Finite Group Databasebeta

← #10823326

Subgroups — #10823326

Order 768.

1334 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5483C211yesmaximal, characteristic
2#82095C211yesmaximal, characteristic
3#82094C212yesmaximal
4C2×C3×#342C211yesmaximal, characteristic
5C2×#4527C212yesmaximal
6#68958C218yesmaximal, complemented
7#23565—31nomaximal, Sylow
8C22×C3×C22⋊C4C2211yescharacteristic
9C22×C3×C22⋊C4C2212yes—
10C22×#217C2211yescharacteristic
11C22×#217C2212yes—
12C22×#217C2211yescharacteristic
13C2×#924—22no—
14#5483—22no—
15#5483—24no—
16#5483C2214yes—
17C2×#878—24no—
18#5212—24no—
19#5210—22no—
20#4950—24no—
21#4950C2218yes—
22#4946C2214yes—
23#4942—22no—
24C3×#342C2214yes—
25#4527C2218yes—
26C2×#543—31no—
27#2315—32no—
28#2315—31no—
29#2062—38no—
30C2×#342—32no—
31C2×#342D311yescharacteristic
32C25×C3C2311yescharacteristic
33C23×C3×C4—22no—
34C23×C3×C4D412yes—
35C23×Dic3—21no—
36C23×Dic3D414yes—
37C23×Dic3C2312yes—
38C22×Dic6—22no—
39C2×C3×C22⋊C4—22no—
40C2×C3×C22⋊C4—22no—
41C2×C3×C22⋊C4—22no—
42C2×C3×C22⋊C4—24no—
43C2×C3×C22⋊C4D412yes—
44C2×C3×C22⋊C4Q818yescomplemented
45C2×C3×C22⋊C4C2312yes—
46C2×C3×C22⋊C4C2314yes—
47C2×#217—21no—
48C2×#217—22no—
49C2×#217—22no—
50C2×#217—22no—
51C2×#217—22no—
52C2×#217—22no—
53C2×#217—21no—
54C2×#217—22no—
55C2×#217—21no—
56C2×#217—22no—
57C2×#217D414yes—
58C2×#217Q814yescomplemented
59C2×#217C2314yes—
60C2×#217C2312yes—
61C2×C4⋊Dic3—22no—
62C2×C4⋊Dic3—28no—
63#924—28no—
64#924—28no—
65C2×Dic3⋊C4—24no—
66C2×Dic3⋊C4—24no—
67#878—216no—
68#878—216no—
69C3⋊#95—24no—
70C3⋊#95—28no—
71#543—34no—
72#543—64no—
73#543—62no—
74C2×C22⋊Q8—64no—
75C2×C22⋊Q8—62no—
76C22×C22⋊C4—31no—
77C22×C22⋊C4—32no—
78C22×C22⋊C4—31no—
79C22×C22⋊C4C2×D311yescharacteristic
80C22×C22⋊C4C2×D312yes—
81#385—34no—
82#385—38no—
83#385—64no—
84#385—62no—
85#380—64no—
86#380—62no—
87#342—38no—
88#342C2×D314yes—
89C24×C3—82no—
90C24×C3C2×D411yescharacteristic
91C24×C3C2×D411yescharacteristic
92C24×C3C2×D412yes—
93C24×C3C2×D412yes—
94C24×C3C2×Q814yes—
95C24×C3C2×Q812yes—
96C24×C3#4112yes—
97C24×C3C2411yescharacteristic
98C22×C3×C4—22no—
99C22×C3×C4—24no—
100C22×C3×C4—22no—
101C22×C3×C4—28no—
102C22×C3×C4—82no—
103C22×C3×C4—84no—
104C22×C3×C4C2×D414yes—
105C22×Dic3—24no—
106C22×Dic3—22no—
107C22×Dic3—24no—
108C22×Dic3—24no—
109C22×Dic3—22no—
110C22×Dic3—24no—
111C22×Dic3—82no—
112C22×Dic3—84no—
113C22×Dic3—82no—
114C22×Dic3C2×D418yes—
115C2×Dic6—28no—
116C2×Dic6—84no—
117C3×C22⋊C4—24no—
118C3×C22⋊C4—28no—
119C3×C22⋊C4—24no—
120C3×C22⋊C4—24no—
121C3×C22⋊C4C2×Q818yescomplemented
122#217—24no—
123#217—24no—
124#217—24no—
125#217—24no—
126#217—24no—
127#217—22no—
128#217—41no—
129#217—42no—
130#217—41no—
