Finite Group Databasebeta

← #10691241

Subgroups — #10691241

Order 768.

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

normal subgroups only

This group has 612 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×#2626C211yesmaximal, characteristic
2#82093C211yesmaximal, characteristic
3#71690C212yesmaximal, complemented
4C2×#4525C211yesmaximal, characteristic
5#67625C212yesmaximal, complemented
6#9201—31nomaximal, Sylow
7C2×C3×C22⋊Q8C2211yescharacteristic
8C22×C3×C22⋊C4C2211yescharacteristic
9C3×#399C412yescomplemented
10C3×#396—22no—
11C3×#393C412yescomplemented
12C3×#385—21no—
13C3×#380C2211yescharacteristic
14C3×#379—21no—
15#5247—21no—
16#5212C2212yes—
17C4×#217—21no—
18C2×#830C2212yes—
19#4951—24no—
20#4538—24no—
21#4525—22no—
22#2626D311yescharacteristic
23#2312—31no—
24#1520—32no—
25C2×#323—31no—
26#1289—32no—
27C25×C3D411yescharacteristic
28C22×C3×Q8C2×C411yescharacteristic
29C23×C3×C4D411yescharacteristic
30C23×C3×C4C2311yescharacteristic
31C3×C22⋊Q8—22no—
32C3×C22⋊Q8—22no—
33C2×C3×C4⋊C4—21no—
34C2×C3×C4⋊C4—22no—
35C2×C3×C4⋊C4—21no—
36C2×C3×C4⋊C4C2×C411yescharacteristic
37C2×C3×C22⋊C4—22no—
38C2×C3×C22⋊C4—22no—
39C2×C3×C22⋊C4—21no—
40C2×C3×C22⋊C4—21no—
41C2×C3×C22⋊C4—22no—
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4C2×C412yes—
44C2×C3×C22⋊C4D412yes—
45C2×#217—22no—
46C2×#217—22no—
47C2×#217—22no—
48C2×C4⋊Dic3—22no—
49C2×C4×Dic3—22no—
50C22×C3⋊C8—22no—
51C3×#95—22no—
52C3×#95C2×C412yes—
53C3⋊#95—22no—
54C3⋊#95D414yes—
55#830—22no—
56#830—22no—
57C2×C22⋊Q8C2×D311yescharacteristic
58C22×C22⋊C4C2×D311yescharacteristic
59C2×C22⋊C8—32no—
60#399Dic312yescomplemented
61#396—22no—
62#393Dic312yescomplemented
63#388—64no—
64#385—21no—
65#380—32no—
66#380C2×D311yescharacteristic
67#379—21no—
68#379—61no—
69C4×C22⋊C4—61no—
70#328—64no—
71#323—62no—
72C24×C3—21no—
73C24×C3—21no—
74C24×C3—22no—
75C24×C3—22no—
76C24×C3—81no—
77C24×C3—81no—
78C24×C3C22⋊C412yes—
79C24×C3C2×D411yescharacteristic
80C2×C3×Q8—22no—
81C2×C3×Q8—81no—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—21no—
85C22×C3×C4—22no—
86C22×C3×C4—21no—
87C22×C3×C4—81no—
88C22×C3×C4—81no—
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—22no—
96C3×C4⋊C4—42no—
97C3×C4⋊C4—81no—
98C3×C4⋊C4—82no—
99C3×C22⋊C4—24no—
100C3×C22⋊C4—24no—
101C3×C22⋊C4—41no—
102C3×C22⋊C4—41no—
103C3×C22⋊C4—42no—
104C3×C22⋊C4—82no—
105C3×C22⋊C4—82no—
106#217—42no—
107#217—42no—
108#217—82no—
109#217—82no—
110C4⋊Dic3—82no—
111C4×Dic3—82no—
112C2×C3⋊C8—42no—
113C2×C3⋊C8—82no—
114C25#6711yescharacteristic
115C22×Q8C2×Dic311yescharacteristic
116C23×C4#6711yescharacteristic
117C23×C4C22×D311yescharacteristic
118C22×C8—62no—
119C22⋊Q8—22no—
120C22⋊Q8—22no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—62no—
125C2×C4⋊C4C2×Dic311yescharacteristic
126C2×C22⋊C4—22no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—62no—
133C2×C22⋊C4—62no—
134C2×C22⋊C4—62no—
135C2×C22⋊C4C2×Dic312yes—
136C2×C22⋊C4#6712yes—
137C2×C42—62no—
138C22⋊C8—62no—
139C22⋊C8—62no—
140#95—22no—
141#95—34no—
142#95—62no—
143#95C2×Dic312yes—
144C23×C3—22no—
145C23×C3—22no—
146C23×C3—22no—
147C23×C3—41no—
148C23×C3—41no—
149C23×C3—42no—
150C23×C3—42no—
151C23×C3—42no—
152C23×C3—42no—
153C23×C3—42no—
154C23×C3—42no—
155C23×C3—81no—
156C23×C3—81no—
157C23×C3—84no—
158C23×C3—84no—
159C23×C3C2×C22⋊C411yescharacteristic
160C23×C3#12011yescharacteristic
161C23×C3#12311yescharacteristic
162C23×C3#13512yes—
