Finite Group Databasebeta

← #10876248

Subgroups — #10876248

Order 768.

954 conjugacy classes of subgroups.

The subgroup lattice has 458 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#69656C211yesmaximal, characteristic, complemented
7#69655C211yesmaximal, characteristic, complemented
8#69641C211yesmaximal, characteristic, complemented
9#69641C211yesmaximal, characteristic, complemented
10#69637C212yesmaximal, complemented
11#69636C212yesmaximal, complemented
12#27594—31nomaximal, Sylow
13C23×C3×D4C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C3×#390—22no—
17C3×#390C2211yescharacteristic
18C3×#390C2211yescharacteristic
19C3×#390C2211yescharacteristic
20C3×#390C2211yescharacteristic
21C3×#379—22no—
22C2×#832C2212yes—
23C2×#832C2212yes—
24C2×#832C2212yes—
25C2×#832C2212yes—
26#5212C2212yes—
27C2×#830C2212yes—
28#4980—22no—
29C2×#808C2212yescomplemented
30C2×#807C2212yescomplemented
31#4951—22no—
32#4936C2211yescharacteristic, complemented
33#4935C2211yescharacteristic, complemented
34C4×C4⋊Dic3C2212yescomplemented
35#4529C2212yescomplemented
36#4529C2212yescomplemented
37#4524C2212yescomplemented
38#4524C2211yescharacteristic, complemented
39#4524C2211yescharacteristic, complemented
40#2314—31no—
41#2309D311yescharacteristic
42#2094—31no—
43#2093—31no—
44#2073—31no—
45#2073—31no—
46#1926—32no—
47#1925—32no—
48#1854—32no—
49#1818—31no—
50#1786—32no—
51C25×C3—21no—
52C22×C3×D4—22no—
53C22×C3×D4—22no—
54C22×C3×D4—21no—
55C22×C3×D4—24no—
56C22×C3×D4—42no—
57C22×C3×D4C2311yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C3×#117—24no—
61C3×#117—42no—
62C2×C3×C4⋊C4D412yes—
63C2×C3×C4⋊C4C2311yescharacteristic
64C2×C3×C4⋊C4C2311yescharacteristic
65C2×C3×C22⋊C4—21no—
66C2×C3×C22⋊C4—22no—
67C2×C3×C22⋊C4—22no—
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4—22no—
70C2×C3×C22⋊C4D412yes—
71C2×C3×C42—21no—
72C2×C3×C42C2311yescharacteristic
73C2×C3×C42C2311yescharacteristic
74C2×#217—22no—
75C2×C4⋊Dic3—22no—
76C2×C4⋊Dic3C2312yes—
77C2×C4⋊Dic3C2312yes—
78C2×C4×Dic3—22no—
79C22×C3⋊C8C2312yes—
80C22×C3⋊C8C2312yes—
81#832—24no—
82#832—24no—
83#832—42no—
84#832—42no—
85C3⋊#95—22no—
86C3⋊#95—22no—
87C3⋊#95—22no—
88#830—24no—
89#808—42no—
90#807—42no—
91C23×D4C2×D311yescharacteristic
92C2×#117C2×D311yescharacteristic
93C2×#117C2×D311yescharacteristic
94C2×#107—32no—
95C2×#106—32no—
96C2×#102—32no—
97C2×#102—32no—
98C2×#102—32no—
99C2×#102—32no—
100C2×C22⋊C8—32no—
101#390—22no—
102#390C2×D311yescharacteristic
103#390C2×D311yescharacteristic
104#390C2×D311yescharacteristic
105#390C2×D311yescharacteristic
106#388—62no—
107#387—62no—
108#384—31no—
109#382—31no—
110#380—32no—
111#379—22no—
112C4×C4⋊C4—32no—
113#340—32no—
114#340—32no—
115#336—31no—
116#336—31no—
117#336—32no—
118C24×C3—24no—
119C24×C3—21no—
120C24×C3—21no—
121C24×C3—42no—
122C24×C3—42no—
123C24×C3—82no—
124C24×C3—82no—
125C24×C3C2×D411yescharacteristic
126C2×C3×D4—24no—
127C2×C3×D4—22no—
128C2×C3×D4—22no—
129C2×C3×D4—21no—
130C2×C3×D4—21no—
131C2×C3×D4—22no—
132C2×C3×D4—22no—
