Finite Group Databasebeta

← #10849240

Subgroups — #10849240

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×C3×#559C211yesmaximal, characteristic
2C2×#5070C211yesmaximal, characteristic
3#82475C211yesmaximal, characteristic, complemented
4C2×#5020C211yesmaximal, characteristic, complemented
5#71761C211yesmaximal, characteristic
6#71722C214yesmaximal, complemented
7#71634C211yesmaximal, characteristic, complemented
8C4×#835C211yesmaximal, characteristic, complemented
9#71607C214yesmaximal, complemented
10#25652—31nomaximal, Sylow
11C3×#559—24no—
12C3×#559—24no—
13C2×C3×#127C2211yescharacteristic
14C2×C3×#124—21no—
15C2×C3×#124C2212yes—
16C2×C3×#123—22no—
17C2×C3×C4×Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×D4—21no—
20C2×C3×C4×D4C2212yes—
21C2×#944—21no—
22C2×#869C2211yescharacteristic
23C2×#835C2212yes—
24C2×#835C2212yes—
25C2×#832C2214yes—
26C2×#832C2214yes—
27#5070—24no—
28#5020—24no—
29#4981—22no—
30#4973—21no—
31C2×#808C2212yes—
32C4×C4⋊Dic3C2211yescharacteristic
33#4928C2214yes—
34C2×C4⋊(C3⋊C8)C2211yescharacteristic
35C2×C4×C3⋊C8C2211yescharacteristic
36#4529C2212yes—
37#4524C2214yes—
38C2×#559D311yescharacteristic
39C2×#486—31no—
40C2×#475—31no—
41#2546—31no—
42#2042—34no—
43#1984—31no—
44#1982—31no—
45#1864—34no—
46C4×#103—31no—
47C22×C3×Q8D411yescharacteristic
48C22×C3×Q8C2311yescharacteristic
49C22×C3×D4—21no—
50C22×C3×D4—21no—
51C22×C3×D4C2312yes—
52C23×C3×C4—22no—
53C22×Dic6—21no—
54C3×#127—22no—
55C3×#127D414yes—
56C3×#124—28no—
57C3×#124—24no—
58C3×#124—24no—
59C3×#123—216no—
60C3×C4×Q8—22no—
61C3×C4×Q8—22no—
62C3×C4×Q8—22no—
63C3×C4×Q8D414yes—
64C3×C4×D4—28no—
65C3×C4×D4—24no—
66C3×C4×D4—24no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4D411yescharacteristic
69C2×C3×C4⋊C4D412yes—
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C22⋊C4—22no—
72C2×C3×C22⋊C4—22no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C42—21no—
75C2×C3×C42C2312yes—
76C2×C3×C42C2311yescharacteristic
77#944—44no—
78C2×C4⋊Dic3—21no—
79C2×C4⋊Dic3C2312yes—
80C2×C4⋊Dic3C2312yes—
81C2×Dic3⋊C4—24no—
82C2×C4×Dic3—21no—
83C22×C3⋊C8C2312yes—
84C22×C3⋊C8C2312yes—
85#869—22no—
86#869—22no—
87#835—24no—
88#835—24no—
89#832—28no—
90#832—28no—
91C3⋊#95—24no—
92C3⋊#95—21no—
93#808—24no—
94C4⋊(C3⋊C8)—22no—
95C4×C3⋊C8—22no—
96#559—24no—
97#559—24no—
98C2×C4⋊Q8—31no—
99C2×C4⋊Q8—61no—
100C2×#127C2×D311yescharacteristic
101C2×#124—21no—
102C2×#124C2×D312yes—
103C2×#123—22no—
104C2×C4×Q8C2×D311yescharacteristic
105C2×C4×Q8C2×D311yescharacteristic
106C2×C4×D4—21no—
107C2×C4×D4C2×D312yes—
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—62no—
118#391—61no—
119#389—34no—
120C4×C4⋊C4—31no—
121#340—32no—
122#336—34no—
123C24×C3—22no—
124C2×C3×Q8—22no—
125C2×C3×Q8—22no—
126C2×C3×Q8—22no—
127C2×C3×Q8—21no—
128C2×C3×Q8—22no—
129C2×C3×Q8—22no—
130C2×C3×Q8C2×D412yes—
131C2×C3×D4—24no—
