Finite Group Databasebeta

← #10876209

Subgroups — #10876209

Order 768.

954 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2314C211yesmaximal, characteristic, complemented
2#82269C211yesmaximal, characteristic
3#71584C211yesmaximal, characteristic, complemented
4#69585C212yesmaximal, complemented
5#69558C212yesmaximal, complemented
6#69521C212yesmaximal
7#69506C212yesmaximal, complemented
8#69463C212yesmaximal, complemented
9#69462C212yesmaximal, complemented
10#27591—31nomaximal, Sylow
11C23×D12C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic
13C2×#926C2211yescharacteristic
14C3×#388—22no—
15C3×#387—22no—
16C3×#384C2211yescharacteristic, complemented
17C3×#382C2211yescharacteristic, complemented
18C3×#380C2212yes—
19C3×C4×C4⋊C4C2212yescomplemented
20C2×#830C2212yes—
21#4988—22no—
22C2×#809C2212yes—
23C2×#809C2212yes—
24C2×#809C2212yes—
25C2×#809C2212yes—
26C2×#808C2212yescomplemented
27C2×#807C2212yescomplemented
28#4955—22no—
29#4939C2212yes—
30#4939C2212yes—
31#4529C2212yescomplemented
32#4529C2212yescomplemented
33#4524C2211yescharacteristic, complemented
34#4524C2211yescharacteristic, complemented
35#4524C2212yescomplemented
36#2314D311yescharacteristic, complemented
37#2309—31no—
38#2092—32no—
39#2071—32no—
40#1926—32no—
41#1925—32no—
42#1854—32no—
43#1818—31no—
44#1786—32no—
45C24×D3—21no—
46C23×C3×C4C2311yescharacteristic
47C22×D12—21no—
48C22×D12—24no—
49C22×D12—24no—
50C22×D12—41no—
51C22×D12—41no—
52C22×D12C2312yes—
53C3×#117—24no—
54C2×C3×C4⋊C4—22no—
55C2×C3×C4⋊C4D412yes—
56C2×C3×C4⋊C4C2312yes—
57C2×C3×C4⋊C4C2312yes—
58C2×C3×C22⋊C4—22no—
59C2×C3×C22⋊C4D412yes—
60C2×C3×C42—22no—
61C2×C3×C42C2311yescharacteristic
62C2×C3×C42C2311yescharacteristic
63C2×#217—21no—
64C2×(C2×D3)⋊C4—22no—
65C2×(C2×D3)⋊C4—22no—
66C2×(C2×D3)⋊C4—22no—
67C2×(C2×D3)⋊C4—22no—
68#926—41no—
69#926—41no—
70C2×C4⋊Dic3C2312yes—
71C2×C4×Dic3—21no—
72C22×C3⋊C8C2312yes—
73C22×C3⋊C8C2312yes—
74C3×#95—22no—
75C3×#95—22no—
76C3×#95—22no—
77#830—24no—
78#809—24no—
79#809—24no—
80#809—42no—
81#809—42no—
82#808—42no—
83#807—42no—
84C23×D4—31no—
85C2×#117—31no—
86C2×#117C2×D311yescharacteristic
87C2×#107—32no—
88C2×#106—32no—
89C2×#102—32no—
90C2×#102—32no—
91C2×#102—32no—
92C2×#102—32no—
93C2×C22⋊C8—32no—
94#390—32no—
95#390—32no—
96#390—62no—
97#388—22no—
98#387—22no—
99#384C2×D311yescharacteristic, complemented
100#382C2×D311yescharacteristic, complemented
101#380C2×D312yes—
102#379—62no—
103C4×C4⋊C4C2×D312yescomplemented
104#340—32no—
105#340—32no—
106#336—32no—
107#336—31no—
108#336—31no—
109C24×C3C2×D411yescharacteristic
110C23×D3—22no—
111C23×D3—24no—
112C23×D3—42no—
113C23×D3—42no—
114C23×D3—82no—
115C23×D3—82no—
116C22×C3×C4—22no—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4C2×D412yes—
122C22×C3×C4C2×D412yes—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4#4114yes—
125C22×C3×C4C2411yescharacteristic
