Finite Group Databasebeta

← #10894940

Subgroups — #10894940

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×C4×C4⋊Q8C211yesmaximal, characteristic
2#73307C211yesmaximal, characteristic
3#71781C212yesmaximal
4#71626C212yesmaximal
5#71626C212yesmaximal
6#69675C214yesmaximal
7#69571C212yesmaximal
8#69391C211yesmaximal, characteristic
9#29132—31nomaximal, Sylow
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×C4×Q8C2212yes—
12C2×#870—21no—
13C3×#391C2212yes—
14C3×#389—21no—
15C3×#389—21no—
16C3×#384C2211yescharacteristic
17C3×C4×C4⋊C4—21no—
18C3×C4×C4⋊C4—21no—
19C3×C43C2211yescharacteristic
20C2×#835C2214yes—
21C2×#835C2214yes—
22#4936—21no—
23#4935C2211yescharacteristic
24#4935C2211yescharacteristic
25#4928—21no—
26#4928—21no—
27#4928C2212yes—
28#4928C2212yes—
29C2×C4⋊(C3⋊C8)C2212yes—
30C2×#803C2212yes—
31#4529C2214yes—
32#4529C2214yes—
33#4524C2212yes—
34#2859—31no—
35C4×C4⋊Q8D311yescharacteristic
36#2099—32no—
37#2043—32no—
38#1935—34no—
39#1803—32no—
40#1803—32no—
41#1767—31no—
42C22×C3×Q8C2312yes—
43C3×C4⋊Q8—24no—
44C3×C4×Q8—24no—
45C3×C4×Q8—24no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—22no—
51C2×C3×C4⋊C4C2312yes—
52C2×C3×C42—22no—
53C2×C3×C42D411yescharacteristic
54C2×C3×C42D411yescharacteristic
55C2×C3×C42D411yescharacteristic
56C2×C3×C42D411yescharacteristic
57C2×C3×C42C2311yescharacteristic
58C2×C3×C42C2311yescharacteristic
59C2×C3×C42C2311yescharacteristic
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3—21no—
64C2×C4⋊Dic3C2312yes—
65C2×C4⋊Dic3C2312yes—
66C2×Dic3⋊C4—22no—
67C2×Dic3⋊C4—22no—
68C22×C3⋊C8C2314yes—
69#870—44no—
70C3×#95—22no—
71C3×#95—22no—
72#835—28no—
73#835—44no—
74C3⋊#95—22no—
75C3⋊#95—22no—
76C4⋊(C3⋊C8)—24no—
77#803—42no—
78C2×C4⋊Q8C2×D311yescharacteristic
79C2×#125—61no—
80C2×C4×Q8C2×D312yes—
81C2×C4⋊C8—32no—
82C2×#103—34no—
83C2×#103—34no—
84C2×#97—32no—
85#391C2×D312yes—
86#389—21no—
87#389—21no—
88#389—32no—
89#389—32no—
90#389—61no—
91#389—61no—
92#384—31no—
93#384—31no—
94#384C2×D311yescharacteristic
95#382—61no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C43C2×D311yescharacteristic
99#340—34no—
100#340—34no—
101#336—32no—
102C2×C3×Q8—22no—
103C2×C3×Q8—22no—
104C2×C3×Q8—22no—
105C2×C3×Q8—22no—
106C2×C3×Q8—22no—
107C2×C3×Q8#4114yes—
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2411yescharacteristic
117C22×Dic3—22no—
118C22×Dic3—22no—
119C3×C4⋊C4—24no—
120C3×C4⋊C4—24no—
121C3×C4⋊C4—22no—
122C3×C4⋊C4—22no—
123C3×C4⋊C4—44no—
124C3×C4⋊C4—42no—
125C3×C4⋊C4—42no—
126C3×C42—22no—
127C3×C42—22no—
128C3×C42—22no—
129C3×C42—22no—
130C3×C42—22no—
131C3×C42—21no—
132C3×C42—21no—
133C3×C42—44no—
134C3×C42#4114yes—
135C4⋊Dic3—24no—
136C4⋊Dic3—22no—
137C4⋊Dic3—22no—
138C4⋊Dic3—42no—
139C4⋊Dic3—42no—
140C4⋊Dic3—42no—
141C4⋊Dic3—42no—
142Dic3⋊C4—44no—
143Dic3⋊C4—44no—
144C2×C3⋊C8—24no—
145C2×C3⋊C8—24no—
146C2×C3⋊C8—24no—
147C22×Q8C22×D312yes—
148C22×C8—34no—
149C4⋊Q8—24no—
150#125—124no—
151C4×Q8—24no—
152C4×Q8—24no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—32no—
159C2×C4⋊C4—32no—
