Finite Group Databasebeta

← #10795130

Subgroups — #10795130

Order 768.

1490 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5486C211yesmaximal, characteristic
2#73820C212yesmaximal, complemented
3C3×#1819C211yesmaximal, characteristic, complemented
4#71542C211yesmaximal, characteristic, complemented
5#69498C212yesmaximal
6#68959C214yesmaximal, complemented
7#68952C214yesmaximal, complemented
8#20724—31nomaximal, Sylow
9C23×#67C2211yescharacteristic
10C22×C3×C22⋊C4C2211yescharacteristic
11C22×#217C2211yescharacteristic
12C2×C4×#67C412yescomplemented
13C2×C4×#67C412yescomplemented
14#5486—24no—
15#5486—24no—
16C2×#67⋊C4C2212yes—
17C2×#67⋊C4C2212yes—
18C2×D3×C22⋊C4C412yescomplemented
19C2×#877C412yescomplemented
20C3×#379C2212yes—
21#5247C2212yes—
22C2×#806C2218yes—
23#4945—24no—
24#4944—24no—
25#4942C2214yescomplemented
26Dic3×C22⋊C4C2214yescomplemented
27C3×#342C2214yescomplemented
28#4526C2214yescomplemented
29C2×#518—31no—
30#2312—32no—
31#1819—31no—
32#1819D311yescharacteristic, complemented
33#1799—34no—
34#1781—32no—
35C4×#99—34no—
36C25×C3C2311yescharacteristic
37C24×D3C2×C411yescharacteristic
38C23×C3×C4C2×C411yescharacteristic
39C23×C3×C4C2×C411yescharacteristic
40C22×#67—22no—
41C22×#67—22no—
42C22×#67—28no—
43C22×#67C2×C412yes—
44C22×#67C2312yes—
45C23×Dic3C2×C411yescharacteristic
46C23×Dic3C2311yescharacteristic
47C23×Dic3C2311yescharacteristic
48C22×C4×D3C2×C412yes—
49C22×C4×D3C2×C412yes—
50C2×C3×C22⋊C4—22no—
51C2×C3×C22⋊C4—22no—
52C2×C3×C22⋊C4—22no—
53C2×C3×C22⋊C4—22no—
54C2×C3×C22⋊C4C2×C412yes—
55C2×C3×C22⋊C4C2314yes—
56C2×C3×C22⋊C4C2312yes—
57C2×#217—22no—
58C2×#217—22no—
59C2×#217C2×C412yes—
60C2×#217C2×C412yes—
61C2×#217D414yes—
62C2×#217D414yes—
63C2×#217C2314yes—
64C4×#67—24no—
65C4×#67—24no—
66C4×#67—24no—
67C4×#67—24no—
68C2×(C2×D3)⋊C4C2×C412yes—
69C2×(C2×D3)⋊C4C2×C412yes—
70C2×Dic3⋊C4—24no—
71C2×Dic3⋊C4C2×C412yes—
72C2×Dic3⋊C4C2×C412yes—
73C2×C4×Dic3—24no—
74C2×C4×Dic3C2×C412yes—
75C2×C4×Dic3C2×C412yes—
76#67⋊C4—28no—
77#67⋊C4—28no—
78D3×C22⋊C4—28no—
79#877—28no—
80C3⋊#95—24no—
81C3⋊#95—24no—
82#806—216no—
83C23×D4—31no—
84#518—64no—
85#518—64no—
86C2×C4×D4—32no—
87C2×C4×D4—32no—
88C2×C4×D4—32no—
89C2×C4×D4—32no—
90C2×#117—32no—
91C22×C22⋊C4—31no—
92C22×C22⋊C4—32no—
93C22×C22⋊C4C2×D311yescharacteristic
94C2×#99—38no—
95#388—64no—
96#388—64no—
97#380—34no—
98#379—32no—
99#379C2×D312yes—
100C4×C22⋊C4—34no—
101#342—34no—
102#342C2×D314yescomplemented
103C24×C3—24no—
104C24×C3—81no—
105C24×C3—81no—
106C24×C3C22×C411yescharacteristic
107C24×C3C22×C411yescharacteristic
108C24×C3C2×D412yes—
109C24×C3C2×D412yes—
110C24×C3C2411yescharacteristic
111C23×D3—22no—
112C23×D3—24no—
113C23×D3—81no—
114C23×D3—81no—
115C23×D3C22×C412yes—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4—21no—
120C22×C3×C4—24no—
121C22×C3×C4—81no—
122C22×C3×C4—81no—
123C22×C3×C4C22×C412yes—
124C22×C3×C4C22×C412yes—
125C2×#67—216no—
126C2×#67—216no—
127C2×#67—22no—
128C2×#67—22no—
129C2×#67—24no—
130C2×#67—24no—
131C2×#67—24no—
132C2×#67—24no—
133C2×#67#4118yes—
134C22×Dic3—22no—
135C22×Dic3—24no—
136C22×Dic3—24no—
137C22×Dic3—81no—
138C22×Dic3—81no—
139C22×Dic3C22×C412yes—
140C22×Dic3C22×C412yes—
141C22×Dic3C22×C412yes—
