Finite Group Databasebeta

← #10944206

Subgroups — #10944206

Order 768.

1352 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×2−1+4C211yesmaximal, characteristic, complemented
2C2×#5246C211yesmaximal, characteristic, complemented
3C2×#5245C211yesmaximal, characteristic
4#82527C212yesmaximal, complemented
5#82474C211yesmaximal, characteristic, complemented
6#71667C212yesmaximal, complemented
7#71666C212yesmaximal, complemented
8#71661C211yesmaximal, characteristic, complemented
9#71660C211yesmaximal, characteristic, complemented
10#71646C212yesmaximal, complemented
11#71637C211yesmaximal, characteristic, complemented
12#32925—31nomaximal, Sylow
13C2×C3×2−1+4C2211yescharacteristic
14C3×#520—22no—
15C3×#520—22no—
16C3×#520C2212yes—
17C3×#519—22no—
18C3×#519C2211yescharacteristic
19C3×C4×#41—22no—
20C3×C4×#41—22no—
21C3×C4×#41C2212yes—
22C2×C3×C4×Q8—22no—
23C2×C3×C4×Q8C2211yescharacteristic
24C2×#951C2212yes—
25C2×#949C2212yes—
26C2×#947C2211yescharacteristic
27C2×#944C2211yescharacteristic
28C2×#875C2211yescharacteristic
29#5246—24no—
30#5245—24no—
31C2×#837C2212yes—
32C2×#837C2212yes—
33C2×#836C2211yescharacteristic
34C2×#836C2211yescharacteristic
35#5028—22no—
36#5028—22no—
37#5016—22no—
38#5016—22no—
39#5004C2212yes—
40#5001C2212yes—
41#4994—22no—
42#4988C2211yescharacteristic
43#4973C2211yescharacteristic
44#4940—22no—
45C4×C4⋊Dic3C2211yescharacteristic
46#4924C2212yes—
47#4532C2212yes—
48#4532C2212yes—
49#4530C2212yes—
50C4×2−1+4D311yescharacteristic, complemented
51#3040—32no—
52C2×#456—31no—
53C2×#454—31no—
54#2545—31no—
55#1888—31no—
56#1886—31no—
57#1883—32no—
58#1882—32no—
59#1806—32no—
60#1801—31no—
61C3×2−1+4—24no—
62C3×2−1+4—22no—
63C3×2−1+4D414yes—
64C2×C3×#41—22no—
65C2×C3×#41—22no—
66C2×C3×#41C2312yes—
67C22×C3×Q8—22no—
68C22×C3×Q8C2311yescharacteristic
69C22×D12C2311yescharacteristic
70C22×Dic6C2311yescharacteristic
71C3×C4×Q8—22no—
72C3×C4×Q8—24no—
73C3×C4×Q8—22no—
74C3×C4×Q8—22no—
75C3×C4×Q8—22no—
76C3×C4×Q8—22no—
77C3×C4×Q8—22no—
78C3×C4×Q8—22no—
79C3×C4×Q8D412yes—
80C3×C4×Q8D412yes—
81C3×C4×D4—22no—
82C3×C4×D4—24no—
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C3×C4×D4—24no—
87C3×C4×D4—22no—
88C3×C4×D4D412yes—
89C3×C4×D4D412yes—
90C3×#117—24no—
91C3×#117—24no—
92C3×#117—22no—
93C3×#117—22no—
94C3×#117—22no—
95C3×#117C2312yes—
96C2×C3×C4⋊C4—24no—
97C2×C3×C4⋊C4—22no—
98C2×C3×C4⋊C4D412yes—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C42—24no—
101C2×C3×C42—22no—
102C2×C3×C42D412yes—
103C2×C3×C42C2311yescharacteristic
104#951—24no—
105#951—24no—
106#949—24no—
107#949—24no—
108#947—22no—
109#947—22no—
110#944—22no—
111#944—22no—
112C2×#216—22no—
113C2×#215—22no—
114C2×#214—22no—
115C2×#213—22no—
116C2×(C2×D3)⋊C4—21no—
117C2×(C2×D3)⋊C4—22no—
118C2×(C2×D3)⋊C4—22no—
