Finite Group Databasebeta

← #10848799

Subgroups — #10848799

Order 768.

1280 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5535C211yesmaximal, characteristic, complemented
2C3×#2619C211yesmaximal, characteristic, complemented
3C2×#5073C211yesmaximal, characteristic, complemented
4C2×#5071C211yesmaximal, characteristic, complemented
5#71684C214yesmaximal, complemented
6#71582C211yesmaximal, characteristic, complemented
7#69657C214yesmaximal, complemented
8#69554C211yesmaximal, characteristic, complemented
9#69553C211yesmaximal, characteristic, complemented
10#25608—31nomaximal, Sylow
11C2×C3×#127C2211yescharacteristic
12C2×C3×#121C2212yes—
13C2×C3×#117C2211yescharacteristic
14C22×#213C2214yes—
15C2×#930C2212yes—
16#5535—24no—
17#5535—24no—
18C2×#894—22no—
19C2×#884—22no—
20C2×#874C2211yescharacteristic
21C2×C4×D12C2212yes—
22C2×#870C2211yescharacteristic
23C3×#394—22no—
24C3×#392—22no—
25C3×#390C2212yes—
26C3×#382C2211yescharacteristic
27C2×#832C2214yes—
28C2×#830C2212yes—
29#5073—24no—
30#5071—24no—
31C2×#809C2214yes—
32C2×#803C2211yescharacteristic
33C2×C4×C3⋊C8C2211yescharacteristic
34#4529C2214yes—
35#4524C2212yes—
36C2×#540—31no—
37C2×#493—31no—
38C2×#490—31no—
39#2619D311yescharacteristic, complemented
40#1999—34no—
41#1991—31no—
42#1961—31no—
43#1879—34no—
44#1750—31no—
45C22×C3×D4—22no—
46C22×C3×D4—22no—
47C22×C3×D4C2312yes—
48C23×C3×C4C2311yescharacteristic
49C22×#67—22no—
50C22×D12—22no—
51C22×D12C2312yes—
52C22×C4×D3—22no—
53C3×#127—22no—
54C3×#127—22no—
55C3×#121—44no—
56C3×#117—22no—
57C3×#117—22no—
58C2×C3×C4⋊C4D412yes—
59C2×C3×C4⋊C4C2312yes—
60C2×C3×C22⋊C4—22no—
61C2×C3×C22⋊C4—24no—
62C2×C3×C22⋊C4D412yes—
63C2×C3×C42C2311yescharacteristic
64C2×C3×C42C2311yescharacteristic
65C2×#213—28no—
66C2×#213—28no—
67C2×#213—44no—
68#930—24no—
69#930—24no—
70C2×(C2×D3)⋊C4—22no—
71C2×(C2×D3)⋊C4—22no—
72C2×C4⋊Dic3C2312yes—
73C2×Dic3⋊C4—22no—
74C22×C3⋊C8C2312yes—
75C22×C3⋊C8C2312yes—
76#894—216no—
77#884—216no—
78#874—22no—
79#874D414yescomplemented
80C4×D12—24no—
81C4×D12—24no—
82#870—22no—
83#870D414yescomplemented
84C3×#95—22no—
85#832—28no—
86#830—24no—
87#809—44no—
88#803—22no—
89C4×C3⋊C8—22no—
90C22×D8—34no—
91#540—64no—
92#540—64no—
93C2×#127—31no—
94C2×#127C2×D311yescharacteristic
95C2×#125—31no—
96C2×#123—62no—
97C2×#121—32no—
98C2×#121—62no—
99C2×#121C2×D312yes—
100C2×C4×D4—32no—
101C2×#117C2×D311yescharacteristic
102#493—64no—
103#490—64no—
104C2×#102—34no—
105C2×#102—34no—
106C2×C22⋊C8—32no—
107C2×#97—31no—
108C2×C4×C8—31no—
109#394—22no—
110#392—22no—
111#390C2×D312yes—
112#382C2×D311yescharacteristic
113#340—34no—
114#336—32no—
115C24×C3—22no—
116C24×C3C2×D411yescharacteristic
117C23×D3—22no—
118C2×C3×D4—22no—
119C2×C3×D4—22no—
120C2×C3×D4—24no—
121C2×C3×D4—24no—
122C2×C3×D4—44no—
123C2×C3×D4—44no—
124C2×C3×D4—84no—
