Finite Group Databasebeta

← #10690972

Subgroups — #10690972

Order 768.

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

normal subgroups only

This group has 648 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
1C2×C3×#383C211yesmaximal, characteristic, complemented
2#82080C211yesmaximal, characteristic
3#71531C211yesmaximal, characteristic, complemented
4C2×#4451C212yesmaximal
5#67612C212yesmaximal, complemented
6#9157—31nomaximal, Sylow
7C22×C3×C42C2211yescharacteristic
8C2×#930C2211yescharacteristic
9C2×#929—21no—
10C3×#383—24no—
11C2×C3×#95—21no—
12C2×C3×#95—21no—
13C2×C3×#95C2212yes—
14#5213—21no—
15#5212C2211yescharacteristic
16C2×#830C2212yes—
17#4940—22no—
18#4939C412yescomplemented
19#4928C412yescomplemented
20#4524—22no—
21#4451—24no—
22#2854—31no—
23C2×#383D311yescharacteristic, complemented
24#1817—31no—
25C2×#327—32no—
26#1294—32no—
27C23×C3×C4—22no—
28C23×C3×C4D411yescharacteristic
29C23×C3×C4D411yescharacteristic
30C23×C3×C4C2311yescharacteristic
31C22×#67—21no—
32C22×D12C2×C411yescharacteristic
33C2×C3×C42—22no—
34C2×C3×C42—22no—
35C2×C3×C42C2×C412yes—
36C2×C3×C42D412yes—
37C2×#217—21no—
38#930—22no—
39#930—22no—
40#929—44no—
41C2×(C2×D3)⋊C4—22no—
42C2×(C2×D3)⋊C4—22no—
43C2×(C2×D3)⋊C4—21no—
44C2×C4⋊Dic3C2×C411yescharacteristic
45C2×C4⋊Dic3C2×C412yes—
46C2×Dic3⋊C4—22no—
47C22×C3⋊C8—22no—
48C3×#95—24no—
49C3×#95—24no—
50C3×#95—22no—
51C3×#95D414yes—
52C3⋊#95—22no—
53C3⋊#95—21no—
54#830—22no—
55#830—22no—
56C2×#123—61no—
57C2×#121—31no—
58C22×C42C2×D311yescharacteristic
59C2×C22⋊C8—32no—
60#390—32no—
61#389—32no—
62#388—62no—
63#386—61no—
64#383—24no—
65#380—31no—
66C2×#95—21no—
67C2×#95—21no—
68C2×#95C2×D312yes—
69#336—62no—
70#327—64no—
71C24×C3C2×D411yescharacteristic
72C23×D3—21no—
73C22×C3×C4—21no—
74C22×C3×C4—21no—
75C22×C3×C4—21no—
76C22×C3×C4—21no—
77C22×C3×C4—22no—
78C22×C3×C4—22no—
79C22×C3×C4—24no—
80C22×C3×C4—42no—
81C22×C3×C4—42no—
82C22×C3×C4—44no—
83C22×C3×C4—44no—
84C22×C3×C4—42no—
85C22×C3×C4—42no—
86C22×C3×C4C22⋊C412yes—
87C22×C3×C4C22⋊C412yes—
88C22×C3×C4C22×C411yescharacteristic
89C22×C3×C4C2×D411yescharacteristic
90C22×C3×C4C2×D412yes—
91C22×C3×C4#4112yes—
92C2×#67—41no—
93C2×#67—41no—
94C2×#67—42no—
95C2×#67—42no—
96C22×Dic3—21no—
97C22×Dic3—22no—
98C2×D12—22no—
99C2×D12—42no—
100C3×C42—24no—
101C3×C42—24no—
102#217—42no—
103(C2×D3)⋊C4—44no—
104(C2×D3)⋊C4—44no—
105(C2×D3)⋊C4—42no—
106C4⋊Dic3—22no—
107C4⋊Dic3—42no—
108Dic3⋊C4—44no—
109C2×C3⋊C8—44no—
110C2×C3⋊C8—42no—
111C22×D4—31no—
112C22×D4—61no—
113C23×C4—22no—
114C23×C4#6711yescharacteristic
115C23×C4#6711yescharacteristic
116C23×C4C22×D311yescharacteristic
117C22×C8—62no—
118#123—124no—
119#121—62no—
120#121—62no—
121C2×C4⋊C4—31no—
122C2×C4⋊C4—32no—
123C2×C4⋊C4—62no—
124C2×C22⋊C4—61no—
125C2×C22⋊C4—62no—
126C2×C22⋊C4—62no—
127C2×C22⋊C4—61no—
128C2×C42—22no—
129C2×C42—22no—
130C2×C42C4×D312yes—
131C2×C42D1212yes—
132C22⋊C8—62no—
133C22⋊C8—62no—
134#95—24no—
135#95—24no—
136#95—22no—
137#95—62no—
138#95—61no—
139#95#6714yes—
140C23×C3—22no—
141C23×C3—22no—
142C23×C3C2×C22⋊C411yescharacteristic
143C23×C3#12011yescharacteristic
144C23×C3#12311yescharacteristic
145C23×C3#13512yes—
146C23×C3AGL(1,ℤ/8)12yes—
147C22×D3—41no—
148C22×D3—42no—
149C22×D3—41no—
150C22×D3—41no—
151C22×D3—42no—
152C2×C3×C4—22no—
153C2×C3×C4—22no—
154C2×C3×C4—22no—
155C2×C3×C4—22no—
156C2×C3×C4—22no—
