Finite Group Databasebeta

← #13166484

Subgroups — #13166484

Order 1728.

488 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 68 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×#88925C211yesmaximal, characteristic, complemented
2C2×D3×#622C211yesmaximal, characteristic, complemented
3D12×C3⋊Dic3C211yesmaximal, characteristic
4#11723102C211yesmaximal, characteristic, complemented
5C3×#63779C211yesmaximal, characteristic, complemented
6(C2×D3)⋊#622C211yesmaximal, characteristic, complemented
7(C2×D3)⋊#622C211yesmaximal, characteristic, complemented
8#10593820—31nomaximal
9C2×C32×D12C411yescharacteristic
10C2×D3×C3⋊Dic3C2211yescharacteristic
11C2×D3×C3⋊Dic3C2211yescharacteristic
12C2×C3⋊#622C2211yescharacteristic
13C2×C3⋊#622C2211yescharacteristic
14C2×C3×#622C2211yescharacteristic
15C2×C3×#622C2211yescharacteristic
16#88925—22no—
17D3×#622—22no—
18C4⋊(C3⋊(C3⋊Dic3))C411yescharacteristic
19C3×C4×C3⋊Dic3C2211yescharacteristic
20(C2×D3)⋊(C3⋊Dic3)C411yescharacteristic
21(C2×D3)⋊(C3⋊Dic3)C411yescharacteristic
22C2×#3792—31no—
23C22×#622—31no—
24D4×C3⋊Dic3—31no—
25Dic3×D12—62no—
26C22⋊#622—31no—
27C22⋊#622—31no—
28#63779D311yescharacteristic, complemented
29#63777—31no—
30C22×C32×D3C2×C411yescharacteristic
31C22×C32×D3C2×C411yescharacteristic
32C2×C33×C4C2×C411yescharacteristic
33C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
34C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
35C2×C3×C3⋊Dic3C2×C411yescharacteristic
36C2×C3×C3⋊Dic3C2311yescharacteristic
37C32×D12—22no—
38D3×C3⋊Dic3—21no—
39D3×C3⋊Dic3—21no—
40D3×C3⋊Dic3—21no—
41D3×C3⋊Dic3—21no—
42C3⋊#622—21no—
43C3⋊#622D412yescomplemented
44C3×#622—21no—
45C3×#622—21no—
46#4836—91nomaximal
47C2×C32×D4—31no—
48C22×C3⋊Dic3—31no—
49C22×C3⋊Dic3—31no—
50C2×C3×D12—22no—
51C2×D3×Dic3—62no—
52C2×D3×Dic3—62no—
53#3792—62no—
54C2×#622—31no—
55C2×#622—31no—
56C2×#622—62no—
57C2×#622C2×D311yescharacteristic
58C2×#622C2×D311yescharacteristic
59#3757—31no—
60#3757—31no—
61C4⋊(C3⋊Dic3)—31no—
62C4⋊(C3⋊Dic3)—62no—
63C4⋊(C3⋊Dic3)—122no—
64C4×C3⋊Dic3C2×D311yescharacteristic
65C3×C4×Dic3—62no—
66(C2×D3)⋊Dic3—62no—
67(C2×D3)⋊Dic3—62no—
68C22×C33C22×C411yescharacteristic
69C2×C32×D3—21no—
70C2×C32×D3—21no—
71C2×C32×D3—21no—
72C2×C32×D3—21no—
73C2×C32×D3—21no—
74C2×C32×D3M1612yescomplemented
75C33×C4—21no—
76C3⋊(C3⋊Dic3)—21no—
77C3⋊(C3⋊Dic3)—21no—
78C3×C3⋊Dic3—21no—
79C3×C3⋊Dic3C2×D411yescharacteristic
80C3×C3⋊Dic3#4111yescharacteristic
81D4×Dic3—182no—
82C2×ΓL(1,𝔽25)—91no—
83C2×D3×C8—91no—
84C4×D12—91no—
85C3×C4⋊C8—91no—
86(C2×D3)⋊C8—91no—
87(C2×D3)⋊C8—91no—
88#803—91no—
