Finite Group Databasebeta

← #10849241

Subgroups — #10849241

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
2C3×#2546C211yesmaximal, characteristic, complemented
3C2×#5094C211yesmaximal, characteristic, complemented
4C2×#5088C211yesmaximal, characteristic
5#71634C211yesmaximal, characteristic, complemented
6#69628C214yesmaximal, complemented
7#69626C214yesmaximal, complemented
8#69487C211yesmaximal, characteristic
9C4×#810C211yesmaximal, characteristic, complemented
10#25652—31nomaximal, Sylow
11C2×C3×C4⋊Q8—21no—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×C4×Q8C2211yescharacteristic
14C2×#947—21no—
15#5585—24no—
16#5585—24no—
17C2×#894—22no—
18C2×#893—21no—
19C2×#875C2212yes—
20C2×#874C2211yescharacteristic
21C2×C4×D12C2212yes—
22C2×C4×Dic6C2211yescharacteristic
23C3×#398—22no—
24C3×#391—21no—
25C3×#389C2214yes—
26C3×C4×C4⋊C4C2211yescharacteristic
27#5094—24no—
28#5088—24no—
29C2×#810C2212yes—
30C2×#810C2212yes—
31C2×#809C2214yes—
32C2×#809C2214yes—
33C2×#808C2212yes—
34C2×C4⋊(C3⋊C8)C2211yescharacteristic
35C2×C4×C3⋊C8C2211yescharacteristic
36#4529C2212yes—
37#4524C2214yes—
38C2×#559—31no—
39C2×#486—31no—
40C2×#475—31no—
41#2546D311yescharacteristic, complemented
42#2042—34no—
43#1984—31no—
44#1982—31no—
45#1864—34no—
46C4×#103—31no—
47C22×C3×Q8—21no—
48C22×C3×Q8D411yescharacteristic
49C22×D12—21no—
50C22×D12—21no—
51C22×D12C2312yes—
52C22×C4×D3—22no—
53C22×Dic6C2311yescharacteristic
54C3×C4⋊Q8—22no—
55C3×C4⋊Q8—22no—
56C3×C4⋊Q8—44no—
57C3×C4×Q8—22no—
58C3×C4×Q8—22no—
59C2×C3×C4⋊C4—24no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4D411yescharacteristic
62C2×C3×C4⋊C4D412yes—
63C2×C3×C4⋊C4C2312yes—
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C42—21no—
66C2×C3×C42C2312yes—
67C2×C3×C42C2311yescharacteristic
68#947—28no—
69C2×(C2×D3)⋊C4—22no—
70C2×(C2×D3)⋊C4—22no—
71C2×(C2×D3)⋊C4—22no—
72C2×C4⋊Dic3C2311yescharacteristic
73C2×Dic3⋊C4—21no—
74C2×C4×Dic3—21no—
75C22×C3⋊C8C2312yes—
76C22×C3⋊C8C2312yes—
77#894—216no—
78#893—28no—
79#875—24no—
80#875—24no—
81#874—22no—
82#874D414yes—
83C4×D12—24no—
84C4×D12—24no—
85C4×Dic6—22no—
86C4×Dic6D414yes—
87C3×#95—21no—
88C3×#95—24no—
89#810—24no—
90#810—24no—
91#809—28no—
92#809—28no—
93#808—24no—
94C4⋊(C3⋊C8)—22no—
95C4×C3⋊C8—22no—
96#559—64no—
97#559—64no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2×D311yescharacteristic
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⋊C8—31no—
112C2×#103—32no—
113C2×#103—32no—
114C2×#102—34no—
115C2×#102—34no—
116C2×C4×C8—31no—
117#398—22no—
118#391—21no—
119#389C2×D314yes—
120C4×C4⋊C4C2×D311yescharacteristic
121#340—32no—
122#336—34no—
123C23×D3—22no—
124C2×C3×Q8—22no—
125C2×C3×Q8—22no—
126C2×C3×Q8—22no—
127C2×C3×Q8—42no—
128C2×C3×Q8—42no—
129C2×C3×Q8—42no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4C2×D412yes—
133C22×C3×C4C2×D412yes—
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2411yescharacteristic
137C22×Dic3—21no—
138C22×Dic3—21no—
139C2×D12—24no—
140C2×D12—24no—
141C2×D12—24no—
142C2×D12—22no—
143C2×D12—24no—
144C2×D12—42no—
145C2×D12—42no—
146C2×D12—82no—
147C2×D12C2×D414yes—
148C2×C4×D3—28no—
149C2×C4×D3—84no—
150C2×Dic6—22no—
151C2×Dic6—22no—
152C2×Dic6—21no—
153C2×Dic6C2×D412yes—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—24no—
156C3×C4⋊C4—22no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—48no—
160C3×C4⋊C4—42no—
161C3×C4⋊C4#4114yes—
162C3×C42—22no—
163C3×C42—22no—
164C3×C42—42no—
165C3×C42C2×D412yes—
166C3×C42C2×D412yes—
167(C2×D3)⋊C4—28no—
168(C2×D3)⋊C4—28no—
169(C2×D3)⋊C4—28no—
170C4⋊Dic3—21no—
171C4⋊Dic3C2×D412yes—
172Dic3⋊C4—24no—
173C4×Dic3—24no—
174C2×C3⋊C8—22no—
175C2×C3⋊C8—22no—
176C2×C3⋊C8—24no—
177C2×C3⋊C8—22no—
178C2×C3⋊C8—22no—
179C22×Q8—21no—
180C22×Q8—31no—
181C22×Q8#6711yescharacteristic
182C22×D4—32no—
183C22×D4—61no—
184C22×D4—61no—
185C23×C4—62no—
186C22×C8—32no—
187C22×C8—32no—
188C4⋊Q8—22no—
189C4⋊Q8—22no—
190C4⋊Q8—44no—
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—24no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—31no—