131#217C2×Q814yescomplemented
132C4⋊Dic3—28no—
133C4⋊Dic3—88no—
134Dic3⋊C4—216no—
135Dic3⋊C4—84no—
136C25C22×D311yescharacteristic
137C22×Q8—62no—
138C23×C4—22no—
139C23×C4—34no—
140C23×C4—32no—
141C23×C4—61no—
142C23×C4D1212yes—
143C22⋊Q8—616no—
144C22⋊Q8—616no—
145C22⋊Q8—68no—
146C22⋊Q8—68no—
147C2×C4⋊C4—64no—
148C2×C4⋊C4—62no—
149C2×C4⋊C4—68no—
150C2×C4⋊C4—64no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—34no—
156C2×C22⋊C4—32no—
157C2×C22⋊C4—34no—
158C2×C22⋊C4—34no—
159C2×C22⋊C4—62no—
160C2×C22⋊C4—62no—
161C2×C22⋊C4—61no—
162C2×C22⋊C4—62no—
163C2×C22⋊C4—62no—
164C2×C22⋊C4—62no—
165C2×C22⋊C4—61no—
166C2×C22⋊C4—61no—
167C2×C22⋊C4—62no—
168C2×C22⋊C4—62no—
169C2×C22⋊C4Dic618yescomplemented
170C2×C22⋊C4D1212yes—
171C2×C22⋊C4C22×D312yes—
172C2×C22⋊C4C22×D314yes—
173#95—68no—
174#95—64no—
175C23×C3—42no—
176C23×C3—44no—
177C23×C3—42no—
178C23×C3—42no—
179C23×C3—44no—
180C23×C3—84no—
181C23×C3—82no—
182C23×C3—82no—
183C23×C3#12011yescharacteristic
184C23×C3#12012yes—
185C23×C3#12011yescharacteristic
186C23×C3C22⋊Q814yes—
187C23×C3C22⋊Q814yes—
188C23×C3C22⋊Q812yes—
189C23×C3C22⋊Q812yes—
190C23×C3C4⋊Q812yes—
191C23×C3C4⋊Q812yes—
192C23×C3C4⋊Q814yes—
193C23×C3C4⋊Q814yes—
194C23×C3C22×D411yescharacteristic
195C23×C3C22×D412yes—
196C23×C3C22×Q812yes—
197C23×C3C22×Q811yescharacteristic
198C23×C3C2×#4111yescharacteristic
199C2×C3×C4—24no—
200C2×C3×C4—28no—
201C2×C3×C4—48no—
202C2×C3×C4—44no—
203C2×C3×C4—44no—
204C2×C3×C4—82no—
205C2×C3×C4—88no—
206C2×Dic3—28no—
207C2×Dic3—24no—
208C2×Dic3—44no—
209C2×Dic3—48no—
210C2×Dic3—48no—
211C2×Dic3—44no—
212C2×Dic3—48no—
213C2×Dic3—42no—
214C2×Dic3—84no—
215C2×Dic3—84no—
216Dic6—88no—
217C24—82no—
218C24C2×Dic614yes—
219C24C2×D1211yescharacteristic
220C24C2×D1211yescharacteristic
221C24D3×D412yes—
222C24D3×D412yes—
223C24#23712yes—
224C24D3×Q812yes—
225C24C23×D311yescharacteristic
226C2×Q8—68no—
227C2×Q8—244no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—28no—
231C22×C4—22no—
232C22×C4—38no—
233C22×C4—64no—
234C22×C4—62no—
235C22×C4—64no—
236C22×C4—64no—
237C22×C4—62no—
238C22×C4—64no—
239C22×C4—82no—
240C22×C4—84no—
241C22×C4—242no—
242C22×C4—244no—
243C22×C4—242no—
244C22×C4C2×D1214yes—
245C4⋊C4—616no—
246C4⋊C4—68no—
247C4⋊C4—244no—
248C4⋊C4—248no—
249C22⋊C4—28no—
250C22⋊C4—24no—
251C22⋊C4—24no—
252C22⋊C4—24no—
253C22⋊C4—34no—
254C22⋊C4—64no—
255C22⋊C4—64no—
256C22⋊C4—64no—
257C22⋊C4—64no—
258C22⋊C4—64no—
259C22⋊C4—62no—
260C22⋊C4—122no—
261C22⋊C4—121no—
262C22⋊C4—121no—
263C22⋊C4C2×Dic618yescomplemented
264C22×C3—22no—
265C22×C3—22no—
266C22×C3—24no—
267C22×C3—24no—
268C22×C3—22no—
269C22×C3—42no—
270C22×C3—44no—
271C22×C3—44no—
272C22×C3—42no—
273C22×C3—42no—
