Finite Group Databasebeta

← #10690126

Subgroups — #10690126

Order 768.

578 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 115 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×#2379C211yesmaximal, characteristic
2C2×#5060C211yesmaximal, characteristic
3C2×#4540C211yesmaximal, characteristic
4#69555C212yesmaximal, complemented
5#67630C212yesmaximal, complemented
6#9003—31nomaximal, Sylow
7C2×C3×#124C2211yescharacteristic
8C22×#216—21no—
9C22×#214—21no—
10C2×#869C2211yescharacteristic
11C3×#391—21no—
12C3×#388—22no—
13C3×#386—21no—
14C3×#385—22no—
15C3×#382C2212yes—
16#5060—22no—
17#5060C414yescomplemented
18C2×#803C2212yes—
19C2×#803C2211yescharacteristic
20#4540—22no—
21#4529—22no—
22#4529—22no—
23C2×#497—31no—
24#2379D311yescharacteristic
25#1993—32no—
26C2×#329—31no—
27#1295—32no—
28C22×C3×Q8D411yescharacteristic
29C22×C3×D4D411yescharacteristic
30C23×C3×C4—21no—
31C22×Dic6—21no—
32C3×#124—22no—
33C3×#124C2×C412yes—
34C3×#124D412yes—
35C2×C3×C4⋊C4—21no—
36C2×C3×C4⋊C4D412yes—
37C2×C3×C4⋊C4D412yes—
38C2×C3×C22⋊C4—22no—
39C2×C3×C22⋊C4—22no—
40C2×C3×C22⋊C4—21no—
41C2×C3×C42—21no—
42C2×C3×C42C2311yescharacteristic
43C2×#216—24no—
44C2×#216—42no—
45C2×#216—42no—
46C2×#214—24no—
47C2×#214—42no—
48C2×#214—42no—
49C2×C4⋊Dic3—22no—
50C22×C3⋊C8—21no—
51C22×C3⋊C8—22no—
52#869—22no—
53#869—21no—
54#869C2×C412yescomplemented
55C3×#95—22no—
56C3×#95—22no—
57C3×#95—22no—
58#803—21no—
59#803—22no—
60#803—22no—
61#803C2×C412yescomplemented
62C22×Q16—61no—
63C22×SD16—61no—
64C2×C4⋊Q8—31no—
65C2×#124C2×D311yescharacteristic
66#497—34no—
67#497—62no—
68C2×#97—31no—
69C2×#97—32no—
70#391—21no—
71#388—22no—
72#386—21no—
73#385—22no—
74#382C2×D312yes—
75#340—62no—
76#340—62no—
77#329—62no—
78C24×C3—21no—
79C2×C3×Q8—21no—
80C2×C3×Q8—41no—
81C2×C3×Q8—41no—
82C2×C3×Q8C22⋊C412yes—
83C2×C3×D4—21no—
84C2×C3×D4—41no—
85C2×C3×D4—41no—
86C2×C3×D4C22⋊C412yes—
87C22×C3×C4—21no—
88C22×C3×C4—22no—
89C22×C3×C4—22no—
90C22×C3×C4—81no—
91C22×C3×C4—82no—
92C22×C3×C4C2×D411yescharacteristic
93C22×C3×C4C2×D412yes—
94C22×Dic3—41no—
95C2×Dic6—22no—
96C2×Dic6—41no—
97C2×Dic6—41no—
98C2×Dic6—41no—
99C2×Dic6—41no—
100C2×Dic6—41no—
101C3×C4⋊C4—42no—
102C3×C4⋊C4—42no—
103C3×C4⋊C4—81no—
104C3×C22⋊C4—24no—
105C3×C22⋊C4—24no—
106C3×C22⋊C4—81no—
107C3×C22⋊C4—82no—
108C3×C42—81no—
109C3×C42C22×C411yescharacteristic
110C3×C42C2×D411yescharacteristic
111C3×C42#4112yes—
112#216—48no—
113#214—48no—
114C4⋊Dic3—24no—
115C4⋊Dic3—42no—
116C2×C3⋊C8—22no—
117C2×C3⋊C8—22no—
118C2×C3⋊C8—22no—
119C2×C3⋊C8—24no—
120C2×C3⋊C8—42no—
121C2×C3⋊C8—42no—
122C22×Q8—61no—
123C22×Q8#6711yescharacteristic
124C22×D4#6711yescharacteristic
125C23×C4—21no—
126C2×Q16—64no—
127C2×Q16—122no—
128C2×Q16—122no—
129C2×SD16—64no—
130C2×SD16—122no—
131C2×SD16—122no—
132C22×C8—61no—
133C22×C8—62no—
134C4⋊Q8—32no—
135C4⋊Q8—61no—
136C4⋊Q8—62no—
137#124—22no—
138#124C4×D312yes—
139#124D1212yes—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—62no—
142C2×C4⋊C4#6712yes—
143C2×C4⋊C4#6712yes—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—21no—
147C2×C42—21no—
148C2×C42C22×D311yescharacteristic
149#97—32no—
150#97—61no—
151#97—62no—
152#97—62no—
153#95—22no—
154#95—22no—
155#95—22no—
156C23×C3—22no—
157C23×C3—41no—
158C23×C3—41no—
159C23×C3—42no—
160C23×C3—42no—
161C23×C3#12011yescharacteristic
162C3×Q8—42no—
163C3×Q8—81no—
164C3×D4—42no—
