Finite Group Databasebeta

← #13166542

Subgroups — #13166542

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×#64287C211yesmaximal, characteristic, complemented
2#11724252C211yesmaximal, characteristic, complemented
3#11723561C211yesmaximal, characteristic
4#11723108C211yesmaximal, characteristic, complemented
5#11722716C211yesmaximal, characteristic, complemented
6#11722712C211yesmaximal, characteristic, complemented
7#11722705C211yesmaximal, characteristic, complemented
8#10593784—31nomaximal
9C3×#3830C2211yescharacteristic
10#89018C2211yescharacteristic
11C4×C3⋊#157C2211yescharacteristic
12#88989C2211yescharacteristic
13C3×#3792—21no—
14C2×C3×#622—21no—
15C3×#3788C2211yescharacteristic
16C3×#3787C2211yescharacteristic
17#88926—21no—
18#88924—21no—
19#88860C411yescharacteristic, complemented
20C32×#214C411yescharacteristic, complemented
21#88798C411yescharacteristic, complemented
22#88797C411yescharacteristic, complemented
23C3⋊D3×C3⋊C8C2211yescharacteristic
24#64289—31no—
25#64287D311yescharacteristic, complemented
26#64125—31no—
27#63935—62no—
28#63785—31no—
29#63783—31no—
30#63771—31no—
31#63767—31no—
32C2×C3⋊#157—21no—
33C33×D4C2×C411yescharacteristic
34C4⋊(C3⋊(C3⋊D3))C2×C411yescharacteristic
35C3×#638—21no—
36C2×C3×C3⋊Dic3—21no—
37C3×C4×C3⋊D3C2311yescharacteristic
38C3×#634C2×C411yescharacteristic
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#5145—91nomaximal
49#3832—31no—
50#3830C2×D311yescharacteristic
51C3×#237—62no—
52#3799—62no—
53#3792—21no—
54C2×#622—21no—
55C4×#157—31no—
56#3788—31no—
57#3788—61no—
58#3788C2×D311yescharacteristic
59#3787—61no—
60#3787C2×D311yescharacteristic
61C32×SD16—31no—
62#3756—31no—
63#3753—31no—
64#3753—62no—
65#3753—122no—
66#3744—31no—
67C8×C3⋊D3—31no—
68C3×#214—22no—
69#3718—62no—
70#3717—62no—
71D3×C3⋊C8—62no—
72C22×C33—21no—
73C2×C3⋊(C3⋊D3)—21no—
74C2×C3×C3⋊D3#4111yescharacteristic
75C3⋊#157—41no—
76C33×C4C22×C411yescharacteristic
77C3×C3⋊Dic3—21no—
78C3×C3⋊Dic3C2×D411yescharacteristic
79C32×Dic3—21no—
80C3×#131—91no—
81#950—91no—
82#916—182no—
83#901—91no—
84#837—91no—
85#825—91no—
86#822—91no—
87#815—91no—
88C2×#157—61no—
89C32×Q8—31no—
90C32×D4—22no—
91C32×D4—42no—
92C32×D4C4×D311yescharacteristic
93#638—21no—
94C2×C3⋊Dic3—21no—
95C4⋊(C3⋊D3)—31no—
96C4⋊(C3⋊D3)—62no—
97C4⋊(C3⋊D3)—61no—
98C4⋊(C3⋊D3)—122no—
99C4×C3⋊D3—61no—
100C4×C3⋊D3C22×D311yescharacteristic
101#634C4×D311yescharacteristic
102C3×#67—122no—
103C2×C3×Dic3—122no—
104C3×C4×D3—62no—
105C3×Dic6—22no—
106C3×Dic6—62no—
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#444—271noSylow
119C2×C33—41no—
120C2×C33C4×D411yescharacteristic
121C3⋊(C3⋊D3)—41no—
122C3×C3⋊D3—21no—
123C3×#41—91no—
124#239—182no—
125#237—62no—
126#235—91no—
127C3×SD16—62no—
128C3×M16—91no—
129C3×M16—181no—
130C2×C3×C8—91no—
131C2×C3×C8—181no—
132#216—91no—
133#216—182no—
134#215—91no—
135#214#15711yescharacteristic, complemented
136#213—91no—
137#213—182no—
138#213—362no—
139C4×Dic3—91no—
140#208—91no—
141C2×C3⋊C8—91no—
142#204—182no—
143ΓL(1,𝔽25)—181no—
144D3×C8—182no—
145D3×C8—181no—
146C22×C32—21no—
147C22×C32—42no—
148C22×C32—82no—
