Finite Group Databasebeta

← #10828618

Subgroups — #10828618

Order 768.

1174 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3492C211yesmaximal, characteristic
2#82914C212yesmaximal, complemented
3C2×#4975C211yesmaximal, characteristic, complemented
4#71900C212yesmaximal, complemented
5#71876C212yesmaximal, complemented
6#71711C212yesmaximal, complemented
7#71664C214yesmaximal, complemented
8#71562C211yesmaximal, characteristic, complemented
9#24076—31nomaximal, Sylow
10C3×#554—22no—
11C3×#553—22no—
12C3×#552—22no—
13C3×#547—22no—
14C3×#544C2211yescharacteristic
15C3×#539—22no—
16C3×#537C2211yescharacteristic
17C3×#525—22no—
18C3×#525C2212yes—
19C3×#520C2212yes—
20C22×C3×C4⋊C4C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C22×Dic3⋊C4—22no—
23C22×C4×Dic3C2211yescharacteristic
24C2×#837C2214yes—
25#5232C2214yes—
26#5207C2212yes—
27#5044—22no—
28#5043—22no—
29#5034—22no—
30#5033—22no—
31#4995C2214yes—
32#4975—22no—
33#4975C2212yes—
34#4975C2212yes—
35#4532C2214yes—
36#4530C2212yes—
37#4530C2212yes—
38#3492D311yescharacteristic
39#3092—32no—
40C2×#384—31no—
41#2060—32no—
42#2039—32no—
43#1896—32no—
44#1859—34no—
45#1821—31no—
46C2×C3×#41—22no—
47C2×C3×#41C2312yes—
48C23×C3×C4D412yes—
49C23×C3×C4C2311yescharacteristic
50C23×Dic3—21no—
51C23×Dic3—21no—
52C3×C4⋊Q8—22no—
53C3×#126—24no—
54C3×#125—24no—
55C3×#125—22no—
56C3×#123—24no—
57C3×#123—24no—
58C3×#123—22no—
59C3×C22⋊Q8—22no—
60C3×C22⋊Q8—24no—
61C3×C22⋊Q8—22no—
62C3×C22⋊Q8—22no—
63C3×C22⋊Q8D412yes—
64C3×C22⋊Q8D412yes—
65C3×#121—22no—
66C3×#121—22no—
67C3×#121—22no—
68C3×#121D412yes—
69C3×#121D412yes—
70C3×#120—22no—
71C3×C4×Q8—22no—
72C3×C4×D4—24no—
73C3×C4×D4—22no—
74C3×C4×D4—24no—
75C3×C4×D4—22no—
76C3×C4×D4—22no—
77C3×#117—22no—
78C3×#117—22no—
79C3×#117C2312yes—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4D412yes—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3—22no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3C2312yes—
90C2×Dic3⋊C4—24no—
91C2×Dic3⋊C4—24no—
92C2×Dic3⋊C4—48no—
93C2×C4×Dic3—21no—
94C2×C4×Dic3—24no—
95C2×C4×Dic3C2312yes—
96C2×#208C2314yes—
97#837—28no—
98#835—24no—
99#832—24no—
100#830—22no—
101#830—22no—
102#808—24no—
103#807—24no—
104#554—22no—
105#553—22no—
106#552—22no—
107#547—22no—
108#544C2×D311yescharacteristic
109#539—22no—
110#537C2×D311yescharacteristic
111#525—22no—
112#525C2×D312yes—
113#520C2×D312yes—
114C22×C4⋊C4—31no—
115C22×C4⋊C4—62no—
116C22×C4⋊C4C2×D311yescharacteristic
117C22×C42—31no—
118#484—62no—
119#483—62no—
120#481—62no—
121#480—62no—
122#428—34no—
123C2×#104—34no—
124#417—34no—
125#407—32no—
126#384—32no—
127#384—32no—
128#384—62no—
129#343—32no—
130#343—32no—
131#339—34no—
132C24×C3C2×D411yescharacteristic
133C3×#41—24no—
134C3×#41—24no—
135C3×#41—84no—
136C2×C3×Q8—22no—
137C2×C3×Q8C2×D412yes—
138C2×C3×D4—22no—
