Finite Group Databasebeta

← #10848614

Subgroups — #10848614

Order 768.

1120 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86204C211yesmaximal, characteristic, complemented
2C3×#2858C211yesmaximal, characteristic, complemented
3#83165C211yesmaximal, characteristic
4#83159C211yesmaximal, characteristic, complemented
5#71789C211yesmaximal, characteristic, complemented
6#71730C214yesmaximal, complemented
7#71724C214yesmaximal, complemented
8#69396C211yesmaximal, characteristic, complemented
9C4×#805C211yesmaximal, characteristic, complemented
10#25591—31nomaximal, Sylow
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×#127C2211yescharacteristic
13C2×C3×#124C2214yes—
14#5816C2212yes—
15#5812C2212yes—
16C2×#932C2212yes—
17#5732—22no—
18#5731—22no—
19#5728—22no—
20#5721—22no—
21#5513—24no—
22C4×#235C2211yescharacteristic
23#5470C2211yescharacteristic
24C3×#390—24no—
25C3×C43C2211yescharacteristic
26C2×#836C2214yes—
27#5215C2212yes—
28#5215C2214yes—
29#5075—22no—
30#5072—22no—
31#5071—22no—
32#5070—22no—
33C2×#805C2212yes—
34#4924C2211yescharacteristic
35C4×#208C2211yescharacteristic
36#4523C2212yes—
37#4523C2214yes—
38#3509—31no—
39#3138—31no—
40#3038—31no—
41#2858D311yescharacteristic, complemented
42#2003—34no—
43#1994—31no—
44#1966—31no—
45#1888—34no—
46C4×#104—31no—
47C22×C3×Q8C2312yes—
48C22×C3×D4—22no—
49C22×C3×D4C2312yes—
50C2×#235C2311yescharacteristic
51C3×C4⋊Q8—22no—
52C3×C4⋊Q8—21no—
53C3×C4⋊Q8—21no—
54C3×#127—21no—
55C3×#127—21no—
56C3×#127—22no—
57C3×#124—28no—
58C3×#124—44no—
59C2×C3×C4⋊C4D414yes—
60C2×C3×C22⋊C4—28no—
61C2×C3×C42—22no—
62C2×C3×C42C2312yes—
63C2×C3×C42C2311yescharacteristic
64#947—22no—
65#946—24no—
66Q8×Dic3—24no—
67Q8×Dic3—24no—
68#944—22no—
69C2×#214—22no—
70#932—44no—
71C2×#213—22no—
72#929—24no—
73C4×#67—22no—
74#926—24no—
75#926C2312yes—
76#926C2311yescharacteristic
77#924—22no—
78C2×#208C2312yes—
79C2×#208C2312yes—
80#898—28no—
81#897—24no—
82#892—24no—
83#890—24no—
84#889—28no—
85#887—24no—
86C4×D12D412yes—
87#872—21no—
88C42×D3—21no—
89#870D412yes—
90#869D412yes—
91C4×Dic6—22no—
92C4×Dic6D412yes—
93#836—44no—
94#832—28no—
95#832—24no—
96#805—24no—
97C4⋊(C3⋊C8)—22no—
98#803—21no—
99C4×C3⋊C8—21no—
100C2×AGL(1,ℤ/8)—32no—
101#557—62no—
102#556—62no—
103#552—64no—
104#541—62no—
105#536—32no—
106#533—31no—
107#533—62no—
108C2×C4⋊Q8C2×D311yescharacteristic
109C2×#127C2×D311yescharacteristic
110C2×#124C2×D314yes—
111#519—32no—
112C4×#41—31no—
113#490—62no—
114#486—62no—
115#460—62no—
116#459—62no—
117#423—31no—
118C2×#104—32no—
119#418—34no—
120#418—32no—
121C2×#101—34no—
122C4×M16—31no—
123#390—24no—
124C43C2×D311yescharacteristic
125#339—32no—
126#339—34no—
127C24×C3—22no—
128C2×C3×Q8—22no—
129C2×C3×Q8—22no—
130C2×C3×Q8—24no—
131C2×C3×Q8—24no—
132C2×C3×D4—22no—
