Finite Group Databasebeta

← #10683676

Subgroups — #10683676

Order 768.

410 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73309C211yesmaximal, characteristic
2C3×C4×C4⋊C8C211yesmaximal, characteristic
3#69391C211yesmaximal, characteristic
4#67480C212yesmaximal
5#67478C212yesmaximal
6#7689—31nomaximal, Sylow
7C2×C3×C4⋊C8—21no—
8C2×C3×C4×C8—21no—
9C3×C43C2211yescharacteristic
10#4937—21no—
11#4933—21no—
12#4933—21no—
13#4933—21no—
14#4929C411yescharacteristic
15#4929C411yescharacteristic
16#4929C2211yescharacteristic
17#4928C411yescharacteristic
18#4928C411yescharacteristic
19#4928C2211yescharacteristic
20C2×C4⋊(C3⋊C8)—21no—
21C2×#803—21no—
22C3×#336C2212yes—
23#4524C2212yes—
24#2863—31no—
25#1767—31no—
26C4×C4⋊C8D311yescharacteristic
27#1272—32no—
28#1271—32no—
29C22×Dic6C2×C411yescharacteristic
30C22×C3×C8—22no—
31C2×C3×C42C2×C411yescharacteristic
32C2×C3×C42C2×C411yescharacteristic
33C2×C3×C42D411yescharacteristic
34C2×C3×C42D411yescharacteristic
35C2×C3×C42D411yescharacteristic
36C2×C3×C42D411yescharacteristic
37C2×C3×C42C2311yescharacteristic
38C2×C4⋊Dic3C2×C411yescharacteristic
39C2×C4⋊Dic3C2×C411yescharacteristic
40C2×C4⋊Dic3C2×C411yescharacteristic
41C2×Dic3⋊C4—21no—
42C2×Dic3⋊C4—21no—
43C2×Dic3⋊C4—21no—
44C22×C3⋊C8—22no—
45C3×C4⋊C8—24no—
46C3×C4×C8—42no—
47C3⋊#95—21no—
48C3⋊#95—21no—
49C3⋊#95—21no—
50C3⋊#95—21no—
51C3⋊#95—21no—
52C3⋊#95—21no—
53C3⋊#95—21no—
54C3⋊#95—21no—
55C3⋊#95—21no—
56C4⋊(C3⋊C8)—24no—
57#803—42no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—61no—
60C2×#97—61no—
61C2×C4×C8—21no—
62#391—31no—
63#391—31no—
64#391—31no—
65#389—31no—
66#389—31no—
67#389—31no—
68#387—61no—
69#387—61no—
70#387—61no—
71#383—61no—
72C43C2×D311yescharacteristic
73#336—32no—
74#336C2×D312yes—
75C22×C3×C4C22⋊C411yescharacteristic
76C22×C3×C4C22⋊C411yescharacteristic
77C22×C3×C4C22⋊C411yescharacteristic
78C22×C3×C4C22⋊C411yescharacteristic
79C22×C3×C4C22×C411yescharacteristic
80C22×C3×C4C2×D411yescharacteristic
81C22×C3×C4C2×D411yescharacteristic
82C22×Dic3—21no—
83C22×Dic3—21no—
84C22×Dic3—21no—
85C22×Dic3—21no—
86C2×Dic6—22no—
87C2×Dic6—41no—
88C2×Dic6—41no—
89C2×C3×C8—24no—
90C2×C3×C8—42no—
91C2×C3×C8—42no—
92C3×C42—22no—
93C3×C42—22no—
94C3×C42—22no—
95C3×C42—21no—
96C3×C42—21no—
97C3×C42—41no—
98C3×C42—41no—
99C3×C42SD1612yes—
100C3×C42Q1612yes—
101C4⋊Dic3—22no—
102C4⋊Dic3—41no—
103C4⋊Dic3—41no—
104Dic3⋊C4—42no—
105Dic3⋊C4—42no—
106Dic3⋊C4—41no—
107Dic3⋊C4—41no—
108C2×C3⋊C8—24no—
109C2×C3⋊C8—42no—
110C2×C3⋊C8—42no—
111C22×Q8—31no—
112C22×C8—22no—
113C22×C8—62no—
114C2×C4⋊C4—31no—
115C2×C4⋊C4—31no—
116C2×C4⋊C4—31no—
117C2×C4⋊C4—61no—
118C2×C4⋊C4—61no—
119C2×C4⋊C4—61no—
120C2×C42C4×D311yescharacteristic
121C2×C42C4×D311yescharacteristic
122C2×C42D1211yescharacteristic
123C2×C42D1211yescharacteristic
124C2×C42#6711yescharacteristic
125C2×C42#6711yescharacteristic
126C2×C42C22×D311yescharacteristic
127C4⋊C8—24no—
128C4⋊C8—64no—
129#97—122no—
130C4×C8—42no—
131#95—61no—
132#95—61no—
133#95—61no—
134#95—61no—
135#95—61no—
136#95—61no—
137#95—61no—
138#95—61no—
139#95—61no—
140C23×C3C2×C22⋊C411yescharacteristic
141C2×C3×C4—21no—
142C2×C3×C4—21no—
143C2×C3×C4—21no—
144C2×C3×C4—21no—
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4—21no—
148C2×C3×C4—21no—
149C2×C3×C4—21no—
150C2×C3×C4—21no—
151C2×C3×C4—21no—
152C2×C3×C4—21no—
153C2×C3×C4—21no—
154C2×C3×C4—21no—
155C2×C3×C4#10312yes—
156C2×C3×C4#10312yes—
157C2×C3×C4#10412yes—
158C2×C3×C4#10412yes—
159C2×C3×C4#13112yes—
