Finite Group Databasebeta

← #10692936

Subgroups — #10692936

Order 768.

554 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 103 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×#2313C211yesmaximal, characteristic, complemented
2#82180C211yesmaximal, characteristic
3#71562C211yesmaximal, characteristic, complemented
4#69531C212yesmaximal
5#67628C212yesmaximal, complemented
6#9465—31nomaximal, Sylow
7C22×C3×C4⋊C4C2211yescharacteristic
8C2×#930C2211yescharacteristic
9C3×#387—24no—
10C3×#381C2212yes—
11C3×#380—21no—
12C3×C4×C22⋊C4—21no—
13#5213—21no—
14#5213—21no—
15C4×#217C2211yescharacteristic
16#5207C2212yes—
17#4992—22no—
18#4988C412yescomplemented
19#4977C412yescomplemented
20#4530—22no—
21#4451—24no—
22#2504—31no—
23#2313D311yescharacteristic, complemented
24#1821—31no—
25#1510—32no—
26#1284—32no—
27C23×C3×C4D411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4C2311yescharacteristic
30C22×#67—21no—
31C22×D12C2×C411yescharacteristic
32C2×C3×C4⋊C4—22no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C4⋊C4—22no—
35C2×C3×C4⋊C4C2×C412yes—
36C2×C3×C4⋊C4D412yes—
37C2×C3×C22⋊C4—22no—
38C2×C3×C42—22no—
39C2×#217—21no—
40#930—22no—
41#930—22no—
42C2×(C2×D3)⋊C4—21no—
43C2×(C2×D3)⋊C4—22no—
44C2×(C2×D3)⋊C4—22no—
45C2×C4⋊Dic3C2×C411yescharacteristic
46C2×C4×Dic3C2×C412yes—
47C2×#208—22no—
48C3×#95—22no—
49C3×#95—22no—
50C3×#95—22no—
51C3×#95D414yes—
52C3⋊#95—22no—
53C3⋊#95—22no—
54C3⋊#95—21no—
55#830—22no—
56#830—22no—
57C2×#121—31no—
58C22×C4⋊C4C2×D311yescharacteristic
59#407—32no—
60#397—62no—
61#390—32no—
62#387—24no—
63#386—61no—
64#386—61no—
65#382—32no—
66#381C2×D312yes—
67#380—21no—
68C4×C22⋊C4—21no—
69C4×C22⋊C4—31no—
70#343—62no—
71#327—64no—
72C24×C3C2×D411yescharacteristic
73C23×D3—21no—
74C22×C3×C4—21no—
75C22×C3×C4—21no—
76C22×C3×C4—21no—
77C22×C3×C4—21no—
78C22×C3×C4—24no—
79C22×C3×C4—42no—
80C22×C3×C4—81no—
81C22×C3×C4—81no—
82C22×C3×C4C22⋊C412yes—
83C22×C3×C4C22⋊C412yes—
84C22×C3×C4C22×C411yescharacteristic
85C22×C3×C4C2×D411yescharacteristic
86C22×C3×C4C2×D412yes—
87C22×C3×C4#4112yes—
88C2×#67—44no—
89C2×#67—41no—
90C2×#67—41no—
91C22×Dic3—21no—
92C22×Dic3—22no—
93C2×D12—22no—
94C2×D12—42no—
95C3×C4⋊C4—24no—
96C3×C4⋊C4—24no—
97C3×C4⋊C4—82no—
98C3×C22⋊C4—44no—
99C3×C42—82no—
100#217—42no—
101(C2×D3)⋊C4—42no—
102(C2×D3)⋊C4—82no—
103(C2×D3)⋊C4—82no—
104C4⋊Dic3—22no—
105C4×Dic3—42no—
106#208—44no—
107C2×C3⋊C8—42no—
108C22×D4—31no—
109C22×D4—61no—
110C23×C4#6711yescharacteristic
111C23×C4#6711yescharacteristic
112C23×C4C22×D311yescharacteristic
113C2×M16—62no—
114#121—62no—
115#121—62no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—31no—
120C2×C4⋊C4C4×D312yes—
121C2×C4⋊C4D1212yes—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—61no—
124C2×C22⋊C4—61no—
125C2×C22⋊C4—62no—
126C2×C22⋊C4—62no—
127C2×C42—22no—
128C2×C42—32no—
129C22⋊C8—62no—
130C22⋊C8—62no—
131#95—22no—
132#95—22no—
133#95—22no—
134#95—62no—
135#95—62no—
136#95—61no—
137#95#6714yes—
138C23×C3—22no—
139C23×C3—22no—
140C23×C3—41no—
141C23×C3C2×C22⋊C411yescharacteristic
142C23×C3#12011yescharacteristic
143C23×C3#12311yescharacteristic
