Finite Group Databasebeta

← #13166568

Subgroups — #13166568

Order 1728.

368 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 56 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#11724517C211yesmaximal, characteristic, complemented
2#11724251C211yesmaximal, characteristic, complemented
3#11723583C211yesmaximal, characteristic
4C3×#63785C211yesmaximal, characteristic, complemented
5#11722716C211yesmaximal, characteristic, complemented
6#11722711C211yesmaximal, characteristic, complemented
7#11722704C211yesmaximal, characteristic, complemented
8#10593784—31nomaximal
9C3×#3832C2211yescharacteristic
10#89014C2211yescharacteristic
11#88989—21no—
12#88989C2211yescharacteristic
13#88988—21no—
14#88988C2211yescharacteristic
15C3×C4×#157C2211yescharacteristic
16C3×#3788C2211yescharacteristic
17#88925—21no—
18D3×#622—21no—
19#88863C411yescharacteristic, complemented
20C32×#215C411yescharacteristic, complemented
21#88796C411yescharacteristic, complemented
22#88789C411yescharacteristic, complemented
23C3⋊D3×C3⋊C8C2211yescharacteristic
24#64289—31no—
25#64287—31no—
26#64125—31no—
27#63947—62no—
28#63785D311yescharacteristic, complemented
29#63783—31no—
30#63771—31no—
31#63767—31no—
32C2×C3×#157—21no—
33C33×Q8C2×C411yescharacteristic
34#6308C2×C411yescharacteristic
35C3×C4⋊(C3⋊D3)—21no—
36C3×C4⋊(C3⋊D3)C2×C411yescharacteristic
37C3×C4×C3⋊D3—21no—
38C3×C4×C3⋊D3C2311yescharacteristic
39C32×D12C2×C411yescharacteristic
40#6290—21no—
41D3×C3⋊Dic3—21no—
42C3⋊#622—21no—
43C3⋊#622D411yescharacteristic
44C3⋊#622D411yescharacteristic
45C3×#622—21no—
46C3⋊(C3⋊(C3⋊C8))C2×C411yescharacteristic
47C32×C3⋊C8C2×C411yescharacteristic
48#4823—91nomaximal
49#3832C2×D311yescharacteristic
50#3830—31no—
51C3×#239—62no—
52#3795—62no—
53#3792—61no—
54C2×#622—61no—
55C4×#157C2×D311yescharacteristic
56#3788—31no—
57#3788—61no—
58#3788C2×D311yescharacteristic
59#3787—31no—
60#3787—61no—
61C32×SD16—31no—
62#3756—31no—
63#3756—62no—
64#3756—122no—
65#3753—31no—
66#3744—31no—
67C8×C3⋊D3—31no—
68C3×#215—22no—
69#3716—62no—
70#3713—62no—
71D3×C3⋊C8—62no—
72C2×C3×C3⋊D3—21no—
73C2×C3×C3⋊D3#4111yescharacteristic
74C2×C32×D3—21no—
75C3×#157—41no—
76C33×C4—21no—
77C33×C4C22×C411yescharacteristic
78C3⋊(C3⋊Dic3)—21no—
79C3×C3⋊Dic3C2×D411yescharacteristic
80#950—91no—
81#948—91no—
82#916—182no—
83#907—91no—
84C3×#104—91no—
85#822—91no—
86#805—91no—
87#803—91no—
88C2×#157—21no—
89C32×Q8—22no—
90C32×Q8—42no—
91C32×Q8C4×D311yescharacteristic
92C32×D4—31no—
93#638—61no—
94C2×C3⋊Dic3—61no—
95C4⋊(C3⋊D3)—21no—
96C4⋊(C3⋊D3)C4×D311yescharacteristic
97C4×C3⋊D3—21no—
98C4×C3⋊D3C22×D311yescharacteristic
99#634—31no—
100#634—62no—
101#634—122no—
102C3×D12—22no—
103C3×D12—62no—
104C3×D12—122no—
105C3×C4×D3—62no—
106C3×C4×D3—122no—
107#625—122no—
108D3×Dic3—122no—
109#622—21no—
110#622—31no—
111#622—31no—
112#622—61no—
113C32×C8—31no—
114C3⋊(C3⋊C8)—31no—
115C3⋊(C3⋊C8)—62no—
116C3⋊(C3⋊C8)—122no—
117C3×C3⋊C8—22no—
118#444—271noSylow
119C2×C33C4×D411yescharacteristic
120C3×C3⋊D3—21no—
121C3×C3⋊D3—41no—
122C32×D3—41no—
123C3×#41—91no—
124#239—62no—
125#237—182no—
126#235—91no—
127C3×SD16—62no—
128C3×M16—91no—
129C3×C42—91no—
130#216—91no—
131#216—182no—
132#216—362no—
133#215#15711yescharacteristic, complemented
134#214—91no—
135#213—91no—
136#213—182no—
137#208—91no—
138#208—181no—
139C2×C3⋊C8—91no—
140C2×C3⋊C8—91no—
