Finite Group Databasebeta

← #10849283

Subgroups — #10849283

Order 768.

1324 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5577C211yesmaximal, characteristic
2#82483C211yesmaximal, characteristic, complemented
3C2×#4847C211yesmaximal, characteristic, complemented
4C2×#4713C211yesmaximal, characteristic, complemented
5C3×#1985C211yesmaximal, characteristic, complemented
6#70457C211yesmaximal, characteristic, complemented
7#69812C211yesmaximal, characteristic, complemented
8#68671C212yesmaximal, complemented
9#68670C212yesmaximal, complemented
10#68629C212yesmaximal, complemented
11#68629C212yesmaximal, complemented
12#25655—31nomaximal, Sylow
13C22×#239—21no—
14C2×C3×C4×Q8C2211yescharacteristic
15#5577—24no—
16#5577—24no—
17C2×#895—22no—
18C2×#895—21no—
19C2×#874C2212yes—
20C2×#874C2211yescharacteristic
21C2×C4×D12C2212yes—
22C2×C4×D12C2211yescharacteristic
23C2×#870C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#97C2211yescharacteristic
26#5230—21no—
27C2×#821C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2212yes—
30C2×#821C2212yes—
31C2×#821C2212yes—
32C2×#821C2212yes—
33C2×#821C2212yes—
34C2×#818C2211yescharacteristic
35C2×#817C2211yescharacteristic
36#4983—21no—
37#4976—22no—
38C4×C4⋊Dic3C2211yescharacteristic
39#4928C2212yes—
40#4928C2212yes—
41#4847—24no—
42#4713—24no—
43C3×#336C2212yes—
44C3×#336C2212yes—
45#4551C2212yes—
46C2×#550—31no—
47C2×#490—31no—
48C2×#478—31no—
49#2584—31no—
50#2041—32no—
51#2041—32no—
52#1985D311yescharacteristic, complemented
53#1982—31no—
54#1864—32no—
55#1862—32no—
56#1762—31no—
57C22×C3×Q8D411yescharacteristic
58C2×#239—28no—
59C2×#239—48no—
60C22×D12—22no—
61C22×D12—21no—
62C22×D12—22no—
63C22×D12—21no—
64C22×D12C2312yes—
65C22×D12C2311yescharacteristic
66C22×C4×D3—22no—
67C22×C4×D3—21no—
68C22×C3×C8C2312yes—
69C22×C3×C8C2311yescharacteristic
70C22×C3×C8C2311yescharacteristic
71C3×C4×Q8—22no—
72C3×C4×Q8—22no—
73C2×C3×C4⋊C4D412yes—
74C2×C3×C4⋊C4D411yescharacteristic
75C2×C3×C42C2312yes—
76C2×C3×C42C2311yescharacteristic
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—21no—
79C2×C4⋊Dic3C2311yescharacteristic
80C2×C4⋊Dic3C2311yescharacteristic
81C2×C4⋊Dic3C2311yescharacteristic
82C2×C4⋊Dic3C2312yes—
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—21no—
85C2×Dic3⋊C4—21no—
86C2×Dic3⋊C4—22no—
87C2×C4×Dic3—21no—
88#895—28no—
89#895—216no—
90#874—24no—
91#874—24no—
92#874—22no—
93#874D414yes—
94C4×D12—24no—
95C4×D12—22no—
96C4×D12—24no—
97C4×D12D414yes—
98#870—22no—
99#870—22no—
100C3×C4⋊C8—22no—
101C3×#97—22no—
102C3⋊#95—22no—
103C3⋊#95—22no—
104C3⋊#95—21no—
