Finite Group Databasebeta

← #10827620

Subgroups — #10827620

Order 768.

1228 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5639C211yesmaximal, characteristic
2C2×C3×#451C211yesmaximal, characteristic, complemented
3#84068C211yesmaximal, characteristic
4C2×#5225C211yesmaximal, characteristic, complemented
5#72071C212yesmaximal, complemented
6#72065C212yesmaximal, complemented
7#71697C211yesmaximal, characteristic, complemented
8#70268C212yesmaximal, complemented
9#70232C212yesmaximal, complemented
10#69953C211yesmaximal, characteristic
11#69926C211yesmaximal, characteristic, complemented
12#23983—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×C3×Q16C2212yes—
15C2×C3×C22⋊Q8C2211yescharacteristic
16C2×#946C2211yescharacteristic
17C2×#946C2212yes—
18C2×Q8×Dic3C2211yescharacteristic
19C2×C4×#67C2211yescharacteristic
20C2×#926C2211yescharacteristic
21C22×#208C2211yescharacteristic
22#5639—24no—
23#5639—24no—
24C2×#892—21no—
25C2×D3×C4⋊C4C2211yescharacteristic
26C2×#890C2211yescharacteristic
27C2×#889—21no—
28C2×#883—21no—
29C2×#879—21no—
30C3×#451—24no—
31C2×C3×#103C2212yes—
32C2×C3×C22⋊C8C2211yescharacteristic
33#5231—21no—
34#5230—21no—
35#5229C2212yes—
36#5225—24no—
37C2×#835C2212yes—
38C2×#835C2212yes—
39C2×#835C2212yes—
40#5213—21no—
41#5212C2211yescharacteristic
42#5210—21no—
43C2×(C2×D3)⋊C8C2212yes—
44#4551C2212yes—
45#4551C2212yes—
46#4550C2212yes—
47#4529C2212yes—
48C2×#552—31no—
49C2×#451D311yescharacteristic, complemented
50C2×#419—31no—
51#2409—31no—
52#2026—32no—
53#2001—32no—
54#1894—31no—
55#1878—32no—
56#1872—32no—
57#1870—31no—
58#1825—31no—
59C22×C3×Q8—21no—
60C22×C3×Q8—24no—
61C22×C3×Q8—22no—
62C22×C3×Q8—22no—
63C22×C3×Q8—42no—
64C22×C3×Q8C2311yescharacteristic
65C22×C3×Q8C2312yes—
66C23×C3×C4—21no—
67C23×C3×C4C2311yescharacteristic
68C22×#67D411yescharacteristic
69C22×C4×D3—21no—
70C22×C4×D3C2311yescharacteristic
71C2×C3×Q16—24no—
72C2×C3×Q16—24no—
73C2×C3×Q16—42no—
74C22×C3×C8C2312yes—
75C3×C22⋊Q8—22no—
76C3×C22⋊Q8D414yes—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C22⋊C4—21no—
80C2×#217—21no—
81C2×#217—21no—
82C2×#217D411yescharacteristic
83#946—22no—
84#946—22no—
85#946—24no—
86#946—42no—
87Q8×Dic3—22no—
88Q8×Dic3—22no—
89C4×#67—22no—
90C4×#67—22no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—22no—
93C2×(C2×D3)⋊C4D411yescharacteristic
94#926—22no—
95#926D414yes—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3C2312yes—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—22no—
105C2×Dic3⋊C4D411yescharacteristic
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3C2311yescharacteristic
109C2×#208—22no—
110C2×#208—42no—
111C22×C3⋊C8C2312yes—
112#892—28no—
113D3×C4⋊C4—22no—
114D3×C4⋊C4—22no—
115#890—22no—
116#890—22no—
117#889—28no—
118#883—28no—
119#879—28no—
120C3×#103—24no—
121C3×C22⋊C8—22no—
122#835—24no—
123#835—24no—
124#835—42no—
125C3⋊#95—21no—
126C3⋊#95—22no—
127C3⋊#95—22no—
128(C2×D3)⋊C8—42no—
129C23×Q8C2×D311yescharacteristic
130C22×Q16C2×D312yes—
131C22×M16—31no—
132#552—64no—
133#552—64no—
134C2×#126—61no—
135C2×#125—31no—
136C2×#125—61no—
137C2×#123—61no—
138C2×C22⋊Q8—31no—
139C2×C22⋊Q8—32no—
140C2×C22⋊Q8C2×D311yescharacteristic
141C2×C4×Q8—31no—
142C2×C4×D4—31no—
143C2×#117—31no—
144C2×#117—61no—
145C22×C4⋊C4—31no—
146#451—24no—
147#419—64no—
148C2×#103—32no—
149C2×#103—32no—
150C2×#103—32no—
151C2×#103C2×D312yes—
152C2×C22⋊C8—32no—
153C2×C22⋊C8C2×D311yescharacteristic
154#395—61no—
155#393—61no—
156#391—32no—
157#386—61no—
158#385—61no—
159#380—31no—
160#340—32no—
161#340—32no—
162#340—32no—
163#336—32no—
164C24×C3C2×D411yescharacteristic
165C23×D3C2×D411yescharacteristic
166C2×C3×Q8—21no—
167C2×C3×Q8—22no—
168C2×C3×Q8—28no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—22no—
174C2×C3×Q8—22no—
175C2×C3×Q8—44no—
176C2×C3×Q8—48no—
177C2×C3×Q8—44no—