142C22×Dic3C2×D414yes—
143C22×Dic3C2×D414yes—
144C2×C4×D3—24no—
145C2×C4×D3—24no—
146C2×C4×D3—82no—
147C2×C4×D3—82no—
148C3×C22⋊C4—24no—
149C3×C22⋊C4—24no—
150C3×C22⋊C4—28no—
151C3×C22⋊C4—82no—
152C3×C22⋊C4—82no—
153#217—28no—
154#217—82no—
155#217—82no—
156#217C22⋊C418yes—
157#217C22⋊C414yes—
158#217C22⋊C414yes—
159(C2×D3)⋊C4—24no—
160(C2×D3)⋊C4—24no—
161Dic3⋊C4—24no—
162Dic3⋊C4—24no—
163Dic3⋊C4—84no—
164C4×Dic3—24no—
165C4×Dic3—24no—
166C4×Dic3—84no—
167C25—31no—
168C25C22×D311yescharacteristic
169C22×D4—32no—
170C22×D4—32no—
171C22×D4—68no—
172C22×D4—62no—
173C22×D4—62no—
174C23×C4—31no—
175C23×C4—31no—
176C23×C4—31no—
177C23×C4—32no—
178C23×C4—32no—
179C23×C4C4×D311yescharacteristic
180C23×C4C4×D311yescharacteristic
181C4×D4—68no—
182C4×D4—68no—
183C4×D4—64no—
184C4×D4—64no—
185C4×D4—64no—
186C4×D4—64no—
187#117—68no—
188C2×C4⋊C4—32no—
189C2×C4⋊C4—32no—
190C2×C4⋊C4—64no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—32no—
196C2×C22⋊C4—32no—
197C2×C22⋊C4—34no—
198C2×C22⋊C4—34no—
199C2×C22⋊C4—32no—
200C2×C22⋊C4—32no—
201C2×C22⋊C4—34no—
202C2×C22⋊C4—68no—
203C2×C22⋊C4—62no—
204C2×C22⋊C4—62no—
205C2×C22⋊C4C4×D312yes—
206C2×C22⋊C4C22×D314yes—
207C2×C22⋊C4C22×D312yes—
208C2×C42—32no—
209C2×C42—32no—
210C2×C42—64no—
211#99—616no—
212#95—64no—
213#95—64no—
214C23×C3—22no—
215C23×C3—22no—
216C23×C3—24no—
217C23×C3—24no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3—42no—
225C23×C3—42no—
226C23×C3—42no—
227C23×C3—42no—
228C23×C3—88no—
229C23×C3C2×C22⋊C412yes—
230C23×C3C2×C22⋊C411yescharacteristic
231C23×C3C2×C22⋊C411yescharacteristic
232C23×C3C23×C411yescharacteristic
233C23×C3C22×D412yes—
234C23×C3C2×#4114yes—
235C22×D3—22no—
236C22×D3—22no—
237C22×D3—28no—
238C22×D3—42no—
239C22×D3—42no—
240C22×D3—82no—
241C22×D3—82no—
242C22×D3—88no—
243C22×D3#11714yes—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4—44no—
249C2×C3×C4—44no—
250C2×C3×C4—44no—
251C2×C3×C4—81no—
252C2×C3×C4—81no—
253C2×C3×C4—81no—
254C2×C3×C4—81no—
255#67—28no—
256#67—28no—
257#67—216no—
258C2×Dic3—28no—
259C2×Dic3—22no—
260C2×Dic3—22no—
261C2×Dic3—44no—
262C2×Dic3—44no—
263C2×Dic3—42no—
264C2×Dic3—42no—
265C2×Dic3—44no—
266C2×Dic3—44no—
267C2×Dic3—44no—
268C2×Dic3C2×C22⋊C414yes—
269C2×Dic3C2×C22⋊C414yes—
270C2×Dic3#11714yes—
271C4×D3—84no—
272C4×D3—84no—
273C24—24no—
274C24—32no—
275C24—64no—
276C24—62no—
277C24—81no—
278C24—81no—
279C24—241no—
280C24—241no—
281C24C2×C4×D311yescharacteristic
282C24C2×C4×D311yescharacteristic
283C24D3×D412yes—
284C24D3×D412yes—
285C24C23×D311yescharacteristic
286C2×D4—38no—
287C2×D4—64no—
288C2×D4—64no—
289C2×D4—616no—
290C2×D4—616no—
291C2×D4—62no—
292C2×D4—62no—
293C2×D4—64no—
294C2×D4—64no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—24no—
300C22×C4—32no—
301C22×C4—32no—
302C22×C4—32no—
303C22×C4—34no—
304C22×C4—34no—
305C22×C4—64no—
306C22×C4—64no—
307C22×C4—64no—
308C22×C4—62no—
309C22×C4—64no—
310C22×C4—81no—
311C22×C4—81no—
312C22×C4—242no—
313C22×C4—242no—
314C22×C4—241no—
315C22×C4—241no—
316C22×C4C2×C4×D312yes—
317C22×C4C2×C4×D312yes—
318C4⋊C4—64no—
319C4⋊C4—64no—
320C4⋊C4—244no—
321C22⋊C4—24no—
322C22⋊C4—24no—
323C22⋊C4—28no—
324C22⋊C4—38no—
325C22⋊C4—34no—
326C22⋊C4—34no—
