Finite Group Databasebeta

← #10828176

Subgroups — #10828176

Order 768.

1344 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5817C211yesmaximal, characteristic
2C22×#835C211yesmaximal, characteristic, complemented
3C2×#4735C211yesmaximal, characteristic, complemented
4#74028C211yesmaximal, characteristic, complemented
5C3×#1787C211yesmaximal, characteristic, complemented
6#70267C212yesmaximal, complemented
7#70245C212yesmaximal, complemented
8#70243C212yesmaximal, complemented
9#70238C212yesmaximal, complemented
10#69952C211yesmaximal, characteristic, complemented
11#69515C211yesmaximal, characteristic, complemented
12#24036—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×#239C2211yescharacteristic
15#5817—24no—
16#5817—24no—
17C2×#947—21no—
18C2×#947C2211yescharacteristic
19C2×#946—21no—
20C2×#946C2211yescharacteristic
21C2×Q8×Dic3C2211yescharacteristic
22C2×#930—21no—
23C2×#930C2211yescharacteristic
24C2×C4×#67—21no—
25C2×C4×#67C2211yescharacteristic
26C22×C4⋊Dic3C2211yescharacteristic
27C23×C3⋊C8C2211yescharacteristic
28C2×C3×#103C2212yes—
29C2×C3×#103C2212yes—
30C2×C3×C22⋊C8C2211yescharacteristic
31C3×#380C2211yescharacteristic
32C2×#835—24no—
33C2×#835C2212yes—
34C2×#835C2212yes—
35C2×#821C2212yes—
36C2×(C2×D3)⋊C8C2212yes—
37C2×#817C2212yes—
38#4987C2212yes—
39#4984C2212yes—
40C2×#809C2212yes—
41#4958—21no—
42#4950—21no—
43#4735—24no—
44#4677—21no—
45#4659—21no—
46#4550C2212yes—
47#4529C2212yes—
48C2×#548—31no—
49C2×#481—31no—
50C22×#103—31no—
51#2487—31no—
52#2042—32no—
53#2035—32no—
54#1893—31no—
55#1869—31no—
56#1865—32no—
57#1788—32no—
58#1787D311yescharacteristic, complemented
59C22×C3×Q8—21no—
60C22×C3×Q8—22no—
61C22×C3×Q8—24no—
62C22×C3×Q8—42no—
63C22×C3×Q8D414yes—
64C22×C3×Q8C2311yescharacteristic
65C22×C3×Q8C2311yescharacteristic
66C23×C3×C4—21no—
67C23×C3×C4C2311yescharacteristic
68C22×#67—21no—
69C22×#67D411yescharacteristic
70C23×Dic3—21no—
71C2×#239—22no—
72C2×#239—22no—
73C2×#239—24no—
74C2×#239—42no—
75C22×D12—21no—
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×C4×D3C2311yescharacteristic
79C22×C3×C8C2312yes—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C22⋊C4—21no—
82C2×#217D411yescharacteristic
83#947—28no—
84#947—22no—
85#947—22no—
86#946—28no—
87#946—22no—
88#946—22no—
89Q8×Dic3—22no—
90Q8×Dic3—22no—
91#930—28no—
92#930—22no—
93#930D414yes—
94C4×#67—28no—
95C4×#67—22no—
96C4×#67—22no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—22no—
102C2×(C2×D3)⋊C4D411yescharacteristic
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—22no—
105C2×C4⋊Dic3—22no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3—22no—
108C2×C4⋊Dic3C2312yes—
109C2×C4⋊Dic3C2311yescharacteristic
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4D411yescharacteristic
112C2×C4×Dic3—21no—
113C2×C4×Dic3C2311yescharacteristic
114C22×C3⋊C8—22no—
115C22×C3⋊C8—42no—
116C22×C3⋊C8C2312yes—
117C3×#103—24no—
118C3×#103—42no—
119C3×C22⋊C8—22no—
120C3×#95—21no—
121#835—24no—
122#835—24no—
123C3⋊#95—22no—
124C3⋊#95—22no—
125#821—24no—
