Finite Group Databasebeta

← #10900918

Subgroups — #10900918

Order 768.

1292 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#4967C212yesmaximal
3#82080C211yesmaximal, characteristic
4#71792C214yesmaximal, complemented
5#71782C214yesmaximal, complemented
6#71553C212yesmaximal, complemented
7#71531C211yesmaximal, characteristic, complemented
8#29638—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#4967—28no—
22C2×#809C2214yes—
23C2×#807C2214yes—
24#4940—22no—
25#4939C2212yes—
26#4928C2212yes—
27#4529C2218yes—
28#4524C2214yes—
29C22×C4⋊Q8D311yescharacteristic, complemented
30C2×#480—32no—
31#2854—31no—
32#2057—34no—
33#2023—34no—
34#1823—32no—
35#1817—31no—
36C22×C3×Q8—28no—
37C22×C3×Q8—24no—
38C22×C3×Q8—44no—
39C22×C3×Q8—44no—
40C23×C3×C4—22no—
41C23×C3×C4D411yescharacteristic
42C23×C3×C4D411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C22×#67—21no—
45C22×D12C2311yescharacteristic
46C3×C4⋊Q8—216no—
47C3×C4⋊Q8—216no—
48C2×C3×C4⋊C4—28no—
49C2×C3×C4⋊C4—24no—
50C2×C3×C4⋊C4—22no—
51C2×C3×C4⋊C4—24no—
52C2×C3×C4⋊C4D418yes—
53C2×C3×C4⋊C4C2314yes—
54C2×C3×C42—24no—
55C2×C3×C42D412yes—
56C2×C3×C42C2312yes—
57C2×#217—21no—
58#930—24no—
59#929—44no—
60C2×(C2×D3)⋊C4—22no—
61C2×(C2×D3)⋊C4—22no—
62C2×(C2×D3)⋊C4—21no—
63C2×C4⋊Dic3C2311yescharacteristic
64C2×C4⋊Dic3C2312yes—
65C2×Dic3⋊C4—22no—
66C22×C3⋊C8C2314yes—
67C3⋊#95—22no—
68C3⋊#95—21no—
69#830—24no—
70#809—28no—
71#807—28no—
72C23×Q8—21no—
73C2×C4⋊Q8—28no—
74C2×C4⋊Q8—22no—
75C2×C4⋊Q8C2×D314yes—
76C2×#123—61no—
77C2×#121—31no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4C2×D312yes—
80C22×C42C2×D311yescharacteristic
81#480—68no—
82C2×#107—34no—
83C2×#102—34no—
84C2×C22⋊C8—32no—
85#390—32no—
86#389—32no—
87#388—62no—
88#386—61no—
89#380—31no—
90#340—38no—
91#336—34no—
92C24×C3C2×D411yescharacteristic
93C23×D3—21no—
94C2×C3×Q8—216no—
95C2×C3×Q8—48no—
96C2×C3×Q8—432no—
97C2×C3×Q8—48no—
98C22×C3×C4—22no—
99C22×C3×C4—22no—
100C22×C3×C4—22no—
101C22×C3×C4—24no—
102C22×C3×C4—24no—
103C22×C3×C4—28no—
104C22×C3×C4—24no—
105C22×C3×C4—44no—
106C22×C3×C4—44no—
107C22×C3×C4D814yes—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D414yes—
110C22×C3×C4C2×D412yes—
111C22×C3×C4C2×D412yes—
112C22×C3×C4C2411yescharacteristic
113C2×#67—42no—
114C2×#67—42no—
115C2×#67—42no—
116C22×Dic3—21no—
117C22×Dic3—22no—
118C2×D12—22no—
119C2×D12—24no—
120C3×C4⋊C4—216no—
121C3×C4⋊C4—28no—
122C3×C4⋊C4—24no—
123C3×C4⋊C4C2×D418yes—
124C3×C42—24no—
125C3×C42—24no—
126#217—42no—
127(C2×D3)⋊C4—42no—
128(C2×D3)⋊C4—44no—
129(C2×D3)⋊C4—44no—
130C4⋊Dic3—22no—
131C4⋊Dic3—24no—
132Dic3⋊C4—44no—
133C2×C3⋊C8—24no—
134C2×C3⋊C8—28no—
135C22×Q8—28no—
136C22×Q8—24no—
137C22×Q8—44no—
138C22×Q8—44no—
139C22×D4—31no—
140C22×D4—61no—
141C23×C4—22no—
142C23×C4#6711yescharacteristic
143C23×C4#6711yescharacteristic
144C23×C4C22×D311yescharacteristic
145C22×C8—34no—
146C4⋊Q8—216no—
147C4⋊Q8—216no—
148#123—124no—
149#121—64no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—31no—
155C2×C4⋊C4—32no—
156C2×C4⋊C4—62no—
157C2×C4⋊C4#6718yes—
158C2×C4⋊C4C22×D314yes—
159C2×C22⋊C4—61no—
160C2×C22⋊C4—61no—
161C2×C22⋊C4—62no—
162C2×C22⋊C4—62no—
163C2×C42—24no—
164C2×C42#6712yes—
165C2×C42C22×D312yes—
166#107—68no—
167#102—68no—
168C22⋊C8—64no—
169#95—61no—
170#95—62no—
171C23×C3—22no—
172C23×C3—22no—
173C23×C3#12012yes—
174C23×C3C2×D812yes—
175C23×C3AGL(1,ℤ/8)12yes—
176C23×C3C22×D411yescharacteristic
177C22×D3—41no—
