Finite Group Databasebeta

← #10732221

Subgroups — #10732221

Order 768.

1750 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊Q8C211yesmaximal, characteristic
2C2×#5229C212yesmaximal
3C2×#5229C212yesmaximal
4C2×C4×Dic3⋊C4C211yesmaximal, characteristic
5#71567C214yesmaximal
6#71566C214yesmaximal
7#71535C211yesmaximal, characteristic
8#15180—31nomaximal, Sylow
9C23×C3×Q8C2212yes—
10C2×C3×C4⋊Q8—28no—
11C2×C3×C4⋊Q8—24no—
12C22×C3×C4⋊C4C2214yes—
13C22×C3×C42C2211yescharacteristic
14C22×C4⋊Dic3C2212yes—
15C22×Dic3⋊C4C2212yes—
16C22×C4×Dic3C2211yescharacteristic
17C22×C4×Dic3C2211yescharacteristic
18#5229—28no—
19#5229—28no—
20C2×C3⋊#95C2214yes—
21C2×C3⋊#95C2218yes—
22C2×C3⋊#95C2214yes—
23C2×C3⋊#95C2214yes—
24C2×C3⋊#95C2212yes—
25C4×Dic3⋊C4—24no—
26C22×C4⋊Q8D311yescharacteristic
27C2×#391—32no—
28C2×#391—32no—
29C2×C4×C4⋊C4—31no—
30#1442—34no—
31#1441—34no—
32C4⋊#95—31no—
33C22×C3×Q8—24no—
34C22×C3×Q8—24no—
35C22×C3×Q8—24no—
36C22×C3×Q8—24no—
37C22×C3×Q8—24no—
38C22×C3×Q8—24no—
39C22×C3×Q8D418yes—
40C23×C3×C4C2314yes—
41C23×C3×C4C2312yes—
42C23×C3×C4C2311yescharacteristic
43C23×Dic3C2314yes—
44C23×Dic3C2311yescharacteristic
45C23×Dic3C2312yes—
46C23×Dic3C2311yescharacteristic
47C3×C4⋊Q8—232no—
48C2×C3×C4⋊C4—216no—
49C2×C3×C4⋊C4—28no—
50C2×C3×C42—24no—
51C2×C3×C42D414yes—
52C2×C4⋊Dic3—24no—
53C2×C4⋊Dic3—24no—
54C2×C4⋊Dic3—24no—
55C2×Dic3⋊C4—24no—
56C2×Dic3⋊C4—24no—
57C2×Dic3⋊C4Q818yescomplemented
58C2×C4×Dic3—22no—
59C2×C4×Dic3—22no—
60C2×C4×Dic3—22no—
61C2×C4×Dic3—24no—
62C2×C4×Dic3D414yes—
63C3⋊#95—24no—
64C3⋊#95—216no—
65C3⋊#95—28no—
66C3⋊#95—28no—
67C3⋊#95—28no—
68C23×Q8C2×D312yes—
69C2×C4⋊Q8—28no—
70C2×C4⋊Q8—24no—
71C22×C4⋊C4—32no—
72C22×C4⋊C4—32no—
73C22×C4⋊C4C2×D314yes—
74C22×C42—31no—
75C22×C42—31no—
76C22×C42C2×D311yescharacteristic
77#391—68no—
78#391—68no—
79C4×C4⋊C4—64no—
80C2×#95—34no—
81C2×#95—34no—
82C2×#95—38no—
83C2×#95—32no—
84C2×#95—34no—
85C24×C3C2411yescharacteristic
86C2×C3×Q8—216no—
87C2×C3×Q8—216no—
88C2×C3×Q8—216no—
89C2×C3×Q8—48no—
90C2×C3×Q8—48no—
91C2×C3×Q8—48no—
92C2×C3×Q8—48no—
93C22×C3×C4—24no—
94C22×C3×C4—24no—
95C22×C3×C4—24no—
96C22×C3×C4—24no—
97C22×C3×C4—24no—
98C22×C3×C4—24no—
99C22×C3×C4—24no—
100C22×C3×C4—24no—
101C22×C3×C4C2×D412yes—
102C22×C3×C4C2×D418yes—
103C22×C3×C4C2×D414yes—
104C22×C3×C4C2×D414yes—
105C22×C3×C4C2×D412yes—
106C22×C3×C4C2×Q814yes—
107C22×C3×C4#4112yes—
108C22×C3×C4#4114yes—
109C22×C3×C4#4114yes—
110C22×Dic3—21no—
111C22×Dic3—21no—
112C22×Dic3—22no—
113C22×Dic3—21no—
114C22×Dic3—24no—
115C22×Dic3—22no—
116C22×Dic3—21no—
117C22×Dic3—22no—
118C22×Dic3—24no—
119C22×Dic3—24no—
120C22×Dic3—24no—
