Finite Group Databasebeta

← #10692832

Subgroups — #10692832

Order 768.

474 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 103 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×#2499C211yesmaximal, characteristic
2#82209C211yesmaximal, characteristic
3#71689C212yesmaximal
4#71594C211yesmaximal, characteristic
5#67616C212yesmaximal
6#9444—31nomaximal, Sylow
7C2×C3×C22⋊Q8C2211yescharacteristic
8C2×C3×#117C2211yescharacteristic
9C3×#397—21no—
10C3×#397—21no—
11C3×#391C411yescharacteristic, complemented
12C3×#391C411yescharacteristic, complemented
13C3×#387—21no—
14C3×#387—21no—
15C3×#384C411yescharacteristic, complemented
16C3×#382C411yescharacteristic, complemented
17C3×C4×C22⋊C4C2211yescharacteristic
18#5211C2212yes—
19#5207C2212yes—
20#5000—24no—
21#4924—21no—
22C4×#208—21no—
23#4538—24no—
24#4526—21no—
25#4526—21no—
26#2921—31no—
27#2499D311yescharacteristic
28#1820—31no—
29#1511—32no—
30#1284—32no—
31C22×C3×Q8C2×C411yescharacteristic
32C23×C3×C4C2311yescharacteristic
33C3×C22⋊Q8—22no—
34C3×C22⋊Q8—22no—
35C3×#117—22no—
36C3×#117—22no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4C2×C411yescharacteristic
41C2×C3×C4⋊C4C2×C411yescharacteristic
42C2×C3×C4⋊C4C2×C411yescharacteristic
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4D411yescharacteristic
46C2×C3×C22⋊C4D411yescharacteristic
47C2×C3×C42C2×C411yescharacteristic
48C2×C3×C42C2×C411yescharacteristic
49C2×C3×C42D411yescharacteristic
50C2×C3×C42D411yescharacteristic
51C2×#217—22no—
52C2×#208—22no—
53C3×#95—21no—
54C3×#95—21no—
55C3×#95—21no—
56C3×#95—21no—
57C3×#95—21no—
58C3×#95—21no—
59C3×#95—21no—
60C3⋊#95—22no—
61C3⋊#95—22no—
62#830—22no—
63#830D414yes—
64C4⋊(C3⋊C8)—24no—
65#803—22no—
66C4×C3⋊C8—22no—
67C2×C22⋊Q8C2×D311yescharacteristic
68C2×#117C2×D311yescharacteristic
69#433—64no—
70#423—61no—
71#407—32no—
72C4×M16—61no—
73#397—21no—
74#397—21no—
75#391Dic311yescharacteristic, complemented
76#391Dic311yescharacteristic, complemented
77#387—21no—
78#387—21no—
79#384Dic311yescharacteristic, complemented
80#382Dic311yescharacteristic, complemented
81#381—32no—
82C4×C22⋊C4C2×D311yescharacteristic
83#342—61no—
84#342—61no—
85#328—64no—
86C24×C3C2×D411yescharacteristic
87C2×C3×Q8—22no—
88C2×C3×Q8—41no—
89C2×C3×Q8—41no—
90C22×C3×C4—21no—
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4—41no—
95C22×C3×C4—41no—
96C22×C3×C4C22⋊C411yescharacteristic
97C22×C3×C4C22⋊C411yescharacteristic
98C22×C3×C4C22⋊C411yescharacteristic
99C22×C3×C4C22⋊C411yescharacteristic
100C22×C3×C4C22×C411yescharacteristic
101C22×C3×C4C2×D412yes—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4#4112yes—
104C22×Dic3—42no—
105C3×C4⋊C4—22no—
106C3×C4⋊C4—22no—
107C3×C4⋊C4—22no—
108C3×C4⋊C4—42no—
109C3×C4⋊C4—42no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—41no—
112C3×C22⋊C4—22no—
113C3×C22⋊C4—22no—
114C3×C22⋊C4—42no—
115C3×C22⋊C4—42no—
116C3×C42—21no—
117C3×C42—21no—
118C3×C42—21no—
119C3×C42—21no—
120C3×C42—41no—
121C3×C42—41no—
122#217—42no—
123#217—42no—
124#208—42no—
125#208—42no—
126C2×C3⋊C8—24no—
127C22×Q8C2×Dic311yescharacteristic
128C23×C4C22×D311yescharacteristic
129C2×M16—62no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132#117—22no—
133#117—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4C2×Dic311yescharacteristic
138C2×C4⋊C4C2×Dic311yescharacteristic
139C2×C4⋊C4C2×Dic311yescharacteristic
140C2×C22⋊C4—21no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—62no—
