Finite Group Databasebeta

← #10848755

Subgroups — #10848755

Order 768.

1178 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5680C211yesmaximal, characteristic
2C2×C3×#497C211yesmaximal, characteristic, complemented
3#83269C211yesmaximal, characteristic, complemented
4C2×#5082C211yesmaximal, characteristic, complemented
5#72071C212yesmaximal, complemented
6#72065C212yesmaximal, complemented
7#72054C212yesmaximal
8#72047C212yesmaximal, complemented
9#71726C211yesmaximal, characteristic, complemented
10#69827C211yesmaximal, characteristic, complemented
11#69807C211yesmaximal, characteristic, complemented
12#25604—31nomaximal, Sylow
13C22×C3×Q16C2212yes—
14C22×C3×SD16C2212yes—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#124C2211yescharacteristic
17C2×#946C2211yescharacteristic
18C2×#946C2212yes—
19C2×Q8×Dic3C2211yescharacteristic
20C2×#938C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22#5680—24no—
23#5680—24no—
24C2×#883—21no—
25C2×#67⋊C4—21no—
26C2×#879—21no—
27C2×#877—21no—
28C2×C42×D3C2211yescharacteristic
29C2×#870C2211yescharacteristic
30C3×#497—24no—
31C2×C3×#97C2211yescharacteristic
32#5229C2212yes—
33C2×#835C2212yes—
34C2×#835C2212yes—
35C2×#835C2212yes—
36C2×#832C2212yes—
37C2×(C2×D3)⋊C8C2212yes—
38#5082—24no—
39#4987—21no—
40#4987—21no—
41#4972—21no—
42#4972—21no—
43#4935C2211yescharacteristic
44C2×#803C2211yescharacteristic
45#4551C2212yes—
46#4551C2212yes—
47#4550C2212yes—
48C2×#529—31no—
49C2×#497D311yescharacteristic, complemented
50C2×#491—31no—
51#2857—31no—
52#2001—32no—
53#2000—32no—
54#1992—31no—
55#1959—31no—
56#1878—32no—
57#1877—32no—
58#1749—31no—
59C22×C3×Q8C2312yes—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4C2311yescharacteristic
62C22×#67—21no—
63C23×Dic3—21no—
64C22×C4×D3D411yescharacteristic
65C22×C4×D3D411yescharacteristic
66C22×C4×D3C2311yescharacteristic
67C2×C3×Q16—24no—
68C2×C3×Q16—24no—
69C2×C3×Q16—42no—
70C2×C3×SD16—24no—
71C2×C3×SD16—24no—
72C2×C3×SD16—42no—
73C22×C3×C8C2312yes—
74C3×C4⋊Q8—22no—
75C3×C4⋊Q8—22no—
76C3×#124—22no—
77C3×#124D414yes—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—21no—
82C2×C3×C42C2311yescharacteristic
83C2×#217—21no—
84C2×#217—21no—
85#946—22no—
86#946—22no—
87#946—42no—
88#946—42no—
89Q8×Dic3—22no—
90Q8×Dic3—22no—
91#938—22no—
92#938—22no—
93D4×Dic3—22no—
94D4×Dic3—22no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—22no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3C2312yes—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—22no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3D411yescharacteristic
109C2×C4×Dic3D411yescharacteristic
110C2×C4×Dic3C2311yescharacteristic
111C22×C3⋊C8C2312yes—
112#883—28no—
113#67⋊C4—28no—
114#879—28no—
115#877—28no—
116C42×D3—22no—
117C42×D3—22no—
118#870—22no—
119#870D414yes—
120C3×#97—22no—
121#835—24no—
122#835—24no—
123#835—42no—
124#832—42no—
125C3⋊#95—22no—
126C3⋊#95—22no—
127(C2×D3)⋊C8—24no—
128#803—22no—
129C22×Q16C2×D312yes—
130C22×SD16C2×D312yes—
131C2×C4⋊Q8C2×D311yescharacteristic
132#529—64no—
133#529—64no—
134C2×#126—61no—
135C2×#125—31no—
136C2×#124C2×D311yescharacteristic
137C2×#123—61no—
138C2×C22⋊Q8—31no—
139C2×C22⋊Q8—32no—
140C2×#121—31no—
141C2×C4×Q8—31no—
142C2×C4×D4—31no—
143C2×C4×D4—61no—
144C2×#117—61no—
145C22×C42—31no—
146#497—24no—
147#491—64no—
148C2×#103—32no—
149C2×#103—32no—
150C2×#103—32no—
151C2×#102—32no—
152C2×C22⋊C8—32no—
153C2×#97—31no—
154C2×#97C2×D311yescharacteristic
155#391—32no—
156#389—61no—
157#389—61no—
158#384—31no—
159#380—61no—
160#380—61no—
161#340—32no—
162#340—32no—
163#336—32no—
164C24×C3—21no—
165C23×D3C2×D411yescharacteristic
166C2×C3×Q8—21no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
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×Q8C2×D412yes—
176C2×C3×D4—21no—
177C2×C3×D4—22no—
178C2×C3×D4—22no—
179C2×C3×D4C2×D412yes—
180C22×C3×C4—21no—
181C22×C3×C4—22no—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2411yescharacteristic
