Finite Group Databasebeta

← #10794595

Subgroups — #10794595

Order 768.

1462 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5827C211yesmaximal, characteristic
2C2×C3×#410C211yesmaximal, characteristic
3C2×#5240C211yesmaximal, characteristic
4#82232C212yesmaximal, complemented
5#71593C212yesmaximal, complemented
6#70028C218yesmaximal, complemented
7#20634—31nomaximal, Sylow
8C22×C3×#41C2211yescharacteristic
9C2×C3×#117C2212yes—
10#5827—24no—
11#5827—24no—
12C2×Q8×Dic3C412yescomplemented
13C2×D4×Dic3C412yescomplemented
14C2×D4×Dic3C2214yes—
15C2×#926C412yescomplemented
16C2×#926C2212yes—
17C2×#926C2211yescharacteristic
18C22×C4⋊Dic3C412yescomplemented
19C22×C4⋊Dic3C2211yescharacteristic
20C3×#410—24no—
21C2×C3×#99C2214yes—
22#5240—24no—
23C2×#834C2214yes—
24#4972—24no—
25#4942C2218yes—
26#4928—24no—
27#4527C2218yes—
28C2×#520—31no—
29C2×#410—31no—
30C2×#410D311yescharacteristic
31#2316—32no—
32#1799—38no—
33#1740—32no—
34C2×C3×#41—22no—
35C2×C3×#41—22no—
36C2×C3×#41—24no—
37C2×C3×#41D414yes—
38C2×C3×#41Q814yes—
39C22×C3×Q8C2×C411yescharacteristic
40C22×C3×D4C2×C411yescharacteristic
41C22×C3×D4C2312yes—
42C23×C3×C4C2×C412yes—
43C23×C3×C4C2311yescharacteristic
44C23×Dic3C2×C414yes—
45C23×Dic3C2312yes—
46C3×#117—24no—
47C3×#117—24no—
48C2×C3×C4⋊C4—22no—
49C2×C3×C22⋊C4C2314yes—
50C2×C3×C42—22no—
51C2×#217C2×C414yes—
52C2×#217C2314yes—
53C2×#217C2312yes—
54Q8×Dic3—28no—
55D4×Dic3—216no—
56D4×Dic3—28no—
57#926—22no—
58#926—22no—
59#926—24no—
60#926—24no—
61#926—24no—
62#926—24no—
63C2×C4⋊Dic3—24no—
64C2×C4⋊Dic3—24no—
65C2×C4⋊Dic3—22no—
66C2×C4⋊Dic3C2×C411yescharacteristic
67C2×C4⋊Dic3C2×C414yes—
68C2×C4⋊Dic3C2×C411yescharacteristic
69C2×C4⋊Dic3C2×C412yes—
70C2×C4⋊Dic3C2×C412yes—
71C2×C4⋊Dic3D414yes—
72C2×C4⋊Dic3D414yes—
73C2×Dic3⋊C4—28no—
74C2×C4×Dic3—22no—
75C2×C4×Dic3C2×C414yes—
76C2×C4×Dic3C2×C412yes—
77C3×#99—28no—
78#834—28no—
79C3⋊#95—28no—
80C22×#41C2×D311yescharacteristic
81#520—64no—
82#520—64no—
83C2×C4×Q8—32no—
84C2×C4×D4—32no—
85C2×C4×D4—34no—
86C2×#117—32no—
87C2×#117—31no—
88C2×#117—32no—
89C2×#117C2×D312yes—
90C22×C4⋊C4—31no—
91C22×C4⋊C4—32no—
92#410—24no—
93#410—64no—
94C2×#99—34no—
95C2×#99C2×D314yes—
96#389—64no—
97#389—64no—
98#380—38no—
99#342—38no—
100C24×C3C22×C412yes—
101C24×C3C2411yescharacteristic
102C3×#41—28no—
103C3×#41—28no—
104C3×#41—28no—
105C3×#41—28no—
106C2×C3×Q8—22no—
107C2×C3×Q8—22no—
108C2×C3×Q8C4⋊C414yescomplemented
109C2×C3×D4—24no—
110C2×C3×D4—24no—
111C2×C3×D4—22no—
112C2×C3×D4—22no—
113C2×C3×D4C4⋊C414yescomplemented
114C2×C3×D4C2×D414yes—
115C2×C3×D4C2×Q814yes—
116C22×C3×C4—22no—
117C22×C3×C4—22no—
118C22×C3×C4—24no—
119C22×C3×C4—24no—
120C22×C3×C4C4⋊C418yescomplemented
121C22×C3×C4C22×C412yes—
122C22×C3×C4C22×C411yescharacteristic
123C22×C3×C4C22×C411yescharacteristic
124C22×C3×C4C2×D412yes—
125C22×C3×C4C2×D412yes—
126C22×C3×C4C2×D412yes—
127C22×C3×C4C2×Q812yes—
128C22×C3×C4#4114yes—
129C22×Dic3—28no—
130C22×Dic3—24no—
131C22×Dic3—82no—
132C22×Dic3—84no—
133C22×Dic3C22×C412yes—
134C22×Dic3C22×C412yes—
135C22×Dic3C22×C414yes—
136C22×Dic3C2×D418yes—
137C3×C4⋊C4—44no—
