Finite Group Databasebeta

← #10849244

Subgroups — #10849244

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
2C2×#5020C211yesmaximal, characteristic, complemented
3C2×#4694C211yesmaximal, characteristic
4#73711C211yesmaximal, characteristic, complemented
5C3×C4×#103C211yesmaximal, characteristic, complemented
6#70245C214yesmaximal, complemented
7#69788C211yesmaximal, characteristic, complemented
8#69487C211yesmaximal, characteristic
9#68895C214yesmaximal, complemented
10#25652—31nomaximal, Sylow
11C2×C3×C4×Q8C2211yescharacteristic
12C2×#947C2212yes—
13#5585—24no—
14#5585—24no—
15C2×#894—22no—
16C2×#893C2212yes—
17C2×#875—21no—
18C2×#874C2211yescharacteristic
19C2×C4×D12—21no—
20C2×#869—21no—
21C2×#869C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×#103C2212yes—
24C2×C3×C4×C8C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26C2×#835C2212yes—
27C2×#821C2214yes—
28#5020—24no—
29#4984C2214yes—
30C2×#809C2214yes—
31C2×#808C2212yes—
32#4929—21no—
33C2×C4⋊(C3⋊C8)C2211yescharacteristic
34#4694—24no—
35#4661—22no—
36C3×#340C2212yes—
37#4550C2214yes—
38C2×#559—31no—
39C2×#486—31no—
40C2×#475—31no—
41#2546—31no—
42#2042—34no—
43#1984—31no—
44#1982—31no—
45#1864—34no—
46C4×#103D311yescharacteristic, complemented
47C22×C3×Q8C2311yescharacteristic
48C22×D12—21no—
49C22×D12—21no—
50C22×D12C2312yes—
51C22×C4×D3—22no—
52C22×Dic6—21no—
53C22×Dic6D411yescharacteristic
54C22×C3×C8C2312yes—
55C3×C4×Q8—22no—
56C3×C4×Q8D414yes—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4C2311yescharacteristic
59C2×C3×C4⋊C4C2312yes—
60C2×C3×C42—21no—
61C2×C3×C42—21no—
62C2×C3×C42C2311yescharacteristic
63#947—24no—
64#947—24no—
65C2×(C2×D3)⋊C4—22no—
66C2×(C2×D3)⋊C4—22no—
67C2×(C2×D3)⋊C4—22no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3—24no—
70C2×C4⋊Dic3D411yescharacteristic
71C2×C4⋊Dic3C2312yes—
72C2×Dic3⋊C4D412yes—
73C2×C4×Dic3C2312yes—
74C22×C3⋊C8C2312yes—
75#894—216no—
76#893—24no—
77#893—24no—
78#875—28no—
79#874—22no—
80#874D414yes—
81C4×D12—28no—
82#869—22no—
83#869—22no—
84#869—44no—
85C4×Dic6—22no—
86C4×Dic6—22no—
87C3×#103—24no—
88C3×C4×C8—22no—
89C3×#95—21no—
90#835—24no—
91C3⋊#95—24no—
92#821—28no—
93#809—28no—
94#808—24no—
95C4⋊(C3⋊C8)—22no—
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⋊C8—31no—
112C2×#103—32no—
113C2×#103C2×D312yes—
114C2×#102—34no—
115C2×#102—34no—
116C2×C4×C8C2×D311yescharacteristic
117#398—62no—
118#391—61no—
119#389—34no—
120C4×C4⋊C4C2×D311yescharacteristic
121#340C2×D312yes—
122#336—34no—
123C23×D3—22no—
124C2×C3×Q8—21no—
125C2×C3×Q8—22no—
126C2×C3×Q8—22no—
127C2×C3×Q8C2×D412yes—
128C22×C3×C4—22no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2411yescharacteristic
134C22×Dic3—22no—
135C22×Dic3C2×D412yes—
136C22×Dic3C2×D412yes—
137C2×D12—24no—
138C2×D12—24no—
139C2×D12—24no—
140C2×D12—22no—
141C2×D12—24no—
142C2×D12—42no—
143C2×D12—42no—
144C2×D12—82no—
145C2×D12C2×D414yes—
146C2×C4×D3—28no—
147C2×C4×D3—84no—
148C2×Dic6—22no—
149C2×Dic6—22no—
150C2×Dic6—22no—
151C2×Dic6—42no—
152C2×Dic6—42no—
153C2×Dic6—42no—
154C2×C3×C8—22no—
155C2×C3×C8—22no—
156C2×C3×C8—22no—
157C3×C4⋊C4—24no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4C2×D412yes—
161C3×C4⋊C4#4114yes—
162C3×C42—24no—
163C3×C42—42no—
164C3×C42C2×D412yes—
165C3×C42C2×D412yes—
166(C2×D3)⋊C4—28no—
167(C2×D3)⋊C4—28no—
168(C2×D3)⋊C4—28no—
169C4⋊Dic3—22no—
170C4⋊Dic3—22no—
171C4⋊Dic3—22no—
172C4⋊Dic3—48no—
173C4⋊Dic3—42no—
174Dic3⋊C4—24no—
175C4×Dic3—22no—
176C4×Dic3—22no—
177C2×C3⋊C8—22no—
178C2×C3⋊C8—24no—
179C22×Q8—31no—
180C22×Q8—61no—
181C22×Q8C22×D311yescharacteristic
182C22×D4—32no—
183C22×D4—61no—
184C22×D4—61no—
185C23×C4—62no—
186C22×C8—32no—
187C22×C8C22×D312yes—
188C4⋊Q8—62no—
189C4⋊Q8—62no—
190C4⋊Q8—124no—
191#127—34no—
192#127—62no—
193#124—68no—
194#124—64no—
195#124—64no—
196#123—616no—
197C4×Q8—22no—
198C4×Q8—62no—
199C4×Q8—62no—
200C4×Q8D1214yes—
201C4×D4—68no—
202C4×D4—64no—
