Finite Group Databasebeta

← #10691266

Subgroups — #10691266

Order 768.

578 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 119 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×#2618C211yesmaximal, characteristic
2#82155C211yesmaximal, characteristic, complemented
3#71675C212yesmaximal
4C2×#4528C211yesmaximal, characteristic, complemented
5#67625C212yesmaximal, complemented
6#9206—31nomaximal, Sylow
7C2×C3×#123—21no—
8C2×C3×#121C2211yescharacteristic
9C22×C3×C4⋊C4C2211yescharacteristic
10C3×#398C412yescomplemented
11C3×#396—22no—
12C3×#394C412yescomplemented
13C3×#386—21no—
14C3×#380C2211yescharacteristic
15#5212C2212yes—
16#5211—21no—
17C4×#217—21no—
18C2×#830C2212yes—
19#4980—24no—
20#4537—24no—
21#4528—22no—
22#2618D311yescharacteristic
23#2313—31no—
24#1519—32no—
25C2×#324—31no—
26#1289—32no—
27C22×C3×D4—21no—
28C22×C3×D4C2×C411yescharacteristic
29C23×C3×C4D411yescharacteristic
30C23×C3×C4D411yescharacteristic
31C23×C3×C4C2311yescharacteristic
32C3×#123—44no—
33C3×#121—22no—
34C3×#121—22no—
35C2×C3×C4⋊C4—22no—
36C2×C3×C4⋊C4—22no—
37C2×C3×C4⋊C4—22no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4C2×C411yescharacteristic
40C2×C3×C4⋊C4C2×C412yes—
41C2×C3×C4⋊C4D412yes—
42C2×C3×C22⋊C4—21no—
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—22no—
46C2×#217—22no—
47C2×C4⋊Dic3—22no—
48C2×C4×Dic3—22no—
49C22×C3⋊C8—22no—
50C3×#95—22no—
51C3×#95C2×C412yes—
52C3⋊#95—22no—
53C3⋊#95—24no—
54C3⋊#95D414yes—
55#830—22no—
56#830—22no—
57C2×#123—21no—
58C2×#121C2×D311yescharacteristic
59C22×C4⋊C4C2×D311yescharacteristic
60C2×C22⋊C8—32no—
61#398Dic312yescomplemented
62#396—22no—
63#394Dic312yescomplemented
64#387—64no—
65#386—21no—
66#381—61no—
67#380—32no—
68#380C2×D311yescharacteristic
69C4×C22⋊C4—61no—
70#327—64no—
71#324—62no—
72C24×C3—21no—
73C24×C3C2×D411yescharacteristic
74C2×C3×D4—22no—
75C2×C3×D4—41no—
76C2×C3×D4—41no—
77C2×C3×D4—42no—
78C2×C3×D4—42no—
79C2×C3×D4—81no—
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4—21no—
83C22×C3×C4—22no—
84C22×C3×C4—22no—
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4—22no—
88C22×C3×C4—81no—
89C22×C3×C4—81no—
90C22×C3×C4C22⋊C412yes—
91C22×C3×C4C22⋊C412yes—
92C22×C3×C4C22×C411yescharacteristic
93C22×C3×C4C2×D412yes—
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4#4112yes—
96C22×Dic3—24no—
97C3×C4⋊C4—24no—
98C3×C4⋊C4—24no—
99C3×C4⋊C4—22no—
100C3×C4⋊C4—44no—
101C3×C4⋊C4—81no—
102C3×C22⋊C4—44no—
103C3×C22⋊C4—42no—
104C3×C22⋊C4—42no—
105C3×C22⋊C4—82no—
106#217—42no—
107#217—42no—
108C4⋊Dic3—82no—
109C4×Dic3—82no—
110C2×C3⋊C8—42no—
111C2×C3⋊C8—82no—
112C22×D4—21no—
113C22×D4C2×Dic311yescharacteristic
114C23×C4#6711yescharacteristic
115C23×C4#6711yescharacteristic
116C23×C4C22×D311yescharacteristic
117C22×C8—62no—
118#123—44no—
119#121—22no—
120#121—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—62no—
126C2×C4⋊C4C2×Dic311yescharacteristic
127C2×C4⋊C4C2×Dic312yes—
128C2×C4⋊C4#6712yes—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—62no—
134C2×C42—62no—
135C22⋊C8—62no—
136C22⋊C8—62no—
137#95—22no—
138#95—34no—
139#95—62no—
140#95—64no—
141#95C2×Dic312yes—
142C23×C3—41no—
143C23×C3—41no—
144C23×C3—42no—
145C23×C3—42no—
146C23×C3—41no—
147C23×C3—41no—
148C23×C3—41no—
149C23×C3C2×C22⋊C411yescharacteristic
150C23×C3#12011yescharacteristic
151C23×C3#12311yescharacteristic
152C23×C3#13512yes—
153C23×C3AGL(1,ℤ/8)12yes—
