Finite Group Databasebeta

← #10950792

Subgroups — #10950792

Order 768.

532 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81909C211yesmaximal, characteristic, complemented
2#81776C211yesmaximal, characteristic, complemented
3C3×#2244C211yesmaximal, characteristic, complemented
4#71354C211yesmaximal, characteristic, complemented
5#71290C211yesmaximal, characteristic, complemented
6#69197C212yesmaximal, complemented
7#69195C212yesmaximal, complemented
8#69177C212yesmaximal, complemented
9#69174C212yesmaximal, complemented
10#69080C211yesmaximal, characteristic, complemented
11#69072C211yesmaximal, characteristic, complemented
12#33740—31nomaximal, Sylow
13#5719—21no—
14#5697—21no—
15C3×#499C2211yescharacteristic
16C3×#493C2211yescharacteristic
17#5159—21no—
18#5157C2212yes—
19#5152—21no—
20Dic3×D8C2212yes—
21C2×#826C2212yes—
22C2×#826C2212yes—
23#5071—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×#357C2212yes—
35C3×#357C2212yes—
36C3×#346C2211yescharacteristic
37#4509C2212yes—
38#4509C2212yes—
39#4502C2211yescharacteristic
40Dic3×C16C2211yescharacteristic
41#4490C2212yes—
42#4489C2212yes—
43C4×C3⋊C16C2211yescharacteristic
44#3176—31no—
45#3176—31no—
46#2253—31no—
47#2246—31no—
48#2244—31no—
49#2244D311yescharacteristic, complemented
50#2218—32no—
51#2215—32no—
52#2177—32no—
53C4×D16—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×D24—21no—
64C2×D24C2312yes—
65C2×ΓL(1,𝔽25)D411yescharacteristic
66C2×D3×C8C2311yescharacteristic
67#895—22no—
68#895—42no—
69#894—42no—
70#893—22no—
71#874—21no—
72#872D411yescharacteristic
73#870—21no—
74C2×C3×C16C2311yescharacteristic
75C2×C3×C16C2311yescharacteristic
76C3×#107C2312yes—
77C3×#106—21no—
78C3×C4×C8C2311yescharacteristic
79#832—22no—
80#832—22no—
81#826—24no—
82#826—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#540—61no—
95#540—61no—
96C2×D16—32no—
97C2×D16—32no—
98C2×M32—31no—
99C22×C16—31no—
100#499—31no—
101#499C2×D311yescharacteristic
102#493—31no—
103#493C2×D311yescharacteristic
104#490—61no—
105#490—61no—
106#469—61no—
107#469—61no—
108#466—32no—
109#466—32no—
110#442—61no—
111#442—61no—
112C4×D8—32no—
113C4×D8—32no—
114#405—31no—
115#367—32no—
116#366—32no—
117#357—32no—
118#357—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×D12—22no—
130C2×D12—42no—
131C2×D12—42no—
132C2×C4×D3—42no—
133C2×C4×D3C2×D411yescharacteristic
134C3×D8—22no—
135C3×D8—22no—
136C3×D8—42no—
137C2×C3×C8C2×D411yescharacteristic
138C2×C3×C8C2411yescharacteristic
139C3×C4⋊C4—22no—
140C3×C4⋊C4—42no—
141C3×C42C2×D411yescharacteristic
142#217—42no—
143#217—42no—
144(C2×D3)⋊C4—42no—
145(C2×D3)⋊C4—42no—
146(C2×D3)⋊C4D812yes—
147C4⋊Dic3—22no—
148Dic3⋊C4—42no—
149Dic3⋊C4D812yes—
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×D8—21no—
164C2×D8—32no—
165C2×D8—61no—
166C2×D8C22×D312yes—
167C2×M16—31no—
168C22×C8—31no—
169#127—21no—
170#127—61no—
171#125—21no—
172#125—61no—
173#123—122no—
174#123—122no—
175#121—62no—
176#121—62no—
177#121—122no—
178#121—122no—
179C4×D4—62no—
180C4×D4—62no—
181#117—31no—
182D16—64no—
183D16—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#102—62no—
195#102—62no—
