Finite Group Databasebeta

← #10713392

Subgroups — #10713392

Order 768.

438 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 115 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#78553C211yesmaximal, characteristic
2C3×#2110C211yesmaximal, characteristic, complemented
3#71437C212yesmaximal, complemented
4#71431C212yesmaximal, complemented
5#69885C211yesmaximal, characteristic, complemented
6#13200—31nomaximal, Sylow
7C2×#901C2211yescharacteristic
8C22×D3×C8C2211yescharacteristic
9#5524C411yescharacteristic
10C2×C3×M32C2212yes—
11C3×#405C2211yescharacteristic
12#5129C411yescharacteristic
13#5126C411yescharacteristic
14C2×#812C2212yes—
15#4999C411yescharacteristic
16#4706—22no—
17C8×D12—22no—
18C3×#347—22no—
19#4508—24no—
20(C2×D3)⋊C16—24no—
21#4463—22no—
22#2929—31no—
23#2153—31no—
24#2117—32no—
25#2112—32no—
26#2110D311yescharacteristic, complemented
27C2×#235C2×C411yescharacteristic
28C22×C4×D3C2×C411yescharacteristic
29C2×C3×M16C2×C411yescharacteristic
30C22×C3×C8C2311yescharacteristic
31C3×#117C2×C411yescharacteristic
32C2×#208C2×C411yescharacteristic
33C22×C3⋊C8C2×C411yescharacteristic
34#901—22no—
35#901—22no—
36C2×ΓL(1,𝔽25)—21no—
37C2×D3×C8—21no—
38C2×D3×C8—21no—
39C2×D3×C8—21no—
40C2×D3×C8—22no—
41C2×D3×C8—22no—
42#897—21no—
43#895—21no—
44#886—21no—
45#882—21no—
46C4×D12—22no—
47C3×M32—22no—
48C3×M32—22no—
49C2×C3×C16D414yescomplemented
50C3×#97D412yes—
51C3×C4×C8D412yes—
52(C2×D3)⋊C8—22no—
53(C2×D3)⋊C8—22no—
54#812—22no—
55#812—22no—
56C2×C3⋊C16—22no—
57C4⋊(C3⋊C8)—22no—
58C2×#131—31no—
59C23×C8—31no—
60#525—31no—
61C2×M32—32no—
62C2×M32C2×D312yes—
63#435—62no—
64C8×D4—62no—
65#424—31no—
66#408—31no—
67#407—31no—
68#405C2×D311yescharacteristic
69#364—62no—
70#350—64no—
71C22⋊C16—64no—
72#347—22no—
73C23×D3M1611yescharacteristic
74C22×C3×C4C22×C411yescharacteristic
75C2×#67—21no—
76C22×Dic3M1611yescharacteristic
77#235—22no—
78#235—22no—
79C2×D12—22no—
80C2×D12M1611yescharacteristic
81C2×C4×D3—21no—
82C2×C4×D3—21no—
83C2×C4×D3—22no—
84C2×C4×D3—21no—
85C2×C4×D3—22no—
86C2×Dic6M1611yescharacteristic
87C3×M16—22no—
88C2×C3×C8C22⋊C412yes—
89C2×C3×C8C2×D412yes—
90C2×C3×C8C2×D411yescharacteristic
91C2×C3×C8C2×D411yescharacteristic
92C2×C3×C8#4112yes—
93C3×C4⋊C4—21no—
94C3×C22⋊C4—21no—
95C3×C42C22⋊C412yes—
96(C2×D3)⋊C4—22no—
97(C2×D3)⋊C4—22no—
98C4⋊Dic3—22no—
99#208—22no—
100C2×C3⋊C8—21no—
101C2×C3⋊C8—21no—
102C2×C3⋊C8—22no—
103ΓL(1,𝔽25)—42no—
104D3×C8—42no—
105D3×C8—41no—
106D3×C8—41no—
107D3×C8—41no—
108D3×C8—41no—
109C3×C16—24no—
110C3⋊C16—42no—
111C2×#41—31no—
112C23×C4—31no—
113#131—62no—
114#131—62no—
115C2×M16—31no—
116C2×M16—61no—
117C2×M16C4×D311yescharacteristic
118C22×C8—31no—
119C22×C8—62no—
120C22×C8—61no—
121C22×C8—61no—
122C22×C8—62no—
123C22×C8—61no—
124C22×C8C22×D311yescharacteristic
125#123—61no—
126C22⋊Q8—61no—
127#121—61no—
128#120—61no—
129C4×D4—62no—
130#117C4×D311yescharacteristic
131M32—22no—
132M32—22no—
133M32—62no—
134M32—62no—
135C2×C16—62no—
136C2×C16D1214yescomplemented
137C4⋊C8—62no—
138C22⋊C8—62no—
139C22⋊C8—62no—
140#97#6712yes—
141C4×C8D1212yes—
142C23×C3C2×M1611yescharacteristic
143C22×D3—21no—
