Finite Group Databasebeta

← #10848781

Subgroups — #10848781

Order 768.

1180 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5535C211yesmaximal, characteristic
2C3×#2621C211yesmaximal, characteristic, complemented
3C2×#5085C211yesmaximal, characteristic, complemented
4C2×#5082C211yesmaximal, characteristic, complemented
5#71699C214yesmaximal, complemented
6#71582C211yesmaximal, characteristic, complemented
7#69687C214yesmaximal, complemented
8#69554C211yesmaximal, characteristic, complemented
9#69553C211yesmaximal, characteristic, complemented
10#25606—31nomaximal, Sylow
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×C22⋊Q8C2212yes—
13C2×C3×#117C2211yescharacteristic
14C22×#215C2214yes—
15C2×#930C2212yes—
16#5535—24no—
17#5535—24no—
18C2×#894—22no—
19C2×#884—22no—
20C2×#874C2211yescharacteristic
21C2×C4×D12C2212yes—
22C2×#870C2211yescharacteristic
23C3×#398—22no—
24C3×#397—22no—
25C3×#391C2212yes—
26C3×#382C2211yescharacteristic
27C2×#835C2214yes—
28C2×#830C2212yes—
29#5085—24no—
30#5082—24no—
31C2×#809C2214yes—
32C2×#803C2211yescharacteristic
33C2×C4×C3⋊C8C2211yescharacteristic
34#4529C2214yes—
35#4524C2212yes—
36C2×#540—31no—
37C2×#492—31no—
38C2×#491—31no—
39#2621D311yescharacteristic, complemented
40#2000—34no—
41#1991—31no—
42#1961—31no—
43#1877—34no—
44#1750—31no—
45C22×C3×Q8C2312yes—
46C23×C3×C4C2311yescharacteristic
47C22×#67—22no—
48C22×D12—22no—
49C22×D12C2312yes—
50C22×C4×D3—22no—
51C3×C4⋊Q8—22no—
52C3×C4⋊Q8—22no—
53C3×C22⋊Q8—44no—
54C3×#117—22no—
55C3×#117—22no—
56C2×C3×C4⋊C4—22no—
57C2×C3×C4⋊C4—22no—
58C2×C3×C4⋊C4D412yes—
59C2×C3×C4⋊C4C2312yes—
60C2×C3×C22⋊C4—22no—
61C2×C3×C22⋊C4D412yes—
62C2×C3×C42C2311yescharacteristic
63C2×C3×C42C2311yescharacteristic
64C2×#215—28no—
65C2×#215—28no—
66C2×#215—44no—
67#930—24no—
68#930—24no—
69C2×(C2×D3)⋊C4—22no—
70C2×(C2×D3)⋊C4—22no—
71C2×C4⋊Dic3C2312yes—
72C2×Dic3⋊C4—22no—
73C22×C3⋊C8C2312yes—
74C22×C3⋊C8C2312yes—
75#894—216no—
76#884—216no—
77#874—22no—
78#874D414yes—
79C4×D12—24no—
80C4×D12—24no—
81#870—22no—
82#870D414yes—
83C3×#95—22no—
84C3×#95—24no—
85#835—28no—
86#830—24no—
87#809—44no—
88#803—22no—
89C4×C3⋊C8—22no—
90C22×SD16—34no—
91#540—64no—
92#540—64no—
93C2×C4⋊Q8C2×D311yescharacteristic
94C2×#127—31no—
95C2×#125—31no—
96C2×#123—62no—
97C2×C22⋊Q8C2×D312yes—
98C2×#121—32no—
99C2×#121—62no—
100C2×C4×D4—32no—
101C2×#117C2×D311yescharacteristic
102#492—64no—
103#491—64no—
104C2×#103—34no—
105C2×#102—34no—
106C2×C22⋊C8—32no—
107C2×#97—31no—
108C2×C4×C8—31no—
109#398—22no—
110#397—22no—
111#391C2×D312yes—
112#382C2×D311yescharacteristic
113#340—34no—
114#336—32no—
115C24×C3C2×D411yescharacteristic
116C23×D3—22no—
117C2×C3×Q8—22no—
118C2×C3×Q8—22no—
119C2×C3×Q8—24no—
120C2×C3×Q8—24no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
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—28no—
132C2×#67—84no—
133C22×Dic3—22no—
134C2×D12—24no—
135C2×D12—24no—
136C2×D12—22no—
137C2×D12—28no—
138C2×D12—84no—
139C2×D12C2×D414yes—
140C2×C4×D3—28no—
141C2×C4×D3—84no—
142C3×C4⋊C4—24no—
143C3×C4⋊C4—24no—
144C3×C4⋊C4—44no—
145C3×C4⋊C4—44no—
146C3×C22⋊C4—24no—
147C3×C22⋊C4—44no—
148C3×C42—21no—
149C3×C42—21no—
150C3×C42C2×D412yes—
151C3×C42C2×D412yes—
152#215—416no—
153(C2×D3)⋊C4—28no—
154(C2×D3)⋊C4—28no—
155C4⋊Dic3—22no—
156C4⋊Dic3C2×D414yes—
157Dic3⋊C4—28no—
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×C4—62no—
169C23×C4C22×D311yescharacteristic
170C2×SD16—68no—
171C2×SD16—68no—
172C2×SD16—124no—
173C22×C8—32no—
174C22×C8—32no—
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177#127—34no—
178#127—62no—
179#125—34no—
180#125—62no—
181#123—616no—
182C22⋊Q8—44no—
183#121—616no—
184#121—64no—
185#121—64no—
186C4×D4—64no—
187C4×D4—64no—
188#117—22no—
189#117—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—32no—
193C2×C4⋊C4—62no—
194C2×C4⋊C4#6712yes—
195C2×C4⋊C4C22×D312yes—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—62no—