149C2×C3⋊D3—61no—
150C2×C3⋊D3—122no—
151C2×C3⋊D3—242no—
152C2×C3⋊D3#23511yescharacteristic
153C2×C3×D3—62no—
154#157—121no—
155C32×C4—22no—
156C32×C4—42no—
157C32×C4—61no—
158C32×C4C2×C4×D311yescharacteristic
159C3⋊Dic3—21no—
160C3⋊Dic3C2×#6711yescharacteristic
161C3×Dic3—42no—
162C3×Dic3—62no—
163C3×Dic3—122no—
164#135—271no—
165#131—91no—
166#131—271no—
167#108—271no—
168#104—271no—
169#104—271no—
170#97—271no—
171C33#44411yescharacteristic, complemented, Sylow
172C3×Q8—62no—
173C3×Q8—91no—
174C3×D4—22no—
175C3×D4—42no—
176C3×D4—181no—
177C3×D4C2×#15711yescharacteristic
178C2×C3×C4—91no—
179C2×C3×C4—181no—
180#67—122no—
181#67—181no—
182C2×Dic3—122no—
183C2×Dic3—181no—
184D12—91no—
185D12—182no—
186D12—362no—
187D12—362no—
188C4×D3—62no—
189C4×D3—181no—
190C4×D3—362no—
191Dic6—62no—
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×C32—41no—
205C2×C32—82no—
206C2×C32—162no—
207C2×C32C4×#6711yescharacteristic
208C3⋊D3—21no—
209C3⋊D3—121no—
210C3⋊D3—242no—
211C3⋊D3—482no—
212C3×D3—122no—
213#41—91no—
214#41—271no—
215Q16—271no—
216SD16—31no—
217SD16—271no—
218D8—271no—
219M16—91no—
220M16—181no—
221M16—271no—
222M16—541no—
223C2×C8—91no—
224C2×C8—181no—
225C2×C8—271no—
226C2×C8—541no—
227C42—271no—
228C22×C3—21no—
229C22×C3—42no—
230C22×C3—82no—
231C22×C3—91no—
232C2×D3—62no—
233C2×D3—181no—
234C2×D3—362no—
235C2×D3—722no—
236C3×C4—22no—
237C3×C4—42no—
238C3×C4—91no—
239C3×C4—122no—
240C3×C4—181no—
241C3×C4C22×#15711yescharacteristic
242Dic3—21no—
243Dic3—62no—
244Dic3—122no—
245Dic3—361no—
246C32—22no—
247C32—42no—
248C32#514511yescharacteristic, complemented
249Q8—31no—
250Q8—91no—
251D4—181no—
252D4—271no—
253D4—541no—
254D4D3×#15711yescharacteristic
255C2×C4—91no—
256C2×C4—181no—
257C2×C4—541no—
258C2×C4—541no—
259C8—31no—
260C8—91no—
261C8—91no—
262C8—181no—
263C8—271no—
264C8—541no—
265C2×C3—22no—
266C2×C3—41no—
267C2×C3—42no—
268C2×C3—82no—
269C2×C3—162no—
270C2×C3—181no—
271C2×C3D4×#15711yescharacteristic
272D3—122no—
273D3—361no—
274D3—722no—
275D3—1442no—
276C22—21no—
277C22—91no—
278C22—541no—
279C4—61no—
280C4—91no—
281C4—181no—
282C4—1081no—
283C4C2×D3×#15711yescharacteristic
284C3—22no—
285C3—42no—
286C3#1059378411yescharacteristic, complemented
287C2—41no—
288C2—181no—
289C2—1081no—
290C2#67×#15711yescharacteristic
subgroupG/H
centerC2#67×#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC33×D4C2×C4
socleC2×C33C4×D4
prime radical#11723561C2
cosocleC3×C4×C3⋊D3C23
hypercenterD4D3×#157
hypocenterC33#444
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
1. C3×#64287C2semidirect
2. #11724252C2semidirect
4. #11723108C2semidirect
5. #11722716C2semidirect
6. #11722712C2semidirect
7. #11722705C2semidirect
19. #88860C4semidirect
20. C32×#214C4semidirect
21. #88798C4semidirect
22. #88797C4semidirect
25. #64287D3semidirect
135. #214#157semidirect
171. C33#444semidirect
248. C32#5145semidirect
286. C3#10593784semidirect

Maximal subgroups

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

subgroupindex
C3×#642872
#117242522
#117235612
#117231082
#117227162
#117227122
#117227052
#105937843
#51459