Finite Group Databasebeta

← #10848617

Subgroups — #10848617

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#85830C211yesmaximal, characteristic, complemented
2#82526C211yesmaximal, characteristic
3#78526C211yesmaximal, characteristic, complemented
4#73313C211yesmaximal, characteristic, complemented
5C3×C4×#104C211yesmaximal, characteristic, complemented
6#70388C214yesmaximal, complemented
7#69877C211yesmaximal, characteristic, complemented
8#69396C211yesmaximal, characteristic, complemented
9#68766C214yesmaximal, complemented
10#25591—31nomaximal, Sylow
11C3×C4×#41C2211yescharacteristic
12#5825C2212yes—
13C2×#949C2212yes—
14#5567—22no—
15#5542—22no—
16#5541—24no—
17#5518—22no—
18#5516C2212yes—
19C2×#875C2214yes—
20C2×#874C2211yescharacteristic
21#5470—22no—
22#5470C2211yescharacteristic
23C2×#869C2211yescharacteristic
24C2×C3×#104C2212yes—
25C3×C4×M16C2211yescharacteristic
26C3×C43C2211yescharacteristic
27C2×#824C2214yes—
28#5131C2214yes—
29#5027—22no—
30#5015—22no—
31#5001C2212yes—
32#4939—24no—
33#4924C2211yescharacteristic
34#4713—22no—
35#4694—22no—
36C3×#339C2212yes—
37#4523C2214yes—
38#3509—31no—
39#3138—31no—
40#3038—31no—
41#2858—31no—
42#2003—34no—
43#1994—31no—
44#1966—31no—
45#1888—34no—
46C4×#104D311yescharacteristic, complemented
47C2×C3×#41C2311yescharacteristic
48C22×D12—22no—
49C22×D12C2312yes—
50C22×Dic6C2312yes—
51C2×C3×M16C2312yes—
52C3×C4×Q8D412yes—
53C3×C4×D4—22no—
54C3×C4×D4D412yes—
55C3×#117—21no—
56C3×#117C2311yescharacteristic
57C2×C3×C42—21no—
58C2×C3×C42—22no—
59C2×C3×C42C2312yes—
60C2×C3×C42C2311yescharacteristic
61#949—44no—
62#944—22no—
63C2×#215—22no—
64#936—22no—
65#935—24no—
66C2×#213—22no—
67C2×(C2×D3)⋊C4—28no—
68#926—24no—
69#926C2312yes—
70C2×C4⋊Dic3D414yes—
71#924—24no—
72#924—22no—
73#924—24no—
74C2×#208C2312yes—
75#889—28no—
76#887—24no—
77#879—28no—
78#877—24no—
79#875—28no—
80#875—44no—
81#874—21no—
82#874—21no—
83#874—22no—
84#870—24no—
85#870D412yes—
86#869—21no—
87#869—21no—
88#869—22no—
89#869D412yes—
90C4×Dic6—24no—
91C4×Dic6—22no—
92C4×Dic6—24no—
93C3×#104—24no—
94C3×#97—21no—
95C3×C4×C8—21no—
96#824—44no—
97#821—28no—
98#809—24no—
99C4⋊(C3⋊C8)—22no—
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⋊Q8—31no—
109C2×#127—31no—
110C2×#124—34no—
111#519—32no—
112C4×#41C2×D311yescharacteristic
113#490—62no—
114#486—62no—
115#460—62no—
116#459—62no—
117#423—31no—
118C2×#104C2×D312yes—
119#418—34no—
120#418—32no—
121C2×#101—34no—
122C4×M16C2×D311yescharacteristic
123#390—64no—
124C43C2×D311yescharacteristic
125#339—34no—
126#339C2×D312yes—
127C23×D3—22no—
128C3×#41—22no—
129C3×#41—22no—
130C2×C3×Q8C2×D411yescharacteristic
131C2×C3×D4—21no—
132C2×C3×D4C2×D411yescharacteristic
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—21no—
136C22×C3×C4C2×D412yes—
