Finite Group Databasebeta

← #13166544

Subgroups — #13166544

Order 1728.

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

normal subgroups only

This group has 372 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#11724515C211yesmaximal, characteristic, complemented
2#11724251C211yesmaximal, characteristic, complemented
3#11723559C211yesmaximal, characteristic
4C3×#63783C211yesmaximal, characteristic, complemented
5#11722717C211yesmaximal, characteristic, complemented
6#11722711C211yesmaximal, characteristic, complemented
7#11722704C211yesmaximal, characteristic, complemented
8#10593780—31nomaximal
9C3×#3830C2211yescharacteristic
10#89014C2211yescharacteristic
11#88993—21no—
12C2×C3⋊#622—21no—
13#88989C2211yescharacteristic
14#88988C2211yescharacteristic
15C3×C4×#157C2211yescharacteristic
16C3×#3788C2211yescharacteristic
17#88925—21no—
18D3×#622—21no—
19#88861C411yescharacteristic, complemented
20C32×#213C411yescharacteristic, complemented
21#88800C411yescharacteristic, complemented
22#88792C411yescharacteristic, complemented
23C3⋊D3×C3⋊C8C2211yescharacteristic
24#64287—31no—
25#64287—31no—
26#64121—31no—
27#63933—62no—
28#63783—31no—
29#63783D311yescharacteristic, complemented
30#63770—31no—
31#63767—31no—
32C2×C3×#157—21no—
33C33×D4C2×C411yescharacteristic
34#6308C2×C411yescharacteristic
35C3×#638—21no—
36C2×C3×C3⋊Dic3—21no—
37C3×C4×C3⋊D3C2311yescharacteristic
38C3×#634C2×C411yescharacteristic
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#3830—31no—
50#3830C2×D311yescharacteristic
51C3×#237—62no—
52#3795—62no—
53#3792—61no—
54#3792—61no—
55C2×#622—61no—
56C2×#622—61no—
57C4×#157C2×D311yescharacteristic
58#3788—31no—
59#3788C2×D311yescharacteristic
60#3787—31no—
61C32×D8—31no—
62#3754—31no—
63#3754—31no—
64#3754—62no—
65#3754—122no—
66#3746—31no—
67C8×C3⋊D3—31no—
68C3×#213—22no—
69#3719—62no—
70#3715—62no—
71D3×C3⋊C8—62no—
72C22×C33—21no—
73C2×C3×C3⋊D3#4111yescharacteristic
74C2×C32×D3—21no—
75C3×#157—41no—
76C33×C4C22×C411yescharacteristic
77C3⋊(C3⋊Dic3)—21no—
78C3×C3⋊Dic3—21no—
79C3×C3⋊Dic3C2×D411yescharacteristic
80#950—91no—
81#948—91no—
82#912—182no—
83#907—91no—
84C3×#104—91no—
85#822—91no—
86#805—91no—
87#803—91no—
88C2×#157—21no—
89C32×D4—22no—
90C32×D4—31no—
91C32×D4—42no—
92C32×D4C4×D311yescharacteristic
93#638—21no—
94#638—61no—
95C2×C3⋊Dic3—21no—
96C2×C3⋊Dic3—61no—
97C4×C3⋊D3C22×D311yescharacteristic
98#634—31no—
99#634—62no—
100#634—122no—
101#634C4×D311yescharacteristic
102C3×#67—122no—
103C2×C3×Dic3—122no—
104C3×D12—22no—
105C3×C4×D3—62no—
106C3×Dic6—62no—
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×C33—41no—
120C2×C33C4×D411yescharacteristic
121C3×C3⋊D3—21no—
122C32×D3—41no—
123C3×#41—91no—
124#237—62no—
125#237—182no—
126#235—91no—
127C3×D8—62no—
128C3×M16—91no—
129C3×C42—91no—
130#216—91no—
131#215—91no—
132#214—91no—
133#214—182no—
134#214—182no—
135#214—362no—
136#213#15711yescharacteristic, complemented
137#208—91no—
138#208—181no—
139C2×C3⋊C8—91no—
140C2×C3⋊C8—91no—
141C2×C3⋊C8—181no—
