Finite Group Databasebeta

← #10900925

Subgroups — #10900925

Order 768.

1292 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#4968C212yesmaximal
3#82080C211yesmaximal, characteristic
4#71792C214yesmaximal, complemented
5#71783C214yesmaximal, complemented
6#71554C212yesmaximal, complemented
7#71531C211yesmaximal, characteristic, complemented
8#29639—31nomaximal, Sylow
9C23×C3×Q8—21no—
10C2×C3×C4⋊Q8—28no—
11C2×C3×C4⋊Q8—22no—
12C2×C3×C4⋊Q8C2214yes—
13C22×C3×C4⋊C4—21no—
14C22×C3×C4⋊C4C2212yes—
15C22×C3×C42C2211yescharacteristic
16C2×#930C2211yescharacteristic
17C2×#929—21no—
18#5213—21no—
19#5212C2211yescharacteristic
20C2×#830C2212yes—
21#4968—28no—
22C2×#809C2214yes—
23C2×#808C2214yes—
24#4940—22no—
25#4939C2212yes—
26#4928C2212yes—
27#4529C2218yes—
28#4524C2214yes—
29C22×C4⋊Q8D311yescharacteristic, complemented
30C2×#481—32no—
31#2854—31no—
32#2058—34no—
33#2023—34no—
34#1824—32no—
35#1817—31no—
36C22×C3×Q8—28no—
37C22×C3×Q8—24no—
38C22×C3×Q8—48no—
39C23×C3×C4—22no—
40C23×C3×C4D411yescharacteristic
41C23×C3×C4D411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C22×#67—21no—
44C22×D12C2311yescharacteristic
45C3×C4⋊Q8—216no—
46C3×C4⋊Q8—216no—
47C2×C3×C4⋊C4—28no—
48C2×C3×C4⋊C4—24no—
49C2×C3×C4⋊C4—22no—
50C2×C3×C4⋊C4—24no—
51C2×C3×C4⋊C4D418yes—
52C2×C3×C4⋊C4C2314yes—
53C2×C3×C42—24no—
54C2×C3×C42D412yes—
55C2×C3×C42C2312yes—
56C2×#217—21no—
57#930—24no—
58#929—44no—
59C2×(C2×D3)⋊C4—22no—
60C2×(C2×D3)⋊C4—22no—
61C2×(C2×D3)⋊C4—21no—
62C2×C4⋊Dic3C2311yescharacteristic
63C2×C4⋊Dic3C2312yes—
64C2×Dic3⋊C4—22no—
65C22×C3⋊C8C2314yes—
66C3⋊#95—22no—
67C3⋊#95—21no—
68#830—24no—
69#809—28no—
70#808—28no—
71C23×Q8—21no—
72C2×C4⋊Q8—28no—
73C2×C4⋊Q8—22no—
74C2×C4⋊Q8C2×D314yes—
75C2×#123—61no—
76C2×#121—31no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4C2×D312yes—
79C22×C42C2×D311yescharacteristic
80#481—68no—
81C2×#106—34no—
82C2×#102—34no—
83C2×C22⋊C8—32no—
84#390—32no—
85#389—32no—
86#388—62no—
87#386—61no—
88#380—31no—
89#340—38no—
90#336—34no—
91C24×C3C2×D411yescharacteristic
92C23×D3—21no—
93C2×C3×Q8—216no—
94C2×C3×Q8—432no—
95C2×C3×Q8—48no—
96C2×C3×Q8—48no—
97C22×C3×C4—22no—
98C22×C3×C4—22no—
99C22×C3×C4—22no—
100C22×C3×C4—24no—
101C22×C3×C4—24no—
102C22×C3×C4—28no—
103C22×C3×C4—24no—
104C22×C3×C4—48no—
105C22×C3×C4SD1614yes—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D414yes—
108C22×C3×C4C2×D412yes—
109C22×C3×C4C2×D412yes—
110C22×C3×C4C2411yescharacteristic
111C2×#67—42no—
112C2×#67—42no—
113C2×#67—42no—
114C22×Dic3—21no—
115C22×Dic3—22no—
116C2×D12—22no—
117C2×D12—24no—
118C3×C4⋊C4—216no—
119C3×C4⋊C4—28no—
120C3×C4⋊C4—24no—
121C3×C4⋊C4C2×D418yes—
122C3×C42—24no—
123C3×C42—24no—
124#217—42no—
125(C2×D3)⋊C4—42no—
126(C2×D3)⋊C4—44no—
127(C2×D3)⋊C4—44no—
128C4⋊Dic3—22no—
129C4⋊Dic3—24no—
130Dic3⋊C4—44no—
131C2×C3⋊C8—24no—
132C2×C3⋊C8—28no—
133C22×Q8—28no—
134C22×Q8—24no—
135C22×Q8—48no—
136C22×D4—31no—
137C22×D4—61no—
138C23×C4—22no—
139C23×C4#6711yescharacteristic
140C23×C4#6711yescharacteristic
141C23×C4C22×D311yescharacteristic
142C22×C8—34no—
143C4⋊Q8—216no—
144C4⋊Q8—216no—
145#123—124no—
146#121—64no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4—32no—
153C2×C4⋊C4—62no—
154C2×C4⋊C4#6718yes—
155C2×C4⋊C4C22×D314yes—
156C2×C22⋊C4—61no—
157C2×C22⋊C4—61no—
158C2×C22⋊C4—62no—
159C2×C22⋊C4—62no—
160C2×C42—24no—
161C2×C42#6712yes—
162C2×C42C22×D312yes—
163#106—68no—
164#102—68no—
165C22⋊C8—64no—
166#95—61no—
167#95—62no—
168C23×C3—22no—
169C23×C3—22no—
170C23×C3#12012yes—
171C23×C3C2×SD1612yes—
172C23×C3#13712yes—