126(C2×D3)⋊C8—42no—
127#817—24no—
128#809—42no—
129C22×#41—31no—
130C23×Q8C2×D311yescharacteristic
131C23×C8—31no—
132#548—64no—
133#548—64no—
134C2×#124—31no—
135C2×#124—61no—
136C2×C22⋊Q8—31no—
137C2×C22⋊Q8—61no—
138C2×#121—31no—
139C2×#121—61no—
140C2×C4×Q8—31no—
141C2×C4×D4—31no—
142C2×C4×D4—61no—
143C22×C4⋊C4—31no—
144#481—64no—
145C2×#106—32no—
146C2×#103—32no—
147C2×#103—32no—
148C2×#103—64no—
149C2×#103C2×D312yes—
150C2×#103C2×D312yes—
151C2×#102—32no—
152C2×#102—32no—
153C2×C22⋊C8—32no—
154C2×C22⋊C8C2×D311yescharacteristic
155#400—61no—
156#399—61no—
157#389—32no—
158#386—61no—
159#385—61no—
160#380—32no—
161#380C2×D311yescharacteristic
162#340—32no—
163#336—32no—
164C24×C3C2×D411yescharacteristic
165C23×D3—21no—
166C23×D3C2×D411yescharacteristic
167C2×C3×Q8—24no—
168C2×C3×Q8—21no—
169C2×C3×Q8—21no—
170C2×C3×Q8—22no—
171C2×C3×Q8—28no—
172C2×C3×Q8—22no—
173C2×C3×Q8—22no—
174C2×C3×Q8—44no—
175C2×C3×Q8—44no—
176C2×C3×Q8—48no—
177C2×C3×Q8C2×D412yes—
178C2×C3×Q8C2×D412yes—
179C2×C3×Q8#4114yes—
180C22×C3×C4—21no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—24no—
184C22×C3×C4—21no—
185C22×C3×C4—22no—
186C22×C3×C4—44no—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D412yes—
190C22×C3×C4C2411yescharacteristic
191C2×#67—22no—
192C2×#67—24no—
193C2×#67—22no—
194C2×#67—22no—
195C2×#67—42no—
196C2×#67—42no—
197C22×Dic3—21no—
198C22×Dic3—22no—
199C22×Dic3—42no—
200C22×Dic3—82no—
201C22×Dic3C2×D411yescharacteristic
202C22×Dic3C2×D411yescharacteristic
203C22×Dic3C2×D411yescharacteristic
204#239—48no—
205#239—44no—
206#239—44no—
207C2×D12—22no—
208C2×D12—22no—
209C2×D12—24no—
210C2×D12—21no—
211C2×D12—44no—
212C2×D12C2×D412yes—
213C2×C4×D3—24no—
214C2×C4×D3—21no—
215C2×C4×D3—21no—
216C2×C4×D3—22no—
217C2×C4×D3—22no—
218C2×C4×D3—44no—
219C2×C3×C8—22no—
220C2×C3×C8—22no—
221C2×C3×C8—22no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—22no—
224C3×C22⋊C4—42no—
225#217—22no—
226(C2×D3)⋊C4—24no—
227(C2×D3)⋊C4—24no—
228(C2×D3)⋊C4—24no—
229(C2×D3)⋊C4—24no—
230(C2×D3)⋊C4—22no—
231(C2×D3)⋊C4—82no—
232C4⋊Dic3—24no—
233C4⋊Dic3—21no—
234C4⋊Dic3—22no—
235C4⋊Dic3—22no—
236C4⋊Dic3—42no—
237C4⋊Dic3—82no—
238C4⋊Dic3C2×D412yes—
239Dic3⋊C4—24no—
240Dic3⋊C4—22no—
241C4×Dic3—24no—
242C4×Dic3—21no—
243C4×Dic3—21no—
244C2×C3⋊C8—22no—
245C2×C3⋊C8—22no—
246C2×C3⋊C8—22no—
247C2×C3⋊C8—44no—
248C2×#41—62no—
249C2×#41—62no—
250C2×#41—64no—
251C2×#41—122no—
252C22×Q8—21no—
253C22×Q8—22no—
254C22×Q8—24no—
255C22×Q8—42no—
256C22×Q8D1214yes—
257C22×Q8C22×D311yescharacteristic
258C22×Q8C22×D311yescharacteristic
259C22×D4—31no—
260C22×D4—31no—
261C22×D4—61no—
262C22×D4—61no—
263C23×C4—21no—
264C23×C4—31no—
265C23×C4—61no—
266C23×C4—61no—
267C23×C4C22×D311yescharacteristic
268C22×C8—32no—
269C22×C8—62no—
270C22×C8—122no—
271C22×C8C22×D312yes—
272#124—62no—
273#124—68no—
274#124—62no—
275C22⋊Q8—62no—
276C22⋊Q8—68no—
277C22⋊Q8—62no—
278#121—34no—
279#121—68no—
280#121—62no—
281C4×Q8—62no—
282C4×Q8—62no—
283C4×D4—68no—
284C4×D4—62no—
