Finite Group Databasebeta

← #10686237

Subgroups — #10686237

Order 768.

436 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79325C211yesmaximal, characteristic
2#69858C211yesmaximal, characteristic
3#69847C211yesmaximal, characteristic, complemented
4#69434C211yesmaximal, characteristic, complemented
5#69431C211yesmaximal, characteristic
6C3×#1300C211yesmaximal, characteristic, complemented
7#67751C211yesmaximal, characteristic, complemented
8#8249—31nomaximal, Sylow
9#5479C2211yescharacteristic
10C3×#432C2211yescharacteristic
11C3×C4×M16C2211yescharacteristic
12C3×#376C2211yescharacteristic
13#4925C2211yescharacteristic
14C2×#803—21no—
15C2×#803C2211yescharacteristic
16C2×C4×C3⋊C8—21no—
17C2×C4×C3⋊C8C2211yescharacteristic
18#4838C411yescharacteristic, complemented
19#4834C411yescharacteristic, complemented
20#4831C411yescharacteristic, complemented
21#4830C411yescharacteristic, complemented
22#4730—21no—
23#67⋊C8—21no—
24#4709—21no—
25#4708—21no—
26#4465C412yescomplemented
27#4464C412yescomplemented
28#4454—21no—
29#4454—21no—
30#4452C412yes—
31#4451C412yes—
32#2966—31no—
33#1726—31no—
34#1524—31no—
35#1500—31no—
36#1451—31no—
37#1302—31no—
38#1300D311yescharacteristic, complemented
39C2×C3×M16—21no—
40C2×C3×C42—21no—
41C2×C3×C42C2311yescharacteristic
42#930C2×C411yescharacteristic
43#929—21no—
44C4×#67—21no—
45#924C2×C411yescharacteristic
46C22×C3⋊C8—21no—
47C22×C3⋊C8D411yescharacteristic
48C22×C3⋊C8D411yescharacteristic
49C2×ΓL(1,𝔽25)—21no—
50C2×D3×C8—21no—
51#876—21no—
52#875C2×C411yescharacteristic
53C4×D12C2×C411yescharacteristic
54#872—21no—
55#870C2×C411yescharacteristic
56C4×Dic6C2×C411yescharacteristic
57C3×C4⋊C8C2×C411yescharacteristic
58C3×C4⋊C8C2×C411yescharacteristic
59C3×C22⋊C8—21no—
60C3×#97—21no—
61C3×#97C2×C412yes—
62C3×C4×C8—21no—
63C3×#95—21no—
64C3×#95C2×C412yes—
65#830C2×C412yes—
66(C2×D3)⋊C8—21no—
67(C2×D3)⋊C8—21no—
68#815—21no—
69Dic3⋊C8—21no—
70Dic3⋊C8—21no—
71C8×Dic3—21no—
72C4⋊(C3⋊C8)—21no—
73#803—24no—
74#803—21no—
75#803C2×C412yes—
76C4×C3⋊C8—24no—
77C4×C3⋊C8—21no—
78C4×C3⋊C8C2×C411yescharacteristic
79C4×C3⋊C8C2×C411yescharacteristic
80C4×C3⋊C8D411yescharacteristic
81C4×C3⋊C8D411yescharacteristic
82#529—31no—
83#446—31no—
84C8×Q8—31no—
85#436—31no—
86#435—61no—
87C8×D4—31no—
88C8×D4—61no—
89#433—61no—
90#432C2×D311yescharacteristic
91#431—31no—
92#405—61no—
93C4×M16C2×D311yescharacteristic
94C2×#97—31no—
95C2×#97—61no—
96C2×C4×C8—31no—
97C2×C4×C8—61no—
98#376C2×D311yescharacteristic
99#330—32no—
100#329—32no—
101#328—32no—
102#327—32no—
103#322—61no—
104#322—61no—
105C22×C3×C4—22no—
106C22×C3×C4C22×C411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C2×#67—21no—