185C2×#67—24no—
186C2×#67—42no—
187C2×#67—42no—
188C22×Dic3—24no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—22no—
192C22×Dic3—42no—
193C22×Dic3—42no—
194C22×Dic3C2×D411yescharacteristic
195C22×Dic3C2×D411yescharacteristic
196C22×Dic3C2×D411yescharacteristic
197C2×C4×D3—22no—
198C2×C4×D3—22no—
199C2×C4×D3—22no—
200C2×C4×D3—21no—
201C2×C4×D3—21no—
202C2×C4×D3—22no—
203C2×C4×D3—22no—
204C2×C4×D3—22no—
205C2×C4×D3—41no—
206C2×C4×D3—41no—
207C3×Q16—48no—
208C3×SD16—48no—
209C2×C3×C8—22no—
210C2×C3×C8—22no—
211C2×C3×C8#4114yes—
212C3×C4⋊C4—42no—
213C3×C4⋊C4—42no—
214C3×C22⋊C4—24no—
215C3×C22⋊C4—24no—
216C3×C42C2×D412yes—
217C3×C42C2×D412yes—
218#217—24no—
219#217—24no—
220(C2×D3)⋊C4—24no—
221(C2×D3)⋊C4—44no—
222C4⋊Dic3—24no—
223C4⋊Dic3—21no—
224C4⋊Dic3—21no—
225C4⋊Dic3—22no—
226C4⋊Dic3—22no—
227C4⋊Dic3—42no—
228C4⋊Dic3—42no—
229C4⋊Dic3C2×D412yes—
230C4⋊Dic3C2×D412yes—
231Dic3⋊C4—24no—
232Dic3⋊C4—24no—
233Dic3⋊C4—24no—
234Dic3⋊C4—44no—
235C4×Dic3—24no—
236C4×Dic3—22no—
237C4×Dic3—22no—
238C4×Dic3—21no—
239C4×Dic3—21no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8—22no—
243C22×Q8C22×D312yes—
244C22×Q8C22×D311yescharacteristic
245C22×D4—61no—
246C22×D4C22×D311yescharacteristic
247C23×C4—31no—
248C23×C4—31no—
249C23×C4—31no—
250C23×C4—61no—
251C2×Q16—24no—
252C2×Q16—24no—
253C2×Q16—42no—
254C2×SD16—24no—
255C2×SD16—24no—
256C2×SD16—42no—
257C22×C8—32no—
258C22×C8C22×D312yes—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261#126—68no—
262#125—34no—
263#125—62no—
264#124—22no—
265#124#6714yes—
266#123—68no—
267C22⋊Q8—62no—
268C22⋊Q8—62no—
269C22⋊Q8—122no—
270C22⋊Q8—122no—
271#121—62no—
272#121—62no—
273C4×Q8—62no—
274C4×Q8—62no—
275C4×D4—68no—
276C4×D4—62no—
277C4×D4—62no—
278#117—68no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—32no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—62no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—62no—
297C2×C42—31no—
298C2×C42—31no—
299C2×C42—31no—
300C2×C42—61no—
301C2×C42—62no—
302C2×C42—62no—
303C2×C42C22×D311yescharacteristic
304#103—64no—
305#103—64no—
306#103—122no—
307#102—122no—
308C22⋊C8—64no—
309#97—22no—
310#97—62no—
311#95—62no—
312#95—62no—
313C23×C3—22no—
314C23×C3—41no—
315C23×C3—41no—
316C23×C3—41no—
317C23×C3—41no—
318C23×C3—41no—
319C23×C3—41no—
320C23×C3C22×D411yescharacteristic
321C22×D3—21no—
322C22×D3—21no—
323C22×D3—21no—
324C22×D3—21no—
325C22×D3—21no—
326C22×D3—21no—
327C22×D3#13712yes—
328C3×Q8—24no—
329C3×Q8—42no—
330C3×Q8—42no—
331C3×Q8—42no—
332C3×Q8—42no—
333C3×Q8—42no—
334C3×D4—24no—
335C3×D4—42no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—22no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4#12012yes—
347C2×C3×C4#12012yes—
348C2×C3×C4C22×D411yescharacteristic
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4C2×#4112yes—
351C2×C3×C4H(5,𝔽2)12yes—
352#67—48no—
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—21no—
356C2×Dic3—21no—
357C2×Dic3—22no—
358C2×Dic3—22no—
359C2×Dic3—21no—
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—48no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3#13712yes—
372C4×D3—44no—
373C4×D3—44no—
374C4×D3—42no—
375C4×D3—42no—
376C3×C8—24no—
377C3⋊C8—42no—
378C24—21no—
379C24—31no—
380C2×Q8—21no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8C2×#6712yes—
390C2×D4—21no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—64no—
394C2×D4—122no—
395C2×D4—122no—
396C2×D4C2×#6712yes—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—31no—
400C22×C4—31no—
401C22×C4—31no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—64no—
405C22×C4—61no—
406C22×C4—61no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4—61no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—61no—
413C22×C4—62no—
414C22×C4—121no—
415C22×C4—121no—
416C22×C4—122no—
417C22×C4—122no—
418C22×C4D3×D411yescharacteristic
419C22×C4D3×D411yescharacteristic
420C22×C4C23×D311yescharacteristic
421Q16—48no—
