Finite Group Databasebeta

← #10848763

Subgroups — #10848763

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×#5691C211yesmaximal, characteristic
2C2×C3×#491C211yesmaximal, characteristic, complemented
3#83269C211yesmaximal, characteristic, complemented
4C2×#5086C211yesmaximal, characteristic, complemented
5#71741C211yesmaximal, characteristic, complemented
6#70281C212yesmaximal, complemented
7#70276C212yesmaximal, complemented
8#70273C212yesmaximal, complemented
9#70272C212yesmaximal
10#69821C211yesmaximal, characteristic, complemented
11#69807C211yesmaximal, characteristic, complemented
12#25604—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×#125C2211yescharacteristic
15C2×#946C2212yes—
16C22×#216C2212yes—
17C22×#215C2212yes—
18#5691—24no—
19#5691—24no—
20C2×#898—21no—
21C2×#897C2211yescharacteristic
22C2×#895C2211yescharacteristic
23C2×#894—21no—
24C2×#893—21no—
25C2×#893C2211yescharacteristic
26C2×#892—21no—
27C2×#887C2211yescharacteristic
28C2×#875C2211yescharacteristic
29C2×C42×D3C2211yescharacteristic
30C3×#491—24no—
31C2×C3×#103C2212yes—
32C2×C3×#103C2212yes—
33C2×C3×#97C2211yescharacteristic
34#5229C2212yes—
35C2×#821C2212yes—
36C2×(C2×D3)⋊C8C2212yes—
37C2×#816C2212yes—
38#5086—24no—
39#4987—21no—
40#4987—21no—
41#4972—21no—
42#4972—21no—
43#4935C2211yescharacteristic
44C2×#803C2211yescharacteristic
45#4550C2212yes—
46#4529C2212yes—
47#4529C2212yes—
48C2×#529—31no—
49C2×#497—31no—
50C2×#491D311yescharacteristic, complemented
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×D12—21no—
61C22×D12C2311yescharacteristic
62C22×C4×D3—21no—
63C22×C4×D3D411yescharacteristic
64C22×C4×D3D411yescharacteristic
65C22×C4×D3C2311yescharacteristic
66C22×Dic6C2311yescharacteristic
67C22×C3×C8C2312yes—
68C3×C4⋊Q8—22no—
69C3×C4⋊Q8—22no—
70C3×#125—22no—
71C3×#125D414yes—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C42C2311yescharacteristic
79#946—42no—
80#946—42no—
81C2×#216—24no—
82C2×#216—24no—
83C2×#216—42no—
84C2×#215—24no—
85C2×#215—24no—
86C2×#215—42no—
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—22no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3C2312yes—
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—22no—
99C2×C4×Dic3—21no—
100C2×C4×Dic3D411yescharacteristic
101C2×C4×Dic3D411yescharacteristic
102C2×C4×Dic3C2311yescharacteristic
103C22×C3⋊C8C2312yes—
104#898—28no—
105#897—22no—
106#897—22no—
107#895—22no—
108#895—22no—
109#894—28no—
110#893—28no—
111#893—22no—
112#893—22no—
113#892—28no—
114#887—22no—
115#887—22no—
116#875—22no—
117#875D414yes—
118C42×D3—22no—
119C42×D3—22no—
120C3×#103—24no—
121C3×#103—24no—
122C3×#97—22no—
123C3⋊#95—22no—
124C3⋊#95—22no—
125#821—42no—
126(C2×D3)⋊C8—24no—
127#816—42no—
128#803—22no—
129C22×Q16—32no—
130C22×SD16—32no—
131C2×C4⋊Q8C2×D311yescharacteristic
132#529—64no—
133#529—64no—
134C2×#126—61no—
135C2×#125C2×D311yescharacteristic
136C2×#124—31no—
137C2×#123—61no—
138C2×C22⋊Q8—32no—
139C2×C22⋊Q8—31no—
140C2×#121—31no—
141C2×C4×Q8—31no—
142C2×C4×D4—31no—
143C2×C4×D4—61no—
144C2×#117—61no—
145C22×C42—31no—
146#497—64no—
147#491—24no—
148C2×#103—32no—
149C2×#103C2×D312yes—
150C2×#103C2×D312yes—
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—
164C23×D3—21no—
165C23×D3C2×D411yescharacteristic
166C2×C3×Q8—22no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—22no—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2411yescharacteristic
