Finite Group Databasebeta

← #10804449

Subgroups — #10804449

Order 768.

1538 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5493C211yesmaximal, characteristic
2C3×#2312C211yesmaximal, characteristic, complemented
3#82093C211yesmaximal, characteristic, complemented
4#69528C214yesmaximal, complemented
5#69498C214yesmaximal, complemented
6#69456C214yesmaximal, complemented
7#21795—31nomaximal, Sylow
8C23×#67C2212yes—
9C22×C3×C22⋊C4C2211yescharacteristic
10C2×#930C2212yes—
11C2×#930C2212yes—
12#5493—24no—
13#5493—24no—
14C2×#886C414yescomplemented
15C2×#882C414yescomplemented
16C3×#388—24no—
17C3×#380C2212yescomplemented
18C3×#379C2212yes—
19C3×C4×C22⋊C4C2212yescomplemented
20#5247C2212yes—
21#5212C2212yescomplemented
22C4×#217C2212yescomplemented
23C2×#806C22116yescomplemented
24#4951—24no—
25C2×#534—31no—
26#2312—31no—
27#2312D311yescharacteristic, complemented
28#1964—34no—
29#1964—34no—
30#1781—34no—
31C25×C3C2311yescharacteristic
32C24×D3C2×C412yes—
33C23×C3×C4C2311yescharacteristic
34C23×C3×C4C2311yescharacteristic
35C22×#67—24no—
36C22×#67—24no—
37C22×#67—48no—
38C22×#67C2×C414yes—
39C22×#67C2314yes—
40C23×Dic3C2×C412yes—
41C22×D12C2×C412yes—
42C22×D12C2×C412yes—
43C2×C3×C4⋊C4—22no—
44C2×C3×C22⋊C4—22no—
45C2×C3×C22⋊C4—22no—
46C2×C3×C22⋊C4C2×C414yes—
47C2×C3×C22⋊C4D414yes—
48C2×C3×C22⋊C4D414yes—
49C2×C3×C22⋊C4C2314yes—
50C2×C3×C42—22no—
51C2×#217—22no—
52C2×#217—22no—
53C2×#217C2314yes—
54#930—24no—
55#930—24no—
56#930—24no—
57#930—24no—
58C2×(C2×D3)⋊C4C2×C414yes—
59C2×(C2×D3)⋊C4C2×C414yes—
60C2×C4⋊Dic3—22no—
61C2×C4⋊Dic3C2×C412yes—
62C2×C4⋊Dic3C2×C412yes—
63C2×C4×Dic3—22no—
64#886—216no—
65#882—216no—
66C3×#95—22no—
67C3×#95—22no—
68C3⋊#95—22no—
69C3⋊#95—22no—
70#806—416no—
71C23×D4—32no—
72#534—64no—
73#534—64no—
74C2×#123—34no—
75C2×#121—32no—
76C2×#121—32no—
77C2×#120—34no—
78C22×C22⋊C4C2×D311yescharacteristic
79C2×#99—316no—
80#388—24no—
81#388—64no—
82#380—32no—
83#380C2×D312yescomplemented
84#379—32no—
85#379C2×D312yes—
86C4×C22⋊C4—32no—
87C4×C22⋊C4C2×D312yescomplemented
88C24×C3—24no—
89C24×C3—81no—
90C24×C3—81no—
91C24×C3C22×C412yes—
92C24×C3C2×D412yes—
93C24×C3C2×D412yes—
94C24×C3C2411yescharacteristic
95C23×D3—24no—
96C23×D3—44no—
97C23×D3—82no—
98C23×D3—82no—
99C23×D3C22×C414yes—
100C22×C3×C4—24no—
101C22×C3×C4—81no—
102C22×C3×C4—81no—
103C22×C3×C4C22×C412yes—
104C22×C3×C4C22×C412yes—
105C22×C3×C4C2×D412yes—
106C22×C3×C4C2×D412yes—
107C22×C3×C4C2×D412yes—
108C22×C3×C4C2×D412yes—
109C2×#67—28no—
110C2×#67—24no—
111C2×#67—24no—
112C2×#67—216no—
113C2×#67—48no—
114C2×#67—416no—
115C2×#67—416no—
116C22×Dic3—24no—
117C22×Dic3—24no—
118C22×Dic3—44no—
119C22×Dic3C22×C414yes—
120C2×D12—24no—
121C2×D12—24no—
122C2×D12—82no—
