Finite Group Databasebeta

← #10894941

Subgroups — #10894941

Order 768.

710 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2859C211yesmaximal, characteristic
2C4×#869C211yesmaximal, characteristic
3#69633C212yesmaximal
4#69616C214yesmaximal
5#69611C212yesmaximal
6#69588C212yesmaximal
7#69588C212yesmaximal
8#69391C211yesmaximal, characteristic
9#29132—31nomaximal, Sylow
10C2×C3×#125—21no—
11C2×#869C2211yescharacteristic
12C2×C4×Dic6C2212yes—
13C3×#389—21no—
14C3×#389—21no—
15C3×#389C2212yes—
16C3×#389C2212yes—
17C3×#384C2211yescharacteristic
18C3×#384C2211yescharacteristic
19C3×#382—21no—
20C3×C43C2211yescharacteristic
21C2×#810C2214yes—
22C2×#810C2214yes—
23#4935C2211yescharacteristic
24C4×C4⋊Dic3—21no—
25C4×C4⋊Dic3—21no—
26#4929C2212yes—
27#4928—21no—
28#4928—21no—
29C2×C4⋊(C3⋊C8)C2212yes—
30C2×#803C2212yes—
31#4529C2214yes—
32#4529C2214yes—
33#4524C2212yes—
34#2859D311yescharacteristic
35C4×C4⋊Q8—31no—
36#2099—32no—
37#2043—32no—
38#1935—34no—
39#1803—32no—
40#1803—32no—
41#1767—31no—
42C22×Dic6C2312yes—
43C3×#125—44no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—22no—
49C2×C3×C4⋊C4—22no—
50C2×C3×C4⋊C4C2312yes—
51C2×C3×C4⋊C4C2312yes—
52C2×C3×C42D411yescharacteristic
53C2×C3×C42D411yescharacteristic
54C2×C3×C42D411yescharacteristic
55C2×C3×C42D411yescharacteristic
56C2×C3×C42C2311yescharacteristic
57C2×C3×C42C2311yescharacteristic
58C2×C3×C42C2311yescharacteristic
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3C2312yes—
64C2×Dic3⋊C4—22no—
65C2×C4×Dic3—22no—
66C22×C3⋊C8C2314yes—
67#869—24no—
68C4×Dic6—24no—
69C4×Dic6—24no—
70C3×#95—22no—
71C3×#95—22no—
72C3⋊#95—22no—
73C3⋊#95—22no—
74#810—28no—
75#810—44no—
76C4⋊(C3⋊C8)—24no—
77#803—42no—
78C2×C4⋊Q8—31no—
79C2×#125—21no—
80C2×C4×Q8—32no—
81C2×C4⋊C8—32no—
82C2×#103—34no—
83C2×#103—34no—
84C2×#97—32no—
85#391—32no—
86#389—21no—
87#389—21no—
88#389—61no—
89#389—61no—
90#389C2×D312yes—
91#389C2×D312yes—
92#384—31no—
93#384C2×D311yescharacteristic
94#384C2×D311yescharacteristic
95#382—21no—
96C4×C4⋊C4—61no—
97C4×C4⋊C4—61no—
98C43C2×D311yescharacteristic
99#340—34no—
100#340—34no—
101#336—32no—
102C22×C3×C4—22no—
103C22×C3×C4—22no—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2411yescharacteristic
111C22×Dic3—22no—
112C22×Dic3—22no—
113C2×Dic6—22no—
114C2×Dic6—22no—
115C2×Dic6—22no—
116C2×Dic6—22no—
117C2×Dic6—22no—
118C2×Dic6#4114yes—
119C3×C4⋊C4—24no—
120C3×C4⋊C4—22no—
121C3×C4⋊C4—22no—
122C3×C4⋊C4—42no—
123C3×C4⋊C4—42no—
124C3×C4⋊C4—44no—
125C3×C4⋊C4—44no—
126C3×C4⋊C4—42no—
127C3×C4⋊C4—42no—
128C3×C42—22no—
129C3×C42—22no—
130C3×C42—22no—
131C3×C42—22no—
132C3×C42—22no—
133C3×C42—21no—
134C3×C42—21no—
135C3×C42#4114yes—
136C4⋊Dic3—24no—
137C4⋊Dic3—24no—
138C4⋊Dic3—22no—
139C4⋊Dic3—22no—
140C4⋊Dic3—42no—
141C4⋊Dic3—42no—
142Dic3⋊C4—44no—
143C4×Dic3—44no—
144C2×C3⋊C8—24no—
145C2×C3⋊C8—24no—
146C2×C3⋊C8—24no—
147C22×Q8—32no—
148C22×C8—34no—
149C4⋊Q8—64no—
150#125—44no—
151C4×Q8—64no—
152C4×Q8—64no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—32no—
160C2×C4⋊C4—61no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4—62no—
163C2×C4⋊C4—61no—
164C2×C4⋊C4—61no—
165C2×C4⋊C4C22×D312yes—
166C2×C4⋊C4C22×D312yes—
167C2×C42—62no—
168C2×C42#6711yescharacteristic
169C2×C42#6711yescharacteristic
170C2×C42#6711yescharacteristic
171C2×C42#6711yescharacteristic
172C2×C42C22×D311yescharacteristic
173C2×C42C22×D311yescharacteristic
174C2×C42C22×D311yescharacteristic
175C4⋊C8—64no—
176#103—68no—
177#103—124no—
178#97—122no—
179#95—22no—
180#95—22no—
181#95—62no—
182#95—62no—
183C23×C3C22×D411yescharacteristic
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—42no—
