Finite Group Databasebeta

← #10848861

Subgroups — #10848861

Order 768.

1216 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×#5682C211yesmaximal, characteristic, complemented
2C2×C3×#496C211yesmaximal, characteristic, complemented
3#83189C211yesmaximal, characteristic, complemented
4C2×#5070C211yesmaximal, characteristic, complemented
5#72054C212yesmaximal, complemented
6#72048C212yesmaximal, complemented
7#72043C212yesmaximal, complemented
8#72037C212yesmaximal, complemented
9#71725C211yesmaximal, characteristic, complemented
10#69830C211yesmaximal, characteristic, complemented
11#69828C211yesmaximal, characteristic, complemented
12#25614—31nomaximal, Sylow
13C22×C3×SD16C2212yes—
14C22×C3×D8C2212yes—
15C2×C3×#127C2211yescharacteristic
16C2×C3×#124C2211yescharacteristic
17C2×#946C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C2×#938C2211yescharacteristic
20C2×#938C2212yes—
21C2×D4×Dic3C2211yescharacteristic
22#5682—24no—
23#5682—24no—
24C2×#886—21no—
25C2×#885—21no—
26C2×#879—21no—
27C2×#878—21no—
28C2×#872C2211yescharacteristic
29C2×#869C2211yescharacteristic
30C3×#496—24no—
31C2×C3×#97C2211yescharacteristic
32C2×#835C2212yes—
33#5223—21no—
34#5223—21no—
35#5222—21no—
36#5222—21no—
37#5221C2212yes—
38C2×#832C2212yes—
39C2×#832C2212yes—
40C2×#832C2212yes—
41C2×(C2×D3)⋊C8C2212yes—
42#5070—24no—
43#4936C2211yescharacteristic
44C2×C4×C3⋊C8C2211yescharacteristic
45#4551C2212yes—
46#4551C2212yes—
47#4550C2212yes—
48C2×#542—31no—
49C2×#496D311yescharacteristic, complemented
50C2×#486—31no—
51#2619—31no—
52#2000—32no—
53#1999—32no—
54#1993—31no—
55#1958—31no—
56#1880—32no—
57#1879—32no—
58#1750—31no—
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4—21no—
61C22×C3×D4—21no—
62C22×C3×D4C2312yes—
63C22×C3×D4C2311yescharacteristic
64C22×#67—21no—
65C22×#67—22no—
66C23×Dic3—21no—
67C22×C4×D3C2311yescharacteristic
68C22×Dic6—21no—
69C2×C3×SD16—24no—
70C2×C3×SD16—24no—
71C2×C3×SD16—42no—
72C2×C3×D8—24no—
73C2×C3×D8—24no—
74C2×C3×D8—42no—
75C22×C3×C8C2312yes—
76C3×#127—22no—
77C3×#127—22no—
78C3×#124—22no—
79C3×#124D414yescomplemented
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—21no—
82C2×C3×C42C2311yescharacteristic
83C2×#217—21no—
84C2×#217—21no—
85C2×#217—22no—
86C2×#217—22no—
87C2×#217—22no—
88#946—22no—
89#946—22no—
90Q8×Dic3—22no—
91Q8×Dic3—22no—
92#938—22no—
93#938—22no—
94#938—42no—
95#938—42no—
96D4×Dic3—22no—
97D4×Dic3—22no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4D411yescharacteristic
100C2×(C2×D3)⋊C4D411yescharacteristic
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3C2312yes—
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4D411yescharacteristic
108C2×Dic3⋊C4D411yescharacteristic
109C2×C4×Dic3—21no—
110C2×C4×Dic3C2311yescharacteristic
111C22×C3⋊C8C2312yes—
112#886—28no—
113#885—28no—
114#879—28no—
115#878—28no—
116#872—22no—
117#872—22no—
118#869—22no—
