Finite Group Databasebeta

← #10696578

Subgroups — #10696578

Order 768.

480 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2642C211yesmaximal, characteristic, complemented
2#83199C211yesmaximal, characteristic
3#83019C211yesmaximal, characteristic
4#71776C211yesmaximal, characteristic, complemented
5#71735C211yesmaximal, characteristic, complemented
6#69432C211yesmaximal, characteristic
7#67615C211yesmaximal, characteristic, complemented
8#10095—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×#125C2211yescharacteristic
11#5477C2211yescharacteristic
12C3×#400—22no—
13C3×#400—22no—
14C3×#391—21no—
15C3×#391—21no—
16C3×#376C2211yescharacteristic
17#5093—21no—
18#5089—22no—
19#5088—21no—
20#5067—21no—
21#5066—22no—
22#5064—21no—
23#4968—21no—
24#4967—21no—
25#4966—22no—
26#4926C2211yescharacteristic
27#4926C2211yescharacteristic
28C4×#208C2211yescharacteristic
29#4547—22no—
30#4543—22no—
31#4451—22no—
32#3234—31no—
33#3141—31no—
34#2642D311yescharacteristic, complemented
35#1556—31no—
36#1528—31no—
37#1505—31no—
38#1300—31no—
39C22×C3×Q8—21no—
40C2×#235—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×#125—21no—
46C3×#125—21no—
47C3×#125—21no—
48C3×#125—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4D412yes—
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C42—21no—
56C2×C3×C42C2311yescharacteristic
57#930—21no—
58#930—21no—
59C2×#208—21no—
60#875—21no—
61#874D411yescharacteristic
62C4×D12—21no—
63C4×D12—21no—
64#869D411yescharacteristic
65C3×#95—24no—
66C3×#95—24no—
67C3×#95—21no—
68C3×#95D412yes—
69#830—21no—
70#830—21no—
71#810—24no—
72#809—24no—
73#809—22no—
74#809—22no—
75#808—22no—
76#807—22no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)D412yes—
79#803—21no—
80#803—21no—
81#803D412yes—
82C4×C3⋊C8—21no—
83#532—31no—
84C2×C4⋊Q8C2×D311yescharacteristic
85C2×#125C2×D311yescharacteristic
86#501—61no—
87#490—61no—
88#489—62no—
89#487—61no—
90#486—61no—
91#481—61no—
92#480—61no—
93#478—62no—
94#432—31no—
95#432—31no—
96#417—62no—
97C4×M16—31no—
98#400—22no—
99#400—22no—
100#391—21no—
101#391—21no—
102#376C2×D311yescharacteristic
103#332—62no—
104#330—62no—
105#327—62no—
106C2×C3×Q8—44no—
107C2×C3×Q8—41no—
108C2×C3×Q8—41no—
109C22×C3×C4—22no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4—22no—
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C2×#67—41no—
117#235—44no—
118C2×D12—21no—
119C2×D12—21no—
120C2×D12—21no—
121C2×C4×D3—41no—
122C2×Dic6—21no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C42—42no—
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42#4111yescharacteristic
140(C2×D3)⋊C4—41no—
141(C2×D3)⋊C4—41no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144#208—42no—
145C2×C3⋊C8—22no—
146C2×C3⋊C8—22no—
147C2×C3⋊C8—41no—
148C2×#41—61no—
149C22×Q8—21no—
150C2×M16—61no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—31no—
156#127—31no—
157#125—21no—
158#125—21no—
159#125—21no—
160#125—21no—
161#124—61no—
162#121—61no—
163#121—61no—
164C4×D4—61no—
165C4×D4—61no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4#6711yescharacteristic
170C2×C4⋊C4#6711yescharacteristic
171C2×C4⋊C4#6712yes—
172C2×C42—21no—
173C2×C42C22×D311yescharacteristic
174#107—62no—
175#106—62no—
176C4⋊C8—32no—
177C4⋊C8—61no—
178#103—64no—
179#102—64no—
180#102—62no—
181#102—62no—
182C22⋊C8—61no—
183C22⋊C8—61no—
184#97—32no—
185#97—61no—
186#97—61no—
187C4×C8—61no—
188#95—24no—
189#95—24no—
190#95—21no—
191#95#6712yes—
192C23×C3#12011yescharacteristic
193C22×D3—41no—
194C3×Q8—82no—
195C3×Q8—82no—
196C2×C3×C4—22no—
197C2×C3×C4—22no—
198C2×C3×C4—41no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4—44no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—42no—
206C2×C3×C4—42no—
207C2×C3×C4—42no—
208C2×C3×C4#12011yescharacteristic
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12711yescharacteristic
214#67—82no—
215C2×Dic3—41no—
216D12—42no—
217D12—42no—
218D12—81no—
219C4×D3—82no—
220Dic6—42no—
221C3⋊C8—42no—
222C3⋊C8—42no—
223C3⋊C8—81no—
224#41—124no—
225C2×Q8—44no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—61no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4—121no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—121no—
238C22×C4C2×#6711yescharacteristic
239C22×C4C2×#6711yescharacteristic
240C22×C4C2×#6711yescharacteristic
241M16—122no—
242C2×C8—62no—
243C2×C8—62no—
244C2×C8—121no—
245C4⋊C4—22no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—22no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C22⋊C4—121no—
260C22⋊C4—121no—
261C42—42no—
262C42D3×D411yescharacteristic
263C42D3×D411yescharacteristic
264C42D3×D411yescharacteristic
265C42#23711yescharacteristic
266C22×C3—21no—
267C22×C3—21no—
268C22×C3#39211yescharacteristic
269C22×C3#45412yes—
270C2×D3—81no—
271C2×D3—81no—
272C2×D3—81no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—82no—
279C3×C4—81no—
280C3×C4—82no—
281C3×C4—81no—
282C3×C4#45412yes—
283Dic3—81no—
284C23—121no—
285C23#95311yescharacteristic
286Q8—82no—
287Q8—82no—
288Q8—122no—
289D4—122no—
290D4—122no—
291D4—241no—
292D4—242no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—44no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—121no—
306C2×C4—242no—
307C2×C4#93711yescharacteristic
308C2×C4#93711yescharacteristic
309C2×C4#93811yescharacteristic
310C2×C4#93911yescharacteristic
311C2×C4#93911yescharacteristic
312C2×C4#94011yescharacteristic
313C8—122no—
314C8—122no—
315C8—241no—
316C2×C3—22no—
317C2×C3#200311yescharacteristic
318C2×C3#200311yescharacteristic
319C2×C3#200911yescharacteristic
320D3—161no—
321C22—21no—
322C22—21no—
323C22—241no—
324C22—241no—
325C22—241no—
326C22#522311yescharacteristic
327C22#524512yes—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—82no—
334C4—82no—
335C4—81no—
336C4—81no—
337C4—241no—
338C4#506212yes—
339C3#1009511yescharacteristic, complemented, Sylow
340C2—22no—
341C2—481no—
342C2#7173011yescharacteristic
343C2#7186711yescharacteristic
344C2#7186911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2642C2
socleC22×C3#392
prime radical#5477C22
hypercenter#2642D3
hypocenterC3#10095
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2642

Complements

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

NH ≅ G/Ntype
1. C3×#2642C2semidirect
4. #71776C2semidirect
5. #71735C2semidirect
7. #67615C2semidirect
34. #2642D3semidirect
339. C3#10095semidirect

Maximal subgroups

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

subgroupindex
C3×#26422
#831992
#830192
#717762
#717352
#694322
#676152
#100953