Finite Group Databasebeta

← #10690932

Subgroups — #10690932

Order 768.

756 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 139 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×#376C211yesmaximal, characteristic, complemented
2#82080C211yesmaximal, characteristic
3#71531C211yesmaximal, characteristic, complemented
4C2×#4451C212yesmaximal
5#67612C212yesmaximal, complemented
6#9149—31nomaximal, Sylow
7C22×C3×C42—21no—
8C22×C3×C42C2211yescharacteristic
9C2×#930C2211yescharacteristic
10C2×#929—21no—
11C3×#376—24no—
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×#376D311yescharacteristic, 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—24no—
34C2×C3×C42—28no—
35C2×C3×C42—22no—
36C2×C3×C42—22no—
37C2×C3×C42C2×C412yes—
38C2×C3×C42D412yes—
39C2×#217—21no—
40#930—22no—
41#930—22no—
42#929—44no—
43C2×(C2×D3)⋊C4—22no—
44C2×(C2×D3)⋊C4—22no—
45C2×(C2×D3)⋊C4—21no—
46C2×C4⋊Dic3C2×C411yescharacteristic
47C2×C4⋊Dic3C2×C412yes—
48C2×Dic3⋊C4—22no—
49C22×C3⋊C8—22no—
50C3×#95—24no—
51C3×#95—22no—
52C3×#95D414yes—
53C3⋊#95—22no—
54C3⋊#95—21no—
55#830—22no—
56#830—22no—
57C2×#123—61no—
58C2×#121—31no—
59C22×C42—21no—
60C22×C42C2×D311yescharacteristic
61C2×C22⋊C8—32no—
62#390—32no—
63#389—32no—
64#388—62no—
65#386—61no—
66#380—31no—
67#376—24no—
68C2×#95—21no—
69C2×#95C2×D312yes—
70#336—62no—
71#327—64no—
72C24×C3C2×D411yescharacteristic
73C23×D3—21no—
74C22×C3×C4—21no—
75C22×C3×C4—21no—
76C22×C3×C4—24no—
77C22×C3×C4—28no—
78C22×C3×C4—21no—
79C22×C3×C4—24no—
80C22×C3×C4—21no—
81C22×C3×C4—22no—
82C22×C3×C4—22no—
83C22×C3×C4—42no—
84C22×C3×C4—44no—
85C22×C3×C4—42no—
86C22×C3×C4—42no—
87C22×C3×C4C22⋊C412yes—
88C22×C3×C4C22⋊C412yes—
89C22×C3×C4C22×C411yescharacteristic
90C22×C3×C4C2×D411yescharacteristic
91C22×C3×C4C2×D412yes—
92C22×C3×C4#4112yes—
93C2×#67—41no—
94C2×#67—41no—
95C2×#67—42no—
96C2×#67—42no—
97C22×Dic3—21no—
98C22×Dic3—22no—
99C2×D12—22no—
100C2×D12—42no—
101C3×C42—216no—
102C3×C42—24no—
103C3×C42—24no—
104#217—42no—
105(C2×D3)⋊C4—44no—
106(C2×D3)⋊C4—44no—
107(C2×D3)⋊C4—42no—
108C4⋊Dic3—22no—
109C4⋊Dic3—42no—
110Dic3⋊C4—44no—
111C2×C3⋊C8—44no—
112C2×C3⋊C8—42no—
113C22×D4—31no—
114C22×D4—61no—
115C23×C4—22no—
116C23×C4#6711yescharacteristic
117C23×C4#6711yescharacteristic
118C23×C4C22×D311yescharacteristic
119C22×C8—62no—
120#123—124no—
121#121—62no—
122#121—62no—
123C2×C4⋊C4—31no—
124C2×C4⋊C4—32no—
125C2×C4⋊C4—62no—
126C2×C22⋊C4—61no—
127C2×C22⋊C4—62no—
128C2×C22⋊C4—62no—
129C2×C22⋊C4—61no—
130C2×C42—24no—
131C2×C42—28no—
132C2×C42—22no—
133C2×C42—22no—
134C2×C42C4×D312yes—
135C2×C42D1212yes—
136C22⋊C8—62no—
137C22⋊C8—62no—
138#95—24no—
139#95—22no—
140#95—62no—
141#95—61no—
142#95#6714yes—
143C23×C3—22no—
144C23×C3—22no—
145C23×C3C2×C22⋊C411yescharacteristic
146C23×C3#12011yescharacteristic
147C23×C3#12311yescharacteristic
148C23×C3#13512yes—
149C23×C3AGL(1,ℤ/8)12yes—
150C22×D3—41no—
151C22×D3—42no—
152C22×D3—41no—
153C22×D3—41no—
154C22×D3—42no—
155C2×C3×C4—24no—
156C2×C3×C4—24no—
157C2×C3×C4—22no—
158C2×C3×C4—22no—
159C2×C3×C4—22no—
160C2×C3×C4—22no—
161C2×C3×C4—28no—
162C2×C3×C4—22no—
163C2×C3×C4—22no—
164C2×C3×C4—22no—
165C2×C3×C4—42no—
166C2×C3×C4—42no—
167C2×C3×C4—44no—
168C2×C3×C4—48no—
169C2×C3×C4—48no—
170C2×C3×C4—42no—
171C2×C3×C4—44no—
172C2×C3×C4—42no—
173C2×C3×C4#10414yes—
174C2×C3×C4C4×D412yes—
175C2×C3×C4#12112yes—
176#67—82no—