125C2×C3×D4—84no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D412yes—
133C22×C3×C4C2×D412yes—
134C22×C3×C4C2411yescharacteristic
135C2×#67—28no—
136C2×#67—84no—
137C22×Dic3—22no—
138C2×D12—24no—
139C2×D12—24no—
140C2×D12—22no—
141C2×D12—28no—
142C2×D12—84no—
143C2×D12C2×D414yes—
144C2×C4×D3—28no—
145C2×C4×D3—84no—
146C3×C4⋊C4—24no—
147C3×C4⋊C4—24no—
148C3×C22⋊C4—24no—
149C3×C22⋊C4—44no—
150C3×C22⋊C4—84no—
151C3×C42—21no—
152C3×C42—21no—
153C3×C42C2×D412yes—
154C3×C42C2×D412yes—
155#213—416no—
156(C2×D3)⋊C4—28no—
157(C2×D3)⋊C4—28no—
158C4⋊Dic3—22no—
159C4⋊Dic3C2×D414yes—
160Dic3⋊C4—28no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8—22no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8#4114yes—
167C22×D4—22no—
168C22×D4—22no—
169C22×D4—32no—
170C22×D4—62no—
171C22×D4—62no—
172C22×D4C22×D312yes—
173C23×C4—62no—
174C23×C4C22×D311yescharacteristic
175C2×D8—68no—
176C2×D8—68no—
177C2×D8—124no—
178C22×C8—32no—
179C22×C8—32no—
180#127—22no—
181#127—22no—
182#127—34no—
183#127—62no—
184#125—34no—
185#125—62no—
186#123—616no—
187#121—44no—
188#121—616no—
189#121—64no—
190#121—64no—
191C4×D4—64no—
192C4×D4—64no—
193#117—22no—
194#117—22no—
195C2×C4⋊C4—32no—
196C2×C4⋊C4—62no—
197C2×C4⋊C4#6712yes—
198C2×C4⋊C4C22×D312yes—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—62no—
202C2×C22⋊C4—62no—
203C2×C22⋊C4#6712yes—
204C2×C42C22×D311yescharacteristic
205C2×C42C22×D311yescharacteristic
206#102—68no—
207#102—124no—
208C22⋊C8—64no—
209#97—62no—
210C4×C8—62no—
211#95—22no—
212C23×C3—21no—
213C23×C3—21no—
214C23×C3—21no—
215C23×C3—24no—
216C23×C3—42no—
217C23×C3—42no—
218C23×C3—42no—
219C23×C3—44no—
220C23×C3—44no—
221C23×C3C22×D411yescharacteristic
222C22×D3—24no—
223C22×D3—42no—
224C22×D3—42no—
225C22×D3—44no—
226C22×D3—44no—
227C3×D4—44no—
228C3×D4—44no—
229C3×D4—88no—
230C3×D4—88no—
231C2×C3×C4—24no—
232C2×C3×C4—21no—
233C2×C3×C4—22no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—24no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—44no—
241C2×C3×C4#12014yes—
242C2×C3×C4C22×D411yescharacteristic
243C2×C3×C4C22×D411yescharacteristic
244C2×C3×C4C2×#4112yes—
245C2×C3×C4H(5,𝔽2)12yes—
246#67—88no—
247C2×Dic3—24no—
248C2×Dic3—44no—
249D12—28no—
250D12—44no—
251D12—88no—
252C4×D3—88no—
253C3⋊C8—24no—
254C3⋊C8—42no—
255C24—22no—
256C24—62no—
257C24C2×#6711yescharacteristic
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—24no—
261C2×D4—24no—
262C2×D4—34no—
263C2×D4—44no—
264C2×D4—44no—
265C2×D4—68no—
266C2×D4—68no—
267C2×D4—62no—
268C2×D4—64no—
269C2×D4—64no—
270C2×D4—84no—
271C2×D4—84no—
272C2×D4—244no—
273C2×D4—244no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—68no—
280C22×C4—62no—
281C22×C4—244no—
282C22×C4C2×#6712yes—
283C22×C4C2×#6711yescharacteristic
284C22×C4C2×#6712yes—
285C22×C4C23×D311yescharacteristic
286D8—1216no—
287C2×C8—34no—
288C2×C8—62no—
