Finite Group Databasebeta

← #11797765

Subgroups — #11797765

Order 1152.

347 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 96 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
1C2×C3×#800C211yesmaximal, characteristic
2#10590818C211yesmaximal, characteristic, complemented
3#10590777C211yesmaximal, characteristic
4#10589368C211yesmaximal, characteristic, complemented
5C3×#4504C211yesmaximal, characteristic, complemented
6#10589117C211yesmaximal, characteristic, complemented
7#10589114C211yesmaximal, characteristic
8#71474—31nomaximal
9#69319—31nomaximal
10C2×C3×Dic12C2211yescharacteristic
11C2×C3×D24C2211yescharacteristic
12#63898—21no—
13#63843—21no—
14C2×C32×C16C2211yescharacteristic
15C8×C3⋊Dic3C2211yescharacteristic
16C2×C3⋊(C3⋊C16)C2211yescharacteristic
17C3×#818C2211yescharacteristic
18C3×#818C2211yescharacteristic
19C3×#800C414yescomplemented
20#63565—21no—
21#63562—21no—
22C2×C3×SD32—31no—
23Dic3×Q16—31no—
24Dic3×D8—31no—
25C2×#800D311yescharacteristic
26#4873—31no—
27#4872—31no—
28C3×#366—31no—
29#4564—31no—
30#4562—31no—
31#4504D311yescharacteristic, complemented
32#4502—31no—
33#4502—31no—
34#4502—61no—
35#4492—31no—
36#4491—31no—
37C2×C3×D12—21no—
38C2×C3×Dic6—21no—
39C2×D3×Dic3—41no—
40C2×C32×C8C2311yescharacteristic
41C4×C3⋊Dic3D411yescharacteristic
42C2×C3⋊(C3⋊C8)D411yescharacteristic
43C3×C4⋊Dic3—21no—
44C3×C4⋊Dic3—21no—
45C3×Dic12—21no—
46C3×Dic12C2×C412yescomplemented
47C3×D24—21no—
48C3×D24C2×C412yes—
49Dic3⋊Dic3—41no—
50(C2×D3)⋊Dic3—41no—
51Dic32—41no—
52C32×C16C2×C412yescomplemented
53C3⋊(C3⋊C16)—21no—
54#2175—91noSylow
55C2×C3×Q16—31no—
56C2×C3×D8—31no—
57Q8×Dic3—61no—
58D4×Dic3—61no—
59C2×Dic12C2×D311yescharacteristic
60C2×D24C2×D311yescharacteristic
61#893—61no—
62#887—61no—
63C3×SD32—34no—
64C2×C3×C16—21no—
65C2×C3×C16C2×D311yescharacteristic
66C2×C3×C16C2×D311yescharacteristic
67C3×#107—31no—
68C3×#107—31no—
69#835—61no—
70#832—61no—
71#818C2×D311yescharacteristic
72#818C2×D311yescharacteristic
73C8×Dic3—31no—
74C8×Dic3—31no—
75C8×Dic3—61no—
76C2×C3⋊C16—31no—
77C2×C3⋊C16—31no—
78C2×C3⋊C16—61no—
79#810—61no—
80#809—61no—
81#800Dic314yescomplemented
82C22×C3×D3—41no—
83C2×C32×C4C2×D411yescharacteristic
84C2×C3⋊Dic3D811yescharacteristic
85C2×C3⋊Dic3D811yescharacteristic
86C2×C3×Dic3—41no—
87C2×C3×Dic3—41no—
88C2×C3×Dic3—41no—
89C3×D12—22no—
90C3×D12—41no—
91C3×Dic6—22no—
92C3×Dic6—41no—
93D3×Dic3—82no—
94C32×C8C22×C411yescharacteristic
95C32×C8#4111yescharacteristic
96C3⋊(C3⋊C8)—21no—
97C2×SD32—31no—
98C4×Q16—91no—
99C4×D8—91no—
100#366—31no—
101#358—91no—
102#357—91no—
103#346—91no—
104C2×C3×Q8—61no—
105C2×C3×D4—61no—
106C22×Dic3—121no—
107C2×D12—21no—
108C2×C4×D3—121no—
109C2×Dic6—21no—
110C3×Q16—32no—
111C3×Q16—61no—
112C3×D8—32no—
113C3×D8—61no—
114C2×C3×C8—21no—
115C2×C3×C8C22×D311yescharacteristic
116C2×C3×C8C22×D311yescharacteristic
117C3×C4⋊C4—61no—
118C3×C4⋊C4—61no—
119#217—121no—
120(C2×D3)⋊C4—121no—
121C4⋊Dic3—21no—
122C4⋊Dic3—21no—
123C4⋊Dic3—121no—
124Dic3⋊C4—121no—
125C4×Dic3—31no—
126C4×Dic3—31no—
127C4×Dic3—61no—
128C4×Dic3—121no—
129C4×Dic3—121no—
130C2×C3⋊C8—31no—
131C2×C3⋊C8—31no—
132C2×C3⋊C8—61no—
133Dic12—21no—
134Dic12C2×Dic312yescomplemented
135D24—21no—
136D24C2×Dic312yes—
137C3×C16—22no—
138C3×C16C4×D312yescomplemented
139C3×C16C2×Dic312yescomplemented
140C3⋊C16—62no—
141C3⋊C16—61no—
142C3⋊C16—61no—
143C22×C32C2×D811yescharacteristic
144C2×C3×D3—42no—
145C2×C3×D3—81no—
146C2×C3×D3—81no—
147C32×C4C4×D411yescharacteristic
148C32×C4#13511yescharacteristic
149C3⋊Dic3—21no—
150C3⋊Dic3—21no—
151C3×Dic3—42no—
152C3×Dic3—81no—
153C3×Dic3—81no—
154C2×Q16—31no—
155C2×D8—31no—
156C4×Q8—181no—
157C4×D4—181no—