119#869D414yescomplemented
120C3×#97—22no—
121#835—42no—
122#832—24no—
123#832—24no—
124#832—42no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127(C2×D3)⋊C8—24no—
128C4×C3⋊C8—22no—
129C22×SD16C2×D312yes—
130C22×D8C2×D312yes—
131#542—64no—
132#542—64no—
133C2×C4⋊Q8—31no—
134C2×#127C2×D311yescharacteristic
135C2×#126—61no—
136C2×#124—61no—
137C2×#124C2×D311yescharacteristic
138C2×#123—61no—
139C2×C22⋊Q8—31no—
140C2×C22⋊Q8—61no—
141C2×#121—31no—
142C2×#121—32no—
143C2×C4×Q8—31no—
144C2×C4×D4—31no—
145C2×#117—31no—
146#496—24no—
147#486—64no—
148C2×#103—32no—
149C2×#102—32no—
150C2×#102—32no—
151C2×#102—32no—
152C2×C22⋊C8—32no—
153C2×#97C2×D311yescharacteristic
154C2×C4×C8—31no—
155#394—61no—
156#394—61no—
157#392—61no—
158#392—61no—
159#390—32no—
160#382—31no—
161#340—32no—
162#340—32no—
163#336—32no—
164C24×C3—21no—
165C24×C3—22no—
166C23×D3C2×D411yescharacteristic
167C2×C3×Q8—21no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8C2×D412yes—
171C2×C3×D4—21no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—42no—
181C2×C3×D4—42no—
182C2×C3×D4—82no—
183C2×C3×D4—82no—
184C2×C3×D4C2×D412yes—
185C22×C3×C4—21no—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2411yescharacteristic
189C2×#67—24no—
190C2×#67—44no—
191C2×#67—82no—
192C2×#67—82no—
193C2×#67—82no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—24no—
197C22×Dic3—22no—
198C22×Dic3—82no—
199C22×Dic3C2×D411yescharacteristic
200C22×Dic3C2×D411yescharacteristic
201C22×Dic3C2×D411yescharacteristic
202C2×C4×D3—21no—
203C2×C4×D3—21no—
204C2×C4×D3—22no—
205C2×C4×D3—22no—
206C2×Dic6—24no—
207C2×Dic6—82no—
208C3×SD16—48no—
209C3×D8—48no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C2×C3×C8#4114yes—
213C3×C22⋊C4—24no—
214C3×C22⋊C4—24no—
215C3×C42C2×D412yes—
216C3×C42C2×D412yes—
217#217—24no—
218#217—24no—
219#217—44no—
220#217—82no—
221#217—82no—
222(C2×D3)⋊C4—24no—
223(C2×D3)⋊C4—22no—
224(C2×D3)⋊C4—22no—
225C4⋊Dic3—24no—
226C4⋊Dic3—24no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229C4⋊Dic3—22no—
230C4⋊Dic3—22no—
231C4⋊Dic3C2×D412yes—
232C4⋊Dic3C2×D412yes—
233Dic3⋊C4—24no—
234Dic3⋊C4—22no—
235Dic3⋊C4—22no—
236C4×Dic3—24no—
237C4×Dic3—21no—
238C4×Dic3—21no—
239C2×C3⋊C8—22no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8—22no—
242C22×Q8—61no—
243C22×Q8C22×D311yescharacteristic
244C22×D4—21no—
245C22×D4—21no—
246C22×D4—61no—
247C22×D4—62no—
248C22×D4C22×D312yes—
249C22×D4C22×D311yescharacteristic
250C23×C4—31no—
251C23×C4—61no—
252C2×SD16—24no—
253C2×SD16—24no—
254C2×SD16—42no—
255C2×D8—24no—
256C2×D8—24no—
257C2×D8—42no—
258C22×C8—32no—
259C22×C8C22×D312yes—
260C4⋊Q8—34no—
261C4⋊Q8—62no—
262#127—22no—
263#127—22no—
264#126—68no—
265#124—22no—
266#124—68no—
267#124#6714yescomplemented
268#123—68no—
269C22⋊Q8—68no—
270C22⋊Q8—62no—
271C22⋊Q8—62no—
272#121—62no—
273#121—62no—
274#121—122no—
275#121—122no—
276C4×Q8—62no—
277C4×Q8—62no—
