Finite Group Databasebeta

← #10950825

Subgroups — #10950825

Order 768.

508 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81909C211yesmaximal, characteristic
2#81793C211yesmaximal, characteristic, complemented
3C3×#2245C211yesmaximal, characteristic, complemented
4#71354C211yesmaximal, characteristic, complemented
5#71313C211yesmaximal, characteristic, complemented
6#69230C212yesmaximal, complemented
7#69228C212yesmaximal, complemented
8#69208C212yesmaximal, complemented
9#69206C212yesmaximal, complemented
10#69080C211yesmaximal, characteristic, complemented
11#69072C211yesmaximal, characteristic, complemented
12#33743—31nomaximal, Sylow
13#5731—21no—
14#5697—21no—
15C3×#499C2211yescharacteristic
16C3×#494C2211yescharacteristic
17#5189—21no—
18#5188C2212yes—
19#5184—21no—
20Dic3×Q16C2212yes—
21C2×#828C2212yes—
22C2×#828C2212yes—
23#5082—21no—
24#5067—21no—
25C2×#798C2211yescharacteristic
26C2×D3×C16C2211yescharacteristic
27#4883C2212yes—
28#4872C2212yes—
29#4869—21no—
30#4865—21no—
31#4695C2211yescharacteristic
32#4689C2211yescharacteristic
33#4683C2211yescharacteristic
34C3×#358C2212yes—
35C3×#358C2212yes—
36C3×#346C2211yescharacteristic
37#4509C2212yes—
38#4509C2212yes—
39#4502C2211yescharacteristic
40Dic3×C16C2211yescharacteristic
41#4490C2212yes—
42#4489C2212yes—
43C4×C3⋊C16C2211yescharacteristic
44#3177—31no—
45#3176—31no—
46#2253—31no—
47#2248—31no—
48#2245D311yescharacteristic, complemented
49#2244—31no—
50#2220—32no—
51#2217—32no—
52#2175—32no—
53C4×SD32—32no—
54#2108—31no—
55C2×C3×Q16—21no—
56C2×C3×Q16C2312yes—
57C3×C4⋊Q8—21no—
58C3×#125—21no—
59#946—22no—
60Q8×Dic3—22no—
61C2×D24—21no—
62C2×D24C2312yes—
63C2×ΓL(1,𝔽25)D411yescharacteristic
64C2×D3×C8C2311yescharacteristic
65#898—42no—
66#895—22no—
67#895—42no—
68#894—42no—
69#893—22no—
70#889—42no—
71#874—21no—
72#872D411yescharacteristic
73#870—21no—
74C2×C3×C16C2311yescharacteristic
75C2×C3×C16C2311yescharacteristic
76C3×#107C2312yes—
77C3×#106—21no—
78C3×C4×C8C2311yescharacteristic
79#835—22no—
80#835—22no—
81#828—24no—
82#828—24no—
83#818C2312yes—
84#817—21no—
85#815D411yescharacteristic
86C8×Dic3C2311yescharacteristic
87C2×C3⋊C16C2311yescharacteristic
88C2×C3⋊C16C2311yescharacteristic
89#809—22no—
90#809—22no—
91#803D411yescharacteristic
92#798—22no—
93D3×C16—22no—
94#541—61no—
95#540—61no—
96C2×SD32—32no—
97C2×SD32—32no—
98C2×M32—31no—
99C22×C16—31no—
100#499—31no—
101#499C2×D311yescharacteristic
102#494C2×D311yescharacteristic
103#493—31no—
104#491—61no—
105#490—61no—
106#470—61no—
107#469—61no—
108#467—32no—
109#466—32no—
110#442—61no—
111#441—61no—
112C4×Q16—32no—
113C4×D8—32no—
114#405—31no—
115#367—32no—
116#366—32no—
117#358C2×D312yes—
118#358C2×D312yes—
119#357—32no—
120#357—32no—
121#346—31no—
122#346C2×D311yescharacteristic
123C4×C16—31no—
124C4×C16—31no—
125C2×C3×Q8—22no—
126C2×D12—22no—
127C2×D12—42no—
128C2×D12—42no—
129C2×C4×D3—42no—
130C2×C4×D3C2×D411yescharacteristic
131C3×Q16—22no—
132C3×Q16—22no—
133C3×Q16—42no—
134C2×C3×C8C2×D411yescharacteristic
135C2×C3×C8C2411yescharacteristic
136C3×C4⋊C4—22no—
137C3×C4⋊C4—42no—
138C3×C4⋊C4—42no—
139C3×C42C2×D411yescharacteristic
140(C2×D3)⋊C4—42no—
141(C2×D3)⋊C4—42no—
142(C2×D3)⋊C4—42no—
143(C2×D3)⋊C4D812yes—
144C4⋊Dic3—22no—
145C4⋊Dic3—42no—
146Dic3⋊C4—42no—
147Dic3⋊C4—42no—
148Dic3⋊C4D812yes—
149C4×Dic3—42no—
150C4×Dic3C2×D411yescharacteristic
151C2×C3⋊C8C2×D411yescharacteristic
152C2×C3⋊C8C2×D411yescharacteristic
153D24—22no—
154D24—22no—
155D24—42no—
156ΓL(1,𝔽25)—22no—
157D3×C8—21no—
158D3×C8—21no—
159C3×C16—21no—
160C3×C16—21no—
161C3⋊C16#4112yes—
162C3⋊C16#4112yes—