105#821—24no—
106#821—24no—
107#821—22no—
108#821—22no—
109#821—24no—
110#821—24no—
111#821—24no—
112#818—22no—
113#817—22no—
114C22×#41—61no—
115#550—64no—
116#550—64no—
117C2×#127—31no—
118C2×#127—32no—
119C2×#125—31no—
120C2×#121—62no—
121C2×#121—61no—
122C2×C4×Q8C2×D311yescharacteristic
123C2×C4×D4—31no—
124C2×C4×D4—32no—
125#490—64no—
126#478—64no—
127C2×#107—31no—
128C2×#106—31no—
129C2×C4⋊C8C2×D311yescharacteristic
130C2×#102—31no—
131C2×#102—31no—
132C2×#102—32no—
133C2×#102—32no—
134C2×#102—32no—
135C2×#102—32no—
136C2×#102—32no—
137C2×#97C2×D311yescharacteristic
138#398—61no—
139#395—61no—
140#389—32no—
141#389—32no—
142#389—62no—
143C4×C4⋊C4—31no—
144#340—32no—
145#336C2×D312yes—
146#336C2×D312yes—
147C23×D3—22no—
148C23×D3—21no—
149C2×C3×Q8—22no—
150C2×C3×Q8—24no—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2411yescharacteristic
156C22×Dic3—22no—
157C22×Dic3—21no—
158C22×Dic3—21no—
159C22×Dic3—21no—
160#239—432no—
161C2×D12—22no—
162C2×D12—22no—
163C2×D12—22no—
164C2×D12—22no—
165C2×D12—22no—
166C2×D12—22no—
167C2×D12—28no—
168C2×D12—24no—
169C2×D12—22no—
170C2×D12—21no—
171C2×D12—28no—
172C2×D12—24no—
173C2×D12—44no—
174C2×D12—44no—
175C2×D12—44no—
176C2×D12—81no—
177C2×D12—81no—
178C2×D12—82no—
179C2×D12—82no—
180C2×D12C2×D414yes—
181C2×D12C2×D412yes—
182C2×C4×D3—28no—
183C2×C4×D3—24no—
184C2×C4×D3—44no—
185C2×C4×D3—44no—
186C2×C4×D3—44no—
187C2×C3×C8—21no—
188C2×C3×C8—21no—
189C2×C3×C8—22no—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—22no—
196C3×C42—22no—
197C3×C42—22no—
198C3×C42C2×D412yes—
199C3×C42C2×D412yes—
200(C2×D3)⋊C4—28no—
201(C2×D3)⋊C4—24no—
202C4⋊Dic3—21no—
203C4⋊Dic3—22no—
204C4⋊Dic3—21no—
205C4⋊Dic3—21no—
206C4⋊Dic3—21no—
207C4⋊Dic3—21no—
208C4⋊Dic3C2×D412yes—
209C4⋊Dic3#4114yescomplemented
210Dic3⋊C4—42no—
211Dic3⋊C4—42no—
212Dic3⋊C4—44no—
213Dic3⋊C4—82no—
214C4×Dic3—81no—
215C2×#41—68no—
216C2×#41—128no—
217C22×Q8D1211yescharacteristic
218C22×D4—31no—
219C22×D4—32no—
220C22×D4—61no—
221C22×D4—62no—
222C22×D4—62no—
223C22×D4—61no—
224C23×C4—62no—
225C23×C4—61no—
226C22×C8C22×D312yes—
227C22×C8C22×D311yescharacteristic
228C22×C8C22×D311yescharacteristic
229#127—34no—
230#127—64no—
231#127—64no—
232#127—62no—
233#125—62no—
234#125—62no—
235#121—68no—
236#121—616no—
237C4×Q8—22no—
238C4×Q8—22no—
239C4×D4—34no—
240C4×D4—64no—
241C4×D4—64no—
242C4×D4—62no—
243C2×C4⋊C4—31no—
244C2×C4⋊C4—32no—
245C2×C4⋊C4—31no—
246C2×C4⋊C4—31no—
247C2×C4⋊C4—61no—
248C2×C4⋊C4—61no—
249C2×C4⋊C4—62no—
250C2×C4⋊C4—62no—
251C2×C4⋊C4D1212yes—
252C2×C4⋊C4D1211yescharacteristic
253C2×C22⋊C4—62no—
254C2×C22⋊C4—61no—