198C2×C22⋊C4—62no—
199C2×C22⋊C4#6712yes—
200C2×C42C22×D311yescharacteristic
201C2×C42C22×D311yescharacteristic
202#103—68no—
203#102—124no—
204C22⋊C8—64no—
205#97—62no—
206C4×C8—62no—
207#95—22no—
208#95—24no—
209C23×C3—21no—
210C23×C3—21no—
211C23×C3—21no—
212C23×C3—24no—
213C23×C3C22×D411yescharacteristic
214C22×D3—24no—
215C22×D3—42no—
216C22×D3—42no—
217C22×D3—44no—
218C22×D3—44no—
219C3×Q8—44no—
220C3×Q8—44no—
221C2×C3×C4—24no—
222C2×C3×C4—21no—
223C2×C3×C4—22no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—24no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4—44no—
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—24no—
241C2×Dic3—44no—
242D12—28no—
243D12—44no—
244D12—88no—
245C4×D3—88no—
246C3⋊C8—24no—
247C3⋊C8—42no—
248C24—62no—
249C24C2×#6711yescharacteristic
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—24no—
253C2×Q8—24no—
254C2×D4—34no—
255C2×D4—68no—
256C2×D4—68no—
257C2×D4—62no—
258C2×D4—64no—
259C2×D4—64no—
260C2×D4—244no—
261C2×D4—244no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—68no—
269C22×C4—62no—
270C22×C4—244no—
271C22×C4C2×#6712yes—
272C22×C4C2×#6711yescharacteristic
273C22×C4C2×#6712yes—
274C22×C4C23×D311yescharacteristic
275SD16—1216no—
276C2×C8—34no—
277C2×C8—62no—
278C2×C8—62no—
279C2×C8—62no—
280C2×C8—62no—
281C2×C8—62no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—34no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—68no—
288C4⋊C4—62no—
289C22⋊C4—24no—
290C22⋊C4—44no—
291C22⋊C4—68no—
292C22⋊C4—68no—
293C42—21no—
294C42—21no—
295C42D3×D412yes—
296C42D3×D412yes—
297C22×C3—42no—
298C22×C3—41no—
299C22×C3—42no—
300C22×C3—42no—
301C22×C3C2×#12011yescharacteristic
302C22×C3#53211yescharacteristic
303C22×C3#53511yescharacteristic
304C22×C3#57714yes—
305C2×D3—44no—
306C2×D3—44no—
307C2×D3—44no—
308C2×D3—84no—
309C2×D3—84no—
310C3×C4—24no—
311C3×C4—41no—
312C3×C4—42no—
313C3×C4—82no—
314C3×C4—82no—
315C3×C4#54612yes—
316C3×C4#55012yes—
317Dic3—44no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—24no—
322C23—64no—
323C23—122no—
324C23—122no—
325C23—124no—
326C23—124no—
327C23C22×#6711yescharacteristic
328Q8—44no—
329Q8—44no—
330D4—68no—
331D4—124no—
332D4—248no—
333D4—248no—
334C2×C4—24no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—24no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—64no—
348C2×C4—124no—
349C2×C4—248no—
350C2×C4#93714yes—
351C2×C4C2×D3×D411yescharacteristic
352C2×C4C2×D3×D411yescharacteristic
353C2×C4#100412yes—
354C2×C4D3×#4112yes—
355C8—64no—
356C8—122no—
357C2×C3—81no—
358C2×C3#243111yescharacteristic
359C2×C3#300912yes—
360C2×C3#317412yes—
361C2×C3#322112yes—
362D3—84no—
363C22—42no—
364C22—42no—
365C22—41no—
366C22—42no—
367C22—124no—
368C22—124no—
369C22—124no—
370C22—244no—
371C22—244no—
372C22C2×#93711yescharacteristic
373C22#580511yescharacteristic
374C22#583211yescharacteristic
375C22#583714yes—
376C4—24no—
377C4—42no—
378C4—41no—
379C4—82no—
380C4—82no—
381C4—124no—
382C4#559912yes—
383C4#561212yes—
384C3#2560611yescharacteristic, complemented, Sylow
385C2—81no—
386C2—244no—
387C2#8265211yescharacteristic
388C2#8340412yes—
389C2#8340912yes—
390C2#8344312yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2621C2
socleC23×C3C22×D4
prime radicalC2×#5535C2
hypercenter#2621D3
hypocenterC3#25606
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2621

Complements

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

NH ≅ G/Ntype
2. C3×#2621C2semidirect
3. C2×#5085C2semidirect
4. C2×#5082C2semidirect
5. #71699C2semidirect
6. #71582C2semidirect
7. #69687C2semidirect
8. #69554C2semidirect
9. #69553C2semidirect
39. #2621D3semidirect
384. C3#25606semidirect

Maximal subgroups

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

subgroupindex
C2×#55352
C3×#26212
C2×#50852
C2×#50822
#716992
#715822
#696872
#695542
#695532
#256063