274C22×C3—42no—
275C22×C3—44no—
276C22×C3—44no—
277C22×C3—42no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3#39512yes—
281C22×C3#39514yes—
282C22×C3#39512yes—
283C22×C3C2×#12011yescharacteristic
284C22×C3C2×C22⋊Q812yes—
285C22×C3C2×C22⋊Q811yescharacteristic
286C22×C3C2×C4⋊Q811yescharacteristic
287C22×C3C2×C4⋊Q812yes—
288C3×C4—48no—
289C3×C4—44no—
290Dic3—44no—
291Dic3—48no—
292Dic3—82no—
293C23—42no—
294C23—44no—
295C23—42no—
296C23—44no—
297C23—42no—
298C23—84no—
299C23—82no—
300C23—82no—
301C23#86912yes—
302C23#86912yes—
303C23#87814yes—
304C23#87814yes—
305C23#88212yes—
306C23#88211yescharacteristic
307C23#88211yescharacteristic
308C23#88814yes—
309C23#88814yes—
310C23#89712yes—
311C23#89712yes—
312C23C22×Dic612yes—
313C23C22×D1211yescharacteristic
314C23C2×D3×D412yes—
315C23C2×#23711yescharacteristic
316C23C2×D3×Q811yescharacteristic
317Q8—248no—
318C2×C4—28no—
319C2×C4—24no—
320C2×C4—48no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—68no—
324C2×C4—64no—
325C2×C4—82no—
326C2×C4—88no—
327C2×C4—124no—
328C2×C4—128no—
329C2×C4—128no—
330C2×C4—124no—
331C2×C4—128no—
332C2×C4—122no—
333C2×C4—244no—
334C2×C4—244no—
335C2×C3—21no—
336C2×C3—22no—
337C2×C3—21no—
338C2×C3—22no—
339C2×C3—22no—
340C2×C3—24no—
341C2×C3—22no—
342C2×C3C2×#39511yescharacteristic
343C2×C3#302612yes—
344C22—22no—
345C22—22no—
346C22—24no—
347C22—22no—
348C22—24no—
349C22—44no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—44no—
354C22—44no—
355C22—44no—
356C22—42no—
357C22—42no—
358C22—41no—
359C22—41no—
360C22#465612yes—
361C22#465614yes—
362C22#465612yes—
363C22C2×#86911yescharacteristic
364C22C2×#87812yes—
365C22C2×#88211yescharacteristic
366C22C2×#88812yes—
367C22C2×#89711yescharacteristic
368C4—48no—
369C4—44no—
370C4—124no—
371C4—128no—
372C4—242no—
373C3#2356511yescharacteristic, complemented, Sylow
374C2—22no—
375C2—21no—
376C2—22no—
377C2—22no—
378C2—24no—
379C2—21no—
380C2—22no—
381C2C2×#465611yescharacteristic
382C2#7868612yes—
subgroupG/H
centerC22C2×#882
commutator, cosocleC24×C3C24
FrattiniC24C23×D3
FittingC2×C3×#342C2
socleC22×C3C2×#120
prime radicalC25×C3C23
hypercenterC2×#342D3
hypocenterC3#23565
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ C2×#342

Complements

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

NH ≅ G/Ntype
6. #68958C2semidirect
44. C2×C3×C22⋊C4Q8semidirect
58. C2×#217Q8semidirect
121. C3×C22⋊C4C2×Q8semidirect
131. #217C2×Q8semidirect
169. C2×C22⋊C4Dic6semidirect
263. C22⋊C4C2×Dic6semidirect
373. C3#23565semidirect

Maximal subgroups

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

subgroupindex
C2×#54832
#820952
#820942
C2×C3×#3422
C2×#45272
#689582
#235653