255C2×C42—61no—
256C2×C42C22×D312yes—
257C2×C42C22×D311yescharacteristic
258#107—62no—
259#106—62no—
260C4⋊C8—22no—
261#102—64no—
262#102—64no—
263#102—64no—
264#102—62no—
265#102—62no—
266#102—64no—
267#102—64no—
268#97—22no—
269#95—61no—
270#95—62no—
271#95—62no—
272C23×C3C22×D411yescharacteristic
273C22×D3—24no—
274C22×D3—22no—
275C22×D3—42no—
276C22×D3—42no—
277C22×D3—42no—
278C22×D3—42no—
279C22×D3—42no—
280C22×D3—42no—
281C22×D3—42no—
282C22×D3—41no—
283C22×D3—42no—
284C22×D3—41no—
285C3×Q8—28no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4#12012yes—
295C2×C3×C4#12012yes—
296C2×C3×C4AGL(1,ℤ/8)14yes—
297C2×C3×C4C22×D411yescharacteristic
298C2×C3×C4C22×D411yescharacteristic
299C2×C3×C4C2×#4112yes—
300C2×C3×C42−1+412yes—
301C2×Dic3—22no—
302C2×Dic3—24no—
303C2×Dic3—42no—
304C2×Dic3—42no—
305C2×Dic3—42no—
306C2×Dic3—41no—
307C2×Dic3—41no—
308C2×Dic3—42no—
309C2×Dic3—41no—
310C2×Dic3—41no—
311D12—22no—
312D12—22no—
313D12—24no—
314D12—24no—
315D12—48no—
316D12—416no—
317D12—42no—
318D12—42no—
319D12—42no—
320D12—84no—
321D12—88no—
322C4×D3—48no—
323C4×D3—416no—
324C3×C8—41no—
325C3×C8—41no—
326C3×C8—42no—
327C24—62no—
328C24—61no—
329#41—1232no—
330C2×Q8—22no—
331C2×Q8—24no—
332C2×D4—32no—
333C2×D4—34no—
334C2×D4—68no—
335C2×D4—68no—
336C2×D4—64no—
337C2×D4—64no—
338C2×D4—62no—
339C2×D4—61no—
340C2×D4—62no—
341C2×D4—62no—
342C2×D4—62no—
343C2×D4—62no—
344C2×D4—62no—
345C2×D4—62no—
346C2×D4—124no—
347C2×D4—124no—
348C2×D4—124no—
349C2×D4—241no—
350C2×D4—241no—
351C2×D4—242no—
352C2×D4—242no—
353C22×C4—68no—
354C22×C4—62no—
355C22×C4—61no—
356C22×C4—64no—
357C22×C4—61no—
358C22×C4—61no—
359C22×C4—124no—
360C22×C4—124no—
361C22×C4—124no—
362C22×C4C2×D1212yes—
363C22×C4C2×D1211yescharacteristic
364C22×C4C2×D1211yescharacteristic
365C22×C4C2×D1212yes—
366C22×C4C23×D311yescharacteristic
367C2×C8—22no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C4⋊C4—24no—
375C4⋊C4—22no—
376C4⋊C4—32no—
377C4⋊C4—34no—
378C4⋊C4—61no—
379C4⋊C4—62no—
380C4⋊C4—61no—
381C4⋊C4—61no—
382C4⋊C4—61no—
383C4⋊C4—61no—
384C4⋊C4—122no—
385C4⋊C4—122no—
386C4⋊C4—124no—
387C4⋊C4—242no—
388C22⋊C4—68no—
389C22⋊C4—64no—
390C42—22no—
391C42—22no—
392C42—241no—
393C42D3×D412yes—
394C42D3×D412yes—
395C22×C3C2×#12011yescharacteristic
396C22×C3#54812yes—
397C22×C3C2×AGL(1,ℤ/8)12yes—
398C22×C3#57611yescharacteristic
399C22×C3#57711yescharacteristic
400C2×D3—42no—
401C2×D3—44no—
402C2×D3—42no—
403C2×D3—44no—
404C2×D3—44no—
405C2×D3—42no—
406C2×D3—82no—
407C2×D3—82no—
408C2×D3—82no—
409C2×D3—82no—
410C2×D3—82no—
411C2×D3—82no—
412C3×C4—22no—
413C3×C4—24no—
