Finite Group Databasebeta

← #13166269

Subgroups — #13166269

Order 1728.

372 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 58 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#11724252C211yesmaximal, characteristic, complemented
2#11724250C211yesmaximal, characteristic, complemented
3#11723338C211yesmaximal, characteristic
4#11723096C211yesmaximal, characteristic, complemented
5C3×#63767C211yesmaximal, characteristic, complemented
6#11722713C211yesmaximal, characteristic, complemented
7#11722711C211yesmaximal, characteristic, complemented
8#10593784—31nomaximal
9#89018C2211yescharacteristic
10#89014C2211yescharacteristic
11#88989C2211yescharacteristic
12#88989C2211yescharacteristic
13C3×C4×#157C2211yescharacteristic
14C3×#3787C2211yescharacteristic
15#88926—21no—
16#88925—21no—
17#88924—21no—
18D3×#622—21no—
19#88851C411yescharacteristic, complemented
20C3×C8×C3⋊D3C2211yescharacteristic
21C32×#204C411yescharacteristic, complemented
22#88799C411yescharacteristic, complemented
23#88790C411yescharacteristic, complemented
24#64289—31no—
25#64287—31no—
26#64125—31no—
27#63806—62no—
28#63785—31no—
29#63783—31no—
30#63771—31no—
31#63767D311yescharacteristic, complemented
32C2×C3×#157—21no—
33C4⋊(C3⋊(C3⋊D3))C2×C411yescharacteristic
34#6308C2×C411yescharacteristic
35C3×C4×C3⋊D3C2311yescharacteristic
36C32×D12C2×C411yescharacteristic
37C32×Dic6C2×C411yescharacteristic
38#6292—21no—
39#6290—21no—
40#6289—21no—
41D3×C3⋊Dic3—21no—
42C3⋊#622—21no—
43C3⋊#622—21no—
44C3×#622D411yescharacteristic
45C3×#622D411yescharacteristic
46C33×C8C2×C411yescharacteristic
47C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
48#4717—91nomaximal
49#3832—31no—
50#3830—31no—
51#3799—62no—
52#3795—62no—
53#3792—61no—
54C2×#622—61no—
55C4×#157C2×D311yescharacteristic
56#3788—31no—
57#3788—31no—
58#3788—61no—
59#3787—61no—
60#3787C2×D311yescharacteristic
61C32×SD16—31no—
62#3756—31no—
63#3753—31no—
64#3744—31no—
65#3744—62no—
66#3744—122no—
67C8×C3⋊D3C2×D311yescharacteristic
68C3×#204—22no—
69C3×D3×C8—62no—
70#3719—62no—
71#3714—62no—
72C2×C3⋊(C3⋊D3)—21no—
73C2×C3×C3⋊D3#4111yescharacteristic
74C2×C32×D3—21no—
75C3×#157—41no—
76C33×C4C22×C411yescharacteristic
77C3⋊(C3⋊Dic3)—21no—
78C3×C3⋊Dic3C2×D411yescharacteristic
79C32×Dic3—21no—
80#916—182no—
81#904—91no—
82#901—91no—
83#901—91no—
84C3×#97—91no—
85#819—91no—
86#805—91no—
87#805—91no—
88C2×#157—21no—
89C32×Q8—31no—
90C32×D4—31no—
91#638—61no—
92C2×C3⋊Dic3—61no—
93C4⋊(C3⋊D3)—31no—
94C4⋊(C3⋊D3)—62no—
95C4⋊(C3⋊D3)—61no—
96C4⋊(C3⋊D3)—122no—
97C4×C3⋊D3—61no—
98C4×C3⋊D3C22×D311yescharacteristic
99#634—31no—
100#634—62no—
101#634—122no—
102C3×D12—22no—
103C3×C4×D3—62no—
104C3×Dic6—22no—
105#626—122no—
106#625—122no—
107#624—122no—
108D3×Dic3—122no—
109#622—61no—
110#622—61no—
111#622D1211yescharacteristic
112#622D1211yescharacteristic
113C32×C8—22no—
114C32×C8—42no—
115C32×C8C4×D311yescharacteristic
116C3⋊(C3⋊C8)C4×D311yescharacteristic
117C3×C3⋊C8—62no—
118#444—271noSylow
119C2×C33C4×D411yescharacteristic
120C3⋊(C3⋊D3)—41no—
121C3×C3⋊D3—21no—
122C32×D3—41no—
123#239—182no—
124#237—182no—
125#235—91no—
126#235—91no—
127C3×SD16—62no—
128C2×C3×C8—91no—
129C2×C3×C8—91no—
130C3×C42—91no—
131#216—182no—
132#213—182no—
133#208—91no—
134#208—91no—
135Dic12—91no—
136D24—91no—
137#204—91no—
138#204—182no—
139#204—362no—
140#204#15711yescharacteristic, complemented