133C2×C3×D4—22no—
134C2×C3×D4—24no—
135C2×C3×D4—24no—
136C2×C3×D4—48no—
137C2×C3×D4—42no—
138C2×C3×D4—42no—
139C22×C3×C4—22no—
140C22×C3×C4C2×D414yes—
141C22×C3×C4C2×D412yes—
142C22×C3×C4C2411yescharacteristic
143C2×#67—21no—
144#235—22no—
145#235—22no—
146C2×D12C2×D411yescharacteristic
147C2×C4×D3—22no—
148C2×C4×D3—21no—
149C2×Dic6—22no—
150C2×Dic6C2×D411yescharacteristic
151C3×C4⋊C4—24no—
152C3×C4⋊C4—24no—
153C3×C22⋊C4—416no—
154C3×C42—22no—
155C3×C42—44no—
156C3×C42C2×D411yescharacteristic
157C3×C42C2×D411yescharacteristic
158C3×C42C2×D412yes—
159C3×C42#4114yes—
160#217—22no—
161#217—21no—
162#214—44no—
163#213—44no—
164(C2×D3)⋊C4—28no—
165(C2×D3)⋊C4—22no—
166C4⋊Dic3—28no—
167C4⋊Dic3—22no—
168C4⋊Dic3C2×D412yes—
169C4⋊Dic3C2×D414yes—
170Dic3⋊C4—28no—
171Dic3⋊C4—24no—
172Dic3⋊C4—22no—
173C4×Dic3—28no—
174C4×Dic3—22no—
175C4×Dic3—22no—
176C4×Dic3—21no—
177#208—22no—
178#208—22no—
179#208—24no—
180C2×C3⋊C8—22no—
181C2×C3⋊C8—22no—
182C2×#41—31no—
183C22×Q8C22×D312yes—
184C22×D4—22no—
185C22×D4C22×D312yes—
186AGL(1,ℤ/8)—124no—
187C2×SD16—62no—
188C2×D8—62no—
189C2×M16—32no—
190C2×M16—32no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—22no—
194C4⋊Q8—32no—
195C4⋊Q8—62no—
196#127—22no—
197#127—21no—
198#127—21no—
199#126—68no—
200#125—32no—
201#125—64no—
202#125—68no—
203#124—28no—
204#124—44no—
205#124—62no—
206#123—64no—
207C22⋊Q8—64no—
208C22⋊Q8—64no—
209C22⋊Q8—62no—
210C4×Q8—32no—
211C4×Q8—64no—
212C4×Q8—64no—
213C4×Q8—64no—
214C4×Q8—62no—
215C4×D4—32no—
216C4×D4—62no—
217#117—31no—
218#117—32no—
219#117—64no—
220#117—64no—
221#117—61no—
222C2×C4⋊C4#6714yes—
223C2×C22⋊C4—28no—
224C2×C42—22no—
225C2×C42—61no—
226C2×C42C22×D312yes—
227C2×C42C22×D311yescharacteristic
228C4⋊C8—62no—
229#104—64no—
230#102—64no—
231#102—68no—
232#101—124no—
233#97—61no—
234C4×C8—61no—
235C23×C3—48no—
236C23×C3—42no—
237C23×C3—44no—
238C23×C3C22×D411yescharacteristic
239C22×D3—21no—
240C3×Q8—42no—
241C3×Q8—42no—
242C3×Q8—44no—
243C3×D4—42no—
244C3×D4—42no—
245C3×D4—44no—
246C3×D4—84no—
247C3×D4—84no—
248C2×C3×C4—24no—
249C2×C3×C4—24no—
250C2×C3×C4—24no—
251C2×C3×C4—28no—
252C2×C3×C4—44no—
253C2×C3×C4#12012yes—
254C2×C3×C4#12012yes—
255C2×C3×C4C22×D411yescharacteristic
256C2×C3×C4C22×D411yescharacteristic
257C2×C3×C4C2×#4112yes—
258C2×C3×C4H(5,𝔽2)12yes—
259#67—42no—
260C2×Dic3—22no—
261C2×Dic3—24no—
262C2×Dic3—21no—
263D12—21no—
264D12—21no—
265C4×D3—42no—
266C4×D3—44no—
267Dic6—21no—
268Dic6—21no—
269Dic6—44no—
270C3⋊C8—42no—
271C3⋊C8—42no—
272C24—22no—
273#41—62no—
274#41—62no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—31no—
280C2×Q8—62no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—24no—
284C2×D4—24no—
285C2×D4—31no—
286C2×D4—48no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—61no—