422SD16—48no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—62no—
426C2×C8—62no—
427C2×C8—62no—
428C2×C8#23714yes—
429C4⋊C4—32no—
430C4⋊C4—32no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—64no—
434C4⋊C4—64no—
435C4⋊C4—64no—
436C4⋊C4—64no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—62no—
440C4⋊C4—62no—
441C4⋊C4—122no—
442C4⋊C4—122no—
443C4⋊C4—124no—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C22⋊C4—64no—
447C22⋊C4—64no—
448C22⋊C4—64no—
449C22⋊C4—124no—
450C42—64no—
451C42—62no—
452C42—62no—
453C42—61no—
454C42—61no—
455C42D3×D412yes—
456C42C2×#6712yes—
457C22×C3—42no—
458C22×C3—42no—
459C22×C3—42no—
460C22×C3—81no—
461C22×C3—81no—
462C22×C3—81no—
463C22×C3—81no—
464C22×C3C2×#12011yescharacteristic
465C22×C3#53211yescharacteristic
466C22×C3#53511yescharacteristic
467C22×C3C2×#13711yescharacteristic
468C22×C3C2×#13711yescharacteristic
469C22×C3#57511yescharacteristic
470C22×C3#57511yescharacteristic
471C2×D3—22no—
472C2×D3—22no—
473C2×D3—22no—
474C2×D3—41no—
475C2×D3—41no—
476C2×D3—41no—
477C2×D3—41no—
478C3×C4—22no—
479C3×C4—22no—
480C3×C4—42no—
481C3×C4—82no—
482C3×C4#54612yes—
483C3×C4#55012yes—
484Dic3—22no—
485Dic3—41no—
486Dic3—41no—
487Dic3—42no—
488Dic3—42no—
489Dic3—82no—
490C23—22no—
491C23—32no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—61no—
499C23—61no—
500C23—61no—
501C23—61no—
502C23—61no—
503C23—61no—
504C23C2×D3×D411yescharacteristic
505Q8—24no—
506Q8—42no—
507Q8—42no—
508Q8—42no—
509Q8—42no—
510Q8—42no—
511D4—24no—
512D4—42no—
513D4—128no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—32no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—61no—
526C2×C4—61no—
527C2×C4—61no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—61no—
533C2×C4—62no—
534C2×C4—62no—
535C2×C4—124no—
536C2×C4—124no—
537C2×C4—128no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—122no—
543C2×C4—122no—
544C2×C4—122no—
545C2×C4—122no—
546C2×C4—122no—
547C2×C4#93712yes—
548C2×C4#93712yes—
549C2×C4C2×D3×D411yescharacteristic
550C2×C4C2×#23712yes—
551C2×C4#100412yes—
552C2×C4C22×#6711yescharacteristic
553C8—24no—
554C8—122no—
555C2×C3—82no—
556C2×C3#243111yescharacteristic
557C2×C3#300711yescharacteristic
558C2×C3#300711yescharacteristic
559C2×C3#317211yescharacteristic
560C2×C3#318111yescharacteristic
561C2×C3#321811yescharacteristic
562C2×C3#321911yescharacteristic
563D3—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—62no—
568C22—62no—
569C22—62no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—121no—
575C22—121no—
576C22—121no—
577C22—121no—
578C22#571511yescharacteristic
579C22#571611yescharacteristic
580C22D3×#13711yescharacteristic
581C22D3×#13711yescharacteristic
582C22#577711yescharacteristic
583C22#577711yescharacteristic
584C22C2×#93711yescharacteristic
585C4—22no—
586C4—22no—
587C4—42no—
588C4—62no—
589C4—82no—
590C4—121no—
591C4—122no—
592C4—122no—
593C4—121no—
594C4—242no—
595C4#571812yes—
596C4#580612yes—
597C3#2560411yescharacteristic, complemented, Sylow
598C2—82no—
599C2—122no—
600C2#8186111yescharacteristic
601C2#8186311yescharacteristic
602C2#8187611yescharacteristic
603C2#8187711yescharacteristic
604C2#8318711yescharacteristic
605C2#8398111yescharacteristic
606C2#8398111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#497C2
socleC23×C3C22×D4
prime radicalC2×#938C22
hypercenterC2×#497D3
hypocenterC3#25604
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#497

Complements

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

NH ≅ G/Ntype
2. C2×C3×#497C2semidirect
3. #83269C2semidirect
4. C2×#5082C2semidirect
5. #72071C2semidirect
6. #72065C2semidirect
8. #72047C2semidirect
9. #71726C2semidirect
10. #69827C2semidirect
11. #69807C2semidirect
49. C2×#497D3semidirect
597. C3#25604semidirect

Maximal subgroups

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

subgroupindex
C2×#56802
C2×C3×#4972
#832692
C2×#50822
#720712
#720652
#720542
#720472
#717262
#698272
#698072
#256043