133C2×C3×D4—48no—
134C2×C3×D4—48no—
135C2×C3×D4—42no—
136C2×C3×D4—42no—
137C2×C3×D4#4112yes—
138C2×C3×D4#4112yes—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—22no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4C2×D412yes—
145C22×C3×C4C2×D412yes—
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4#4114yes—
148C22×C3×C4C2411yescharacteristic
149C22×Dic3—22no—
150C22×Dic3—22no—
151C3×C4⋊C4—24no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—22no—
154C3×C22⋊C4—24no—
155C3×C22⋊C4—42no—
156C3×C22⋊C4—82no—
157C3×C22⋊C4—82no—
158C3×C22⋊C4—82no—
159C3×C22⋊C4—82no—
160C3×C42—22no—
161C3×C42—22no—
162C3×C42—42no—
163#217—82no—
164C4⋊Dic3—22no—
165C4⋊Dic3—22no—
166C4⋊Dic3—22no—
167C4⋊Dic3—22no—
168C4⋊Dic3—82no—
169C4×Dic3—82no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—22no—
172C2×C3⋊C8—22no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—22no—
175C2×C3⋊C8—22no—
176C25—21no—
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—24no—
180C22×D4—21no—
181C22×D4—42no—
182C22×D4C22×D311yescharacteristic
183C22×D4C22×D311yescharacteristic
184C23×C4C22×D311yescharacteristic
185C22×C8—32no—
186C22×C8—32no—
187#117—24no—
188#117—42no—
189C2×C4⋊C4—32no—
190C2×C4⋊C4—32no—
191C2×C4⋊C4—62no—
192C2×C4⋊C4#6712yes—
193C2×C4⋊C4C22×D311yescharacteristic
194C2×C4⋊C4C22×D311yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—62no—
201C2×C22⋊C4#6712yes—
202C2×C42—21no—
203C2×C42—62no—
204C2×C42C22×D311yescharacteristic
205C2×C42C22×D311yescharacteristic
206#107—122no—
207#106—122no—
208#102—64no—
209#102—64no—
210#102—122no—
211#102—122no—
212C22⋊C8—64no—
213#95—62no—
214#95—62no—
215#95—62no—
216C23×C3—21no—
217C23×C3—21no—
218C23×C3—22no—
219C23×C3—22no—
220C23×C3—22no—
221C23×C3—22no—
222C23×C3—48no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3—42no—
228C23×C3—42no—
229C23×C3—42no—
230C23×C3—42no—
231C23×C3—88no—
232C23×C3—88no—
233C23×C3—84no—
234C23×C3—84no—
235C23×C3#12312yes—
236C23×C3C22×D411yescharacteristic
237C3×D4—24no—
238C3×D4—24no—
239C3×D4—48no—
240C3×D4—42no—
241C3×D4—42no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—24no—
245C2×C3×C4—24no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4#12312yes—
257C2×C3×C4C2×#4111yescharacteristic
258C2×C3×C4C2×#4111yescharacteristic
259C2×C3×C4C2×#4112yes—
260C2×C3×C4H(5,𝔽2)12yes—
261C2×Dic3—42no—
262C2×Dic3—42no—
263C2×Dic3—42no—
264C2×Dic3—42no—
265C2×Dic3—42no—
266C2×Dic3—42no—
267C3⋊C8—42no—
268C3⋊C8—42no—
269C24—24no—
270C24—21no—
271C24—21no—
272C24—42no—
273C24—42no—
274C24—82no—
275C24—82no—
276C24C2×#6711yescharacteristic
277C2×D4—24no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—48no—
285C2×D4—48no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4#23512yes—
289C2×D4#23512yes—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—62no—
296C22×C4—62no—
297C22×C4#23714yes—
298C22×C4C2×#6712yes—
299C22×C4C2×#6712yes—
300C22×C4C2×#6711yescharacteristic
301C22×C4C23×D311yescharacteristic
302C2×C8—62no—
303C2×C8—62no—
304C2×C8—62no—
305C2×C8—62no—
