Finite Group Databasebeta

← #10708693

Subgroups — #10708693

Order 768.

378 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 71 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
1#79910C211yesmaximal, characteristic, complemented
2C3×#1668C211yesmaximal, characteristic, complemented
3#71293C211yesmaximal, characteristic, complemented
4#71290C211yesmaximal, characteristic, complemented
5#67768C211yesmaximal, characteristic, complemented
6#67473C211yesmaximal, characteristic, complemented
7#67471C211yesmaximal, characteristic, complemented
8#12328—31nomaximal, Sylow
9#5577—21no—
10C3×#493C2211yescharacteristic
11C3×#460—21no—
12C3×C8×Q8C2211yescharacteristic
13C2×#826—22no—
14C2×#826—21no—
15#4843—22no—
16#4843—21no—
17#4695—21no—
18#4695C2211yescharacteristic
19C4×D24—21no—
20C4×D24C2211yescharacteristic
21C3×#335C2211yescharacteristic
22C3×#331—21no—
23#4562—21no—
24C4⋊(C3⋊C16)C2211yescharacteristic
25C4×C3⋊C16C2211yescharacteristic
26#3391—31no—
27#2246—31no—
28#2206—31no—
29#1668D311yescharacteristic, complemented
30#1361—31no—
31Q8⋊C16—31no—
32#1331—31no—
33C2×#239—41no—
34C2×C3×SD16—41no—
35C2×C3×D8—21no—
36C2×C3×D8—21no—
37C3×#127—21no—
38C3×C4×Q8D411yescharacteristic
39C2×D24—21no—
40C2×D24—22no—
41C2×D24D411yescharacteristic
42C2×D24D412yes—
43#895—42no—
44#895—41no—
45#874—22no—
46#874—21no—
47C4×D12—22no—
48C4×D12—21no—
49C3×C4⋊C8—21no—
50C3×C4⋊C8—21no—
51C3×C4⋊C8D411yescharacteristic
52C3×#102—41no—
53C3×C4×C8—21no—
54C3×C4×C8C2311yescharacteristic
55#826—44no—
56#826—42no—
57#821—22no—
58#821—21no—
59#818D411yescharacteristic
60C2×C3⋊C16—21no—
61C2×C3⋊C16—21no—
62#550—61no—
63C2×D16—61no—
64C2×D16—62no—
65#493—31no—
66#493—61no—
67#493C2×D311yescharacteristic
68#460—21no—
69#459—62no—
70#459—61no—
71C8×Q8C2×D311yescharacteristic
72C4×D8—31no—
73C4×D8—61no—
74C4⋊C16—31no—
75#357—61no—
76C4×C16—31no—
77#335C2×D311yescharacteristic
78#331—21no—
79C2×C3×Q8D811yescharacteristic
80C2×C3×D4—41no—
81C2×C3×D4—41no—
82C2×C3×D4—41no—
83#239—82no—
84#239—82no—
85C2×D12—21no—
86C2×D12—22no—
87C2×D12—42no—
88C2×D12—41no—
89C2×D12—42no—
90C2×D12—41no—
91C2×C4×D3—42no—
92C2×C4×D3—41no—
93C3×SD16—82no—
94C3×D8—42no—
95C3×D8—41no—
96C3×D8—41no—
97C2×C3×C8—21no—
98C2×C3×C8—41no—
99C2×C3×C8C2×D411yescharacteristic
100C2×C3×C8C2×D411yescharacteristic
101C3×C4⋊C4—21no—
102C3×C4⋊C4D811yescharacteristic
103C3×C42—21no—
104C3×C42C2×D411yescharacteristic
105(C2×D3)⋊C4—42no—
106(C2×D3)⋊C4—41no—
107C4⋊Dic3—21no—
108D24—21no—
109D24—21no—
110D24—42no—
111D24—41no—
112D24—41no—
113D24—82no—
114C3⋊C16—22no—
115C3⋊C16—41no—
116C2×#41—121no—
117C2×SD16—41no—
118C2×D8—21no—
119C2×D8—21no—
120C2×D8—31no—
121C2×D8—32no—
122C2×D8—62no—
123C2×D8—61no—
124#127—21no—
125#127—62no—
126#127—61no—
127#121—122no—
128#121—121no—
129C4×Q8#6711yescharacteristic
130C4×D4—62no—
131C4×D4—61no—
132D16—124no—
133D16—122no—
134C2×C16—61no—
135C2×C16—61no—
136#107—31no—
137C4⋊C8—21no—
138C4⋊C8—21no—
139C4⋊C8#6711yescharacteristic
140#102—41no—
141#102—62no—
142#102—61no—
143C4×C8—21no—
144C4×C8C22×D311yescharacteristic
145C23×C3—81no—
146C22×D3—42no—
147C22×D3—41no—
148C3×Q8—21no—
149C3×Q8D1612yescomplemented
150C3×D4—81no—
151C3×D4—81no—
152C3×D4—81no—
153C3×D4—81no—
154C3×D4—161no—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4#12011yescharacteristic
