Finite Group Databasebeta

← #13178796

Subgroups — #13178796

Order 1728.

1532 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1532 conjugacy classes of subgroups — too many to draw. Only its 167 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
1C3×#64162C213yesmaximal, complemented
2#11723728C211yesmaximal, characteristic, complemented
3#11723725C216yesmaximal, complemented
4#11723724C213yesmaximal, complemented
5#11723716C212yesmaximal, complemented
6#10595505—33nomaximal
7C24×C33C2211yescharacteristic
8C2×C3×#638C2213yes—
9C2×C3×#638C2216yes—
10C2×#6295C2213yescomplemented
11C2×#6294C2213yescomplemented
12C3×#3757C2216yes—
13C3×#3757C2213yes—
14#88815C2216yescomplemented
15#88814C2213yescomplemented
16#88813C2211yescharacteristic, complemented
17#64162D313yescomplemented
18C3×#953—36no—
19C3×#953—63no—
20#63979—33no—
21#63975—33no—
22#63971—36no—
23#63970—36no—
24#63969—36no—
25#63958—36no—
26#63956—36no—
27C23×C33—43no—
28C23×C33C2312yes—
29C23×C33C2311yescharacteristic
30C22×C3×C3⋊D3C2313yes—
31C3×#638—46no—
32C3×#638—43no—
33C2×C3×C3⋊Dic3C2316yes—
34C2×C3×C3⋊Dic3C2313yes—
35#6295—43no—
36#6294—43no—
37#5595—93no—
38C24×C32—23no—
39C24×C32—41no—
40C24×C32C2×D313yes—
41C2×#638C2×D316yes—
42C2×#638C2×D313yes—
43C2×C3×#67—36no—
44C2×C3×#67—36no—
45C2×C3×#67—36no—
46C2×C3×#67—63no—
47C2×C3×#67—66no—
48C2×#626—36no—
49C2×#625—33no—
50C2×#624—33no—
51C2×D3×Dic3—36no—
52#3757C2×D316yes—
53#3757C2×D313yes—
54C3×#217—36no—
55C3×#217—36no—
56C3×#217—36no—
57C3×#217—66no—
58C3×#217—63no—
59#3724—36no—
60Dic3⋊Dic3—36no—
61Dic3⋊Dic3—36no—
62#3722—33no—
63(C2×D3)⋊Dic3—36no—
64Dic32—33no—
65C22×C33—26no—
66C22×C33—23no—
67C22×C33—44no—
68C22×C33C2411yescharacteristic
69C2×C3×C3⋊D3—23no—
70C2×C3×C3⋊D3—26no—
71C3×C3⋊Dic3—26no—
72C3×C3⋊Dic3—23no—
73C3×#120—93no—
74#953—36no—
75#953—63no—
76#940—93no—
77#939—96no—
78#938—96no—
79#935—93no—
80#884—96no—
81#883—96no—
82#879—96no—
83C23×C32—26no—
84C23×C32—23no—
85C23×C32—42no—
86C23×C32—41no—
87C23×C32—46no—
88C23×C32—43no—
89C23×C32—412no—
90C23×C32—46no—
91C23×C32—412no—
92C23×C32C22×D316yes—
93C23×C32C22×D313yes—
94C22×C3⋊D3C22×D313yes—
95C22×C3×D3—36no—
96C22×C3×D3—63no—
97#638—46no—
98#638—43no—
99C2×C3⋊Dic3C22×D313yes—
100C2×C3⋊Dic3C22×D316yes—
101C3×#67—126no—
102C3×#67—126no—
103C3×#67—126no—
104C3×#67—1212no—
105C3×#67—126no—
106C2×C3×Dic3—36no—
107C2×C3×Dic3—36no—
108C2×C3×Dic3—36no—
109C2×C3×Dic3—63no—
110C2×C3×Dic3—66no—
111#626—126no—
112#625—123no—
113#624—123no—
114D3×Dic3—126no—
115#560—271noSylow
116C2×C33—42no—
117C2×C33—41no—
118C2×C33H(5,𝔽2)13yes—
119C3×C3⋊D3—43no—
120C24×C3—23no—
121C24×C3—41no—
122C24×C3D3213yes—
123C2×C3×D4—93no—
124C2×C3×D4—96no—
125C2×#67—36no—
126C2×#67—36no—
127C2×#67—36no—
128C2×#67—66no—
129C2×#67—63no—
130C2×#67—96no—
131C22×Dic3—93no—
132C2×D12—93no—
133C2×C4×D3—96no—
134C3×C22⋊C4—96no—
135C3×C22⋊C4—93no—
136#217—36no—
137#217—36no—
138#217—36no—
139#217—66no—
140#217—63no—
141#217—93no—
142(C2×D3)⋊C4—96no—
143C4⋊Dic3—96no—
144Dic3⋊C4—96no—
145Dic3⋊C4—96no—
146C4×Dic3—93no—
147C22×C32—26no—
148C22×C32—23no—
149C22×C32—212no—
150C22×C32—26no—
151C22×C32—26no—
152C22×C32—26no—
153C22×C32—26no—
154C22×C32—23no—
155C22×C32—41no—
156C22×C32—46no—
157C22×C32—46no—
158C22×C32—46no—
159C22×C32—412no—
