Finite Group Databasebeta

← #11797934

Subgroups — #11797934

Order 1152.

311 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 311 conjugacy classes of subgroups — too many to draw. Only its 88 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×#5184C211yesmaximal, characteristic, complemented
2C3×#5179C211yesmaximal, characteristic
3#10590916C211yesmaximal, characteristic
4#10589431C211yesmaximal, characteristic, complemented
5#10589136C211yesmaximal, characteristic, complemented
6#10589123C211yesmaximal, characteristic
7#10589105C211yesmaximal, characteristic, complemented
8#71521—31nomaximal
9#70917—31nomaximal
10C2×C32×Q16C2211yescharacteristic
11C3×Q8×Dic3—21no—
12C3×#904C2211yescharacteristic
13#63814—21no—
14#63646C2211yescharacteristic
15#63639C2211yescharacteristic
16C3×#835—21no—
17C3×#829C412yescomplemented
18C3×#828C412yescomplemented
19C3×#817C2211yescharacteristic
20C3×#815C2211yescharacteristic
21C3×#812C2211yescharacteristic
22#63569—21no—
23C3×#510—31no—
24C3×#441—31no—
25#5197—31no—
26#5184D311yescharacteristic, complemented
27#5179D311yescharacteristic
28#4717—31no—
29#4566—31no—
30#4565—31no—
31#4565—31no—
32#4565—61no—
33#4513—31no—
34#4512—31no—
35#4488—31no—
36#4487—31no—
37#4466—31no—
38C2×C32×Q8—21no—
39C3×#235—21no—
40C32×Q16—21no—
41C32×Q16C2×C412yes—
42C2×C32×C8C2311yescharacteristic
43#3748—21no—
44C3×C4⋊Dic3—21no—
45C3×C4⋊Dic3—41no—
46C3×C4×Dic3—41no—
47C3×C4×Dic3D411yescharacteristic
48C2×C3×C3⋊C8D411yescharacteristic
49C3×Dic12C2×C411yescharacteristic
50C3×D24C2×C411yescharacteristic
51C3×#204—21no—
52#3711—41no—
53D3×C3⋊C8—41no—
54C3⋊(C3⋊C16)—21no—
55C3×C3⋊C16C2×C412yes—
56#2180—91noSylow
57C3×#135—31no—
58C2×C3×Q16—21no—
59C2×C3×Q16C2×D311yescharacteristic
60C2×C3×Q16C2×D311yescharacteristic
61C3×C4×Q8—61no—
62#948—61no—
63Q8×Dic3—21no—
64#904C2×D311yescharacteristic
65#901—61no—
66C3×Q32—32no—
67C3×SD32—32no—
68C3×M32—31no—
69C3×#106—31no—
70C3×#103—61no—
71C3×#97—31no—
72#837—61no—
73#835—21no—
74#829Dic312yescomplemented
75#828Dic312yescomplemented
76#819—31no—
77#819—31no—
78#819—61no—
79#817C2×D311yescharacteristic
80#815C2×D311yescharacteristic
81#812—31no—
82#812—31no—
83#812—61no—
84#812C2×D311yescharacteristic
85#805—61no—
86C32×Q8—22no—
87C32×Q8—41no—
88C2×C32×C4—41no—
89C2×C32×C4C2×D411yescharacteristic
90C3×#67—41no—
91C2×C3×Dic3—21no—
92C2×C3×Dic3—41no—
93C3×D12—21no—
94C3×C4×D3—41no—
95C3×Dic6—21no—
96C32×C8C22×C411yescharacteristic
97C32×C8#4111yescharacteristic
98C3⋊(C3⋊C8)—41no—
99C3×C3⋊C8—21no—
100#510—31no—
101#444—91no—
102#441—31no—
103#365—91no—
104#362—91no—
105#360—91no—
106#347—91no—
107C3×#41—61no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—41no—
111#235—21no—
112C3×Q16—21no—
113C3×Q16—21no—
114C3×Q16—22no—
115C3×Q16—31no—
116C3×Q16—41no—
117C3×Q16C4×D312yes—
118C3×Q16C2×Dic312yes—
119C3×SD16—61no—
120C3×D8—31no—
121C2×C3×C8—21no—
122C2×C3×C8—31no—
123C2×C3×C8C22×D311yescharacteristic
124C2×C3×C8C22×D311yescharacteristic
125C3×C4⋊C4—61no—
126C3×C4⋊C4—121no—
127C3×C42—31no—
128C3×C42—121no—
129C4⋊Dic3—21no—
130C4⋊Dic3—41no—
131C4×Dic3—41no—
132C4×Dic3#6711yescharacteristic
133#208—61no—
134#208—61no—
135#208—121no—
136#208—121no—
137C2×C3⋊C8—121no—
138C2×C3⋊C8#6711yescharacteristic
139Dic12C2×Dic311yescharacteristic
140D24C2×Dic311yescharacteristic
141#204—21no—
142ΓL(1,𝔽25)—121no—
143D3×C8—121no—
144C3×C16—32no—
145C3⋊C16—61no—
146C3⋊C16—61no—
147C3⋊C16—121no—
148C3⋊C16C2×Dic312yes—