138C3×C22⋊C4—24no—
139C3×C22⋊C4—24no—
140C3×C42—44no—
141#217—24no—
142#217—28no—
143#217—24no—
144#217—24no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—28no—
150C4⋊Dic3—44no—
151C4⋊Dic3C22⋊C414yes—
152C4⋊Dic3C22⋊C414yes—
153C4⋊Dic3C22⋊C418yes—
154Dic3⋊C4—88no—
155C4×Dic3—24no—
156C4×Dic3—28no—
157C4×Dic3—44no—
158C2×#41—22no—
159C2×#41—22no—
160C2×#41—24no—
161C2×#41Dic614yes—
162C2×#41#6714yes—
163C22×Q8C4×D311yescharacteristic
164C22×D4C4×D311yescharacteristic
165C22×D4C22×D312yes—
166C23×C4—32no—
167C23×C4—34no—
168C23×C4C4×D312yes—
169C23×C4C22×D311yescharacteristic
170C4×Q8—68no—
171C4×D4—68no—
172C4×D4—616no—
173#117—24no—
174#117—24no—
175#117—64no—
176#117—64no—
177#117—62no—
178#117—62no—
179#117—64no—
180#117—64no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—32no—
183C2×C4⋊C4—32no—
184C2×C4⋊C4—34no—
185C2×C4⋊C4—34no—
186C2×C4⋊C4—31no—
187C2×C4⋊C4—34no—
188C2×C4⋊C4—31no—
189C2×C4⋊C4—64no—
190C2×C4⋊C4—64no—
191C2×C4⋊C4—62no—
192C2×C4⋊C4—68no—
193C2×C22⋊C4—34no—
194C2×C22⋊C4—32no—
195C2×C22⋊C4—34no—
196C2×C22⋊C4C22×D314yes—
197C2×C42—22no—
198C2×C42—34no—
199C2×C42—32no—
200C2×C42—62no—
201#99—28no—
202#99—68no—
203#95—68no—
204C23×C3—22no—
205C23×C3—22no—
206C23×C3—41no—
207C23×C3—41no—
208C23×C3—42no—
209C23×C3—42no—
210C23×C3C2×C4⋊C414yes—
211C23×C3C23×C411yescharacteristic
212C23×C3C22×D412yes—
213C23×C3C22×D411yescharacteristic
214C23×C3C22×Q811yescharacteristic
215C23×C3C2×#4112yes—
216C3×Q8—28no—
217C3×D4—28no—
218C3×D4—28no—
219C3×D4—28no—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—28no—
223C2×C3×C4—28no—
224C2×C3×C4—44no—
225C2×C3×C4—44no—
226C2×C3×C4—44no—
227C2×C3×C4C2×C22⋊C412yes—
228C2×C3×C4C2×C22⋊C411yescharacteristic
229C2×C3×C4C2×C22⋊C411yescharacteristic
230C2×C3×C4C2×C4⋊C414yes—
231C2×C3×C4C2×C4⋊C412yes—
232C2×C3×C4C2×C4⋊C412yes—
233C2×C3×C4#11712yes—
234C2×C3×C4#11712yes—
235C2×C3×C4#12114yes—
236C2×C3×C4#12114yes—
237C2×C3×C4C22⋊Q814yes—
238C2×C3×C4C22⋊Q814yes—
239C2×Dic3—22no—
240C2×Dic3—22no—
241C2×Dic3—48no—
242C2×Dic3—44no—
243C2×Dic3—42no—
244C2×Dic3—42no—
245C2×Dic3—44no—
246C2×Dic3—44no—
247C2×Dic3—44no—
248C2×Dic3—88no—
249C2×Dic3C2×C22⋊C418yes—
250C24C2×C4×D312yes—
251C24C23×D311yescharacteristic
252#41—28no—
253#41—28no—
254#41—28no—
255#41—28no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8Dic3⋊C414yescomplemented
259C2×D4—24no—
260C2×D4—24no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4Dic3⋊C414yescomplemented
264C2×D4C2×Dic614yes—
265C2×D4C2×#6714yes—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—34no—
271C22×C4—38no—
272C22×C4—32no—
273C22×C4—32no—
274C22×C4—68no—
275C22×C4—64no—
276C22×C4—242no—
277C22×C4—244no—
278C22×C4Dic3⋊C418yescomplemented
279C22×C4C2×Dic612yes—
280C22×C4C2×C4×D312yes—
281C22×C4C2×C4×D311yescharacteristic
282C22×C4C2×C4×D311yescharacteristic
283C22×C4D3×D412yes—
284C22×C4D3×D412yes—
285C22×C4#23714yes—
286C22×C4C2×#6712yes—
287C4⋊C4—34no—
288C4⋊C4—34no—
289C4⋊C4—38no—
290C4⋊C4—44no—
291C4⋊C4—68no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—124no—
297C4⋊C4—248no—
298C22⋊C4—24no—
