Finite Group Databasebeta

← #10712071

Subgroups — #10712071

Order 768.

488 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 135 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×#3407C211yesmaximal, characteristic, complemented
2#83794C211yesmaximal, characteristic
3#72007C211yesmaximal, characteristic, complemented
4#71431C212yesmaximal, complemented
5#67813C212yesmaximal, complemented
6#12981—31nomaximal, Sylow
7C2×C3×#131C2211yescharacteristic
8#5828C411yescharacteristic
9C22×D3×C8C2211yescharacteristic
10C3×#504—24no—
11C2×C3×M32—22no—
12C2×C3×M32C2212yes—
13#5234C411yescharacteristic
14#5127—22no—
15#5126C411yescharacteristic
16#5119C411yescharacteristic
17C8×#67—22no—
18#5101C2211yescharacteristic
19C2×#812C2212yes—
20#4553—22no—
21(C2×D3)⋊C16—24no—
22#3407D311yescharacteristic, complemented
23#2929—31no—
24#2115—31no—
25#2112—32no—
26#1378—32no—
27C2×C3×#41C2×C411yescharacteristic
28C22×C4×D3C2×C411yescharacteristic
29C3×#131—22no—
30C3×#131—22no—
31C2×C3×M16C2×C411yescharacteristic
32C2×C3×M16D412yes—
33C22×C3×C8D412yes—
34C22×C3×C8C2311yescharacteristic
35#946—21no—
36#938—21no—
37#937C812yes—
38#935C812yes—
39C4×#67—22no—
40#926C2×C411yescharacteristic
41C2×#208C2×C411yescharacteristic
42C22×C3⋊C8C2×C411yescharacteristic
43C2×D3×C8—21no—
44C2×D3×C8—22no—
45C2×D3×C8—21no—
46C2×D3×C8—22no—
47C3×M32—24no—
48C3×M32—22no—
49C3×M32—22no—
50C2×C3×C16—24no—
51C2×C3×C16D414yescomplemented
52#830—22no—
53(C2×D3)⋊C8—22no—
54#815—21no—
55Dic3⋊C8—22no—
56C8×Dic3—21no—
57#812—22no—
58#812—22no—
59C2×C3⋊C16—22no—
60C2×#131C2×D311yescharacteristic
61C23×C8—31no—
62#525—31no—
63#504—24no—
64C2×M32—22no—
65C2×M32—32no—
66C2×M32C2×D312yes—
67#435—62no—
68C8×D4—62no—
69#424—31no—
70#408—31no—
71#407—31no—
72#405—31no—
73#349—62no—
74C22⋊C16—64no—
75C23×D3C2×C811yescharacteristic
76C3×#41—22no—
77C3×#41—22no—
78C2×C3×Q8M1611yescharacteristic
79C2×C3×D4C2×C812yes—
80C2×C3×D4M1611yescharacteristic
81C22×C3×C4C22⋊C412yes—
82C22×C3×C4C22×C411yescharacteristic
83C2×#67—22no—
84C22×Dic3C2×C811yescharacteristic
85C2×C4×D3—21no—
86C2×C4×D3—22no—
87C2×C4×D3—21no—
88C2×C4×D3—22no—
89C3×M16—22no—
90C3×M16—24no—
91C2×C3×C8—22no—
92C2×C3×C8—22no—
93C2×C3×C8C22⋊C412yes—
94C2×C3×C8C2×D411yescharacteristic
95C2×C3×C8C2×D412yes—
96C2×C3×C8C2×D411yescharacteristic
97C2×C3×C8#4112yes—
98#217—22no—
99#217C2×C812yes—
100(C2×D3)⋊C4—22no—
101C4⋊Dic3—21no—
102Dic3⋊C4—22no—
103C4×Dic3—21no—
104#208—22no—
105C2×C3⋊C8—21no—
106C2×C3⋊C8—21no—
107C2×C3⋊C8—22no—
108D3×C8—41no—
109D3×C8—42no—
110D3×C8—41no—
111C3×C16—24no—
112C3⋊C16—42no—
113C2×#41C4×D311yescharacteristic
114C23×C4—31no—
115#131—22no—
116#131—22no—
117C2×M16—31no—
118C2×M16C4×D311yescharacteristic
119C2×M16D1212yes—
120C22×C8—31no—
121C22×C8—61no—
122C22×C8—62no—
123C22×C8—62no—
124C22×C8—61no—
125C22×C8#6712yes—
126C22×C8C22×D311yescharacteristic
127#123—32no—
128C22⋊Q8—61no—
129#121—61no—
130#120—32no—
131C4×D4—62no—
132#117—31no—
133M32—24no—
134M32—22no—
135M32—22no—
136M32—62no—
137M32—62no—
138C2×C16—24no—
139C2×C16—62no—
140C2×C16#6714yescomplemented
141C4⋊C8—62no—
142C22⋊C8—62no—
143C22⋊C8—62no—
144#97—61no—
145C4×C8—61no—
146C23×C3C22⋊C812yes—
147C23×C3C22×C811yescharacteristic
148C22×D3—21no—
