Finite Group Databasebeta

← #10796274

Subgroups — #10796274

Order 768.

1502 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3428C211yesmaximal, characteristic
2#84086C211yesmaximal, characteristic
3C2×#5240C211yesmaximal, characteristic
4#82226C212yesmaximal, complemented
5#71657C218yesmaximal, complemented
6#71592C212yesmaximal, complemented
7#20888—31nomaximal, Sylow
8C22×C3×#41C2211yescharacteristic
9C3×#520—24no—
10C3×#519—21no—
11C3×#518—21no—
12C3×#518C2214yes—
13C3×C4×#41—24no—
14C2×C3×#117—22no—
15C2×C3×#117C2212yes—
16C2×#926C2211yescharacteristic
17C2×#926C2212yes—
18C22×C4×Dic3C2211yescharacteristic
19#5240C418yescomplemented
20#5229—21no—
21#5221—21no—
22#5220C2214yes—
23C2×#834C2214yes—
24C4×#217—22no—
25#4972—24no—
26#4946C2218yes—
27C4×Dic3⋊C4—24no—
28#4526C2218yes—
29#3428D311yescharacteristic
30C2×#410—31no—
31#2297—31no—
32#2296—32no—
33#1867—38no—
34#1740—32no—
35C2×C3×#41—22no—
36C2×C3×#41—22no—
37C2×C3×#41—24no—
38C2×C3×#41C2×C414yes—
39C2×C3×#41Q814yes—
40C22×C3×Q8D411yescharacteristic
41C22×C3×D4D411yescharacteristic
42C22×C3×D4C2312yes—
43C23×C3×C4D412yes—
44C23×C3×C4C2311yescharacteristic
45C23×Dic3C2312yes—
46C3×C4×Q8—28no—
47C3×C4×D4—28no—
48C3×C4×D4—216no—
49C3×#117—24no—
50C3×#117—24no—
51C3×#117—22no—
52C3×#117—24no—
53C3×#117—22no—
54C3×#117—22no—
55C3×#117—22no—
56C2×C3×C4⋊C4—24no—
57C2×C3×C4⋊C4D414yes—
58C2×C3×C22⋊C4—24no—
59C2×C3×C22⋊C4C2314yes—
60C2×C3×C42—24no—
61C2×C3×C42D414yes—
62C2×#217—24no—
63C2×#217C2312yes—
64C2×#217C2314yes—
65#926—24no—
66#926—24no—
67#926C2×C418yescomplemented
68C2×C4⋊Dic3—22no—
69C2×C4⋊Dic3—21no—
70C2×Dic3⋊C4—28no—
71C2×C4×Dic3—22no—
72C2×C4×Dic3—22no—
73C2×C4×Dic3—21no—
74C2×C4×Dic3—21no—
75C2×C4×Dic3—21no—
76C2×C4×Dic3—22no—
77#834C2×C4116yescomplemented
78C3⋊#95—28no—
79C3⋊#95—24no—
80C22×#41C2×D311yescharacteristic
81#520—24no—
82#519—21no—
83#518—21no—
84#518C2×D314yes—
85C4×#41—24no—
86C2×#117—22no—
87C2×#117—32no—
88C2×#117—31no—
89C2×#117C2×D312yes—
90C22×C42—31no—
91#410—38no—
92C2×#99—34no—
93#391—61no—
94#390—61no—
95#389—64no—
96#386—34no—
97#385—38no—
98C4×C4⋊C4—64no—
99C4×C22⋊C4—62no—
100#342—38no—
101C24×C3C2×D412yes—
102C24×C3C2411yescharacteristic
103C3×#41—28no—
104C3×#41—28no—
105C3×#41—28no—
106C3×#41—28no—
107C2×C3×Q8—22no—
108C2×C3×Q8—22no—
109C2×C3×Q8C4⋊C414yescomplemented
110C2×C3×D4—24no—
111C2×C3×D4—24no—
112C2×C3×D4—22no—
113C2×C3×D4—22no—
114C2×C3×D4C4⋊C414yescomplemented
115C2×C3×D4C22×C414yes—
116C2×C3×D4C2×Q814yes—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—24no—
120C22×C3×C4—28no—
121C22×C3×C4C4⋊C418yescomplemented
122C22×C3×C4C22×C412yes—
123C22×C3×C4C2×D412yes—
124C22×C3×C4C2×D418yes—
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D412yes—
128C22×C3×C4C2×D412yes—
129C22×C3×C4C2×Q812yes—
130C22×C3×C4#4114yes—
131C22×Dic3—24no—
132C22×Dic3—22no—
133C22×Dic3—24no—
134C22×Dic3—22no—
135C22×Dic3—44no—
136C3×C4⋊C4—28no—
137C3×C4⋊C4—24no—
138C3×C4⋊C4—24no—
