Finite Group Databasebeta

← #10692693

Subgroups — #10692693

Order 768.

424 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 89 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#83199C211yesmaximal, characteristic
2#79337C211yesmaximal, characteristic
3C3×#1556C211yesmaximal, characteristic, complemented
4#71777C211yesmaximal, characteristic, complemented
5#70811C211yesmaximal, characteristic
6#67754C211yesmaximal, characteristic, complemented
7#67753C211yesmaximal, characteristic, complemented
8#9419—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#432C2211yescharacteristic
13#5093C412yes—
14#5089—22no—
15#5088C412yes—
16#4966—22no—
17#4926C2211yescharacteristic
18#4926C2211yescharacteristic
19C4×#208C2211yescharacteristic
20#4836—22no—
21#4727—22no—
22#4706—22no—
23C3×#332—22no—
24#4546—22no—
25#4464—22no—
26#3141—31no—
27#2980—31no—
28#1558—31no—
29#1556D311yescharacteristic, complemented
30#1536—31no—
31#1291—31no—
32#1291—31no—
33C22×C3×Q8—21no—
34C2×#235D412yes—
35C3×C4⋊Q8—21no—
36C3×C4⋊Q8—21no—
37C3×C4⋊Q8—21no—
38C3×C4⋊Q8C2×C412yes—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C42C2311yescharacteristic
42#930—21no—
43#930—21no—
44C2×#208—21no—
45C2×ΓL(1,𝔽25)—22no—
46C2×D3×C8—22no—
47#875D412yes—
48#874C2×C411yescharacteristic
49C4×D12—21no—
50C4×D12—21no—
51#869C2×C411yescharacteristic
52C3×C4⋊C8—21no—
53C3×C4⋊C8D412yes—
54C3×C22⋊C8—21no—
55C3×C22⋊C8—21no—
56C3×#97—21no—
57C3×#97—21no—
58#830—21no—
59#830—21no—
60(C2×D3)⋊C8—24no—
61#810—24no—
62#809—24no—
63C4⋊(C3⋊C8)—21no—
64C4⋊(C3⋊C8)D412yes—
65#803—21no—
66#803—21no—
67#803—21no—
68C4×C3⋊C8C2×C412yes—
69#532—31no—
70C2×C4⋊Q8C2×D311yescharacteristic
71#489—62no—
72#487—32no—
73#486—32no—
74#435—62no—
75#435—62no—
76#432—31no—
77#432—31no—
78#432C2×D311yescharacteristic
79#432C2×D311yescharacteristic
80#417—62no—
81#408—62no—
82C4×M16—31no—
83#333—62no—
84#332—22no—
85#329—62no—
86C2×C3×Q8—22no—
87C2×C3×Q8—41no—
88C2×C3×Q8—82no—
89C22×C3×C4—41no—
90C22×C3×C4C2×D411yescharacteristic
91C22×C3×C4#4111yescharacteristic
92C22×C3×C4#4111yescharacteristic
93C2×#67—22no—
94#235—42no—
95#235—42no—
96C2×D12—21no—
97C2×D12—21no—
98C2×D12C22⋊C412yes—
99C2×C4×D3—22no—
100C2×Dic6C22⋊C412yes—
101C2×C3×C8—22no—
102C2×C3×C8—41no—
103C3×C4⋊C4—22no—
104C3×C4⋊C4—22no—
105C3×C4⋊C4—41no—
106C3×C4⋊C4—41no—
107C3×C42C22×C411yescharacteristic
108C3×C42C2×D411yescharacteristic
109C3×C42C2×D411yescharacteristic
110C3×C42C2×D411yescharacteristic
111(C2×D3)⋊C4—22no—
112(C2×D3)⋊C4—22no—
113C4⋊Dic3—21no—
114C4⋊Dic3—21no—
115#208—42no—
116C2×C3⋊C8—22no—
117C2×C3⋊C8—22no—
118C2×C3⋊C8—41no—
119ΓL(1,𝔽25)—82no—
120D3×C8—82no—
121C2×#41—32no—
122C22×Q8—21no—
123C2×M16—62no—
124C2×M16—61no—
125C22×C8—62no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8—31no—
130C4⋊Q8C4×D312yes—
131#127—31no—
132#124—32no—
133#121—61no—
134#121—61no—
135C4×D4—61no—
136C4×D4—61no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C42C22×D311yescharacteristic
140C4⋊C8—21no—
141C4⋊C8—32no—
142C4⋊C8—61no—
143C4⋊C8#6712yes—
144#103—64no—
145#102—64no—
146C22⋊C8—21no—
147C22⋊C8—21no—
148C22⋊C8—64no—
149C22⋊C8—61no—
150C22⋊C8—61no—