203C4×D4—64no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—31no—
206C2×C4⋊C4—32no—
207C2×C4⋊C4—32no—
208C2×C4⋊C4—64no—
209C2×C4⋊C4—61no—
210C2×C4⋊C4C22×D311yescharacteristic
211C2×C4⋊C4C22×D312yes—
212C2×C22⋊C4—62no—
213C2×C22⋊C4—62no—
214C2×C22⋊C4—62no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—32no—
218C2×C42C22×D311yescharacteristic
219#106—64no—
220C4⋊C8—62no—
221#103—24no—
222#103—64no—
223#102—68no—
224#102—68no—
225C4×C8—22no—
226#95—21no—
227#95—64no—
228C23×C3C22×D411yescharacteristic
229C22×D3—24no—
230C22×D3—44no—
231C22×D3—42no—
232C22×D3—42no—
233C22×D3—44no—
234C3×Q8—22no—
235C3×Q8—22no—
236C3×Q8—42no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—24no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4#12012yes—
246C2×C3×C4#12012yes—
247C2×C3×C4#13512yes—
248C2×C3×C4AGL(1,ℤ/8)12yes—
249C2×C3×C4C22×D411yescharacteristic
250C2×C3×C4C22×D411yescharacteristic
251C2×C3×C4C2×#4112yes—
252C2×C3×C42−1+412yes—
253C2×Dic3—22no—
254C2×Dic3—22no—
255C2×Dic3—22no—
256C2×Dic3—22no—
257C2×Dic3—24no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261D12—28no—
262D12—42no—
263D12—42no—
264D12—48no—
265D12—84no—
266C4×D3—88no—
267Dic6—44no—
268Dic6—84no—
269C3×C8—42no—
270C3⋊C8—42no—
271C24—62no—
272C2×Q8—21no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—62no—
276C2×Q8—62no—
277C2×Q8—62no—
278C2×Q8—122no—
279C2×Q8—122no—
280C2×Q8—122no—
281C2×Q8C2×D1212yes—
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—21no—
293C22×C4—21no—
294C22×C4—32no—
295C22×C4—32no—
296C22×C4—68no—
297C22×C4—62no—
298C22×C4—244no—
299C22×C4D3×D411yescharacteristic
300C22×C4D3×D411yescharacteristic
301C22×C4C23×D311yescharacteristic
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—62no—
306C2×C8—64no—
307C4⋊C4—24no—
308C4⋊C4—21no—
309C4⋊C4—22no—
310C4⋊C4—62no—
311C4⋊C4—64no—
312C4⋊C4—62no—
313C4⋊C4—62no—
314C4⋊C4—128no—
315C4⋊C4—122no—
316C4⋊C4C2×D1212yes—
317C4⋊C4#23514yes—
318C22⋊C4—68no—
319C22⋊C4—68no—
320C22⋊C4—68no—
321C42—24no—
322C42—42no—
323C42—62no—
324C42—62no—
325C42C2×D1212yes—
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#44914yes—
342C3×C4#53211yescharacteristic
343C3×C4#53611yescharacteristic
344C3×C4#54712yes—
345Dic3—42no—
346Dic3—42no—
347Dic3—82no—
348C23—64no—
349C23—122no—
350C23—122no—
351C23—124no—
352C23—124no—
353C23C2×D3×D411yescharacteristic
354Q8—22no—
355Q8—22no—
356Q8—42no—
357Q8—124no—
358Q8—244no—
359D4—68no—
360D4—122no—
361D4—122no—
362D4—128no—
363D4—244no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—62no—
373C2×C4—62no—
374C2×C4—62no—
375C2×C4—62no—
376C2×C4—64no—
377C2×C4—122no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—248no—
381C2×C4#88212yes—
382C2×C4#88212yes—
383C2×C4#91412yes—
384C2×C4#91912yes—
385C2×C4C22×D1211yescharacteristic
386C2×C4C2×#23512yes—
387C2×C4C2×D3×D411yescharacteristic
388C2×C4#101012yes—
389C8—42no—
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×#88211yescharacteristic
403C22#555212yes—
404C22C2×#91411yescharacteristic
405C22C2×#91911yescharacteristic
406C22#577012yes—
407C4—22no—
408C4—44no—
409C4—42no—
410C4—42no—
411C4—122no—
412C4—122no—
413C4—242no—
414C4#481014yes—
415C4#547711yescharacteristic
416C4#552611yescharacteristic
417C4#555112yes—
418C3#2565211yescharacteristic, complemented, Sylow
419C2—244no—
420C2#7348911yescharacteristic
421C2C2×#481011yescharacteristic
422C2#7883511yescharacteristic
423C2#7962912yes—
424C2#7970112yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#103C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenterC4×#103D3
hypocenterC3#25652
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
2. C2×#5020C2semidirect
4. #73711C2semidirect
5. C3×C4×#103C2semidirect
6. #70245C2semidirect
7. #69788C2semidirect
9. #68895C2semidirect
46. C4×#103D3semidirect
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
C2×#50202
C2×#46942
#737112
C3×C4×#1032
#702452
#697882
#694872
#688952
#256523