Finite Group Databasebeta

← #10773129

Subgroups — #10773129

Order 768.

834 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 834 conjugacy classes of subgroups — too many to draw. Only its 163 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
1#86270C211yesmaximal, characteristic, complemented
2C3×#3131C211yesmaximal, characteristic
3#83321C212yesmaximal
4#71834C211yesmaximal, characteristic, complemented
5#71236C214yesmaximal, complemented
6#71204C214yesmaximal, complemented
7#69738C212yesmaximal, complemented
8#18698—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C3×#532C2211yescharacteristic
11#5721C2214yes—
12C3⋊#564—22no—
13#5625—24no—
14#5570—22no—
15#5567—22no—
16#5518—22no—
17#5470C2212yes—
18C3×#488—22no—
19C3×#487C2212yes—
20C3×#486C2212yes—
21C3×#416—22no—
22C2×C3×C4×C8C2211yescharacteristic
23#5091C2214yes—
24#5075C2214yes—
25#5042—24no—
26#5019—22no—
27#5007—22no—
28#4924C2212yes—
29#4547C2214yes—
30#4456C2218yes—
31#3556—31no—
32#3254—32no—
33#3131D311yescharacteristic
34#1636—34no—
35#1626—34no—
36#1561—31no—
37#1493—32no—
38C2×C3×#41—21no—
39C22×C3×C8—21no—
40C3×C4⋊Q8C2311yescharacteristic
41C3×C4⋊Q8C2312yes—
42C3×#127C2311yescharacteristic
43C3×#125D412yes—
44C3×#124—21no—
45C3×C22⋊Q8—22no—
46C3×#121—22no—
47C3×C4×Q8—22no—
48C3×C4×D4—22no—
49C3×#117D412yes—
50C2×C3×C42C2311yescharacteristic
51Q8×Dic3—24no—
52#944—24no—
53C2×#214—24no—
54#926—22no—
55#924—22no—
56#924D414yes—
57C2×#208—22no—
58#890—28no—
59#890—24no—
60#889—28no—
61#889—28no—
62#887—24no—
63#879—28no—
64#870—22no—
65#870—22no—
66#869—22no—
67#869D412yes—
68#869D412yes—
69C4×Dic6—24no—
70C4×Dic6—22no—
71C4×Dic6C2314yes—
72C3×#103—24no—
73C3×#102—24no—
74C3×C4×C8—21no—
75C3×C4×C8C2312yes—
76#832—24no—
77#810—24no—
78#807—24no—
79C4⋊(C3⋊C8)—22no—
80C4⋊(C3⋊C8)C2314yes—
81#564—62no—
82#557—34no—
83#557—62no—
84#557—62no—
85#541—64no—
86#541—62no—
87#533—32no—
88#533C2×D311yescharacteristic
89#532C2×D311yescharacteristic
90#488—22no—
91#487C2×D312yes—
92#486C2×D312yes—
93#477—34no—
94#474—62no—
95#468—64no—
96#461—62no—
97#460—34no—
98#423—32no—
99#416—22no—
100C2×C4×C8C2×D311yescharacteristic
101#332—34no—
102Q8⋊C8—38no—
103C3×#41—44no—
104C2×C3×Q8—22no—
105C2×C3×Q8—21no—
106C2×C3×D4—22no—
107C2×C3×D4—21no—
108C2×C3×D4—41no—
109C22×C3×C4—41no—
110C22×C3×C4C2×D412yes—
111C22×C3×C4C2×D411yescharacteristic
112C2×Dic6—22no—
113C2×Dic6C2×D414yes—
114C2×C3×C8—22no—
115C2×C3×C8—42no—
116C3×C4⋊C4—24no—
117C3×C4⋊C4—22no—
118C3×C4⋊C4—22no—
119C3×C4⋊C4C2×D414yes—
120C3×C22⋊C4—22no—
121C3×C22⋊C4—42no—
122C3×C42—22no—
123C3×C42C2×D412yes—
124C3×C42C2×D411yescharacteristic
125C3×C42C2411yescharacteristic
126#217—24no—
127#214—84no—
128C4⋊Dic3—28no—
129C4⋊Dic3—22no—
130C4⋊Dic3—22no—
131C4⋊Dic3—22no—
132C4⋊Dic3C2×D414yes—
133Dic3⋊C4—28no—
134Dic3⋊C4—28no—
135Dic3⋊C4—24no—
136Dic3⋊C4—24no—
137C4×Dic3—28no—
138C4×Dic3—24no—
139#208—82no—
140C2×C3⋊C8—24no—
141C2×#41—21no—
142C2×SD16—64no—
143C2×M16—62no—
144C22×C8—21no—
145C4⋊Q8—32no—
146C4⋊Q8—32no—
147C4⋊Q8—62no—
148C4⋊Q8—64no—
149C4⋊Q8C22×D311yescharacteristic
150C4⋊Q8C22×D312yes—
151#127C22×D311yescharacteristic
152#126—68no—
153#125—62no—
154#125—64no—
155#125—68no—
156#125—68no—
157#125—68no—
158#125—62no—
159#125#6712yes—