141C2×C3⋊C8—181no—
142#204—182no—
143ΓL(1,𝔽25)—181no—
144D3×C8—182no—
145D3×C8—181no—
146C22×C32—61no—
147C2×C3⋊D3—21no—
148C2×C3⋊D3#23711yescharacteristic
149C2×C3×D3—42no—
150C2×C3×D3—62no—
151C2×C3×D3—122no—
152#157—41no—
153C32×C4—22no—
154C32×C4—21no—
155C32×C4—42no—
156C32×C4—42no—
157C32×C4—82no—
158C32×C4C2×C4×D311yescharacteristic
159C3⋊Dic3—61no—
160C3⋊Dic3—122no—
161C3⋊Dic3—242no—
162C3⋊Dic3D3×D411yescharacteristic
163C3×Dic3—62no—
164#135—271no—
165#131—271no—
166#131—271no—
167#108—271no—
168#104—91no—
169#104—271no—
170#97—271no—
171C33#44411yescharacteristic, complemented, Sylow
172C3×Q8—22no—
173C3×Q8—42no—
174C3×Q8C2×#15711yescharacteristic
175C3×D4—62no—
176C3×D4—91no—
177C3×D4—181no—
178C2×C3×C4—91no—
179C2×C3×C4—181no—
180C2×C3×C4—181no—
181#67—181no—
182#67—362no—
183C2×Dic3—362no—
184D12—62no—
185D12—122no—
186D12C2×#15711yescharacteristic
187C4×D3—62no—
188C4×D3—122no—
189C4×D3—181no—
190Dic6—91no—
191Dic6—182no—
192Dic6—362no—
193C3×C8—62no—
194C3×C8—181no—
195C3⋊C8—91no—
196C3⋊C8—91no—
197C3⋊C8—91no—
198C3⋊C8—182no—
199C3⋊C8—181no—
200C3⋊C8—362no—
201C3⋊C8C2×#15711yescharacteristic
202C2×C32—22no—
203C2×C32—42no—
204C2×C32—121no—
205C2×C32#67⋊C411yescharacteristic
206C3⋊D3—21no—
207C3⋊D3—41no—
208C3×D3—82no—
209C3×D3—122no—
210C3×D3—242no—
211#41—91no—
212#41—271no—
213Q16—271no—
214SD16—31no—
215SD16—271no—
216D8—271no—
217M16—91no—
218M16—271no—
219M16—541no—
220M16—541no—
221C2×C8—271no—
222C2×C8—271no—
223C2×C8—541no—
224C2×C8—541no—
225C42—91no—
226C22×C3—91no—
227C22×C3—122no—
228C22×C3—181no—
229C2×D3—21no—
230C2×D3—62no—
231C2×D3—122no—
232C3×C4—21no—
233C3×C4—22no—
234C3×C4—42no—
235C3×C4—42no—
236C3×C4—82no—
237C3×C4—91no—
238C3×C4—361no—
239C3×C4C22×#15711yescharacteristic
240Dic3—62no—
241Dic3—181no—
242Dic3—362no—
243Dic3—722no—
244C32—22no—
245C32—42no—
246C32#482311yescharacteristic, complemented
247Q8—271no—
248Q8D3×#15711yescharacteristic
249D4—31no—
250D4—91no—
251D4—181no—
252D4—541no—
253C2×C4—91no—
254C2×C4—181no—
255C2×C4—181no—
256C2×C4—541no—
257C8—31no—
258C8—181no—
259C8—271no—
260C8—271no—
261C8—271no—
262C8—541no—
263C2×C3—22no—
264C2×C3—42no—
265C2×C3—181no—
266C2×C3—242no—
267C2×C3—361no—
268C2×C3D4×#15711yescharacteristic
269D3—41no—
270D3—122no—
271D3—242no—
272C22—61no—
273C22—91no—
274C22—181no—
275C4—21no—
276C4—91no—
277C4—361no—
278C4—541no—
279C4C2×D3×#15711yescharacteristic
280C3—22no—
281C3—42no—
282C3#1059378411yescharacteristic, complemented
283C2—121no—
284C2—181no—
285C2—361no—
286C2#67⋊#15711yescharacteristic
subgroupG/H
centerC2#67⋊#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC33×Q8C2×C4
socleC2×C33C4×D4
prime radical#11723583C2
cosocleC3×C4×C3⋊D3C23
hypercenterQ8D3×#157
hypocenterC33#444
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. #11724517C2semidirect
2. #11724251C2semidirect
4. C3×#63785C2semidirect
5. #11722716C2semidirect
6. #11722711C2semidirect
7. #11722704C2semidirect
19. #88863C4semidirect
20. C32×#215C4semidirect
21. #88796C4semidirect
22. #88789C4semidirect
28. #63785D3semidirect
133. #215#157semidirect
171. C33#444semidirect
246. C32#4823semidirect
282. C3#10593784semidirect

Maximal subgroups

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

subgroupindex
#117245172
#117242512
#117235832
C3×#637852
#117227162
#117227112
#117227042
#105937843
#48239