Finite Group Databasebeta

← #10713393

Subgroups — #10713393

Order 768.

438 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 123 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×#2929C211yesmaximal, characteristic
2#72006C212yesmaximal, complemented
3#71963C212yesmaximal, complemented
4#69885C211yesmaximal, characteristic, complemented
5#69884C211yesmaximal, characteristic, complemented
6#13200—31nomaximal, Sylow
7C2×C3×#131C2211yescharacteristic
8C23×C3×C8C2211yescharacteristic
9C3×#525C411yescharacteristic
10C3×#435—22no—
11C3×C8×D4—22no—
12C3×#424C411yescharacteristic
13C3×#408C411yescharacteristic
14C3×#407C411yescharacteristic
15C3×#405C2211yescharacteristic
16C2×#812C2212yes—
17C2×#812C2212yes—
18#4561—24no—
19#4549—24no—
20#4466—22no—
21#4463—22no—
22#2929D311yescharacteristic
23#2153—31no—
24#2117—32no—
25#2112—32no—
26#2110—31no—
27C2×C3×#41C2×C411yescharacteristic
28C23×C3×C4C2×C411yescharacteristic
29C3×#131—22no—
30C3×#131—22no—
31C2×C3×M16—21no—
32C2×C3×M16C2×C411yescharacteristic
33C2×C3×M16C2×C411yescharacteristic
34C22×C3×C8—21no—
35C22×C3×C8—21no—
36C22×C3×C8—22no—
37C22×C3×C8—21no—
38C22×C3×C8—22no—
39C22×C3×C8C2×C411yescharacteristic
40C22×C3×C8C2311yescharacteristic
41C3×#123—21no—
42C3×C22⋊Q8—21no—
43C3×#121—21no—
44C3×#120—21no—
45C3×C4×D4—22no—
46C3×#117C2×C411yescharacteristic
47C3×C4⋊C8—22no—
48C3×C22⋊C8—22no—
49C3×C22⋊C8—22no—
50C3×#97D412yes—
51C3×C4×C8D412yes—
52#812—22no—
53#812—22no—
54#812—22no—
55#812—22no—
56C2×C3⋊C16—22no—
57C2×C3⋊C16D414yescomplemented
58C2×#131C2×D311yescharacteristic
59C23×C8C2×D311yescharacteristic
60#525Dic311yescharacteristic
61C2×M32—32no—
62C2×M32—32no—
63#435—22no—
64C8×D4—22no—
65#424Dic311yescharacteristic
66#408Dic311yescharacteristic
67#407Dic311yescharacteristic
68#405C2×D311yescharacteristic
69#364—62no—
70#350—64no—
71C22⋊C16—64no—
72#347—62no—
73C24×C3M1611yescharacteristic
74C3×#41—22no—
75C3×#41—22no—
76C2×C3×Q8M1611yescharacteristic
77C2×C3×D4—22no—
78C2×C3×D4—21no—
79C2×C3×D4M1611yescharacteristic
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4—22no—
83C22×C3×C4—21no—
84C22×C3×C4—22no—
85C22×C3×C4M1611yescharacteristic
86C22×C3×C4C22×C411yescharacteristic
87C3×M16—22no—
88C3×M16—22no—
89C3×M16—42no—
90C2×C3×C8—21no—
91C2×C3×C8—21no—
92C2×C3×C8—22no—
93C2×C3×C8—42no—
94C2×C3×C8—41no—
95C2×C3×C8—41no—
96C2×C3×C8—41no—
97C2×C3×C8—41no—
98C2×C3×C8C22⋊C412yes—
99C2×C3×C8C2×D412yes—
100C2×C3×C8C2×D411yescharacteristic
101C2×C3×C8C2×D411yescharacteristic
102C2×C3×C8#4112yes—
103C3×C4⋊C4—22no—
104C3×C4⋊C4—21no—
105C3×C22⋊C4—22no—
106C3×C22⋊C4—22no—
107C3×C22⋊C4—21no—
108C3×C42C22⋊C412yes—
109C3⋊C16—24no—
110C3⋊C16—42no—
111C2×#41C2×Dic311yescharacteristic
112C23×C4C2×Dic311yescharacteristic
113#131—22no—
114#131—22no—
115C2×M16—21no—
116C2×M16C2×Dic311yescharacteristic
117C2×M16C2×Dic311yescharacteristic
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—21no—
121C22×C8—21no—
122C22×C8—21no—
123C22×C8C2×Dic311yescharacteristic
124C22×C8C22×D311yescharacteristic
125#123—21no—
126C22⋊Q8—21no—
127#121—21no—
128#120—21no—
129C4×D4—22no—
130#117C2×Dic311yescharacteristic
131M32—62no—
132M32—62no—
133M32—62no—
134M32—62no—