142Dic12—182no—
143ΓL(1,𝔽25)—181no—
144D3×C8—182no—
145D3×C8—181no—
146C22×C32—21no—
147C22×C32—42no—
148C22×C32—61no—
149C22×C32—82no—
150C2×C3⋊D3#23711yescharacteristic
151C2×C3×D3—42no—
152C2×C3×D3—62no—
153#157—41no—
154C32×C4—22no—
155C32×C4—42no—
156C32×C4C2×C4×D311yescharacteristic
157C3⋊Dic3—21no—
158C3⋊Dic3—61no—
159C3⋊Dic3—122no—
160C3⋊Dic3—242no—
161C3⋊Dic3D3×D411yescharacteristic
162C3×Dic3—62no—
163C3×Dic3—122no—
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—91no—
173C3×D4—22no—
174C3×D4—42no—
175C3×D4—62no—
176C3×D4—181no—
177C3×D4C2×#15711yescharacteristic
178C2×C3×C4—91no—
179C2×C3×C4—181no—
180C2×C3×C4—181no—
181#67—122no—
182#67—181no—
183#67—362no—
184C2×Dic3—122no—
185C2×Dic3—362no—
186D12C2×#15711yescharacteristic
187C4×D3—62no—
188C4×D3—181no—
189Dic6—62no—
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—41no—
205C2×C32—82no—
206C2×C32—121no—
207C2×C32—162no—
208C2×C32#67⋊C411yescharacteristic
209C3⋊D3—21no—
210C3×D3—82no—
211C3×D3—122no—
212#41—91no—
213#41—271no—
214Q16—271no—
215SD16—271no—
216SD16—271no—
217D8—31no—
218M16—91no—
219M16—271no—
220M16—541no—
221M16—541no—
222C2×C8—271no—
223C2×C8—271no—
224C2×C8—541no—
225C2×C8—541no—
226C42—91no—
227C22×C3—21no—
228C22×C3—42no—
229C22×C3—82no—
230C22×C3—91no—
231C22×C3—122no—
232C2×D3—21no—
233C2×D3—62no—
234C3×C4—22no—
235C3×C4—42no—
236C3×C4—91no—
237C3×C4—181no—
238C3×C4—361no—
239C3×C4C22×#15711yescharacteristic
240Dic3—62no—
241Dic3—122no—
242Dic3—181no—
243Dic3—362no—
244Dic3—722no—
245C32—22no—
246C32—42no—
247C32#482311yescharacteristic, complemented
248Q8—91no—
249Q8—271no—
250D4—31no—
251D4—181no—
252D4—541no—
253D4D3×#15711yescharacteristic
254C2×C4—91no—
255C2×C4—181no—
256C2×C4—181no—
257C2×C4—541no—
258C8—31no—
259C8—181no—
260C8—271no—
261C8—271no—
262C8—271no—
263C8—541no—
264C2×C3—22no—
265C2×C3—41no—
266C2×C3—42no—
267C2×C3—82no—
268C2×C3—162no—
269C2×C3—181no—
270C2×C3—242no—
271C2×C3D4×#15711yescharacteristic
272D3—41no—
273D3—122no—
274C22—21no—
275C22—61no—
276C22—91no—
277C4—91no—
278C4—181no—
279C4—361no—
280C4—541no—
281C4C2×D3×#15711yescharacteristic
282C3—22no—
283C3—42no—
284C3#1059378011yescharacteristic, complemented
285C2—41no—
286C2—121no—
287C2—181no—
288C2#67⋊#15711yescharacteristic
subgroupG/H
centerC2#67⋊#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC33×D4C2×C4
socleC2×C33C4×D4
prime radical#11723559C2
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. #11724515C2semidirect
2. #11724251C2semidirect
4. C3×#63783C2semidirect
5. #11722717C2semidirect
6. #11722711C2semidirect
7. #11722704C2semidirect
19. #88861C4semidirect
20. C32×#213C4semidirect
21. #88800C4semidirect
22. #88792C4semidirect
29. #63783D3semidirect
136. #213#157semidirect
171. C33#444semidirect
247. C32#4823semidirect
284. C3#10593780semidirect

Maximal subgroups

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

subgroupindex
#117245152
#117242512
#117235592
C3×#637832
#117227172
#117227112
#117227042
#105937803
#48239