Finite Group Databasebeta

← #10693041

Subgroups — #10693041

Order 768.

542 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 131 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×#2982C211yesmaximal, characteristic, complemented
2#82499C211yesmaximal, characteristic
3#71637C211yesmaximal, characteristic, complemented
4#70762C212yesmaximal
5#67738C212yesmaximal, complemented
6#9486—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C2×#946—21no—
9C2×C4×D12C2211yescharacteristic
10C3×C8×Q8—24no—
11C3×#431—22no—
12C3×#431C2212yes—
13C2×#836—21no—
14#4991C412yes—
15#4987—21no—
16#4981C412yes—
17#4940—22no—
18#4936C2211yescharacteristic
19#4924C2212yes—
20#4530—21no—
21#4459—24no—
22#2982D311yescharacteristic, complemented
23#2536—31no—
24#1801—31no—
25#1486—32no—
26#1286—32no—
27C22×C3×Q8D411yescharacteristic
28C22×D12C2×C411yescharacteristic
29C22×C4×D3—21no—
30C3×C4×Q8—22no—
31C3×C4×Q8—22no—
32C2×C3×C4⋊C4C2×C412yes—
33C2×C3×C4⋊C4D411yescharacteristic
34C2×C3×C42D412yes—
35C2×C3×C42C2311yescharacteristic
36#946—42no—
37#946—42no—
38C2×(C2×D3)⋊C4—22no—
39C2×(C2×D3)⋊C4—22no—
40C2×(C2×D3)⋊C4—21no—
41C2×C4⋊Dic3—21no—
42C2×C4⋊Dic3C2×C411yescharacteristic
43C2×Dic3⋊C4—22no—
44C2×#208—22no—
45C4×D12—22no—
46C4×D12—22no—
47C3×C4⋊C8—24no—
48C3×C4⋊C8—22no—
49C3×C22⋊C8—22no—
50C3×C4×C8—24no—
51C3×C4×C8D414yes—
52#836—42no—
53C3⋊#95—22no—
54C3⋊#95—21no—
55C3⋊#95C2×C412yes—
56C4⋊(C3⋊C8)—22no—
57C4⋊(C3⋊C8)—22no—
58C2×C22⋊Q8—61no—
59C2×C4×Q8C2×D311yescharacteristic
60C2×C4×D4—31no—
61C8×Q8—24no—
62#431—22no—
63#431C2×D312yes—
64#423—32no—
65C2×#101—61no—
66#398—32no—
67#394—32no—
68#388—62no—
69#382—31no—
70#380—61no—
71#343—61no—
72D4⋊C8—64no—
73C23×D3—21no—
74C2×C3×Q8—22no—
75C2×C3×Q8—22no—
76C2×C3×Q8D812yes—
77C2×C3×Q8SD1612yes—
78C22×C3×C4C22⋊C412yes—
79C22×C3×C4C22⋊C412yes—
80C22×C3×C4C22×C411yescharacteristic
81C22×C3×C4C2×D411yescharacteristic
82C22×C3×C4C2×D411yescharacteristic
83C22×Dic3—22no—
84C22×Dic3—21no—
85C2×D12—22no—
86C2×D12—42no—
87C2×C4×D3—42no—
88C2×C4×D3—41no—
89C2×C4×D3—41no—
90C2×C4×D3—42no—
91C2×C3×C8—24no—
92C3×C4⋊C4—24no—
93C3×C4⋊C4—22no—
94C3×C42—22no—
95C3×C42—22no—
96C3×C42C2×D412yes—
97C3×C42#4112yes—
98(C2×D3)⋊C4—44no—
99(C2×D3)⋊C4—44no—
100(C2×D3)⋊C4—41no—
101(C2×D3)⋊C4—41no—
102C4⋊Dic3—22no—
103C4⋊Dic3—41no—
104C4⋊Dic3—41no—
105Dic3⋊C4—44no—
106#208—42no—
107#208—42no—
108C2×C3⋊C8—42no—
109C22×Q8D1211yescharacteristic
110C22×D4—31no—
111C23×C4—61no—
112C2×M16—62no—
113C22⋊Q8—122no—
114C22⋊Q8—122no—
115C4×Q8—22no—
116C4×Q8—22no—
117C4×D4—62no—
118C4×D4—62no—
119C2×C4⋊C4—31no—
120C2×C4⋊C4—62no—
121C2×C4⋊C4—61no—
122C2×C4⋊C4C4×D312yes—
123C2×C4⋊C4D1211yescharacteristic
124C2×C22⋊C4—62no—
125C2×C22⋊C4—62no—
126C2×C22⋊C4—61no—
127C2×C42#6712yes—
128C2×C42C22×D311yescharacteristic
129C4⋊C8—24no—
130C4⋊C8—22no—
131C4⋊C8—62no—
132C4⋊C8—62no—
133#101—122no—
134C22⋊C8—22no—
135C4×C8—24no—
136C4×C8#6714yes—
137#95—32no—
138#95—62no—
139#95—61no—
140C23×C3C2×C22⋊C411yescharacteristic
141C22×D3—22no—
142C22×D3—41no—
143C22×D3—41no—
144C22×D3—41no—
145C22×D3—41no—
146C22×D3—41no—
147C22×D3—41no—
148C3×Q8—24no—
149C3×Q8—24no—
150C2×C3×C4—22no—
151C2×C3×C4—22no—
152C2×C3×C4—22no—
153C2×C3×C4—41no—
154C2×C3×C4—41no—