290C22×C4—22no—
291C22×C4—62no—
292C22×C4—61no—
293C22×C4C2×#6714yes—
294C22×C4C2×#6712yes—
295C22×C4C23×D311yescharacteristic
296SD16—124no—
297D8—124no—
298M16—62no—
299M16—62no—
300M16—64no—
301C2×C8—62no—
302C2×C8—62no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—32no—
306C4⋊C4—34no—
307C4⋊C4—68no—
308C4⋊C4—68no—
309C4⋊C4—62no—
310C4⋊C4—62no—
311C4⋊C4—64no—
312C22⋊C4—416no—
313C22⋊C4—68no—
314C22⋊C4—61no—
315C22⋊C4—62no—
316C22⋊C4—62no—
317C42—22no—
318C42—44no—
319C42—68no—
320C42—62no—
321C42—61no—
322C42—62no—
323C42D3×D411yescharacteristic
324C42D3×D411yescharacteristic
325C42D3×D412yes—
326C42#23714yes—
327C22×C3—22no—
328C22×C3—88no—
329C22×C3—82no—
330C22×C3—84no—
331C22×C3C2×#12011yescharacteristic
332C22×C3#54612yes—
333C2×D3—41no—
334C2×D3—42no—
335C3×C4—22no—
336C3×C4—44no—
337C3×C4—44no—
338C3×C4#45414yes—
339C3×C4#53211yescharacteristic
340C3×C4#53511yescharacteristic
341C3×C4#55012yes—
342Dic3—41no—
343Dic3—44no—
344Dic3—42no—
345C23—48no—
346C23—42no—
347C23—44no—
348C23—61no—
349C23C22×#6711yescharacteristic
350Q8—42no—
351Q8—42no—
352Q8—44no—
353Q8—61no—
354Q8—61no—
355Q8—124no—
356D4—42no—
357D4—42no—
358D4—44no—
359D4—61no—
360D4—61no—
361D4—84no—
362D4—84no—
363D4—122no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—28no—
368C2×C4—44no—
369C2×C4—62no—
370C2×C4—64no—
371C2×C4—61no—
372C2×C4—122no—
373C2×C4—124no—
374C2×C4#93712yes—
375C2×C4#93712yes—
376C2×C4C2×D3×D411yescharacteristic
377C2×C4C2×D3×D411yescharacteristic
378C2×C4C2×#23712yes—
379C2×C4#100412yes—
380C8—122no—
381C8—122no—
382C2×C3—22no—
383C2×C3—162no—
384C2×C3#243111yescharacteristic
385C2×C3C2×#45412yes—
386D3—81no—
387C22—22no—
388C22—88no—
389C22—84no—
390C22—82no—
391C22—122no—
392C22—121no—
393C22C2×#93711yescharacteristic
394C22#580612yes—
395C4—22no—
396C4—44no—
397C4—44no—
398C4—122no—
399C4—124no—
400C4—121no—
401C4#506214yes—
402C4#571511yescharacteristic
403C4#571611yescharacteristic
404C4#571812yes—
405C3#2559111yescharacteristic, complemented, Sylow
406C2—22no—
407C2—162no—
408C2—241no—
409C2C2×#506212yes—
410C2#8318711yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2858C2
socleC22×C3C2×#120
prime radical#83165C2
hypercenter#2858D3
hypocenterC3#25591
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2858

Complements

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

NH ≅ G/Ntype
1. #86204C2semidirect
2. C3×#2858C2semidirect
4. #83159C2semidirect
5. #71789C2semidirect
6. #71730C2semidirect
7. #71724C2semidirect
8. #69396C2semidirect
9. C4×#805C2semidirect
41. #2858D3semidirect
405. C3#25591semidirect

Maximal subgroups

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

subgroupindex
#862042
C3×#28582
#831652
#831592
#717892
#717302
#717242
#693962
C4×#8052
#255913