Finite Group Databasebeta

← #10848758

Subgroups — #10848758

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×#5479C211yesmaximal, characteristic
2#82076C211yesmaximal, characteristic, complemented
3C2×#4715C211yesmaximal, characteristic, complemented
4C2×#4703C211yesmaximal, characteristic, complemented
5C3×#1749C211yesmaximal, characteristic, complemented
6#69964C212yesmaximal, complemented
7#69962C212yesmaximal
8#69827C211yesmaximal, characteristic, complemented
9#69821C211yesmaximal, characteristic, complemented
10#68685C212yesmaximal, complemented
11#68672C212yesmaximal, complemented
12#25604—31nomaximal, Sylow
13C22×C3×C42C2211yescharacteristic
14C2×#930C2211yescharacteristic
15C2×#929—21no—
16C2×C4×#67—21no—
17C2×#924C2211yescharacteristic
18C2×#924C2212yes—
19C22×Dic12C2212yes—
20C22×#204C2212yes—
21#5479—24no—
22#5479—24no—
23C2×#876—21no—
24C2×#875C2211yescharacteristic
25C2×C4×D12C2211yescharacteristic
26C2×#872—21no—
27C2×#870C2211yescharacteristic
28C2×#869C2211yescharacteristic
29C2×C4×Dic6C2211yescharacteristic
30C2×C3×C22⋊C8C2212yes—
31C2×C3×#97C2211yescharacteristic
32C2×C3×#97C2211yescharacteristic
33#5212—21no—
34#5212—21no—
35C2×#821C2212yes—
36C2×#816C2212yes—
37C2×#816C2212yes—
38C2×#816C2212yes—
39#4935C2211yescharacteristic
40#4929C2212yes—
41#4928—21no—
42#4928—21no—
43#4715—24no—
44#4703—24no—
45C3×#336C2212yes—
46#4551C2212yes—
47#4551C2212yes—
48C2×#529—31no—
49C2×#497—31no—
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#1749D311yescharacteristic, complemented
59C23×C3×C4D411yescharacteristic
60C23×C3×C4D411yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×#67—21no—
63C22×D12C2311yescharacteristic
64C22×C4×D3—21no—
65C22×Dic6C2312yes—
66C22×Dic6C2311yescharacteristic
67C22×C3×C8C2312yes—
68C22×C3×C8C2312yes—
69C2×C3×C42—22no—
70C2×C3×C42—22no—
71C2×C3×C42D411yescharacteristic
72C2×C3×C42D411yescharacteristic
73C2×C3×C42C2311yescharacteristic
74C2×C3×C42C2311yescharacteristic
75C2×#217—21no—
76C2×#217—22no—
77#930—22no—
78#930—22no—
79#929—28no—
80C4×#67—28no—
81C2×(C2×D3)⋊C4—21no—
82C2×(C2×D3)⋊C4—21no—
83C2×(C2×D3)⋊C4—21no—
84C2×(C2×D3)⋊C4—21no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3C2312yes—
90C2×C4⋊Dic3C2311yescharacteristic
91C2×C4⋊Dic3C2311yescharacteristic
92#924—22no—
93#924—22no—
94#924—42no—
95#924—42no—
96C2×Dic3⋊C4—21no—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—22no—
101C2×C4×Dic3—21no—
102C2×Dic12—24no—
103C2×Dic12—24no—
104C2×Dic12—42no—
105C2×#204—24no—
106C2×#204—24no—
107C2×#204—42no—
108#876—28no—
109#875—22no—
110#875D414yes—
111C4×D12—22no—
112C4×D12—22no—
113#872—28no—
114#870—22no—
115#870D414yes—
116#869—22no—
117#869—22no—
118C4×Dic6—22no—
119C4×Dic6—22no—
120C3×C22⋊C8—24no—
121C3×#97—22no—
122C3×#97—22no—
123C3⋊#95—22no—
124C3⋊#95—22no—
125#821—42no—
126#816—24no—
127#816—24no—
128#816—42no—
129C22×Q16—32no—
130C22×SD16—32no—
131C2×C4⋊Q8—31no—
132#529—64no—
133#529—64no—
134C2×#126—61no—
135C2×#125—31no—
136C2×#124—31no—
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×C42C2×D311yescharacteristic
146#497—64no—
147#491—64no—
148C2×#103—32no—
149C2×#103—32no—
150C2×#103—32no—
151C2×#102—32no—
152C2×C22⋊C8C2×D312yes—
153C2×#97C2×D311yescharacteristic
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#336C2×D312yes—
164C24×C3C2×D411yescharacteristic
165C23×D3—21no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—41no—
175C22×C3×C4—41no—
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2411yescharacteristic
182C2×#67—24no—
183C2×#67—42no—
184C2×#67—42no—
185C22×Dic3—21no—
186C22×Dic3—21no—
