Finite Group Databasebeta

← #10848618

Subgroups — #10848618

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
2C3×#3038C211yesmaximal, characteristic
3#83184C211yesmaximal, characteristic, complemented
4#83159C211yesmaximal, characteristic, complemented
5#72096C214yesmaximal, complemented
6C4×#837C211yesmaximal, characteristic, complemented
7#70522C211yesmaximal, characteristic, complemented
8#70514C211yesmaximal, characteristic, complemented
9#70325C214yesmaximal, complemented
10#25591—31nomaximal, Sylow
11C2×C3×AGL(1,ℤ/8)C2212yes—
12C2×C3×#127C2211yescharacteristic
13C3×C4×#41C2211yescharacteristic
14#5825C2212yes—
15C2×#936C2214yes—
16#5567—22no—
17#5542—22no—
18#5541—24no—
19#5518—22no—
20#5516C2212yes—
21#5470—22no—
22#5470C2211yescharacteristic
23C2×#869C2211yescharacteristic
24C3×#460—22no—
25C3×#459—22no—
26C3×#423C2211yescharacteristic
27C3×#418C2212yes—
28C2×#837C2212yes—
29#5221—24no—
30#5215C2214yes—
31C2×#824C2214yes—
32#5071—22no—
33#5070—22no—
34C42×Dic3C2211yescharacteristic
35C4×#208C2211yescharacteristic
36#4555C2214yes—
37#4532C2212yes—
38#3509—31no—
39#3138—31no—
40#3038D311yescharacteristic
41#2858—31no—
42#2003—34no—
43#1994—31no—
44#1966—31no—
45#1888—34no—
46C4×#104—31no—
47C2×C3×#41C2311yescharacteristic
48C22×C3×D4—22no—
49C22×C3×D4C2312yes—
50C22×Dic6C2312yes—
51C3×AGL(1,ℤ/8)—44no—
52C2×C3×SD16—22no—
53C2×C3×D8—22no—
54C2×C3×M16C2312yes—
55C3×#127—21no—
56C3×#127—21no—
57C3×#127—22no—
58C3×C4×Q8D412yes—
59C3×C4×D4—22no—
60C3×C4×D4D412yes—
61C3×#117—21no—
62C3×#117C2311yescharacteristic
63C2×C3×C42—21no—
64C2×C3×C42C2311yescharacteristic
65C2×#217—28no—
66#944—22no—
67#936—22no—
68#936—28no—
69#936—44no—
70#935—24no—
71#926—24no—
72#926C2312yes—
73C2×C4⋊Dic3D414yes—
74#924—24no—
75#924—22no—
76#924—24no—
77C2×C4×Dic3—22no—
78C2×C4×Dic3C2312yes—
79C2×#208C2312yes—
80#889—28no—
81#887—24no—
82#879—28no—
83#877—24no—
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×C4⋊C8—22no—
94C3×#102—24no—
95#837—24no—
96#832—28no—
97#824—44no—
98#803—21no—
99C4×C3⋊C8—21no—
100C2×AGL(1,ℤ/8)C2×D312yes—
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×#127C2×D311yescharacteristic
110C2×#124—34no—
111#519—32no—
112C4×#41C2×D311yescharacteristic
113#490—62no—
114#486—62no—
115#460—22no—
116#459—22no—
117#423C2×D311yescharacteristic
118C2×#104—32no—
119#418—34no—
120#418C2×D312yes—
121C2×#101—34no—
122C4×M16—31no—
123#390—64no—
124C43—31no—
125#339—34no—
126#339—32no—
127C24×C3—22no—
128C3×#41—22no—
129C3×#41—22no—
130C2×C3×Q8C2×D411yescharacteristic
131C2×C3×D4—21no—
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—
139C2×C3×D4C2×D411yescharacteristic
140C22×C3×C4—22no—
141C22×C3×C4—21no—
142C22×C3×C4C2×D412yes—
143C22×C3×C4C2411yescharacteristic
144C22×Dic3—22no—
145C22×Dic3C2×D414yes—
146C2×Dic6—22no—
147C2×Dic6—22no—
148C2×Dic6—22no—
149C2×Dic6—24no—
150C2×Dic6—24no—
151C3×SD16—44no—
152C3×D8—44no—
153C3×M16—24no—
154C2×C3×C8—22no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4C2×D412yes—
157C3×C22⋊C4—21no—
158C3×C22⋊C4—22no—
159C3×C42—22no—
160C3×C42—21no—
161C3×C42C2×D411yescharacteristic
162C3×C42C2×D411yescharacteristic
163C3×C42C2×D412yes—
164#217—28no—
165#217—22no—
166#217—416no—
167C4⋊Dic3—24no—
168C4⋊Dic3—22no—
169C4⋊Dic3—24no—
170C4⋊Dic3C2×D414yes—
171Dic3⋊C4—28no—
172Dic3⋊C4—28no—
173Dic3⋊C4—24no—
174C4×Dic3—28no—
175C4×Dic3—22no—
176C4×Dic3—22no—
177C4×Dic3—44no—
178C4×Dic3#4114yes—
179#208—22no—
180#208—22no—
181C2×C3⋊C8—22no—
182C2×#41C22×D311yescharacteristic
183C22×Q8—32no—
184C22×D4—22no—
185C22×D4C22×D312yes—
186AGL(1,ℤ/8)—44no—
187C2×SD16—22no—
188C2×D8—22no—
189C2×M16—32no—
190C2×M16C22×D312yes—
191C4⋊Q8—32no—
192C4⋊Q8—62no—
193C4⋊Q8—61no—
194C4⋊Q8—61no—
195C4⋊Q8—62no—
196#127—21no—
197#127—21no—
198#127—22no—
199#126—68no—
200#125—32no—
201#125—64no—
202#125—68no—
203#124—68no—
204#124—62no—
