Finite Group Databasebeta

← #10823300

Subgroups — #10823300

Order 768.

1082 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85725C211yesmaximal, characteristic
2#83752C212yesmaximal
3#82224C211yesmaximal, characteristic
4C3×#1742C211yesmaximal, characteristic
5C2×#4555C212yesmaximal
6#68979C214yesmaximal, complemented
7#68978C214yesmaximal, complemented
8#23560—31nomaximal, Sylow
9C22×C3×M16C2212yes—
10C2×C3×#117C2211yescharacteristic
11C2×#926C2212yes—
12C2×#926C2211yescharacteristic
13C22×C4×Dic3C2211yescharacteristic
14#5519—24no—
15#5517—24no—
16#5500C2214yes—
17#5483—22no—
18#5470—22no—
19#5121—24no—
20#5121C2218yes—
21#5106—24no—
22#4977C2212yes—
23#4975C2212yes—
24#4942—22no—
25C4×Dic3⋊C4—22no—
26C3×#343C2214yes—
27#4555C2214yes—
28#4555C2214yes—
29#3458—31no—
30#2911—32no—
31#2285—31no—
32#2061—34no—
33#2060—34no—
34#1742D311yescharacteristic
35C2×#339—32no—
36C23×C3×C4C2311yescharacteristic
37C23×Dic3—21no—
38C2×C3×M16—22no—
39C2×C3×M16—24no—
40C2×C3×M16Q818yescomplemented
41C2×C3×M16C2314yes—
42C22×C3×C8—22no—
43C3×#117—22no—
44C3×#117—22no—
45C2×C3×C4⋊C4C2312yes—
46C2×C3×C22⋊C4D412yes—
47C2×C3×C42D412yes—
48C2×#217—22no—
49C2×#217—21no—
50#926—22no—
51#926—22no—
52#926—22no—
53#926—22no—
54#926D414yes—
55#926C2314yes—
56C2×C4⋊Dic3D414yes—
57C2×C4⋊Dic3C2312yes—
58#924—28no—
59C2×Dic3⋊C4—24no—
60C2×C4×Dic3—21no—
61C2×C4×Dic3—22no—
62C2×C4×Dic3—22no—
63C2×C4×Dic3—21no—
64C2×C4×Dic3Q814yescomplemented
65C2×C4×Dic3C2311yescharacteristic
66C2×C4×Dic3C2311yescharacteristic
67#890—28no—
68#888—28no—
69#878—216no—
70#870—24no—
71#869—24no—
72C4×Dic6—28no—
73C3⋊#95—24no—
74#818—28no—
75#817—28no—
76C22×M16C2×D312yes—
77#544—34no—
78#544—64no—
79#543—62no—
80#533—64no—
81#533—62no—
82C2×#117—32no—
83C2×#117—31no—
84C2×#117C2×D311yescharacteristic
85C22×C42—31no—
86#428—38no—
87#428—64no—
88#427—64no—
89#384—32no—
90#382—32no—
91#380—62no—
92C4×C4⋊C4—62no—
93#343C2×D314yes—
94#339—34no—
95#339—34no—
96C24×C3C2×D411yescharacteristic
97C22×C3×C4C2×D414yes—
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D412yes—
100C22×C3×C4C2×D412yes—
101C22×C3×C4C2×Q814yes—
102C22×C3×C4C2×Q812yes—
103C22×C3×C4#4112yes—
104C22×C3×C4C2411yescharacteristic
105C22×Dic3—21no—
106C22×Dic3—24no—
107C22×Dic3—21no—
108C22×Dic3—24no—
109C22×Dic3—22no—
110C22×Dic3—44no—
111C2×Dic6—28no—
112C3×M16—28no—
113C3×M16—24no—
114C3×M16C2×Q818yes—
115C2×C3×C8—28no—
116C2×C3×C8—24no—
117C3×C4⋊C4—24no—
118C3×C22⋊C4—24no—
119C3×C42—22no—
120C3×C42—22no—
121#217—24no—
122#217—24no—
123#217—42no—
124C4⋊Dic3—28no—
125C4⋊Dic3—24no—
126C4⋊Dic3—24no—
127C4⋊Dic3C2×D418yes—
128Dic3⋊C4—216no—
129Dic3⋊C4—48no—
130C4×Dic3—24no—
131C4×Dic3—24no—
132C4×Dic3—24no—
133C4×Dic3—21no—
134C4×Dic3—21no—
135C4×Dic3—41no—
136C4×Dic3—41no—
137C4×Dic3C2×Q812yes—
138C4×Dic3C2×Q812yes—
139C23×C4—61no—
140C23×C4C22×D311yescharacteristic
141C2×M16—22no—
142C2×M16—24no—
143C2×M16Dic618yescomplemented
144C2×M16C22×D314yes—
145C22×C8—22no—
146C4⋊Q8—64no—
147C4⋊Q8—68no—
148#125—64no—
149#125—68no—
150C22⋊Q8—616no—
151C22⋊Q8—68no—
152C4×Q8—68no—
153#117—22no—
154#117—22no—
155#117—34no—
156#117—34no—
157#117—62no—
158#117—62no—
159#117—62no—
160#117—62no—
161C2×C4⋊C4—34no—
162C2×C4⋊C4—32no—
163C2×C4⋊C4—64no—
164C2×C4⋊C4C22×D312yes—
165C2×C22⋊C4—62no—
166C2×C22⋊C4—61no—
167C2×C22⋊C4D1212yes—
168C2×C42—34no—
169C2×C42—31no—
170C2×C42—31no—
171C2×C42—61no—
172C2×C42—62no—
173C2×C42—62no—
174C2×C42—61no—
175C2×C42D1212yes—
176#107—68no—
177#106—68no—
178#95—64no—
179C23×C3—42no—