155C2×C3×C4—42no—
156C2×C3×C4#10214yes—
157C2×C3×C4C4×D412yes—
158C2×C3×C4#12011yescharacteristic
159C2×C3×C4#12112yes—
160C2×C3×C4#12311yescharacteristic
161C2×C3×C4C2×D811yescharacteristic
162C2×C3×C4C2×SD1611yescharacteristic
163C2×C3×C4#13512yes—
164C2×Dic3—42no—
165C2×Dic3—42no—
166C2×Dic3—41no—
167C2×Dic3—41no—
168C2×Dic3—42no—
169C2×Dic3—41no—
170D12—24no—
171D12—42no—
172C4×D3—82no—
173C4×D3—82no—
174C3×C8—44no—
175C3⋊C8—82no—
176C24—61no—
177C2×Q8—22no—
178C2×Q8—22no—
179C2×Q8#20412yes—
180C2×Q8D2412yes—
181C2×D4—62no—
182C2×D4—122no—
183C22×C4—62no—
184C22×C4—61no—
185C22×C4—121no—
186C22×C4—121no—
187C22×C4—122no—
188C22×C4—122no—
189C22×C4(C2×D3)⋊C412yes—
190C22×C4(C2×D3)⋊C412yes—
191C22×C4C2×C4×D311yescharacteristic
192C22×C4C2×D1211yescharacteristic
193C22×C4C2×#6711yescharacteristic
194M16—122no—
195M16—122no—
196C2×C8—24no—
197C2×C8—122no—
198C4⋊C4—24no—
199C4⋊C4—22no—
200C4⋊C4—62no—
201C4⋊C4—124no—
202C4⋊C4—121no—
203C4⋊C4—121no—
204C22⋊C4—124no—
205C22⋊C4—124no—
206C22⋊C4—121no—
207C22⋊C4—121no—
208C42—22no—
209C42—22no—
210C42#23512yes—
211C42C2×#6712yes—
212C22×C3—21no—
213C22×C3—21no—
214C22×C3#38611yescharacteristic
215C22×C3C2×#10211yescharacteristic
216C22×C3#41611yescharacteristic
217C2×D3—42no—
218C2×D3—42no—
219C2×D3—42no—
220C2×D3—81no—
221C2×D3—81no—
222C2×D3—81no—
223C2×D3—81no—
224C3×C4—22no—
225C3×C4—24no—
226C3×C4—21no—
227C3×C4—21no—
228C3×C4—42no—
229C3×C4#46512yes—
230C3×C4#46612yes—
231Dic3—82no—
232Dic3—81no—
233C23—62no—
234C23—121no—
235C23—121no—
236C23—121no—
237C23—121no—
238C23—121no—
239C23—121no—
240C23C2×(C2×D3)⋊C411yescharacteristic
241Q8—24no—
242Q8—24no—
243D4—64no—
244D4—122no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—122no—
252C2×C4—122no—
253C2×C4—121no—
254C2×C4—122no—
255C2×C4—121no—
256C2×C4—121no—
257C2×C4—242no—
258C2×C4—242no—
259C2×C4#82114yes—
260C2×C4C2×#20411yescharacteristic
261C2×C4C2×D2411yescharacteristic
262C2×C4#90412yes—
263C2×C4C4×#6712yes—
264C2×C4#92911yescharacteristic
265C2×C4#93012yes—
266C2×C4#95311yescharacteristic
267C8—44no—
268C8—242no—
269C2×C3—21no—
270C2×C3—21no—
271C2×C3#189111yescharacteristic
272C2×C3#189311yescharacteristic
273C2×C3#191211yescharacteristic
274D3—82no—
275C22—21no—
276C22—21no—
277C22—122no—
278C22—122no—
279C22—122no—
280C22—241no—
281C22—241no—
282C22—241no—
283C22—241no—
284C22C2×#82111yescharacteristic
285C22#511311yescharacteristic
286C22#521311yescharacteristic
287C4—24no—
288C4—22no—
289C4—21no—
290C4—21no—
291C4—42no—
292C4—242no—
293C4—241no—
294C4#511412yes—
295C4#511512yes—
296C3#948611yescharacteristic, complemented, Sylow
297C2—21no—
298C2—21no—
299C2—242no—
300C2#7166311yescharacteristic
301C2#7197311yescharacteristic
302C2#7202911yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2982C2
socleC22×C3#386
prime radicalC22×D12C2×C4
cosocleC2×C3×C42C23
hypercenter#2982D3
hypocenterC3#9486
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
1. C3×#2982C2semidirect
3. #71637C2semidirect
5. #67738C2semidirect
22. #2982D3semidirect
296. C3#9486semidirect

Maximal subgroups

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

subgroupindex
C3×#29822
#824992
#716372
#707622
#677382
#94863