Finite Group Databasebeta

← #11797937

Subgroups — #11797937

Order 1152.

311 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 311 conjugacy classes of subgroups — too many to draw. Only its 78 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×#4894C211yesmaximal, characteristic
2#10590810C211yesmaximal, characteristic, complemented
3#10590638C211yesmaximal, characteristic
4#10589362C211yesmaximal, characteristic, complemented
5C3×#4494C211yesmaximal, characteristic, complemented
6#10589123C211yesmaximal, characteristic
7#10589105C211yesmaximal, characteristic, complemented
8#70928—31nomaximal
9#69309—31nomaximal
10C2×#3746C2211yescharacteristic
11C3×#907—21no—
12C3×#904C2211yescharacteristic
13#63844—21no—
14C32×#108C2211yescharacteristic
15#63639C2211yescharacteristic
16C3×#825—21no—
17C3×#817C2211yescharacteristic
18C3×#815C2211yescharacteristic
19C3×#812C2211yescharacteristic
20#63567—21no—
21#63551C412yescomplemented
22#63548C412yescomplemented
23C3×#444—31no—
24#5195—31no—
25#4909—31no—
26#4894D311yescharacteristic
27#4853—31no—
28#4716—31no—
29C3×#365—31no—
30#4566—31no—
31#4512—31no—
32#4496—31no—
33#4496—31no—
34#4496—61no—
35#4494D311yescharacteristic, complemented
36#4488—31no—
37#4466—31no—
38C2×#634—21no—
39C3×#235—21no—
40C32×M16—21no—
41C2×C32×C8C2311yescharacteristic
42#3746—21no—
43#3746C2×C412yes—
44C3×C4⋊Dic3—21no—
45C3×C4×Dic3D411yescharacteristic
46C2×C3×C3⋊C8D411yescharacteristic
47C3×Dic12C2×C411yescharacteristic
48C3×D24C2×C411yescharacteristic
49C3×#204—21no—
50C3×ΓL(1,𝔽25)—41no—
51C3×D3×C8—41no—
52#3724—41no—
53Dic32—41no—
54C3⋊(C3⋊C16)—21no—
55C3×C3⋊C16C2×C412yes—
56#2180—91noSylow
57C3×#135—31no—
58C3×#131—61no—
59#907—21no—
60C2×Dic12—31no—
61C2×Dic12—31no—
62C2×Dic12—61no—
63#904C2×D311yescharacteristic
64#887—61no—
65C4×Dic6—61no—
66C3×M32—31no—
67C3×#108—21no—
68C3×#108C2×D311yescharacteristic
69C3×#108C2×D311yescharacteristic
70C3×#106—31no—
71C3×#104—61no—
72C3×#97—31no—
73#829—32no—
74#827—32no—
75#825—21no—
76#817C2×D311yescharacteristic
77#816—61no—
78#815C2×D311yescharacteristic
79#812—31no—
80#812—31no—
81#812—61no—
82#812C2×D311yescharacteristic
83#810—61no—
84Dic24—32no—
85#800—32no—
86C2×C32×C4C2×D411yescharacteristic
87C2×C3⋊Dic3—41no—
88#634—22no—
89#634—41no—
90C3×#67—41no—
91C2×C3×Dic3—21no—
92C2×C3×Dic3—41no—
93C3×D12—21no—
94C3×C4×D3—41no—
95C3×Dic6—21no—
96C32×C8—41no—
97C32×C8C22×C411yescharacteristic
98C32×C8#4111yescharacteristic
99C3×C3⋊C8—21no—
100#510—91no—
101#444—31no—
102#441—91no—
103#365—31no—
104#362—91no—
105#360—91no—
106#347—91no—
107C3×#41—61no—
108#235—21no—
109C2×Dic6—61no—
110C2×Dic6—61no—
111C2×Dic6—121no—
112C3×Q16—31no—
113C3×SD16—61no—
114C3×D8—31no—
115C3×M16—21no—
116C3×M16—21no—
117C3×M16—41no—
118C3×M16—121no—
119C2×C3×C8—21no—
120C2×C3×C8—31no—
121C2×C3×C8—121no—
122C2×C3×C8C22×D311yescharacteristic
123C2×C3×C8C22×D311yescharacteristic
124C3×C4⋊C4—61no—
125C3×C42—31no—
126C4⋊Dic3—21no—
127Dic3⋊C4—121no—
128Dic3⋊C4—121no—
129C4×Dic3—121no—
130C4×Dic3—121no—
131C4×Dic3#6711yescharacteristic
132C2×C3⋊C8#6711yescharacteristic
133Dic12—32no—
134Dic12—32no—
135Dic12—62no—
136Dic12—61no—
137Dic12—61no—
138Dic12—121no—
139Dic12C4×D311yescharacteristic
140D24C4×D311yescharacteristic
141#204—21no—
142ΓL(1,𝔽25)—41no—
143D3×C8—41no—
144C3×C16—32no—
145C3⋊C16—61no—
146C3⋊C16—61no—
147C3⋊C16—121no—
148C3⋊C16C4×D312yes—
