Finite Group Databasebeta

← #10693151

Subgroups — #10693151

Order 768.

542 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 107 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#82373C211yesmaximal, characteristic
3#71602C211yesmaximal, characteristic
4#69973C212yesmaximal, complemented
5#67684C212yesmaximal, complemented
6#9508—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#935C412yes—
9C22×C4⋊Dic3C2211yescharacteristic
10C3×#436—24no—
11C3×#431—21no—
12C3×#407—21no—
13C3×#407C2212yes—
14#5212—21no—
15#5212C2211yescharacteristic
16#5207C2212yes—
17#4981—21no—
18#4953—21no—
19#4943C412yes—
20#4928—21no—
21#4554—24no—
22#4528—21no—
23#4525—21no—
24#2975D311yescharacteristic
25#2486—31no—
26#1821—32no—
27#1465—31no—
28#1305—32no—
29C22×C3×D4C2×C411yescharacteristic
30C23×C3×C4D411yescharacteristic
31C23×C3×C4C2311yescharacteristic
32C23×Dic3C2×C412yes—
33C2×C3×M16—22no—
34C2×C3×M16D414yes—
35C3×C4×D4—22no—
36C3×C4×D4—22no—
37C2×C3×C4⋊C4D411yescharacteristic
38C2×C3×C22⋊C4C2×C411yescharacteristic
39C2×C3×C22⋊C4D411yescharacteristic
40C2×C3×C42D411yescharacteristic
41C2×#217—22no—
42C2×#217C2×C412yes—
43#935—44no—
44C2×C4⋊Dic3—21no—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3—22no—
48C2×C4⋊Dic3—22no—
49C2×Dic3⋊C4—22no—
50C2×#208—22no—
51C3×C4⋊C8—22no—
52C3×C22⋊C8—22no—
53C3×C22⋊C8—22no—
54C3×C22⋊C8—22no—
55C3×C4×C8—22no—
56C3⋊#95—22no—
57C3⋊#95—22no—
58C3⋊#95—21no—
59#830—22no—
60#830—22no—
61C2×#123—32no—
62C2×C4×D4C2×D311yescharacteristic
63C22×C4⋊C4—31no—
64#436—24no—
65#431—21no—
66#407—21no—
67#407—32no—
68#407C2×D312yes—
69#398—61no—
70#397—61no—
71#389—61no—
72#381—32no—
73#380—31no—
74#380—61no—
75#343—64no—
76#324—61no—
77#323—61no—
78C24×C3C22⋊C411yescharacteristic
79C24×C3C22×C411yescharacteristic
80C2×C3×D4—22no—
81C2×C3×D4—22no—
82C2×C3×D4—41no—
83C22×C3×C4—21no—
84C22×C3×C4—21no—
85C22×C3×C4—42no—
86C22×C3×C4—42no—
87C22×C3×C4C22⋊C411yescharacteristic
88C22×C3×C4C22⋊C411yescharacteristic
89C22×C3×C4C22⋊C411yescharacteristic
90C22×C3×C4C2×D411yescharacteristic
91C22×C3×C4C2×D412yes—
92C22×C3×C4C2×D411yescharacteristic
93C22×C3×C4#4112yes—
94C22×Dic3—22no—
95C22×Dic3—22no—
96C22×Dic3—24no—
97C22×Dic3—42no—
98C22×Dic3—42no—
99C3×M16—44no—
100C3×M16—82no—
101C2×C3×C8—24no—
102C3×C4⋊C4—22no—
103C3×C22⋊C4—22no—
104C3×C22⋊C4—22no—
105C3×C42—21no—
106C3×C42—21no—
107#217—44no—
108#217—42no—
109C4⋊Dic3—42no—
110C4⋊Dic3—41no—
111C4⋊Dic3—42no—
112C4⋊Dic3—41no—
113Dic3⋊C4—44no—
114#208—44no—
115C2×C3⋊C8—42no—
116C22×D4C4×D311yescharacteristic
117C23×C4—32no—
118C23×C4#6711yescharacteristic
119C23×C4C22×D311yescharacteristic
120C2×M16—22no—
121C2×M16—62no—
122C2×M16#6714yes—
123#123—124no—
124C4×D4—22no—
125C4×D4—22no—
126C2×C4⋊C4—62no—
127C2×C4⋊C4—61no—
128C2×C4⋊C4—62no—
129C2×C4⋊C4—62no—
130C2×C4⋊C4—61no—
131C2×C4⋊C4—61no—
132C2×C4⋊C4D1211yescharacteristic
133C2×C22⋊C4—32no—
134C2×C22⋊C4—62no—
135C2×C22⋊C4C4×D311yescharacteristic
136C2×C22⋊C4D1211yescharacteristic
137C2×C42#6711yescharacteristic
138C4⋊C8—22no—
139C22⋊C8—22no—
140C22⋊C8—22no—
141C22⋊C8—22no—
142C22⋊C8—62no—
143C22⋊C8—62no—
144C4×C8—22no—
145#95—62no—
146#95—62no—
147#95—61no—
148C23×C3—21no—
149C23×C3—21no—
150C23×C3—21no—
151C23×C3—21no—
152C23×C3—21no—
153C23×C3—41no—
154C23×C3—41no—
155C23×C3—41no—
156C23×C3—41no—
157C23×C3C2×C22⋊C411yescharacteristic
158C23×C3C4×D412yes—
