Finite Group Databasebeta

← #10876242

Subgroups — #10876242

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×#2314C211yesmaximal, characteristic, complemented
2#82269C211yesmaximal, characteristic
3#71584C211yesmaximal, characteristic, complemented
4#69587C211yesmaximal, characteristic, complemented
5#69586C211yesmaximal, characteristic, complemented
6#69557C211yesmaximal, characteristic, complemented
7#69557C211yesmaximal, characteristic, complemented
8#69521C212yesmaximal
9#69506C212yesmaximal, complemented
10#69463C212yesmaximal, complemented
11#69462C212yesmaximal, complemented
12#27594—31nomaximal, Sylow
13C23×D12C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C2×#926C2211yescharacteristic
16C3×#388—22no—
17C3×#387—22no—
18C3×#384C2211yescharacteristic, complemented
19C3×#382C2211yescharacteristic, complemented
20C3×#380C2212yes—
21C3×C4×C4⋊C4C2212yescomplemented
22C2×#830C2212yes—
23#4988—22no—
24C2×#809C2212yes—
25C2×#809C2212yes—
26C2×#809C2212yes—
27C2×#809C2212yes—
28C2×#808C2212yescomplemented
29C2×#807C2212yescomplemented
30#4955—22no—
31#4939C2211yescharacteristic
32#4939C2211yescharacteristic
33#4939C2211yescharacteristic
34#4939C2211yescharacteristic
35#4529C2212yescomplemented
36#4529C2212yescomplemented
37#4524C2212yescomplemented
38#4524C2211yescharacteristic, complemented
39#4524C2211yescharacteristic, complemented
40#2314D311yescharacteristic, complemented
41#2309—31no—
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—
51C24×D3—21no—
52C23×C3×C4C2311yescharacteristic
53C22×D12—21no—
54C22×D12—22no—
55C22×D12—22no—
56C22×D12—24no—
57C22×D12—42no—
58C22×D12C2311yescharacteristic
59C22×D12C2311yescharacteristic
60C3×#117—24no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4D412yes—
63C2×C3×C4⋊C4C2312yes—
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C22⋊C4—22no—
66C2×C3×C22⋊C4D412yes—
67C2×C3×C42—22no—
68C2×C3×C42C2311yescharacteristic
69C2×C3×C42C2311yescharacteristic
70C2×#217—21no—
71C2×(C2×D3)⋊C4—22no—
72C2×(C2×D3)⋊C4—22no—
73C2×(C2×D3)⋊C4—22no—
74C2×(C2×D3)⋊C4—22no—
75#926—42no—
76C2×C4⋊Dic3C2311yescharacteristic
77C2×C4⋊Dic3C2311yescharacteristic
78C2×C4×Dic3—21no—
79C22×C3⋊C8C2312yes—
80C22×C3⋊C8C2312yes—
81C3×#95—22no—
82C3×#95—22no—
83C3×#95—22no—
84#830—24no—
85#809—24no—
86#809—24no—
87#809—42no—
88#809—42no—
89#808—42no—
90#807—42no—
91C23×D4—31no—
92C2×#117—31no—
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—31no—
102#390—31no—
103#390—31no—
104#390—31no—
105#390—62no—
106#388—22no—
107#387—22no—
108#384C2×D311yescharacteristic, complemented
109#382C2×D311yescharacteristic, complemented
110#380C2×D312yes—
111#379—62no—
112C4×C4⋊C4C2×D312yescomplemented
113#340—32no—
114#340—32no—
115#336—31no—
116#336—31no—
117#336—32no—
118C24×C3C2×D411yescharacteristic
119C23×D3—21no—
120C23×D3—21no—
121C23×D3—24no—
122C23×D3—42no—
123C23×D3—42no—
124C23×D3—82no—
125C23×D3—82no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
