Finite Group Databasebeta

← #10848743

Subgroups — #10848743

Order 768.

1172 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#533C211yesmaximal, characteristic, complemented
2C2×#5086C211yesmaximal, characteristic, complemented
3C2×#5089C211yesmaximal, characteristic, complemented
4#82077C211yesmaximal, characteristic
5#71788C211yesmaximal, characteristic, complemented
6#71786C211yesmaximal, characteristic, complemented
7#71700C212yesmaximal, complemented
8#71699C212yesmaximal
9#71633C212yesmaximal, complemented
10#71632C212yesmaximal, complemented
11#71529C211yesmaximal, characteristic, complemented
12#25603—31nomaximal, Sylow
13C3×#533—24no—
14C3×#533—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#125—21no—
18C2×C3×C22⋊Q8—21no—
19C2×C3×C22⋊Q8C2212yes—
20C2×C3×C4×Q8—21no—
21C2×C3×C4×Q8C2212yes—
22C2×C3×#117—21no—
23C22×C3×C42C2211yescharacteristic
24C22×#216C2212yes—
25C22×#215C2212yes—
26C2×#930C2211yescharacteristic
27C2×#924C2211yescharacteristic
28C2×#875C2211yescharacteristic
29C2×#835C2212yes—
30C2×#835C2212yes—
31#5213—21no—
32#5213—21no—
33C2×#830C2212yes—
34#5089—24no—
35#5086—24no—
36C2×#810C2212yes—
37C2×#809C2212yes—
38#4940—21no—
39#4940—21no—
40#4939C2211yescharacteristic
41#4936C2211yescharacteristic
42#4929C2211yescharacteristic
43C2×#803C2211yescharacteristic
44C2×#803C2211yescharacteristic
45#4529C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#533D311yescharacteristic, complemented
49C2×#497—31no—
50C2×#489—31no—
51#2850—31no—
52#2001—32no—
53#2000—32no—
54#1993—31no—
55#1960—31no—
56#1880—32no—
57#1878—32no—
58#1749—31no—
59C22×C3×Q8—21no—
60C22×C3×Q8C2312yes—
61C23×C3×C4D411yescharacteristic
62C23×C3×C4D411yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C22×#67—21no—
65C22×D12C2311yescharacteristic
66C22×Dic6C2311yescharacteristic
67C3×C4⋊Q8—22no—
68C3×C4⋊Q8—22no—
69C3×C4⋊Q8D414yescomplemented
70C3×C4⋊Q8D414yescomplemented
71C3×#125—28no—
72C3×C22⋊Q8—28no—
73C3×C22⋊Q8—24no—
74C3×C22⋊Q8—24no—
75C3×C4×Q8—28no—
76C3×C4×Q8—24no—
77C3×C4×Q8—24no—
78C3×#117—28no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C22⋊C4—22no—
86C2×C3×C42—22no—
87C2×C3×C42—22no—
88C2×C3×C42—22no—
89C2×C3×C42D411yescharacteristic
90C2×C3×C42D411yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C2×#217—21no—
94C2×#216—24no—
95C2×#216—24no—
96C2×#216—42no—
97C2×#215—24no—
98C2×#215—24no—
99C2×#215—42no—
100#930—42no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108#924—42no—
109C2×Dic3⋊C4—21no—
110C22×C3⋊C8C2312yes—
111C22×C3⋊C8C2312yes—
112#875—22no—
113#875—22no—
114#835—42no—
115#835—42no—
116C3⋊#95—21no—
117C3⋊#95—21no—
118C3⋊#95—21no—
119C3⋊#95—21no—
120#830—24no—
121#810—24no—
122#809—24no—
123#803—22no—
124#803—22no—
125C22×Q16—32no—
126C22×SD16—32no—
127#533—24no—
128#533—24no—
129C2×C4⋊Q8C2×D311yescharacteristic
130C2×C4⋊Q8C2×D311yescharacteristic
131C2×#125—21no—
132C2×#124—31no—
133C2×C22⋊Q8—21no—
134C2×C22⋊Q8—31no—
135C2×C22⋊Q8C2×D312yes—
136C2×#121—31no—
137C2×C4×Q8—21no—
138C2×C4×Q8C2×D312yes—
139C2×#117—21no—
140C22×C42C2×D311yescharacteristic
141#497—64no—
142#489—64no—
143C2×#103—32no—
144C2×#103—32no—
145C2×#103—32no—
146C2×#102—32no—
147C2×C22⋊C8—32no—
148C2×#97—31no—
149C2×#97—31no—
150#391—31no—
151#390—31no—
152#388—61no—
153#388—61no—
154#386—61no—
155#386—61no—
156#382—31no—
157#340—32no—
158#340—32no—
159#336—32no—
160C24×C3C2×D411yescharacteristic
161C23×D3—21no—
162C2×C3×Q8—24no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—22no—
168C2×C3×Q8—42no—
169C2×C3×Q8—42no—