306C2×C8—62no—
307C2×C8—62no—
308C4⋊C4—24no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—62no—
312C4⋊C4—62no—
313C4⋊C4—62no—
314C4⋊C4—62no—
315C4⋊C4—242no—
316C22⋊C4—24no—
317C22⋊C4—42no—
318C22⋊C4—82no—
319C22⋊C4—82no—
320C22⋊C4—82no—
321C22⋊C4—82no—
322C22⋊C4—242no—
323C42—22no—
324C42—22no—
325C42—42no—
326C42—242no—
327C22×C3—24no—
328C22×C3—22no—
329C22×C3—42no—
330C22×C3—41no—
331C22×C3—41no—
332C22×C3—44no—
333C22×C3—44no—
334C22×C3—42no—
335C22×C3—42no—
336C22×C3—816no—
337C22×C3—82no—
338C22×C3—82no—
339C22×C3—81no—
340C22×C3—81no—
341C22×C3—81no—
342C22×C3—81no—
343C22×C3C2×#12311yescharacteristic
344C22×C3#52511yescharacteristic
345C22×C3#53511yescharacteristic
346C22×C3#57612yes—
347C22×C3#57712yes—
348C3×C4—22no—
349C3×C4—42no—
350C3×C4—42no—
351C3×C4—81no—
352C3×C4—81no—
353C3×C4#55112yes—
354C3×C4#55112yes—
355Dic3—82no—
356Dic3—82no—
357C23—21no—
358C23—21no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—48no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—88no—
373C23—88no—
374C23—84no—
375C23—84no—
376C23#93512yes—
377C23C22×#6711yescharacteristic
378D4—24no—
379D4—24no—
380D4—48no—
381D4—42no—
382D4—42no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—122no—
398C2×C4—122no—
399C2×C4—122no—
400C2×C4—122no—
401C2×C4—122no—
402C2×C4—122no—
403C2×C4#93512yes—
404C2×C4C2×#23511yescharacteristic
405C2×C4C2×#23511yescharacteristic
406C2×C4C2×#23712yes—
407C2×C4#100412yes—
408C8—122no—
409C8—122no—
410C2×C3—42no—
411C2×C3—82no—
412C2×C3—82no—
413C2×C3#248211yescharacteristic
414C2×C3#309612yes—
415C2×C3#311812yes—
416C2×C3#322011yescharacteristic
417C2×C3#322211yescharacteristic
418C22—24no—
419C22—22no—
420C22—42no—
421C22—41no—
422C22—41no—
423C22—44no—
424C22—44no—
425C22—42no—
426C22—42no—
427C22—816no—
428C22—82no—
429C22—82no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22#579111yescharacteristic
435C22C2×#93511yescharacteristic
436C22#580511yescharacteristic
437C22#583512yes—
438C22#583612yes—
439C4—22no—
440C4—42no—
441C4—42no—
442C4—81no—
443C4—81no—
444C4—242no—
445C4—242no—
446C4#549512yes—
447C4#549512yes—
448C3#2759411yescharacteristic, complemented, Sylow
449C2—42no—
450C2—82no—
451C2—82no—
452C2#8211311yescharacteristic
453C2#8254112yes—
454C2#8337612yes—
455C2#8338811yescharacteristic
456C2#8339311yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2309C2
socleC23×C3C22×D4
prime radicalC23×C3×D4C22
hypercenter#2309D3
hypocenterC3#27594
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. #69656C2semidirect
7. #69655C2semidirect
8. #69641C2semidirect
9. #69641C2semidirect
10. #69637C2semidirect
11. #69636C2semidirect
29. C2×#808C22semidirect
30. C2×#807C22semidirect
32. #4936C22semidirect
33. #4935C22semidirect
34. C4×C4⋊Dic3C22semidirect
35. #4529C22semidirect
36. #4529C22semidirect
37. #4524C22semidirect
38. #4524C22semidirect
39. #4524C22semidirect
448. C3#27594semidirect

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
#696562
#696552
#696412
#696372
#696362
#275943