289C2×C8—62no—
290C2×C8—62no—
291C2×C8—62no—
292C2×C8—62no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—34no—
296C4⋊C4—68no—
297C4⋊C4—62no—
298C22⋊C4—24no—
299C22⋊C4—44no—
300C22⋊C4—68no—
301C22⋊C4—68no—
302C22⋊C4—84no—
303C42—21no—
304C42—21no—
305C42D3×D412yes—
306C42D3×D412yes—
307C22×C3—42no—
308C22×C3—41no—
309C22×C3—42no—
310C22×C3—42no—
311C22×C3—84no—
312C22×C3—82no—
313C22×C3—82no—
314C22×C3—82no—
315C22×C3—84no—
316C22×C3C2×#12011yescharacteristic
317C22×C3#53211yescharacteristic
318C22×C3#53511yescharacteristic
319C22×C3#57614yes—
320C2×D3—44no—
321C2×D3—44no—
322C2×D3—44no—
323C2×D3—84no—
324C2×D3—84no—
325C3×C4—24no—
326C3×C4—41no—
327C3×C4—42no—
328C3×C4—82no—
329C3×C4#54612yes—
330C3×C4#55012yes—
331Dic3—44no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—24no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—44no—
340C23—44no—
341C23—64no—
342C23—122no—
343C23—122no—
344C23—124no—
345C23—124no—
346C23C22×#6711yescharacteristic
347D4—44no—
348D4—44no—
349D4—68no—
350D4—88no—
351D4—88no—
352D4—124no—
353D4—248no—
354D4—248no—
355C2×C4—24no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—24no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—64no—
366C2×C4—124no—
367C2×C4—248no—
368C2×C4#93714yes—
369C2×C4C2×D3×D411yescharacteristic
370C2×C4C2×D3×D411yescharacteristic
371C2×C4#100412yes—
372C2×C4D3×#4112yes—
373C8—64no—
374C8—122no—
375C2×C3—81no—
376C2×C3—162no—
377C2×C3#243111yescharacteristic
378C2×C3#300812yes—
379C2×C3#317612yes—
380C2×C3#322012yes—
381D3—84no—
382C22—42no—
383C22—42no—
384C22—41no—
385C22—42no—
386C22—84no—
387C22—82no—
388C22—82no—
389C22—82no—
390C22—84no—
391C22—124no—
392C22—124no—
393C22—124no—
394C22—244no—
395C22—244no—
396C22C2×#93711yescharacteristic
397C22#580511yescharacteristic
398C22#583211yescharacteristic
399C22#583514yes—
400C4—24no—
401C4—42no—
402C4—41no—
403C4—82no—
404C4—124no—
405C4#559912yes—
406C4#561212yes—
407C3#2560811yescharacteristic, complemented, Sylow
408C2—81no—
409C2—162no—
410C2—244no—
411C2#8265211yescharacteristic
412C2#8338312yes—
413C2#8338812yes—
414C2#8343212yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2619C2
socleC23×C3C22×D4
hypercenter#2619D3
hypocenterC3#25608
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2619

Complements

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

NH ≅ G/Ntype
1. C2×#5535C2semidirect
2. C3×#2619C2semidirect
3. C2×#5073C2semidirect
4. C2×#5071C2semidirect
5. #71684C2semidirect
6. #71582C2semidirect
7. #69657C2semidirect
8. #69554C2semidirect
9. #69553C2semidirect
39. #2619D3semidirect
79. #874D4semidirect
83. #870D4semidirect
407. C3#25608semidirect

Maximal subgroups

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

subgroupindex
C2×#55352
C3×#26192
C2×#50732
C2×#50712
#716842
#715822
#696572
#695542
#695532
#256083