165C3×D4—81no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—22no—
169C2×C3×C4—22no—
170C2×C3×C4—22no—
171C2×C3×C4—41no—
172C2×C3×C4—41no—
173C2×C3×C4—42no—
174C2×C3×C4—42no—
175C2×C3×C4—42no—
176C2×C3×C4—42no—
177C2×C3×C4—42no—
178C2×C3×C4—42no—
179C2×C3×C4—84no—
180C2×C3×C4C2×C22⋊C411yescharacteristic
181C2×C3×C4C4×D412yes—
182C2×C3×C4#12112yes—
183C2×C3×C4#12311yescharacteristic
184C2×Dic3—42no—
185C2×Dic3—81no—
186C2×Dic3—81no—
187Dic6—42no—
188Dic6—42no—
189Dic6—81no—
190Dic6—81no—
191C3⋊C8—24no—
192C3⋊C8—44no—
193C24—21no—
194C2×Q8—21no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8—62no—
198C2×Q8—121no—
199C2×Q8—121no—
200C2×Q8—121no—
201C2×Q8—121no—
202C2×Q8—121no—
203C2×Q8(C2×D3)⋊C412yes—
204C2×D4—21no—
205C2×D4—41no—
206C2×D4—41no—
207C2×D4(C2×D3)⋊C412yes—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—81no—
212C22×C4—82no—
213C22×C4—121no—
214C22×C4C2×#6711yescharacteristic
215C22×C4C2×#6712yes—
216Q16—128no—
217SD16—128no—
218C2×C8—62no—
219C2×C8—62no—
220C2×C8—62no—
221C2×C8—64no—
222C2×C8—122no—
223C2×C8—122no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—64no—
227C4⋊C4—81no—
228C4⋊C4—122no—
229C22⋊C4—24no—
230C22⋊C4—24no—
231C22⋊C4—81no—
232C22⋊C4—82no—
233C42—81no—
234C42C2×C4×D311yescharacteristic
235C42C2×D1211yescharacteristic
236C42#23512yes—
237C22×C3—44no—
238C22×C3—42no—
239C22×C3—81no—
240C22×C3—81no—
241C22×C3—82no—
242C22×C3#38611yescharacteristic
243C22×C3#45311yescharacteristic
244C22×C3#45311yescharacteristic
245C22×C3#45512yes—
246C22×C3#45612yes—
247C3×C4—22no—
248C3×C4—22no—
249C3×C4—21no—
250C3×C4—21no—
251C3×C4—42no—
252C3×C4—82no—
253C3×C4—82no—
254Dic3—82no—
255C23—22no—
256C23—41no—
257C23—41no—
258C23—42no—
259C23—42no—
260C23#95311yescharacteristic
261Q8—42no—
262Q8—81no—
263Q8—122no—
264Q8—122no—
265Q8—241no—
266Q8—241no—
267D4—42no—
268D4—81no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—84no—
283C2×C4—122no—
284C2×C4—241no—
285C2×C4—241no—
286C2×C4C2×(C2×D3)⋊C411yescharacteristic
287C2×C4C4×#6712yes—
288C2×C4#92911yescharacteristic
289C2×C4#93012yes—
290C8—64no—
291C8—124no—
292C2×C3—82no—
293C2×C3#165911yescharacteristic
294C2×C3#166011yescharacteristic
295C2×C3#166112yes—
296C2×C3#188212yes—
297C2×C3#188711yescharacteristic
298C22—42no—
299C22—44no—
300C22—81no—
301C22—82no—
302C22—81no—
303C22#521311yescharacteristic
304C22#524111yescharacteristic
305C22#524111yescharacteristic
306C22#524212yes—
307C22#524612yes—
308C4—21no—
309C4—21no—
310C4—22no—
311C4—22no—
312C4—42no—
313C4—82no—
314C4—82no—
315C4—242no—
316C3#900311yescharacteristic, complemented, Sylow
317C2—82no—
318C2#7165011yescharacteristic
319C2#7166612yes—
320C2#7194911yescharacteristic
321C2#7195312yes—
322C2#7195611yescharacteristic
subgroupG/H
centerC23#953
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2379C2
socleC23×C3#120
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenter#2379D3
hypocenterC3#9003
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2379

Complements

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

NH ≅ G/Ntype
4. #69555C2semidirect
5. #67630C2semidirect
17. #5060C4semidirect
54. #869C2×C4semidirect
61. #803C2×C4semidirect
316. C3#9003semidirect

Maximal subgroups

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

subgroupindex
C3×#23792
C2×#50602
C2×#45402
#695552
#676302
#90033