Finite Group Databasebeta

← #10795535

Subgroups — #10795535

Order 768.

758 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#117C211yesmaximal, characteristic
2#71584C212yesmaximal
3#71582C212yesmaximal
4#71530C212yesmaximal
5#69437C212yesmaximal, complemented
6#69436C212yesmaximal, complemented
7#69393C212yesmaximal, complemented
8#69389C211yesmaximal, characteristic, complemented
9#69388C211yesmaximal, characteristic, complemented
10#20785—31nomaximal, Sylow
11C2×C3×#117C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic
13C22×C3×C42C2211yescharacteristic
14C3×C4×C4⋊C4C412yes—
15C3×C4×C4⋊C4C412yes—
16C3×C4×C22⋊C4C412yes—
17C3×C4×C22⋊C4C412yes—
18C3×#376C2211yescharacteristic
19C3×#376C2211yescharacteristic
20C3×C43C2211yescharacteristic
21C3×C43C2211yescharacteristic
22C2×#830C2214yes—
23C2×C4⋊(C3⋊C8)C2214yes—
24C2×#803C2212yes—
25C2×C4×C3⋊C8C2212yes—
26#4524C2212yes—
27#4524C2212yes—
28#4524C2212yes—
29#4524C2212yes—
30#4524C2214yes—
31#4524C2214yes—
32C4×#117D311yescharacteristic
33#1839—32no—
34#1832—32no—
35#1818—32no—
36#1797—32no—
37#1768—32no—
38#1750—32no—
39#1745—31no—
40#1744—31no—
41C23×C3×C4C2×C412yes—
42C23×C3×C4C2311yescharacteristic
43C3×#117—24no—
44C3×#117—24no—
45C2×C3×C4⋊C4C2×C412yes—
46C2×C3×C4⋊C4C2×C411yescharacteristic
47C2×C3×C4⋊C4C2×C411yescharacteristic
48C2×C3×C22⋊C4C2×C412yes—
49C2×C3×C22⋊C4C2×C411yescharacteristic
50C2×C3×C22⋊C4C2×C411yescharacteristic
51C2×C3×C42—24no—
52C2×C3×C42C2×C412yes—
53C2×C3×C42C2×C412yes—
54C2×C3×C42C2×C412yes—
55C2×C3×C42C2×C412yes—
56C2×C3×C42C2×C412yes—
57C2×C3×C42C2311yescharacteristic
58C2×C3×C42C2311yescharacteristic
59C2×C3×C42C2311yescharacteristic
60C2×C3×C42C2311yescharacteristic
61C2×C3×C42C2311yescharacteristic
62C2×C3×C42C2311yescharacteristic
63C22×C3⋊C8C2314yes—
64C22×C3⋊C8C2314yes—
65C3×#95C2×C412yes—
66C3×#95C2×C412yes—
67C3×#95C2×C412yes—
68C3×#95C2×C412yes—
69#830—28no—
70C4⋊(C3⋊C8)—28no—
71#803—24no—
72C4×C3⋊C8—24no—
73C2×#117C2×D311yescharacteristic
74C2×#117C2×D311yescharacteristic
75C22×C42C2×D311yescharacteristic
76C2×C4⋊C8—34no—
77C2×C22⋊C8—34no—
78C2×#97—32no—
79C2×C4×C8—32no—
80C4×C4⋊C4Dic312yes—
81C4×C4⋊C4Dic312yes—
82C4×C22⋊C4Dic312yes—
83C4×C22⋊C4Dic312yes—
84#376C2×D311yescharacteristic
85#376C2×D311yescharacteristic
86C43C2×D311yescharacteristic
87C43C2×D311yescharacteristic
88#336—32no—
89#336—32no—
90#336—32no—
91#336—32no—
92#336—34no—
93#336—34no—
94C24×C3C22×C411yescharacteristic
95C22×C3×C4—22no—
96C22×C3×C4—22no—
97C22×C3×C4—24no—
98C22×C3×C4—24no—
99C22×C3×C4—42no—
100C22×C3×C4C22×C411yescharacteristic
101C22×C3×C4C22×C412yes—
102C22×C3×C4C22×C412yes—
103C22×C3×C4C22×C411yescharacteristic
104C22×C3×C4C22×C412yes—
105C22×C3×C4C22×C411yescharacteristic
106C22×C3×C4C22×C411yescharacteristic
107C22×C3×C4C22×C411yescharacteristic
108C22×C3×C4C22×C412yes—
109C22×C3×C4#4114yes—
110C22×C3×C4C2411yescharacteristic
111C3×C4⋊C4—22no—
112C3×C4⋊C4—22no—
113C3×C4⋊C4—24no—
114C3×C22⋊C4—24no—
115C3×C22⋊C4—22no—
116C3×C22⋊C4—22no—
117C3×C42—24no—
118C3×C42—24no—
119C3×C42—24no—
120C3×C42—22no—
121C3×C42—22no—
122C3×C42—22no—
123C3×C42—22no—
124C3×C42—42no—
125C3×C42—42no—
126C3×C42#4114yes—
127C3×C42#4114yes—
128C2×C3⋊C8—24no—
129C2×C3⋊C8—24no—
130C2×C3⋊C8—24no—
131C2×C3⋊C8—24no—
132C2×C3⋊C8—24no—
133C2×C3⋊C8—24no—
134C23×C4C2×Dic312yes—
135C23×C4C22×D311yescharacteristic
136C22×C8—34no—
137C22×C8—34no—
138#117—24no—
139#117—24no—
140C2×C4⋊C4C2×Dic312yes—
141C2×C4⋊C4C2×Dic311yescharacteristic
142C2×C4⋊C4C2×Dic311yescharacteristic
143C2×C22⋊C4C2×Dic312yes—
144C2×C22⋊C4C2×Dic311yescharacteristic
145C2×C22⋊C4C2×Dic311yescharacteristic
146C2×C42—24no—
147C2×C42C2×Dic312yes—
148C2×C42C2×Dic312yes—
149C2×C42C2×Dic312yes—
150C2×C42C2×Dic312yes—
151C2×C42C2×Dic312yes—
152C2×C42C22×D311yescharacteristic
153C2×C42C22×D311yescharacteristic