187C22×Dic3—21no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C2×D12—22no—
191C2×D12—22no—
192C2×D12—21no—
193C2×D12C2×D412yes—
194C2×C4×D3—24no—
195C2×C4×D3—42no—
196C2×C4×D3—42no—
197C2×Dic6—22no—
198C2×Dic6—22no—
199C2×Dic6—21no—
200C2×Dic6—22no—
201C2×Dic6—22no—
202C2×Dic6—22no—
203C2×Dic6—22no—
204C2×Dic6—22no—
205C2×Dic6—22no—
206C2×Dic6C2×D412yes—
207C2×C3×C8—22no—
208C2×C3×C8—22no—
209C2×C3×C8—22no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C2×C3×C8#4114yes—
213C3×C42—22no—
214C3×C42—22no—
215C3×C42—21no—
216C3×C42—21no—
217C3×C42C2×D412yes—
218C3×C42C2×D412yes—
219#217—24no—
220#217—44no—
221(C2×D3)⋊C4—24no—
222(C2×D3)⋊C4—24no—
223(C2×D3)⋊C4—24no—
224(C2×D3)⋊C4—24no—
225C4⋊Dic3—21no—
226C4⋊Dic3—21no—
227C4⋊Dic3—22no—
228C4⋊Dic3—22no—
229C4⋊Dic3—42no—
230C4⋊Dic3—42no—
231C4⋊Dic3—42no—
232C4⋊Dic3—42no—
233C4⋊Dic3C2×D412yes—
234C4⋊Dic3C2×D412yes—
235Dic3⋊C4—24no—
236Dic3⋊C4—24no—
237Dic3⋊C4—24no—
238Dic3⋊C4—24no—
239Dic3⋊C4—44no—
240C4×Dic3—24no—
241Dic12—48no—
242#204—48no—
243C22×Q8—31no—
244C22×Q8—32no—
245C22×D4—31no—
246C22×D4—61no—
247C23×C4—61no—
248C23×C4D1211yescharacteristic
249C23×C4D1211yescharacteristic
250C23×C4C22×D311yescharacteristic
251C2×Q16—64no—
252C2×Q16—64no—
253C2×Q16—122no—
254C2×SD16—64no—
255C2×SD16—64no—
256C2×SD16—122no—
257C22×C8C22×D312yes—
258C22×C8C22×D312yes—
259C4⋊Q8—62no—
260C4⋊Q8—62no—
261#126—68no—
262#125—34no—
263#125—62no—
264#124—34no—
265#124—62no—
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—62no—
276C4×D4—62no—
277C4×D4—68no—
278#117—68no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—32no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—62no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—62no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—61no—
300C2×C42D1211yescharacteristic
301C2×C42D1211yescharacteristic
302C2×C42C22×D311yescharacteristic
303C2×C42C22×D311yescharacteristic
304#103—64no—
305#103—64no—
306#103—122no—
307#102—122no—
308C22⋊C8—24no—
309#97—22no—
310#97—22no—
311#95—62no—
312#95—62no—
313C23×C3—21no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3#13712yes—
320C23×C3C22×D411yescharacteristic
321C22×D3—22no—
322C22×D3—41no—
323C22×D3—41no—
324C22×D3—41no—
325C22×D3—41no—
326C22×D3—41no—
327C22×D3—41no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—44no—
341C2×C3×C4—44no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4#12012yes—
345C2×C3×C4#12012yes—
346C2×C3×C4#13712yes—
347C2×C3×C4C22×D411yescharacteristic
348C2×C3×C4C22×D411yescharacteristic
349C2×C3×C4C2×#4112yes—
350C2×C3×C4H(5,𝔽2)12yes—
351#67—48no—
352C2×Dic3—22no—
353C2×Dic3—22no—
354C2×Dic3—22no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—42no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C2×Dic3—42no—
365C2×Dic3—42no—
366C2×Dic3—42no—
367D12—24no—
368D12—42no—
369C4×D3—48no—
370Dic6—24no—
371Dic6—42no—
372Dic6—42no—
373Dic6—42no—
374Dic6—42no—
375Dic6—42no—
376C3×C8—24no—
377C3×C8—42no—
378C24—61no—
379C24C2×D1211yescharacteristic
380C2×Q8—32no—
381C2×Q8—61no—
382C2×Q8—62no—
383C2×Q8—62no—
384C2×Q8—62no—
385C2×Q8—62no—
386C2×Q8—62no—
387C2×Q8—62no—
388C2×Q8—62no—
389C2×Q8—62no—
390C2×D4—32no—
391C2×D4—64no—
392C2×D4—62no—
393C2×D4—61no—
394C2×D4—62no—
395C2×D4—122no—
396C2×D4—122no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—64no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—122no—
414C22×C4—122no—
415C22×C4C2×D1211yescharacteristic
416C22×C4C2×D1211yescharacteristic
417C22×C4C2×D1211yescharacteristic
418C22×C4C2×D1211yescharacteristic
419C22×C4C2×D1211yescharacteristic
420C22×C4C23×D311yescharacteristic
421Q16—128no—