160C2×C3×C4C2×SD1611yescharacteristic
161C2×C3×C4C2×Q1611yescharacteristic
162C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
163C2×C3×C4#13711yescharacteristic
164C2×Dic3—41no—
165C2×Dic3—41no—
166C2×Dic3—41no—
167C2×Dic3—41no—
168C2×Dic3—41no—
169C2×Dic3—41no—
170C2×Dic3—41no—
171C2×Dic3—41no—
172C2×Dic3—41no—
173C2×Dic3—41no—
174C2×Dic3—41no—
175C2×Dic3—41no—
176Dic6—82no—
177C3×C8—44no—
178C3⋊C8—44no—
179C2×Q8—62no—
180C2×Q8—121no—
181C2×Q8—121no—
182C22×C4—61no—
183C22×C4—61no—
184C22×C4—61no—
185C22×C4—61no—
186C22×C4(C2×D3)⋊C411yescharacteristic
187C22×C4(C2×D3)⋊C411yescharacteristic
188C22×C4(C2×D3)⋊C411yescharacteristic
189C22×C4(C2×D3)⋊C411yescharacteristic
190C22×C4C2×C4×D311yescharacteristic
191C22×C4C2×D1211yescharacteristic
192C22×C4C2×#6711yescharacteristic
193C2×C8—24no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—64no—
197C2×C8—122no—
198C2×C8—122no—
199C4⋊C4—62no—
200C4⋊C4—121no—
201C4⋊C4—121no—
202C4⋊C4—122no—
203C4⋊C4—122no—
204C4⋊C4—121no—
205C4⋊C4—121no—
206C42—22no—
207C42—22no—
208C42—22no—
209C42—21no—
210C42—21no—
211C42—41no—
212C42—41no—
213C42#20412yes—
214C42Dic1212yes—
215C22×C3#40811yescharacteristic
216C22×C3#41011yescharacteristic
217C22×C3#41311yescharacteristic
218C22×C3C2×#10311yescharacteristic
219C22×C3#41711yescharacteristic
220C22×C3C2×#10411yescharacteristic
221C22×C3C2×#10411yescharacteristic
222C3×C4—22no—
223C3×C4—22no—
224C3×C4—22no—
225C3×C4—22no—
226C3×C4—21no—
227C3×C4—21no—
228C3×C4—21no—
229C3×C4—21no—
230C3×C4—22no—
231Dic3—81no—
232Dic3—81no—
233Dic3—81no—
234Dic3—81no—
235C23C2×(C2×D3)⋊C411yescharacteristic
236Q8—242no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—121no—
252C2×C4—121no—
253C2×C4—121no—
254C2×C4—121no—
255C2×C4—121no—
256C2×C4—121no—
257C2×C4—121no—
258C2×C4—121no—
259C2×C4—121no—
260C2×C4—121no—
261C2×C4—121no—
262C2×C4—121no—
263C2×C4#80512yes—
264C2×C4#81612yes—
265C2×C4#81612yes—
266C2×C4#82512yes—
267C2×C4#90112yes—
268C2×C4C2×#20411yescharacteristic
269C2×C4C2×Dic1211yescharacteristic
270C2×C4#93211yescharacteristic
271C2×C4#94211yescharacteristic
272C8—44no—
273C8—124no—
274C2×C3#147311yescharacteristic
275C2×C3#148311yescharacteristic
276C2×C3#149311yescharacteristic
277C2×C3#150311yescharacteristic
278C2×C3#150911yescharacteristic
279C2×C3#152411yescharacteristic
280C2×C3#155011yescharacteristic
281C22C2×#80511yescharacteristic
282C22#496611yescharacteristic
283C22#500611yescharacteristic
284C22C2×#81611yescharacteristic
285C22#511011yescharacteristic
286C22#513211yescharacteristic
287C22C2×#82511yescharacteristic
288C4—22no—
289C4—22no—
290C4—21no—
291C4—21no—
292C4—22no—
293C4—22no—
294C4—21no—
295C4—21no—
296C4—22no—
297C4—241no—
298C4—241no—
299C4—241no—
300C4—241no—
301C3#768911yescharacteristic, complemented, Sylow
302C2#6947011yescharacteristic
303C2#6973811yescharacteristic
304C2#6985811yescharacteristic
305C2#7042711yescharacteristic
306C2#7043811yescharacteristic
307C2#7078511yescharacteristic
308C2#7079211yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×C4×C4⋊C8C2
socle, prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenterC4×C4⋊C8D3
hypocenterC3#7689
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C43 ◁ C4×C4⋊C8

Complements

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

NH ≅ G/Ntype
301. C3#7689semidirect

Maximal subgroups

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

subgroupindex
#733092
C3×C4×C4⋊C82
#693912
#674802
#674782
#76893