119C2×(C2×D3)⋊C4—21no—
120C2×C4⋊Dic3—21no—
121C2×Dic3⋊C4—22no—
122C2×C4×Dic3C2311yescharacteristic
123C2×C4×Dic3C2311yescharacteristic
124C2×#208C2312yes—
125C2×#208C2312yes—
126#875—22no—
127#875—22no—
128#837—24no—
129#837—24no—
130#836—22no—
131#836—22no—
132C3⋊#95—21no—
133C3⋊#95—22no—
134#810—22no—
135#810—22no—
136#809—22no—
137#809—22no—
138C4⋊(C3⋊C8)—22no—
139C4⋊(C3⋊C8)—22no—
140C2×2−1+4C2×D311yescharacteristic
141C2×#137—32no—
142C2×AGL(1,ℤ/8)—32no—
143C2×C4⋊Q8—31no—
144C2×#124—31no—
145C2×#124—31no—
146#520—22no—
147#520—22no—
148#520C2×D312yes—
149#519—22no—
150#519C2×D311yescharacteristic
151C4×#41—22no—
152C4×#41—22no—
153C4×#41C2×D312yes—
154C2×C4×Q8—22no—
155C2×C4×Q8C2×D311yescharacteristic
156#464—62no—
157#464—62no—
158#463—62no—
159#463—62no—
160#456—64no—
161#454—64no—
162#423—32no—
163C2×#104—32no—
164C2×#104—32no—
165#419—32no—
166#418—32no—
167C2×#101—31no—
168C2×#101—31no—
169#396—62no—
170#391—31no—
171#390—31no—
172#388—62no—
173C4×C4⋊C4—31no—
174#343—32no—
175#339—32no—
176#339—32no—
177C23×D3—21no—
178C3×#41—28no—
179C3×#41—28no—
180C3×#41—28no—
181C3×#41—28no—
182C3×#41—22no—
183C3×#41—22no—
184C3×#41C2×D414yes—
185C3×#41#4114yes—
186C2×C3×Q8—22no—
187C2×C3×Q8—28no—
188C2×C3×Q8—28no—
189C2×C3×Q8—22no—
190C2×C3×Q8—22no—
191C2×C3×Q8C2×D412yes—
192C2×C3×Q8C2×D412yes—
193C2×C3×Q8#4112yes—
194C2×C3×D4—22no—
195C2×C3×D4—22no—
196C2×C3×D4C2×D412yes—
197C22×C3×C4—24no—
198C22×C3×C4—22no—
199C22×C3×C4—24no—
200C22×C3×C4—22no—
201C22×C3×C4—22no—
202C22×C3×C4C2×D412yes—
203C22×C3×C4C2×D412yes—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C2×D12—22no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—22no—
214C2×Dic6—22no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6—22no—
218C3×C4⋊C4—24no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—24no—
223C3×C4⋊C4—22no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4C2×D412yes—
226C3×C4⋊C4C2×D412yes—
227C3×C22⋊C4—24no—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—22no—
230C3×C22⋊C4—22no—
231C3×C42—24no—
232C3×C42—24no—
233C3×C42—22no—
234C3×C42—22no—
235C3×C42—22no—
236C3×C42—22no—
237C3×C42—22no—
238C3×C42C2×D412yes—
239C3×C42C2×D412yes—
240#216—44no—
241#215—44no—
242#214—44no—
243#213—44no—
244(C2×D3)⋊C4—44no—
245(C2×D3)⋊C4—41no—
246(C2×D3)⋊C4—41no—
247(C2×D3)⋊C4—42no—
248(C2×D3)⋊C4—44no—
249C4⋊Dic3—42no—
250Dic3⋊C4—44no—
251C4×Dic3—21no—
252C4×Dic3—21no—
253C4×Dic3—21no—
254C4×Dic3—21no—
255#208—22no—
256#208—22no—
257#208—22no—
258#208—22no—
259C2×C3⋊C8—22no—
260C2×C3⋊C8—22no—
2612−1+4—24no—
2622−1+4—22no—
2632−1+4D1214yes—
264C2×#41—22no—
265C2×#41—22no—
266C2×#41C22×D312yes—
267C22×Q8—22no—
268C22×Q8—31no—
269C22×Q8C22×D311yescharacteristic
270C22×D4—31no—
271#137—64no—
272#137—64no—
273AGL(1,ℤ/8)—64no—
274AGL(1,ℤ/8)—64no—