151#97—21no—
152#97—21no—
153#97—61no—
154#97—61no—
155#97—61no—
156C4×C8—32no—
157C23×C3#12311yescharacteristic
158C22×D3—22no—
159C3×Q8—82no—
160C3×Q8—82no—
161C2×C3×C4—22no—
162C2×C3×C4—41no—
163C2×C3×C4—41no—
164C2×C3×C4—42no—
165C2×C3×C4—82no—
166C2×C3×C4C2×C22⋊C411yescharacteristic
167C2×C3×C4C4×D411yescharacteristic
168C2×C3×C4C4×D411yescharacteristic
169C2×C3×C4#12011yescharacteristic
170C2×C3×C4#12111yescharacteristic
171C2×C3×C4#12111yescharacteristic
172#67—82no—
173C2×Dic3—22no—
174D12—42no—
175D12—81no—
176D12—81no—
177C4×D3—42no—
178C4×D3—42no—
179Dic6—42no—
180C3×C8—42no—
181C3×C8—81no—
182C3⋊C8—42no—
183C3⋊C8—42no—
184C3⋊C8—81no—
185#41—122no—
186#41—122no—
187C2×Q8—22no—
188C2×Q8—32no—
189C2×Q8—41no—
190C2×Q8—82no—
191C2×D4—32no—
192C2×D4—62no—
193C2×D4—61no—
194C2×D4—61no—
195C22×C4—41no—
196C22×C4—62no—
197C22×C4#23511yescharacteristic
198C22×C4D3×D411yescharacteristic
199C22×C4#23711yescharacteristic
200M16—122no—
201M16—242no—
202C2×C8—22no—
203C2×C8—41no—
204C2×C8—62no—
205C2×C8—62no—
206C2×C8—121no—
207C2×C8—242no—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—61no—
213C4⋊C4—61no—
214C22⋊C4—62no—
215C22⋊C4—62no—
216C42C2×C4×D311yescharacteristic
217C42D3×D411yescharacteristic
218C42D3×D411yescharacteristic
219C42C2×#6711yescharacteristic
220C22×C3—21no—
221C22×C3—21no—
222C22×C3—21no—
223C22×C3#38611yescharacteristic
224C2×D3—42no—
225C2×D3—42no—
226C2×D3—42no—
227C3×C4—21no—
228C3×C4—21no—
229C3×C4—21no—
230C3×C4—21no—
231C3×C4—21no—
232C3×C4—82no—
233C3×C4#45512yes—
234Dic3—42no—
235C23—62no—
236C23#88311yescharacteristic
237Q8—82no—
238Q8—82no—
239Q8—122no—
240D4—122no—
241D4—241no—
242D4—241no—
243D4—242no—
244C2×C4—22no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—62no—
249C2×C4—82no—
250C2×C4—122no—
251C2×C4—122no—
252C2×C4D3×C22⋊C411yescharacteristic
253C2×C4#67⋊C411yescharacteristic
254C2×C4#88411yescharacteristic
255C2×C4C4×#6711yescharacteristic
256C2×C4#93711yescharacteristic
257C2×C4#93811yescharacteristic
258C8—42no—
259C8—81no—
260C8—122no—
261C8—122no—
262C8—241no—
263C2×C3—41no—
264C2×C3#188511yescharacteristic
265C2×C3#190211yescharacteristic
266C2×C3#190311yescharacteristic
267D3—82no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—122no—
272C22—122no—
273C22—122no—
274C22#495611yescharacteristic
275C4—21no—
276C4—21no—
277C4—21no—
278C4—21no—
279C4—21no—
280C4—82no—
281C4—122no—
282C4#520212yes—
283C3#941911yescharacteristic, complemented, Sylow
284C2—41no—
285C2—242no—
286C2#7012211yescharacteristic
287C2#7032911yescharacteristic
288C2#7034311yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1556C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1556D3
hypocenterC3#9419
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1556

Complements

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

NH ≅ G/Ntype
3. C3×#1556C2semidirect
4. #71777C2semidirect
6. #67754C2semidirect
7. #67753C2semidirect
29. #1556D3semidirect
283. C3#9419semidirect

Maximal subgroups

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

subgroupindex
#831992
#793372
C3×#15562
#717772
#708112
#677542
#677532
#94193