178C22×D3—42no—
179C22×D3—41no—
180C22×D3—41no—
181C22×D3—42no—
182C3×Q8—432no—
183C2×C3×C4—28no—
184C2×C3×C4—22no—
185C2×C3×C4—22no—
186C2×C3×C4—22no—
187C2×C3×C4—22no—
188C2×C3×C4—22no—
189C2×C3×C4—28no—
190C2×C3×C4—28no—
191C2×C3×C4—28no—
192C2×C3×C4—44no—
193C2×C3×C4—44no—
194C2×C3×C4—416no—
195C2×C3×C4#12012yes—
196C2×C3×C4C2×D814yes—
197C2×C3×C4#13714yes—
198C2×C3×C4C22×D412yes—
199C2×C3×C42−1+414yes—
200#67—84no—
201C2×Dic3—41no—
202C2×Dic3—42no—
203C2×Dic3—41no—
204C2×Dic3—42no—
205C2×Dic3—41no—
206C2×Dic3—42no—
207D12—44no—
208C3⋊C8—44no—
209C24—61no—
210C24C2×#6711yescharacteristic
211C2×Q8—216no—
212C2×Q8—432no—
213C2×Q8—48no—
214C2×Q8—48no—
215C2×D4—64no—
216C2×D4—62no—
217C2×D4—122no—
218C2×D4—122no—
219C2×D4—122no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—28no—
226C22×C4—24no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—62no—
230C22×C4—61no—
231C22×C4#21314yes—
232C22×C4C2×#6712yes—
233C22×C4C2×#6712yes—
234C22×C4C2×#6711yescharacteristic
235C22×C4C2×#6714yes—
236C22×C4C23×D311yescharacteristic
237C2×C8—68no—
238C2×C8—64no—
239C4⋊C4—216no—
240C4⋊C4—28no—
241C4⋊C4—24no—
242C4⋊C4—64no—
243C4⋊C4—62no—
244C4⋊C4—124no—
245C4⋊C4C2×#6718yes—
246C22⋊C4—122no—
247C22⋊C4—124no—
248C22⋊C4—124no—
249C22⋊C4—122no—
250C42—24no—
251C42—24no—
252C22×C3—22no—
253C22×C3—24no—
254C22×C3—24no—
255C22×C3—22no—
256C22×C3#44714yes—
257C22×C3C2×#12011yescharacteristic
258C22×C3#53612yes—
259C22×C3C22×D811yescharacteristic
260C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
261C22×C3C2×#13712yes—
262C2×D3—81no—
263C2×D3—82no—
264C2×D3—82no—
265C2×D3—81no—
266C2×D3—81no—
267C3×C4—22no—
268C3×C4—24no—
269C3×C4—24no—
270C3×C4—48no—
271C3×C4#53614yes—
272Dic3—82no—
273Dic3—81no—
274C23—22no—
275C23—22no—
276C23—121no—
277C23—121no—
278C23—122no—
279C23—122no—
280C23—121no—
281C23C2×#21312yes—
282C23#93212yes—
283C23#95312yes—
284C23C22×#6711yescharacteristic
285Q8—432no—
286D4—124no—
287D4—244no—
288C2×C4—28no—
289C2×C4—28no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—28no—
296C2×C4—28no—
297C2×C4—416no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—122no—
301C2×C4—121no—
302C2×C4—121no—
303C2×C4—122no—
304C2×C4—122no—
305C2×C4—121no—
306C2×C4C2×#21314yes—
307C2×C4#94214yes—
308C2×C4#95312yes—
309C2×C4#100714yes—
310C2×C4C22×#6712yes—
311C8—124no—
312C2×C3—24no—
313C2×C3#261411yescharacteristic
314C2×C3C2×#44711yescharacteristic
315C2×C3#300411yescharacteristic
316C2×C3#322612yes—
317C2×C3#323812yes—
318D3—161no—
319C22—24no—
320C22—24no—
321C22—22no—
322C22—22no—
323C22—242no—
324C22—242no—
325C22—241no—
326C22—241no—
327C22—241no—
328C22#521714yes—
329C22C22×#21311yescharacteristic
330C22C2×#93211yescharacteristic
331C22C2×#94212yes—
332C22#581612yes—
333C22C2×#95311yescharacteristic
334C4—24no—
335C4—24no—
336C4—22no—
337C4—48no—
338C4—242no—
339C4—241no—
340C4#581614yes—
341C3#2963811yescharacteristic, complemented, Sylow
342C2—24no—
343C2—481no—
344C2#8321512yes—
345C2#8321712yes—
346C2C2×#521711yescharacteristic
347C2#8405911yescharacteristic
348C2#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#29638
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. #71782C2semidirect
6. #71553C2semidirect
7. #71531C2semidirect
29. C22×C4⋊Q8D3semidirect
341. C3#29638semidirect

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×#49672
#820802
#717922
#717822
#715532
#715312
#296383