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—21no—
137C22×Dic3—21no—
138C2×D12—22no—
139C2×D12—22no—
140C2×D12—22no—
141C2×D12—22no—
142C2×D12—24no—
143C2×D12—21no—
144C2×D12—21no—
145C2×D12—48no—
146C2×D12—48no—
147C2×D12—42no—
148C2×D12—42no—
149C2×D12#4112yes—
150C2×D12#4112yes—
151C3×C4⋊C4—24no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—82no—
157C3×C22⋊C4—24no—
158C3×C22⋊C4—82no—
159C3×C42—22no—
160C3×C42—22no—
161C3×C42—82no—
162#217—42no—
163(C2×D3)⋊C4—82no—
164(C2×D3)⋊C4—82no—
165(C2×D3)⋊C4—82no—
166(C2×D3)⋊C4—82no—
167C4⋊Dic3—22no—
168C4⋊Dic3—22no—
169C4×Dic3—42no—
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—61no—
177C22×D4—31no—
178C22×D4—31no—
179C22×D4—62no—
180C22×D4—62no—
181C22×D4—64no—
182C22×D4—61no—
183C22×D4—122no—
184C23×C4C22×D311yescharacteristic
185C22×C8—32no—
186C22×C8—32no—
187#117—24no—
188#117—122no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—31no—
191C2×C4⋊C4—31no—
192C2×C4⋊C4#6712yes—
193C2×C4⋊C4C22×D312yes—
194C2×C4⋊C4C22×D312yes—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—61no—
197C2×C22⋊C4—62no—
198C2×C22⋊C4—62no—
199C2×C22⋊C4—62no—
200C2×C22⋊C4—62no—
201C2×C22⋊C4#6712yes—
202C2×C42—22no—
203C2×C42—61no—
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—22no—
214#95—22no—
215#95—22no—
216C23×C3—21no—
217C23×C3—21no—
218C23×C3—22no—
219C23×C3—22no—
220C23×C3#12312yes—
221C23×C3C22×D411yescharacteristic
222C22×D3—22no—
223C22×D3—22no—
224C22×D3—48no—
225C22×D3—41no—
226C22×D3—41no—
227C22×D3—41no—
228C22×D3—41no—
229C22×D3—42no—
230C22×D3—42no—
231C22×D3—42no—
232C22×D3—42no—
233C22×D3—88no—
234C22×D3—88no—
235C22×D3—84no—
236C22×D3—84no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—24no—
240C2×C3×C4—24no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4#12312yes—
254C2×C3×C4C2×#4112yes—
255C2×C3×C4C2×#4111yescharacteristic
256C2×C3×C4C2×#4111yescharacteristic
257C2×C3×C4H(5,𝔽2)12yes—
258C2×Dic3—41no—
259C2×Dic3—41no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262D12—24no—
263D12—24no—
264D12—48no—
265D12—42no—
266D12—42no—
267C3⋊C8—42no—
268C3⋊C8—42no—
269C24—64no—
270C24—61no—
271C24—61no—
272C24—122no—
273C24—122no—
274C24—242no—
275C24—242no—
276C24C2×#6711yescharacteristic
277C2×D4—32no—
278C2×D4—32no—
279C2×D4—64no—
280C2×D4—62no—
281C2×D4—62no—
282C2×D4—61no—
283C2×D4—61no—
284C2×D4—62no—
285C2×D4—62no—
286C2×D4—128no—
287C2×D4—128no—
288C2×D4—122no—
289C2×D4—122no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—61no—
296C22×C4—61no—
297C22×C4#23514yes—
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—22no—
312C4⋊C4—22no—
313C4⋊C4—62no—
314C4⋊C4—62no—
315C4⋊C4—82no—
316C22⋊C4—24no—
317C22⋊C4—82no—
318C22⋊C4—122no—
319C22⋊C4—242no—
320C22⋊C4—242no—
321C22⋊C4—242no—
322C22⋊C4—242no—
323C42—22no—
324C42—22no—
325C42—82no—
326C42—122no—
327C22×C3—24no—
328C22×C3—22no—
329C22×C3—42no—
330C22×C3—41no—
331C22×C3—41no—
