Finite Group Databasebeta

← #10849243

Subgroups — #10849243

Order 768.

1118 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5585C211yesmaximal, characteristic
2#82475C211yesmaximal, characteristic, complemented
3C2×#4851C211yesmaximal, characteristic, complemented
4C2×#4694C211yesmaximal, characteristic
5C3×#1984C211yesmaximal, characteristic, complemented
6#70457C211yesmaximal, characteristic
7C4×#816C211yesmaximal, characteristic, complemented
8#68671C214yesmaximal, complemented
9#68630C214yesmaximal, complemented
10#25652—31nomaximal, Sylow
11C2×C3×C4×Q8C2211yescharacteristic
12C2×#947—21no—
13C2×#944—21no—
14#5585—24no—
15#5585—24no—
16C2×#894—22no—
17C2×#893—21no—
18C2×#875C2212yes—
19C2×#874C2211yescharacteristic
20C2×C4×D12C2212yes—
21C2×#869C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C4×C8C2211yescharacteristic
25C2×#821C2214yes—
26C2×#821C2214yes—
27C2×#817C2212yes—
28C2×#816C2212yes—
29C2×#816C2212yes—
30#4981—22no—
31#4973—21no—
32C4×C4⋊Dic3C2211yescharacteristic
33#4928C2214yes—
34#4851—24no—
35#4694—24no—
36C3×#336C2214yes—
37#4551C2212yes—
38C2×#559—31no—
39C2×#486—31no—
40C2×#475—31no—
41#2546—31no—
42#2042—34no—
43#1984D311yescharacteristic, complemented
44#1982—31no—
45#1864—34no—
46C4×#103—31no—
47C22×C3×Q8D411yescharacteristic
48C22×D12—21no—
49C22×D12—21no—
50C22×D12C2312yes—
51C22×C4×D3—22no—
52C22×Dic6—21no—
53C22×Dic6C2311yescharacteristic
54C22×C3×C8C2312yes—
55C22×C3×C8C2312yes—
56C3×C4×Q8—22no—
57C3×C4×Q8—22no—
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C4⋊C4D412yes—
60C2×C3×C42C2312yes—
61C2×C3×C42C2311yescharacteristic
62#947—28no—
63#944—44no—
64C2×(C2×D3)⋊C4—22no—
65C2×(C2×D3)⋊C4—22no—
66C2×(C2×D3)⋊C4—22no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3C2312yes—
69C2×C4⋊Dic3C2311yescharacteristic
70C2×C4⋊Dic3C2312yes—
71C2×Dic3⋊C4—21no—
72C2×Dic3⋊C4—24no—
73C2×C4×Dic3—21no—
74C2×C4×Dic3—21no—
75#894—216no—
76#893—28no—
77#875—24no—
78#875—24no—
79#874—22no—
80#874D414yes—
81C4×D12—24no—
82C4×D12—24no—
83#869—22no—
84#869—22no—
85C4×Dic6—22no—
86C4×Dic6D414yes—
87C3×C4⋊C8—22no—
88C3×C4×C8—22no—
89C3⋊#95—24no—
90C3⋊#95—21no—
91#821—28no—
92#821—28no—
93#817—24no—
94#816—24no—
95#816—24no—
96#559—64no—
97#559—64no—
98C2×C4⋊Q8—31no—
99C2×C4⋊Q8—61no—
100C2×#127—31no—
101C2×#124—32no—
102C2×#124—61no—
103C2×#123—62no—
104C2×C4×Q8—31no—
105C2×C4×Q8C2×D311yescharacteristic
106C2×C4×D4—32no—
107C2×C4×D4—61no—
108#486—64no—
109#475—64no—
110C2×#106—32no—
111C2×C4⋊C8C2×D311yescharacteristic
112C2×#103—32no—
113C2×#103—32no—
114C2×#102—34no—
115C2×#102—34no—
116C2×C4×C8C2×D311yescharacteristic
117#398—62no—
118#391—61no—
119#389—34no—
120C4×C4⋊C4—31no—
121#340—32no—
122#336C2×D314yes—
123C23×D3—22no—
124C2×C3×Q8—22no—
125C2×C3×Q8—22no—
126C2×C3×Q8—22no—
127C22×C3×C4C2×D412yes—
128C22×C3×C4C2×D412yes—
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2411yescharacteristic
132C22×Dic3—22no—
133C22×Dic3—21no—