178C2×C3×Q8—42no—
179C2×C3×Q8C2×D412yes—
180C2×C3×Q8#4114yes—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—44no—
187C22×C3×C4—42no—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D412yes—
190C22×C3×C4C2×D412yes—
191C22×C3×C4C2411yescharacteristic
192C2×#67—22no—
193C2×#67—22no—
194C2×#67—41no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—22no—
198C22×Dic3C2×D411yescharacteristic
199C22×Dic3C2×D411yescharacteristic
200C22×Dic3C2×D411yescharacteristic
201C2×C4×D3—24no—
202C2×C4×D3—21no—
203C2×C4×D3—21no—
204C2×C4×D3—22no—
205C2×C4×D3—22no—
206C2×C4×D3—42no—
207C2×C4×D3—42no—
208C3×Q16—48no—
209C2×C3×C8—22no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C3×C4⋊C4—24no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4C2×D412yes—
215C3×C22⋊C4—24no—
216#217—22no—
217#217—24no—
218#217—42no—
219(C2×D3)⋊C4—24no—
220(C2×D3)⋊C4—22no—
221(C2×D3)⋊C4—44no—
222C4⋊Dic3—24no—
223C4⋊Dic3—24no—
224C4⋊Dic3—21no—
225C4⋊Dic3—21no—
226C4⋊Dic3—22no—
227C4⋊Dic3—22no—
228C4⋊Dic3C2×D412yes—
229C4⋊Dic3C2×D412yes—
230Dic3⋊C4—24no—
231Dic3⋊C4—24no—
232Dic3⋊C4—24no—
233Dic3⋊C4—22no—
234Dic3⋊C4—44no—
235C4×Dic3—24no—
236C4×Dic3—24no—
237C4×Dic3—21no—
238C4×Dic3—21no—
239#208—44no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8—22no—
243C22×Q8—21no—
244C22×Q8—24no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—42no—
248C22×Q8C22×D312yes—
249C22×Q8C22×D311yescharacteristic
250C22×D4—31no—
251C23×C4—21no—
252C23×C4—31no—
253C23×C4—61no—
254C23×C4C22×D311yescharacteristic
255C2×Q16—24no—
256C2×Q16—24no—
257C2×Q16—42no—
258C2×M16—62no—
259C2×M16—122no—
260C22×C8—32no—
261C22×C8C22×D312yes—
262#126—68no—
263#125—62no—
264#125—68no—
265#125—62no—
266#123—68no—
267C22⋊Q8—22no—
268C22⋊Q8—62no—
269C22⋊Q8—62no—
270C22⋊Q8—64no—
271C22⋊Q8—122no—
272C22⋊Q8#6714yes—
273C4×Q8—62no—
274C4×Q8—62no—
275C4×D4—62no—
276C4×D4—62no—
277#117—34no—
278#117—68no—
279#117—62no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—32no—
285C2×C4⋊C4—62no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—62no—
292C2×C4⋊C4—62no—
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C22⋊C4—21no—
295C2×C22⋊C4—31no—
296C2×C22⋊C4—31no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4—62no—
300C2×C22⋊C4—61no—
301C2×C42—31no—
302C2×C42—61no—
303C2×C42—61no—
304#103—24no—
305#103—64no—
306#103—64no—
307#103—122no—
308C22⋊C8—22no—
309C22⋊C8—122no—
310#95—62no—
311#95—62no—
312#95—61no—
313C23×C3—22no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—22no—
317C23×C3#12012yes—
318C23×C3C22×D411yescharacteristic
319C22×D3—21no—
320C22×D3—22no—
321C22×D3—21no—
322C22×D3—22no—
323C22×D3#13712yes—
324C3×Q8—24no—
325C3×Q8—24no—
326C3×Q8—24no—
327C3×Q8—48no—
328C3×Q8—41no—
329C3×Q8—41no—
330C3×Q8—42no—
331C3×Q8—42no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—24no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—24no—
339C2×C3×C4—48no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4#12012yes—
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4C22×D411yescharacteristic
348C2×C3×C4C2×#4112yes—
349C2×C3×C4H(5,𝔽2)12yes—
350#67—44no—
351C2×Dic3—21no—
352C2×Dic3—22no—
353C2×Dic3—21no—
354C2×Dic3—22no—
355C2×Dic3—21no—
356C2×Dic3—22no—
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—22no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C2×Dic3—42no—
365C2×Dic3—42no—
366C4×D3—48no—
367C4×D3—41no—
368C4×D3—41no—
369C4×D3—42no—
370C3×C8—42no—
371C3⋊C8—42no—
372C24—31no—
373C24D3×D411yescharacteristic
374C2×Q8—21no—
375C2×Q8—22no—
376C2×Q8—28no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—44no—
384C2×Q8—48no—
385C2×Q8—44no—
386C2×Q8—42no—
387C2×Q8#23714yes—
388C2×Q8C2×#6712yes—
389C2×D4—62no—
390C2×D4—62no—
391C2×D4—121no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—31no—
398C22×C4—31no—
399C22×C4—31no—
400C22×C4—44no—