285C4×D4—62no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—32no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—62no—
292C2×C4⋊C4—62no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—62no—
295C2×C4⋊C4C22×D312yes—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—31no—
298C2×C22⋊C4—31no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4—61no—
303C2×C22⋊C4—62no—
304C2×C42—31no—
305C2×C42—61no—
306#106—64no—
307#103—24no—
308#103—42no—
309#103—64no—
310#103—64no—
311#102—64no—
312#102—122no—
313C22⋊C8—22no—
314C22⋊C8—122no—
315#95—21no—
316#95—62no—
317#95—62no—
318C23×C3—22no—
319C23×C3—22no—
320C23×C3#12012yes—
321C23×C3#13512yes—
322C23×C3#13712yes—
323C23×C3C22×D411yescharacteristic
324C22×D3—21no—
325C22×D3—22no—
326C22×D3—22no—
327C22×D3—21no—
328C22×D3—21no—
329C22×D3—22no—
330C22×D3—42no—
331C22×D3—41no—
332C22×D3—41no—
333C22×D3—42no—
334C3×Q8—22no—
335C3×Q8—24no—
336C3×Q8—22no—
337C3×Q8—22no—
338C3×Q8—48no—
339C3×Q8—42no—
340C3×Q8#12314yes—
341C2×C3×C4—22no—
342C2×C3×C4—24no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—24no—
347C2×C3×C4—22no—
348C2×C3×C4—48no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4#12012yes—
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4C22×D411yescharacteristic
356C2×C3×C4C2×#4112yes—
357C2×C3×C4H(5,𝔽2)12yes—
358#67—44no—
359#67—48no—
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—22no—
363C2×Dic3—21no—
364C2×Dic3—22no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—22no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3—84no—
374D12—24no—
375D12—48no—
376D12—41no—
377D12—41no—
378C4×D3—48no—
379C4×D3—41no—
380C4×D3—41no—
381C4×D3—42no—
382C3×C8—42no—
383C3⋊C8—42no—
384C24—31no—
385C24—61no—
386C24D3×D411yescharacteristic
387#41—128no—
388#41—124no—
389#41—124no—
390C2×Q8—24no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—28no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—48no—
400C2×Q8C2×D1212yes—
401C2×Q8C2×D1212yes—
402C2×Q8#23714yes—
403C2×D4—32no—
404C2×D4—64no—
405C2×D4—64no—
406C2×D4—62no—
407C2×D4—62no—
408C2×D4—62no—
409C2×D4—61no—
410C2×D4—62no—
411C2×D4—62no—
412C2×D4—124no—
413C2×D4—122no—
414C2×D4—122no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—31no—
422C22×C4—31no—
423C22×C4—31no—
424C22×C4—44no—
425C22×C4—64no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—62no—
430C22×C4—62no—
431C22×C4—62no—
432C22×C4—124no—
433C22×C4—122no—
434C22×C4—242no—
435C22×C4C2×D1212yes—
436C22×C4D3×D411yescharacteristic
437C22×C4D3×D412yes—
438C22×C4C23×D311yescharacteristic
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—62no—
443C2×C8—62no—
444C2×C8—62no—
445C2×C8—124no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—32no—
449C4⋊C4—64no—
450C4⋊C4—64no—
451C4⋊C4—62no—
452C4⋊C4—61no—
453C4⋊C4—62no—
454C4⋊C4—62no—
455C4⋊C4—122no—
456C4⋊C4—242no—
457C22⋊C4—42no—
458C22⋊C4—64no—
459C22⋊C4—64no—
460C22⋊C4—64no—
461C22⋊C4—62no—
462C22⋊C4—62no—
463C22⋊C4—64no—
464C22⋊C4—242no—
465C42—64no—
466C42—61no—
467C42—61no—
468C22×C3—22no—
469C22×C3—22no—
470C22×C3—24no—
471C22×C3—24no—
472C22×C3#45214yes—