110C2×D12C4211yescharacteristic
111C2×C4×D3—21no—
112C2×Dic6C4211yescharacteristic
113C3×M16—42no—
114C2×C3×C8—21no—
115C2×C3×C8—21no—
116C2×C3×C8—22no—
117C3×C42—22no—
118C3×C42—22no—
119C3×C42C22×C411yescharacteristic
120C3×C42C22×C411yescharacteristic
121C3×C42C2×D411yescharacteristic
122C3×C42#4111yescharacteristic
123#217—21no—
124(C2×D3)⋊C4—21no—
125(C2×D3)⋊C4—21no—
126(C2×D3)⋊C4—21no—
127(C2×D3)⋊C4—21no—
128C4⋊Dic3C4211yescharacteristic
129C4⋊Dic3C4211yescharacteristic
130Dic3⋊C4—21no—
131Dic3⋊C4—21no—
132Dic3⋊C4—21no—
133Dic3⋊C4—21no—
134C4×Dic3—21no—
135C2×C3⋊C8—22no—
136C2×C3⋊C8—22no—
137C2×C3⋊C8—22no—
138C2×C3⋊C8—22no—
139C2×C3⋊C8—22no—
140C2×C3⋊C8—22no—
141C2×C3⋊C8C22⋊C411yescharacteristic
142C2×C3⋊C8C22⋊C411yescharacteristic
143C2×C3⋊C8C22⋊C411yescharacteristic
144C2×C3⋊C8C22⋊C411yescharacteristic
145ΓL(1,𝔽25)—42no—
146D3×C8—42no—
147C2×M16—21no—
148C2×M16—61no—
149C22×C8—31no—
150C22×C8—31no—
151C22×C8—61no—
152C22×C8—61no—
153#126—61no—
154#125—31no—
155#124—31no—
156#123—61no—
157C22⋊Q8—31no—
158#121—31no—
159C4×Q8—31no—
160C4×D4—31no—
161C4×D4—61no—
162#117—61no—
163C2×C42—21no—
164C2×C42C22×D311yescharacteristic
165C4⋊C8—61no—
166C4⋊C8—61no—
167C4⋊C8—61no—
168C4⋊C8C4×D311yescharacteristic
169C4⋊C8C4×D311yescharacteristic
170C22⋊C8—21no—
171C22⋊C8—32no—
172C22⋊C8—61no—
173C22⋊C8—61no—
174#97—21no—
175#97—32no—
176#97—64no—
177#97—61no—
178#97—61no—
179#97C4×D312yes—
180C4×C8—21no—
181C4×C8—31no—
182C4×C8—31no—
183C4×C8—31no—
184C4×C8—31no—
185C4×C8—64no—
186C4×C8—61no—
187C4×C8—61no—
188#95—21no—
189#95C4×D312yes—
190C23×C3C4×D411yescharacteristic
191C22×D3—21no—
192C2×C3×C4—22no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—24no—
196C2×C3×C4—41no—
197C2×C3×C4—42no—
198C2×C3×C4—42no—
199C2×C3×C4C2×C4211yescharacteristic
200C2×C3×C4C2×C22⋊C411yescharacteristic
201C2×C3×C4#11711yescharacteristic
202C2×C3×C4C4×D411yescharacteristic
203C2×C3×C4C4×D411yescharacteristic
204C2×C3×C4C4×D411yescharacteristic
205#67—42no—
206C2×Dic3—21no—
207C2×Dic3—21no—
208C2×Dic3—21no—
209D12—41no—
210C4×D3—41no—
211C4×D3—41no—
212Dic6—41no—
213C3×C8—42no—
214C3×C8—41no—
215C3×C8—41no—
216C3⋊C8—24no—
217C3⋊C8—21no—
218C3⋊C8—21no—
219C3⋊C8—42no—
220C3⋊C8#10412yes—
221C3⋊C8#10412yes—
222C2×Q8—31no—
223C2×D4—31no—
224C2×D4—61no—
225C22×C4—22no—
226C22×C4—61no—
227C22×C4C2×C4×D311yescharacteristic
228C22×C4D3×D411yescharacteristic
229C22×C4#23711yescharacteristic
230M16—42no—
231M16—122no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—22no—
235C2×C8—31no—
236C2×C8—31no—
237C2×C8—31no—
238C2×C8—31no—
239C2×C8—62no—
240C2×C8—62no—
