Finite Group Databasebeta

← #13166565

Subgroups — #13166565

Order 1728.

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

normal subgroups only

This group has 376 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
1C3×#64289C211yesmaximal, characteristic, complemented
2#11724252C211yesmaximal, characteristic, complemented
3#11723585C211yesmaximal, characteristic
4#11723110C211yesmaximal, characteristic, complemented
5#11722716C211yesmaximal, characteristic, complemented
6#11722712C211yesmaximal, characteristic, complemented
7#11722703C211yesmaximal, characteristic, complemented
8#10593787—31nomaximal
9C3×#3832C2211yescharacteristic
10#89018C2211yescharacteristic
11C4×C3⋊#157C2211yescharacteristic
12#88989C2211yescharacteristic
13C3×#3788—21no—
14C3×#3788C2211yescharacteristic
15C3×#3787—21no—
16C3×#3787C2211yescharacteristic
17#88926—21no—
18#88924—21no—
19#88862C411yescharacteristic, complemented
20C32×#216C411yescharacteristic, complemented
21#88793C411yescharacteristic, complemented
22#88791C411yescharacteristic, complemented
23C3⋊D3×C3⋊C8C2211yescharacteristic
24#64289—31no—
25#64289D311yescharacteristic, complemented
26#64128—31no—
27#63949—62no—
28#63785—31no—
29#63785—31no—
30#63771—31no—
31C8×#157—31no—
32C2×C3⋊#157—21no—
33C33×Q8C2×C411yescharacteristic
34C4⋊(C3⋊(C3⋊D3))C2×C411yescharacteristic
35C3×C4⋊(C3⋊D3)—21no—
36C3×C4⋊(C3⋊D3)C2×C411yescharacteristic
37C3×C4×C3⋊D3—21no—
38C3×C4×C3⋊D3C2311yescharacteristic
39C32×Dic6C2×C411yescharacteristic
40#6292—21no—
41#6289—21no—
42C3⋊#622—21no—
43C3×#622—21no—
44C3×#622D411yescharacteristic
45C3×#622D411yescharacteristic
46C3⋊(C3⋊(C3⋊C8))C2×C411yescharacteristic
47C32×C3⋊C8C2×C411yescharacteristic
48#5144—91nomaximal
49#3832—31no—
50#3832C2×D311yescharacteristic
51C3×#239—62no—
52#3799—62no—
53C4×#157—31no—
54#3788—21no—
55#3788—31no—
56#3788—61no—
57#3788C2×D311yescharacteristic
58#3787—21no—
59#3787—61no—
60#3787C2×D311yescharacteristic
61C32×Q16—31no—
62#3755—31no—
63#3755—31no—
64#3755—62no—
65#3755—122no—
66#3745—31no—
67C8×C3⋊D3—31no—
68C3×#216—22no—
69#3715—62no—
70#3714—62no—
71D3×C3⋊C8—62no—
72C2×C3⋊(C3⋊D3)—21no—
73C2×C3×C3⋊D3—21no—
74C2×C3×C3⋊D3#4111yescharacteristic
75C3⋊#157—22no—
76C33×C4—21no—
77C33×C4C22×C411yescharacteristic
78C3×C3⋊Dic3C2×D411yescharacteristic
79C32×Dic3—21no—
80C3×#131—91no—
81#950—91no—
82#919—182no—
83#901—91no—
84#837—91no—
85#825—91no—
86#822—91no—
87C8×Dic3—91no—
88C2×#157—61no—
89C32×Q8—22no—
90C32×Q8—31no—
91C32×Q8—42no—
92C32×Q8C4×D311yescharacteristic
93C4⋊(C3⋊D3)—21no—
94C4⋊(C3⋊D3)—31no—
95C4⋊(C3⋊D3)—62no—
96C4⋊(C3⋊D3)—61no—
97C4⋊(C3⋊D3)—122no—
98C4⋊(C3⋊D3)C4×D311yescharacteristic
99C4×C3⋊D3—21no—
100C4×C3⋊D3—61no—
101C4×C3⋊D3C22×D311yescharacteristic
102C3×D12—62no—
103C3×D12—122no—
104C3×C4×D3—62no—
105C3×C4×D3—122no—
106C3×Dic6—22no—
107#626—122no—
108#624—122no—
109#622—21no—
110#622—61no—
111#622#6711yescharacteristic
112#622#6711yescharacteristic
113C32×C8—31no—
114C3⋊(C3⋊C8)—31no—
115C3⋊(C3⋊C8)—62no—
116C3⋊(C3⋊C8)—122no—
117C3×C3⋊C8—22no—
118#443—271noSylow
119C2×C33C4×D411yescharacteristic
120C3⋊(C3⋊D3)—41no—
121C3×C3⋊D3—21no—
122C3×C3⋊D3—41no—
123C3×#41—91no—
124#239—62no—
125#239—182no—
126#235—91no—
127C3×Q16—62no—
128C3×M16—91no—
129C3×M16—181no—
130C2×C3×C8—91no—
131C2×C3×C8—181no—
132#216#15711yescharacteristic, complemented
133#215—91no—
134#215—182no—
135#215—182no—
136#215—362no—
137#214—91no—