170C2×C3×Q8C2×D414yes—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—41no—
182C22×C3×C4—41no—
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2411yescharacteristic
189C2×#67—42no—
190C2×#67—42no—
191C2×#67—42no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C2×D12—21no—
196C2×D12—21no—
197C2×D12—22no—
198C2×D12—22no—
199C2×Dic6—21no—
200C2×Dic6—21no—
201C2×Dic6—22no—
202C2×Dic6—22no—
203C3×C4⋊C4—28no—
204C3×C4⋊C4—28no—
205C3×C4⋊C4—24no—
206C3×C4⋊C4—24no—
207C3×C4⋊C4—24no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4C2×D414yes—
210C3×C22⋊C4—28no—
211C3×C42—28no—
212C3×C42—22no—
213C3×C42—22no—
214C3×C42—21no—
215C3×C42—21no—
216C3×C42C2×D412yes—
217C3×C42C2×D412yes—
218#217—42no—
219#216—48no—
220#215—48no—
221(C2×D3)⋊C4—42no—
222(C2×D3)⋊C4—42no—
223(C2×D3)⋊C4—42no—
224(C2×D3)⋊C4—42no—
225(C2×D3)⋊C4—42no—
226C4⋊Dic3—22no—
227C4⋊Dic3—22no—
228Dic3⋊C4—42no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8—22no—
232C2×C3⋊C8—22no—
233C2×C3⋊C8—22no—
234C2×C3⋊C8#4114yes—
235C22×Q8—21no—
236C22×Q8—31no—
237C22×Q8C22×D312yes—
238C22×D4—31no—
239C22×D4—61no—
240C23×C4#6711yescharacteristic
241C23×C4#6711yescharacteristic
242C23×C4C22×D311yescharacteristic
243C2×Q16—64no—
244C2×Q16—64no—
245C2×Q16—122no—
246C2×SD16—64no—
247C2×SD16—64no—
248C2×SD16—122no—
249C22×C8—32no—
250C22×C8—32no—
251C4⋊Q8—22no—
252C4⋊Q8—22no—
253C4⋊Q8#6714yescomplemented
254C4⋊Q8#6714yescomplemented
255#125—28no—
256#124—62no—
257#124—62no—
258C22⋊Q8—28no—
259C22⋊Q8—24no—
260C22⋊Q8—24no—
261C22⋊Q8—122no—
262#121—122no—
263C4×Q8—28no—
264C4×Q8—24no—
265C4×Q8—24no—
266#117—28no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4C22×D312yes—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—61no—
279C2×C22⋊C4—61no—
280C2×C22⋊C4—61no—
281C2×C22⋊C4—61no—
282C2×C22⋊C4—61no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42#6711yescharacteristic
287C2×C42#6711yescharacteristic
288C2×C42C22×D311yescharacteristic
289C2×C42C22×D311yescharacteristic
290#103—64no—
291#103—122no—
292#103—122no—
293#102—64no—
294C22⋊C8—64no—
295#97—62no—
296#97—62no—
297#95—61no—
298#95—61no—
299#95—61no—
300#95—61no—
301C23×C3—21no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3#13712yes—
308C23×C3C22×D411yescharacteristic
309C22×D3—41no—
310C22×D3—41no—
311C22×D3—41no—
312C22×D3—41no—
313C22×D3—41no—
314C22×D3—41no—
315C22×D3—41no—
316C3×Q8—24no—
317C3×Q8—24no—
318C3×Q8—48no—
319C3×Q8—42no—
320C3×Q8—42no—
321C2×C3×C4—28no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—24no—
331C2×C3×C4—24no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—44no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4#12012yes—
344C2×C3×C4#12012yes—
345C2×C3×C4#13712yes—
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4C22×D411yescharacteristic
348C2×C3×C4C2×#4112yes—
349C2×C3×C4H(5,𝔽2)12yes—
350#67—84no—
351C2×Dic3—41no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354C2×Dic3—41no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360D12—42no—
361D12—42no—
362Dic6—42no—
363Dic6—42no—
364C3⋊C8—24no—
365C3⋊C8—42no—
366C24—61no—
367C24C2×#6711yescharacteristic
368C2×Q8—24no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—42no—
375C2×Q8—42no—
376C2×Q8—62no—
377C2×Q8—62no—
378C2×Q8—61no—
379C2×Q8—61no—
380C2×Q8C2×#6714yes—
381C2×D4—62no—
382C2×D4—62no—
383C2×D4—61no—
384C2×D4—61no—
385C2×D4—122no—
386C2×D4—122no—
387C2×D4—122no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4C2×#6711yescharacteristic
404C22×C4C2×#6711yescharacteristic
405C22×C4C2×#6711yescharacteristic
406C22×C4C2×#6711yescharacteristic