160#124—21no—
161C22⋊Q8—22no—
162C22⋊Q8—34no—
163C22⋊Q8—62no—
164#121—22no—
165C4×Q8—22no—
166C4×Q8—34no—
167C4×Q8—64no—
168C4×Q8—64no—
169C4×Q8—64no—
170C4×Q8—62no—
171C4×D4—22no—
172#117—62no—
173#117#6712yes—
174C2×C42C22×D311yescharacteristic
175#107—64no—
176C4⋊C8—34no—
177C4⋊C8—62no—
178#103—24no—
179#103—64no—
180#102—24no—
181#102—64no—
182C4×C8—21no—
183C4×C8C22×D312yes—
184C23×C3—41no—
185C23×C3#12011yescharacteristic
186C3×Q8—42no—
187C3×Q8—42no—
188C3×Q8—42no—
189C3×D4—42no—
190C3×D4—42no—
191C3×D4—42no—
192C3×D4—82no—
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4—22no—
196C2×C3×C4—24no—
197C2×C3×C4—41no—
198C2×C3×C4—82no—
199C2×C3×C4#12012yes—
200C2×C3×C4#12011yescharacteristic
201C2×C3×C4C22×D412yes—
202C2×C3×C4C22×D411yescharacteristic
203C2×Dic3—24no—
204C2×Dic3—24no—
205Dic6—24no—
206Dic6—24no—
207Dic6—44no—
208C3×C8—42no—
209C3⋊C8—44no—
210#41—44no—
211C2×Q8—22no—
212C2×Q8—21no—
213C2×Q8—34no—
214C2×Q8—62no—
215C2×D4—22no—
216C2×D4—21no—
217C2×D4—41no—
218C22×C4—41no—
219C22×C4D3×D412yes—
220C22×C4C2×#6711yescharacteristic
221SD16—244no—
222M16—242no—
223C2×C8—22no—
224C2×C8—42no—
225C2×C8—64no—
226C4⋊C4—24no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—34no—
230C4⋊C4—68no—
231C4⋊C4—68no—
232C4⋊C4—68no—
233C4⋊C4—62no—
234C4⋊C4—62no—
235C4⋊C4—62no—
236C4⋊C4—64no—
237C4⋊C4—64no—
238C4⋊C4C2×#6714yes—
239C22⋊C4—22no—
240C22⋊C4—42no—
241C22⋊C4—64no—
242C42—22no—
243C42—68no—
244C42—64no—
245C42D3×D412yes—
246C42C2×#6711yescharacteristic
247C42C23×D311yescharacteristic
248C22×C3—22no—
249C22×C3—21no—
250C22×C3—82no—
251C22×C3—81no—
252C22×C3C2×#12011yescharacteristic
253C3×C4—22no—
254C3×C4—21no—
255C3×C4—44no—
256C3×C4—81no—
257C3×C4#45412yes—
258C3×C4#57712yes—
259C3×C4#57812yes—
260Dic3—44no—
261Dic3—44no—
262C23—41no—
263C23#93711yescharacteristic
264Q8—42no—
265Q8—42no—
266Q8—42no—
267Q8—64no—
268Q8—64no—
269Q8—124no—
270D4—42no—
271D4—42no—
272D4—42no—
273D4—82no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—24no—
278C2×C4—41no—
279C2×C4—64no—
280C2×C4—64no—
281C2×C4—82no—
282C2×C4#93712yes—
283C2×C4#93711yescharacteristic
284C2×C4C2×D3×D412yes—
285C2×C4C22×#6711yescharacteristic
286C8—42no—
287C8—124no—
288C2×C3—41no—
289C2×C3—161no—
290C2×C3#301312yes—
291C2×C3C2×#45411yescharacteristic
292C22—21no—
293C22—22no—
294C22—81no—
295C22—82no—
296C22C2×#93711yescharacteristic
297C4—21no—
298C4—22no—
299C4—44no—
300C4—81no—
301C4—124no—
302C4—124no—
303C4#506212yes—
304C4#577012yes—
305C4#577112yes—
306C3#1869811yescharacteristic, complemented, Sylow
307C2—41no—
308C2—161no—
309C2C2×#506211yescharacteristic
310C2#8391312yes—
subgroupG/H
centerC22C2×#937
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3131C2
socleC22×C3C2×#120
prime radicalC3×#532C22
hypercenter#3131D3
hypocenterC3#18698
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3131

Complements

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

NH ≅ G/Ntype
1. #86270C2semidirect
4. #71834C2semidirect
5. #71236C2semidirect
6. #71204C2semidirect
7. #69738C2semidirect
306. C3#18698semidirect

Maximal subgroups

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

subgroupindex
#862702
C3×#31312
#833212
#718342
#712362
#712042
#697382
#186983