207C2×C4⋊C4—61no—
208C2×C4⋊C4#6711yescharacteristic
209C2×C4⋊C4#6712yes—
210C2×C4⋊C4C22×D312yes—
211C2×C4⋊C4C22×D312yes—
212C2×C22⋊C4—62no—
213C2×C22⋊C4—62no—
214C2×C22⋊C4—62no—
215C2×C42—21no—
216C2×C42—61no—
217C2×C42C22×D312yes—
218C2×C42C22×D311yescharacteristic
219#106—64no—
220C4⋊C8—62no—
221#103—64no—
222#103—64no—
223#102—68no—
224#102—68no—
225C4×C8—62no—
226#95—21no—
227#95—24no—
228C23×C3C22×D411yescharacteristic
229C22×D3—24no—
230C22×D3—44no—
231C22×D3—42no—
232C22×D3—42no—
233C22×D3—44no—
234C3×Q8—44no—
235C3×Q8—84no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—24no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—24no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4#12012yes—
249C2×C3×C4#12012yes—
250C2×C3×C4#13512yes—
251C2×C3×C4AGL(1,ℤ/8)12yes—
252C2×C3×C4C22×D411yescharacteristic
253C2×C3×C4C22×D411yescharacteristic
254C2×C3×C4C2×#4112yes—
255C2×C3×C42−1+412yes—
256C2×Dic3—22no—
257C2×Dic3—22no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260D12—28no—
261D12—42no—
262D12—42no—
263D12—48no—
264D12—84no—
265C4×D3—88no—
266Dic6—22no—
267Dic6—22no—
268Dic6—42no—
269C3⋊C8—42no—
270C3⋊C8—42no—
271C24—62no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—32no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—61no—
280C2×Q8—62no—
281C2×Q8—62no—
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—22no—
292C22×C4—22no—
293C22×C4—68no—
294C22×C4—61no—
295C22×C4—61no—
296C22×C4—244no—
297C22×C4C2×#6712yes—
298C22×C4C2×#6712yes—
299C22×C4C2×#6711yescharacteristic
300C22×C4C2×#6711yescharacteristic
301C22×C4C23×D311yescharacteristic
302C2×C8—62no—
303C2×C8—62no—
304C2×C8—64no—
305C2×C8—62no—
306C2×C8—62no—
307C4⋊C4—22no—
308C4⋊C4—24no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—32no—
313C4⋊C4—48no—
314C4⋊C4—42no—
315C4⋊C4—64no—
316C4⋊C4—61no—
317C4⋊C4#23914yes—
318C22⋊C4—68no—
319C22⋊C4—68no—
320C22⋊C4—68no—
321C42—22no—
322C42—22no—
323C42—42no—
324C42—64no—
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—44no—
339C3×C4—42no—
340C3×C4—42no—
341C3×C4—82no—
342C3×C4#44914yes—
343C3×C4#53211yescharacteristic
344C3×C4#53611yescharacteristic
345C3×C4#54712yes—
346Dic3—42no—
347Dic3—42no—
348C23—64no—
349C23—122no—
350C23—124no—
351C23—122no—
352C23—124no—
353C23C22×#6711yescharacteristic
354Q8—44no—
355Q8—62no—
356Q8—62no—
357Q8—84no—
358Q8—122no—
359D4—68no—
360D4—122no—
361D4—122no—
362D4—128no—
363D4—244no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—62no—
377C2×C4—62no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—248no—
381C2×C4#93212yes—
382C2×C4#93712yes—
383C2×C4#93712yes—
384C2×C4#95012yes—
385C2×C4C2×D3×D411yescharacteristic
386C2×C4C2×D3×D411yescharacteristic
387C2×C4C2×#23912yes—
388C2×C4#100712yes—
389C8—122no—
390C8—122no—
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×#93211yescharacteristic
403C22C2×#93711yescharacteristic
404C22#581712yes—
405C22C2×#95011yescharacteristic
406C22#583712yes—
407C4—22no—
408C4—44no—
409C4—42no—
410C4—42no—
411C4—82no—
412C4—122no—
413C4—122no—
414C4#503714yes—
415C4#572411yescharacteristic
416C4#572511yescharacteristic
417C4#572612yes—
418C3#2565211yescharacteristic, complemented, Sylow
419C2—244no—
420C2C2×#503711yescharacteristic
421C2#8325511yescharacteristic
422C2#8340411yescharacteristic
423C2#8356912yes—
424C2#8369712yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2546C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenter#2546D3
hypocenterC3#25652
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2546

Complements

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

NH ≅ G/Ntype
2. C3×#2546C2semidirect
3. C2×#5094C2semidirect
5. #71634C2semidirect
6. #69628C2semidirect
7. #69626C2semidirect
9. C4×#810C2semidirect
41. #2546D3semidirect
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
C3×#25462
C2×#50942
C2×#50882
#716342
#696282
#696262
#694872
C4×#8102
#256523