121C22×Dic3—24no—
122C22×Dic3—28no—
123C22×Dic3—24no—
124C22×Dic3—28no—
125C22×Dic3C2×D412yes—
126C22×Dic3C2×D412yes—
127C22×Dic3C2×Q814yes—
128C22×Dic3C2×Q814yes—
129C3×C4⋊C4—232no—
130C3×C42—28no—
131C4⋊Dic3—48no—
132Dic3⋊C4—216no—
133C4×Dic3—28no—
134C4×Dic3—44no—
135C22×Q8—24no—
136C22×Q8—24no—
137C22×Q8—24no—
138C22×Q8—24no—
139C22×Q8—24no—
140C22×Q8—24no—
141C22×Q8#6718yes—
142C23×C4—31no—
143C23×C4—32no—
144C23×C4—31no—
145C23×C4—34no—
146C23×C4C22×D314yes—
147C23×C4C22×D312yes—
148C23×C4C22×D311yescharacteristic
149C4⋊Q8—232no—
150C2×C4⋊C4—216no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—38no—
153C2×C4⋊C4—64no—
154C2×C4⋊C4—64no—
155C2×C4⋊C4—64no—
156C2×C4⋊C4—64no—
157C2×C4⋊C4—64no—
158C2×C42—24no—
159C2×C42—34no—
160C2×C42—62no—
161C2×C42—62no—
162C2×C42—62no—
163C2×C42—64no—
164C2×C42#6714yes—
165#95—68no—
166#95—68no—
167#95—68no—
168#95—64no—
169#95—616no—
170C23×C3C22×D411yescharacteristic
171C23×C3C22×D411yescharacteristic
172C23×C3C22×D412yes—
173C23×C3C22×Q812yes—
174C23×C3C2×#4111yescharacteristic
175C23×C3C2×#4112yes—
176C23×C3C2×#4112yes—
177C23×C32−1+414yes—
178C3×Q8—432no—
179C2×C3×C4—24no—
180C2×C3×C4—24no—
181C2×C3×C4—28no—
182C2×C3×C4—28no—
183C2×C3×C4—28no—
184C2×C3×C4—28no—
185C2×C3×C4—28no—
186C2×C3×C4—44no—
187C2×C3×C4—44no—
188C2×C3×C4—44no—
189C2×C3×C4—44no—
190C2×C3×C4#12014yes—
191C2×C3×C4#12114yes—
192C2×C3×C4#12118yes—
193C2×C3×C4C22⋊Q818yes—
194C2×C3×C4#12318yes—
195C2×C3×C4#12414yes—
196C2×Dic3—22no—
197C2×Dic3—24no—
198C2×Dic3—22no—
199C2×Dic3—24no—
200C2×Dic3—22no—
201C2×Dic3—24no—
202C2×Dic3—24no—
203C2×Dic3—28no—
204C2×Dic3—42no—
205C2×Dic3—42no—
206C2×Dic3—42no—
207C2×Dic3—42no—
208C2×Dic3—44no—
209C2×Dic3—44no—
210C2×Dic3—44no—
211C2×Dic3—44no—
212C2×Dic3—48no—
213C2×Dic3—48no—
214C2×Dic3C4⋊Q814yescomplemented
215C24C23×D311yescharacteristic
216C2×Q8—216no—
217C2×Q8—216no—
218C2×Q8—216no—
219C2×Q8—48no—
220C2×Q8—48no—
221C2×Q8—48no—
222C2×Q8—48no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—32no—
232C22×C4—34no—
233C22×C4—32no—
234C22×C4—34no—
235C22×C4—61no—
236C22×C4—62no—
237C22×C4—61no—
238C22×C4—61no—
239C22×C4—62no—
240C22×C4—64no—
241C22×C4—64no—
242C22×C4—61no—
243C22×C4—64no—
244C22×C4—62no—
245C22×C4—64no—
246C22×C4—64no—
247C22×C4—68no—
248C22×C4—68no—
249C22×C4—64no—
250C22×C4D3×D412yes—
251C22×C4#23712yes—
252C22×C4#23714yes—
253C22×C4D3×Q814yes—
254C22×C4#23914yes—
255C22×C4C2×#6718yes—
256C22×C4C2×#6714yes—
257C22×C4C2×#6714yes—
258C22×C4C2×#6712yes—
259C4⋊C4—232no—
260C4⋊C4—616no—
261C4⋊C4—128no—
262C42—28no—
263C42—68no—
264C42—124no—
265C22×C3C2×#12011yescharacteristic
266C22×C3C2×#12111yescharacteristic
