Finite Group Databasebeta

← #10848803

Subgroups — #10848803

Order 768.

1052 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#541C211yesmaximal, characteristic
2C2×#5092C211yesmaximal, characteristic, complemented
3C2×#5069C211yesmaximal, characteristic, complemented
4#82262C211yesmaximal, characteristic
5#71786C211yesmaximal, characteristic, complemented
6#71740C211yesmaximal, characteristic, complemented
7#71700C214yesmaximal
8#71632C214yesmaximal, complemented
9#71582C211yesmaximal, characteristic
10#25609—31nomaximal, Sylow
11C3×#541—24no—
12C3×#541—24no—
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×#126—22no—
15C2×C3×#125—22no—
16C2×C3×#125C2211yescharacteristic
17C2×C3×C22⋊Q8C2212yes—
18C2×C3×C4×Q8C2212yes—
19C2×C3×#117C2211yescharacteristic
20C22×#216C2214yes—
21C2×#924C2212yes—
22C2×#869C2211yescharacteristic
23C2×#835C2214yes—
24C2×#830C2212yes—
25#5092—24no—
26#5069—24no—
27#4981—22no—
28C2×#810C2214yes—
29#4953—22no—
30#4936C2211yescharacteristic
31#4929C2212yes—
32C2×#803C2211yescharacteristic
33C2×C4×C3⋊C8C2211yescharacteristic
34#4529C2214yes—
35#4524C2212yes—
36C2×#541D311yescharacteristic
37C2×#494—31no—
38C2×#491—31no—
39#2621—31no—
40#2001—34no—
41#1991—31no—
42#1960—31no—
43#1878—34no—
44#1750—31no—
45C22×C3×Q8C2312yes—
46C23×C3×C4C2311yescharacteristic
47C22×Dic6C2312yes—
48C3×C4⋊Q8—22no—
49C3×C4⋊Q8D414yes—
50C3×#126—216no—
51C3×#125—216no—
52C3×#125—22no—
53C3×#125D414yes—
54C3×C22⋊Q8—24no—
55C3×C22⋊Q8—24no—
56C3×C4×Q8—24no—
57C3×C4×Q8—24no—
58C3×#117—22no—
59C3×#117—22no—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4D412yes—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C22⋊C4—22no—
67C2×C3×C22⋊C4D412yes—
68C2×C3×C42—22no—
69C2×C3×C42C2311yescharacteristic
70C2×C3×C42C2311yescharacteristic
71C2×#217—22no—
72C2×#216—28no—
73C2×#216—28no—
74C2×#216—44no—
75C2×C4⋊Dic3—22no—
76C2×C4⋊Dic3C2312yes—
77#924—44no—
78C2×Dic3⋊C4—22no—
79C22×C3⋊C8C2312yes—
80C22×C3⋊C8C2312yes—
81#869—22no—
82#869—22no—
83#835—44no—
84C3⋊#95—24no—
85C3⋊#95—22no—
86#830—24no—
87#810—28no—
88#803—22no—
89C4×C3⋊C8—22no—
90C22×Q16—34no—
91#541—24no—
92#541—24no—
93C2×C4⋊Q8—31no—
94C2×C4⋊Q8C2×D311yescharacteristic
95C2×#126—22no—
96C2×#125—22no—
97C2×#125C2×D311yescharacteristic
98C2×C22⋊Q8—32no—
99C2×C22⋊Q8C2×D312yes—
100C2×C4×Q8C2×D312yes—
101C2×#117C2×D311yescharacteristic
102#494—64no—
103#491—64no—
104C2×#103—34no—
105C2×#103—34no—
106C2×C22⋊C8—32no—
107C2×#97—31no—
108C2×C4×C8—31no—
109#398—62no—
110#397—62no—
111#391—32no—
112#382—31no—
113#340—34no—
114#336—32no—
115C24×C3C2×D411yescharacteristic
116C2×C3×Q8—22no—
117C2×C3×Q8—24no—
118C2×C3×Q8—24no—
119C2×C3×Q8C2×D414yes—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4C2×D412yes—
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D412yes—
129C22×C3×C4C2411yescharacteristic
130C22×Dic3—22no—
131C22×Dic3—22no—
132C2×Dic6—22no—
133C2×Dic6—22no—
134C2×Dic6—24no—
135C2×Dic6—24no—
136C3×C4⋊C4—28no—
137C3×C4⋊C4—28no—
138C3×C4⋊C4—28no—
139C3×C4⋊C4—28no—
140C3×C4⋊C4—24no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4C2×D414yes—
143C3×C22⋊C4—28no—
144C3×C22⋊C4—24no—
145C3×C42—28no—
146C3×C42—21no—
147C3×C42—21no—
148C3×C42C2×D412yes—
149C3×C42C2×D412yes—
150#217—44no—
151#216—416no—
152C4⋊Dic3—24no—
153C4⋊Dic3—44no—
154Dic3⋊C4—44no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—22no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8#4114yes—
161C22×Q8—32no—
162C22×Q8C22×D312yes—
163C23×C4C22×D311yescharacteristic
164C2×Q16—68no—
165C2×Q16—68no—
166C2×Q16—124no—
167C22×C8—32no—
168C22×C8—32no—
169C4⋊Q8—22no—
170C4⋊Q8—62no—
171C4⋊Q8—62no—
172C4⋊Q8#6714yes—
173#126—216no—
174#125—216no—
175#125—22no—
176#125#6714yes—
177C22⋊Q8—24no—
178C22⋊Q8—24no—
179C22⋊Q8—124no—
180C4×Q8—24no—
181C4×Q8—24no—
182#117—22no—