178C22×Dic3—21no—
179C22×Dic3—22no—
180C22×Dic3C2×D411yescharacteristic
181C22×Dic3C2×D411yescharacteristic
182C22×Dic3C2×D411yescharacteristic
183C2×D12—22no—
184C2×D12—22no—
185C2×D12—24no—
186C2×D12—21no—
187C2×D12—42no—
188C2×D12—42no—
189C2×D12C2×D412yes—
190C2×C4×D3—24no—
191C2×C4×D3—22no—
192C2×C4×D3—22no—
193C2×C4×D3—22no—
194C2×C4×D3—21no—
195C2×C4×D3—21no—
196C2×C4×D3—22no—
197C2×C4×D3—22no—
198C2×C4×D3—22no—
199C2×C4×D3—41no—
200C2×C4×D3—41no—
201C2×C4×D3—42no—
202C2×C4×D3—42no—
203C2×Dic6—22no—
204C2×Dic6—22no—
205C2×Dic6—21no—
206C2×Dic6C2×D412yes—
207C2×C3×C8—22no—
208C2×C3×C8—22no—
209C2×C3×C8—22no—
210C3×C4⋊C4—24no—
211C3×C4⋊C4—24no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—42no—
215C3×C4⋊C4—42no—
216C3×C4⋊C4C2×D412yes—
217C3×C4⋊C4C2×D412yes—
218C3×C42C2×D412yes—
219C3×C42C2×D412yes—
220#216—48no—
221#215—48no—
222(C2×D3)⋊C4—24no—
223(C2×D3)⋊C4—24no—
224(C2×D3)⋊C4—24no—
225(C2×D3)⋊C4—24no—
226(C2×D3)⋊C4—24no—
227(C2×D3)⋊C4—44no—
228C4⋊Dic3—24no—
229C4⋊Dic3—22no—
230C4⋊Dic3—22no—
231C4⋊Dic3—42no—
232C4⋊Dic3—42no—
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#4114yes—
243C22×Q8—31no—
244C22×Q8C22×D312yes—
245C22×D4—31no—
246C22×D4—61no—
247C23×C4—31no—
248C23×C4—31no—
249C23×C4—31no—
250C23×C4—61no—
251C2×Q16—64no—
252C2×Q16—64no—
253C2×Q16—122no—
254C2×SD16—64no—
255C2×SD16—64no—
256C2×SD16—122no—
257C22×C8—32no—
258C22×C8C22×D312yes—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261#126—68no—
262#125—22no—
263#125#6714yes—
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—68no—
277C4×D4—62no—
278#117—68no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—32no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—62no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C4⋊C4C22×D311yescharacteristic
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—31no—
298C2×C42—31no—
299C2×C42—31no—
300C2×C42—61no—
301C2×C42—62no—
302C2×C42—62no—
303C2×C42C22×D311yescharacteristic
304#103—24no—
305#103—24no—
306#103—122no—
307#102—122no—
308C22⋊C8—64no—
309#97—22no—
310#97—62no—
311#95—62no—
312#95—62no—
313C23×C3C22×D411yescharacteristic
314C22×D3—21no—
315C22×D3—21no—
316C22×D3—21no—
317C22×D3—21no—
318C22×D3—21no—
319C22×D3—21no—
320C22×D3—22no—
321C22×D3—41no—
322C22×D3—41no—
323C22×D3—41no—
324C22×D3—41no—
325C22×D3—41no—
326C22×D3—41no—
327C22×D3#13712yes—
328C3×Q8—42no—
329C3×Q8—42no—
330C3×Q8—42no—
331C3×Q8—42no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
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—
351C2×Dic3—21no—
352C2×Dic3—21no—
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—22no—
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—42no—
363C2×Dic3—42no—
364C2×Dic3—42no—
365C2×Dic3#13712yes—
366D12—24no—
367D12—48no—
368D12—42no—
369C4×D3—44no—
370C4×D3—48no—
371C4×D3—44no—
372C4×D3—42no—
373C4×D3—42no—
374Dic6—24no—
375Dic6—42no—
376C3×C8—42no—
377C3⋊C8—24no—
378C24—31no—
379C24—61no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—32no—
387C2×Q8—61no—
388C2×Q8—62no—
389C2×Q8—62no—
390C2×D4—32no—
391C2×D4—64no—
392C2×D4—62no—
393C2×D4—62no—
394C2×D4—61no—
395C2×D4—122no—
396C2×D4—122no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—31no—
401C22×C4—31no—
402C22×C4—31no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4—64no—
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—128no—