278C4×D4—62no—
279C4×D4—62no—
280#117—62no—
281#117—62no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—32no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—31no—
293C2×C22⋊C4—31no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—62no—
298C2×C22⋊C4—62no—
299C2×C22⋊C4—62no—
300C2×C42—31no—
301C2×C42—61no—
302C2×C42C22×D311yescharacteristic
303#103—122no—
304#102—64no—
305#102—64no—
306#102—122no—
307C22⋊C8—64no—
308#97—22no—
309C4×C8—62no—
310#95—61no—
311#95—61no—
312C23×C3—22no—
313C23×C3—41no—
314C23×C3—41no—
315C23×C3—42no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3—42no—
319C23×C3—42no—
320C23×C3—42no—
321C23×C3—42no—
322C23×C3—42no—
323C23×C3—42no—
324C23×C3C22×D411yescharacteristic
325C22×D3—21no—
326C22×D3—21no—
327C22×D3—21no—
328C22×D3—22no—
329C22×D3—22no—
330C3×Q8—24no—
331C3×Q8—42no—
332C3×D4—24no—
333C3×D4—42no—
334C3×D4—42no—
335C3×D4—42no—
336C3×D4—42no—
337C3×D4—42no—
338C3×D4—84no—
339C3×D4—84no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4#12012yes—
346C2×C3×C4#12012yes—
347C2×C3×C4C22×D411yescharacteristic
348C2×C3×C4C22×D411yescharacteristic
349C2×C3×C4C2×#4112yes—
350C2×C3×C4H(5,𝔽2)12yes—
351#67—84no—
352#67—88no—
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—22no—
356C2×Dic3—21no—
357C2×Dic3—22no—
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—22no—
361C2×Dic3—22no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C2×Dic3—42no—
365C2×Dic3—42no—
366C2×Dic3—42no—
367C2×Dic3—84no—
368C4×D3—42no—
369C4×D3—42no—
370Dic6—84no—
371C3×C8—24no—
372C3⋊C8—42no—
373C24—21no—
374C24—22no—
375C24—31no—
376C2×Q8—21no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—64no—
380C2×Q8—242no—
381C2×Q8C2×#6712yes—
382C2×D4—21no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4—64no—
394C2×D4—82no—
395C2×D4—82no—
396C2×D4—124no—
397C2×D4—242no—
398C2×D4—242no—
399C2×D4—242no—
400C2×D4C2×#6712yes—
401C22×C4—21no—
402C22×C4—31no—
403C22×C4—31no—
404C22×C4—31no—
405C22×C4—61no—
406C22×C4—61no—
407C22×C4—64no—
408C22×C4—61no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—61no—
412C22×C4—62no—
413C22×C4—242no—
414C22×C4D3×D411yescharacteristic
415C22×C4D3×D411yescharacteristic
416C22×C4C23×D311yescharacteristic
417SD16—48no—
418D8—48no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—62no—
422C2×C8—62no—
423C2×C8—62no—
424C2×C8#23714yes—
425C4⋊C4—32no—
426C4⋊C4—32no—
427C4⋊C4—64no—
428C4⋊C4—64no—
429C4⋊C4—62no—
430C4⋊C4—62no—
431C4⋊C4—64no—
432C4⋊C4—61no—
433C4⋊C4—61no—
434C4⋊C4—62no—
435C4⋊C4—62no—
436C22⋊C4—24no—
437C22⋊C4—24no—
438C22⋊C4—64no—
439C22⋊C4—64no—
440C22⋊C4—64no—
441C22⋊C4—62no—
442C22⋊C4—62no—
443C22⋊C4—124no—
444C22⋊C4—242no—
445C22⋊C4—242no—
446C42—64no—
447C42—61no—
448C42—61no—
449C42D3×D412yes—
450C42C2×#6712yes—
451C22×C3—42no—
452C22×C3—42no—
453C22×C3—42no—
454C22×C3—82no—
455C22×C3—82no—
456C22×C3—82no—
457C22×C3—82no—
458C22×C3—82no—