157C2×C3×C4—22no—
158C2×C3×C4—22no—
159C2×C3×C4—22no—
160C2×C3×C4—42no—
161C2×C3×C4—42no—
162C2×C3×C4—42no—
163C2×C3×C4—42no—
164C2×C3×C4—42no—
165C2×C3×C4—42no—
166C2×C3×C4—44no—
167C2×C3×C4—44no—
168C2×C3×C4—44no—
169C2×C3×C4—84no—
170C2×C3×C4—84no—
171C2×C3×C4C4×D412yes—
172C2×C3×C4#12112yes—
173#67—82no—
174#67—82no—
175C2×Dic3—41no—
176C2×Dic3—42no—
177C2×Dic3—41no—
178C2×Dic3—41no—
179C2×Dic3—42no—
180C2×Dic3—42no—
181D12—82no—
182C3⋊C8—82no—
183C24—61no—
184C24C2×#6711yescharacteristic
185C2×D4—62no—
186C2×D4—122no—
187C2×D4—121no—
188C2×D4—122no—
189C2×D4—122no—
190C2×D4—121no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—24no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—62no—
205C22×C4—61no—
206C22×C4(C2×D3)⋊C412yes—
207C22×C4(C2×D3)⋊C412yes—
208C22×C4C2×C4×D311yescharacteristic
209C22×C4C2×D1211yescharacteristic
210C22×C4#23512yes—
211C22×C4C2×#6712yes—
212C2×C8—124no—
213C2×C8—122no—
214C4⋊C4—62no—
215C4⋊C4—122no—
216C4⋊C4—124no—
217C22⋊C4—124no—
218C22⋊C4—124no—
219C22⋊C4—122no—
220C22⋊C4—122no—
221C42—24no—
222C42—24no—
223C22×C3—21no—
224C22×C3—21no—
225C22×C3—21no—
226C22×C3—21no—
227C22×C3—24no—
228C22×C3—24no—
229C22×C3#38611yescharacteristic
230C22×C3#41012yes—
231C22×C3#41611yescharacteristic
232C22×C3#41811yescharacteristic
233C22×C3#42112yes—
234C22×C3#44912yes—
235C22×C3#48212yes—
236C2×D3—81no—
237C2×D3—81no—
238C2×D3—81no—
239C2×D3—82no—
240C2×D3—82no—
241C3×C4—22no—
242C3×C4—22no—
243C3×C4—42no—
244C3×C4—42no—
245C3×C4—84no—
246Dic3—81no—
247Dic3—82no—
248C23—22no—
249C23—22no—
250C23—121no—
251C23—121no—
252C23—121no—
253C23—122no—
254C23—122no—
255C23C2×(C2×D3)⋊C411yescharacteristic
256C23#92911yescharacteristic
257C23#94912yes—
258C23#95012yes—
259C23#95311yescharacteristic
260D4—242no—
261D4—242no—
262D4—242no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—84no—
281C2×C4—84no—
282C2×C4—121no—
283C2×C4—122no—
284C2×C4—121no—
285C2×C4—121no—
286C2×C4—122no—
287C2×C4—122no—
288C2×C4C4×#6712yes—
289C2×C4#93012yes—
290C8—242no—
291C2×C3—22no—
292C2×C3—22no—
293C2×C3#151412yes—
294C2×C3#152112yes—
295C2×C3#187511yescharacteristic
296C2×C3#189711yescharacteristic
297C2×C3#189911yescharacteristic
298D3—161no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—24no—
304C22—24no—
305C22—241no—
306C22—241no—
307C22—241no—
308C22—242no—
309C22—242no—
310C22#500111yescharacteristic
311C22#500211yescharacteristic
312C22#500312yes—
313C22#500512yes—
314C22#500612yes—
315C22#521311yescharacteristic
316C22#523812yes—
317C4—22no—
318C4—22no—
319C4—42no—
320C4—42no—
321C4—84no—
322C4—241no—
323C4—242no—
324C3#915711yescharacteristic, complemented, Sylow
325C2—22no—
326C2—22no—
327C2—481no—
328C2#6956412yes—
329C2#6956512yes—
330C2#7158611yescharacteristic
331C2#7159111yescharacteristic
332C2#7159511yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#383C2
socleC23×C3C2×C22⋊C4
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenterC2×#383D3
hypocenterC3#9157
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#383

Complements

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

NH ≅ G/Ntype
1. C2×C3×#383C2semidirect
3. #71531C2semidirect
5. #67612C2semidirect
18. #4939C4semidirect
19. #4928C4semidirect
23. C2×#383D3semidirect
324. C3#9157semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3832
#820802
#715312
C2×#44512
#676122
#91573