126C22×Dic3—22no—
127C2×D12—22no—
128C2×D12—22no—
129C2×D12—22no—
130C2×D12—22no—
131C2×D12—24no—
132C2×D12—22no—
133C2×D12—48no—
134C2×D12—48no—
135C2×D12—42no—
136C2×D12—42no—
137C2×D12#4114yes—
138C3×C4⋊C4—24no—
139C3×C4⋊C4—22no—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—22no—
143C3×C4⋊C4—82no—
144C3×C22⋊C4—24no—
145C3×C22⋊C4—82no—
146C3×C42—22no—
147C3×C42—22no—
148C3×C42—82no—
149#217—42no—
150(C2×D3)⋊C4—82no—
151(C2×D3)⋊C4—82no—
152(C2×D3)⋊C4—82no—
153(C2×D3)⋊C4—82no—
154C4⋊Dic3—22no—
155C4⋊Dic3—22no—
156C4×Dic3—42no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—22no—
163C25—61no—
164C22×D4—32no—
165C22×D4—64no—
166C22×D4—64no—
167C22×D4—61no—
168C22×D4—121no—
169C22×D4—121no—
170C23×C4C22×D311yescharacteristic
171C22×C8—32no—
172C22×C8—32no—
173#117—24no—
174#117—121no—
175#117—121no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—32no—
178C2×C4⋊C4#6712yes—
179C2×C4⋊C4C22×D312yes—
180C2×C4⋊C4C22×D312yes—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—61no—
183C2×C22⋊C4—62no—
184C2×C22⋊C4—62no—
185C2×C22⋊C4—62no—
186C2×C22⋊C4—62no—
187C2×C22⋊C4#6712yes—
188C2×C42—22no—
189C2×C42—61no—
190C2×C42C22×D311yescharacteristic
191C2×C42C22×D311yescharacteristic
192#107—122no—
193#106—122no—
194#102—64no—
195#102—64no—
196#102—122no—
197#102—122no—
198C22⋊C8—64no—
199#95—22no—
200#95—22no—
201#95—22no—
202C23×C3—21no—
203C23×C3—21no—
204C23×C3—22no—
205C23×C3—22no—
206C23×C3#12312yes—
207C23×C3C22×D411yescharacteristic
208C22×D3—24no—
209C22×D3—48no—
210C22×D3—42no—
211C22×D3—42no—
212C22×D3—42no—
213C22×D3—42no—
214C22×D3—42no—
215C22×D3—42no—
216C22×D3—84no—
217C22×D3—84no—
218C22×D3—88no—
219C22×D3—88no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—24no—
223C2×C3×C4—24no—
224C2×C3×C4—22no—
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4#12312yes—
237C2×C3×C4C2×#4112yes—
238C2×C3×C4C2×#4112yes—
239C2×C3×C4H(5,𝔽2)11yescharacteristic
240C2×C3×C4H(5,𝔽2)11yescharacteristic
241C2×Dic3—42no—
242C2×Dic3—42no—
243C2×Dic3—42no—
244D12—24no—
245D12—24no—
246D12—48no—
247D12—42no—
248D12—42no—
249C3⋊C8—42no—
250C3⋊C8—42no—
251C24—64no—
252C24—62no—
253C24—122no—
254C24—122no—
255C24—242no—
256C24—242no—
257C24C2×#6711yescharacteristic
258C2×D4—34no—
259C2×D4—64no—
260C2×D4—62no—
261C2×D4—62no—
262C2×D4—62no—
263C2×D4—62no—
264C2×D4—62no—
265C2×D4—128no—
266C2×D4—128no—
267C2×D4—122no—
268C2×D4—122no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—62no—
275C22×C4#23514yes—
276C22×C4C2×#6712yes—
277C22×C4C2×#6712yes—
278C22×C4C2×#6711yescharacteristic
279C22×C4C23×D311yescharacteristic
280C2×C8—62no—
281C2×C8—62no—
282C2×C8—62no—
283C2×C8—62no—
284C2×C8—62no—
285C2×C8—62no—
286C4⋊C4—24no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—62no—
292C4⋊C4—62no—
293C4⋊C4—82no—
294C22⋊C4—24no—
295C22⋊C4—82no—
296C22⋊C4—122no—
297C22⋊C4—242no—
298C22⋊C4—242no—
299C22⋊C4—242no—
300C22⋊C4—242no—