89C23×C32—31no—
90C23×C32—31no—
91C22×C3×D3—22no—
92C22×C3×D3—22no—
93C32×D4—62no—
94C2×C32×C4—22no—
95C2×C32×C4—42no—
96C2×C32×C4C4×D311yescharacteristic
97C2×C3⋊Dic3—31no—
98C2×C3⋊Dic3—31no—
99C2×C3⋊Dic3—62no—
100C2×C3⋊Dic3—62no—
101C2×C3⋊Dic3—61no—
102C2×C3⋊Dic3—61no—
103C2×C3⋊Dic3—61no—
104C2×C3⋊Dic3—61no—
105C2×C3⋊Dic3—122no—
106C2×C3⋊Dic3—122no—
107C2×C3⋊Dic3C4×D311yescharacteristic
108C2×C3⋊Dic3C22×D311yescharacteristic
109C2×C3×Dic3—62no—
110C2×C3×Dic3—62no—
111C3×D12—28no—
112D3×Dic3—122no—
113D3×Dic3—122no—
114D3×Dic3—122no—
115D3×Dic3—122no—
116#622—21no—
117#622—21no—
118#622—32no—
119#622—61no—
120#435—271noSylow
121C2×C33C4×D411yescharacteristic
122C2×C33C2×M1611yescharacteristic
123C2×C33#13111yescharacteristic
124C32×D3—22no—
125C32×D3—41no—
126C2×C3×D4—62no—
127C22×Dic3—182no—
128C22×Dic3—182no—
129C2×D12#15711yescharacteristic
130C2×C4×D3—91no—
131C2×C4×D3—91no—
132C2×C3×C8—91no—
133C2×C3×C8—91no—
134C3×C42—91no—
135#217—182no—
136#217—182no—
137(C2×D3)⋊C4—91no—
138(C2×D3)⋊C4—91no—
139C4⋊Dic3—91no—
140C4⋊Dic3—182no—
141C4⋊Dic3—362no—
142C4×Dic3—62no—
143C2×C3⋊C8—91no—
144C2×C3⋊C8—91no—
145ΓL(1,𝔽25)—182no—
146D3×C8—182no—
147C22×C32—22no—
148C22×C32—32no—
149C22×C32—42no—
150C22×C32—61no—
151C22×C32—61no—
152C22×C32—61no—
153C22×C32—61no—
154C22×C32—61no—
155C22×C32C2×C4×D311yescharacteristic
156C2×C3×D3—24no—
157C2×C3×D3—24no—
158C2×C3×D3—42no—
159C2×C3×D3—42no—
160C2×C3×D3—42no—
161C2×C3×D3—42no—
162C32×C4—21no—
163C32×C4—24no—
164C32×C4—44no—
165C3⋊Dic3—21no—
166C3⋊Dic3—61no—
167C3⋊Dic3—61no—
168C3⋊Dic3—122no—
169C3⋊Dic3—122no—
170C3⋊Dic3—242no—
171C3⋊Dic3—242no—
172C3⋊Dic3D3×D411yescharacteristic
173C3⋊Dic3#23911yescharacteristic
174C3×Dic3—62no—
175C3×Dic3—62no—
176C3×Dic3—122no—
177C2×M16—271no—
178C22×C8—271no—
179C4×D4—271no—
180C4⋊C8—91no—
181C22⋊C8—271no—
182C22⋊C8—271no—
183#97—271no—
184C33#43511yescharacteristic, complemented, Sylow
185C23×C3—62no—
186C23×C3—62no—
187C22×D3C2×#15711yescharacteristic
188C22×D3C2×#15711yescharacteristic
189C3×D4—68no—
190C2×C3×C4—22no—
191C2×C3×C4—42no—
192C2×C3×C4—91no—
193C2×C3×C4—91no—
194C2×C3×C4C2×#15711yescharacteristic
195C2×Dic3—62no—
196C2×Dic3—62no—
197C2×Dic3—91no—
198C2×Dic3—91no—
199C2×Dic3—182no—
200C2×Dic3—182no—
201C2×Dic3—362no—
202C2×Dic3—362no—
203C2×Dic3—362no—
204C2×Dic3—362no—
205C2×Dic3—362no—
206C2×Dic3—362no—
207D12—22no—
208C4×D3—181no—
209C4×D3—181no—
210C4×D3—181no—
211C4×D3—181no—
212C3×C8—181no—
213C3×C8—181no—
214C3⋊C8—92no—
215C3⋊C8—181no—
216C2×C32—22no—
217C2×C32—22no—
218C2×C32—22no—
219C2×C32—42no—
220C2×C32—42no—