143C2×C22⋊C4#6711yescharacteristic
144C2×C22⋊C4#6711yescharacteristic
145C2×C42C2×Dic311yescharacteristic
146C2×C42C2×Dic311yescharacteristic
147C2×C42#6711yescharacteristic
148C2×C42#6711yescharacteristic
149C4⋊C8—64no—
150C22⋊C8—34no—
151C22⋊C8—62no—
152#97—62no—
153C4×C8—62no—
154#95—21no—
155#95—21no—
156#95—21no—
157#95—21no—
158#95—21no—
159#95—21no—
160#95—21no—
161#95—62no—
162#95—62no—
163C23×C3—21no—
164C23×C3—21no—
165C23×C3—41no—
166C23×C3—41no—
167C23×C3C2×C22⋊C411yescharacteristic
168C23×C3#12112yes—
169C3×Q8—82no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—22no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—42no—
180C2×C3×C4—42no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4C4×D412yes—
197C2×C3×C4#12011yescharacteristic
198C2×C3×C4#12311yescharacteristic
199C2×Dic3—82no—
200C2×Dic3—82no—
201C2×Dic3—82no—
202C3⋊C8—44no—
203C24C2×#6711yescharacteristic
204C2×Q8—22no—
205C2×Q8—41no—
206C2×Q8—41no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—122no—
214C22×C4#21711yescharacteristic
215C22×C4#21711yescharacteristic
216C22×C4#21711yescharacteristic
217C22×C4#21711yescharacteristic
218C22×C4D3×D412yes—
219C22×C4#23712yes—
220C22×C4C22×Dic311yescharacteristic
221C22×C4C2×#6711yescharacteristic
222M16—122no—
223M16—122no—
224C2×C8—64no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—122no—
237C22⋊C4—122no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—41no—
243C42—41no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—22no—
247C22×C3—22no—
248C22×C3—22no—
249C22×C3—81no—
250C22×C3—81no—
251C22×C3#38611yescharacteristic
252C22×C3#41011yescharacteristic
253C22×C3#41011yescharacteristic
254C3×C4—21no—
255C3×C4—22no—
256C3×C4—21no—
257C3×C4—42no—
258C3×C4—41no—
259C3×C4—41no—
260C3×C4—41no—
261C3×C4—41no—
262C3×C4—81no—
263C3×C4—81no—
264C3×C4—81no—
265Dic3—162no—
266C23—21no—
267C23—21no—
268C23—41no—
269C23—41no—
270C23#93912yes—
271C23C2×#21711yescharacteristic
272Q8—82no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—242no—
300C2×C4—242no—
301C2×C4—242no—
302C2×C4D4×Dic312yes—
303C2×C4#93511yescharacteristic
304C2×C4#93711yescharacteristic
305C8—124no—
306C2×C3—21no—
307C2×C3—21no—
308C2×C3—42no—
309C2×C3#188411yescharacteristic
310C2×C3#189911yescharacteristic
311C2×C3#190911yescharacteristic
312C22—21no—
313C22—21no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—81no—
318C22—81no—
319C22#522011yescharacteristic
320C22#524011yescharacteristic
321C22#524011yescharacteristic
322C4—21no—
323C4—22no—
324C4—21no—
325C4—42no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—81no—
331C4—81no—
332C4—81no—
333C4—482no—
334C3#944411yescharacteristic, complemented, Sylow
335C2—21no—
336C2—21no—
337C2—42no—
338C2#7170511yescharacteristic
339C2#7181711yescharacteristic
340C2#7186511yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2499C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#2499D3
hypocenterC3#9444
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2499

Complements

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

NH ≅ G/Ntype
11. C3×#391C4semidirect
12. C3×#391C4semidirect
15. C3×#384C4semidirect
16. C3×#382C4semidirect
75. #391Dic3semidirect
76. #391Dic3semidirect
79. #384Dic3semidirect
80. #382Dic3semidirect
334. C3#9444semidirect

Maximal subgroups

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

subgroupindex
C3×#24992
#822092
#716892
#715942
#676162
#94443