Finite Group Databasebeta

← #10848795

Subgroups — #10848795

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×#5719C211yesmaximal, characteristic, complemented
2C2×C3×#493C211yesmaximal, characteristic, complemented
3#83189C211yesmaximal, characteristic, complemented
4C2×#5071C211yesmaximal, characteristic, complemented
5#72043C214yesmaximal, complemented
6#72037C214yesmaximal, complemented
7#71765C211yesmaximal, characteristic, complemented
8#69778C211yesmaximal, characteristic, complemented
9#69753C211yesmaximal, characteristic, complemented
10#25608—31nomaximal, Sylow
11C22×C3×D8C2214yes—
12C2×C3×#127C2211yescharacteristic
13C2×C3×#127C2211yescharacteristic
14C2×#939—22no—
15C2×#938C2212yes—
16C2×#938C2212yes—
17C2×#935—22no—
18C2×D4×Dic3C2212yes—
19#5719—24no—
20#5719—24no—
21C2×#872C2211yescharacteristic
22C2×#870C2211yescharacteristic
23C3×#493—24no—
24C2×C3×C4×C8C2211yescharacteristic
25#5223—22no—
26#5222—22no—
27#5221C2212yes—
28C2×#832C2214yes—
29C2×#832C2214yes—
30C2×(C2×D3)⋊C8C2212yes—
31#5071—24no—
32#4936C2211yescharacteristic
33C2×#803C2211yescharacteristic
34#4551C2214yes—
35#4550C2212yes—
36C2×#540—31no—
37C2×#493D311yescharacteristic, complemented
38C2×#490—31no—
39#2619—31no—
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—
48C22×C3×D4C2312yes—
49C22×#67—22no—
50C22×#67—22no—
51C23×Dic3—22no—
52C22×C4×D3C2311yescharacteristic
53C2×C3×D8—28no—
54C2×C3×D8—28no—
55C2×C3×D8—44no—
56C22×C3×C8C2312yes—
57C3×#127—22no—
58C3×#127—22no—
59C3×#127—22no—
60C3×#127D414yescomplemented
61C2×C3×C42C2311yescharacteristic
62C2×#217—22no—
63C2×#217—22no—
64C2×#217—22no—
65C2×#217—24no—
66#939—216no—
67#938—24no—
68#938—24no—
69#938—44no—
70#935—216no—
71D4×Dic3—24no—
72D4×Dic3—24no—
73C2×(C2×D3)⋊C4D412yes—
74C2×C4⋊Dic3C2312yes—
75C2×C4⋊Dic3C2312yes—
76C2×Dic3⋊C4—22no—
77C2×Dic3⋊C4D412yes—
78C2×C4×Dic3C2311yescharacteristic
79C22×C3⋊C8C2312yes—
80#872—22no—
81#872—22no—
82#870—22no—
83#870D414yescomplemented
84C3×C4×C8—22no—
85#832—28no—
86#832—44no—
87C3⋊#95—22no—
88(C2×D3)⋊C8—24no—
89#803—22no—
90C22×D8C2×D314yes—
91#540—64no—
92#540—64no—
93C2×#127C2×D311yescharacteristic
94C2×#127C2×D311yescharacteristic
95C2×#125—31no—
96C2×#123—62no—
97C2×#121—32no—
98C2×#121—32no—
99C2×#121—62no—
100C2×C4×D4—32no—
101C2×#117—31no—
102#493—24no—
103#490—64no—
104C2×#102—34no—
105C2×#102—34no—
106C2×C22⋊C8—32no—
107C2×#97—31no—
108C2×C4×C8C2×D311yescharacteristic
109#394—62no—
110#392—62no—
111#390—32no—
112#382—31no—
113#340—34no—
114#336—32no—
115C24×C3—22no—
116C24×C3—22no—
117C23×D3C2×D411yescharacteristic
118C2×C3×D4—28no—
119C2×C3×D4—22no—
120C2×C3×D4—22no—
121C2×C3×D4—22no—
122C2×C3×D4—24no—
123C2×C3×D4—24no—
124C2×C3×D4—24no—
125C2×C3×D4—24no—
126C2×C3×D4—44no—
127C2×C3×D4—84no—
128C2×C3×D4—84no—
129C2×C3×D4C2×D414yes—
130C22×C3×C4C2×D412yes—