163C2×Q16—21no—
164C2×Q16C22×D312yes—
165C2×D8—32no—
166C2×D8—61no—
167C2×M16—31no—
168C22×C8—31no—
169C4⋊Q8—21no—
170#127—61no—
171#126—122no—
172#125—21no—
173#125—61no—
174#125—122no—
175#123—122no—
176C22⋊Q8—62no—
177#121—62no—
178#121—122no—
179C4×Q8—62no—
180C4×D4—62no—
181#117—31no—
182SD32—64no—
183SD32—64no—
184M32—62no—
185C2×C16—31no—
186C2×C16—31no—
187C2×C16—62no—
188C2×C16C22×D311yescharacteristic
189C2×C16C22×D311yescharacteristic
190#107—32no—
191#107C22×D312yes—
192#106—21no—
193#106—61no—
194#103—62no—
195#103—62no—
196#102—62no—
197#102—62no—
198#97—31no—
199#97—31no—
200C4×C8—31no—
201C4×C8C22×D311yescharacteristic
202C22×D3—42no—
203C22×D3C2×D811yescharacteristic
204C3×Q8—42no—
205C3×Q8—42no—
206C2×C3×C4—42no—
207C2×C3×C4—42no—
208C2×C3×C4C2×D812yes—
209C2×C3×C4C22×D411yescharacteristic
210C2×Dic3—42no—
211C2×Dic3C2×D812yes—
212C2×Dic3C2×D811yescharacteristic
213D12—42no—
214D12—42no—
215D12—162no—
216D12—162no—
217C4×D3—21no—
218C4×D3—21no—
219C4×D3—162no—
220C3×C8#13512yes—
221C3×C8C2×#4111yescharacteristic
222C3×C8C2×#4111yescharacteristic
223C3⋊C8—21no—
224C3⋊C8—21no—
225C2×Q8—22no—
226C2×D4—62no—
227C2×D4—122no—
228C2×D4—122no—
229C22×C4—31no—
230C22×C4—122no—
231Q16—22no—
232Q16—22no—
233Q16—42no—
234D8—62no—
235D8—62no—
236D8—122no—
237M16—62no—
238C2×C8—31no—
239C2×C8—31no—
240C2×C8—61no—
241C2×C8—61no—
242C2×C8D3×D411yescharacteristic
243C2×C8C23×D311yescharacteristic
244C4⋊C4—22no—
245C4⋊C4—32no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—62no—
249C4⋊C4—122no—
250C4⋊C4—122no—
251C4⋊C4—122no—
252C22⋊C4—32no—
253C22⋊C4—122no—
254C22⋊C4—122no—
255C22⋊C4—122no—
256C42—31no—
257C42—122no—
258C42D3×D411yescharacteristic
259C16—21no—
260C16—21no—
261C16—32no—
262C16—32no—
263C22×C3C22×D811yescharacteristic
264C2×D3—22no—
265C2×D3—21no—
266C2×D3—82no—
267C2×D3—82no—
268C2×D3—82no—
269C3×C4—22no—
270C3×C4—82no—
271C3×C4—82no—
272C3×C4#53211yescharacteristic
273C3×C4C2×#13511yescharacteristic
274Dic3—22no—
275Dic3—21no—
276Dic3—82no—
277C23—31no—
278C23—122no—
279Q8—42no—
280Q8—42no—
281D4—122no—
282D4—122no—
283D4—482no—
284D4—482no—
285C2×C4—32no—
286C2×C4—31no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—61no—
290C2×C4—61no—
291C2×C4—122no—
292C2×C4—482no—
293C2×C4D3×D812yes—
294C2×C4C2×D3×D411yescharacteristic
295C8—61no—
296C8—61no—
297C8#91612yes—
298C8D3×#4111yescharacteristic
299C8D3×#4111yescharacteristic
300C2×C3#316911yescharacteristic
301C2×C3#341612yes—
302D3—41no—
303D3—162no—
304C22—62no—
305C22—61no—
306C22—242no—
307C22—242no—
308C22—242no—
309C22C2×D3×D811yescharacteristic
310C4—22no—
311C4—62no—
312C4—61no—
313C4—82no—
314C4—82no—
315C4—242no—
316C4#566811yescharacteristic
317C4C2×#91611yescharacteristic
318C3#3374311yescharacteristic, complemented, Sylow
319C2—121no—
320C2—482no—
321C2#8155711yescharacteristic
322C2#8204012yes—
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2245C2
socleC22×C3C22×D8
prime radical#81909C2
hypercenter#2245D3
hypocenterC3#33743
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2245

Complements

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

NH ≅ G/Ntype
2. #81793C2semidirect
3. C3×#2245C2semidirect
4. #71354C2semidirect
5. #71313C2semidirect
6. #69230C2semidirect
7. #69228C2semidirect
8. #69208C2semidirect
9. #69206C2semidirect
10. #69080C2semidirect
11. #69072C2semidirect
48. #2245D3semidirect
318. C3#33743semidirect

Maximal subgroups

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

subgroupindex
#819092
#817932
C3×#22452
#713542
#713132
#692302
#692282
#692082
#692062
#690802
#690722
#337433