414C3×C4—22no—
415C3×C4—42no—
416C3×C4#53211yescharacteristic
417C3×C4#53611yescharacteristic
418C3×C4#54712yes—
419Dic3—44no—
420Dic3—42no—
421Dic3—81no—
422Dic3—81no—
423C23—64no—
424C23—62no—
425C23—122no—
426C23—122no—
427C23—122no—
428C23—122no—
429C23—121no—
430C23—122no—
431C23—121no—
432C23—122no—
433C23—122no—
434C23—122no—
435C23C22×D1211yescharacteristic
436Q8—28no—
437D4—62no—
438D4—62no—
439D4—64no—
440D4—64no—
441D4—128no—
442D4—1216no—
443D4—122no—
444D4—122no—
445D4—122no—
446D4—244no—
447D4—248no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—62no—
457C2×C4—64no—
458C2×C4—128no—
459C2×C4—1216no—
460C2×C4—122no—
461C2×C4—122no—
462C2×C4—122no—
463C2×C4—121no—
464C2×C4—121no—
465C2×C4—122no—
466C2×C4—121no—
467C2×C4—121no—
468C2×C4#88212yes—
469C2×C4#88212yes—
470C2×C4#90814yes—
471C2×C4C2×D3×D411yescharacteristic
472C2×C4C2×D3×D411yescharacteristic
473C2×C4#100712yes—
474C2×C4D3×#4112yes—
475C8—41no—
476C8—41no—
477C8—42no—
478C2×C3#243211yescharacteristic
479C2×C3#300311yescharacteristic
480C2×C3#301211yescharacteristic
481C2×C3#336612yes—
482C2×C3#336812yes—
483D3—82no—
484D3—84no—
485C22—122no—
486C22—124no—
487C22—124no—
488C22—122no—
489C22—122no—
490C22—124no—
491C22—242no—
492C22—242no—
493C22—242no—
494C22—242no—
495C22—242no—
496C22—242no—
497C22C2×#88211yescharacteristic
498C22#557612yes—
499C22C2×#90812yes—
500C22#575111yescharacteristic
501C22#575211yescharacteristic
502C4—22no—
503C4—24no—
504C4—22no—
505C4—42no—
506C4—122no—
507C4—124no—
508C4—241no—
509C4—241no—
510C4#563112yes—
511C4#566811yescharacteristic
512C4#572411yescharacteristic
513C3#2565511yescharacteristic, complemented, Sylow
514C2—244no—
515C2—242no—
516C2#7649811yescharacteristic
517C2#7863711yescharacteristic
518C2#7867211yescharacteristic
519C2#7996912yes—
520C2#7998012yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1985C2
socleC23×C3C22×D4
prime radicalC2×#5577C2
hypercenter#1985D3
hypocenterC3#25655
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #1985

Complements

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

NH ≅ G/Ntype
2. #82483C2semidirect
3. C2×#4847C2semidirect
4. C2×#4713C2semidirect
5. C3×#1985C2semidirect
6. #70457C2semidirect
7. #69812C2semidirect
8. #68671C2semidirect
9. #68670C2semidirect
10. #68629C2semidirect
11. #68629C2semidirect
52. #1985D3semidirect
209. C4⋊Dic3#41semidirect
513. C3#25655semidirect

Maximal subgroups

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

subgroupindex
C2×#55772
#824832
C2×#48472
C2×#47132
C3×#19852
#704572
#698122
#686712
#686702
#686292
#256553