159C3×D4—44no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
162C2×C3×C4—24no—
163C2×C3×C4—22no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—41no—
167C2×C3×C4—42no—
168C2×C3×C4—42no—
169C2×C3×C4—41no—
170C2×C3×C4—41no—
171C2×C3×C4—81no—
172C2×C3×C4#10012yes—
173C2×C3×C4#12011yescharacteristic
174C2×C3×C4#12112yes—
175C2×C3×C4#12311yescharacteristic
176C2×Dic3—42no—
177C2×Dic3—42no—
178C2×Dic3—42no—
179C2×Dic3—42no—
180C2×Dic3—42no—
181C2×Dic3—42no—
182C2×Dic3—84no—
183C3×C8—44no—
184C3⋊C8—82no—
185C24(C2×D3)⋊C411yescharacteristic
186C24C2×C4×D311yescharacteristic
187C2×D4—22no—
188C2×D4—22no—
189C2×D4—41no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—62no—
195C22×C4—62no—
196C22×C4—64no—
197C22×C4—122no—
198C22×C4—122no—
199C22×C4(C2×D3)⋊C411yescharacteristic
200C22×C4(C2×D3)⋊C411yescharacteristic
201C22×C4(C2×D3)⋊C411yescharacteristic
202C22×C4C2×D1211yescharacteristic
203C22×C4#23512yes—
204C22×C4C2×#6711yescharacteristic
205C22×C4C2×#6712yes—
206M16—44no—
207M16—82no—
208M16—124no—
209C2×C8—24no—
210C2×C8—122no—
211C4⋊C4—22no—
212C4⋊C4—122no—
213C4⋊C4—124no—
214C4⋊C4—121no—
215C4⋊C4—122no—
216C4⋊C4—121no—
217C22⋊C4—22no—
218C22⋊C4—22no—
219C22⋊C4—124no—
220C22⋊C4—122no—
221C42—21no—
222C42—21no—
223C22×C3—21no—
224C22×C3—21no—
225C22×C3—22no—
226C22×C3—22no—
227C22×C3—22no—
228C22×C3—42no—
229C22×C3—42no—
230C22×C3—42no—
231C22×C3—41no—
232C22×C3—41no—
233C22×C3—41no—
234C22×C3—81no—
235C22×C3#38611yescharacteristic
236C22×C3C2×#10011yescharacteristic
237C22×C3#41311yescharacteristic
238C3×C4—22no—
239C3×C4—21no—
240C3×C4—22no—
241C3×C4—41no—
242C3×C4—41no—
243C3×C4#45812yes—
244Dic3—82no—
245Dic3—82no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23C2×(C2×D3)⋊C411yescharacteristic
256C23C4×#6712yes—
257D4—44no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—81no—
270C2×C4—122no—
271C2×C4—122no—
272C2×C4—122no—
273C2×C4—122no—
274C2×C4—122no—
275C2×C4—122no—
276C2×C4—244no—
277C2×C4#82312yes—
278C2×C4#92911yescharacteristic
279C2×C4#93012yes—
280C2×C4#95311yescharacteristic
281C8—44no—
282C8—242no—
283C2×C3—21no—
284C2×C3—21no—
285C2×C3—42no—
286C2×C3—81no—
287C2×C3#189011yescharacteristic
288C2×C3#190011yescharacteristic
289C2×C3#191511yescharacteristic
290C22—21no—
291C22—21no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—42no—
296C22—42no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—81no—
302C22C2×#82311yescharacteristic
303C22#513211yescharacteristic
304C22#521311yescharacteristic
305C4—22no—
306C4—22no—
307C4—21no—
308C4—41no—
309C4—41no—
310C4—242no—
311C4—242no—
312C4#524312yes—
313C3#950811yescharacteristic, complemented, Sylow
314C2—21no—
315C2—21no—
316C2—42no—
317C2—81no—
318C2#7165511yescharacteristic
319C2#7200211yescharacteristic
320C2#7203111yescharacteristic
subgroupG/H
centerC22#5213
commutatorC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2975C2
socleC22×C3#386
prime radicalC22×C3×D4C2×C4
cosocleC23×C3×C4C23
hypercenter#2975D3
hypocenterC3#9508
LCSG ▷ C24×C3 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2975

Complements

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

NH ≅ G/Ntype
4. #69973C2semidirect
5. #67684C2semidirect
313. C3#9508semidirect

Maximal subgroups

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

subgroupindex
C3×#29752
#823732
#716022
#699732
#676842
#95083