131C22×C3×C4C2411yescharacteristic
132C2×#67—28no—
133C2×#67—44no—
134C2×#67—84no—
135C2×#67—84no—
136C22×Dic3—22no—
137C22×Dic3—28no—
138C22×Dic3—22no—
139C22×Dic3—84no—
140C22×Dic3C2×D412yes—
141C22×Dic3C2×D411yescharacteristic
142C2×C4×D3—21no—
143C2×C4×D3—21no—
144C2×C4×D3—22no—
145C2×C4×D3—22no—
146C3×D8—416no—
147C2×C3×C8—22no—
148C2×C3×C8—22no—
149C2×C3×C8#4114yes—
150C3×C42C2×D412yes—
151C3×C42C2×D412yes—
152#217—28no—
153#217—28no—
154#217—44no—
155#217—84no—
156(C2×D3)⋊C4—24no—
157C4⋊Dic3—22no—
158C4⋊Dic3—24no—
159C4⋊Dic3C2×D414yes—
160Dic3⋊C4—28no—
161Dic3⋊C4—24no—
162C4×Dic3—21no—
163C4×Dic3—21no—
164C2×C3⋊C8—22no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—22no—
167C22×D4—22no—
168C22×D4—22no—
169C22×D4—62no—
170C22×D4—62no—
171C22×D4C22×D312yes—
172C22×D4C22×D312yes—
173C23×C4—31no—
174C23×C4—62no—
175C2×D8—28no—
176C2×D8—28no—
177C2×D8—44no—
178C22×C8—32no—
179C22×C8C22×D312yes—
180#127—22no—
181#127—22no—
182#127—22no—
183#127#6714yescomplemented
184#125—34no—
185#125—62no—
186#123—616no—
187#121—64no—
188#121—616no—
189#121—64no—
190#121—124no—
191C4×D4—64no—
192C4×D4—64no—
193#117—62no—
194#117—62no—
195C2×C4⋊C4—32no—
196C2×C4⋊C4—32no—
197C2×C4⋊C4—32no—
198C2×C4⋊C4—62no—
199C2×C22⋊C4—32no—
200C2×C22⋊C4—62no—
201C2×C22⋊C4—62no—
202C2×C22⋊C4—62no—
203C2×C22⋊C4—64no—
204C2×C42—31no—
205C2×C42C22×D311yescharacteristic
206#102—68no—
207#102—124no—
208C22⋊C8—64no—
209#97—62no—
210C4×C8—22no—
211#95—62no—
212C23×C3—24no—
213C23×C3—42no—
214C23×C3—42no—
215C23×C3—44no—
216C23×C3—44no—
217C23×C3—42no—
218C23×C3—42no—
219C23×C3—42no—
220C23×C3—44no—
221C23×C3—44no—
222C23×C3C22×D411yescharacteristic
223C22×D3—21no—
224C22×D3—21no—
225C22×D3—21no—
226C22×D3—24no—
227C3×D4—28no—
228C3×D4—44no—
229C3×D4—44no—
230C3×D4—44no—
231C3×D4—88no—
232C3×D4—88no—
233C2×C3×C4—24no—
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—
240#67—88no—
241C2×Dic3—22no—
242C2×Dic3—24no—
243C2×Dic3—21no—
244C2×Dic3—21no—
245C2×Dic3—21no—
246C2×Dic3—24no—
247C2×Dic3—44no—
248C2×Dic3—42no—
249C2×Dic3—44no—
250C2×Dic3—88no—
251C4×D3—42no—
252C4×D3—42no—
253C3×C8—24no—
254C3⋊C8—42no—
255C24—22no—
256C24—22no—
257C24—31no—
258C2×D4—28no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—24no—
262C2×D4—24no—
263C2×D4—24no—
264C2×D4—24no—
265C2×D4—22no—
266C2×D4—44no—
267C2×D4—68no—
268C2×D4—84no—
269C2×D4—84no—
270C2×D4—124no—
271C2×D4—244no—
272C2×D4—244no—
273C2×D4C2×#6714yes—
274C22×C4—32no—
275C22×C4—31no—
276C22×C4—62no—
277C22×C4—68no—
278C22×C4—61no—
279C22×C4—62no—
280C22×C4—62no—
281C22×C4—61no—
282C22×C4—62no—
283C22×C4—244no—
284C22×C4D3×D412yes—
285C22×C4C23×D311yescharacteristic
286D8—416no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—62no—
290C2×C8—62no—
291C2×C8—62no—
292C2×C8#23714yes—
293C4⋊C4—34no—
294C4⋊C4—64no—
