Finite Group Databasebeta

← #10692937

Subgroups — #10692937

Order 768.

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

normal subgroups only

This group has 554 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×#2504C211yesmaximal, characteristic
2#82155C211yesmaximal, characteristic, complemented
3#71674C212yesmaximal
4#71562C211yesmaximal, characteristic, complemented
5#67616C212yesmaximal, complemented
6#9465—31nomaximal, Sylow
7C2×C3×#121C2211yescharacteristic
8C22×C3×C4⋊C4C2211yescharacteristic
9C3×#397—22no—
10C3×#390C412yescomplemented
11C3×#386—21no—
12C3×#386—21no—
13C3×#382C412yescomplemented
14C3×C4×C22⋊C4C2211yescharacteristic
15#5212—21no—
16#5211C2212yes—
17C4×#217—21no—
18#5207C2212yes—
19#4980—24no—
20#4537—24no—
21#4530—22no—
22#2504D311yescharacteristic
23#2313—31no—
24#1821—31no—
25#1510—32no—
26#1284—32no—
27C22×C3×D4—21no—
28C22×C3×D4C2×C411yescharacteristic
29C23×C3×C4D411yescharacteristic
30C23×C3×C4D411yescharacteristic
31C23×C3×C4C2311yescharacteristic
32C3×#121—22no—
33C3×#121—22no—
34C2×C3×C4⋊C4—22no—
35C2×C3×C4⋊C4—22no—
36C2×C3×C4⋊C4C2×C411yescharacteristic
37C2×C3×C4⋊C4C2×C412yes—
38C2×C3×C4⋊C4D412yes—
39C2×C3×C22⋊C4—21no—
40C2×C3×C22⋊C4—21no—
41C2×C3×C22⋊C4—22no—
42C2×C3×C22⋊C4—22no—
43C2×C3×C42C2×C412yes—
44C2×#217—22no—
45C2×C4⋊Dic3—22no—
46C2×C4×Dic3—22no—
47C2×#208—22no—
48C3×#95—22no—
49C3×#95—22no—
50C3×#95—21no—
51C3⋊#95—22no—
52C3⋊#95—22no—
53C3⋊#95—22no—
54C3⋊#95D414yes—
55#830—22no—
56#830—22no—
57C2×#121C2×D311yescharacteristic
58C22×C4⋊C4C2×D311yescharacteristic
59#407—32no—
60#397—22no—
61#390Dic312yescomplemented
62#387—64no—
63#386—21no—
64#386—21no—
65#382Dic312yescomplemented
66#381—32no—
67#380—61no—
68C4×C22⋊C4—61no—
69C4×C22⋊C4C2×D311yescharacteristic
70#343—62no—
71#327—64no—
72C24×C3—21no—
73C24×C3C2×D411yescharacteristic
74C2×C3×D4—22no—
75C2×C3×D4—42no—
76C2×C3×D4—44no—
77C2×C3×D4—41no—
78C2×C3×D4—41no—
79C22×C3×C4—21no—
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—22no—
85C22×C3×C4—42no—
86C22×C3×C4—81no—
87C22×C3×C4—81no—
88C22×C3×C4C22⋊C412yes—
89C22×C3×C4C22⋊C412yes—
90C22×C3×C4C22×C411yescharacteristic
91C22×C3×C4C2×D412yes—
92C22×C3×C4C2×D411yescharacteristic
93C22×C3×C4#4112yes—
94C22×Dic3—24no—
95C3×C4⋊C4—24no—
96C3×C4⋊C4—24no—
97C3×C4⋊C4—22no—
98C3×C22⋊C4—42no—
99C3×C22⋊C4—42no—
100C3×C22⋊C4—82no—
101C3×C22⋊C4—82no—
102C3×C42—42no—
103#217—44no—
104C4⋊Dic3—82no—
105C4×Dic3—82no—
106#208—44no—
107C2×C3⋊C8—42no—
108C22×D4—21no—
109C22×D4C2×Dic311yescharacteristic
110C23×C4#6711yescharacteristic
111C23×C4#6711yescharacteristic
112C23×C4C22×D311yescharacteristic
113C2×M16—62no—
114#121—22no—
115#121—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—62no—
119C2×C4⋊C4C2×Dic311yescharacteristic
120C2×C4⋊C4C2×Dic312yes—
121C2×C4⋊C4#6712yes—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—62no—
127C2×C42—62no—
128C2×C42C2×Dic312yes—
129C22⋊C8—62no—
130C22⋊C8—62no—
131#95—22no—
132#95—22no—
133#95—21no—
134#95—34no—
135#95—62no—
136#95—62no—
137#95—62no—
138C23×C3—22no—
139C23×C3—22no—
140C23×C3—41no—
141C23×C3—42no—
142C23×C3—42no—
143C23×C3—41no—
144C23×C3—42no—