197C2×C3×C4—42no—
198C2×C3×C4—42no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4#12412yes—
203C2×C3×C4#12412yes—
204C2×C3×C4AGL(1,ℤ/8)12yes—
205C2×C3×C4#13712yes—
206C2×C3×C4#13712yes—
207C2×C3×C4#13712yes—
208C2×C3×C4C2×#4112yes—
209C2×C3×C4C2×#4112yes—
210C2×C3×C4H(5,𝔽2)11yescharacteristic
211C2×C3×C42−1+411yescharacteristic
212C2×Dic3—24no—
213C2×Dic3—42no—
214C2×Dic3—42no—
215C2×Dic3—42no—
216C2×Dic3—42no—
217C2×Dic3—42no—
218Dic6—24no—
219Dic6—24no—
220Dic6—42no—
221Dic6—42no—
222C3⋊C8—44no—
223C2×Q8—34no—
224C2×Q8—62no—
225C2×Q8—62no—
226C2×Q8—62no—
227C2×Q8—62no—
228C2×Q8—62no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—62no—
232C22×C4—62no—
233C22×C4C2×#6711yescharacteristic
234C22×C4C2×#6711yescharacteristic
235C22×C4C2×#6711yescharacteristic
236C22×C4C2×#6711yescharacteristic
237C22×C4C2×#6711yescharacteristic
238C22×C4C2×#6711yescharacteristic
239C22×C4C23×D311yescharacteristic
240C2×C8—64no—
241C2×C8—64no—
242C2×C8—64no—
243C4⋊C4—24no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—44no—
249C4⋊C4—44no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—64no—
253C4⋊C4—64no—
254C4⋊C4—62no—
255C4⋊C4—62no—
256C4⋊C4—122no—
257C4⋊C4—122no—
258C4⋊C4—124no—
259C42—22no—
260C42—22no—
261C42—22no—
262C42—22no—
263C42—22no—
264C42—21no—
265C42—21no—
266C42—124no—
267C42#23914yes—
268C22×C3#52511yescharacteristic
269C22×C3C2×#12411yescharacteristic
270C22×C3#53711yescharacteristic
271C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
272C22×C3C2×#13711yescharacteristic
273C22×C3C2×#13711yescharacteristic
274C22×C3C2×#13711yescharacteristic
275C3×C4—22no—
276C3×C4—22no—
277C3×C4—22no—
278C3×C4—41no—
279C3×C4—41no—
280C3×C4—82no—
281C3×C4—82no—
282C3×C4#49114yes—
283C3×C4#55212yes—
284C3×C4#55312yes—
285Dic3—44no—
286Dic3—82no—
287C23C22×#6711yescharacteristic
288Q8—64no—
289Q8—64no—
290Q8—122no—
291Q8—122no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—64no—
311C2×C4—122no—
312C2×C4—122no—
313C2×C4—122no—
314C2×C4—122no—
315C2×C4—122no—
316C2×C4#93212yes—
317C2×C4#94212yes—
318C2×C4#94712yes—
319C2×C4#94712yes—
320C2×C4#95112yes—
321C2×C4#95112yes—
322C2×C4C2×#23912yes—
323C2×C4D3×#4112yes—
324C2×C4#100911yescharacteristic
325C2×C4#101011yescharacteristic
326C8—124no—
327C2×C3#256211yescharacteristic
328C2×C3#311211yescharacteristic
329C2×C3#311311yescharacteristic
330C2×C3C2×#49111yescharacteristic
331C2×C3#314111yescharacteristic
332C2×C3#325411yescharacteristic
333C2×C3#325511yescharacteristic
334C22C2×#93211yescharacteristic
335C22C2×#94211yescharacteristic
336C22C2×#94711yescharacteristic
337C22C2×#95111yescharacteristic
338C22C2×#95111yescharacteristic
339C22#582811yescharacteristic
340C22#583311yescharacteristic
341C4—22no—
342C4—22no—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—82no—
347C4—82no—
348C4—124no—
349C4—242no—
350C4#506914yes—
351C4#569412yes—
352C4#569512yes—
353C3#2913211yescharacteristic, complemented, Sylow
354C2C2×#506911yescharacteristic
355C2#8306311yescharacteristic
356C2#8319911yescharacteristic
357C2#8331611yescharacteristic
358C2#8332111yescharacteristic
359C2#8336911yescharacteristic
360C2#8337411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2859C2
socle, prime radicalC23×C3C22×D4
hypercenter#2859D3
hypocenterC3#29132
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C43 ◁ #2859

Complements

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

NH ≅ G/Ntype
353. C3#29132semidirect

Maximal subgroups

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

subgroupindex
C3×#28592
C4×#8692
#696332
#696162
#696112
#695882
#693912
#291323