295C4⋊C4—68no—
296C4⋊C4—62no—
297C4⋊C4—64no—
298C22⋊C4—68no—
299C22⋊C4—68no—
300C22⋊C4—64no—
301C22⋊C4—124no—
302C22⋊C4—244no—
303C42—61no—
304C42—61no—
305C42D3×D412yes—
306C42C2×#6712yes—
307C22×C3—44no—
308C22×C3—44no—
309C22×C3—44no—
310C22×C3—84no—
311C22×C3—84no—
312C22×C3—84no—
313C22×C3—82no—
314C22×C3—82no—
315C22×C3—82no—
316C22×C3—84no—
317C22×C3C2×#12011yescharacteristic
318C22×C3#53211yescharacteristic
319C22×C3#53511yescharacteristic
320C22×C3#57614yes—
321C2×D3—42no—
322C2×D3—41no—
323C2×D3—42no—
324C2×D3—42no—
325C3×C4—24no—
326C3×C4#54612yes—
327C3×C4#55012yes—
328Dic3—41no—
329Dic3—42no—
330Dic3—44no—
331Dic3—82no—
332C23—24no—
333C23—42no—
334C23—42no—
335C23—44no—
336C23—44no—
337C23—42no—
338C23—42no—
339C23—44no—
340C23—44no—
341C23—42no—
342C23—61no—
343C23—61no—
344C23—64no—
345C23—61no—
346C23C2×D3×D411yescharacteristic
347D4—28no—
348D4—44no—
349D4—44no—
350D4—44no—
351D4—88no—
352D4—88no—
353D4—248no—
354D4—248no—
355C2×C4—24no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4—64no—
359C2×C4—62no—
360C2×C4—61no—
361C2×C4—64no—
362C2×C4—124no—
363C2×C4—122no—
364C2×C4—122no—
365C2×C4—122no—
366C2×C4—124no—
367C2×C4—248no—
368C2×C4#93714yes—
369C2×C4C2×D3×D411yescharacteristic
370C2×C4C2×#23712yes—
371C2×C4#100412yes—
372C2×C4C22×#6711yescharacteristic
373C8—24no—
374C8—122no—
375C2×C3—84no—
376C2×C3—162no—
377C2×C3#243111yescharacteristic
378C2×C3#300812yes—
379C2×C3#317612yes—
380C2×C3#322012yes—
381D3—81no—
382C22—44no—
383C22—44no—
384C22—44no—
385C22—84no—
386C22—84no—
387C22—84no—
388C22—84no—
389C22—82no—
390C22—82no—
391C22—82no—
392C22—122no—
393C22—122no—
394C22—122no—
395C22—121no—
396C22#571511yescharacteristic
397C22#571611yescharacteristic
398C22#575714yes—
399C22C2×#93711yescharacteristic
400C4—24no—
401C4—122no—
402C4—124no—
403C4—121no—
404C4—242no—
405C4#571812yes—
406C4#580612yes—
407C3#2560811yescharacteristic, complemented, Sylow
408C2—84no—
409C2—162no—
410C2—241no—
411C2#8177612yes—
412C2#8178712yes—
413C2#8318711yescharacteristic
414C2#8383312yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#493C2
socleC23×C3C22×D4
hypercenterC2×#493D3
hypocenterC3#25608
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#493

Complements

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

NH ≅ G/Ntype
1. C2×#5719C2semidirect
2. C2×C3×#493C2semidirect
3. #83189C2semidirect
4. C2×#5071C2semidirect
5. #72043C2semidirect
6. #72037C2semidirect
7. #71765C2semidirect
8. #69778C2semidirect
9. #69753C2semidirect
37. C2×#493D3semidirect
60. C3×#127D4semidirect
83. #870D4semidirect
183. #127#67semidirect
407. C3#25608semidirect

Maximal subgroups

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

subgroupindex
C2×#57192
C2×C3×#4932
#831892
C2×#50712
#720432
#720372
#717652
#697782
#697532
#256083