144C22×D3—22no—
145C22×D3—21no—
146C22×D3—42no—
147C2×C3×C4—22no—
148C2×C3×C4C2×C22⋊C411yescharacteristic
149C2×C3×C4C4×D412yes—
150C2×C3×C4C2×M1611yescharacteristic
151C2×C3×C4#13112yes—
152#67—42no—
153C2×Dic3—21no—
154C2×Dic3—21no—
155C2×Dic3—22no—
156D12—22no—
157D12—44no—
158C4×D3—41no—
159C4×D3—41no—
160C4×D3—41no—
161C4×D3—41no—
162C4×D3—42no—
163Dic6—22no—
164C3×C8—22no—
165C3×C8#12112yes—
166C3×C8#12411yescharacteristic
167C3×C8#12711yescharacteristic
168C3⋊C8—41no—
169C3⋊C8—41no—
170C24—31no—
171#41—62no—
172#41—62no—
173C2×Q8—31no—
174C2×D4—31no—
175C2×D4—62no—
176C2×D4—61no—
177C22×C4—31no—
178C22×C4—62no—
179C22×C4—61no—
180C22×C4—62no—
181C22×C4—61no—
182C22×C4—61no—
183C22×C4C2×C4×D311yescharacteristic
184M16—22no—
185M16—62no—
186M16—122no—
187C2×C8—62no—
188C2×C8—61no—
189C2×C8—61no—
190C2×C8—121no—
191C2×C8—121no—
192C2×C8—121no—
193C2×C8—121no—
194C2×C8—122no—
195C2×C8(C2×D3)⋊C412yes—
196C2×C8C2×D1211yescharacteristic
197C2×C8C2×D1212yes—
198C2×C8#23512yes—
199C2×C8C2×#6711yescharacteristic
200C4⋊C4—21no—
201C4⋊C4—62no—
202C22⋊C4—21no—
203C22⋊C4—62no—
204C22⋊C4—62no—
205C42(C2×D3)⋊C412yes—
206C16—24no—
207C16—122no—
208C22×C3—41no—
209C22×C3#40711yescharacteristic
210C22×C3#43512yes—
211C2×D3—22no—
212C2×D3—24no—
213C2×D3—41no—
214C2×D3—42no—
215C2×D3—41no—
216C2×D3—42no—
217C2×D3—41no—
218C3×C4—42no—
219C3×C4#39011yescharacteristic
220C3×C4#40811yescharacteristic
221C3×C4#43512yes—
222Dic3—41no—
223Dic3—41no—
224C23—61no—
225C23—62no—
226C23—61no—
227C23—122no—
228C23D3×M1611yescharacteristic
229Q8—62no—
230D4—62no—
231D4—124no—
232D4—122no—
233C2×C4—22no—
234C2×C4—61no—
235C2×C4—61no—
236C2×C4—62no—
237C2×C4—121no—
238C2×C4—121no—
239C2×C4—121no—
240C2×C4—121no—
241C2×C4—122no—
242C2×C4C4×D1212yes—
243C2×C4D3×M1611yescharacteristic
244C2×C4#90712yes—
245C2×C4C2×(C2×D3)⋊C411yescharacteristic
246C8—22no—
247C8—121no—
248C8—121no—
249C8#87411yescharacteristic
250C8#87511yescharacteristic
251C8#93012yes—
252C2×C3—21no—
253C2×C3—22no—
254C2×C3#195511yescharacteristic
255D3—42no—
256D3—81no—
257C22—41no—
258C22—62no—
259C22—64no—
260C22—121no—
261C22—122no—
262C22—122no—
263C22—121no—
264C22—121no—
265C22#470612yes—
266C22#512611yescharacteristic
267C4—42no—
268C4—121no—
269C4—121no—
270C4#470612yes—
271C4#493911yescharacteristic
272C4#512911yescharacteristic
273C3#1320011yescharacteristic, complemented, Sylow
274C2—21no—
275C2—22no—
276C2—122no—
277C2—241no—
278C2#6983711yescharacteristic
subgroupG/H
centerC8#874
commutatorC22×C3×C4C22×C4
FrattiniC22×C8C22×D3
FittingC3×#2110C2
socleC2×C3#1955
prime radical#5524C4
cosocleC22×C3×C8C23
hypercenter#2110D3
hypocenterC3#13200
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2110

Complements

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

NH ≅ G/Ntype
2. C3×#2110C2semidirect
3. #71437C2semidirect
4. #71431C2semidirect
5. #69885C2semidirect
26. #2110D3semidirect
49. C2×C3×C16D4semidirect
136. C2×C16D12semidirect
273. C3#13200semidirect

Maximal subgroups

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

subgroupindex
#785532
C3×#21102
#714372
#714312
#698852
#132003