Finite Group Databasebeta

← #10950821

Subgroups — #10950821

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#81910C211yesmaximal, characteristic, complemented
2#81776C211yesmaximal, characteristic
3C3×#2244C211yesmaximal, characteristic, complemented
4#71354C211yesmaximal, characteristic, complemented
5#71291C211yesmaximal, characteristic, complemented
6#69198C212yesmaximal, complemented
7#69196C212yesmaximal, complemented
8#69176C212yesmaximal, complemented
9#69175C212yesmaximal, complemented
10#69083C211yesmaximal, characteristic, complemented
11#69072C211yesmaximal, characteristic, complemented
12#33743—31nomaximal, Sylow
13#5719—21no—
14#5698—21no—
15C3×#499C2211yescharacteristic
16C3×#493C2211yescharacteristic
17#5159—21no—
18#5157C2212yes—
19#5152—21no—
20Dic3×D8C2212yes—
21C2×#827C2212yes—
22C2×#827C2212yes—
23#5071—21no—
24#5069—21no—
25C2×#798C2211yescharacteristic
26C2×D3×C16C2211yescharacteristic
27#4887C2212yes—
28#4873C2212yes—
29#4867—21no—
30#4853—21no—
31#4699C2211yescharacteristic
32#4689C2211yescharacteristic
33#4683C2211yescharacteristic
34C3×#357C2212yes—
35C3×#357C2212yes—
36C3×#346C2211yescharacteristic
37#4503C2212yes—
38#4503C2212yes—
39#4502C2211yescharacteristic
40Dic3×C16C2211yescharacteristic
41#4490C2212yes—
42#4489C2212yes—
43C4×C3⋊C16C2211yescharacteristic
44#3177—31no—
45#3176—31no—
46#2253—31no—
47#2248—31no—
48#2245—31no—
49#2244D311yescharacteristic, complemented
50#2220—32no—
51#2217—32no—
52#2175—32no—
53C4×SD32—32no—
54#2108—31no—
55C2×C3×D8—21no—
56C2×C3×D8C2312yes—
57C3×#127—21no—
58C3×#125—21no—
59#939—42no—
60#938—22no—
61#935—42no—
62D4×Dic3—22no—
63C2×Dic12—21no—
64C2×Dic12C2312yes—
65C2×ΓL(1,𝔽25)D411yescharacteristic
66C2×D3×C8C2311yescharacteristic
67#898—42no—
68#897—22no—
69#890—42no—
70#887—22no—
71#872D411yescharacteristic
72#870—21no—
73#869—21no—
74C2×C3×C16C2311yescharacteristic
75C2×C3×C16C2311yescharacteristic
76C3×#107C2312yes—
77C3×#106—21no—
78C3×C4×C8C2311yescharacteristic
79#832—22no—
80#832—22no—
81#827—24no—
82#827—24no—
83#818C2312yes—
84#817—21no—
85#815D411yescharacteristic
86C8×Dic3C2311yescharacteristic
87C2×C3⋊C16C2311yescharacteristic
88C2×C3⋊C16C2311yescharacteristic
89#810—22no—
90#810—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#494—31no—
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#357C2×D312yes—
120#357C2×D312yes—
121#346—31no—
122#346C2×D311yescharacteristic
123C4×C16—31no—
124C4×C16—31no—
125C2×C3×D4—22no—
126C2×C3×D4—42no—
127C2×#67—42no—
128C22×Dic3—42no—
129C2×C4×D3C2×D411yescharacteristic
130C2×Dic6—22no—
131C3×D8—22no—
132C3×D8—22no—
133C3×D8—42no—
134C2×C3×C8C2×D411yescharacteristic
135C2×C3×C8C2411yescharacteristic
136C3×C4⋊C4—22no—
137C3×C4⋊C4—42no—
138C3×C42C2×D411yescharacteristic
139#217—42no—
140#217—42no—
141(C2×D3)⋊C4—42no—
142(C2×D3)⋊C4D812yes—
143C4⋊Dic3—22no—
144C4⋊Dic3—42no—
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
153Dic12—22no—
154Dic12—22no—
155Dic12—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—32no—
164C2×Q16—61no—
165C2×D8—21no—
166C2×D8C22×D312yes—
167C2×M16—31no—
168C22×C8—31no—
169C4⋊Q8—61no—
170#127—21no—
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
202C23×C3—42no—
203C22×D3C2×D811yescharacteristic
204C3×D4—42no—
205C3×D4—42no—
206C3×D4—162no—
207C2×C3×C4—42no—
208C2×C3×C4C2×D812yes—
209C2×C3×C4C22×D411yescharacteristic
210#67—162no—
211C2×Dic3—42no—
212C2×Dic3—42no—
213C2×Dic3—162no—
214C2×Dic3C2×D812yes—
215C2×Dic3C2×D811yescharacteristic
216C4×D3—21no—
217C4×D3—21no—
218Dic6—42no—
219Dic6—42no—
220C3×C8#13512yes—
221C3×C8C2×#4111yescharacteristic
222C3×C8C2×#4111yescharacteristic
223C3⋊C8—21no—
224C3⋊C8—21no—
225C2×Q8—62no—
226C2×D4—22no—
227C2×D4—42no—
228C2×D4—122no—
229C22×C4—31no—
230C22×C4—122no—
231Q16—62no—
232Q16—62no—
233Q16—122no—
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—22no—
245C4⋊C4—32no—
246C4⋊C4—42no—
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—21no—
260C16—21no—
261C16—32no—
262C16—32no—
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—122no—
280Q8—122no—
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#91612yes—
298C8D3×#4111yescharacteristic
299C8D3×#4111yescharacteristic
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#566811yescharacteristic
317C4C2×#91611yescharacteristic
318C3#3374311yescharacteristic, complemented, Sylow
319C2—121no—
320C2—162no—
321C2#8155711yescharacteristic
322C2#8203712yes—
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2244C2
socleC22×C3C22×D8
prime radical#81776C2
hypercenter#2244D3
hypocenterC3#33743
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2244

Complements

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

NH ≅ G/Ntype
1. #81910C2semidirect
3. C3×#2244C2semidirect
4. #71354C2semidirect
5. #71291C2semidirect
6. #69198C2semidirect
7. #69196C2semidirect
8. #69176C2semidirect
9. #69175C2semidirect
10. #69083C2semidirect
11. #69072C2semidirect
49. #2244D3semidirect
318. C3#33743semidirect

Maximal subgroups

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

subgroupindex
#819102
#817762
C3×#22442
#713542
#712912
#691982
#691962
#691762
#691752
#690832
#690722
#337433