144C22×D3—42no—
145C22×D3—42no—
146C22×D3—41no—
147C22×D3—42no—
148C2×C3×C4—22no—
149C2×C3×C4—22no—
150C2×C3×C4—22no—
151C2×C3×C4—22no—
152C2×C3×C4—42no—
153C2×C3×C4—42no—
154C2×C3×C4—42no—
155C2×C3×C4—42no—
156C2×C3×C4—42no—
157C2×C3×C4—42no—
158C2×C3×C4—44no—
159C2×C3×C4—44no—
160C2×C3×C4—44no—
161C2×C3×C4—81no—
162C2×C3×C4—81no—
163C2×C3×C4C4×D412yes—
164C2×C3×C4#12112yes—
165#67—82no—
166#67—82no—
167C2×Dic3—42no—
168C2×Dic3—42no—
169C2×Dic3—42no—
170C2×Dic3—41no—
171C2×Dic3—42no—
172D12—82no—
173C3⋊C8—82no—
174C24—61no—
175C24C2×#6711yescharacteristic
176C2×D4—62no—
177C2×D4—122no—
178C2×D4—124no—
179C2×D4—121no—
180C2×D4—121no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—24no—
186C22×C4—42no—
187C22×C4—61no—
188C22×C4—62no—
189C22×C4—81no—
190C22×C4—81no—
191C22×C4(C2×D3)⋊C412yes—
192C22×C4(C2×D3)⋊C412yes—
193C22×C4C2×C4×D311yescharacteristic
194C22×C4C2×D1211yescharacteristic
195C22×C4#23512yes—
196C22×C4C2×#6712yes—
197M16—124no—
198C2×C8—122no—
199C4⋊C4—24no—
200C4⋊C4—24no—
201C4⋊C4—62no—
202C4⋊C4—82no—
203C22⋊C4—44no—
204C22⋊C4—122no—
205C22⋊C4—122no—
206C22⋊C4—242no—
207C22⋊C4—242no—
208C42—82no—
209C42—122no—
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#38611yescharacteristic
219C22×C3#41012yes—
220C22×C3#45412yes—
221C2×D3—84no—
222C2×D3—81no—
223C2×D3—81no—
224C2×D3—81no—
225C3×C4—22no—
226C3×C4—22no—
227C3×C4—42no—
228C3×C4—42no—
229C3×C4—84no—
230Dic3—82no—
231Dic3—81no—
232C23—22no—
233C23—22no—
234C23—41no—
235C23—122no—
236C23—122no—
237C23—121no—
238C23—122no—
239C23C2×(C2×D3)⋊C411yescharacteristic
240C23#92911yescharacteristic
241C23#95311yescharacteristic
242D4—242no—
243D4—242no—
244D4—242no—
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—44no—
256C2×C4—44no—
257C2×C4—44no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—122no—
261C2×C4—122no—
262C2×C4—122no—
263C2×C4—121no—
264C2×C4—122no—
265C2×C4C4×#6712yes—
266C2×C4#93012yes—
267C8—242no—
268C2×C3—22no—
269C2×C3—22no—
270C2×C3—41no—
271C2×C3#188811yescharacteristic
272C2×C3#189911yescharacteristic
273C2×C3#191111yescharacteristic
274D3—161no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—44no—
283C22—244no—
284C22—241no—
285C22—241no—
286C22—241no—
287C22#500612yes—
288C22#521311yescharacteristic
289C22#524512yes—
290C4—22no—
291C4—22no—
292C4—42no—
293C4—42no—
294C4—84no—
295C4—242no—
296C4—241no—
297C3#946511yescharacteristic, complemented, Sylow
298C2—22no—
299C2—22no—
300C2—41no—
301C2—481no—
302C2#7159511yescharacteristic
303C2#7165211yescharacteristic
304C2#7166111yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2313C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#2313D3
hypocenterC3#9465
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2313

Complements

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

NH ≅ G/Ntype
1. C3×#2313C2semidirect
3. #71562C2semidirect
5. #67628C2semidirect
18. #4988C4semidirect
19. #4977C4semidirect
23. #2313D3semidirect
297. C3#9465semidirect

Maximal subgroups

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

subgroupindex
C3×#23132
#821802
#715622
#695312
#676282
#94653