Finite Group Databasebeta

← #10829970

Subgroups — #10829970

Order 768.

1394 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5577C211yesmaximal, characteristic
2C3×#2582C211yesmaximal, characteristic, complemented
3C2×#5027C211yesmaximal, characteristic, complemented
4C2×C4×#215C211yesmaximal, characteristic, complemented
5#71634C211yesmaximal, characteristic, complemented
6#69688C212yesmaximal, complemented
7#69687C212yesmaximal, complemented
8#69627C212yesmaximal, complemented
9#69626C212yesmaximal, complemented
10#69584C211yesmaximal, characteristic, complemented
11#69580C211yesmaximal, characteristic, complemented
12#24188—31nomaximal, Sylow
13C22×#239—21no—
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C22×#215C2211yescharacteristic
17C22×#215C2212yes—
18#5577—24no—
19#5577—24no—
20C2×#895—21no—
21C2×#895—22no—
22C2×#874C2212yes—
23C2×#874C2211yescharacteristic
24C2×C4×D12C2212yes—
25C2×C4×D12C2211yescharacteristic
26C3×#398—22no—
27C3×#391—21no—
28C3×#391C2212yes—
29C3×#389—21no—
30C3×#389C2212yes—
31C3×#384C2211yescharacteristic
32C2×#835C2211yescharacteristic
33#5027—24no—
34C4×#215—24no—
35C2×#809C2212yes—
36C2×#809C2212yes—
37C2×#809C2211yescharacteristic
38C2×#809C2212yes—
39C2×#809C2212yes—
40C2×#808C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×C4×C3⋊C8C2211yescharacteristic
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#550—31no—
47C2×#460—31no—
48C2×C4×SD16—31no—
49#2582D311yescharacteristic, complemented
50#2041—32no—
51#2014—32no—
52#1984—31no—
53#1957—31no—
54#1877—32no—
55#1864—32no—
56#1802—31no—
57C22×C3×Q8D411yescharacteristic
58C22×C3×Q8C2311yescharacteristic
59C2×#239—28no—
60C2×#239—44no—
61C2×#239—44no—
62C22×D12—22no—
63C22×D12—21no—
64C22×D12—22no—
65C22×D12—21no—
66C22×D12C2312yes—
67C22×D12C2311yescharacteristic
68C22×C4×D3—22no—
69C22×C4×D3—21no—
70C3×C4×Q8—22no—
71C3×C4×Q8—22no—
72C3×C4×Q8—22no—
73C3×C4×Q8—22no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4D411yescharacteristic
79C2×C3×C4⋊C4D412yes—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C42—21no—
83C2×C3×C42C2312yes—
84C2×C3×C42C2311yescharacteristic
85C2×#215—22no—
86C2×#215—22no—
87C2×#215—24no—
88C2×#215—24no—
89C2×#215—22no—
90C2×#215—42no—
91C2×(C2×D3)⋊C4—22no—
92C2×(C2×D3)⋊C4—21no—
93C2×C4⋊Dic3C2311yescharacteristic
94C22×C3⋊C8C2311yescharacteristic
95C22×C3⋊C8C2312yes—
96C22×C3⋊C8C2311yescharacteristic
97#895—28no—
98#895—216no—
99#874—24no—
100#874—24no—
101#874—22no—
102#874D414yes—
103C4×D12—24no—
104C4×D12—22no—
105C4×D12—24no—
106C4×D12D414yes—
107C3×#95—22no—
108C3×#95—22no—
109C3×#95—22no—
110#835—22no—
111#809—22no—
112#809—24no—
113#809—24no—
114#809—42no—
115#809—42no—
116#808—22no—
117C4⋊(C3⋊C8)—22no—
118C4×C3⋊C8—22no—
119C22×#41—61no—
120C22×SD16—31no—
121C22×SD16—32no—
122#550—64no—
123#550—64no—
124C2×#127—31no—
125C2×#127—32no—
126C2×#121—61no—
127C2×#121—62no—
128C2×C4×Q8C2×D311yescharacteristic
129C2×C4×Q8C2×D311yescharacteristic