180C23×C3#12012yes—
181C23×C3C22⋊Q812yes—
182C23×C3C4⋊Q812yes—
183C23×C3C22×D411yescharacteristic
184C2×C3×C4—24no—
185C2×C3×C4—44no—
186C2×C3×C4#12011yescharacteristic
187C2×C3×C4#12011yescharacteristic
188C2×C3×C4C22⋊Q814yes—
189C2×C3×C4C22⋊Q814yes—
190C2×C3×C4C22⋊Q812yes—
191C2×C3×C4C4⋊Q812yes—
192C2×C3×C4C4⋊Q814yes—
193C2×C3×C4C4⋊Q814yes—
194C2×C3×C4C22×D412yes—
195C2×C3×C4C22×Q812yes—
196C2×C3×C4C22×Q811yescharacteristic
197C2×C3×C4C2×#4111yescharacteristic
198C2×Dic3—24no—
199C2×Dic3—22no—
200C2×Dic3—28no—
201C2×Dic3—22no—
202C2×Dic3—48no—
203C2×Dic3—42no—
204C2×Dic3—44no—
205C2×Dic3—42no—
206C2×Dic3—84no—
207Dic6—88no—
208C3×C8—28no—
209C24C2×D1211yescharacteristic
210C2×Q8—68no—
211C22×C4—61no—
212C22×C4—64no—
213C22×C4—61no—
214C22×C4—64no—
215C22×C4—62no—
216C22×C4—124no—
217C22×C4C2×Dic614yes—
218C22×C4C2×D1214yes—
219C22×C4C2×D1211yescharacteristic
220C22×C4D3×D412yes—
221C22×C4D3×D412yes—
222C22×C4#23712yes—
223C22×C4D3×Q812yes—
224C22×C4C23×D311yescharacteristic
225M16—28no—
226M16—24no—
227M16C2×Dic618yes—
228C2×C8—28no—
229C2×C8—24no—
230C4⋊C4—24no—
231C4⋊C4—38no—
232C4⋊C4—616no—
233C4⋊C4—68no—
234C4⋊C4—64no—
235C4⋊C4—64no—
236C4⋊C4—128no—
237C22⋊C4—24no—
238C22⋊C4—64no—
239C22⋊C4—64no—
240C22⋊C4—122no—
241C42—22no—
242C42—22no—
243C42—32no—
244C42—32no—
245C42—64no—
246C42—64no—
247C42—64no—
248C42—61no—
249C42—61no—
250C42—121no—
251C42—121no—
252C22×C3—22no—
253C22×C3—24no—
254C22×C3—42no—
255C22×C3—42no—
256C22×C3#39514yes—
257C22×C3C2×#12011yescharacteristic
258C22×C3C2×C22⋊Q811yescharacteristic
259C22×C3C2×C4⋊Q811yescharacteristic
260C3×C4—44no—
261C3×C4#39512yes—
262C3×C4#39512yes—
263C3×C4C2×C22⋊Q812yes—
264C3×C4C2×C4⋊Q812yes—
265Dic3—24no—
266Dic3—42no—
267Dic3—48no—
268Dic3—82no—
269C23—42no—
270C23#86912yes—
271C23#88212yes—
272C23#89712yes—
273C23C22×D1211yescharacteristic
274Q8—248no—
275C2×C4—24no—
276C2×C4—44no—
277C2×C4—64no—
278C2×C4—62no—
279C2×C4—68no—
280C2×C4—62no—
281C2×C4—128no—
282C2×C4—122no—
283C2×C4—124no—
284C2×C4—122no—
285C2×C4—244no—
286C2×C4#86912yes—
287C2×C4#87814yes—
288C2×C4#87814yes—
289C2×C4#88211yescharacteristic
290C2×C4#88211yescharacteristic
291C2×C4#88814yes—
292C2×C4#88814yes—
293C2×C4#89712yes—
294C2×C4C22×Dic612yes—
295C2×C4C2×D3×D412yes—
296C2×C4C2×#23711yescharacteristic
297C2×C4C2×D3×Q811yescharacteristic
298C8—28no—
299C2×C3—21no—
300C2×C3—21no—
301C2×C3—24no—
302C2×C3C2×#39511yescharacteristic
303C2×C3#301812yes—
304C22—22no—
305C22—24no—
306C22—42no—
307C22—42no—
308C22#465614yes—
309C22C2×#86911yescharacteristic
310C22C2×#88211yescharacteristic
311C22C2×#89711yescharacteristic
312C4—44no—
313C4—64no—
314C4—122no—
315C4—128no—
316C4—242no—
317C4#465612yes—
318C4#465612yes—
319C4C2×#87812yes—
320C4C2×#88812yes—
321C3#2356011yescharacteristic, complemented, Sylow
322C2—21no—
323C2—24no—
324C2—21no—
325C2C2×#465611yescharacteristic
326C2#7871612yes—
subgroupG/H
centerC2×C4#882
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1742C2
socleC22×C3C2×#120
prime radicalC24×C3C2×D4
hypercenter#1742D3
hypocenterC3#23560
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1742

Complements

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

NH ≅ G/Ntype
6. #68979C2semidirect
7. #68978C2semidirect
40. C2×C3×M16Q8semidirect
64. C2×C4×Dic3Q8semidirect
143. C2×M16Dic6semidirect
321. C3#23560semidirect

Maximal subgroups

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

subgroupindex
#857252
#837522
#822242
C3×#17422
C2×#45552
#689792
#689782
#235603