473C22×C3C2×#12011yescharacteristic
474C22×C3#54612yes—
475C22×C3C2×#13511yescharacteristic
476C22×C3C2×#13711yescharacteristic
477C22×C3#57712yes—
478C2×D3—42no—
479C2×D3—41no—
480C2×D3—41no—
481C2×D3—42no—
482C2×D3—41no—
483C2×D3—42no—
484C2×D3—42no—
485C2×D3—42no—
486C2×D3—82no—
487C2×D3—82no—
488C3×C4—24no—
489C3×C4—22no—
490C3×C4—42no—
491C3×C4—82no—
492C3×C4C2×#12311yescharacteristic
493C3×C4#53411yescharacteristic
494C3×C4#54612yes—
495Dic3—41no—
496Dic3—41no—
497Dic3—42no—
498Dic3—41no—
499Dic3—82no—
500C23—22no—
501C23—22no—
502C23—62no—
503C23—61no—
504C23—61no—
505C23—61no—
506C23—62no—
507C23—62no—
508C23—121no—
509C23—121no—
510C23—122no—
511C23—122no—
512C23#88212yes—
513C23#91612yes—
514C23#91812yes—
515C23C2×D3×D411yescharacteristic
516Q8—22no—
517Q8—24no—
518Q8—22no—
519Q8—22no—
520Q8—48no—
521Q8—42no—
522Q8#88614yes—
523D4—64no—
524D4—128no—
525D4—124no—
526D4—128no—
527D4—121no—
528D4—121no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—48no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—61no—
542C2×C4—62no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—62no—
548C2×C4—62no—
549C2×C4—128no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—122no—
555C2×C4—122no—
556C2×C4—122no—
557C2×C4—122no—
558C2×C4—244no—
559C2×C4#88212yes—
560C2×C4C22×D1211yescharacteristic
561C2×C4C2×D3×D411yescharacteristic
562C2×C4C2×#23712yes—
563C2×C4#100412yes—
564C8—42no—
565C8—122no—
566C2×C3—24no—
567C2×C3#240511yescharacteristic
568C2×C3C2×#45211yescharacteristic
569C2×C3#301011yescharacteristic
570C2×C3#330112yes—
571C2×C3#331612yes—
572D3—81no—
573D3—82no—
574C22—24no—
575C22—22no—
576C22—22no—
577C22—24no—
578C22—122no—
579C22—122no—
580C22—121no—
581C22—121no—
582C22—122no—
583C22—122no—
584C22—121no—
585C22—122no—
586C22—242no—
587C22—242no—
588C22#480814yes—
589C22C2×#88211yescharacteristic
590C22#559112yes—
591C22C2×#91611yescharacteristic
592C22C2×#91811yescharacteristic
593C22#577812yes—
594C4—22no—
595C4—24no—
596C4—42no—
597C4—82no—
598C4—121no—
599C4—121no—
600C4—121no—
601C4—122no—
602C4—242no—
603C4C2×#88611yescharacteristic
604C4#549311yescharacteristic
605C4#559112yes—
606C3#2403611yescharacteristic, complemented, Sylow
607C2—24no—
608C2—241no—
609C2—242no—
610C2#7382311yescharacteristic
611C2C2×#480811yescharacteristic
612C2#7896011yescharacteristic
613C2#8005012yes—
614C2#8019812yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1787C2
socleC23×C3C22×D4
prime radicalC2×#5817C2
hypercenter#1787D3
hypocenterC3#24036
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
2. C22×#835C2semidirect
3. C2×#4735C2semidirect
4. #74028C2semidirect
5. C3×#1787C2semidirect
6. #70267C2semidirect
7. #70245C2semidirect
8. #70243C2semidirect
9. #70238C2semidirect
10. #69952C2semidirect
11. #69515C2semidirect
58. #1787D3semidirect
606. C3#24036semidirect

Maximal subgroups

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

subgroupindex
C2×#58172
C22×#8352
C2×#47352
#740282
C3×#17872
#702672
#702452
#702432
#702382
#699522
#695152
#240363