177#67—82no—
178C2×Dic3—41no—
179C2×Dic3—42no—
180C2×Dic3—41no—
181C2×Dic3—41no—
182C2×Dic3—42no—
183C2×Dic3—42no—
184D12—82no—
185C3⋊C8—82no—
186C24—61no—
187C24C2×#6711yescharacteristic
188C2×D4—62no—
189C2×D4—122no—
190C2×D4—121no—
191C2×D4—122no—
192C2×D4—122no—
193C2×D4—121no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—24no—
197C22×C4—28no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—24no—
203C22×C4—42no—
204C22×C4—44no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—62no—
208C22×C4—61no—
209C22×C4(C2×D3)⋊C412yes—
210C22×C4(C2×D3)⋊C412yes—
211C22×C4C2×C4×D311yescharacteristic
212C22×C4C2×D1211yescharacteristic
213C22×C4#23512yes—
214C22×C4C2×#6712yes—
215C2×C8—124no—
216C2×C8—122no—
217C4⋊C4—62no—
218C4⋊C4—122no—
219C4⋊C4—124no—
220C22⋊C4—124no—
221C22⋊C4—122no—
222C22⋊C4—124no—
223C22⋊C4—122no—
224C42—216no—
225C42—24no—
226C42—24no—
227C22×C3—21no—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—21no—
231C22×C3—24no—
232C22×C3—24no—
233C22×C3#38611yescharacteristic
234C22×C3#41012yes—
235C22×C3#41611yescharacteristic
236C22×C3#41811yescharacteristic
237C22×C3C2×#10412yes—
238C22×C3#44912yes—
239C22×C3#48212yes—
240C2×D3—81no—
241C2×D3—81no—
242C2×D3—81no—
243C2×D3—82no—
244C2×D3—82no—
245C3×C4—24no—
246C3×C4—22no—
247C3×C4—22no—
248C3×C4—48no—
249C3×C4—42no—
250Dic3—81no—
251Dic3—82no—
252C23—22no—
253C23—22no—
254C23—122no—
255C23—121no—
256C23—121no—
257C23—121no—
258C23—122no—
259C23C2×(C2×D3)⋊C411yescharacteristic
260C23#92911yescharacteristic
261C23#94912yes—
262C23#95012yes—
263C23#95311yescharacteristic
264D4—242no—
265D4—242no—
266D4—242no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—28no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—48no—
280C2×C4—48no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—44no—
284C2×C4—42no—
285C2×C4—122no—
286C2×C4—121no—
287C2×C4—122no—
288C2×C4—121no—
289C2×C4—121no—
290C2×C4—122no—
291C2×C4#80514yes—
292C2×C4C4×#6712yes—
293C2×C4#93012yes—
294C8—242no—
295C2×C3—22no—
296C2×C3—22no—
297C2×C3#150012yes—
298C2×C3#150512yes—
299C2×C3#187511yescharacteristic
300C2×C3#189611yescharacteristic
301C2×C3#189911yescharacteristic
302D3—161no—
303C22—24no—
304C22—24no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—242no—
310C22—242no—
311C22—241no—
312C22—241no—
313C22—241no—
314C22C2×#80512yes—
315C22#500111yescharacteristic
316C22#500211yescharacteristic
317C22#500312yes—
318C22#500612yes—
319C22#521311yescharacteristic
320C22#523812yes—
321C4—24no—
322C4—22no—
323C4—22no—
324C4—48no—
325C4—42no—
326C4—242no—
327C4—241no—
328C3#914911yescharacteristic, complemented, Sylow
329C2—22no—
330C2—22no—
331C2—481no—
332C2#6943112yes—
333C2#6943212yes—
334C2#7153211yescharacteristic
335C2#7158611yescharacteristic
336C2#7159511yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#376C2
socleC23×C3C2×C22⋊C4
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenterC2×#376D3
hypocenterC3#9149
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#376

Complements

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

NH ≅ G/Ntype
1. C2×C3×#376C2semidirect
3. #71531C2semidirect
5. #67612C2semidirect
18. #4939C4semidirect
19. #4928C4semidirect
23. C2×#376D3semidirect
328. C3#9149semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3762
#820802
#715312
C2×#44512
#676122
#91493