123C2×D12—82no—
124C3×C4⋊C4—82no—
125C3×C22⋊C4—28no—
126C3×C22⋊C4—82no—
127C3×C22⋊C4—82no—
128C3×C22⋊C4C22⋊C4116yes—
129C3×C42—82no—
130#217—28no—
131#217—82no—
132#217—82no—
133(C2×D3)⋊C4—28no—
134(C2×D3)⋊C4—28no—
135C4⋊Dic3—24no—
136C4⋊Dic3—24no—
137C4⋊Dic3—82no—
138C4×Dic3—82no—
139C25—32no—
140C25C22×D311yescharacteristic
141C22×D4—34no—
142C22×D4—34no—
143C22×D4—32no—
144C22×D4—32no—
145C22×D4—64no—
146C22×D4—64no—
147C22×D4—128no—
148C23×C4—32no—
149C23×C4C22×D311yescharacteristic
150C23×C4C22×D311yescharacteristic
151#123—616no—
152#121—64no—
153#121—64no—
154#121—64no—
155#121—64no—
156#120—616no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—32no—
159C2×C4⋊C4—32no—
160C2×C4⋊C4—62no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—34no—
164C2×C22⋊C4—34no—
165C2×C22⋊C4—34no—
166C2×C22⋊C4—62no—
167C2×C22⋊C4—62no—
168C2×C22⋊C4C4×D314yes—
169C2×C22⋊C4D1214yes—
170C2×C22⋊C4#6714yes—
171C2×C22⋊C4C22×D314yes—
172C2×C42—22no—
173C2×C42—62no—
174#99—1216no—
175#95—22no—
176#95—22no—
177#95—62no—
178#95—62no—
179C23×C3—24no—
180C23×C3—24no—
181C23×C3—24no—
182C23×C3—41no—
183C23×C3—41no—
184C23×C3—42no—
185C23×C3—42no—
186C23×C3—42no—
187C23×C3—42no—
188C23×C3—42no—
189C23×C3—42no—
190C23×C3—88no—
191C23×C3C2×C22⋊C414yes—
192C23×C3C23×C411yescharacteristic
193C23×C3C22×D411yescharacteristic
194C23×C3C22×D411yescharacteristic
195C23×C3H(5,𝔽2)14yes—
196C22×D3—24no—
197C22×D3—28no—
198C22×D3—48no—
199C22×D3—44no—
200C22×D3—44no—
201C22×D3—84no—
202C22×D3—84no—
203C22×D3—816no—
204C2×C3×C4—42no—
205C2×C3×C4—42no—
206C2×C3×C4—42no—
207C2×C3×C4—42no—
208C2×C3×C4—42no—
209C2×C3×C4—42no—
210C2×C3×C4—44no—
211C2×C3×C4—44no—
212C2×C3×C4—44no—
213C2×C3×C4C2×C22⋊C414yes—
214C2×C3×C4C2×C22⋊C414yes—
215#67—416no—
216#67—48no—
217#67—48no—
218C2×Dic3—24no—
219C2×Dic3—28no—
220C2×Dic3—48no—
221C2×Dic3—44no—
222C2×Dic3—44no—
223C2×Dic3—44no—
224D12—84no—
225D12—84no—
226C24—24no—
227C24—34no—
228C24—64no—
229C24—81no—
230C24—81no—
231C24—124no—
232C24—242no—
233C24—242no—
234C24C2×C4×D312yes—
235C24C2×D1212yes—
236C24C2×#6712yes—
237C24C23×D311yescharacteristic
238C2×D4—68no—
239C2×D4—64no—
240C2×D4—616no—
241C2×D4—64no—
242C2×D4—64no—
243C2×D4—64no—
244C2×D4—128no—
245C2×D4—1216no—
246C2×D4—1216no—
247C2×D4—242no—
248C2×D4—242no—
249C22×C4—24no—
250C22×C4—34no—
251C22×C4—64no—
252C22×C4—64no—
253C22×C4—81no—
254C22×C4—81no—
255C22×C4—124no—
256C22×C4C2×C4×D312yes—
257C22×C4C2×C4×D312yes—
258C22×C4C2×D1212yes—
259C22×C4C2×D1212yes—
260C22×C4C2×#6712yes—
261C22×C4C2×#6712yes—
262C4⋊C4—64no—
263C4⋊C4—64no—
264C4⋊C4—82no—
265C4⋊C4—242no—
266C22⋊C4—28no—
267C22⋊C4—68no—
268C22⋊C4—68no—
269C22⋊C4—68no—
270C22⋊C4—82no—
271C22⋊C4—82no—
272C22⋊C4—242no—
273C22⋊C4—242no—
