23#63841—36no—
24#63841—63no—
25#63838—36no—
26C22×C3×C3⋊D3C2313yes—
27C2×C33×C4C2311yescharacteristic
28C2×C3⋊(C3⋊Dic3)C2312yes—
29C2×C3×C3⋊Dic3C2313yes—
30C3×C4×C3⋊D3C2×C4112yescomplemented
31C2×C32×Dic3C2316yes—
32#6296C2×C414yescomplemented
33#6295C2×C4112yescomplemented
34C3×#624—46no—
35#5504—93no—
36C2×C4×C3⋊D3C2×D313yes—
37C2×C3×C4×D3—36no—
38C2×C3×C4×D3—312no—
39C2×C3×C4×D3—63no—
40C2×#627—33no—
41C2×#626—36no—
42C2×#625—33no—
43C2×#624C2×D316yes—
44#3798—324no—
45C32×C4⋊C4—33no—
46#3752—36no—
47C4⋊(C3⋊Dic3)—33no—
48C4⋊(C3⋊Dic3)—63no—
49C4⋊(C3⋊Dic3)—121no—
50C4×C3⋊Dic3—36no—
51C3×C4⋊Dic3—23no—
52C3×C4⋊Dic3C2×D316yes—
53C3×Dic3⋊C4—312no—
54#3724C2×D316yes—
55(C2×D3)⋊Dic3—312no—
56(C2×D3)⋊Dic3—66no—
57Dic32—312no—
58Dic32—66no—
59C22×C33C2411yescharacteristic
60C2×C3×C3⋊D3—23no—
61C2×C3×C3⋊D3—23no—
62C2×C3×C3⋊D3C22×C416yes—
63C33×C4C22×C412yes—
64C3⋊(C3⋊Dic3)—22no—
65C3×C3⋊Dic3C22×C416yes—
66C32×Dic3—26no—
67C2×#235—93no—
68C2×C3×C4⋊C4—93no—
69C4×#67—912no—
70#926—93no—
71#893—96no—
72#892—96no—
73#892—183no—
74D3×C4⋊C4—36no—
75#887—96no—
76C22×C3⋊D3C22×D313yes—
77C22×C3×D3—36no—
78C22×C3×D3—63no—
79C2×C32×C4—23no—
80C2×C32×C4—36no—
81C2×C32×C4—41no—
82C2×C32×C4C22×D313yes—
83C2×C3⋊Dic3—36no—
84C2×C3⋊Dic3—66no—
85C2×C3⋊Dic3—122no—
86C2×C3⋊Dic3C22×D313yes—
87C4×C3⋊D3C4×D3112yescomplemented
88C2×C3×Dic3—26no—
89C2×C3×Dic3—36no—
90C2×C3×Dic3—63no—
91C2×C3×Dic3C22×D3112yes—
92C3×C4×D3—324no—
93C3×C4×D3—612no—
94C3×C4×D3—1212no—
95#627—312no—
96#626—324no—
97#625—312no—
98#624—46no—
99#520—271noSylow
100C2×C33C23×C411yescharacteristic
101C2×C33H(5,𝔽2)11yescharacteristic
102C2×C332−1+411yescharacteristic
103C3×C3⋊D3—26no—
104C22×C3×C4—96no—
105C22×C3×C4—93no—
106C2×#67—96no—
107#235—924no—
108C2×D12—93no—
109C2×C4×D3—36no—
110C2×C4×D3—312no—
111C2×C4×D3—63no—
112C2×Dic6—93no—
113C3×C4⋊C4—36no—
114C3×C4⋊C4—63no—
115C3×C4⋊C4—96no—
116#217—96no—
117(C2×D3)⋊C4—912no—
118(C2×D3)⋊C4—186no—
119C4⋊Dic3—93no—
120C4⋊Dic3—183no—
121C4⋊Dic3—361no—
122C4⋊Dic3D3213yes—
123Dic3⋊C4—312no—
124C4×Dic3—912no—
125C4×Dic3—96no—
126C4×Dic3—186no—
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—42no—
141C32×C4—66no—
142C32×C4C2×C4×D316yes—
143C3⋊Dic3—66no—
144C3⋊Dic3—126no—
145C3⋊Dic3—242no—
146C3⋊Dic3C2×C4×D316yes—
147C3×Dic3—212no—
148C3×Dic3—312no—
149C3×Dic3—46no—
150C3×Dic3—66no—
151C2×#41—271no—
152C4×Q8—272no—
153C4×D4—276no—
154#117—273no—
155C2×C4⋊C4—93no—
156C33#52011yescharacteristic, complemented, Sylow
157C23×C3—93no—
158C22×D3—36no—
159C22×D3—63no—
160C2×C3×C4—23no—
161C2×C3×C4—312no—
162C2×C3×C4—41no—
163C2×C3×C4—66no—
164C2×C3×C4—93no—
165C2×C3×C4—912no—
166C2×C3×C4—366no—
167C2×C3×C4C2×D3213yes—
168#67—924no—
169C2×Dic3—36no—
170C2×Dic3—63no—
171C2×Dic3—96no—
172C2×Dic3—186no—
173C2×Dic3—362no—
174C2×Dic3C2×D3216yes—
175D12—912no—
176C4×D3—324no—
177C4×D3—612no—
178C4×D3—1212no—
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#100413yes—
188C2×C32#100713yes—
189C3⋊D3—26no—
190C3×D3—612no—
191C3×D3—126no—
192#41—278no—
193C2×Q8—271no—
194C2×D4—273no—
195C22×C4—96no—
196C22×C4—93no—
197C4⋊C4—33no—
198C4⋊C4—96no—
199C4⋊C4—271no—
200C22⋊C4—276no—
201C42—276no—
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—42no—
216C3×C4—612no—
217C3×C4—96no—
218C3×C4—126no—
219C3×C4C4×D3216yes—
220Dic3—26no—
221Dic3—312no—
222Dic3—66no—
223Dic3—186no—
224Dic3—366no—
225Dic3—722no—
226C32—23no—
227C32—41no—
228C32#550413yescomplemented
229C23—93no—
230Q8—274no—
231D4—2712no—
232C2×C4—36no—
233C2×C4—93no—
234C2×C4—912no—
235C2×C4—272no—
236C2×C4—366no—
237C2×C4D3311yescharacteristic
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#6431313yes—
247C2×C3#6431613yes—
248D3—612no—
249D3—126no—
250C22—96no—
251C22—183no—
252C22—183no—
253C22C2×D3311yescharacteristic
254C4—66no—
255C4—96no—
256C4—542no—
257C4#8894912yes—
258C3—23no—
259C3—41no—
260C3#1059450913yescomplemented
261C2—186no—
262C2#1172438211yescharacteristic
263C2#1172439911yescharacteristic
264C2C2×#8894911yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutatorC2×C33C23×C4
FittingC2×C33×C4C23
socle, cosocleC22×C33C24
prime radicalC2×#89042C2
hypercenterC2×C4D33
hypocenterC33#520
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723404C2semidirect
3. C3×#63900C2semidirect
4. #11722945C2semidirect
5. #11722931C2semidirect
10. #89042C4semidirect
22. #63900D3semidirect
30. C3×C4×C3⋊D3C2×C4semidirect
32. #6296C2×C4semidirect
33. #6295C2×C4semidirect
87. C4×C3⋊D3C4×D3semidirect
156. C33#520semidirect
228. C32#5504semidirect
260. C3#10594509semidirect

Maximal subgroups

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

subgroupindex
C2×#890422
#117234042
C3×#639002
#117229452
#117229312
#105945093