401C22×C4—42no—
402C22×C4—64no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—61no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—61no—
409C22×C4—62no—
410C22×C4—122no—
411C22×C4—122no—
412C22×C4D3×D411yescharacteristic
413C22×C4D3×D412yes—
414C22×C4C2×#6712yes—
415C22×C4C23×D311yescharacteristic
416Q16—48no—
417M16—124no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—62no—
422C2×C8—62no—
423C2×C8—62no—
424C4⋊C4—24no—
425C4⋊C4—21no—
426C4⋊C4—32no—
427C4⋊C4—32no—
428C4⋊C4—64no—
429C4⋊C4—64no—
430C4⋊C4—64no—
431C4⋊C4—64no—
432C4⋊C4—64no—
433C4⋊C4—62no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—62no—
437C4⋊C4—62no—
438C4⋊C4—124no—
439C4⋊C4C2×#6712yes—
440C22⋊C4—24no—
441C22⋊C4—64no—
442C22⋊C4—62no—
443C22⋊C4—64no—
444C22⋊C4—62no—
445C22⋊C4—124no—
446C22⋊C4—122no—
447C42—64no—
448C42—64no—
449C42—61no—
450C42—61no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—22no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3C2×#12011yescharacteristic
457C22×C3#54612yes—
458C22×C3C2×#13711yescharacteristic
459C22×C3#57511yescharacteristic
460C22×C3#57812yes—
461C2×D3—22no—
462C2×D3—24no—
463C2×D3—41no—
464C2×D3—41no—
465C2×D3—42no—
466C3×C4—22no—
467C3×C4—44no—
468C3×C4—42no—
469C3×C4—42no—
470C3×C4#52511yescharacteristic
471C3×C4#53511yescharacteristic
472C3×C4#54612yes—
473Dic3—41no—
474Dic3—42no—
475Dic3—41no—
476Dic3—41no—
477Dic3—42no—
478Dic3—82no—
479C23—22no—
480C23—21no—
481C23—21no—
482C23—22no—
483C23—32no—
484C23—62no—
485C23—61no—
486C23—61no—
487C23—62no—
488C23#93712yes—
489C23C2×D3×D411yescharacteristic
490Q8—24no—
491Q8—24no—
492Q8—24no—
493Q8—48no—
494Q8—42no—
495Q8—41no—
496Q8—42no—
497Q8—41no—
498D4—124no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—48no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—62no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—62no—
517C2×C4—62no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—62no—
521C2×C4—128no—
522C2×C4—122no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—122no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4—122no—
529C2×C4—122no—
530C2×C4—122no—
531C2×C4#93712yes—
532C2×C4C2×D3×D411yescharacteristic
533C2×C4C2×#23712yes—
534C2×C4#100412yes—
535C2×C4C22×#6711yescharacteristic
536C8—42no—
537C8—122no—
538C2×C3—42no—
539C2×C3#240411yescharacteristic
540C2×C3#300711yescharacteristic
541C2×C3#301111yescharacteristic
542C2×C3#331212yes—
543C2×C3#332712yes—
544D3—42no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—42no—
549C22—42no—
550C22—62no—
551C22—64no—
552C22—122no—
553C22—121no—
554C22—121no—
555C22#559112yes—
556C22#576612yes—
557C22D3×#13711yescharacteristic
558C22#577711yescharacteristic
559C22C2×#93711yescharacteristic
560C4—22no—
561C4—42no—
562C4—44no—
563C4—42no—
564C4—121no—
565C4—122no—
566C4—121no—
567C4—121no—
568C4—122no—
569C4—242no—
570C4#558611yescharacteristic
571C4#559211yescharacteristic
572C4#580612yes—
573C3#2398311yescharacteristic, complemented, Sylow
574C2—42no—
575C2—122no—
576C2#8019612yes—
577C2#8019912yes—
578C2#8258111yescharacteristic
579C2#8388411yescharacteristic
580C2#8398111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#451C2
socleC23×C3C22×D4
prime radicalC22×#67D4
hypercenterC2×#451D3
hypocenterC3#23983
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#451

Complements

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

NH ≅ G/Ntype
2. C2×C3×#451C2semidirect
4. C2×#5225C2semidirect
5. #72071C2semidirect
6. #72065C2semidirect
7. #71697C2semidirect
8. #70268C2semidirect
9. #70232C2semidirect
11. #69926C2semidirect
49. C2×#451D3semidirect
573. C3#23983semidirect

Maximal subgroups

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

subgroupindex
C2×#56392
C2×C3×#4512
#840682
C2×#52252
#720712
#720652
#716972
#702682
#702322
#699532
#699262
#239833