145C23×C3C2×C22⋊C411yescharacteristic
146C23×C3#12011yescharacteristic
147C23×C3#12311yescharacteristic
148C3×D4—82no—
149C3×D4—82no—
150C3×D4—82no—
151C2×C3×C4—22no—
152C2×C3×C4—22no—
153C2×C3×C4—22no—
154C2×C3×C4—22no—
155C2×C3×C4—42no—
156C2×C3×C4—42no—
157C2×C3×C4—42no—
158C2×C3×C4—42no—
159C2×C3×C4—42no—
160C2×C3×C4—42no—
161C2×C3×C4—42no—
162C2×C3×C4—42no—
163C2×C3×C4—42no—
164C2×C3×C4—41no—
165C2×C3×C4—42no—
166C2×C3×C4—81no—
167C2×C3×C4—81no—
168C2×C3×C4C4×D412yes—
169C2×C3×C4#12112yes—
170C2×Dic3—44no—
171C2×Dic3—44no—
172C2×Dic3—44no—
173C3⋊C8—82no—
174C24—21no—
175C24C2×#6711yescharacteristic
176C2×D4—22no—
177C2×D4—42no—
178C2×D4—44no—
179C2×D4—41no—
180C2×D4—41no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—42no—
188C22×C4—64no—
189C22×C4—81no—
190C22×C4—81no—
191C22×C4#21712yes—
192C22×C4#21712yes—
193C22×C4D3×D412yes—
194C22×C4#23712yes—
195C22×C4C22×Dic311yescharacteristic
196C22×C4C2×#6711yescharacteristic
197M16—124no—
198C2×C8—122no—
199C4⋊C4—24no—
200C4⋊C4—24no—
201C4⋊C4—22no—
202C4⋊C4—242no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—82no—
206C22⋊C4—82no—
207C22⋊C4—124no—
208C42—42no—
209C42—242no—
210C22×C3—21no—
211C22×C3—21no—
212C22×C3—21no—
213C22×C3—21no—
214C22×C3—21no—
215C22×C3—21no—
216C22×C3—21no—
217C22×C3—44no—
218C22×C3—84no—
219C22×C3—81no—
220C22×C3—81no—
221C22×C3—81no—
222C22×C3#38611yescharacteristic
223C22×C3#41012yes—
224C22×C3#45412yes—
225C3×C4—22no—
226C3×C4—22no—
227C3×C4—42no—
228C3×C4—42no—
229C3×C4—82no—
230C3×C4—81no—
231Dic3—84no—
232C23—22no—
233C23—22no—
234C23—41no—
235C23—42no—
236C23—42no—
237C23—41no—
238C23—42no—
239C23#93511yescharacteristic
240C23#93711yescharacteristic
241C23C2×#21711yescharacteristic
242D4—82no—
243D4—82no—
244D4—82no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—124no—
263C2×C4—124no—
264C2×C4—124no—
265C2×C4D4×Dic312yes—
266C2×C4#93912yes—
267C8—242no—
268C2×C3—22no—
269C2×C3—22no—
270C2×C3—41no—
271C2×C3—161no—
272C2×C3#188811yescharacteristic
273C2×C3#189911yescharacteristic
274C2×C3#191111yescharacteristic
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—44no—
283C22—84no—
284C22—81no—
285C22—81no—
286C22—81no—
287C22#506212yes—
288C22#522011yescharacteristic
289C22#524012yes—
290C4—22no—
291C4—22no—
292C4—42no—
293C4—42no—
294C4—82no—
295C4—81no—
296C4—244no—
297C3#946511yescharacteristic, complemented, Sylow
298C2—22no—
299C2—22no—
300C2—41no—
301C2—161no—
302C2#7170511yescharacteristic
303C2#7172411yescharacteristic
304C2#7185311yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2504C2
socleC22×C3#386
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenter#2504D3
hypocenterC3#9465
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2504

Complements

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

NH ≅ G/Ntype
2. #82155C2semidirect
4. #71562C2semidirect
5. #67616C2semidirect
10. C3×#390C4semidirect
13. C3×#382C4semidirect
61. #390Dic3semidirect
65. #382Dic3semidirect
297. C3#9465semidirect

Maximal subgroups

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

subgroupindex
C3×#25042
#821552
#716742
#715622
#676162
#94653