459C22×C3—82no—
460C22×C3—82no—
461C22×C3—82no—
462C22×C3—82no—
463C22×C3C2×#12011yescharacteristic
464C22×C3#53211yescharacteristic
465C22×C3#53511yescharacteristic
466C22×C3#57612yes—
467C22×C3#57712yes—
468C2×D3—42no—
469C2×D3—41no—
470C2×D3—42no—
471C2×D3—41no—
472C2×D3—41no—
473C3×C4—22no—
474C3×C4—22no—
475C3×C4—42no—
476C3×C4#54612yes—
477C3×C4#55012yes—
478Dic3—41no—
479Dic3—42no—
480Dic3—42no—
481Dic3—41no—
482Dic3—41no—
483Dic3—82no—
484C23—22no—
485C23—41no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—61no—
497C23—61no—
498C23—62no—
499C23—62no—
500C23—61no—
501C23C2×D3×D411yescharacteristic
502Q8—24no—
503Q8—42no—
504Q8—244no—
505D4—24no—
506D4—42no—
507D4—42no—
508D4—42no—
509D4—42no—
510D4—42no—
511D4—84no—
512D4—84no—
513D4—244no—
514D4—248no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—61no—
521C2×C4—61no—
522C2×C4—62no—
523C2×C4—62no—
524C2×C4—61no—
525C2×C4—62no—
526C2×C4—61no—
527C2×C4—62no—
528C2×C4—61no—
529C2×C4—122no—
530C2×C4—122no—
531C2×C4—122no—
532C2×C4—122no—
533C2×C4—122no—
534C2×C4—122no—
535C2×C4—122no—
536C2×C4—244no—
537C2×C4#93712yes—
538C2×C4#93712yes—
539C2×C4C2×D3×D411yescharacteristic
540C2×C4C2×#23712yes—
541C2×C4#100412yes—
542C2×C4C22×#6711yescharacteristic
543C8—24no—
544C8—122no—
545C2×C3—82no—
546C2×C3—162no—
547C2×C3#243111yescharacteristic
548C2×C3#300811yescharacteristic
549C2×C3#301011yescharacteristic
550C2×C3#317912yes—
551C2×C3#322412yes—
552D3—81no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—122no—
566C22—122no—
567C22—121no—
568C22—121no—
569C22—121no—
570C22#571511yescharacteristic
571C22#571611yescharacteristic
572C22#577412yes—
573C22#577512yes—
574C22C2×#93711yescharacteristic
575C4—22no—
576C4—22no—
577C4—42no—
578C4—121no—
579C4—122no—
580C4—122no—
581C4—121no—
582C4—121no—
583C4—242no—
584C4#571812yes—
585C4#580612yes—
586C3#2561411yescharacteristic, complemented, Sylow
587C2—82no—
588C2—162no—
589C2—241no—
590C2#8183712yes—
591C2#8185512yes—
592C2#8318711yescharacteristic
593C2#8394711yescharacteristic
594C2#8394811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#496C2
socleC23×C3C22×D4
hypercenterC2×#496D3
hypocenterC3#25614
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#496

Complements

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

NH ≅ G/Ntype
1. C2×#5682C2semidirect
2. C2×C3×#496C2semidirect
3. #83189C2semidirect
4. C2×#5070C2semidirect
5. #72054C2semidirect
6. #72048C2semidirect
7. #72043C2semidirect
8. #72037C2semidirect
9. #71725C2semidirect
10. #69830C2semidirect
11. #69828C2semidirect
49. C2×#496D3semidirect
79. C3×#124D4semidirect
119. #869D4semidirect
267. #124#67semidirect
586. C3#25614semidirect

Maximal subgroups

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

subgroupindex
C2×#56822
C2×C3×#4962
#831892
C2×#50702
#720542
#720482
#720432
#720372
#717252
#698302
#698282
#256143