241C2×C8—62no—
242C2×C8—62no—
243C2×C8—62no—
244C2×C8—62no—
245C2×C8—122no—
246C4⋊C4—31no—
247C4⋊C4—31no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C22⋊C4—61no—
253C22⋊C4—61no—
254C22⋊C4—61no—
255C22⋊C4—61no—
256C22⋊C4—61no—
257C42—22no—
258C42—22no—
259C42—61no—
260C42C2×C4×D311yescharacteristic
261C42C2×C4×D311yescharacteristic
262C42D3×D411yescharacteristic
263C42#23911yescharacteristic
264C22×C3—21no—
265C22×C3—21no—
266C22×C3C4×C22⋊C411yescharacteristic
267C22×C3#44412yes—
268C2×D3—41no—
269C2×D3—41no—
270C2×D3—41no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—44no—
277C3×C4C2×#10411yescharacteristic
278C3×C4C2×#10411yescharacteristic
279Dic3—41no—
280Dic3—41no—
281Dic3—41no—
282C23—61no—
283C23#67⋊C411yescharacteristic
284Q8—121no—
285D4—121no—
286D4—122no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—24no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—61no—
295C2×C4—61no—
296C2×C4—61no—
297C2×C4—121no—
298C2×C4—121no—
299C2×C4C42×D311yescharacteristic
300C2×C4D3×C22⋊C411yescharacteristic
301C2×C4#67⋊C411yescharacteristic
302C2×C4#89211yescharacteristic
303C2×C4#89311yescharacteristic
304C2×C4#89311yescharacteristic
305C8—32no—
306C8—32no—
307C8—42no—
308C8—41no—
309C8—41no—
310C8—64no—
311C8—61no—
312C8—61no—
313C8—122no—
314C2×C3—22no—
315C2×C3#175711yescharacteristic
316C2×C3C4×#10411yescharacteristic
317C2×C3#176511yescharacteristic
318D3—81no—
319C22—21no—
320C22—21no—
321C22—121no—
322C22—121no—
323C22—121no—
324C22#466611yescharacteristic
325C22#482312yes—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—44no—
332C4—121no—
333C4—121no—
334C4—121no—
335C4D3×#10411yescharacteristic
336C4D3×#10411yescharacteristic
337C3#824911yescharacteristic, complemented, Sylow
338C2—22no—
339C2—241no—
340C2#6872811yescharacteristic
341C2#6874011yescharacteristic
342C2#6901411yescharacteristic
subgroupG/H
centerC2×C4D3×C22⋊C4
commutatorC2×C3×C4C2×C42
FrattiniC2×C42C22×D3
FittingC3×#1300C2
socleC22×C3C4×C22⋊C4
prime radical#930C2×C4
cosocleC2×C3×C42C23
hypercenter#1300D3
hypocenterC3#8249
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1300

Complements

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

NH ≅ G/Ntype
3. #69847C2semidirect
4. #69434C2semidirect
6. C3×#1300C2semidirect
7. #67751C2semidirect
18. #4838C4semidirect
19. #4834C4semidirect
20. #4831C4semidirect
21. #4830C4semidirect
26. #4465C4semidirect
27. #4464C4semidirect
38. #1300D3semidirect
337. C3#8249semidirect

Maximal subgroups

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

subgroupindex
#793252
#698582
#698472
#694342
#694312
C3×#13002
#677512
#82493