135C2×C16—34no—
136C2×C16—62no—
137C4⋊C8—22no—
138C22⋊C8—22no—
139C22⋊C8—22no—
140#97#6712yes—
141C4×C8#6712yes—
142C23×C3—21no—
143C23×C3—22no—
144C23×C3—21no—
145C23×C3—42no—
146C23×C3C2×M1611yescharacteristic
147C3×Q8—22no—
148C3×D4—22no—
149C3×D4—44no—
150C3×D4—42no—
151C2×C3×C4—22no—
152C2×C3×C4—21no—
153C2×C3×C4—21no—
154C2×C3×C4—22no—
155C2×C3×C4—41no—
156C2×C3×C4—41no—
157C2×C3×C4—41no—
158C2×C3×C4—41no—
159C2×C3×C4—42no—
160C2×C3×C4C2×C22⋊C411yescharacteristic
161C2×C3×C4C4×D412yes—
162C2×C3×C4C2×M1611yescharacteristic
163C2×C3×C4#13112yes—
164C3×C8—22no—
165C3×C8—41no—
166C3×C8—41no—
167C3×C8#12112yes—
168C3×C8#12411yescharacteristic
169C3×C8#12711yescharacteristic
170C24#20811yescharacteristic
171#41—22no—
172#41—22no—
173C2×Q8#20811yescharacteristic
174C2×D4—22no—
175C2×D4—21no—
176C2×D4#20811yescharacteristic
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4#20811yescharacteristic
183C22×C4C22×Dic311yescharacteristic
184M16—22no—
185M16—22no—
186M16—42no—
187C2×C8—22no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—42no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8#21712yes—
196C2×C8D3×D412yes—
197C2×C8#23712yes—
198C2×C8C2×#6711yescharacteristic
199C2×C8C2×#6711yescharacteristic
200C4⋊C4—22no—
201C4⋊C4—21no—
202C22⋊C4—22no—
203C22⋊C4—22no—
204C22⋊C4—21no—
205C42#21712yes—
206C16—64no—
207C16—122no—
208C22×C3—22no—
209C22×C3—24no—
210C22×C3—41no—
211C22×C3—41no—
212C22×C3—42no—
213C22×C3—42no—
214C22×C3—41no—
215C22×C3—41no—
216C22×C3#40711yescharacteristic
217C22×C3#43512yes—
218C3×C4—42no—
219C3×C4—41no—
220C3×C4—41no—
221C3×C4#39011yescharacteristic
222C3×C4#40811yescharacteristic
223C3×C4#43512yes—
224C23—21no—
225C23—22no—
226C23—21no—
227C23—42no—
228C23C2×#20811yescharacteristic
229Q8—22no—
230D4—22no—
231D4—44no—
232D4—42no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4C2×#20811yescharacteristic
243C2×C4D4×Dic312yes—
244C2×C4#94812yes—
245C2×C4C2×#21711yescharacteristic
246C8—22no—
247C8—41no—
248C8—41no—
249C8#93611yescharacteristic
250C8#93812yes—
251C8#94011yescharacteristic
252C2×C3—21no—
253C2×C3—22no—
254C2×C3—42no—
255C2×C3—81no—
256C2×C3#195511yescharacteristic
257C22—22no—
258C22—24no—
259C22—41no—
260C22—41no—
261C22—42no—
262C22—42no—
263C22—41no—
264C22—41no—
265C22#501112yes—
266C22#520711yescharacteristic
267C4—42no—
268C4—41no—
269C4—41no—
270C4#501112yes—
271C4#522111yescharacteristic
272C4#523411yescharacteristic
273C3#1320011yescharacteristic, complemented, Sylow
274C2—21no—
275C2—22no—
276C2—42no—
277C2—81no—
278C2#7161511yescharacteristic
subgroupG/H
centerC8#940
commutatorC22×C3×C4C22×C4
FrattiniC22×C8C22×D3
FittingC3×#2929C2
socleC2×C3#1955
prime radicalC3×#525C4
cosocleC22×C3×C8C23
hypercenter#2929D3
hypocenterC3#13200
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2929

Complements

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

NH ≅ G/Ntype
2. #72006C2semidirect
3. #71963C2semidirect
4. #69885C2semidirect
5. #69884C2semidirect
57. C2×C3⋊C16D4semidirect
273. C3#13200semidirect

Maximal subgroups

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

subgroupindex
C3×#29292
#720062
#719632
#698852
#698842
#132003