160C22×C32—46no—
161C22×C32—424no—
162C22×C32—416no—
163C22×C32—412no—
164C22×C32C23×D313yes—
165C2×C3⋊D3—23no—
166C2×C3⋊D3—26no—
167C2×C3×D3—66no—
168C2×C3×D3—66no—
169C2×C3×D3—66no—
170C2×C3×D3—123no—
171C2×C3×D3—126no—
172C3⋊Dic3—26no—
173C3⋊Dic3—23no—
174C3×Dic3—66no—
175C3×Dic3—66no—
176C3×Dic3—66no—
177C3×Dic3—126no—
178C3×Dic3—123no—
179#127—271no—
180#126—272no—
181#123—273no—
182#121—276no—
183#120—93no—
184C33#56011yescharacteristic, complemented, Sylow
185C23×C3—26no—
186C23×C3—23no—
187C23×C3—46no—
188C23×C3—43no—
189C23×C3—412no—
190C23×C3—46no—
191C23×C3—412no—
192C23×C3—42no—
193C23×C3—41no—
194C23×C3—93no—
195C23×C3C2×D3216yes—
196C23×C3C2×D3213yes—
197C22×D3—36no—
198C22×D3—63no—
199C3×D4—366no—
200C3×D4—363no—
201C2×C3×C4—93no—
202C2×C3×C4—96no—
203#67—126no—
204#67—126no—
205#67—126no—
206#67—1212no—
207#67—126no—
208#67—366no—
209C2×Dic3—36no—
210C2×Dic3—36no—
211C2×Dic3—36no—
212C2×Dic3—63no—
213C2×Dic3—66no—
214C2×Dic3—363no—
215D12—363no—
216C4×D3—366no—
217C2×C32—26no—
218C2×C32—23no—
219C2×C32—43no—
220C2×C32—412no—
221C2×C32—46no—
222C2×C32—48no—
223C2×C32—44no—
224C2×C32—46no—
225C2×C32—43no—
226C2×C32#100416yes—
227C2×C32#100413yes—
228C3⋊D3—43no—
229C3×D3—126no—
230C3×D3—243no—
231C24#111311yescharacteristic
232C2×D4—96no—
233C2×D4—93no—
234C2×D4—273no—
235C22×C4—273no—
236C4⋊C4—276no—
237C22⋊C4—96no—
238C22⋊C4—93no—
239C22⋊C4—273no—
240C42—271no—
241C22×C3—212no—
242C22×C3—26no—
243C22×C3—26no—
244C22×C3—26no—
245C22×C3—26no—
246C22×C3—26no—
247C22×C3—23no—
248C22×C3—23no—
249C22×C3—46no—
250C22×C3—46no—
251C22×C3—46no—
252C22×C3—412no—
253C22×C3—46no—
254C22×C3—424no—
255C22×C3—416no—
256C22×C3—412no—
257C22×C3—41no—
258C22×C3—183no—
259C22×C3—186no—
260C22×C3C22×D3213yes—
261C2×D3—66no—
262C2×D3—66no—
263C2×D3—66no—
264C2×D3—123no—
265C2×D3—126no—
266C3×C4—186no—
267C3×C4—183no—
268Dic3—66no—
269Dic3—66no—
270Dic3—66no—
271Dic3—126no—
272Dic3—123no—
273C32—23no—
274C32—41no—
275C32#559513yescomplemented
276C23—43no—
277C23—93no—
278C23C2×#111312yes—
279C23C2×#111311yescharacteristic
280D4—366no—
281D4—363no—
282D4—1083no—
283C2×C4—93no—
284C2×C4—96no—
285C2×C4—1083no—
286C2×C3—26no—
287C2×C3—23no—
288C2×C3—412no—
289C2×C3—46no—
290C2×C3—48no—
291C2×C3—44no—
292C2×C3—46no—
293C2×C3—43no—
294C2×C3—43no—
295C2×C3—363no—
296C2×C3#6433016yes—
297C2×C3#6433013yes—
298D3—126no—
299D3—243no—
300C22—26no—
301C22—23no—
302C22—44no—
303C22—183no—
304C22—186no—
305C22C22×#111311yescharacteristic
306C4—186no—
307C4—183no—
308C3—23no—
309C3—41no—
310C3#1059550513yescomplemented
311C2—42no—
312C2—41no—
313C2—363no—
314C2#1172463313yes—
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC24×C33C22
hypercenterC24#1113
hypocenterC33#560
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C3×#64162C2semidirect
2. #11723728C2semidirect
3. #11723725C2semidirect
4. #11723724C2semidirect
5. #11723716C2semidirect
10. C2×#6295C22semidirect
11. C2×#6294C22semidirect
14. #88815C22semidirect
15. #88814C22semidirect
16. #88813C22semidirect
17. #64162D3semidirect
184. C33#560semidirect
275. C32#5595semidirect
310. C3#10595505semidirect

Maximal subgroups

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

subgroupindex
C3×#641622
#117237282
#117237252
#117237242
#117237162
#105955053