158SD32—34no—
159C2×C16—91no—
160C2×C16D3211yescharacteristic
161#107—31no—
162#107—31no—
163#103—181no—
164#102—181no—
165C4×C8—91no—
166C23×C3—121no—
167C22×D3—41no—
168C3×Q8—62no—
169C3×Q8—121no—
170C3×D4—62no—
171C3×D4—121no—
172C2×C3×C4—21no—
173C2×C3×C4—121no—
174C2×C3×C4—121no—
175C2×C3×C4—121no—
176C2×C3×C4D3×D411yescharacteristic
177C2×C3×C4D3×D411yescharacteristic
178C2×Dic3—31no—
179C2×Dic3—31no—
180C2×Dic3—31no—
181C2×Dic3—31no—
182C2×Dic3—41no—
183C2×Dic3—41no—
184C2×Dic3—41no—
185C2×Dic3—61no—
186C2×Dic3—61no—
187C2×Dic3—242no—
188D12—22no—
189D12—41no—
190C4×D3—242no—
191Dic6—22no—
192Dic6—41no—
193C3×C8—21no—
194C3×C8—21no—
195C3×C8C2×C4×D311yescharacteristic
196C3×C8#23711yescharacteristic
197C3×C8#23911yescharacteristic
198C3×C8C22×Dic311yescharacteristic
199C3⋊C8—62no—
200C3⋊C8—61no—
201C3⋊C8—61no—
202C2×C32C4×D811yescharacteristic
203C2×C32#50911yescharacteristic
204C2×C32#51011yescharacteristic
205C3×D3—82no—
206C2×Q8—61no—
207C2×D4—61no—
208C22×C4—361no—
209Q16—32no—
210Q16—61no—
211D8—32no—
212D8—61no—
213C2×C8—91no—
214C2×C8C2×D3211yescharacteristic
215C4⋊C4—61no—
216C4⋊C4—61no—
217C4⋊C4—361no—
218C22⋊C4—361no—
219C42—91no—
220C42—361no—
221C16—181no—
222C16D3×Dic312yescomplemented
223C22×C3—21no—
224C22×C3—122no—
225C22×C3—241no—
226C22×C3—241no—
227C22×C3D3×D811yescharacteristic
228C22×C3D3×D811yescharacteristic
229C2×D3—42no—
230C2×D3—81no—
231C2×D3—81no—
232C3×C4—21no—
233C3×C4—21no—
234C3×C4—122no—
235C3×C4—241no—
236C3×C4—241no—
237C3×C4#89311yescharacteristic
238C3×C4#91211yescharacteristic
239C3×C4#91911yescharacteristic
240C3×C4D4×Dic311yescharacteristic
241Dic3—42no—
242Dic3—61no—
243Dic3—61no—
244Dic3—61no—
245Dic3—61no—
246Dic3—81no—
247Dic3—81no—
248Dic3—121no—
249Dic3—121no—
250C32#217511yescharacteristic, complemented, Sylow
251C23—121no—
252Q8—62no—
253Q8—121no—
254D4—62no—
255D4—121no—
256C2×C4—91no—
257C2×C4—91no—
258C2×C4—121no—
259C2×C4—121no—
260C2×C4—121no—
261C2×C4—722no—
262C2×C4#380211yescharacteristic
263C8—181no—
264C8#379611yescharacteristic
265C8C2×D3×Dic311yescharacteristic
266C2×C3—21no—
267C2×C3—21no—
268C2×C3—21no—
269C2×C3—242no—
270C2×C3#487211yescharacteristic
271C2×C3#491111yescharacteristic
272C2×C3#491411yescharacteristic
273C2×C3#491511yescharacteristic
274C2×C3#491811yescharacteristic
275C2×C3Dic3×D811yescharacteristic
276D3—82no—
277C22—122no—
278C22—241no—
279C22—241no—
280C22#6379711yescharacteristic
281C4—122no—
282C4—181no—
283C4—181no—
284C4—241no—
285C4—241no—
286C4#6381011yescharacteristic
287C4#6389811yescharacteristic
288C3—21no—
289C3#6931911yescharacteristic, complemented
290C3#7147411yescharacteristic, complemented
291C2—242no—
292C2#1059023611yescharacteristic
293C2#1059024611yescharacteristic
294C2#1059077711yescharacteristic
subgroupG/H
centerC22#63797
commutatorC32×C8C22×C4
FrattiniC2×C8C2×D32
FittingC2×C32×C16C22
socleC22×C32C2×D8
prime radicalC2×C3×D24C22
cosocleC2×C32×C8C23
hypercenterC2×C16D32
hypocenterC32#2175
LCSG ▷ C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16

Complements

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

NH ≅ G/Ntype
2. #10590818C2semidirect
4. #10589368C2semidirect
5. C3×#4504C2semidirect
6. #10589117C2semidirect
19. C3×#800C4semidirect
31. #4504D3semidirect
46. C3×Dic12C2×C4semidirect
52. C32×C16C2×C4semidirect
81. #800Dic3semidirect
134. Dic12C2×Dic3semidirect
138. C3×C16C4×D3semidirect
139. C3×C16C2×Dic3semidirect
222. C16D3×Dic3semidirect
250. C32#2175semidirect
289. C3#69319semidirect
290. C3#71474semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#8002
#105908182
#105907772
#105893682
C3×#45042
#105891172
#105891142
#714743
#693193