Finite Group Databasebeta

← #10848785

Subgroups — #10848785

Order 768.

1152 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#541C211yesmaximal, characteristic, complemented
2C2×#5085C211yesmaximal, characteristic, complemented
3C2×#5067C211yesmaximal, characteristic, complemented
4#82286C211yesmaximal, characteristic
5#71786C211yesmaximal, characteristic, complemented
6#71740C211yesmaximal, characteristic, complemented
7#71699C214yesmaximal, complemented
8#71633C214yesmaximal, complemented
9#71582C211yesmaximal, characteristic, complemented
10#25607—31nomaximal, Sylow
11C3×#541—24no—
12C3×#541—24no—
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×#126—22no—
15C2×C3×#125—22no—
16C2×C3×#125C2211yescharacteristic
17C2×C3×C22⋊Q8C2212yes—
18C2×C3×C4×Q8C2212yes—
19C2×C3×#117C2211yescharacteristic
20C22×#215C2214yes—
21C2×#930C2212yes—
22C2×#874C2211yescharacteristic
23C2×#835C2214yes—
24C2×#830C2212yes—
25#5085—24no—
26#5067—24no—
27#4991—22no—
28C2×#809C2214yes—
29#4960—22no—
30#4939C2212yes—
31#4936C2211yescharacteristic
32C2×#803C2211yescharacteristic
33C2×C4×C3⋊C8C2211yescharacteristic
34#4529C2214yes—
35#4524C2212yes—
36C2×#541D311yescharacteristic, complemented
37C2×#492—31no—
38C2×#490—31no—
39#2619—31no—
40#2000—34no—
41#1991—31no—
42#1960—31no—
43#1880—34no—
44#1750—31no—
45C22×C3×Q8C2312yes—
46C23×C3×C4C2311yescharacteristic
47C22×#67—22no—
48C22×D12—22no—
49C22×D12C2312yes—
50C3×C4⋊Q8—22no—
51C3×C4⋊Q8D414yescomplemented
52C3×#126—216no—
53C3×#125—216no—
54C3×#125—22no—
55C3×#125D414yescomplemented
56C3×C22⋊Q8—24no—
57C3×C22⋊Q8—24no—
58C3×C4×Q8—24no—
59C3×C4×Q8—24no—
60C3×#117—22no—
61C3×#117—22no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4D412yes—
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4D412yes—
70C2×C3×C42—22no—
71C2×C3×C42C2311yescharacteristic
72C2×C3×C42C2311yescharacteristic
73C2×#215—28no—
74C2×#215—28no—
75C2×#215—44no—
76#930—44no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—24no—
79C2×C4⋊Dic3C2312yes—
80C22×C3⋊C8C2312yes—
81C22×C3⋊C8C2312yes—
82#874—22no—
83#874—22no—
84#835—44no—
85C3⋊#95—22no—
86#830—24no—
87#809—28no—
88#803—22no—
89C4×C3⋊C8—22no—
90C22×SD16—34no—
91#541—24no—
92#541—24no—
93C2×C4⋊Q8C2×D311yescharacteristic
94C2×#127—31no—
95C2×#126—22no—
96C2×#125—22no—
97C2×#125C2×D311yescharacteristic
98C2×C22⋊Q8C2×D312yes—
99C2×#121—32no—
100C2×C4×Q8C2×D312yes—
101C2×#117C2×D311yescharacteristic
102#492—64no—
103#490—64no—
104C2×#103—34no—
105C2×#102—34no—
106C2×C22⋊C8—32no—
107C2×#97—31no—
108C2×C4×C8—31no—
109#394—62no—
110#392—62no—
111#390—32no—
112#382—31no—
113#340—34no—
114#336—32no—
115C24×C3C2×D411yescharacteristic
116C23×D3—22no—
117C2×C3×Q8—22no—
118C2×C3×Q8—24no—
119C2×C3×Q8—24no—
120C2×C3×Q8C2×D414yes—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4C2×D412yes—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D412yes—
130C22×C3×C4C2411yescharacteristic
131C2×#67—44no—
132C2×#67—84no—
133C22×Dic3—22no—
134C2×D12—22no—
135C2×D12—22no—
136C2×D12—24no—
137C2×D12—24no—
138C2×D12—44no—
139C2×D12—84no—
140C3×C4⋊C4—28no—
141C3×C4⋊C4—28no—
142C3×C4⋊C4—28no—
143C3×C4⋊C4—28no—
144C3×C4⋊C4—24no—
145C3×C4⋊C4—22no—
146C3×C4⋊C4C2×D414yes—
147C3×C22⋊C4—28no—
148C3×C22⋊C4—24no—
149C3×C42—28no—
150C3×C42—21no—
151C3×C42—21no—
152C3×C42C2×D412yes—
153C3×C42C2×D412yes—
154#215—416no—
155(C2×D3)⋊C4—44no—
156(C2×D3)⋊C4—84no—
157C4⋊Dic3—24no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8#4114yes—
164C22×Q8C22×D312yes—
165C22×D4—32no—
166C22×D4—62no—
167C22×D4—62no—
168C23×C4C22×D311yescharacteristic
169C2×SD16—68no—
170C2×SD16—68no—
171C2×SD16—124no—
172C22×C8—32no—
173C22×C8—32no—
174C4⋊Q8—22no—
175C4⋊Q8#6714yescomplemented
176#127—62no—
177#127—62no—
178#126—216no—
179#125—216no—
180#125—22no—
181#125#6714yescomplemented
182C22⋊Q8—24no—
183C22⋊Q8—24no—
184#121—124no—
185C4×Q8—24no—
186C4×Q8—24no—
187#117—22no—
188#117—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—32no—
194C2×C4⋊C4#6712yes—
195C2×C4⋊C4C22×D312yes—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—62no—