267C22×C3C2×#12112yes—
268C22×C3C2×C22⋊Q812yes—
269C22×C3C2×#12312yes—
270C22×C3C2×#12411yescharacteristic
271C22×C3C2×C4⋊Q811yescharacteristic
272C22×C3#53211yescharacteristic
273C22×C3#53312yes—
274C22×C3#53612yes—
275C22×C3#53612yes—
276C22×C3#54112yes—
277C22×C3#54814yes—
278C22×C3D4×Q814yes—
279C22×C3#55614yes—
280C22×C3#55714yes—
281C3×C4—28no—
282C3×C4—48no—
283C3×C4#39418yes—
284Dic3—24no—
285Dic3—42no—
286Dic3—44no—
287Dic3—84no—
288C23C2×D3×D411yescharacteristic
289C23C2×#23712yes—
290C23C2×#23711yescharacteristic
291C23C2×D3×Q812yes—
292C23C2×#23912yes—
293C23#100714yes—
294C23C22×#6712yes—
295C23C22×#6711yescharacteristic
296Q8—432no—
297C2×C4—28no—
298C2×C4—24no—
299C2×C4—28no—
300C2×C4—24no—
301C2×C4—28no—
302C2×C4—28no—
303C2×C4—28no—
304C2×C4—34no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—62no—
310C2×C4—64no—
311C2×C4—62no—
312C2×C4—64no—
313C2×C4—64no—
314C2×C4—62no—
315C2×C4—64no—
316C2×C4—68no—
317C2×C4—122no—
318C2×C4—122no—
319C2×C4—122no—
320C2×C4—124no—
321C2×C4—122no—
322C2×C4—124no—
323C2×C4—124no—
324C2×C4—128no—
325C2×C4—124no—
326C2×C4—128no—
327C2×C4#93518yes—
328C2×C4#93614yes—
329C2×C4#93814yes—
330C2×C4#93918yes—
331C2×C4#94618yes—
332C2×C4#95314yes—
333C2×C3C2×#39411yescharacteristic
334C2×C3#240912yes—
335C2×C3#244614yes—
336C2×C3#254512yes—
337C2×C3#254612yes—
338C2×C3#256812yes—
339C2×C3#262111yescharacteristic
340C2×C3#285711yescharacteristic
341C22#572114yes—
342C22D3×C4⋊Q811yescharacteristic
343C22#572412yes—
344C22#572511yescharacteristic
345C22#572812yes—
346C22#573112yes—
347C22#573214yes—
348C22C2×#93512yes—
349C22C2×#93611yescharacteristic
350C22C2×#93811yescharacteristic
351C22C2×#93912yes—
352C22C2×#94612yes—
353C22Q8×#6714yes—
354C22#581612yes—
355C22#581714yes—
356C22C2×#95311yescharacteristic
357C4—28no—
358C4—48no—
359C4—64no—
360C4—122no—
361C4—124no—
362C4—244no—
363C4#522218yes—
364C3#1518011yescharacteristic, complemented, Sylow
365C2#8323414yes—
366C2#8323712yes—
367C2#8323812yes—
368C2#8324412yes—
369C2#8326111yescharacteristic
370C2#8326911yescharacteristic
371C2C2×#522211yescharacteristic
372C2#8406812yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, prime radical, cosocleC24×C3C24
FittingC22×C3×C4⋊Q8C2
hypercenterC22×C4⋊Q8D3
hypocenterC3#15180
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×C4⋊Q8

Complements

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

NH ≅ G/Ntype
57. C2×Dic3⋊C4Q8semidirect
214. C2×Dic3C4⋊Q8semidirect
364. C3#15180semidirect

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×#52292
C2×C4×Dic3⋊C42
#715672
#715662
#715352
#151803