422SD16—128no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—34no—
427C2×C8—62no—
428C2×C8—62no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—64no—
436C4⋊C4—64no—
437C4⋊C4—62no—
438C4⋊C4—62no—
439C4⋊C4—122no—
440C4⋊C4—122no—
441C4⋊C4—124no—
442C4⋊C4C2×#6712yes—
443C4⋊C4C2×#6712yes—
444C22⋊C4—64no—
445C22⋊C4—64no—
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×C3C2×#12011yescharacteristic
458C22×C3#53211yescharacteristic
459C22×C3#53511yescharacteristic
460C22×C3C2×#13711yescharacteristic
461C22×C3C2×#13711yescharacteristic
462C22×C3#57511yescharacteristic
463C22×C3#57511yescharacteristic
464C2×D3—22no—
465C2×D3—22no—
466C2×D3—22no—
467C2×D3—41no—
468C2×D3—41no—
469C2×D3—41no—
470C2×D3—41no—
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—42no—
481C3×C4—42no—
482C3×C4—82no—
483C3×C4#54612yes—
484C3×C4#55012yes—
485Dic3—22no—
486Dic3—42no—
487Dic3—41no—
488Dic3—41no—
489Dic3—82no—
490C23—32no—
491C23—61no—
492C23—61no—
493C23—61no—
494C23—61no—
495C23—61no—
496C23—61no—
497C23—62no—
498C23—121no—
499C23—121no—
500C23—121no—
501C23—121no—
502C23—121no—
503C23—121no—
504C23C2×D3×D411yescharacteristic
505Q8—42no—
506Q8—42no—
507Q8—42no—
508Q8—42no—
509Q8—64no—
510Q8—122no—
511D4—64no—
512D4—128no—
513D4—122no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—32no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—62no—
537C2×C4—124no—
538C2×C4—124no—
539C2×C4—128no—
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×C4D3×#4112yes—
551C2×C4#100912yes—
552C2×C4C22×#6711yescharacteristic
553C8—42no—
554C8—64no—
555C2×C3#243111yescharacteristic
556C2×C3#300711yescharacteristic
557C2×C3#300711yescharacteristic
558C2×C3#317211yescharacteristic
559C2×C3#318111yescharacteristic
560C2×C3#321811yescharacteristic
561C2×C3#321911yescharacteristic
562D3—42no—
563D3—82no—
564C22—62no—
565C22—62no—
566C22—62no—
567C22—121no—
568C22—121no—
569C22—121no—
570C22—121no—
571C22—122no—
572C22—122no—
573C22—122no—
574C22—241no—
575C22—241no—
576C22—241no—
577C22—241no—
578C22#566811yescharacteristic
579C22#567411yescharacteristic
580C22D3×#13711yescharacteristic
581C22D3×#13711yescharacteristic
582C22#577711yescharacteristic
583C22#577711yescharacteristic
584C22C2×#93711yescharacteristic
585C4—22no—
586C4—22no—
587C4—42no—
588C4—42no—
589C4—62no—
590C4—82no—
591C4—121no—
592C4—121no—
593C4—122no—
594C4—242no—
595C4#567612yes—
596C4#583412yes—
597C3#2560411yescharacteristic, complemented, Sylow
598C2—122no—
599C2—242no—
600C2#8172811yescharacteristic
601C2#8173111yescharacteristic
602C2#8174111yescharacteristic
603C2#8174211yescharacteristic
604C2#8295811yescharacteristic
605C2#8398111yescharacteristic
606C2#8398111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#491C2
socleC23×C3C22×D4
prime radicalC2×#895C22
hypercenterC2×#491D3
hypocenterC3#25604
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#491

Complements

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

NH ≅ G/Ntype
2. C2×C3×#491C2semidirect
3. #83269C2semidirect
4. C2×#5086C2semidirect
5. #71741C2semidirect
6. #70281C2semidirect
7. #70276C2semidirect
8. #70273C2semidirect
10. #69821C2semidirect
11. #69807C2semidirect
50. C2×#491D3semidirect
597. C3#25604semidirect

Maximal subgroups

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

subgroupindex
C2×#56912
C2×C3×#4912
#832692
C2×#50862
#717412
#702812
#702762
#702732
#702722
#698212
#698072
#256043