274C22⋊C4(C2×D3)⋊C4116yes—
275C42—82no—
276C42—242no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—24no—
284C22×C3—24no—
285C22×C3—24no—
286C22×C3—44no—
287C22×C3—48no—
288C22×C3—84no—
289C22×C3—84no—
290C22×C3C22×C22⋊C411yescharacteristic
291C22×C3#51812yes—
292C22×C3#53512yes—
293C22×C3#53512yes—
294C2×D3—44no—
295C2×D3—44no—
296C2×D3—44no—
297C2×D3—816no—
298C2×D3—88no—
299C3×C4—24no—
300C3×C4—24no—
301C3×C4—84no—
302Dic3—44no—
303Dic3—84no—
304C23—24no—
305C23—24no—
306C23—24no—
307C23—41no—
308C23—41no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—64no—
316C23—68no—
317C23—88no—
318C23—128no—
319C23—124no—
320C23—124no—
321C23—244no—
322C23—244no—
323C23—2416no—
324C23C2×(C2×D3)⋊C414yes—
325C23C22×C4×D311yescharacteristic
326C23C22×D1211yescharacteristic
327C23#100914yes—
328C23C22×#6711yescharacteristic
329D4—1216no—
330D4—128no—
331D4—128no—
332D4—244no—
333D4—244no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—68no—
344C2×C4—64no—
345C2×C4—128no—
346C2×C4—124no—
347C2×C4—124no—
348C2×C4—124no—
349C2×C4C2×(C2×D3)⋊C414yes—
350C2×C4C2×(C2×D3)⋊C414yes—
351C2×C3—22no—
352C2×C3—22no—
353C2×C3—22no—
354C2×C3—44no—
355C2×C3#230911yescharacteristic
356C2×C3#288311yescharacteristic
357C2×C3#288311yescharacteristic
358D3—84no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—24no—
366C22—24no—
367C22—24no—
368C22—44no—
369C22—48no—
370C22—84no—
371C22—84no—
372C22—124no—
373C22—124no—
374C22—124no—
375C22—248no—
376C22—2416no—
377C22#552112yes—
378C22#552512yes—
379C22C22×(C2×D3)⋊C411yescharacteristic
380C22#583012yes—
381C4—24no—
382C4—24no—
383C4—84no—
384C4—124no—
385C4—244no—
386C3#2179511yescharacteristic, complemented, Sylow
387C2—22no—
388C2—22no—
389C2—22no—
390C2—44no—
391C2—244no—
392C2#8226911yescharacteristic
393C2#8254611yescharacteristic
394C2#8254611yescharacteristic
subgroupG/H
centerC22C22×(C2×D3)⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#2312C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#5493C2
cosocleC24×C3C24
hypercenter#2312D3
hypocenterC3#21795
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C22⋊C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
2. C3×#2312C2semidirect
3. #82093C2semidirect
4. #69528C2semidirect
5. #69498C2semidirect
6. #69456C2semidirect
14. C2×#886C4semidirect
15. C2×#882C4semidirect
17. C3×#380C22semidirect
19. C3×C4×C22⋊C4C22semidirect
21. #5212C22semidirect
22. C4×#217C22semidirect
23. C2×#806C22semidirect
27. #2312D3semidirect
83. #380C2×D3semidirect
87. C4×C22⋊C4C2×D3semidirect
386. C3#21795semidirect

Maximal subgroups

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

subgroupindex
C2×#54932
C3×#23122
#820932
#695282
#694982
#694562
#217953