139C2×C3×D4—22no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4C2×D412yes—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—24no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—44no—
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2411yescharacteristic
156C22×Dic3—24no—
157C22×Dic3—24no—
158C22×Dic3—24no—
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3—44no—
162C22×Dic3—41no—
163C22×Dic3—41no—
164C3×C4⋊C4—24no—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—24no—
168C3×C4⋊C4—24no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4C2×D412yes—
174C3×C4⋊C4C2×D412yes—
175C3×C22⋊C4—24no—
176C3×C22⋊C4—24no—
177C3×C22⋊C4—22no—
178C3×C22⋊C4—22no—
179C3×C22⋊C4—22no—
180C3×C22⋊C4—22no—
181C3×C22⋊C4—22no—
182C3×C42—24no—
183C3×C42—22no—
184C4⋊Dic3—24no—
185C4⋊Dic3—42no—
186Dic3⋊C4—48no—
187Dic3⋊C4—48no—
188C4×Dic3—24no—
189C4×Dic3—22no—
190C4×Dic3#4114yescomplemented
191#208—24no—
192#208—24no—
193C2×C3⋊C8—24no—
194C2×#41—22no—
195C2×#41C22×D312yes—
196C23×C4—61no—
197C23×C4—61no—
198C23×C4#6712yes—
199C23×C4C22×D311yescharacteristic
200C2×M16—34no—
201C4⋊Q8—22no—
202#126—24no—
203#125—24no—
204#125—22no—
205#123—24no—
206#123—24no—
207#123—22no—
208C22⋊Q8—22no—
209C22⋊Q8—24no—
210C22⋊Q8—22no—
211C22⋊Q8—22no—
212C22⋊Q8#6712yes—
213C22⋊Q8#6712yes—
214#121—22no—
215#121—22no—
216#121—22no—
217#121#6712yes—
218#121#6712yes—
219#120—22no—
220C4×Q8—22no—
221C4×D4—24no—
222C4×D4—22no—
223C4×D4—24no—
224C4×D4—22no—
225C4×D4—22no—
226#117—22no—
227#117—22no—
228#117C22×D312yes—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—32no—
233C2×C4⋊C4—62no—
234C2×C4⋊C4—62no—
235C2×C4⋊C4—61no—
236C2×C4⋊C4—64no—
237C2×C4⋊C4—64no—
238C2×C4⋊C4—128no—
239C2×C4⋊C4#6712yes—
240C2×C4⋊C4C22×D311yescharacteristic
241C2×C4⋊C4C22×D311yescharacteristic
242C2×C42—32no—
243C2×C42—64no—
244C2×C42—61no—
245#107—64no—
246#106—64no—
247#104—68no—
248#103—64no—
249#102—64no—
250C22⋊C8—62no—
251C22⋊C8—62no—
252C23×C3—21no—
253C23×C3—21no—
254C23×C3—22no—
255C23×C3—22no—
256C23×C3—22no—
257C23×C3#12011yescharacteristic
258C23×C3#12011yescharacteristic
259C23×C3C22×D411yescharacteristic
260C3×Q8—24no—
261C3×Q8—82no—
262C3×D4—24no—
263C3×D4—42no—
264C3×D4—42no—
265C3×D4—44no—
266C3×D4—44no—
267C3×D4—82no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—24no—
271C2×C3×C4—24no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—44no—
278C2×C3×C4—44no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—82no—
282C2×C3×C4—84no—
283C2×C3×C4#12012yes—
284C2×C3×C4C22×D412yes—
285C2×C3×C4C2×#4112yes—
286C2×C3×C42−1+412yes—
287C2×Dic3—24no—
288C2×Dic3—216no—
289C2×Dic3—24no—
290C2×Dic3—24no—
291C2×Dic3—44no—
292C2×Dic3—42no—
293C2×Dic3—82no—
294C2×Dic3—82no—
295C3⋊C8—44no—
296C24C2×#6711yescharacteristic
297#41—24no—
298#41—24no—
299#41—84no—
300C2×Q8—22no—
301C2×Q8C2×#6712yes—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4C2×#6712yes—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—44no—
314C22×C4—64no—
315C22×C4—64no—
316C22×C4—64no—
