Finite Group Databasebeta

← #10876214

Subgroups — #10876214

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×#2309C211yesmaximal, characteristic
2#82231C211yesmaximal, characteristic, complemented
3#72128C212yesmaximal
4#71584C211yesmaximal, characteristic, complemented
5#71576C212yesmaximal, complemented
6#69654C212yesmaximal, complemented
7#69640C212yesmaximal, complemented
8#69637C212yesmaximal, complemented
9#69636C212yesmaximal, complemented
10#27591—31nomaximal, Sylow
11C23×C3×D4C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic
13C2×C3×#117C2211yescharacteristic
14C3×#390—22no—
15C3×#390C2212yes—
16C3×#390C2212yes—
17C3×#379—22no—
18C2×#832C2212yes—
19C2×#832C2212yes—
20C2×#832C2212yes—
21C2×#832C2212yes—
22#5212C2212yes—
23C2×#830C2212yes—
24#4980—22no—
25C2×#808C2212yescomplemented
26C2×#807C2212yescomplemented
27#4951—22no—
28#4936C2211yescharacteristic, complemented
29#4935C2211yescharacteristic, complemented
30C4×C4⋊Dic3C2212yescomplemented
31#4529C2212yescomplemented
32#4529C2212yescomplemented
33#4524C2211yescharacteristic, complemented
34#4524C2211yescharacteristic, complemented
35#4524C2212yescomplemented
36#2314—31no—
37#2309D311yescharacteristic
38#2092—32no—
39#2071—32no—
40#1926—32no—
41#1925—32no—
42#1854—32no—
43#1818—31no—
44#1786—32no—
45C25×C3—21no—
46C22×C3×D4—24no—
47C22×C3×D4—21no—
48C22×C3×D4—24no—
49C22×C3×D4—41no—
50C22×C3×D4—41no—
51C22×C3×D4C2312yes—
52C23×C3×C4C2311yescharacteristic
53C3×#117—24no—
54C3×#117—41no—
55C3×#117—41no—
56C2×C3×C4⋊C4D412yes—
57C2×C3×C4⋊C4C2312yes—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—22no—
60C2×C3×C22⋊C4—22no—
61C2×C3×C22⋊C4—22no—
62C2×C3×C22⋊C4—22no—
63C2×C3×C22⋊C4D412yes—
64C2×C3×C42—21no—
65C2×C3×C42C2311yescharacteristic
66C2×C3×C42C2311yescharacteristic
67C2×#217—22no—
68C2×C4⋊Dic3—22no—
69C2×C4⋊Dic3C2312yes—
70C2×C4⋊Dic3C2312yes—
71C2×C4×Dic3—22no—
72C22×C3⋊C8C2312yes—
73C22×C3⋊C8C2312yes—
74#832—24no—
75#832—24no—
76#832—42no—
77#832—42no—
78C3⋊#95—22no—
79C3⋊#95—22no—
80C3⋊#95—22no—
81#830—24no—
82#808—42no—
83#807—42no—
84C23×D4C2×D311yescharacteristic
85C2×#117C2×D311yescharacteristic
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—22no—
95#390C2×D312yes—
96#390C2×D312yes—
97#388—62no—
98#387—62no—
99#384—31no—
100#382—31no—
101#380—32no—
102#379—22no—
103C4×C4⋊C4—32no—
104#340—32no—
105#340—32no—
106#336—32no—
107#336—31no—
108#336—31no—
109C24×C3—24no—
110C24×C3—22no—
111C24×C3—42no—
112C24×C3—42no—
113C24×C3—82no—
114C24×C3—82no—
115C24×C3C2×D411yescharacteristic
116C2×C3×D4—24no—
117C2×C3×D4—22no—
118C2×C3×D4—22no—
119C2×C3×D4—22no—
120C2×C3×D4—22no—
121C2×C3×D4—22no—
122C2×C3×D4—48no—
123C2×C3×D4—48no—
124C2×C3×D4—42no—
125C2×C3×D4—42no—
126C2×C3×D4#4114yes—
127C22×C3×C4—22no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—22no—
131C22×C3×C4C2×D412yes—
132C22×C3×C4C2×D412yes—
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4#4114yes—
135C22×C3×C4C2411yescharacteristic
136C22×Dic3—22no—
137C22×Dic3—22no—
138C3×C4⋊C4—24no—
139C3×C4⋊C4—22no—
140C3×C4⋊C4—22no—
141C3×C22⋊C4—24no—
142C3×C22⋊C4—42no—
143C3×C22⋊C4—82no—
144C3×C22⋊C4—82no—
145C3×C22⋊C4—82no—
146C3×C22⋊C4—82no—
147C3×C42—22no—
148C3×C42—22no—
149C3×C42—42no—
150#217—82no—
151C4⋊Dic3—22no—
152C4⋊Dic3—22no—
153C4⋊Dic3—22no—
154C4⋊Dic3—22no—
155C4⋊Dic3—82no—
156C4×Dic3—82no—
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—21no—
164C22×D4—24no—
165C22×D4—24no—
166C22×D4—21no—
167C22×D4—41no—
168C22×D4—41no—
169C22×D4C22×D312yes—
170C23×C4C22×D311yescharacteristic
171C22×C8—32no—
172C22×C8—32no—
173#117—24no—
174#117—41no—
175#117—41no—
176C2×C4⋊C4—32no—
177C2×C4⋊C4—32no—
178C2×C4⋊C4—62no—
179C2×C4⋊C4#6712yes—
180C2×C4⋊C4C22×D312yes—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—62no—
187C2×C22⋊C4#6712yes—
188C2×C42—21no—
189C2×C42—62no—
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—62no—
200#95—62no—
201#95—62no—
202C23×C3—21no—
203C23×C3—21no—
204C23×C3—22no—
205C23×C3—22no—
206C23×C3—24no—
207C23×C3—48no—
208C23×C3—42no—
209C23×C3—42no—
210C23×C3—42no—
211C23×C3—42no—