332C22×C3C2×#12311yescharacteristic
333C22×C3#52511yescharacteristic
334C22×C3#53511yescharacteristic
335C22×C3#57612yes—
336C22×C3#57712yes—
337C2×D3—44no—
338C2×D3—44no—
339C2×D3—42no—
340C2×D3—42no—
341C2×D3—816no—
342C2×D3—82no—
343C2×D3—82no—
344C2×D3—81no—
345C2×D3—81no—
346C2×D3—81no—
347C2×D3—81no—
348C3×C4—22no—
349C3×C4—42no—
350C3×C4—42no—
351C3×C4—82no—
352C3×C4—82no—
353C3×C4#55112yes—
354C3×C4#55112yes—
355Dic3—81no—
356Dic3—81no—
357C23—21no—
358C23—21no—
359C23—22no—
360C23—22no—
361C23—62no—
362C23—62no—
363C23—128no—
364C23—121no—
365C23—121no—
366C23—121no—
367C23—121no—
368C23—122no—
369C23—122no—
370C23—122no—
371C23—122no—
372C23—248no—
373C23—248no—
374C23—244no—
375C23—244no—
376C23#92912yes—
377C23C22×#6711yescharacteristic
378D4—64no—
379D4—64no—
380D4—128no—
381D4—122no—
382D4—122no—
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—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—121no—
400C2×C4—121no—
401C2×C4—122no—
402C2×C4—122no—
403C2×C4#92912yes—
404C2×C4C2×#23512yes—
405C2×C4D3×#4111yescharacteristic
406C2×C4D3×#4111yescharacteristic
407C2×C4#100912yes—
408C8—122no—
409C8—122no—
410C2×C3—42no—
411C2×C3#248211yescharacteristic
412C2×C3#309612yes—
413C2×C3#311812yes—
414C2×C3#322011yescharacteristic
415C2×C3#322211yescharacteristic
416D3—82no—
417D3—82no—
418C22—22no—
419C22—24no—
420C22—42no—
421C22—41no—
422C22—41no—
423C22—124no—
424C22—124no—
425C22—122no—
426C22—122no—
427C22—2416no—
428C22—242no—
429C22—242no—
430C22—241no—
431C22—241no—
432C22—241no—
433C22—241no—
434C22C2×#92911yescharacteristic
435C22#582811yescharacteristic
436C22#583011yescharacteristic
437C22#583512yes—
438C22#583712yes—
439C4—22no—
440C4—42no—
441C4—42no—
442C4—82no—
443C4—82no—
444C4—241no—
445C4—241no—
446C4#552712yes—
447C4#552712yes—
448C3#2759411yescharacteristic, complemented, Sylow
449C2—42no—
450C2—242no—
451C2—242no—
452C2#8228211yescharacteristic
453C2#8256612yes—
454C2#8343611yescharacteristic
455C2#8344811yescharacteristic
456C2#8346012yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2314C2
socleC23×C3C22×D4
prime radicalC23×D12C22
hypercenter#2314D3
hypocenterC3#27594
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. #69587C2semidirect
5. #69586C2semidirect
6. #69557C2semidirect
7. #69557C2semidirect
9. #69506C2semidirect
10. #69463C2semidirect
11. #69462C2semidirect
18. C3×#384C22semidirect
19. C3×#382C22semidirect
21. C3×C4×C4⋊C4C22semidirect
28. C2×#808C22semidirect
29. C2×#807C22semidirect
35. #4529C22semidirect
36. #4529C22semidirect
37. #4524C22semidirect
38. #4524C22semidirect
39. #4524C22semidirect
40. #2314D3semidirect
108. #384C2×D3semidirect
109. #382C2×D3semidirect
112. C4×C4⋊C4C2×D3semidirect
448. C3#27594semidirect

Maximal subgroups

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

subgroupindex
C3×#23142
#822692
#715842
#695872
#695862
#695572
#695212
#695062
#694632
#694622
#275943