301C42—22no—
302C42—22no—
303C42—82no—
304C42—122no—
305C22×C3—24no—
306C22×C3—22no—
307C22×C3—41no—
308C22×C3—41no—
309C22×C3—42no—
310C22×C3C2×#12311yescharacteristic
311C22×C3#52511yescharacteristic
312C22×C3#53511yescharacteristic
313C22×C3#57612yes—
314C22×C3#57712yes—
315C2×D3—44no—
316C2×D3—44no—
317C2×D3—44no—
318C2×D3—816no—
319C2×D3—82no—
320C2×D3—82no—
321C2×D3—82no—
322C2×D3—82no—
323C3×C4—22no—
324C3×C4—42no—
325C3×C4—42no—
326C3×C4—82no—
327C3×C4—82no—
328C3×C4#55112yes—
329C3×C4#55112yes—
330Dic3—82no—
331C23—21no—
332C23—21no—
333C23—22no—
334C23—22no—
335C23—64no—
336C23—128no—
337C23—122no—
338C23—122no—
339C23—122no—
340C23—122no—
341C23—122no—
342C23—122no—
343C23—248no—
344C23—244no—
345C23—244no—
346C23—248no—
347C23#92912yes—
348C23C22×#6711yescharacteristic
349D4—64no—
350D4—64no—
351D4—128no—
352D4—122no—
353D4—122no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—122no—
371C2×C4—122no—
372C2×C4—122no—
373C2×C4#92912yes—
374C2×C4C2×#23512yes—
375C2×C4D3×#4112yes—
376C2×C4#100911yescharacteristic
377C2×C4#100911yescharacteristic
378C8—122no—
379C8—122no—
380C2×C3—42no—
381C2×C3#248211yescharacteristic
382C2×C3#309612yes—
383C2×C3#312211yescharacteristic
384C2×C3#312311yescharacteristic
385C2×C3#322412yes—
386D3—84no—
387C22—22no—
388C22—24no—
389C22—42no—
390C22—41no—
391C22—41no—
392C22—124no—
393C22—124no—
394C22—124no—
395C22—2416no—
396C22—242no—
397C22—242no—
398C22—242no—
399C22—242no—
400C22C2×#92911yescharacteristic
401C22#582811yescharacteristic
402C22#583011yescharacteristic
403C22#583512yes—
404C22#583712yes—
405C4—22no—
406C4—42no—
407C4—42no—
408C4—82no—
409C4—82no—
410C4—242no—
411C4#552712yes—
412C4#552712yes—
413C3#2759111yescharacteristic, complemented, Sylow
414C2—42no—
415C2—244no—
416C2#8228211yescharacteristic
417C2#8256612yes—
418C2#8343511yescharacteristic
419C2#8344711yescharacteristic
420C2#8346312yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2314C2
socleC23×C3C22×D4
prime radicalC23×D12C22
hypercenter#2314D3
hypocenterC3#27591
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2314

Complements

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

NH ≅ G/Ntype
1. C3×#2314C2semidirect
3. #71584C2semidirect
4. #69585C2semidirect
5. #69558C2semidirect
7. #69506C2semidirect
8. #69463C2semidirect
9. #69462C2semidirect
16. C3×#384C22semidirect
17. C3×#382C22semidirect
19. C3×C4×C4⋊C4C22semidirect
26. C2×#808C22semidirect
27. C2×#807C22semidirect
31. #4529C22semidirect
32. #4529C22semidirect
33. #4524C22semidirect
34. #4524C22semidirect
35. #4524C22semidirect
36. #2314D3semidirect
99. #384C2×D3semidirect
100. #382C2×D3semidirect
103. C4×C4⋊C4C2×D3semidirect
413. C3#27591semidirect

Maximal subgroups

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

subgroupindex
C3×#23142
#822692
#715842
#695852
#695582
#695212
#695062
#694632
#694622
#275913