Finite Group Databasebeta

← #10950819

Subgroups — #10950819

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#81793C211yesmaximal, characteristic, complemented
2#81776C211yesmaximal, characteristic
3C3×#2248C211yesmaximal, characteristic, complemented
4#71488C212yesmaximal, complemented
5#71487C212yesmaximal, complemented
6#71474C212yesmaximal, complemented
7Dic3×SD32C212yesmaximal, complemented
8#71311C211yesmaximal, characteristic, complemented
9#71289C211yesmaximal, characteristic, complemented
10#69051C211yesmaximal, characteristic, complemented
11#69045C211yesmaximal, characteristic, complemented
12#33743—31nomaximal, Sylow
13#5731—21no—
14#5719—21no—
15C2×C3×SD32C2212yes—
16C2×C3×SD32C2212yes—
17C3×#494C2211yescharacteristic
18C3×#493C2211yescharacteristic
19#5189—21no—
20#5188C2212yes—
21#5184—21no—
22Dic3×Q16C2212yes—
23#5159—21no—
24#5157C2212yes—
25#5152—21no—
26Dic3×D8C2212yes—
27#5082—21no—
28#5071—21no—
29C2×#798C2211yescharacteristic
30C2×D3×C16C2211yescharacteristic
31#4689C2211yescharacteristic
32#4683C2211yescharacteristic
33#4683C2211yescharacteristic
34C3×C4×C16C2211yescharacteristic
35#4564C2212yes—
36#4564C2212yes—
37#4562C2212yes—
38#4562C2212yes—
39#4505C2212yes—
40#4504C2212yes—
41#4502C2211yescharacteristic
42Dic3×C16C2211yescharacteristic
43#4461C2211yescharacteristic
44#3177—31no—
45#3176—31no—
46#2253—31no—
47#2248D311yescharacteristic, complemented
48#2245—31no—
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—
57C2×C3×D8—21no—
58C2×C3×D8C2312yes—
59C3×C4⋊Q8—21no—
60C3×#127—21no—
61#946—22no—
62Q8×Dic3—22no—
63#939—42no—
64#938—22no—
65#935—42no—
66D4×Dic3—22no—
67C2×ΓL(1,𝔽25)D411yescharacteristic
68C2×D3×C8C2311yescharacteristic
69#898—42no—
70#889—42no—
71#872D411yescharacteristic
72#870—21no—
73#870—21no—
74C3×SD32—24no—
75C3×SD32—24no—
76C2×C3×C16C2311yescharacteristic
77C2×C3×C16C2311yescharacteristic
78C3×C4×C8C2311yescharacteristic
79#835—22no—
80#835—22no—
81#832—22no—
82#832—22no—
83#818C2312yes—
84#818C2312yes—
85#817—21no—
86#817—21no—
87#815D411yescharacteristic
88C8×Dic3C2311yescharacteristic
89C2×C3⋊C16C2311yescharacteristic
90C2×C3⋊C16C2311yescharacteristic
91#803D411yescharacteristic
92#798—22no—
93D3×C16—22no—
94#541—61no—
95#540—61no—
96C2×SD32C2×D312yes—
97C2×SD32C2×D312yes—
98C2×M32—31no—
99C22×C16—31no—
100#499—31no—
101#499—31no—
102#494C2×D311yescharacteristic
103#493C2×D311yescharacteristic
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#358—32no—
118#358—32no—
119#357—32no—
120#357—32no—
121#346—31no—
122#346—31no—
123C4×C16—31no—
124C4×C16C2×D311yescharacteristic
125C2×C3×Q8—22no—
126C2×C3×D4—22no—
127C2×C3×D4—42no—
128C2×#67—42no—
129C22×Dic3—42no—
130C2×C4×D3C2×D411yescharacteristic
131C3×Q16—22no—
132C3×Q16—22no—
133C3×Q16—42no—
134C3×D8—22no—
135C3×D8—22no—
136C3×D8—42no—
137C2×C3×C8C2×D411yescharacteristic
138C2×C3×C8C2411yescharacteristic
139C3×C4⋊C4—42no—
140C3×C42C2×D411yescharacteristic
141#217—42no—
142#217—42no—
143(C2×D3)⋊C4—42no—
144(C2×D3)⋊C4D812yes—
145C4⋊Dic3—22no—
146C4⋊Dic3—22no—
147C4⋊Dic3—42no—
148Dic3⋊C4—42no—
149Dic3⋊C4—42no—
150Dic3⋊C4—42no—
151Dic3⋊C4D812yes—
152C4×Dic3—42no—
153C4×Dic3C2×D411yescharacteristic
154C2×C3⋊C8C2×D411yescharacteristic
155C2×C3⋊C8C2×D411yescharacteristic
156ΓL(1,𝔽25)—22no—
157D3×C8—21no—
158D3×C8—21no—
159C3×C16#4112yes—