154C3×D4—82no—
155C3×D4—82no—
156C3×D4—82no—
157C2×C3×C4—22no—
158C2×C3×C4—22no—
159C2×C3×C4—22no—
160C2×C3×C4—22no—
161C2×C3×C4—42no—
162C2×C3×C4—42no—
163C2×C3×C4—41no—
164C2×C3×C4—41no—
165C2×C3×C4—42no—
166C2×C3×C4—42no—
167C2×C3×C4—42no—
168C2×C3×C4—42no—
169C2×C3×C4—42no—
170C2×C3×C4—42no—
171C2×C3×C4—41no—
172C2×C3×C4—41no—
173C2×C3×C4—42no—
174C2×C3×C4—41no—
175C2×C3×C4—81no—
176C2×C3×C4C4×D412yes—
177C2×C3×C4#12112yes—
178C2×Dic3—44no—
179C2×Dic3—44no—
180C2×Dic3—44no—
181C3⋊C8—82no—
182C24—21no—
183C24C2×#6711yescharacteristic
184C2×D4—22no—
185C2×D4—42no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4—42no—
189C2×D4—81no—
190C22×C4—22no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—64no—
199C22×C4—81no—
200C22×C4—81no—
201C22×C4#21712yes—
202C22×C4#21712yes—
203C22×C4D3×D412yes—
204C22×C4#23712yes—
205C22×C4C22×Dic311yescharacteristic
206C22×C4C2×#6711yescharacteristic
207C2×C8—122no—
208C2×C8—242no—
209C4⋊C4—24no—
210C4⋊C4—24no—
211C4⋊C4—22no—
212C4⋊C4—44no—
213C4⋊C4—81no—
214C4⋊C4—242no—
215C22⋊C4—44no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—82no—
219C22⋊C4—122no—
220C22⋊C4—122no—
221C42—242no—
222C22×C3—21no—
223C22×C3—21no—
224C22×C3—21no—
225C22×C3—21no—
226C22×C3—21no—
227C22×C3—21no—
228C22×C3—42no—
229C22×C3—42no—
230C22×C3—82no—
231C22×C3—82no—
232C22×C3—81no—
233C22×C3—81no—
234C22×C3—81no—
235C22×C3#38611yescharacteristic
236C22×C3#41611yescharacteristic
237C22×C3#41811yescharacteristic
238C22×C3#44912yes—
239C22×C3#48212yes—
240C3×C4—22no—
241C3×C4—22no—
242C3×C4—42no—
243C3×C4—42no—
244C3×C4—82no—
245C3×C4—81no—
246Dic3—84no—
247C23—41no—
248C23—41no—
249C23—42no—
250C23—42no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23#93212yes—
255C23#93511yescharacteristic
256C23#93711yescharacteristic
257C23#95012yes—
258C23C2×#21711yescharacteristic
259D4—82no—
260D4—82no—
261D4—82no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—81no—
281C2×C4—124no—
282C2×C4—124no—
283C2×C4—124no—
284C2×C4D4×Dic312yes—
285C2×C4#93912yes—
286C8—242no—
287C2×C3—22no—
288C2×C3—21no—
289C2×C3—21no—
290C2×C3—22no—
291C2×C3—161no—
292C2×C3#187511yescharacteristic
293C2×C3#191211yescharacteristic
294C2×C3#191311yescharacteristic
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—42no—
302C22—42no—
303C22—82no—
304C22—82no—
305C22—81no—
306C22—81no—
307C22—81no—
308C22#503512yes—
309C22#503712yes—
310C22#521511yescharacteristic
311C22#522011yescharacteristic
312C22#523311yescharacteristic
313C4—22no—
314C4—22no—
315C4—42no—
316C4—42no—
317C4—82no—
318C4—81no—
319C4—244no—
320C3#920611yescharacteristic, complemented, Sylow
321C2—21no—
322C2—22no—
323C2—21no—
324C2—22no—
325C2—161no—
326C2#7167811yescharacteristic
327C2#7184811yescharacteristic
328C2#7186411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2618C2
socleC22×C3#386
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenter#2618D3
hypocenterC3#9206
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2618

Complements

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

NH ≅ G/Ntype
2. #82155C2semidirect
4. C2×#4528C2semidirect
5. #67625C2semidirect
10. C3×#398C4semidirect
12. C3×#394C4semidirect
61. #398Dic3semidirect
63. #394Dic3semidirect
320. C3#9206semidirect

Maximal subgroups

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

subgroupindex
C3×#26182
#821552
#716752
C2×#45282
#676252
#92063