138#213—91no—
139C4×Dic3—91no—
140#208—91no—
141C2×C3⋊C8—91no—
142D24—182no—
143ΓL(1,𝔽25)—181no—
144D3×C8—182no—
145D3×C8—181no—
146C2×C3⋊D3—21no—
147C2×C3⋊D3—61no—
148C2×C3⋊D3—122no—
149C2×C3⋊D3—242no—
150C2×C3⋊D3#23511yescharacteristic
151C2×C3×D3—62no—
152C2×C3×D3—122no—
153#157—62no—
154C32×C4—22no—
155C32×C4—21no—
156C32×C4—42no—
157C32×C4—42no—
158C32×C4—61no—
159C32×C4—82no—
160C32×C4C2×C4×D311yescharacteristic
161C3⋊Dic3C2×#6711yescharacteristic
162C3×Dic3—42no—
163C3×Dic3—62no—
164#135—271no—
165#131—91no—
166#131—271no—
167#108—271no—
168#104—271no—
169#104—271no—
170C4×C8—271no—
171C33#44311yescharacteristic, complemented, Sylow
172C3×Q8—22no—
173C3×Q8—42no—
174C3×Q8—62no—
175C3×Q8C2×#15711yescharacteristic
176C3×D4—91no—
177C3×D4—181no—
178C2×C3×C4—91no—
179C2×C3×C4—181no—
180#67—181no—
181C2×Dic3—181no—
182D12—62no—
183D12—91no—
184D12—122no—
185D12—182no—
186D12—362no—
187D12—362no—
188C4×D3—62no—
189C4×D3—122no—
190C4×D3—181no—
191C4×D3—362no—
192Dic6C2×#15711yescharacteristic
193C3×C8—62no—
194C3×C8—91no—
195C3×C8—91no—
196C3×C8—181no—
197C3⋊C8—91no—
198C3⋊C8—182no—
199C3⋊C8—181no—
200C3⋊C8—362no—
201C3⋊C8C2×#15711yescharacteristic
202C2×C32—22no—
203C2×C32—42no—
204C2×C32C4×#6711yescharacteristic
205C3⋊D3—21no—
206C3⋊D3—41no—
207C3⋊D3—121no—
208C3⋊D3—242no—
209C3⋊D3—482no—
210C3×D3—122no—
211C3×D3—242no—
212#41—91no—
213#41—271no—
214Q16—31no—
215SD16—271no—
216SD16—271no—
217D8—271no—
218M16—91no—
219M16—181no—
220M16—271no—
221M16—541no—
222C2×C8—91no—
223C2×C8—181no—
224C2×C8—271no—
225C2×C8—541no—
226C42—271no—
227C22×C3—91no—
228C22×C3—181no—
229C2×D3—62no—
230C2×D3—122no—
231C2×D3—181no—
232C2×D3—362no—
233C2×D3—722no—
234C3×C4—21no—
235C3×C4—22no—
236C3×C4—42no—
237C3×C4—42no—
238C3×C4—82no—
239C3×C4—91no—
240C3×C4—122no—
241C3×C4C22×#15711yescharacteristic
242Dic3—21no—
243Dic3—62no—
244Dic3—182no—
245C32—22no—
246C32—42no—
247C32#514411yescharacteristic, complemented
248Q8—31no—
249Q8D3×#15711yescharacteristic
250D4—91no—
251D4—181no—
252D4—271no—
253D4—541no—
254C2×C4—91no—
255C2×C4—181no—
256C2×C4—541no—
257C2×C4—541no—
258C8—31no—
259C8—91no—
260C8—91no—
261C8—181no—
262C8—271no—
263C8—541no—
264C2×C3—22no—
265C2×C3—42no—
266C2×C3—181no—
267C2×C3—361no—
268C2×C3D4×#15711yescharacteristic
269D3—122no—
270D3—242no—
271D3—361no—
272D3—722no—
273D3—1442no—
274C22—91no—
275C22—181no—
276C22—541no—
277C4—21no—
278C4—61no—
279C4—91no—
280C4—542no—
281C4C2×D3×#15711yescharacteristic
282C3—22no—
283C3—42no—
284C3#1059378711yescharacteristic, complemented
285C2—181no—
286C2—361no—
287C2—1081no—
288C2#67×#15711yescharacteristic
subgroupG/H
centerC2#67×#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC33×Q8C2×C4
socleC2×C33C4×D4
prime radical#11723585C2
cosocleC3×C4×C3⋊D3C23
hypercenterQ8D3×#157
hypocenterC33#443
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. C3×#64289C2semidirect
2. #11724252C2semidirect
4. #11723110C2semidirect
5. #11722716C2semidirect
6. #11722712C2semidirect
7. #11722703C2semidirect
19. #88862C4semidirect
20. C32×#216C4semidirect
21. #88793C4semidirect
22. #88791C4semidirect
25. #64289D3semidirect
132. #216#157semidirect
171. C33#443semidirect
247. C32#5144semidirect
284. C3#10593787semidirect

Maximal subgroups

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

subgroupindex
C3×#642892
#117242522
#117235852
#117231102
#117227162
#117227122
#117227032
#105937873
#51449