139C3×C22⋊C4—28no—
140C3×C22⋊C4—24no—
141C3×C22⋊C4—24no—
142C3×C42—28no—
143C3×C42—24no—
144C3×C42—24no—
145#217—24no—
146#217—24no—
147#217—48no—
148#217C22×C418yes—
149C4⋊Dic3—28no—
150C4⋊Dic3—42no—
151Dic3⋊C4—416no—
152C4×Dic3—28no—
153C4×Dic3—42no—
154C4×Dic3—41no—
155C4×Dic3—41no—
156C4×Dic3—42no—
157C2×#41—24no—
158C2×#41—22no—
159C2×#41—22no—
160C2×#41Dic614yes—
161C2×#41C4×D314yes—
162C22×Q8#6711yescharacteristic
163C22×D4#6711yescharacteristic
164C22×D4C22×D312yes—
165C23×C4—32no—
166C23×C4#6712yes—
167C23×C4C22×D311yescharacteristic
168C4×Q8—28no—
169C4×D4—28no—
170C4×D4—216no—
171#117—24no—
172#117—24no—
173#117—22no—
174#117—24no—
175#117—22no—
176#117—22no—
177#117—22no—
178#117—38no—
179#117—64no—
180#117—64no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—62no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—68no—
185C2×C4⋊C4#6714yes—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—34no—
188C2×C22⋊C4—32no—
189C2×C22⋊C4—64no—
190C2×C22⋊C4C22×D314yes—
191C2×C42—24no—
192C2×C42—62no—
193C2×C42—61no—
194C2×C42—61no—
195C2×C42—62no—
196C2×C42—62no—
197C2×C42—61no—
198C2×C42#6714yes—
199#99—316no—
200#95—64no—
201#95—68no—
202C23×C3—22no—
203C23×C3—22no—
204C23×C3—42no—
205C23×C3—44no—
206C23×C3C2×C4⋊C414yes—
207C23×C3C23×C411yescharacteristic
208C23×C3C22×D411yescharacteristic
209C23×C3C22×D412yes—
210C23×C3C22×Q811yescharacteristic
211C23×C3C2×#4112yes—
212C3×Q8—28no—
213C3×D4—28no—
214C3×D4—28no—
215C3×D4—28no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—28no—
219C2×C3×C4—28no—
220C2×C3×C4—28no—
221C2×C3×C4—48no—
222C2×C3×C4—88no—
223C2×C3×C4C2×C4⋊C414yes—
224C2×C3×C4C2×C4⋊C412yes—
225C2×C3×C4C2×C4⋊C412yes—
226C2×C3×C4C4×D414yes—
227C2×C3×C4C4×D414yes—
228C2×C3×C4#12012yes—
229C2×C3×C4#12011yescharacteristic
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4C22⋊Q814yes—
232C2×C3×C4C22⋊Q814yes—
233C2×C3×C4#12312yes—
234C2×C3×C4#12312yes—
235C2×Dic3—28no—
236C2×Dic3—24no—
237C2×Dic3—28no—
238C2×Dic3—48no—
239C2×Dic3—44no—
240C2×Dic3—42no—
241C2×Dic3—42no—
242C2×Dic3—44no—
243C2×Dic3—42no—
244C24C2×#6712yes—
245C24C23×D311yescharacteristic
246#41—28no—
247#41—28no—
248#41—28no—
249#41—28no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8Dic3⋊C414yescomplemented
253C2×D4—24no—
254C2×D4—24no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4Dic3⋊C414yescomplemented
258C2×D4C2×Dic614yes—
259C2×D4C2×C4×D314yes—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—28no—
264C22×C4—62no—
265C22×C4—64no—
266C22×C4—64no—
267C22×C4—62no—
268C22×C4—124no—
269C22×C4Dic3⋊C418yescomplemented
270C22×C4C2×Dic612yes—
271C22×C4C2×C4×D312yes—
272C22×C4#23514yes—
273C22×C4C2×#6712yes—
274C22×C4C2×#6718yes—
275C22×C4C2×#6711yescharacteristic
276C22×C4C2×#6711yescharacteristic
277C22×C4C2×#6712yes—
278C22×C4C2×#6712yes—
279C4⋊C4—28no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—68no—
283C4⋊C4—1216no—
284C4⋊C4—122no—