221C2×C32—42no—
222C2×C32—62no—
223C2×C32—121no—
224C2×C32#89311yescharacteristic
225C2×C32#90111yescharacteristic
226C2×C32D3×M1611yescharacteristic
227C3×D3—44no—
228C3×D3—44no—
229C2×D4—31no—
230C22×C4—271no—
231C22×C4—271no—
232M16—542no—
233C2×C8—91no—
234C2×C8—91no—
235C2×C8—271no—
236C2×C8—271no—
237C2×C8—542no—
238C4⋊C4—271no—
239C22⋊C4—271no—
240C22⋊C4—271no—
241C42—91no—
242C22×C3—22no—
243C22×C3—42no—
244C22×C3—64no—
245C22×C3—64no—
246C22×C3—122no—
247C22×C3—122no—
248C22×C3—122no—
249C22×C3—122no—
250C22×C3C22×#15711yescharacteristic
251C2×D3—21no—
252C2×D3—21no—
253C2×D3—21no—
254C2×D3—21no—
255C2×D3—21no—
256C2×D3#379212yescomplemented
257C3×C4—24no—
258C3×C4—21no—
259C3×C4—44no—
260C3×C4—91no—
261C3×C4—91no—
262C3×C4—181no—
263Dic3—62no—
264Dic3—62no—
265Dic3—122no—
266Dic3—181no—
267Dic3—181no—
268Dic3—362no—
269Dic3—362no—
270Dic3—722no—
271Dic3—722no—
272C32—22no—
273C32—42no—
274C32#483611yescharacteristic, complemented
275C23—31no—
276C23—31no—
277D4—62no—
278C2×C4—91no—
279C2×C4—91no—
280C2×C4—271no—
281C2×C4—271no—
282C2×C4—541no—
283C2×C4—541no—
284C2×C4—541no—
285C2×C4—541no—
286C2×C4D3×#15711yescharacteristic
287C8—181no—
288C8—181no—
289C8—272no—
290C8—541no—
291C2×C3—22no—
292C2×C3—22no—
293C2×C3—22no—
294C2×C3—42no—
295C2×C3—42no—
296C2×C3—42no—
297C2×C3—124no—
298C2×C3—124no—
299C2×C3#6428711yescharacteristic
300C2×C3D4×#15711yescharacteristic
301C2×C3C2×#379211yescharacteristic
302D3—22no—
303D3—41no—
304C22—32no—
305C22—61no—
306C22—61no—
307C22—61no—
308C22—61no—
309C22—61no—
310C22C2×D3×#15711yescharacteristic
311C4—21no—
312C4—91no—
313C4—91no—
314C4—181no—
315C4—541no—
316C4—541no—
317C3—22no—
318C3—42no—
319C3#1059382011yescharacteristic, complemented
320C2—62no—
321C2—121no—
322C2#1172425011yescharacteristic
323C2#1172425911yescharacteristic
324C2D3×#379211yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC2×C33×C4C2×C4
prime radicalC2×C32×D12C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D3×#157
hypocenterC33#435
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88925C2semidirect
2. C2×D3×#622C2semidirect
4. #11723102C2semidirect
5. C3×#63779C2semidirect
6. (C2×D3)⋊#622C2semidirect
7. (C2×D3)⋊#622C2semidirect
28. #63779D3semidirect
43. C3⋊#622D4semidirect
74. C2×C32×D3M16semidirect
184. C33#435semidirect
256. C2×D3#3792semidirect
274. C32#4836semidirect
319. C3#10593820semidirect

Maximal subgroups

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

subgroupindex
C2×#889252
C2×D3×#6222
D12×C3⋊Dic32
#117231022
C3×#637792
(C2×D3)⋊#6222
#105938203
#48369