154C2×C42C22×D311yescharacteristic
155C2×C42C22×D311yescharacteristic
156C2×C42C22×D311yescharacteristic
157C2×C42C22×D311yescharacteristic
158C4⋊C8—68no—
159C22⋊C8—68no—
160#97—64no—
161C4×C8—64no—
162#95C2×Dic312yes—
163#95C2×Dic312yes—
164#95C2×Dic312yes—
165#95C2×Dic312yes—
166C23×C3—21no—
167C23×C3—21no—
168C23×C3—22no—
169C23×C3—22no—
170C23×C3#11712yes—
171C23×C3C23×C411yescharacteristic
172C2×C3×C4—22no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—22no—
176C2×C3×C4—22no—
177C2×C3×C4—22no—
178C2×C3×C4—22no—
179C2×C3×C4—22no—
180C2×C3×C4—28no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—22no—
184C2×C3×C4—22no—
185C2×C3×C4—22no—
186C2×C3×C4—22no—
187C2×C3×C4—22no—
188C2×C3×C4—22no—
189C2×C3×C4—22no—
190C2×C3×C4—22no—
191C2×C3×C4—48no—
192C2×C3×C4#11714yes—
193C2×C3×C4#11712yes—
194C2×C3×C4#11712yes—
195C2×C3×C4#11712yes—
196C2×C3×C4#13114yes—
197C2×C3×C4C2×#4112yes—
198C2×C3×C4C2×#4112yes—
199C2×C3×C4C2×#4112yes—
200C3⋊C8—44no—
201C3⋊C8—44no—
202C24C22×Dic311yescharacteristic
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—42no—
208C22×C4#23514yes—
209C22×C4C22×Dic311yescharacteristic
210C22×C4C22×Dic312yes—
211C22×C4C22×Dic312yes—
212C22×C4C22×Dic311yescharacteristic
213C22×C4C22×Dic312yes—
214C22×C4C22×Dic311yescharacteristic
215C22×C4C22×Dic311yescharacteristic
216C22×C4C22×Dic311yescharacteristic
217C22×C4C22×Dic312yes—
218C22×C4C23×D311yescharacteristic
219C2×C8—64no—
220C2×C8—64no—
221C2×C8—64no—
222C2×C8—64no—
223C2×C8—64no—
224C2×C8—64no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—24no—
228C22⋊C4—24no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C42—24no—
232C42—24no—
233C42—24no—
234C42—22no—
235C42—22no—
236C42—22no—
237C42—22no—
238C42—42no—
239C42—42no—
240C42#23514yes—
241C42#23514yes—
242C22×C3—24no—
243C22×C3—22no—
244C22×C3—41no—
245C22×C3—41no—
246C22×C3—42no—
247C22×C3C2×#11711yescharacteristic
248C22×C3C2×#11711yescharacteristic
249C22×C3C2×#11711yescharacteristic
250C22×C3C2×#13112yes—
251C22×C3#56812yes—
252C3×C4—22no—
253C3×C4—22no—
254C3×C4—24no—
255C3×C4—24no—
256C3×C4—22no—
257C3×C4—41no—
258C3×C4—41no—
259C3×C4—42no—
260C3×C4—42no—
261C3×C4#52914yes—
262C23—21no—
263C23—21no—
264C23—22no—
265C23—22no—
266C23#92612yes—
267C23C23×Dic311yescharacteristic
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—28no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—48no—
288C2×C4#92614yes—
289C2×C4#92612yes—
290C2×C4#92612yes—
291C2×C4#92612yes—
292C2×C4#94814yes—
293C2×C4C2×#23512yes—
294C2×C4C2×#23512yes—
295C2×C4C2×#23512yes—
296C8—124no—
297C8—124no—
298C2×C3—42no—
299C2×C3#228311yescharacteristic
300C2×C3#292211yescharacteristic
301C2×C3#292311yescharacteristic
302C2×C3#292412yes—
303C2×C3#292512yes—
304C22—24no—
305C22—22no—
306C22—42no—
307C22—41no—
308C22—41no—
309C22C2×#92611yescharacteristic
310C22C2×#92611yescharacteristic
311C22C2×#92611yescharacteristic
312C22C2×#94812yes—
313C22#581912yes—
314C4—24no—
315C4—24no—
316C4—22no—
317C4—22no—
318C4—22no—
319C4—41no—
320C4—41no—
321C4—42no—
322C4—42no—
323C4#547914yes—
324C3#2078511yescharacteristic, complemented, Sylow
325C2—42no—
326C2#8206011yescharacteristic
327C2#8221012yes—
328C2#8251311yescharacteristic
329C2#8251411yescharacteristic
330C2#8251512yes—
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC3×C4×#117C2
prime radicalC24×C3C22×C4
cosocleC22×C3×C4C24
hypercenterC4×#117D3
hypocenterC3#20785
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C4×#117

Complements

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

NH ≅ G/Ntype
5. #69437C2semidirect
6. #69436C2semidirect
7. #69393C2semidirect
8. #69389C2semidirect
9. #69388C2semidirect
324. C3#20785semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#1172
#715842
#715822
#715302
#694372
#694362
#693932
#693892
#693882
#207853