285C22⋊C4—28no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—38no—
289C22⋊C4—64no—
290C22⋊C4—64no—
291C22⋊C4—128no—
292C42—28no—
293C42—24no—
294C42—24no—
295C42—68no—
296C42—122no—
297C42—122no—
298C42—121no—
299C42—121no—
300C22×C3—24no—
301C22×C3—22no—
302C22×C3—24no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—42no—
306C22×C3—44no—
307C22×C3—44no—
308C22×C3—82no—
309C22×C3C22×C4⋊C411yescharacteristic
310C22×C3C2×C4×D412yes—
311C22×C3C2×#12011yescharacteristic
312C22×C3C2×C22⋊Q812yes—
313C22×C3C2×#12311yescharacteristic
314C3×C4—24no—
315C3×C4—48no—
316C3×C4#38512yes—
317C3×C4#38512yes—
318C3×C4#38514yes—
319Dic3—48no—
320Dic3—44no—
321Dic3—82no—
322C23—22no—
323C23—22no—
324C23—42no—
325C23—44no—
326C23C2×Dic3⋊C414yes—
327C23C22×Dic611yescharacteristic
328C23C22×C4×D311yescharacteristic
329C23C2×#23512yes—
330C23C22×#6711yescharacteristic
331C23C22×#6712yes—
332Q8—28no—
333D4—28no—
334D4—28no—
335D4—28no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—28no—
339C2×C4—28no—
340C2×C4—28no—
341C2×C4—48no—
342C2×C4—64no—
343C2×C4—68no—
344C2×C4—68no—
345C2×C4—88no—
346C2×C4—128no—
347C2×C4—124no—
348C2×C4—122no—
349C2×C4—122no—
350C2×C4—124no—
351C2×C4—122no—
352C2×C4C2×Dic3⋊C414yes—
353C2×C4C2×Dic3⋊C412yes—
354C2×C4C2×Dic3⋊C412yes—
355C2×C4#92414yes—
356C2×C4#92414yes—
357C2×C4C4×#6714yes—
358C2×C4C4×#6714yes—
359C2×C4#92912yes—
360C2×C4#92912yes—
361C2×C4#95312yes—
362C2×C4#95311yescharacteristic
363C2×C4#95311yescharacteristic
364C2×C3—22no—
365C2×C3—21no—
366C2×C3—21no—
367C2×C3—22no—
368C2×C3—44no—
369C2×C3C2×#38511yescharacteristic
370C2×C3#302512yes—
371C22—24no—
372C22—22no—
373C22—24no—
374C22—41no—
375C22—41no—
376C22—42no—
377C22—44no—
378C22—44no—
379C22—82no—
380C22C22×Dic3⋊C411yescharacteristic
381C22C2×#92412yes—
382C22C2×C4×#6712yes—
383C22C2×#92911yescharacteristic
384C22C2×#95311yescharacteristic
385C4—24no—
386C4—48no—
387C4—128no—
388C4—124no—
389C4—242no—
390C4#521012yes—
391C4#521012yes—
392C4#521014yes—
393C3#2088811yescharacteristic, complemented, Sylow
394C2—22no—
395C2—21no—
396C2—21no—
397C2—22no—
398C2—44no—
399C2C2×#521011yescharacteristic
400C2#8412512yes—
subgroupG/H
centerC2×C4#953
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#3428C2
socleC22×C3C2×#120
prime radicalC22×C3×#41C22
cosocleC24×C3C24
hypercenter#3428D3
hypocenterC3#20888
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #3428

Complements

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

NH ≅ G/Ntype
4. #82226C2semidirect
5. #71657C2semidirect
6. #71592C2semidirect
19. #5240C4semidirect
67. #926C2×C4semidirect
77. #834C2×C4semidirect
109. C2×C3×Q8C4⋊C4semidirect
114. C2×C3×D4C4⋊C4semidirect
121. C22×C3×C4C4⋊C4semidirect
252. C2×Q8Dic3⋊C4semidirect
257. C2×D4Dic3⋊C4semidirect
269. C22×C4Dic3⋊C4semidirect
393. C3#20888semidirect

Maximal subgroups

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

subgroupindex
C3×#34282
#840862
C2×#52402
#822262
#716572
#715922
#208883