134C22×Dic3—21no—
135C22×Dic3—22no—
136C2×D12—24no—
137C2×D12—24no—
138C2×D12—24no—
139C2×D12—22no—
140C2×D12—24no—
141C2×D12—42no—
142C2×D12—42no—
143C2×D12—82no—
144C2×D12C2×D414yes—
145C2×C4×D3—28no—
146C2×C4×D3—84no—
147C2×Dic6—22no—
148C2×Dic6—22no—
149C2×Dic6—21no—
150C2×Dic6—42no—
151C2×Dic6—42no—
152C2×Dic6—42no—
153C2×Dic6C2×D412yes—
154C2×C3×C8—22no—
155C2×C3×C8—22no—
156C2×C3×C8—22no—
157C2×C3×C8—24no—
158C2×C3×C8—22no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—24no—
161C3×C42—22no—
162C3×C42—22no—
163C3×C42C2×D412yes—
164C3×C42C2×D412yes—
165(C2×D3)⋊C4—28no—
166(C2×D3)⋊C4—28no—
167(C2×D3)⋊C4—28no—
168C4⋊Dic3—21no—
169C4⋊Dic3—22no—
170C4⋊Dic3—22no—
171C4⋊Dic3—22no—
172C4⋊Dic3—42no—
173C4⋊Dic3C2×D412yes—
174C4⋊Dic3#4114yes—
175Dic3⋊C4—24no—
176Dic3⋊C4—48no—
177C4×Dic3—24no—
178C4×Dic3—42no—
179C22×Q8—31no—
180C22×Q8—61no—
181C22×Q8D1211yescharacteristic
182C22×D4—32no—
183C22×D4—61no—
184C22×D4—61no—
185C23×C4—62no—
186C22×C8C22×D312yes—
187C22×C8C22×D312yes—
188C4⋊Q8—62no—
189C4⋊Q8—62no—
190C4⋊Q8—124no—
191#127—34no—
192#127—62no—
193#124—64no—
194#124—64no—
195#124—68no—
196#123—616no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—34no—
200C4×Q8—62no—
201C4×D4—64no—
202C4×D4—64no—
203C4×D4—68no—
204C2×C4⋊C4—31no—
205C2×C4⋊C4—32no—
206C2×C4⋊C4—32no—
207C2×C4⋊C4—61no—
208C2×C4⋊C4—61no—
209C2×C4⋊C4—64no—
210C2×C4⋊C4D1211yescharacteristic
211C2×C4⋊C4D1212yes—
212C2×C22⋊C4—62no—
213C2×C22⋊C4—62no—
214C2×C22⋊C4—62no—
215C2×C42—61no—
216C2×C42—61no—
217C2×C42C22×D312yes—
218C2×C42C22×D311yescharacteristic
219#106—64no—
220C4⋊C8—22no—
221#103—64no—
222#103—64no—
223#102—68no—
224#102—68no—
225C4×C8—22no—
226#95—61no—
227#95—64no—
228C23×C3C22×D411yescharacteristic
229C22×D3—24no—
230C22×D3—44no—
231C22×D3—42no—
232C22×D3—42no—
233C22×D3—44no—
234C3×Q8—44no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—24no—
241C2×C3×C4#12012yes—
242C2×C3×C4#12012yes—
243C2×C3×C4#13512yes—
244C2×C3×C4AGL(1,ℤ/8)12yes—
245C2×C3×C4C22×D411yescharacteristic
246C2×C3×C4C22×D411yescharacteristic
247C2×C3×C4C2×#4112yes—
248C2×C3×C42−1+412yes—
249C2×Dic3—22no—
250C2×Dic3—22no—
251C2×Dic3—24no—
252C2×Dic3—42no—
253C2×Dic3—42no—
254C2×Dic3—42no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—42no—
258C2×Dic3—42no—
259D12—28no—
260D12—42no—
261D12—42no—
262D12—48no—
263D12—84no—
264C4×D3—88no—
265Dic6—22no—
266Dic6—22no—
267Dic6—42no—
268Dic6—84no—
269C3×C8—42no—
270C3×C8—42no—
271C24—62no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—32no—
276C2×Q8—61no—
277C2×Q8—62no—
278C2×Q8—62no—
279C2×Q8—122no—
280C2×Q8—122no—
281C2×Q8—122no—
282C2×D4—34no—
283C2×D4—64no—
284C2×D4—64no—
285C2×D4—62no—
286C2×D4—64no—
287C2×D4—64no—
288C2×D4—122no—
289C2×D4—122no—
290C2×D4—242no—
291C22×C4—68no—
292C22×C4—62no—
293C22×C4—61no—
294C22×C4—61no—
295C22×C4—62no—
296C22×C4—244no—
297C22×C4C2×D1212yes—
298C22×C4C2×D1212yes—
299C22×C4C2×D1211yescharacteristic
300C22×C4C2×D1211yescharacteristic