132C2×C3×D4—24no—
133C2×C3×D4—22no—
134C2×C3×D4—24no—
135C2×C3×D4—24no—
136C2×C3×D4—42no—
137C2×C3×D4—42no—
138C2×C3×D4—82no—
139C2×C3×D4C2×D414yes—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—28no—
143C22×C3×C4—84no—
144C22×C3×C4C2×D412yes—
145C22×C3×C4C2×D412yes—
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2411yescharacteristic
149C22×Dic3—22no—
150C22×Dic3—22no—
151C2×Dic6—42no—
152C2×Dic6—42no—
153C2×Dic6—42no—
154C3×C4⋊C4—24no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—24no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4C2×D412yes—
159C3×C22⋊C4—28no—
160C3×C22⋊C4—28no—
161C3×C22⋊C4—28no—
162C3×C42—24no—
163C3×C42—22no—
164C3×C42—22no—
165C3×C42C2×D412yes—
166C3×C42C2×D412yes—
167C4⋊Dic3—22no—
168C4⋊Dic3—22no—
169C4⋊Dic3—22no—
170C4⋊Dic3—42no—
171C4⋊Dic3#4114yes—
172Dic3⋊C4—48no—
173C4×Dic3—42no—
174C2×C3⋊C8—22no—
175C2×C3⋊C8—22no—
176C2×C3⋊C8—24no—
177C2×C3⋊C8—22no—
178C2×C3⋊C8—22no—
179C22×Q8—61no—
180C22×Q8#6711yescharacteristic
181C22×Q8C22×D311yescharacteristic
182C22×D4—21no—
183C22×D4—21no—
184C22×D4C22×D312yes—
185C23×C4—22no—
186C22×C8—32no—
187C22×C8—32no—
188C4⋊Q8—62no—
189C4⋊Q8—62no—
190C4⋊Q8—124no—
191#127—22no—
192#127#6714yes—
193#124—28no—
194#124—24no—
195#124—24no—
196#123—216no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8#6714yes—
201C4×D4—28no—
202C4×D4—24no—
203C4×D4—24no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—32no—
206C2×C4⋊C4—32no—
207C2×C4⋊C4—61no—
208C2×C4⋊C4—64no—
209C2×C4⋊C4#6711yescharacteristic
210C2×C4⋊C4#6712yes—
211C2×C4⋊C4C22×D311yescharacteristic
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
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—61no—
227#95—64no—
228C23×C3—24no—
229C23×C3—44no—
230C23×C3—42no—
231C23×C3—42no—
232C23×C3—44no—
233C23×C3C22×D411yescharacteristic
234C3×Q8—22no—
235C3×Q8—22no—
236C3×Q8—42no—
237C3×Q8—44no—
238C3×D4—28no—
239C3×D4—42no—
240C3×D4—42no—
241C3×D4—48no—
242C3×D4—84no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—24no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—88no—
254C2×C3×C4#12012yes—
255C2×C3×C4#12012yes—
256C2×C3×C4#13512yes—
257C2×C3×C4AGL(1,ℤ/8)12yes—
258C2×C3×C4C22×D411yescharacteristic
259C2×C3×C4C22×D411yescharacteristic
260C2×C3×C4C2×#4112yes—
261C2×C3×C42−1+412yes—
262C2×Dic3—24no—
263C2×Dic3—42no—
264C2×Dic3—42no—
265C2×Dic3—42no—
266C2×Dic3—42no—
267C2×Dic3—42no—
268Dic6—84no—
269C3⋊C8—42no—
270C3⋊C8—42no—
271C24—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—21no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—122no—
279C2×Q8—122no—
280C2×Q8—122no—
281C2×Q8C2×#6712yes—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—22no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—82no—
290C2×D4C2×#6714yes—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—28no—
294C22×C4—62no—