137C22×C3×C4C2411yescharacteristic
138C22×Dic3C2×D414yes—
139C2×D12—22no—
140C2×D12—22no—
141C2×D12—24no—
142C2×D12—24no—
143C2×D12—48no—
144C2×D12—42no—
145C2×D12—42no—
146C2×Dic6—22no—
147C2×Dic6—22no—
148C2×Dic6—22no—
149C2×Dic6—24no—
150C2×Dic6—24no—
151C3×M16—22no—
152C3×M16—22no—
153C2×C3×C8—22no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4C2×D412yes—
156C3×C22⋊C4—21no—
157C3×C22⋊C4—22no—
158C3×C42—22no—
159C3×C42—21no—
160C3×C42—22no—
161C3×C42—44no—
162C3×C42C2×D412yes—
163C3×C42C2×D411yescharacteristic
164C3×C42C2×D411yescharacteristic
165C3×C42#4114yes—
166#217—28no—
167#217—22no—
168#215—44no—
169#213—44no—
170(C2×D3)⋊C4—416no—
171C4⋊Dic3—24no—
172C4⋊Dic3—22no—
173C4⋊Dic3—24no—
174C4⋊Dic3C2×D414yes—
175Dic3⋊C4—28no—
176Dic3⋊C4—28no—
177Dic3⋊C4—24no—
178C4×Dic3—28no—
179C4×Dic3—22no—
180#208—24no—
181C2×C3⋊C8—22no—
182C2×#41C22×D311yescharacteristic
183C22×Q8—32no—
184C22×D4—32no—
185C22×D4—62no—
186AGL(1,ℤ/8)—124no—
187C2×SD16—62no—
188C2×D8—62no—
189C2×M16—32no—
190C2×M16C22×D312yes—
191C4⋊Q8—32no—
192C4⋊Q8—62no—
193C4⋊Q8—62no—
194C4⋊Q8—61no—
195C4⋊Q8—61no—
196#127—62no—
197#127—61no—
198#127—61no—
199#126—68no—
200#125—32no—
201#125—64no—
202#125—68no—
203#124—62no—
204#124—68no—
205#124—124no—
206#123—64no—
207C22⋊Q8—64no—
208C22⋊Q8—64no—
209C22⋊Q8—62no—
210C4×Q8—64no—
211C4×Q8—64no—
212C4×Q8—64no—
213C4×Q8—62no—
214C4×Q8D1212yes—
215C4×D4—22no—
216C4×D4D1212yes—
217#117—21no—
218#117—32no—
219#117—64no—
220#117—64no—
221#117C22×D311yescharacteristic
222C2×C4⋊C4—34no—
223C2×C22⋊C4—68no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42C22×D312yes—
227C2×C42C22×D311yescharacteristic
228C4⋊C8—62no—
229#104—24no—
230#102—68no—
231#102—64no—
232#101—124no—
233#97—21no—
234C4×C8—21no—
235C23×C3—21no—
236C23×C3C22×D411yescharacteristic
237C22×D3—48no—
238C22×D3—42no—
239C22×D3—44no—
240C3×Q8—21no—
241C3×Q8—21no—
242C3×D4—21no—
243C3×D4—21no—
244C3×D4—42no—
245C2×C3×C4—24no—
246C2×C3×C4—22no—
247C2×C3×C4—21no—
248C2×C3×C4—24no—
249C2×C3×C4—44no—
250C2×C3×C4—42no—
251C2×C3×C4—44no—
252C2×C3×C4#12012yes—
253C2×C3×C4#12012yes—
254C2×C3×C4C22×D411yescharacteristic
255C2×C3×C4C22×D411yescharacteristic
256C2×C3×C4C2×#4112yes—
257C2×C3×C4H(5,𝔽2)12yes—
258C2×Dic3—24no—
259C2×Dic3—24no—
260C2×Dic3—28no—
261D12—42no—
262D12—42no—
263D12—44no—
264D12—84no—
265D12—84no—
266Dic6—42no—
267Dic6—42no—
268Dic6—44no—
269Dic6—44no—
270C3×C8—42no—
271C3⋊C8—42no—
272C24—62no—
273#41—22no—
274#41—22no—
275C2×Q8—62no—
276C2×Q8—62no—
277C2×Q8—64no—
278C2×Q8—62no—
279C2×Q8—64no—
280C2×Q8C2×D1211yescharacteristic
281C2×D4—21no—
282C2×D4—62no—
283C2×D4—64no—
284C2×D4—62no—
285C2×D4—64no—
286C2×D4—128no—
287C2×D4—122no—
288C2×D4—122no—
289C2×D4C2×D1211yescharacteristic
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—34no—