149C22×C32AGL(1,ℤ/8)11yescharacteristic
150C2×C3×D3—41no—
151C32×C4—42no—
152C32×C4C4×D411yescharacteristic
153C32×C4AGL(1,ℤ/8)11yescharacteristic
154C3×Dic3—41no—
155C3×Dic3—41no—
156C3×Dic3—81no—
157#135—31no—
158C2×Q16D3211yescharacteristic
159#131—181no—
160C4×Q8—61no—
161Q32—32no—
162SD32—32no—
163M32—31no—
164M32—91no—
165#108—91no—
166#106—31no—
167#104—181no—
168#103—61no—
169#97—31no—
170C3×Q8—22no—
171C3×Q8—22no—
172C3×Q8—42no—
173C3×Q8—41no—
174C3×Q8—41no—
175C3×Q8—61no—
176C3×Q8—81no—
177C3×D4—61no—
178C3×D4—121no—
179C2×C3×C4—21no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4—61no—
183C2×C3×C4—81no—
184C2×C3×C4—121no—
185C2×C3×C4—121no—
186C2×C3×C4D3×D411yescharacteristic
187C2×C3×C4C2×#6711yescharacteristic
188#67—41no—
189C2×Dic3—21no—
190C2×Dic3—41no—
191D12—21no—
192C4×D3—41no—
193Dic6—21no—
194C3×C8—21no—
195C3×C8—21no—
196C3×C8—61no—
197C3×C8C2×C4×D311yescharacteristic
198C3×C8#23511yescharacteristic
199C3×C8#23711yescharacteristic
200C3×C8C22×Dic311yescharacteristic
201C3⋊C8—21no—
202C3⋊C8—121no—
203C3⋊C8—121no—
204C3⋊C8—241no—
205C2×C32—21no—
206C2×C32#44211yescharacteristic
207C3×D3—81no—
208#41—61no—
209C2×Q8—21no—
210Q16—21no—
211Q16—31no—
212Q16D3×Dic312yes—
213SD16—61no—
214D8—31no—
215M16—181no—
216M16—361no—
217C2×C8—31no—
218C2×C8—361no—
219C2×C8C2×D3211yescharacteristic
220C4⋊C4—61no—
221C4⋊C4—121no—
222C42—31no—
223C42—121no—
224C16—32no—
225C16—181no—
226C22×C3—21no—
227C22×C3—121no—
228C22×C3#91111yescharacteristic
229C22×C3#93211yescharacteristic
230C2×D3—41no—
231C3×C4—21no—
232C3×C4—21no—
233C3×C4—42no—
234C3×C4—42no—
235C3×C4—82no—
236C3×C4—121no—
237C3×C4—121no—
238C3×C4—241no—
239C3×C4#91111yescharacteristic
240C3×C4C4×#6711yescharacteristic
241C3×C4D4×Dic311yescharacteristic
242C3×C4#94911yescharacteristic
243Dic3—41no—
244Dic3—41no—
245Dic3—81no—
246C32#218011yescharacteristic, complemented, Sylow
247Q8—22no—
248Q8—41no—
249Q8—61no—
250D4—61no—
251D4—121no—
252C2×C4—41no—
253C2×C4—61no—
254C2×C4—121no—
255C2×C4—121no—
256C2×C4D3×#6711yescharacteristic
257C8—61no—
258C8—361no—
259C8C2×D3×Dic311yescharacteristic
260C8#380411yescharacteristic
261C2×C3—21no—
262C2×C3—21no—
263C2×C3—21no—
264C2×C3—41no—
265C2×C3—241no—
266C2×C3#501411yescharacteristic
267C2×C3#515211yescharacteristic
268D3—81no—
269C22—121no—
270C22#6393211yescharacteristic
271C4—42no—
272C4—121no—
273C4—121no—
274C4—241no—
275C4#6392511yescharacteristic
276C4Dic3×#6711yescharacteristic
277C3—21no—
278C3#7091711yescharacteristic, complemented
279C3#7152111yescharacteristic, complemented
280C2—21no—
281C2—241no—
282C2#1059143011yescharacteristic
subgroupG/H
centerC2#10591430
commutatorC32×C8C22×C4
FrattiniC2×C8C2×D32
FittingC2×C32×Q16C22
socleC2×C32#442
prime radicalC3×#904C22
cosocleC2×C32×C8C23
hypercenterC2×Q16D32
hypocenterC32#2180
LCSG ▷ C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×Q16

Complements

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

NH ≅ G/Ntype
1. C3×#5184C2semidirect
4. #10589431C2semidirect
5. #10589136C2semidirect
7. #10589105C2semidirect
17. C3×#829C4semidirect
18. C3×#828C4semidirect
26. #5184D3semidirect
74. #829Dic3semidirect
75. #828Dic3semidirect
246. C32#2180semidirect
278. C3#70917semidirect
279. C3#71521semidirect

Maximal subgroups

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

subgroupindex
C3×#51842
C3×#51792
#105909162
#105894312
#105891362
#105891232
#105891052
#715213
#709173