160C3×C16#4112yes—
161C3⋊C16—21no—
162C3⋊C16—21no—
163C2×Q16—21no—
164C2×Q16C22×D312yes—
165C2×D8—21no—
166C2×D8C22×D312yes—
167C2×M16—31no—
168C22×C8—31no—
169C4⋊Q8—21no—
170#127—21no—
171#126—122no—
172#125—61no—
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—24no—
183SD32—24no—
184M32—62no—
185C2×C16—31no—
186C2×C16—31no—
187C2×C16—62no—
188C2×C16C22×D311yescharacteristic
189C2×C16C22×D311yescharacteristic
190#107—32no—
191#107—32no—
192#106—61no—
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
202C23×C3—42no—
203C22×D3C2×D811yescharacteristic
204C3×Q8—42no—
205C3×Q8—42no—
206C3×D4—42no—
207C3×D4—42no—
208C3×D4—162no—
209C2×C3×C4—42no—
210C2×C3×C4C2×D812yes—
211C2×C3×C4C22×D411yescharacteristic
212#67—162no—
213C2×Dic3—42no—
214C2×Dic3—42no—
215C2×Dic3—162no—
216C2×Dic3C2×D812yes—
217C2×Dic3C2×D811yescharacteristic
218C4×D3—21no—
219C4×D3—21no—
220C3×C8#13512yes—
221C3×C8C2×#4111yescharacteristic
222C3×C8C2×#4111yescharacteristic
223C3⋊C8—21no—
224C3⋊C8—21no—
225C2×Q8—22no—
226C2×D4—22no—
227C2×D4—42no—
228C2×D4—122no—
229C22×C4—31no—
230C22×C4—122no—
231Q16—22no—
232Q16—22no—
233Q16—42no—
234D8—22no—
235D8—22no—
236D8—42no—
237M16—62no—
238C2×C8—31no—
239C2×C8—31no—
240C2×C8—61no—
241C2×C8—61no—
242C2×C8D3×D411yescharacteristic
243C2×C8C23×D311yescharacteristic
244C4⋊C4—32no—
245C4⋊C4—42no—
246C4⋊C4—62no—
247C4⋊C4—62no—
248C4⋊C4—122no—
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—61no—
260C16—61no—
261C16#23712yes—
262C16#23712yes—
263C22×C3—82no—
264C22×C3—82no—
265C22×C3—82no—
266C22×C3C22×D811yescharacteristic
267C2×D3—22no—
268C2×D3—21no—
269C3×C4—22no—
270C3×C4—82no—
271C3×C4#53211yescharacteristic
272C3×C4C2×#13511yescharacteristic
273Dic3—22no—
274Dic3—21no—
275Dic3—82no—
276Dic3—82no—
277C23—31no—
278C23—42no—
279Q8—42no—
280Q8—42no—
281D4—42no—
282D4—42no—
283D4—162no—
284D4—482no—
285C2×C4—32no—
286C2×C4—31no—
287C2×C4—42no—
288C2×C4—61no—
289C2×C4—61no—
290C2×C4—122no—
291C2×C4—122no—
292C2×C4—482no—
293C2×C4D3×D812yes—
294C2×C4C2×D3×D411yescharacteristic
295C8—61no—
296C8—61no—
297C8#91212yes—
298C8C2×#23711yescharacteristic
299C8C2×#23711yescharacteristic
300C2×C3—162no—
301C2×C3#316911yescharacteristic
302C2×C3#341612yes—
303D3—41no—
304C22—62no—
305C22—61no—
306C22—82no—
307C22—82no—
308C22—82no—
309C22C2×D3×D811yescharacteristic
310C4—22no—
311C4—62no—
312C4—61no—
313C4—82no—
314C4—242no—
315C4—242no—
316C4#571611yescharacteristic
317C4C2×#91211yescharacteristic
318C3#3374311yescharacteristic, complemented, Sylow
319C2—121no—
320C2—162no—
321C2#8177511yescharacteristic
322C2#8202412yes—
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2248C2
socleC22×C3C22×D8
prime radical#81776C2
hypercenter#2248D3
hypocenterC3#33743
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2248

Complements

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

NH ≅ G/Ntype
1. #81793C2semidirect
3. C3×#2248C2semidirect
4. #71488C2semidirect
5. #71487C2semidirect
6. #71474C2semidirect
7. Dic3×SD32C2semidirect
8. #71311C2semidirect
9. #71289C2semidirect
10. #69051C2semidirect
11. #69045C2semidirect
47. #2248D3semidirect
318. C3#33743semidirect

Maximal subgroups

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

subgroupindex
#817932
#817762
C3×#22482
#714882
#714872
#714742
Dic3×SD322
#713112
#712892
#690512
#690452
#337433