Finite Group Databasebeta

← #10828622

Subgroups — #10828622

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
1#85718C211yesmaximal, characteristic
2C2×C3×#384C211yesmaximal, characteristic, complemented
3#82139C212yesmaximal, complemented
4#71562C211yesmaximal, characteristic, complemented
5#71532C212yesmaximal, complemented
6#69593C212yesmaximal, complemented
7#69592C214yesmaximal, complemented
8#69573C212yesmaximal, complemented
9#24076—31nomaximal, Sylow
10C22×C3×C4⋊C4—22no—
11C22×C3×C4⋊C4C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C3×C42C2211yescharacteristic
14#5791—22no—
15#5791C2212yes—
16#5515—22no—
17#5514—22no—
18#5513—22no—
19#5511—22no—
20#5510C2211yescharacteristic
21#5507—22no—
22#5504C2212yes—
23#5500C2211yescharacteristic
24C3×#384—22no—
25C3×#384C2212yes—
26C3×#384C2212yes—
27#5207C2212yes—
28#4971—22no—
29#4970—22no—
30#4968—22no—
31#4967—22no—
32#4966C2214yes—
33#4964C2214yes—
34C2×#805C2214yes—
35#4530C2212yes—
36#4530C2212yes—
37#4523C2214yes—
38#3492—31no—
39#3092—32no—
40C2×#384D311yescharacteristic, complemented
41#2060—32no—
42#2039—32no—
43#1896—32no—
44#1859—34no—
45#1821—31no—
46C23×C3×C4—21no—
47C23×C3×C4—21no—
48C23×C3×C4D412yes—
49C23×C3×C4C2311yescharacteristic
50C2×#235—22no—
51C2×#235C2312yes—
52C2×C3×C4⋊C4—22no—
53C2×C3×C4⋊C4—22no—
54C2×C3×C4⋊C4—22no—
55C2×C3×C4⋊C4—22no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4—24no—
58C2×C3×C4⋊C4—24no—
59C2×C3×C4⋊C4—48no—
60C2×C3×C4⋊C4D412yes—
61C2×C3×C4⋊C4C2312yes—
62C2×C3×C4⋊C4C2311yescharacteristic
63C2×C3×C4⋊C4C2311yescharacteristic
64C2×C3×C42—21no—
65C2×C3×C42—24no—
66C2×C3×C42C2312yes—
67#953—22no—
68#930—22no—
69#930—22no—
70#930D412yes—
71#930D412yes—
72#929—24no—
73#929—22no—
74C4×#67—24no—
75C4×#67—24no—
76C4×#67—22no—
77C4×#67—22no—
78#926—22no—
79#926C2312yes—
80#924—22no—
81#924—22no—
82#924D412yes—
83#924D412yes—
84C2×#208C2314yes—
85#898—24no—
86#897—22no—
87#896—24no—
88#895—22no—
89#894—24no—
90#893—22no—
91#892—22no—
92D3×C4⋊C4—22no—
93#890—22no—
94#889—24no—
95#888—22no—
96#887—22no—
97#830—22no—
98#830—22no—
99#810—24no—
100#809—24no—
101#808—24no—
102#807—24no—
103#805—28no—
104#554—62no—
105#553—62no—
106#552—62no—
107#547—62no—
108#544—31no—
109#539—62no—
110#537—31no—
111#525—32no—
112#525—62no—
113#520—32no—
114C22×C4⋊C4—22no—
115C22×C4⋊C4C2×D311yescharacteristic
116C22×C4⋊C4C2×D311yescharacteristic
117C22×C42C2×D311yescharacteristic
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—22no—
127#384C2×D312yes—
128#384C2×D312yes—
129#343—32no—
130#343—32no—
131#339—34no—
132C24×C3C2×D411yescharacteristic
133C22×C3×C4—24no—
134C22×C3×C4—24no—
135C22×C3×C4—24no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—44no—
143C22×C3×C4—44no—
144C22×C3×C4—41no—
145C22×C3×C4—41no—
146C22×C3×C4C2×D412yes—
147C22×C3×C4C2×D412yes—