317C22×C4—62no—
318C22×C4—62no—
319C22×C4—124no—
320C22×C4—121no—
321C22×C4—121no—
322C22×C4C2×#6711yescharacteristic
323C22×C4C2×#6712yes—
324C22×C4C2×#6712yes—
325C22×C4C2×#6712yes—
326C22×C4C2×#6712yes—
327C22×C4C23×D311yescharacteristic
328M16—64no—
329M16—64no—
330C2×C8—64no—
331C4⋊C4—24no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—64no—
341C4⋊C4—128no—
342C4⋊C4—128no—
343C4⋊C4—122no—
344C4⋊C4C2×#6712yes—
345C4⋊C4C2×#6712yes—
346C22⋊C4—24no—
347C22⋊C4—24no—
348C22⋊C4—22no—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C42—24no—
354C42—22no—
355C42—34no—
356C42—64no—
357C42—62no—
358C22×C3—24no—
359C22×C3—24no—
360C22×C3—21no—
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—42no—
367C22×C3—42no—
368C22×C3—42no—
369C22×C3#45312yes—
370C22×C3#45612yes—
371C22×C3C2×#12011yescharacteristic
372C22×C3#52511yescharacteristic
373C22×C3#53611yescharacteristic
374C3×C4—22no—
375C3×C4—42no—
376C3×C4—42no—
377C3×C4—42no—
378C3×C4—42no—
379C3×C4—42no—
380C3×C4#54714yes—
381Dic3—28no—
382Dic3—82no—
383C23—22no—
384C23—22no—
385C23—21no—
386C23—21no—
387C23—22no—
388C23#95311yescharacteristic
389C23#95311yescharacteristic
390C23C22×#6711yescharacteristic
391Q8—24no—
392Q8—82no—
393D4—24no—
394D4—44no—
395D4—44no—
396D4—42no—
397D4—42no—
398D4—82no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—64no—
413C2×C4—616no—
414C2×C4—64no—
415C2×C4—64no—
416C2×C4—82no—
417C2×C4—84no—
418C2×C4—124no—
419C2×C4—122no—
420C2×C4—242no—
421C2×C4—242no—
422C2×C4#95312yes—
423C2×C4D3×#4112yes—
424C2×C4#101012yes—
425C2×C4C22×#6712yes—
426C8—124no—
427C2×C3—21no—
428C2×C3—21no—
429C2×C3—22no—
430C2×C3—22no—
431C2×C3—82no—
432C2×C3#240611yescharacteristic
433C2×C3C2×#45311yescharacteristic
434C2×C3C2×#45611yescharacteristic
435C22—24no—
436C22—24no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22#524112yes—
447C22#524612yes—
448C22#582511yescharacteristic
449C22#582811yescharacteristic
450C22C2×#95311yescharacteristic
451C4—22no—
452C4—42no—
453C4—42no—
454C4—42no—
455C4—42no—
456C4—42no—
457C4—68no—
458C4—242no—
459C4#582414yes—
460C3#2407611yescharacteristic, complemented, Sylow
461C2—21no—
462C2—22no—
463C2—22no—
464C2—21no—
465C2—82no—
466C2#8409011yescharacteristic
467C2C2×#524111yescharacteristic
468C2C2×#524611yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3492C2
socleC22×C3C2×#120
hypercenter#3492D3
hypocenterC3#24076
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3492

Complements

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

NH ≅ G/Ntype
2. #82914C2semidirect
3. C2×#4975C2semidirect
4. #71900C2semidirect
5. #71876C2semidirect
6. #71711C2semidirect
7. #71664C2semidirect
8. #71562C2semidirect
190. C4×Dic3#41semidirect
460. C3#24076semidirect

Maximal subgroups

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

subgroupindex
C3×#34922
#829142
C2×#49752
#719002
#718762
#717112
#716642
#715622
#240763