158C2×C3×C4C2×D811yescharacteristic
159C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
160C2×Dic3—41no—
161D12—41no—
162D12—42no—
163D12—42no—
164D12—41no—
165D12—84no—
166D12—81no—
167D12—81no—
168D12—161no—
169D12—162no—
170C4×D3—84no—
171C4×D3—81no—
172C4×D3—81no—
173C3×C8—21no—
174C3×C8—41no—
175C3×C8—81no—
176C3×C8#13511yescharacteristic
177C3×C8AGL(1,ℤ/8)11yescharacteristic
178#41—242no—
179#41—242no—
180C2×Q8#21311yescharacteristic
181C2×D4—41no—
182C2×D4—41no—
183C2×D4—41no—
184C2×D4—61no—
185C2×D4—62no—
186C2×D4—122no—
187C2×D4—122no—
188C2×D4—121no—
189C2×D4—121no—
190C22×C4—122no—
191C22×C4—121no—
192SD16—82no—
193D8—42no—
194D8—41no—
195D8—41no—
196D8—61no—
197D8—61no—
198D8—122no—
199D8—121no—
200D8—121no—
201D8—242no—
202C2×C8—21no—
203C2×C8—41no—
204C2×C8D3×D411yescharacteristic
205C2×C8D3×D411yescharacteristic
206C4⋊C4—21no—
207C4⋊C4—61no—
208C4⋊C4#21311yescharacteristic
209C22⋊C4—122no—
210C22⋊C4—121no—
211C42—21no—
212C42C2×#6711yescharacteristic
213C16—62no—
214C16—121no—
215C22×C3—161no—
216C22×C3—161no—
217C22×C3—161no—
218C22×C3#44711yescharacteristic
219C2×D3—81no—
220C2×D3—82no—
221C2×D3—82no—
222C2×D3—81no—
223C2×D3—82no—
224C2×D3—81no—
225C3×C4—22no—
226C3×C4—21no—
227C3×C4—41no—
228C3×C4#44911yescharacteristic
229C3×C4#45311yescharacteristic
230C3×C4C2×D1611yescharacteristic
231C3×C4#50911yescharacteristic
232Dic3—81no—
233C23—81no—
234C23—122no—
235C23—121no—
236Q8—21no—
237Q8#82612yescomplemented
238D4—81no—
239D4—81no—
240D4—81no—
241D4—81no—
242D4—121no—
243D4—122no—
244D4—122no—
245D4—121no—
246D4—161no—
247D4—244no—
248D4—241no—
249D4—241no—
250D4—481no—
251D4—482no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—121no—
255C2×C4—244no—
256C2×C4—241no—
257C2×C4—241no—
258C2×C4C2×#21311yescharacteristic
259C2×C4#93711yescharacteristic
260C2×C4#94911yescharacteristic
261C8—21no—
262C8—41no—
263C8—81no—
264C8#91411yescharacteristic
265C8#91611yescharacteristic
266C2×C3—321no—
267C2×C3#162111yescharacteristic
268C2×C3#218411yescharacteristic
269C2×C3#219211yescharacteristic
270D3—161no—
271D3—162no—
272C22—161no—
273C22—161no—
274C22—161no—
275C22—241no—
276C22—242no—
277C22—242no—
278C22—241no—
279C22—242no—
280C22—241no—
281C22#503611yescharacteristic
282C4—22no—
283C4—21no—
284C4—41no—
285C4—241no—
286C4C2×#82611yescharacteristic
287C4#517211yescharacteristic
288C4#519311yescharacteristic
289C4#519811yescharacteristic
290C3#1232811yescharacteristic, complemented, Sylow
291C2—321no—
292C2—481no—
293C2—482no—
294C2#7103211yescharacteristic
295C2#7107511yescharacteristic
296C2#7109411yescharacteristic
subgroupG/H
centerC22#5036
commutator, cosocleC3×C4×C8C23
FrattiniC4×C8C22×D3
FittingC3×#1668C2
socleC22×C3#447
hypercenter#1668D3
hypocenterC3#12328
LCSG ▷ C3×C4×C8 ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #1668

Complements

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

NH ≅ G/Ntype
1. #79910C2semidirect
2. C3×#1668C2semidirect
3. #71293C2semidirect
4. #71290C2semidirect
5. #67768C2semidirect
6. #67473C2semidirect
7. #67471C2semidirect
29. #1668D3semidirect
149. C3×Q8D16semidirect
237. Q8#826semidirect
290. C3#12328semidirect

Maximal subgroups

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

subgroupindex
#799102
C3×#16682
#712932
#712902
#677682
#674732
#674712
#123283