183#117—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—32no—
189C2×C4⋊C4—62no—
190C2×C4⋊C4—62no—
191C2×C4⋊C4#6712yes—
192C2×C4⋊C4C22×D312yes—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—62no—
195C2×C22⋊C4#6712yes—
196C2×C42—22no—
197C2×C42C22×D311yescharacteristic
198C2×C42C22×D311yescharacteristic
199#103—68no—
200#103—124no—
201C22⋊C8—64no—
202#97—62no—
203C4×C8—62no—
204#95—62no—
205#95—64no—
206C23×C3—21no—
207C23×C3—21no—
208C23×C3—21no—
209C23×C3—24no—
210C23×C3C22×D411yescharacteristic
211C3×Q8—28no—
212C3×Q8—44no—
213C2×C3×C4—24no—
214C2×C3×C4—22no—
215C2×C3×C4—24no—
216C2×C3×C4—24no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—24no—
221C2×C3×C4—42no—
222C2×C3×C4—42no—
223C2×C3×C4—44no—
224C2×C3×C4—42no—
225C2×C3×C4—42no—
226C2×C3×C4#12014yes—
227C2×C3×C4C22×D411yescharacteristic
228C2×C3×C4C22×D411yescharacteristic
229C2×C3×C4C2×#4112yes—
230C2×C3×C4H(5,𝔽2)12yes—
231C2×Dic3—42no—
232C2×Dic3—42no—
233C2×Dic3—42no—
234C2×Dic3—42no—
235C2×Dic3—44no—
236Dic6—44no—
237Dic6—44no—
238C3⋊C8—24no—
239C3⋊C8—42no—
240C24C2×#6711yescharacteristic
241C2×Q8—22no—
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—64no—
245C2×Q8—64no—
246C2×Q8—62no—
247C2×Q8—62no—
248C2×Q8C2×#6714yes—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—62no—
256C22×C4—62no—
257C22×C4C2×#6712yes—
258C22×C4C2×#6711yescharacteristic
259C22×C4C2×#6712yes—
260C22×C4C23×D311yescharacteristic
261Q16—1216no—
262C2×C8—34no—
263C2×C8—62no—
264C2×C8—62no—
265C2×C8—62no—
266C2×C8—62no—
267C2×C8—62no—
268C4⋊C4—28no—
269C4⋊C4—28no—
270C4⋊C4—28no—
271C4⋊C4—28no—
272C4⋊C4—24no—
273C4⋊C4—22no—
274C4⋊C4—64no—
275C4⋊C4—124no—
276C4⋊C4—124no—
277C4⋊C4C2×#6714yes—
278C22⋊C4—28no—
279C22⋊C4—24no—
280C22⋊C4—124no—
281C42—28no—
282C42—21no—
283C42—21no—
284C42C2×#6712yes—
285C42C2×#6712yes—
286C22×C3—42no—
287C22×C3—41no—
288C22×C3—42no—
289C22×C3—42no—
290C22×C3C2×#12011yescharacteristic
291C22×C3#53211yescharacteristic
292C22×C3#53511yescharacteristic
293C22×C3#57814yes—
294C3×C4—24no—
295C3×C4—41no—
296C3×C4—44no—
297C3×C4—42no—
298C3×C4—44no—
299C3×C4#54612yes—
300C3×C4#55012yes—
301Dic3—82no—
302Dic3—82no—
303C23—21no—
304C23—21no—
305C23—24no—
306C23—21no—
307C23C22×#6711yescharacteristic
308Q8—28no—
309Q8—44no—
310Q8—124no—
311Q8—124no—
312C2×C4—24no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—24no—
316C2×C4—22no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—21no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—122no—
326C2×C4—122no—
327C2×C4—122no—
328C2×C4—124no—
329C2×C4—122no—
330C2×C4#95314yes—
331C2×C4D3×#4112yes—
332C2×C4#100912yes—
333C2×C4C22×#6711yescharacteristic
334C2×C4C22×#6711yescharacteristic
335C8—64no—
336C8—122no—
337C2×C3—81no—
338C2×C3#243111yescharacteristic
339C2×C3#301112yes—
340C2×C3#317712yes—
341C2×C3#322312yes—
342C22—41no—
343C22—42no—
344C22—42no—
345C22—42no—
346C22#583011yescharacteristic
347C22#583211yescharacteristic
348C22#583814yes—
349C22C2×#95311yescharacteristic
350C4—24no—
351C4—41no—
352C4—44no—
353C4—42no—
354C4—44no—
355C4—242no—
356C4—242no—
357C4#582912yes—
358C4#583412yes—
359C3#2560911yescharacteristic, complemented, Sylow
360C2—81no—
361C2#8344412yes—
362C2#8345212yes—
363C2#8409211yescharacteristic
364C2#8411712yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#541C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC2×#541D3
hypocenterC3#25609
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#541

Complements

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

NH ≅ G/Ntype
2. C2×#5092C2semidirect
3. C2×#5069C2semidirect
5. #71786C2semidirect
6. #71740C2semidirect
8. #71632C2semidirect
359. C3#25609semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5412
C2×#50922
C2×#50692
#822622
#717862
#717402
#717002
#716322
#715822
#256093