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×Q8#6712yes—
215C4×D4—22no—
216C4×D4#6712yes—
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—32no—
226C2×C42—62no—
227C2×C42C22×D311yescharacteristic
228C4⋊C8—22no—
229#104—64no—
230#102—24no—
231#102—68no—
232#101—124no—
233#97—61no—
234C4×C8—61no—
235C23×C3—21no—
236C23×C3—48no—
237C23×C3—42no—
238C23×C3—44no—
239C23×C3C22×D411yescharacteristic
240C3×Q8—21no—
241C3×Q8—21no—
242C3×D4—21no—
243C3×D4—21no—
244C3×D4—42no—
245C3×D4—42no—
246C3×D4—42no—
247C3×D4—44no—
248C3×D4—84no—
249C3×D4—84no—
250C2×C3×C4—24no—
251C2×C3×C4—22no—
252C2×C3×C4—21no—
253C2×C3×C4—42no—
254C2×C3×C4—44no—
255C2×C3×C4#12012yes—
256C2×C3×C4#12012yes—
257C2×C3×C4C22×D411yescharacteristic
258C2×C3×C4C22×D411yescharacteristic
259C2×C3×C4C2×#4112yes—
260C2×C3×C4H(5,𝔽2)12yes—
261C2×Dic3—24no—
262C2×Dic3—24no—
263C2×Dic3—24no—
264C2×Dic3—28no—
265C2×Dic3—44no—
266Dic6—42no—
267Dic6—42no—
268Dic6—44no—
269Dic6—44no—
270C3×C8—42no—
271C3⋊C8—42no—
272C24—22no—
273#41—22no—
274#41—22no—
275C2×Q8—62no—
276C2×Q8—62no—
277C2×Q8—64no—
278C2×Q8—64no—
279C2×Q8—62no—
280C2×Q8C2×#6711yescharacteristic
281C2×D4—21no—
282C2×D4—22no—
283C2×D4—24no—
284C2×D4—24no—
285C2×D4—22no—
286C2×D4—48no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4C2×#6711yescharacteristic
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—34no—
293C22×C4—62no—
294C22×C4D3×D412yes—
295C22×C4C23×D311yescharacteristic
296SD16—44no—
297D8—44no—
298M16—24no—
299M16—62no—
300M16—62no—
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×#6712yes—
312C22⋊C4—21no—
313C22⋊C4—22no—
314C22⋊C4—68no—
315C22⋊C4—62no—
316C22⋊C4—1216no—
317C42—22no—
318C42—21no—
319C42—34no—
320C42—68no—
321C42—62no—
322C42—62no—
323C42—124no—
324C42D3×D411yescharacteristic
325C42D3×D411yescharacteristic
326C42C2×#6712yes—
327C22×C3—22no—
328C22×C3—41no—
329C22×C3—42no—
330C22×C3—88no—
331C22×C3—82no—
332C22×C3—84no—
333C22×C3C2×#12011yescharacteristic
334C22×C3#54612yes—
335C3×C4—22no—
336C3×C4—41no—
337C3×C4—42no—
338C3×C4#45414yes—
339C3×C4#53211yescharacteristic
340C3×C4#53511yescharacteristic
341C3×C4#55012yes—
342Dic3—44no—
343Dic3—44no—
344Dic3—44no—
345C23—21no—
346C23—48no—
347C23—42no—
348C23—44no—
349C23C2×D3×D411yescharacteristic
350Q8—21no—
351Q8—21no—
352Q8—124no—
353Q8—122no—
354Q8—122no—
355Q8—124no—
356D4—21no—
357D4—21no—
358D4—42no—
359D4—42no—
360D4—42no—
361D4—44no—
362D4—84no—
363D4—84no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—64no—
370C2×C4—64no—
371C2×C4—64no—
372C2×C4—68no—
373C2×C4—124no—
374C2×C4#93712yes—
375C2×C4#93712yes—
376C2×C4C2×D3×D411yescharacteristic
377C2×C4#100412yes—
378C2×C4D3×#4112yes—
379C2×C4C22×#6711yescharacteristic
380C8—42no—
381C8—122no—
382C2×C3—22no—
383C2×C3—81no—
384C2×C3—162no—
385C2×C3#243111yescharacteristic
386C2×C3C2×#45412yes—
387C22—22no—
388C22—41no—
389C22—42no—
390C22—88no—
391C22—82no—
392C22—84no—
393C22#561212yes—
394C22C2×#93711yescharacteristic
395C4—22no—
396C4—41no—
397C4—42no—
398C4—124no—
399C4—124no—
400C4—124no—
401C4#520114yes—
402C4#559912yes—
403C4#580511yescharacteristic
404C4#583211yescharacteristic
405C3#2559111yescharacteristic, complemented, Sylow
406C2—22no—
407C2—81no—
408C2—162no—
409C2#8265211yescharacteristic
410C2C2×#520112yes—
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3038C2
socleC22×C3C2×#120
hypercenter#3038D3
hypocenterC3#25591
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3038

Complements

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

NH ≅ G/Ntype
1. #85830C2semidirect
3. #83184C2semidirect
4. #83159C2semidirect
5. #72096C2semidirect
6. C4×#837C2semidirect
7. #70522C2semidirect
8. #70514C2semidirect
9. #70325C2semidirect
405. C3#25591semidirect

Maximal subgroups

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

subgroupindex
#858302
C3×#30382
#831842
#831592
#720962
C4×#8372
#705222
#705142
#703252
#255913