299C22⋊C4—24no—
300C22⋊C4—68no—
301C22⋊C4—64no—
302C22⋊C4—64no—
303C22⋊C4—64no—
304C42—44no—
305C42—68no—
306C42—64no—
307C42—124no—
308C22×C3—22no—
309C22×C3—21no—
310C22×C3—22no—
311C22×C3—21no—
312C22×C3—24no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—42no—
316C22×C3—44no—
317C22×C3—44no—
318C22×C3—42no—
319C22×C3—42no—
320C22×C3C22×C22⋊C411yescharacteristic
321C22×C3C22×C4⋊C411yescharacteristic
322C22×C3C2×#11711yescharacteristic
323C22×C3C2×#12112yes—
324C22×C3C2×C22⋊Q812yes—
325C3×C4—24no—
326C3×C4—84no—
327C3×C4#38012yes—
328C3×C4#38012yes—
329C3×C4#38014yes—
330Dic3—28no—
331Dic3—42no—
332Dic3—42no—
333Dic3—84no—
334C23—22no—
335C23—22no—
336C23—41no—
337C23—41no—
338C23—42no—
339C23—42no—
340C23C2×Dic3⋊C414yes—
341C23C22×Dic611yescharacteristic
342C23C22×C4×D311yescharacteristic
343C23C2×D3×D412yes—
344C23C2×#23712yes—
345C23C22×#6711yescharacteristic
346Q8—28no—
347D4—28no—
348D4—28no—
349D4—28no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—28no—
353C2×C4—28no—
354C2×C4—38no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—62no—
359C2×C4—62no—
360C2×C4—128no—
361C2×C4—124no—
362C2×C4—122no—
363C2×C4—122no—
364C2×C4—124no—
365C2×C4—124no—
366C2×C4—124no—
367C2×C4—248no—
368C2×C4#87712yes—
369C2×C4#87712yes—
370C2×C4#87814yes—
371C2×C4#87814yes—
372C2×C4D3×C22⋊C412yes—
373C2×C4D3×C22⋊C411yescharacteristic
374C2×C4D3×C22⋊C411yescharacteristic
375C2×C4C2×Dic3⋊C414yes—
376C2×C4C2×Dic3⋊C412yes—
377C2×C4C2×Dic3⋊C412yes—
378C2×C4#93914yes—
379C2×C4#93914yes—
380C2×C3—22no—
381C2×C3—21no—
382C2×C3—21no—
383C2×C3—22no—
384C2×C3—44no—
385C2×C3C2×#38011yescharacteristic
386C2×C3#288311yescharacteristic
387C2×C3#288411yescharacteristic
388C22—22no—
389C22—21no—
390C22—22no—
391C22—21no—
392C22—24no—
393C22—41no—
394C22—42no—
395C22—41no—
396C22—44no—
397C22—44no—
398C22—42no—
399C22—42no—
400C22C2×#87711yescharacteristic
401C22C2×#87812yes—
402C22C2×D3×C22⋊C411yescharacteristic
403C22C22×Dic3⋊C411yescharacteristic
404C22C2×#93912yes—
405C4—24no—
406C4—68no—
407C4—84no—
408C4—122no—
409C4—122no—
410C4—244no—
411C4#494212yes—
412C4#494212yes—
413C4#494214yes—
414C3#2063411yescharacteristic, complemented, Sylow
415C2—22no—
416C2—21no—
417C2—21no—
418C2—22no—
419C2—44no—
420C2#7868911yescharacteristic
421C2#7869411yescharacteristic
422C2C2×#494211yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC2×C3×#410C2
socleC22×C3C22×C22⋊C4
prime radicalC22×C3×#41C22
cosocleC24×C3C24
hypercenterC2×#410D3
hypocenterC3#20634
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
4. #82232C2semidirect
5. #71593C2semidirect
6. #70028C2semidirect
12. C2×Q8×Dic3C4semidirect
13. C2×D4×Dic3C4semidirect
15. C2×#926C4semidirect
18. C22×C4⋊Dic3C4semidirect
108. C2×C3×Q8C4⋊C4semidirect
113. C2×C3×D4C4⋊C4semidirect
120. C22×C3×C4C4⋊C4semidirect
258. C2×Q8Dic3⋊C4semidirect
263. C2×D4Dic3⋊C4semidirect
278. C22×C4Dic3⋊C4semidirect
414. C3#20634semidirect

Maximal subgroups

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

subgroupindex
C2×#58272
C2×C3×#4102
C2×#52402
#822322
#715932
#700282
#206343