163C23×C3#13712yes—
164C3×Q8—82no—
165C2×C3×C4—22no—
166C2×C3×C4—42no—
167C2×C3×C4—42no—
168C2×C3×C4—41no—
169C2×C3×C4—41no—
170C2×C3×C4—42no—
171C2×C3×C4—42no—
172C2×C3×C4—42no—
173C2×C3×C4—42no—
174C2×C3×C4—42no—
175C2×C3×C4—42no—
176C2×C3×C4—41no—
177C2×C3×C4—41no—
178C2×C3×C4—42no—
179C2×C3×C4—81no—
180C2×C3×C4C4×D412yes—
181C2×C3×C4#12112yes—
182C2×Dic3—44no—
183C2×Dic3—44no—
184C2×Dic3—44no—
185C3⋊C8—82no—
186C24—21no—
187C24—21no—
188C24—22no—
189C24—22no—
190C24—81no—
191C24—81no—
192C24#21712yes—
193C24C2×#6711yescharacteristic
194C2×Q8—22no—
195C2×Q8—81no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—64no—
202C22×C4—81no—
203C22×C4—81no—
204C22×C4#21712yes—
205C22×C4D3×D412yes—
206C22×C4#23712yes—
207C22×C4C22×Dic311yescharacteristic
208C22×C4C2×#6711yescharacteristic
209C2×C8—122no—
210C2×C8—242no—
211C4⋊C4—22no—
212C4⋊C4—42no—
213C4⋊C4—81no—
214C4⋊C4—82no—
215C4⋊C4—242no—
216C22⋊C4—24no—
217C22⋊C4—24no—
218C22⋊C4—41no—
219C22⋊C4—41no—
220C22⋊C4—42no—
221C22⋊C4—82no—
222C22⋊C4—82no—
223C22⋊C4—122no—
224C22⋊C4—122no—
225C22⋊C4—242no—
226C22⋊C4—242no—
227C42—242no—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—21no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3—21no—
234C22×C3—42no—
235C22×C3—42no—
236C22×C3—44no—
237C22×C3—42no—
238C22×C3—42no—
239C22×C3—42no—
240C22×C3—82no—
241C22×C3—84no—
242C22×C3—84no—
243C22×C3#38611yescharacteristic
244C22×C3#41611yescharacteristic
245C22×C3#41911yescharacteristic
246C22×C3#45212yes—
247C22×C3#48512yes—
248C3×C4—22no—
249C3×C4—22no—
250C3×C4—42no—
251C3×C4—82no—
252C3×C4—81no—
253C3×C4—81no—
254Dic3—84no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—41no—
259C23—41no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—81no—
267C23—81no—
268C23—84no—
269C23—84no—
270C23#93511yescharacteristic
271C23#93711yescharacteristic
272C23#94212yes—
273C23#95012yes—
274C23C2×#21711yescharacteristic
275Q8—82no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—81no—
291C2×C4—124no—
292C2×C4—124no—
293C2×C4—124no—
294C2×C4D4×Dic312yes—
295C2×C4#93912yes—
296C8—242no—
297C2×C3—22no—
298C2×C3—21no—
299C2×C3—21no—
300C2×C3—22no—
301C2×C3—82no—
302C2×C3#187311yescharacteristic
303C2×C3#191111yescharacteristic
304C2×C3#191311yescharacteristic
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—42no—
312C22—42no—
313C22—44no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—82no—
318C22—84no—
319C22—84no—
320C22#504512yes—
321C22#504712yes—
322C22#522011yescharacteristic
323C22#522511yescharacteristic
324C22#523311yescharacteristic
325C4—22no—
326C4—22no—
327C4—42no—
328C4—82no—
329C4—81no—
330C4—81no—
331C4—244no—
332C3#920111yescharacteristic, complemented, Sylow
333C2—21no—
334C2—22no—
335C2—21no—
336C2—22no—
337C2—82no—
338C2#7169311yescharacteristic
339C2#7184811yescharacteristic
340C2#7185311yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2626C2
socleC22×C3#386
prime radicalC25×C3D4
cosocleC23×C3×C4C23
hypercenter#2626D3
hypocenterC3#9201
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2626

Complements

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

NH ≅ G/Ntype
3. #71690C2semidirect
5. #67625C2semidirect
9. C3×#399C4semidirect
11. C3×#393C4semidirect
60. #399Dic3semidirect
62. #393Dic3semidirect
332. C3#9201semidirect

Maximal subgroups

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

subgroupindex
C3×#26262
#820932
#716902
C2×#45252
#676252
#92013