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D412yes—
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2411yescharacteristic
152C2×#67—22no—
153C2×#67—22no—
154C2×#67—22no—
155#235—24no—
156#235—24no—
157#235—84no—
158C2×D12—22no—
159C2×D12C2×D412yes—
160C2×C4×D3—22no—
161C2×C4×D3—24no—
162C2×Dic6—22no—
163C2×Dic6C2×D412yes—
164C3×C4⋊C4—24no—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—24no—
168C3×C4⋊C4—48no—
169C3×C4⋊C4—48no—
170C3×C4⋊C4—42no—
171C3×C42—24no—
172C3×C42—22no—
173C3×C42#4114yescomplemented
174#217—22no—
175#217—22no—
176#217—22no—
177(C2×D3)⋊C4—24no—
178(C2×D3)⋊C4—24no—
179(C2×D3)⋊C4—22no—
180(C2×D3)⋊C4—22no—
181C4⋊Dic3—22no—
182C4⋊Dic3—22no—
183C4⋊Dic3C2×D412yes—
184C4⋊Dic3C2×D412yes—
185Dic3⋊C4—24no—
186Dic3⋊C4—24no—
187Dic3⋊C4—22no—
188Dic3⋊C4—22no—
189C4×Dic3—24no—
190C4×Dic3—22no—
191#208—24no—
192#208—24no—
193C2×C3⋊C8—24no—
194C2×#41—32no—
195C2×#41—62no—
196C23×C4—21no—
197C23×C4—21no—
198C23×C4#6712yes—
199C23×C4C22×D311yescharacteristic
200C2×M16—34no—
201C4⋊Q8—62no—
202#126—64no—
203#125—62no—
204#125—64no—
205#123—64no—
206#123—64no—
207#123—62no—
208C22⋊Q8—32no—
209C22⋊Q8—32no—
210C22⋊Q8—62no—
211C22⋊Q8—64no—
212C22⋊Q8—62no—
213C22⋊Q8—62no—
214#121—32no—
215#121—32no—
216#121—62no—
217#121—62no—
218#121—62no—
219#120—62no—
220C4×Q8—62no—
221C4×D4—62no—
222C4×D4—64no—
223C4×D4—62no—
224C4×D4—62no—
225C4×D4—64no—
226#117—32no—
227#117—62no—
228#117—62no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—48no—
237C2×C4⋊C4—62no—
238C2×C4⋊C4#6712yes—
239C2×C4⋊C4C22×D312yes—
240C2×C4⋊C4C22×D311yescharacteristic
241C2×C4⋊C4C22×D311yescharacteristic
242C2×C42—21no—
243C2×C42—24no—
244C2×C42C22×D312yes—
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#12011yescharacteristic
257C23×C3#12011yescharacteristic
258C23×C3C22×D411yescharacteristic
259C22×D3—22no—
260C2×C3×C4—24no—
261C2×C3×C4—216no—
262C2×C3×C4—24no—
263C2×C3×C4—21no—
264C2×C3×C4—24no—
265C2×C3×C4—21no—
266C2×C3×C4—24no—
267C2×C3×C4—24no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—44no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—44no—
274C2×C3×C4—42no—
275C2×C3×C4—82no—
276C2×C3×C4—82no—
277C2×C3×C4—82no—
278C2×C3×C4#12012yes—
279C2×C3×C4C22×D412yes—
280C2×C3×C4C2×#4112yes—
281C2×C3×C42−1+412yes—
282#67—42no—
283#67—42no—
284#67—44no—
285#67—44no—
286C2×Dic3—22no—
287C2×Dic3—22no—
288C2×Dic3—22no—
289D12—24no—
290D12—82no—
291C4×D3—44no—
292C4×D3—84no—
293Dic6—24no—
294Dic6—82no—
295C3⋊C8—44no—
296C24C2×#6711yescharacteristic
297#41—64no—
298#41—64no—
299#41—244no—
300C2×Q8—32no—
301C2×Q8—62no—
302C2×D4—32no—
303C2×D4—62no—
304C2×D4—62no—
305C2×D4—62no—
306C2×D4—62no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—64no—
321C22×C4—62no—
322C22×C4D3×D412yes—
323C22×C4D3×D412yes—
324C22×C4C2×#6711yescharacteristic