130C2×C4×D4—31no—
131C2×C4×D4—32no—
132#460—64no—
133C4×SD16—64no—
134C2×#106—31no—
135C2×C4⋊C8—31no—
136C2×#103—31no—
137C2×#102—32no—
138C2×#102—32no—
139C2×#102—31no—
140C2×#102—32no—
141C2×#102—32no—
142C2×C4×C8—31no—
143#398—22no—
144#391—21no—
145#391C2×D312yes—
146#389—21no—
147#389C2×D312yes—
148#384C2×D311yescharacteristic
149#340—32no—
150#336—32no—
151#336—32no—
152C23×D3—22no—
153C23×D3—21no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—21no—
159C2×C3×Q8—21no—
160C2×C3×Q8—21no—
161C2×C3×Q8SD1614yescomplemented
162C2×C3×Q8#4112yescomplemented
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—21no—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2411yescharacteristic
171C22×Dic3—21no—
172#239—432no—
173C2×D12—22no—
174C2×D12—21no—
175C2×D12—22no—
176C2×D12—21no—
177C2×D12—21no—
178C2×D12—22no—
179C2×D12—21no—
180C2×D12—22no—
181C2×D12—28no—
182C2×D12—24no—
183C2×D12—22no—
184C2×D12—21no—
185C2×D12—28no—
186C2×D12—24no—
187C2×D12—44no—
188C2×D12—44no—
189C2×D12—42no—
190C2×D12—42no—
191C2×D12—81no—
192C2×D12—81no—
193C2×D12—82no—
194C2×D12—82no—
195C2×D12C2×D414yes—
196C2×D12C2×D412yes—
197C2×C4×D3—28no—
198C2×C4×D3—24no—
199C2×C4×D3—44no—
200C2×C4×D3—44no—
201C2×C4×D3—42no—
202C2×C4×D3—42no—
203C3×C4⋊C4—24no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—42no—
210C3×C4⋊C4—42no—
211C3×C4⋊C4—81no—
212C3×C4⋊C4—82no—
213C3×C42—22no—
214C3×C42—22no—
215C3×C42—42no—
216C3×C42C2×D412yes—
217C3×C42C2×D412yes—
218#215—28no—
219#215—48no—
220(C2×D3)⋊C4—28no—
221(C2×D3)⋊C4—24no—
222C4⋊Dic3—21no—
223C4⋊Dic3C2×D412yes—
224C2×C3⋊C8—21no—
225C2×C3⋊C8—21no—
226C2×C3⋊C8—21no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—21no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—21no—
232C2×C3⋊C8#4112yescomplemented
233C2×#41—68no—
234C2×#41—124no—
235C2×#41—124no—
236C22×Q8#6711yescharacteristic
237C22×Q8C22×D311yescharacteristic
238C22×D4—31no—
239C22×D4—32no—
240C22×D4—62no—
241C22×D4—61no—
242C22×D4—62no—
243C22×D4—61no—
244C23×C4—62no—
245C23×C4—61no—
246C2×SD16—62no—
247C2×SD16—62no—
248C2×SD16—62no—
249C2×SD16—64no—
250C2×SD16—64no—
251C2×SD16—122no—
252C22×C8—31no—
253C22×C8—32no—
254C22×C8—31no—
255#127—34no—
256#127—64no—
257#127—64no—
258#127—62no—
259#121—68no—
260#121—616no—
261C4×Q8—22no—
262C4×Q8—22no—
263C4×Q8—22no—
264C4×Q8—22no—
265C4×D4—34no—
266C4×D4—64no—
267C4×D4—64no—
268C4×D4—62no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4#6711yescharacteristic
275C2×C4⋊C4#6712yes—
276C2×C4⋊C4C22×D311yescharacteristic
277C2×C4⋊C4C22×D312yes—
278C2×C22⋊C4—62no—
279C2×C22⋊C4—61no—
280C2×C42—21no—
281C2×C42C22×D312yes—
282C2×C42C22×D311yescharacteristic
283#106—62no—
284C4⋊C8—62no—
285#103—62no—
286#102—64no—
287#102—62no—
288#102—64no—
289#102—122no—
290#102—122no—
291C4×C8—62no—
292#95—22no—
293#95—22no—
294#95—22no—
295C23×C3C22×D411yescharacteristic
296C22×D3—24no—
297C22×D3—22no—
298C22×D3—41no—
299C22×D3—42no—