173C23×C3C22×D411yescharacteristic
174C22×D3—41no—
175C22×D3—42no—
176C22×D3—41no—
177C22×D3—42no—
178C22×D3—41no—
179C3×Q8—432no—
180C2×C3×C4—28no—
181C2×C3×C4—22no—
182C2×C3×C4—22no—
183C2×C3×C4—22no—
184C2×C3×C4—22no—
185C2×C3×C4—22no—
186C2×C3×C4—28no—
187C2×C3×C4—28no—
188C2×C3×C4—28no—
189C2×C3×C4—416no—
190C2×C3×C4—48no—
191C2×C3×C4#12012yes—
192C2×C3×C4C2×SD1614yes—
193C2×C3×C4AGL(1,ℤ/8)14yes—
194C2×C3×C4C22×D412yes—
195C2×C3×C42−1+414yes—
196#67—84no—
197C2×Dic3—41no—
198C2×Dic3—42no—
199C2×Dic3—41no—
200C2×Dic3—42no—
201C2×Dic3—42no—
202C2×Dic3—41no—
203D12—44no—
204C3⋊C8—44no—
205C24—61no—
206C24C2×#6711yescharacteristic
207C2×Q8—216no—
208C2×Q8—432no—
209C2×Q8—48no—
210C2×Q8—48no—
211C2×D4—64no—
212C2×D4—62no—
213C2×D4—122no—
214C2×D4—122no—
215C2×D4—122no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—28no—
222C22×C4—24no—
223C22×C4—48no—
224C22×C4—62no—
225C22×C4—61no—
226C22×C4#21514yes—
227C22×C4C2×#6712yes—
228C22×C4C2×#6712yes—
229C22×C4C2×#6711yescharacteristic
230C22×C4C2×#6714yes—
231C22×C4C23×D311yescharacteristic
232C2×C8—68no—
233C2×C8—64no—
234C4⋊C4—216no—
235C4⋊C4—28no—
236C4⋊C4—24no—
237C4⋊C4—64no—
238C4⋊C4—62no—
239C4⋊C4—124no—
240C4⋊C4C2×#6718yes—
241C22⋊C4—122no—
242C22⋊C4—124no—
243C22⋊C4—124no—
244C22⋊C4—122no—
245C42—24no—
246C42—24no—
247C22×C3—22no—
248C22×C3—24no—
249C22×C3—24no—
250C22×C3—22no—
251C22×C3#44814yes—
252C22×C3C2×#12011yescharacteristic
253C22×C3#53612yes—
254C22×C3C22×SD1611yescharacteristic
255C22×C3C2×AGL(1,ℤ/8)12yes—
256C22×C3C2×#13711yescharacteristic
257C2×D3—81no—
258C2×D3—82no—
259C2×D3—82no—
260C2×D3—81no—
261C2×D3—81no—
262C3×C4—22no—
263C3×C4—24no—
264C3×C4—24no—
265C3×C4—48no—
266C3×C4#53614yes—
267Dic3—82no—
268Dic3—81no—
269C23—22no—
270C23—22no—
271C23—122no—
272C23—121no—
273C23—121no—
274C23—122no—
275C23—121no—
276C23C2×#21512yes—
277C23#94212yes—
278C23#95312yes—
279C23C22×#6711yescharacteristic
280Q8—432no—
281D4—124no—
282D4—244no—
283C2×C4—28no—
284C2×C4—28no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—28no—
291C2×C4—28no—
292C2×C4—416no—
293C2×C4—48no—
294C2×C4—122no—
295C2×C4—122no—
296C2×C4—121no—
297C2×C4—121no—
298C2×C4—122no—
299C2×C4—121no—
300C2×C4#93214yes—
301C2×C4C2×#21514yes—
302C2×C4#95312yes—
303C2×C4#100714yes—
304C2×C4C22×#6712yes—
305C8—124no—
306C2×C3—24no—
307C2×C3#261411yescharacteristic
308C2×C3C2×#44811yescharacteristic
309C2×C3#300311yescharacteristic
310C2×C3#322712yes—
311C2×C3#323912yes—
312D3—161no—
313C22—24no—
314C22—24no—
315C22—22no—
316C22—22no—
317C22—242no—
318C22—242no—
319C22—241no—
320C22—241no—
321C22—241no—
322C22#522714yes—
323C22C2×#93212yes—
324C22C22×#21511yescharacteristic
325C22C2×#94211yescharacteristic
326C22#581612yes—
327C22C2×#95311yescharacteristic
328C4—24no—
329C4—24no—
330C4—22no—
331C4—48no—
332C4—242no—
333C4—241no—
334C4#581614yes—
335C3#2963911yescharacteristic, complemented, Sylow
336C2—24no—
337C2—481no—
338C2#8321612yes—
339C2#8321812yes—
340C2#8403811yescharacteristic
341C2C2×#522711yescharacteristic
342C2#8406911yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×C4⋊Q8C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenterC22×C4⋊Q8D3
hypocenterC3#29639
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4⋊Q8C2semidirect
4. #71792C2semidirect
5. #71783C2semidirect
6. #71554C2semidirect
7. #71531C2semidirect
29. C22×C4⋊Q8D3semidirect
335. C3#29639semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊Q82
C2×#49682
#820802
#717922
#717832
#715542
#715312
#296393