422SD16—128no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8#23514yes—
429C4⋊C4—32no—
430C4⋊C4—32no—
431C4⋊C4—64no—
432C4⋊C4—64no—
433C4⋊C4—64no—
434C4⋊C4—64no—
435C4⋊C4—61no—
436C4⋊C4—61no—
437C4⋊C4—62no—
438C4⋊C4—62no—
439C4⋊C4—124no—
440C4⋊C4—122no—
441C4⋊C4—122no—
442C4⋊C4—122no—
443C4⋊C4—122no—
444C22⋊C4—64no—
445C22⋊C4—64no—
446C22⋊C4—64no—
447C22⋊C4—64no—
448C22⋊C4—64no—
449C22⋊C4—124no—
450C42—22no—
451C42—22no—
452C42—21no—
453C42—21no—
454C42—64no—
455C42D3×D412yes—
456C42D3×D412yes—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—22no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3—41no—
464C22×C3C2×#12011yescharacteristic
465C22×C3#53211yescharacteristic
466C22×C3#53511yescharacteristic
467C22×C3C2×#13711yescharacteristic
468C22×C3C2×#13711yescharacteristic
469C22×C3#57511yescharacteristic
470C22×C3#57511yescharacteristic
471C2×D3—42no—
472C2×D3—42no—
473C2×D3—42no—
474C2×D3—81no—
475C2×D3—81no—
476C2×D3—81no—
477C2×D3—81no—
478C3×C4—22no—
479C3×C4—22no—
480C3×C4—22no—
481C3×C4—41no—
482C3×C4—41no—
483C3×C4#54612yes—
484C3×C4#55012yes—
485Dic3—42no—
486Dic3—42no—
487Dic3—42no—
488Dic3—82no—
489Dic3—82no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—62no—
497C23—121no—
498C23—121no—
499C23—121no—
500C23—121no—
501C23—121no—
502C23—121no—
503C23#90912yes—
504C23C22×D1211yescharacteristic
505Q8—64no—
506Q8—122no—
507Q8—122no—
508Q8—122no—
509Q8—122no—
510Q8—122no—
511D4—64no—
512D4—128no—
513D4—122no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—62no—
531C2×C4—62no—
532C2×C4—62no—
533C2×C4—128no—
534C2×C4—121no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—122no—
541C2×C4—122no—
542C2×C4—122no—
543C2×C4—122no—
544C2×C4—122no—
545C2×C4—122no—
546C2×C4#88212yes—
547C2×C4#88212yes—
548C2×C4#90912yes—
549C2×C4C2×#23512yes—
550C2×C4C2×D3×D411yescharacteristic
551C2×C4C2×D3×D411yescharacteristic
552C2×C4#100912yes—
553C8—24no—
554C8—42no—
555C2×C3—42no—
556C2×C3#243111yescharacteristic
557C2×C3#300711yescharacteristic
558C2×C3#300711yescharacteristic
559C2×C3#317211yescharacteristic
560C2×C3#318111yescharacteristic
561C2×C3#321811yescharacteristic
562C2×C3#321911yescharacteristic
563D3—82no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—122no—
572C22—122no—
573C22—122no—
574C22—241no—
575C22—241no—
576C22—241no—
577C22—241no—
578C22#547711yescharacteristic
579C22C2×#88211yescharacteristic
580C22#552511yescharacteristic
581C22C2×#90911yescharacteristic
582C22C2×#90911yescharacteristic
583C22#574811yescharacteristic
584C22#574811yescharacteristic
585C4—22no—
586C4—22no—
587C4—22no—
588C4—41no—
589C4—41no—
590C4—122no—
591C4—122no—
592C4—122no—
593C4—242no—
594C4—242no—
595C4#555012yes—
596C4#555312yes—
597C3#2560411yescharacteristic, complemented, Sylow
598C2—42no—
599C2—242no—
600C2#7358611yescharacteristic
601C2#7850411yescharacteristic
602C2#7853911yescharacteristic
603C2#7864011yescharacteristic
604C2#7864011yescharacteristic
605C2#7935411yescharacteristic
606C2#7935811yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1749C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1749D3
hypocenterC3#25604
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
2. #82076C2semidirect
3. C2×#4715C2semidirect
4. C2×#4703C2semidirect
5. C3×#1749C2semidirect
6. #69964C2semidirect
8. #69827C2semidirect
9. #69821C2semidirect
10. #68685C2semidirect
11. #68672C2semidirect
58. #1749D3semidirect
597. C3#25604semidirect

Maximal subgroups

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

subgroupindex
C2×#54792
#820762
C2×#47152
C2×#47032
C3×#17492
#699642
#699622
#698272
#698212
#686852
#686722
#256043