149C22×C32AGL(1,ℤ/8)11yescharacteristic
150C2×C3×D3—41no—
151C32×C4C4×D411yescharacteristic
152C32×C4AGL(1,ℤ/8)11yescharacteristic
153C3⋊Dic3—42no—
154C3×Dic3—41no—
155C3×Dic3—41no—
156C3×Dic3—81no—
157#135—31no—
158C2×Q16—91no—
159#131—61no—
160C4×Q8—181no—
161Q32—92no—
162SD32—92no—
163M32—31no—
164M32—91no—
165#108D3211yescharacteristic
166#106—31no—
167#104—61no—
168#103—181no—
169#97—31no—
170C3×Q8—61no—
171C3×D4—61no—
172C3×D4—121no—
173C2×C3×C4—21no—
174C2×C3×C4—61no—
175C2×C3×C4—121no—
176C2×C3×C4—121no—
177C2×C3×C4D3×D411yescharacteristic
178C2×C3×C4C2×#6711yescharacteristic
179#67—41no—
180C2×Dic3—21no—
181C2×Dic3—41no—
182C2×Dic3—121no—
183C2×Dic3—121no—
184C2×Dic3—241no—
185D12—21no—
186C4×D3—41no—
187Dic6—21no—
188Dic6—62no—
189Dic6—62no—
190Dic6—122no—
191Dic6—121no—
192Dic6—121no—
193Dic6—241no—
194C3×C8—21no—
195C3×C8—21no—
196C3×C8—41no—
197C3×C8—41no—
198C3×C8—61no—
199C3×C8—81no—
200C3×C8C2×C4×D311yescharacteristic
201C3×C8C2×C4×D311yescharacteristic
202C3×C8#23511yescharacteristic
203C3×C8#23911yescharacteristic
204C3⋊C8—21no—
205C2×C32—21no—
206C2×C32#44211yescharacteristic
207C3×D3—81no—
208#41—61no—
209C2×Q8—181no—
210Q16—31no—
211Q16—92no—
212Q16—181no—
213SD16—61no—
214D8—31no—
215M16—21no—
216M16—121no—
217C2×C8—31no—
218C2×C8—121no—
219C2×C8C2×D3211yescharacteristic
220C4⋊C4—61no—
221C4⋊C4—361no—
222C42—31no—
223C42—361no—
224C16—32no—
225C16—181no—
226C22×C3—21no—
227C22×C3—121no—
228C22×C3#91411yescharacteristic
229C22×C3#94911yescharacteristic
230C2×D3—41no—
231C3×C4—21no—
232C3×C4—21no—
233C3×C4—121no—
234C3×C4—121no—
235C3×C4—241no—
236C3×C4#89311yescharacteristic
237C3×C4#91411yescharacteristic
238C3×C4C4×#6711yescharacteristic
239C3×C4#93211yescharacteristic
240Dic3—41no—
241Dic3—41no—
242Dic3—81no—
243Dic3—122no—
244Dic3—122no—
245Dic3—242no—
246C32#218011yescharacteristic, complemented, Sylow
247Q8—61no—
248Q8—182no—
249Q8—361no—
250D4—61no—
251D4—121no—
252C2×C4—61no—
253C2×C4—121no—
254C2×C4—121no—
255C2×C4—361no—
256C2×C4D3×#6711yescharacteristic
257C8—41no—
258C8—61no—
259C8#379911yescharacteristic
260C8C4×D3211yescharacteristic
261C2×C3—21no—
262C2×C3—21no—
263C2×C3—21no—
264C2×C3—41no—
265C2×C3—241no—
266C2×C3#486911yescharacteristic
267C2×C3#501411yescharacteristic
268D3—81no—
269C22—121no—
270C22#6393911yescharacteristic
271C4—121no—
272C4—121no—
273C4—241no—
274C4—362no—
275C4#6390411yescharacteristic
276C4#6393411yescharacteristic
277C3—21no—
278C3#6930911yescharacteristic, complemented
279C3#7092811yescharacteristic, complemented
280C2—21no—
281C2—241no—
282C2#1059058911yescharacteristic
subgroupG/H
centerC2#10590589
commutatorC32×C8C22×C4
FrattiniC2×C8C2×D32
FittingC32×#108C22
socleC2×C32#442
prime radicalC3×#904C22
cosocleC2×C32×C8C23
hypercenter#108D32
hypocenterC32#2180
LCSG ▷ C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #108

Complements

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

NH ≅ G/Ntype
2. #10590810C2semidirect
4. #10589362C2semidirect
5. C3×#4494C2semidirect
7. #10589105C2semidirect
21. #63551C4semidirect
22. #63548C4semidirect
35. #4494D3semidirect
246. C32#2180semidirect
278. C3#69309semidirect
279. C3#70928semidirect

Maximal subgroups

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

subgroupindex
C3×#48942
#105908102
#105906382
#105893622
C3×#44942
#105891232
#105891052
#709283
#693093