198C2×C22⋊C4—64no—
199C2×C22⋊C4#6712yes—
200C2×C42—22no—
201C2×C42C22×D311yescharacteristic
202C2×C42C22×D311yescharacteristic
203#103—124no—
204#102—68no—
205C22⋊C8—64no—
206#97—62no—
207C4×C8—62no—
208#95—62no—
209C23×C3—21no—
210C23×C3—21no—
211C23×C3—21no—
212C23×C3—24no—
213C23×C3C22×D411yescharacteristic
214C22×D3—42no—
215C22×D3—42no—
216C22×D3—42no—
217C22×D3—44no—
218C22×D3—44no—
219C3×Q8—28no—
220C3×Q8—44no—
221C2×C3×C4—24no—
222C2×C3×C4—22no—
223C2×C3×C4—24no—
224C2×C3×C4—24no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—24no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—44no—
232C2×C3×C4—42no—
233C2×C3×C4—42no—
234C2×C3×C4#12014yes—
235C2×C3×C4C22×D411yescharacteristic
236C2×C3×C4C22×D411yescharacteristic
237C2×C3×C4C2×#4112yes—
238C2×C3×C4H(5,𝔽2)12yes—
239#67—88no—
240C2×Dic3—42no—
241C2×Dic3—44no—
242D12—44no—
243D12—44no—
244D12—88no—
245C3⋊C8—24no—
246C3⋊C8—42no—
247C24—62no—
248C24C2×#6711yescharacteristic
249C2×Q8—22no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8C2×#6714yes—
253C2×D4—64no—
254C2×D4—64no—
255C2×D4—62no—
256C2×D4—62no—
257C2×D4—124no—
258C2×D4—124no—
259C2×D4—244no—
260C2×D4—244no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—62no—
268C22×C4C2×#6712yes—
269C22×C4C2×#6711yescharacteristic
270C22×C4C2×#6712yes—
271C22×C4C23×D311yescharacteristic
272SD16—1216no—
273C2×C8—34no—
274C2×C8—62no—
275C2×C8—62no—
276C2×C8—62no—
277C2×C8—62no—
278C2×C8—62no—
279C4⋊C4—28no—
280C4⋊C4—28no—
281C4⋊C4—28no—
282C4⋊C4—28no—
283C4⋊C4—24no—
284C4⋊C4—22no—
285C4⋊C4—64no—
286C4⋊C4C2×#6714yes—
287C22⋊C4—28no—
288C22⋊C4—24no—
289C22⋊C4—124no—
290C22⋊C4—244no—
291C42—28no—
292C42—21no—
293C42—21no—
294C42C2×#6712yes—
295C42C2×#6712yes—
296C22×C3—42no—
297C22×C3—41no—
298C22×C3—42no—
299C22×C3—42no—
300C22×C3C2×#12011yescharacteristic
301C22×C3#53211yescharacteristic
302C22×C3#53511yescharacteristic
303C22×C3#57714yes—
304C2×D3—84no—
305C2×D3—82no—
306C2×D3—82no—
307C2×D3—82no—
308C2×D3—84no—
309C3×C4—24no—
310C3×C4—41no—
311C3×C4—44no—
312C3×C4—42no—
313C3×C4—44no—
314C3×C4#54612yes—
315C3×C4#55012yes—
316Dic3—82no—
317C23—21no—
318C23—21no—
319C23—24no—
320C23—21no—
321C23—122no—
322C23—122no—
323C23—124no—
324C23—124no—
325C23—122no—
326C23C22×#6711yescharacteristic
327Q8—28no—
328Q8—44no—
329D4—124no—
330D4—124no—
331D4—248no—
332D4—248no—
333C2×C4—24no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—24no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—21no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—122no—
347C2×C4—124no—
348C2×C4#95314yes—
349C2×C4D3×#4112yes—
350C2×C4#100912yes—
351C2×C4C22×#6711yescharacteristic
352C2×C4C22×#6711yescharacteristic
353C8—64no—
354C8—122no—
355C2×C3—81no—
356C2×C3#243111yescharacteristic
357C2×C3#301012yes—
358C2×C3#317412yes—
359C2×C3#322212yes—
360D3—162no—
361C22—41no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—242no—
366C22—244no—
367C22—244no—
368C22—242no—
369C22—242no—
370C22#583011yescharacteristic
371C22#583211yescharacteristic
372C22#583714yes—
373C22C2×#95311yescharacteristic
374C4—24no—
375C4—41no—
376C4—44no—
377C4—42no—
378C4—44no—
379C4—242no—
380C4#582912yes—
381C4#583412yes—
382C3#2560711yescharacteristic, complemented, Sylow
383C2—81no—
384C2—482no—
385C2#8344312yes—
386C2#8344812yes—
387C2#8409211yescharacteristic
388C2#8411312yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#541C2
socleC23×C3C22×D4
prime radical#82286C2
hypercenterC2×#541D3
hypocenterC3#25607
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#541

Complements

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

NH ≅ G/Ntype
1. C2×C3×#541C2semidirect
2. C2×#5085C2semidirect
3. C2×#5067C2semidirect
5. #71786C2semidirect
6. #71740C2semidirect
7. #71699C2semidirect
8. #71633C2semidirect
9. #71582C2semidirect
36. C2×#541D3semidirect
51. C3×C4⋊Q8D4semidirect
55. C3×#125D4semidirect
175. C4⋊Q8#67semidirect
181. #125#67semidirect
382. C3#25607semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5412
C2×#50852
C2×#50672
#822862
#717862
#717402
#716992
#716332
#715822
#256073