141ΓL(1,𝔽25)—181no—
142ΓL(1,𝔽25)—181no—
143D3×C8—62no—
144D3×C8—181no—
145D3×C8—181no—
146C22×C32—61no—
147C2×C3⋊D3—61no—
148C2×C3⋊D3—122no—
149C2×C3⋊D3—242no—
150C2×C3⋊D3#23511yescharacteristic
151C2×C3×D3—42no—
152C2×C3×D3—62no—
153#157—41no—
154C32×C4—22no—
155C32×C4—42no—
156C32×C4—61no—
157C32×C4C2×C4×D311yescharacteristic
158C3⋊Dic3—61no—
159C3⋊Dic3—122no—
160C3⋊Dic3—242no—
161C3⋊Dic3C2×D1211yescharacteristic
162C3×Dic3—42no—
163C3×Dic3—62no—
164#135—271no—
165#131—271no—
166#131—271no—
167#108—271no—
168#104—271no—
169#104—271no—
170#97—91no—
171C33#44411yescharacteristic, complemented, Sylow
172C3×Q8—62no—
173C3×D4—62no—
174C2×C3×C4—91no—
175C2×C3×C4—181no—
176#67—181no—
177#67—181no—
178#67—362no—
179C2×Dic3—362no—
180D12—91no—
181D12—182no—
182D12—362no—
183D12—362no—
184D12C2×#15711yescharacteristic
185C4×D3—62no—
186C4×D3—181no—
187C4×D3—181no—
188C4×D3—362no—
189Dic6—91no—
190Dic6—182no—
191Dic6—362no—
192Dic6C2×#15711yescharacteristic
193C3×C8—22no—
194C3×C8—42no—
195C3×C8—91no—
196C3×C8—91no—
197C3×C8—91no—
198C3×C8C2×#15711yescharacteristic
199C3⋊C8—62no—
200C3⋊C8—181no—
201C3⋊C8—181no—
202C2×C32—22no—
203C2×C32—42no—
204C2×C32—121no—
205C2×C32C4×D1211yescharacteristic
206C3⋊D3—21no—
207C3⋊D3—121no—
208C3⋊D3—242no—
209C3⋊D3—482no—
210C3×D3—82no—
211C3×D3—122no—
212#41—271no—
213#41—271no—
214Q16—271no—
215SD16—31no—
216SD16—271no—
217D8—271no—
218M16—271no—
219M16—271no—
220M16—541no—
221M16—541no—
222C2×C8—91no—
223C2×C8—91no—
224C2×C8—541no—
225C2×C8—541no—
226C42—91no—
227C22×C3—91no—
228C22×C3—122no—
229C2×D3—21no—
230C2×D3—62no—
231C2×D3—181no—
232C2×D3—362no—
233C2×D3—722no—
234C3×C4—22no—
235C3×C4—42no—
236C3×C4—91no—
237C3×C4—122no—
238C3×C4—361no—
239C3×C4C22×#15711yescharacteristic
240Dic3—21no—
241Dic3—62no—
242Dic3—181no—
243Dic3—362no—
244Dic3—722no—
245C32—22no—
246C32—42no—
247C32#471711yescharacteristic, complemented
248Q8—31no—
249Q8—271no—
250D4—31no—
251D4—271no—
252D4—541no—
253D4—541no—
254C2×C4—91no—
255C2×C4—181no—
256C2×C4—541no—
257C2×C4—541no—
258C8—91no—
259C8—91no—
260C8—91no—
261C8—541no—
262C8—541no—
263C8D3×#15711yescharacteristic
264C2×C3—22no—
265C2×C3—42no—
266C2×C3—181no—
267C2×C3—242no—
268C2×C3D4×#15711yescharacteristic
269D3—41no—
270D3—122no—
271D3—361no—
272D3—722no—
273D3—1442no—
274C22—61no—
275C22—91no—
276C22—541no—
277C4—61no—
278C4—91no—
279C4—361no—
280C4—541no—
281C4C2×D3×#15711yescharacteristic
282C3—22no—
283C3—42no—
284C3#1059378411yescharacteristic, complemented
285C2—121no—
286C2—181no—
287C2—1081no—
288C2D12×#15711yescharacteristic
subgroupG/H
centerC2D12×#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC33×C8C2×C4
socleC2×C33C4×D4
prime radical#11723338C2
cosocleC3×C4×C3⋊D3C23
hypercenterC8D3×#157
hypocenterC33#444
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
1. #11724252C2semidirect
2. #11724250C2semidirect
4. #11723096C2semidirect
5. C3×#63767C2semidirect
6. #11722713C2semidirect
7. #11722711C2semidirect
19. #88851C4semidirect
21. C32×#204C4semidirect
22. #88799C4semidirect
23. #88790C4semidirect
31. #63767D3semidirect
140. #204#157semidirect
171. C33#444semidirect
247. C32#4717semidirect
284. C3#10593784semidirect

Maximal subgroups

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

subgroupindex
#117242522
#117242502
#117233382
#117230962
C3×#637672
#117227132
#117227112
#105937843
#47179