300C22×D3—42no—
301C22×D3—41no—
302C22×D3—42no—
303C22×D3—41no—
304C22×D3—42no—
305C22×D3—41no—
306C22×D3—42no—
307C22×D3—41no—
308C22×D3—42no—
309C22×D3—41no—
310C3×Q8—28no—
311C3×Q8—22no—
312C3×Q8—22no—
313C3×Q8—41no—
314C3×Q8—41no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4#12012yes—
332C2×C3×C4#12012yes—
333C2×C3×C4C2×SD1614yes—
334C2×C3×C4C2×SD1612yes—
335C2×C3×C4AGL(1,ℤ/8)12yes—
336C2×C3×C4C22×D411yescharacteristic
337C2×C3×C4C22×D411yescharacteristic
338C2×C3×C4C2×#4111yescharacteristic
339C2×C3×C4C2×#4111yescharacteristic
340C2×C3×C4H(5,𝔽2)11yescharacteristic
341C2×C3×C42−1+411yescharacteristic
342C2×Dic3—22no—
343C2×Dic3—41no—
344C2×Dic3—41no—
345D12—22no—
346D12—22no—
347D12—24no—
348D12—24no—
349D12—416no—
350D12—42no—
351D12—42no—
352D12—41no—
353D12—41no—
354D12—48no—
355D12—84no—
356D12—88no—
357C4×D3—416no—
358C4×D3—48no—
359C3⋊C8—22no—
360C3⋊C8—41no—
361C3⋊C8—42no—
362C24—62no—
363C24—61no—
364#41—1232no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8#21514yescomplemented
373C2×Q8#23912yescomplemented
374C2×D4—32no—
375C2×D4—34no—
376C2×D4—68no—
377C2×D4—68no—
378C2×D4—64no—
379C2×D4—64no—
380C2×D4—62no—
381C2×D4—61no—
382C2×D4—61no—
383C2×D4—62no—
384C2×D4—61no—
385C2×D4—62no—
386C2×D4—62no—
387C2×D4—61no—
388C2×D4—61no—
389C2×D4—62no—
390C2×D4—124no—
391C2×D4—124no—
392C2×D4—122no—
393C2×D4—122no—
394C2×D4—241no—
395C2×D4—241no—
396C2×D4—242no—
397C2×D4—242no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—68no—
402C22×C4—61no—
403C22×C4—64no—
404C22×C4—124no—
405C22×C4—124no—
406C22×C4—122no—
407C22×C4—122no—
408C22×C4C2×#6712yes—
409C22×C4C2×#6711yescharacteristic
410C22×C4C2×#6712yes—
411C22×C4C2×#6711yescharacteristic
412C22×C4C23×D311yescharacteristic
413SD16—68no—
414SD16—128no—
415C2×C8—32no—
416C2×C8—61no—
417C2×C8—61no—
418C2×C8—61no—
419C2×C8—62no—
420C2×C8—62no—
421C2×C8—61no—
422C2×C8—61no—
423C2×C8—62no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—32no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—61no—
434C4⋊C4—81no—
435C4⋊C4—82no—
436C22⋊C4—68no—
437C22⋊C4—64no—
438C42—22no—
439C42—22no—
440C42—42no—
441C42D3×D412yes—
442C42D3×D412yes—
443C22×C3C2×#12011yescharacteristic
444C22×C3#54811yescharacteristic
445C22×C3#55011yescharacteristic
446C22×C3C22×SD1611yescharacteristic
447C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
448C22×C3#57611yescharacteristic
449C22×C3#57711yescharacteristic
450C2×D3—42no—
451C2×D3—44no—
452C2×D3—44no—
453C2×D3—42no—
454C2×D3—44no—
455C2×D3—42no—
456C2×D3—81no—
457C2×D3—82no—
458C2×D3—82no—
459C2×D3—81no—
460C2×D3—82no—
461C2×D3—81no—
462C2×D3—82no—
463C2×D3—81no—
464C3×C4—24no—
465C3×C4—22no—
466C3×C4—42no—
467C3×C4—42no—
468C3×C4—82no—
469C3×C4—81no—
470C3×C4#45014yes—
471C3×C4#52511yescharacteristic
472C3×C4#53711yescharacteristic
473C3×C4#54611yescharacteristic
474C3×C4#54711yescharacteristic
475Dic3—42no—
476C23—64no—
477C23—62no—
478C23—122no—
479C23—121no—
