Finite Group Databasebeta

← #10692602

Subgroups — #10692602

Order 768.

570 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 119 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×#2975C211yesmaximal, characteristic
2#82379C211yesmaximal, characteristic
3#71617C211yesmaximal, characteristic
4#70780C212yesmaximal, complemented
5#67746C212yesmaximal, complemented
6#9403—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#924—22no—
9C2×#869C2211yescharacteristic
10C3×#436—24no—
11C3×#431C2212yes—
12C3×#407—22no—
13#5221—21no—
14C2×#833—21no—
15#4975—21no—
16#4949C414yes—
17C4×C4⋊Dic3C2211yescharacteristic
18#4924C2212yes—
19#4530—21no—
20#4482—24no—
21#2975D311yescharacteristic
22#2542—31no—
23#1800—31no—
24#1549—32no—
25#1303—32no—
26C22×C3×D4D411yescharacteristic
27C23×C3×C4D412yes—
28C22×Dic6—21no—
29C2×C3×M16—24no—
30C3×C4×D4—22no—
31C3×C4×D4—22no—
32C2×C3×C4⋊C4D411yescharacteristic
33C2×C3×C22⋊C4C2×C412yes—
34C2×C3×C42C2311yescharacteristic
35C2×#217—24no—
36C2×C4⋊Dic3—21no—
37C2×C4⋊Dic3—21no—
38C2×C4⋊Dic3C2×C412yes—
39#924—48no—
40C2×Dic3⋊C4—24no—
41C2×C4×Dic3—21no—
42C2×#208—22no—
43#869—22no—
44#869—22no—
45C3×C4⋊C8D414yes—
46C3×C22⋊C8—24no—
47C3×C22⋊C8—22no—
48C3×C4×C8—22no—
49#833—42no—
50C3⋊#95C2×C412yes—
51C4⋊(C3⋊C8)—22no—
52C4⋊(C3⋊C8)—22no—
53C2×C4⋊Q8—31no—
54C2×C22⋊Q8—62no—
55C2×C4×D4C2×D311yescharacteristic
56#436—24no—
57#431C2×D312yes—
58#423—32no—
59C2×#100—61no—
60#407—22no—
61#399—34no—
62#390—61no—
63#384—61no—
64C4×C4⋊C4—31no—
65#343—61no—
66#332—64no—
67C24×C3C22⋊C412yes—
68C2×C3×D4—22no—
69C2×C3×D4—21no—
70C2×C3×D4—22no—
71C2×C3×D4—21no—
72C22×C3×C4—22no—
73C22×C3×C4—22no—
74C22×C3×C4—44no—
75C22×C3×C4C22⋊C412yes—
76C22×C3×C4C22×C411yescharacteristic
77C22×C3×C4C2×D411yescharacteristic
78C22×C3×C4C2×D411yescharacteristic
79C22×Dic3—22no—
80C22×Dic3—22no—
81C2×Dic6—44no—
82C2×Dic6—41no—
83C2×Dic6—41no—
84C3×M16—84no—
85C2×C3×C8—24no—
86C3×C4⋊C4—22no—
87C3×C22⋊C4—24no—
88C3×C42C2×D412yes—
89C3×C42#4112yes—
90#217—48no—
91C4⋊Dic3—24no—
92C4⋊Dic3—24no—
93C4⋊Dic3—42no—
94Dic3⋊C4—48no—
95C4×Dic3—41no—
96C4×Dic3—41no—
97#208—42no—
98#208—42no—
99C2×C3⋊C8—42no—
100C22×Q8—61no—
101C22×D4D1211yescharacteristic
102C23×C4#6712yes—
103C2×M16—24no—
104C2×M16—62no—
105C4⋊Q8—62no—
106C4⋊Q8—62no—
107C22⋊Q8—128no—
108C4×D4—22no—
109C4×D4—22no—
110C2×C4⋊C4—32no—
111C2×C4⋊C4—61no—
112C2×C4⋊C4—64no—
113C2×C4⋊C4—61no—
114C2×C4⋊C4D1211yescharacteristic
115C2×C22⋊C4—64no—
116C2×C22⋊C4C4×D312yes—
117C2×C42—61no—
118C2×C42C22×D311yescharacteristic
119C4⋊C8—62no—
120C4⋊C8—62no—
121C4⋊C8#6714yes—
122#100—122no—
123C22⋊C8—24no—
124C22⋊C8—22no—
125C4×C8—22no—
126#95—32no—
127C23×C3—22no—
128C23×C3—22no—
129C23×C3—22no—
130C23×C3—42no—
131C23×C3—42no—
132C23×C3C2×C22⋊C411yescharacteristic
133C3×D4—42no—
134C3×D4—42no—
135C2×C3×C4—22no—
136C2×C3×C4—22no—
137C2×C3×C4—41no—
138C2×C3×C4—44no—
139C2×C3×C4—42no—
140C2×C3×C4—82no—
141C2×C3×C4C4×D412yes—