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—32no—
336C4⋊C4—32no—
337C4⋊C4—48no—
338C4⋊C4—48no—
339C4⋊C4—42no—
340C4⋊C4—64no—
341C4⋊C4—62no—
342C4⋊C4—62no—
343C4⋊C4—64no—
344C4⋊C4—62no—
345C4⋊C4—62no—
346C22⋊C4—64no—
347C22⋊C4—64no—
348C22⋊C4—62no—
349C22⋊C4—62no—
350C22⋊C4—62no—
351C22⋊C4—62no—
352C22⋊C4—62no—
353C42—24no—
354C42—22no—
355C42—64no—
356C42—62no—
357C42#23514yescomplemented
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#45312yes—
367C22×C3#45612yes—
368C22×C3C2×#12011yescharacteristic
369C22×C3#52511yescharacteristic
370C22×C3#53611yescharacteristic
371C2×D3—42no—
372C2×D3—42no—
373C2×D3—42no—
374C3×C4—28no—
375C3×C4—22no—
376C3×C4—42no—
377C3×C4—42no—
378C3×C4—82no—
379C3×C4#54714yes—
380Dic3—42no—
381Dic3—42no—
382Dic3—42no—
383C23—21no—
384C23—21no—
385C23—22no—
386C23—22no—
387C23—62no—
388C23#93711yescharacteristic
389C23#93711yescharacteristic
390C23C22×#6711yescharacteristic
391Q8—64no—
392Q8—242no—
393D4—64no—
394D4—122no—
395D4—122no—
396D4—124no—
397D4—124no—
398D4—242no—
399C2×C4—24no—
400C2×C4—216no—
401C2×C4—24no—
402C2×C4—21no—
403C2×C4—24no—
404C2×C4—21no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—62no—
415C2×C4—62no—
416C2×C4—62no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—124no—
421C2×C4—244no—
422C2×C4#93712yes—
423C2×C4C2×#23512yes—
424C2×C4C2×D3×D412yes—
425C2×C4#100712yes—
426C8—124no—
427C2×C3—21no—
428C2×C3—21no—
429C2×C3—22no—
430C2×C3—22no—
431C2×C3#240611yescharacteristic
432C2×C3C2×#45311yescharacteristic
433C2×C3C2×#45611yescharacteristic
434D3—82no—
435C22—24no—
436C22—24no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—122no—
444C22—122no—
445C22—122no—
446C22#507712yes—
447C22#509512yes—
448C22#579111yescharacteristic
449C22C2×#93711yescharacteristic
450C22#581611yescharacteristic
451C4—28no—
452C4—22no—
453C4—42no—
454C4—42no—
455C4—82no—
456C4—122no—
457C4—122no—
458C4—122no—
459C4#550714yes—
460C3#2407611yescharacteristic, complemented, Sylow
461C2—21no—
462C2—21no—
463C2—22no—
464C2—22no—
465C2—242no—
466C2#8219211yescharacteristic
467C2C2×#507711yescharacteristic
468C2C2×#509511yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#384C2
socleC22×C3C2×#120
prime radical#85718C2
hypercenterC2×#384D3
hypocenterC3#24076
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#384

Complements

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

NH ≅ G/Ntype
2. C2×C3×#384C2semidirect
3. #82139C2semidirect
4. #71562C2semidirect
5. #71532C2semidirect
6. #69593C2semidirect
7. #69592C2semidirect
8. #69573C2semidirect
40. C2×#384D3semidirect
173. C3×C42#41semidirect
357. C42#235semidirect
460. C3#24076semidirect

Maximal subgroups

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

subgroupindex
#857182
C2×C3×#3842
#821392
#715622
#715322
#695932
#695922
#695732
#240763