480C23—121no—
481C23—122no—
482C23—122no—
483C23—121no—
484C23—122no—
485C23—121no—
486C23—122no—
487C23—121no—
488C23—122no—
489C23—121no—
490C23C22×#6711yescharacteristic
491Q8—28no—
492Q8—22no—
493Q8—22no—
494Q8—41no—
495Q8—41no—
496D4—62no—
497D4—62no—
498D4—64no—
499D4—64no—
500D4—1216no—
501D4—128no—
502D4—122no—
503D4—122no—
504D4—121no—
505D4—121no—
506D4—244no—
507D4—248no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—62no—
525C2×C4—1216no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4—128no—
529C2×C4#93712yes—
530C2×C4#93712yes—
531C2×C4C2×#21514yes—
532C2×C4C2×#21512yes—
533C2×C4#94912yes—
534C2×C4C2×D3×D411yescharacteristic
535C2×C4C2×D3×D411yescharacteristic
536C2×C4C2×#23911yescharacteristic
537C2×C4#100711yescharacteristic
538C2×C4D3×#4111yescharacteristic
539C2×C4#100911yescharacteristic
540C8—62no—
541C8—121no—
542C8—122no—
543C2×C3#240711yescharacteristic
544C2×C3C2×#45011yescharacteristic
545C2×C3#301211yescharacteristic
546C2×C3#336211yescharacteristic
547C2×C3#336811yescharacteristic
548C2×C3#339211yescharacteristic
549C2×C3#339911yescharacteristic
550D3—82no—
551D3—84no—
552C22—124no—
553C22—122no—
554C22—122no—
555C22—124no—
556C22—124no—
557C22—122no—
558C22—242no—
559C22—241no—
560C22—242no—
561C22—241no—
562C22—242no—
563C22—241no—
564C22—242no—
565C22—241no—
566C22C2×#93711yescharacteristic
567C22C22×#21511yescharacteristic
568C22#581711yescharacteristic
569C22C2×#94911yescharacteristic
570C22#583411yescharacteristic
571C22#583511yescharacteristic
572C22#583711yescharacteristic
573C4—24no—
574C4—22no—
575C4—42no—
576C4—42no—
577C4—81no—
578C4—82no—
579C4—122no—
580C4#504614yes—
581C4#562711yescharacteristic
582C4#562911yescharacteristic
583C4#563011yescharacteristic
584C4#563111yescharacteristic
585C3#2418811yescharacteristic, complemented, Sylow
586C2—244no—
587C2—242no—
588C2C2×#504611yescharacteristic
589C2#8281211yescharacteristic
590C2#8342411yescharacteristic
591C2#8368511yescharacteristic
592C2#8369011yescharacteristic
593C2#8370311yescharacteristic
594C2#8371111yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2582C2
socleC23×C3C22×D4
prime radicalC2×#5577C2
hypercenter#2582D3
hypocenterC3#24188
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2582

Complements

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

NH ≅ G/Ntype
2. C3×#2582C2semidirect
3. C2×#5027C2semidirect
4. C2×C4×#215C2semidirect
5. #71634C2semidirect
6. #69688C2semidirect
7. #69687C2semidirect
8. #69627C2semidirect
9. #69626C2semidirect
10. #69584C2semidirect
11. #69580C2semidirect
49. #2582D3semidirect
161. C2×C3×Q8SD16semidirect
162. C2×C3×Q8#41semidirect
232. C2×C3⋊C8#41semidirect
372. C2×Q8#215semidirect
373. C2×Q8#239semidirect
585. C3#24188semidirect

Maximal subgroups

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

subgroupindex
C2×#55772
C3×#25822
C2×#50272
C2×C4×#2152
#716342
#696882
#696872
#696272
#696262
#695842
#695802
#241883