142C2×C3×C4#12011yescharacteristic
143C2×C3×C4#12112yes—
144C2×C3×C4#12311yescharacteristic
145C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
146C2×C3×C4#13712yes—
147C2×C3×C4#13711yescharacteristic
148C2×Dic3—24no—
149C2×Dic3—42no—
150C2×Dic3—42no—
151C2×Dic3—44no—
152C2×Dic3—42no—
153Dic6—82no—
154Dic6—82no—
155C3×C8—44no—
156C3⋊C8—82no—
157C24(C2×D3)⋊C412yes—
158C2×Q8—124no—
159C2×Q8—121no—
160C2×Q8—121no—
161C2×D4—21no—
162C2×D4—22no—
163C2×D4—21no—
164C2×D4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—44no—
168C22×C4—62no—
169C22×C4—62no—
170C22×C4(C2×D3)⋊C412yes—
171C22×C4C2×C4×D311yescharacteristic
172C22×C4C2×D1211yescharacteristic
173C22×C4C2×#6711yescharacteristic
174M16—84no—
175M16—122no—
176M16—122no—
177C2×C8—24no—
178C2×C8—122no—
179C4⋊C4—22no—
180C4⋊C4—64no—
181C4⋊C4—64no—
182C4⋊C4—128no—
183C4⋊C4—122no—
184C22⋊C4—24no—
185C22⋊C4—128no—
186C42—121no—
187C42—121no—
188C42#23512yes—
189C42C2×#6712yes—
190C22×C3—21no—
191C22×C3—21no—
192C22×C3—44no—
193C22×C3—42no—
194C22×C3—42no—
195C22×C3—42no—
196C22×C3—82no—
197C22×C3#38611yescharacteristic
198C22×C3#41711yescharacteristic
199C22×C3#41911yescharacteristic
200C3×C4—21no—
201C3×C4—22no—
202C3×C4—42no—
203C3×C4#45312yes—
204C3×C4#46812yes—
205C3×C4#47012yes—
206Dic3—44no—
207Dic3—82no—
208C23—22no—
209C23—22no—
210C23—22no—
211C23—42no—
212C23—42no—
213C23C2×(C2×D3)⋊C411yescharacteristic
214Q8—242no—
215Q8—242no—
216D4—42no—
217D4—42no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—41no—
221C2×C4—44no—
222C2×C4—42no—
223C2×C4—64no—
224C2×C4—82no—
225C2×C4—122no—
226C2×C4—122no—
227C2×C4—124no—
228C2×C4—122no—
229C2×C4#90811yescharacteristic
230C2×C4#90912yes—
231C2×C4#90911yescharacteristic
232C2×C4C4×#6712yes—
233C2×C4#92911yescharacteristic
234C2×C4#93012yes—
235C2×C4#95311yescharacteristic
236C8—44no—
237C8—242no—
238C2×C3—21no—
239C2×C3—21no—
240C2×C3—82no—
241C2×C3#188711yescharacteristic
242C2×C3#189411yescharacteristic
243C2×C3#191511yescharacteristic
244C22—21no—
245C22—21no—
246C22—44no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—82no—
251C22#512011yescharacteristic
252C22#513311yescharacteristic
253C22#521311yescharacteristic
254C4—22no—
255C4—21no—
256C4—42no—
257C4—124no—
258C4—242no—
259C4#513412yes—
260C4#513712yes—
261C4#524112yes—
262C3#940311yescharacteristic, complemented, Sylow
263C2—21no—
264C2—21no—
265C2—82no—
266C2#7165011yescharacteristic
267C2#7198911yescharacteristic
268C2#7203111yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2975C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenter#2975D3
hypocenterC3#9403
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2975

Complements

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

NH ≅ G/Ntype
4. #70780C2semidirect
5. #67746C2semidirect
262. C3#9403semidirect

Maximal subgroups

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

subgroupindex
C3×#29752
#823792
#716172
#707802
#677462
#94033