294C22×C4D3×D412yes—
295C22×C4C23×D311yescharacteristic
296SD16—124no—
297D8—124no—
298M16—22no—
299M16—22no—
300M16—64no—
301C2×C8—22no—
302C2×C8—62no—
303C4⋊C4—22no—
304C4⋊C4—34no—
305C4⋊C4—68no—
306C4⋊C4—68no—
307C4⋊C4—64no—
308C4⋊C4—64no—
309C4⋊C4—62no—
310C4⋊C4—64no—
311C4⋊C4C2×D1212yes—
312C22⋊C4—21no—
313C22⋊C4—22no—
314C22⋊C4—68no—
315C22⋊C4—62no—
316C22⋊C4—1216no—
317C42—22no—
318C42—21no—
319C42—22no—
320C42—44no—
321C42—68no—
322C42—62no—
323C42C2×D1212yes—
324C42#23514yes—
325C42D3×D411yescharacteristic
326C42D3×D411yescharacteristic
327C22×C3—22no—
328C22×C3—41no—
329C22×C3—42no—
330C22×C3C2×#12011yescharacteristic
331C22×C3#54612yes—
332C2×D3—88no—
333C2×D3—82no—
334C2×D3—84no—
335C3×C4—22no—
336C3×C4—44no—
337C3×C4—41no—
338C3×C4—42no—
339C3×C4#45414yes—
340C3×C4#53211yescharacteristic
341C3×C4#53511yescharacteristic
342C3×C4#55012yes—
343Dic3—44no—
344Dic3—44no—
345C23—21no—
346C23—128no—
347C23—122no—
348C23—124no—
349C23C2×D3×D411yescharacteristic
350Q8—21no—
351Q8—21no—
352Q8—122no—
353Q8—122no—
354Q8—124no—
355Q8—124no—
356D4—21no—
357D4—21no—
358D4—42no—
359D4—122no—
360D4—122no—
361D4—124no—
362D4—244no—
363D4—244no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—24no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—64no—
372C2×C4—64no—
373C2×C4—68no—
374C2×C4#88212yes—
375C2×C4#88212yes—
376C2×C4C22×D1211yescharacteristic
377C2×C4C2×#23512yes—
378C2×C4C2×D3×D411yescharacteristic
379C2×C4#100912yes—
380C8—42no—
381C8—122no—
382C2×C3—22no—
383C2×C3—81no—
384C2×C3#243111yescharacteristic
385C2×C3C2×#45412yes—
386D3—162no—
387C22—22no—
388C22—41no—
389C22—42no—
390C22—248no—
391C22—242no—
392C22—244no—
393C22C2×#88211yescharacteristic
394C22#555012yes—
395C4—22no—
396C4—44no—
397C4—41no—
398C4—42no—
399C4—124no—
400C4—124no—
401C4#482714yes—
402C4#547711yescharacteristic
403C4#552511yescharacteristic
404C4#555312yes—
405C3#2559111yescharacteristic, complemented, Sylow
406C2—22no—
407C2—81no—
408C2—482no—
409C2#7358611yescharacteristic
410C2C2×#482712yes—
subgroupG/H
centerC22C2×#882
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#104C2
socleC22×C3C2×#120
prime radical#82526C2
hypercenterC4×#104D3
hypocenterC3#25591
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#104

Complements

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

NH ≅ G/Ntype
1. #85830C2semidirect
3. #78526C2semidirect
4. #73313C2semidirect
5. C3×C4×#104C2semidirect
6. #70388C2semidirect
7. #69877C2semidirect
8. #69396C2semidirect
9. #68766C2semidirect
46. C4×#104D3semidirect
405. C3#25591semidirect

Maximal subgroups

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

subgroupindex
#858302
#825262
#785262
#733132
C3×C4×#1042
#703882
#698772
#693962
#687662
#255913