295C22×C4—62no—
296C22×C4—84no—
297C22×C4C2×#6712yes—
298C22×C4C2×#6712yes—
299C22×C4C2×#6711yescharacteristic
300C22×C4C2×#6711yescharacteristic
301C22×C4C23×D311yescharacteristic
302C2×C8—64no—
303C2×C8—62no—
304C2×C8—62no—
305C2×C8—62no—
306C2×C8—62no—
307C4⋊C4—24no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—21no—
311C4⋊C4—34no—
312C4⋊C4—62no—
313C4⋊C4—62no—
314C4⋊C4—62no—
315C4⋊C4—128no—
316C4⋊C4—122no—
317C4⋊C4C2×#6712yes—
318C22⋊C4—28no—
319C22⋊C4—28no—
320C22⋊C4—28no—
321C42—24no—
322C42—22no—
323C42—22no—
324C42—122no—
325C42C2×#6712yes—
326C42C2×#6712yes—
327C22×C3—44no—
328C22×C3—44no—
329C22×C3—44no—
330C22×C3—84no—
331C22×C3—84no—
332C22×C3C2×#12011yescharacteristic
333C22×C3#54812yes—
334C22×C3C2×#13511yescharacteristic
335C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
336C22×C3#57712yes—
337C3×C4—22no—
338C3×C4—42no—
339C3×C4—42no—
340C3×C4—42no—
341C3×C4—42no—
342C3×C4#44914yes—
343C3×C4#53211yescharacteristic
344C3×C4#53611yescharacteristic
345C3×C4#54712yes—
346Dic3—44no—
347Dic3—82no—
348C23—24no—
349C23—42no—
350C23—42no—
351C23—44no—
352C23—44no—
353C23C22×#6711yescharacteristic
354Q8—22no—
355Q8—22no—
356Q8—44no—
357Q8—42no—
358Q8—244no—
359D4—28no—
360D4—48no—
361D4—42no—
362D4—42no—
363D4—84no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—64no—
375C2×C4—88no—
376C2×C4—122no—
377C2×C4—122no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—122no—
381C2×C4#94912yes—
382C2×C4#95012yes—
383C2×C4#95312yes—
384C2×C4#95312yes—
385C2×C4D3×#4112yes—
386C2×C4#101012yes—
387C2×C4C22×#6711yescharacteristic
388C2×C4C22×#6711yescharacteristic
389C8—122no—
390C8—122no—
391C2×C3—84no—
392C2×C3#243211yescharacteristic
393C2×C3C2×#44911yescharacteristic
394C2×C3#300911yescharacteristic
395C2×C3#336512yes—
396C2×C3#337312yes—
397C22—44no—
398C22—44no—
399C22—44no—
400C22—84no—
401C22—84no—
402C22C2×#94911yescharacteristic
403C22C2×#95011yescharacteristic
404C22#583112yes—
405C22#583612yes—
406C22C2×#95311yescharacteristic
407C4—22no—
408C4—42no—
409C4—42no—
410C4—42no—
411C4—42no—
412C4—124no—
413C4—242no—
414C4#523814yes—
415C4#582412yes—
416C4#582511yescharacteristic
417C4#583211yescharacteristic
418C3#2565211yescharacteristic, complemented, Sylow
419C2—84no—
420C2#8371512yes—
421C2#8371712yes—
422C2C2×#523811yescharacteristic
423C2#8409311yescharacteristic
424C2#8410511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#559C2
socleC23×C3C22×D4
prime radicalC2×C3×#127C22
hypercenterC2×#559D3
hypocenterC3#25652
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
3. #82475C2semidirect
4. C2×#5020C2semidirect
6. #71722C2semidirect
7. #71634C2semidirect
8. C4×#835C2semidirect
9. #71607C2semidirect
418. C3#25652semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5592
C2×#50702
#824752
C2×#50202
#717612
#717222
#716342
C4×#8352
#716072
#256523