196#102—62no—
197#102—62no—
198#97—31no—
199#97—31no—
200C4×C8—31no—
201C4×C8C22×D311yescharacteristic
202C23×C3—42no—
203C22×D3—42no—
204C22×D3C2×D811yescharacteristic
205C3×D4—42no—
206C3×D4—42no—
207C3×D4—162no—
208C2×C3×C4—42no—
209C2×C3×C4C2×D812yes—
210C2×C3×C4C22×D411yescharacteristic
211#67—162no—
212C2×Dic3—42no—
213C2×Dic3—162no—
214C2×Dic3C2×D812yes—
215C2×Dic3C2×D811yescharacteristic
216D12—42no—
217D12—42no—
218D12—162no—
219D12—162no—
220C4×D3—21no—
221C4×D3—21no—
222C4×D3—162no—
223C3×C8#13512yes—
224C3×C8C2×#4111yescharacteristic
225C3×C8C2×#4111yescharacteristic
226C3⋊C8—21no—
227C3⋊C8—21no—
228C2×D4—22no—
229C2×D4—42no—
230C2×D4—62no—
231C2×D4—122no—
232C2×D4—122no—
233C2×D4—122no—
234C22×C4—31no—
235C22×C4—122no—
236C22×C4—122no—
237D8—22no—
238D8—22no—
239D8—42no—
240D8—62no—
241D8—62no—
242D8—122no—
243M16—62no—
244C2×C8—31no—
245C2×C8—31no—
246C2×C8—61no—
247C2×C8—61no—
248C2×C8D3×D411yescharacteristic
249C2×C8C23×D311yescharacteristic
250C4⋊C4—22no—
251C4⋊C4—32no—
252C4⋊C4—42no—
253C4⋊C4—62no—
254C4⋊C4—122no—
255C22⋊C4—32no—
256C22⋊C4—122no—
257C22⋊C4—122no—
258C22⋊C4—122no—
259C22⋊C4—122no—
260C42—31no—
261C42D3×D411yescharacteristic
262C16—21no—
263C16—21no—
264C16—32no—
265C16—32no—
266C22×C3—82no—
267C22×C3—82no—
268C22×C3—82no—
269C22×C3C22×D811yescharacteristic
270C2×D3—22no—
271C2×D3—21no—
272C2×D3—82no—
273C2×D3—82no—
274C2×D3—82no—
275C3×C4—22no—
276C3×C4—82no—
277C3×C4#53211yescharacteristic
278C3×C4C2×#13511yescharacteristic
279Dic3—22no—
280Dic3—21no—
281Dic3—82no—
282C23—31no—
283C23—42no—
284C23—122no—
285D4—42no—
286D4—42no—
287D4—122no—
288D4—122no—
289D4—162no—
290D4—482no—
291D4—482no—
292D4—482no—
293C2×C4—32no—
294C2×C4—31no—
295C2×C4—42no—
296C2×C4—61no—
297C2×C4—61no—
298C2×C4—122no—
299C2×C4—482no—
300C2×C4—482no—
301C2×C4D3×D812yes—
302C2×C4C2×D3×D411yescharacteristic
303C8—61no—
304C8—61no—
305C8#91612yes—
306C8D3×#4111yescharacteristic
307C8D3×#4111yescharacteristic
308C2×C3—162no—
309C2×C3#316911yescharacteristic
310C2×C3#341512yes—
311D3—41no—
312D3—162no—
313C22—62no—
314C22—61no—
315C22—82no—
316C22—82no—
317C22—82no—
318C22—242no—
319C22—242no—
320C22—242no—
321C22C2×D3×D811yescharacteristic
322C4—22no—
323C4—62no—
324C4—61no—
325C4—82no—
326C4—242no—
327C4#566811yescharacteristic
328C4C2×#91611yescharacteristic
329C3#3374011yescharacteristic, complemented, Sylow
330C2—121no—
331C2—162no—
332C2—482no—
333C2#8155711yescharacteristic
334C2#8203612yes—
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2244C2
socleC22×C3C22×D8
hypercenter#2244D3
hypocenterC3#33740
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. #81909C2semidirect
2. #81776C2semidirect
3. C3×#2244C2semidirect
4. #71354C2semidirect
5. #71290C2semidirect
6. #69197C2semidirect
7. #69195C2semidirect
8. #69177C2semidirect
9. #69174C2semidirect
10. #69080C2semidirect
11. #69072C2semidirect
49. #2244D3semidirect
329. C3#33740semidirect

Maximal subgroups

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

subgroupindex
#819092
#817762
C3×#22442
#713542
#712902
#691972
#691952
#691772
#691742
#690802
#690722
#337403