149C22×D3—22no—
150C22×D3—42no—
151C3×Q8—22no—
152C3×D4—24no—
153C3×D4—22no—
154C2×C3×C4—22no—
155C2×C3×C4—22no—
156C2×C3×C4C22⋊C812yes—
157C2×C3×C4C2×C22⋊C411yescharacteristic
158C2×C3×C4C4×D412yes—
159C2×C3×C4C2×M1611yescharacteristic
160C2×C3×C4#13112yes—
161#67—44no—
162C2×Dic3—22no—
163C2×Dic3—21no—
164C2×Dic3—22no—
165C4×D3—41no—
166C4×D3—41no—
167C4×D3—42no—
168C3×C8—22no—
169C3×C8#12011yescharacteristic
170C3×C8#12112yes—
171C3×C8#12311yescharacteristic
172C3⋊C8—41no—
173C3⋊C8—41no—
174C24—31no—
175#41—22no—
176#41—22no—
177C2×Q8ΓL(1,𝔽25)11yescharacteristic
178C2×D4—62no—
179C2×D4D3×C812yes—
180C2×D4ΓL(1,𝔽25)11yescharacteristic
181C22×C4—31no—
182C22×C4—61no—
183C22×C4—62no—
184C22×C4—61no—
185C22×C4—62no—
186C22×C4(C2×D3)⋊C412yes—
187C22×C4C2×C4×D311yescharacteristic
188M16—22no—
189M16—24no—
190M16—62no—
191C2×C8—22no—
192C2×C8—22no—
193C2×C8—61no—
194C2×C8—61no—
195C2×C8—62no—
196C2×C8—121no—
197C2×C8—122no—
198C2×C8—121no—
199C2×C8(C2×D3)⋊C412yes—
200C2×C8C2×D1211yescharacteristic
201C2×C8#23512yes—
202C2×C8C2×#6711yescharacteristic
203C2×C8C2×#6712yes—
204C4⋊C4—62no—
205C4⋊C4—61no—
206C22⋊C4—32no—
207C22⋊C4—62no—
208C22⋊C4—62no—
209C42—61no—
210C16—24no—
211C16—122no—
212C22×C3—22no—
213C22×C3—41no—
214C22×C3—42no—
215C22×C3C2×C22⋊C811yescharacteristic
216C22×C3C8×D412yes—
217C2×D3—22no—
218C2×D3—24no—
219C2×D3—41no—
220C2×D3—42no—
221C2×D3—41no—
222C3×C4—22no—
223C3×C4#38611yescharacteristic
224C3×C4#40811yescharacteristic
225C3×C4#43512yes—
226Dic3—42no—
227Dic3—41no—
228C23—61no—
229C23—62no—
230C23—122no—
231C23(C2×D3)⋊C812yes—
232C23C2×D3×C811yescharacteristic
233Q8—22no—
234D4—24no—
235D4—22no—
236D4—124no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—62no—
240C2×C4—61no—
241C2×C4—62no—
242C2×C4—121no—
243C2×C4—121no—
244C2×C4—122no—
245C2×C4(C2×D3)⋊C812yes—
246C2×C4C2×ΓL(1,𝔽25)11yescharacteristic
247C2×C4#90112yes—
248C2×C4C2×(C2×D3)⋊C411yescharacteristic
249C2×C4C4×#6712yes—
250C8—22no—
251C8—121no—
252C8—121no—
253C8#92911yescharacteristic
254C8#93012yes—
255C8#95311yescharacteristic
256C2×C3—21no—
257C2×C3—22no—
258C2×C3—42no—
259C2×C3#186911yescharacteristic
260D3—42no—
261C22—22no—
262C22—41no—
263C22—42no—
264C22—62no—
265C22—64no—
266C22—121no—
267C22—122no—
268C22—121no—
269C22C2×(C2×D3)⋊C811yescharacteristic
270C22C8×#6712yes—
271C4—22no—
272C4—122no—
273C4—121no—
274C4#511011yescharacteristic
275C4#511112yes—
276C4#521311yescharacteristic
277C3#1298111yescharacteristic, complemented, Sylow
278C2—21no—
279C2—22no—
280C2—42no—
281C2—122no—
282C2#7197511yescharacteristic
subgroupG/H
centerC8#953
commutatorC23×C3C22×C8
FrattiniC22×C8C22×D3
FittingC3×#3407C2
socleC2×C3#1869
prime radical#5828C4
cosocleC22×C3×C8C23
hypercenter#3407D3
hypocenterC3#12981
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C8 ◁ C22×C8 ◁ #3407

Complements

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

NH ≅ G/Ntype
1. C3×#3407C2semidirect
3. #72007C2semidirect
4. #71431C2semidirect
5. #67813C2semidirect
22. #3407D3semidirect
51. C2×C3×C16D4semidirect
140. C2×C16#67semidirect
277. C3#12981semidirect

Maximal subgroups

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

subgroupindex
C3×#34072
#837942
#720072
#714312
#678132
#129813