327C22⋊C4—64no—
328C22⋊C4—64no—
329C22⋊C4—68no—
330C22⋊C4—82no—
331C22⋊C4—82no—
332C22⋊C4—242no—
333C22⋊C4—242no—
334C42—64no—
335C42—64no—
336C42—244no—
337C22×C3—22no—
338C22×C3—22no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—22no—
346C22×C3—22no—
347C22×C3—22no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—44no—
351C22×C3—44no—
352C22×C3—44no—
353C22×C3—84no—
354C22×C3—84no—
355C22×C3C22×C22⋊C411yescharacteristic
356C22×C3C2×#11712yes—
357C22×C3#52514yes—
358C2×D3—24no—
359C2×D3—44no—
360C2×D3—42no—
361C2×D3—42no—
362C2×D3—44no—
363C2×D3—88no—
364C3×C4—42no—
365C3×C4—42no—
366C3×C4—84no—
367Dic3—24no—
368Dic3—24no—
369Dic3—24no—
370Dic3—84no—
371C23—22no—
372C23—22no—
373C23—24no—
374C23—24no—
375C23—34no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—68no—
387C23—62no—
388C23—62no—
389C23—88no—
390C23—122no—
391C23—122no—
392C23—242no—
393C23—242no—
394C23—248no—
395C23D3×C22⋊C412yes—
396C23D3×C22⋊C411yescharacteristic
397C23D3×C22⋊C411yescharacteristic
398C23C22×C4×D311yescharacteristic
399C23C2×D3×D412yes—
400C23D3×#4114yes—
401D4—68no—
402D4—68no—
403D4—616no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—34no—
407C2×C4—34no—
408C2×C4—34no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—68no—
415C2×C4—62no—
416C2×C4—62no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—124no—
422C2×C4—124no—
423C2×C4—122no—
424C2×C4—122no—
425C2×C4—124no—
426C2×C4—124no—
427C2×C4—124no—
428C2×C4—244no—
429C2×C4—244no—
430C2×C3—24no—
431C2×C3—22no—
432C2×C3—42no—
433C2×C3—41no—
434C2×C3—41no—
435C2×C3#228011yescharacteristic
436C2×C3#288311yescharacteristic
437C2×C3#288311yescharacteristic
438D3—44no—
439C22—22no—
440C22—22no—
441C22—21no—
442C22—21no—
443C22—22no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—22no—
451C22—22no—
452C22—44no—
453C22—44no—
454C22—44no—
455C22—64no—
456C22—84no—
457C22—84no—
458C22—124no—
459C22—124no—
460C22—122no—
461C22—122no—
462C22—248no—
463C22C2×D3×C22⋊C411yescharacteristic
464C22D3×#11712yes—
465C22#565614yes—
466C4—42no—
467C4—42no—
468C4—64no—
469C4—64no—
470C4—64no—
471C4—84no—
472C4—244no—
473C3#2072411yescharacteristic, complemented, Sylow
474C2—24no—
475C2—22no—
476C2—42no—
477C2—41no—
478C2—41no—
479C2—124no—
480C2#7405011yescharacteristic
481C2#7868811yescharacteristic
482C2#7868811yescharacteristic
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#1819C2
socleC22×C3C22×C22⋊C4
prime radicalC23×#67C22
cosocleC24×C3C24
hypercenter#1819D3
hypocenterC3#20724
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C25 ◁ #1819

Complements

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

NH ≅ G/Ntype
2. #73820C2semidirect
3. C3×#1819C2semidirect
4. #71542C2semidirect
6. #68959C2semidirect
7. #68952C2semidirect
12. C2×C4×#67C4semidirect
13. C2×C4×#67C4semidirect
18. C2×D3×C22⋊C4C4semidirect
19. C2×#877C4semidirect
25. #4942C22semidirect
26. Dic3×C22⋊C4C22semidirect
27. C3×#342C22semidirect
28. #4526C22semidirect
32. #1819D3semidirect
102. #342C2×D3semidirect
473. C3#20724semidirect

Maximal subgroups

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

subgroupindex
C2×#54862
#738202
C3×#18192
#715422
#694982
#689592
#689522
#207243