407C22×C4C2×#6711yescharacteristic
408C22×C4C23×D311yescharacteristic
409Q16—128no—
410SD16—128no—
411C2×C8—34no—
412C2×C8—62no—
413C2×C8—62no—
414C2×C8—62no—
415C2×C8—62no—
416C2×C8—62no—
417C4⋊C4—28no—
418C4⋊C4—28no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—62no—
424C4⋊C4—62no—
425C4⋊C4—122no—
426C4⋊C4C2×#6714yes—
427C22⋊C4—28no—
428C22⋊C4—122no—
429C22⋊C4—122no—
430C22⋊C4—122no—
431C22⋊C4—122no—
432C22⋊C4—122no—
433C22⋊C4—122no—
434C42—28no—
435C42—22no—
436C42—22no—
437C42—21no—
438C42—21no—
439C42C2×#6712yes—
440C42C2×#6712yes—
441C22×C3—22no—
442C22×C3—22no—
443C22×C3—22no—
444C22×C3—41no—
445C22×C3—41no—
446C22×C3—41no—
447C22×C3—41no—
448C22×C3C2×#12011yescharacteristic
449C22×C3#53211yescharacteristic
450C22×C3#53511yescharacteristic
451C22×C3C2×#13711yescharacteristic
452C22×C3C2×#13711yescharacteristic
453C22×C3#57511yescharacteristic
454C22×C3#57511yescharacteristic
455C2×D3—81no—
456C2×D3—81no—
457C2×D3—81no—
458C2×D3—81no—
459C2×D3—81no—
460C2×D3—81no—
461C2×D3—81no—
462C3×C4—22no—
463C3×C4—22no—
464C3×C4—22no—
465C3×C4—41no—
466C3×C4—44no—
467C3×C4—41no—
468C3×C4—44no—
469C3×C4#54612yes—
470C3×C4#55012yes—
471Dic3—81no—
472Dic3—81no—
473Dic3—81no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—121no—
481C23—121no—
482C23—121no—
483C23—121no—
484C23—121no—
485C23—121no—
486C23—121no—
487C23#94212yes—
488C23C22×#6711yescharacteristic
489Q8—24no—
490Q8—24no—
491Q8—48no—
492Q8—42no—
493Q8—42no—
494Q8—122no—
495Q8—122no—
496D4—122no—
497D4—122no—
498D4—244no—
499C2×C4—28no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—121no—
530C2×C4#94212yes—
531C2×C4#95312yes—
532C2×C4#95312yes—
533C2×C4D3×#4112yes—
534C2×C4#100912yes—
535C2×C4C22×#6711yescharacteristic
536C2×C4C22×#6711yescharacteristic
537C8—64no—
538C8—122no—
539C2×C3—42no—
540C2×C3#243111yescharacteristic
541C2×C3#300411yescharacteristic
542C2×C3#300511yescharacteristic
543C2×C3#317211yescharacteristic
544C2×C3#318111yescharacteristic
545C2×C3#321511yescharacteristic
546C2×C3#321611yescharacteristic
547D3—161no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—241no—
556C22—241no—
557C22—241no—
558C22—241no—
559C22—241no—
560C22—241no—
561C22—241no—
562C22C2×#94211yescharacteristic
563C22C2×#94211yescharacteristic
564C22#582211yescharacteristic
565C22#582211yescharacteristic
566C22#583011yescharacteristic
567C22#583211yescharacteristic
568C22C2×#95311yescharacteristic
569C4—22no—
570C4—22no—
571C4—22no—
572C4—41no—
573C4—44no—
574C4—41no—
575C4—44no—
576C4—241no—
577C4—241no—
578C4—241no—
579C4#582912yes—
580C4#583412yes—
581C3#2560311yescharacteristic, complemented, Sylow
582C2—42no—
583C2—481no—
584C2#8334511yescharacteristic
585C2#8334611yescharacteristic
586C2#8335711yescharacteristic
587C2#8335811yescharacteristic
588C2#8405911yescharacteristic
589C2#8408111yescharacteristic
590C2#8409211yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#533C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenterC2×#533D3
hypocenterC3#25603
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
1. C2×C3×#533C2semidirect
2. C2×#5086C2semidirect
3. C2×#5089C2semidirect
5. #71788C2semidirect
6. #71786C2semidirect
7. #71700C2semidirect
9. #71633C2semidirect
10. #71632C2semidirect
11. #71529C2semidirect
48. C2×#533D3semidirect
69. C3×C4⋊Q8D4semidirect
70. C3×C4⋊Q8D4semidirect
253. C4⋊Q8#67semidirect
254. C4⋊Q8#67semidirect
581. C3#25603semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5332
C2×#50862
C2×#50892
#820772
#717882
#717862
#717002
#716992
#716332
#716322
#715292
#256033