212C23×C3—42no—
213C23×C3—42no—
214C23×C3—88no—
215C23×C3—84no—
216C23×C3—84no—
217C23×C3—88no—
218C23×C3#12312yes—
219C23×C3C22×D411yescharacteristic
220C3×D4—24no—
221C3×D4—24no—
222C3×D4—48no—
223C3×D4—42no—
224C3×D4—42no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—24no—
228C2×C3×C4—24no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—22no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4#12312yes—
239C2×C3×C4C2×#4112yes—
240C2×C3×C4C2×#4112yes—
241C2×C3×C4H(5,𝔽2)11yescharacteristic
242C2×C3×C4H(5,𝔽2)11yescharacteristic
243C2×Dic3—42no—
244C2×Dic3—42no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3—42no—
248C2×Dic3—42no—
249C3⋊C8—42no—
250C3⋊C8—42no—
251C24—24no—
252C24—22no—
253C24—42no—
254C24—42no—
255C24—82no—
256C24—82no—
257C24C2×#6711yescharacteristic
258C2×D4—24no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—48no—
265C2×D4—48no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4#23514yes—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—62no—
274C22×C4—62no—
275C22×C4#23714yes—
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—62no—
290C4⋊C4—62no—
291C4⋊C4—62no—
292C4⋊C4—62no—
293C4⋊C4—242no—
294C22⋊C4—24no—
295C22⋊C4—42no—
296C22⋊C4—82no—
297C22⋊C4—82no—
298C22⋊C4—82no—
299C22⋊C4—82no—
300C22⋊C4—242no—
301C42—22no—
302C42—22no—
303C42—42no—
304C42—242no—
305C22×C3—24no—
306C22×C3—22no—
307C22×C3—41no—
308C22×C3—41no—
309C22×C3—42no—
310C22×C3—44no—
311C22×C3—44no—
312C22×C3—44no—
313C22×C3—816no—
314C22×C3—82no—
315C22×C3—82no—
316C22×C3—82no—
317C22×C3—82no—
318C22×C3C2×#12311yescharacteristic
319C22×C3#52511yescharacteristic
320C22×C3#53511yescharacteristic
321C22×C3#57612yes—
322C22×C3#57712yes—
323C3×C4—22no—
324C3×C4—42no—
325C3×C4—42no—
326C3×C4—82no—
327C3×C4#55112yes—
328C3×C4#55112yes—
329Dic3—82no—
330Dic3—82no—
331C23—21no—
332C23—21no—
333C23—22no—
334C23—22no—
335C23—24no—
336C23—48no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—88no—
344C23—84no—
345C23—84no—
346C23—88no—
347C23#93512yes—
348C23C22×#6711yescharacteristic
349D4—24no—
350D4—24no—
351D4—48no—
352D4—42no—
353D4—42no—
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—122no—
368C2×C4—122no—
369C2×C4—122no—
370C2×C4—122no—
371C2×C4—122no—
372C2×C4—122no—
373C2×C4#93512yes—
374C2×C4C2×#23512yes—
375C2×C4C2×#23712yes—
376C2×C4#100411yescharacteristic
377C2×C4#100411yescharacteristic
378C8—122no—
379C8—122no—
380C2×C3—42no—
381C2×C3—84no—
382C2×C3#248211yescharacteristic
383C2×C3#309612yes—
384C2×C3#312211yescharacteristic
385C2×C3#312311yescharacteristic
386C2×C3#322412yes—
387C22—24no—
388C22—22no—
389C22—41no—
390C22—41no—
391C22—42no—
392C22—44no—
393C22—44no—
394C22—44no—
395C22—816no—
396C22—82no—
397C22—82no—
398C22—82no—
399C22—82no—
400C22#579111yescharacteristic
401C22C2×#93511yescharacteristic
402C22#580511yescharacteristic
403C22#583512yes—
404C22#583612yes—
405C4—22no—
406C4—42no—
407C4—42no—
408C4—82no—
409C4—242no—
410C4—242no—
411C4#549512yes—
412C4#549512yes—
413C3#2759111yescharacteristic, complemented, Sylow
414C2—42no—
415C2—84no—
416C2#8211311yescharacteristic
417C2#8254211yescharacteristic
418C2#8254511yescharacteristic
419C2#8337612yes—
420C2#8338912yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2309C2
socleC23×C3C22×D4
prime radicalC23×C3×D4C22
hypercenter#2309D3
hypocenterC3#27591
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2309

Complements

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

NH ≅ G/Ntype
2. #82231C2semidirect
4. #71584C2semidirect
5. #71576C2semidirect
6. #69654C2semidirect
7. #69640C2semidirect
8. #69637C2semidirect
9. #69636C2semidirect
25. C2×#808C22semidirect
26. C2×#807C22semidirect
28. #4936C22semidirect
29. #4935C22semidirect
30. C4×C4⋊Dic3C22semidirect
31. #4529C22semidirect
32. #4529C22semidirect
33. #4524C22semidirect
34. #4524C22semidirect
35. #4524C22semidirect
413. C3#27591semidirect

Maximal subgroups

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

subgroupindex
C3×#23092
#822312
#721282
#715842
#715762
#696542
#696402
#696372
#696362
#275913