301C22×C4C23×D311yescharacteristic
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—24no—
306C2×C8—22no—
307C4⋊C4—22no—
308C4⋊C4—24no—
309C4⋊C4—32no—
310C4⋊C4—34no—
311C4⋊C4—64no—
312C4⋊C4—61no—
313C4⋊C4—62no—
314C4⋊C4—62no—
315C4⋊C4—62no—
316C4⋊C4—128no—
317C4⋊C4—122no—
318C22⋊C4—68no—
319C22⋊C4—68no—
320C22⋊C4—68no—
321C42—22no—
322C42—22no—
323C42—64no—
324C42—122no—
325C42D3×D412yes—
326C42D3×D412yes—
327C22×C3C2×#12011yescharacteristic
328C22×C3#54812yes—
329C22×C3C2×#13511yescharacteristic
330C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
331C22×C3#57712yes—
332C2×D3—44no—
333C2×D3—44no—
334C2×D3—44no—
335C2×D3—84no—
336C2×D3—84no—
337C3×C4—22no—
338C3×C4—42no—
339C3×C4—42no—
340C3×C4#44914yes—
341C3×C4#53211yescharacteristic
342C3×C4#53611yescharacteristic
343C3×C4#54712yes—
344Dic3—44no—
345Dic3—42no—
346Dic3—42no—
347Dic3—82no—
348C23—64no—
349C23—122no—
350C23—122no—
351C23—124no—
352C23—124no—
353C23C22×D1211yescharacteristic
354Q8—44no—
355Q8—62no—
356Q8—62no—
357Q8—122no—
358Q8—244no—
359D4—68no—
360D4—122no—
361D4—122no—
362D4—128no—
363D4—244no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—62no—
371C2×C4—62no—
372C2×C4—64no—
373C2×C4—122no—
374C2×C4—122no—
375C2×C4—122no—
376C2×C4—122no—
377C2×C4—122no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—248no—
381C2×C4#88212yes—
382C2×C4#88212yes—
383C2×C4#90412yes—
384C2×C4#90812yes—
385C2×C4C2×D3×D411yescharacteristic
386C2×C4C2×D3×D411yescharacteristic
387C2×C4#100712yes—
388C2×C4D3×#4112yes—
389C8—42no—
390C8—42no—
391C2×C3#243211yescharacteristic
392C2×C3C2×#44911yescharacteristic
393C2×C3#300911yescharacteristic
394C2×C3#336512yes—
395C2×C3#337312yes—
396D3—84no—
397C22—124no—
398C22—124no—
399C22—124no—
400C22—244no—
401C22—244no—
402C22C2×#88211yescharacteristic
403C22#557612yes—
404C22C2×#90411yescharacteristic
405C22C2×#90811yescharacteristic
406C22#575112yes—
407C4—22no—
408C4—42no—
409C4—42no—
410C4—122no—
411C4—124no—
412C4—122no—
413C4—242no—
414C4#473414yes—
415C4#563112yes—
416C4#566811yescharacteristic
417C4#572411yescharacteristic
418C3#2565211yescharacteristic, complemented, Sylow
419C2—244no—
420C2#7649811yescharacteristic
421C2C2×#473411yescharacteristic
422C2#7867111yescharacteristic
423C2#7993512yes—
424C2#7997312yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1984C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenter#1984D3
hypocenterC3#25652
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #1984

Complements

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

NH ≅ G/Ntype
2. #82475C2semidirect
3. C2×#4851C2semidirect
5. C3×#1984C2semidirect
7. C4×#816C2semidirect
8. #68671C2semidirect
9. #68630C2semidirect
43. #1984D3semidirect
418. C3#25652semidirect

Maximal subgroups

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

subgroupindex
C2×#55852
#824752
C2×#48512
C2×#46942
C3×#19842
#704572
C4×#8162
#686712
#686302
#256523