160C2×C4⋊C4—61no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4—62no—
163C2×C4⋊C4—62no—
164C2×C4⋊C4—61no—
165C2×C4⋊C4—61no—
166C2×C4⋊C4C22×D312yes—
167C2×C42—22no—
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
184C3×Q8—24no—
185C3×Q8—24no—
186C3×Q8—42no—
187C3×Q8—42no—
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—24no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—42no—
202C2×C3×C4—42no—
203C2×C3×C4—42no—
204C2×C3×C4—42no—
205C2×C3×C4—42no—
206C2×C3×C4#12412yes—
207C2×C3×C4#12412yes—
208C2×C3×C4AGL(1,ℤ/8)12yes—
209C2×C3×C4#13712yes—
210C2×C3×C4#13712yes—
211C2×C3×C4#13712yes—
212C2×C3×C4C2×#4112yes—
213C2×C3×C4C2×#4112yes—
214C2×C3×C4H(5,𝔽2)11yescharacteristic
215C2×C3×C42−1+411yescharacteristic
216C2×Dic3—42no—
217C2×Dic3—42no—
218C2×Dic3—42no—
219C2×Dic3—42no—
220C2×Dic3—42no—
221C2×Dic3—42no—
222C3⋊C8—44no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8#23514yes—
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—24no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—44no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—64no—
251C4⋊C4—62no—
252C4⋊C4—62no—
253C4⋊C4—124no—
254C4⋊C4—122no—
255C4⋊C4—122no—
256C4⋊C4—122no—
257C4⋊C4—122no—
258C4⋊C4—124no—
259C42—22no—
260C42—22no—
261C42—22no—
262C42—22no—
263C42—22no—
264C42—21no—
265C42—21no—
266C42—44no—
267C42#23714yes—
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—44no—
279C3×C4—41no—
280C3×C4—41no—
281C3×C4—82no—
282C3×C4#49114yes—
283C3×C4#55212yes—
284C3×C4#55312yes—
285Dic3—82no—
286Dic3—82no—
287C23C22×#6711yescharacteristic
288Q8—24no—
289Q8—24no—
290Q8—42no—
291Q8—42no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—24no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—122no—
311C2×C4—122no—
312C2×C4—122no—
313C2×C4—122no—
314C2×C4—122no—
315C2×C4—122no—
316C2×C4#93612yes—
317C2×C4#93612yes—
318C2×C4#94212yes—
319C2×C4#94212yes—
320C2×C4#94912yes—
321C2×C4#95112yes—
322C2×C4C2×#23512yes—
323C2×C4C2×#23712yes—
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
334C22#579111yescharacteristic
335C22C2×#93611yescharacteristic
336C22C2×#94211yescharacteristic
337C22C2×#94211yescharacteristic
338C22C2×#94911yescharacteristic
339C22C2×#95111yescharacteristic
340C22#583311yescharacteristic
341C4—22no—
342C4—22no—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—44no—
347C4—82no—
348C4—242no—
349C4—242no—
350C4#508214yes—
351C4#554112yes—
352C4#556512yes—
353C3#2913211yescharacteristic, complemented, Sylow
354C2#8238311yescharacteristic
355C2#8244411yescharacteristic
356C2#8244811yescharacteristic
357C2C2×#508211yescharacteristic
358C2#8328311yescharacteristic
359C2#8336611yescharacteristic
360C2#8337411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×C4⋊Q8C2
socle, prime radicalC23×C3C22×D4
hypercenterC4×C4⋊Q8D3
hypocenterC3#29132
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C43 ◁ C4×C4⋊Q8

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×C4×C4⋊Q82
#733072
#717812
#716262
#696752
#695712
#693912
#291323