275C2×Q16—62no—
276C2×SD16—62no—
277C2×SD16—62no—
278C2×D8—62no—
279C2×M16—32no—
280C2×M16—32no—
281C4⋊Q8—62no—
282C4⋊Q8—62no—
283#124—62no—
284#124—62no—
285#124—62no—
286#124—62no—
287C4×Q8—22no—
288C4×Q8—24no—
289C4×Q8—22no—
290C4×Q8—22no—
291C4×Q8—22no—
292C4×Q8—22no—
293C4×Q8—22no—
294C4×Q8—22no—
295C4×Q8#6712yes—
296C4×Q8#6712yes—
297C4×D4—22no—
298C4×D4—24no—
299C4×D4—22no—
300C4×D4—22no—
301C4×D4—22no—
302C4×D4—24no—
303C4×D4—22no—
304C4×D4#6712yes—
305C4×D4#6712yes—
306#117—24no—
307#117—24no—
308#117—22no—
309#117—22no—
310#117—22no—
311#117C22×D312yes—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—62no—
315C2×C4⋊C4—61no—
316C2×C4⋊C4#6712yes—
317C2×C4⋊C4C22×D311yescharacteristic
318C2×C22⋊C4—61no—
319C2×C22⋊C4—62no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—62no—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42—31no—
325C2×C42—31no—
326C2×C42#6712yes—
327C2×C42C22×D311yescharacteristic
328C4⋊C8—62no—
329C4⋊C8—62no—
330#104—64no—
331#104—64no—
332#103—62no—
333#103—62no—
334#102—62no—
335#102—62no—
336#101—62no—
337#101—62no—
338#95—61no—
339#95—62no—
340C23×C3—22no—
341C23×C3C22×D411yescharacteristic
342C22×D3—42no—
343C22×D3—42no—
344C22×D3—41no—
345C22×D3—41no—
346C22×D3—41no—
347C3×Q8—28no—
348C3×Q8—24no—
349C3×Q8—22no—
350C3×Q8—24no—
351C3×Q8#12114yes—
352C3×D4—28no—
353C3×D4—24no—
354C3×D4—24no—
355C3×D4—22no—
356C3×D4#12114yes—
357C2×C3×C4—216no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—24no—
362C2×C3×C4—24no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—44no—
369C2×C3×C4—42no—
370C2×C3×C4—82no—
371C2×C3×C4—84no—
372C2×C3×C4#12011yescharacteristic
373C2×C3×C4#12011yescharacteristic
374C2×C3×C4#12012yes—
375C2×C3×C4#12114yes—
376C2×C3×C4C22×D411yescharacteristic
377C2×C3×C4C22×D412yes—
378C2×C3×C4C2×#4112yes—
379C2×C3×C4C2×#4111yescharacteristic
380C2×Dic3—22no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389D12—42no—
390D12—42no—
391Dic6—42no—
392Dic6—42no—
393C3⋊C8—42no—
394C3⋊C8—42no—
395C24—61no—
396#41—28no—
397#41—28no—
398#41—28no—
399#41—28no—
400#41—22no—
401#41—22no—
402#41C2×D1214yes—
403#41#23514yes—
404C2×Q8—22no—
405C2×Q8—28no—
406C2×Q8—28no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—61no—
410C2×Q8—61no—
411C2×Q8—62no—
412C2×Q8—62no—
413C2×Q8C2×D1212yes—
414C2×Q8#23512yes—
415C2×Q8C2×#6712yes—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—61no—
419C2×D4—61no—
420C2×D4—62no—
421C2×D4—62no—
422C2×D4C2×#6712yes—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4C2×#6712yes—
432C22×C4C2×#6712yes—
433C22×C4C2×#6711yescharacteristic
434C22×C4C2×#6711yescharacteristic
435C22×C4C23×D311yescharacteristic
436Q16—124no—
437SD16—124no—
438SD16—124no—
439D8—124no—
440M16—62no—
441M16—62no—
442M16—62no—
443M16—62no—
444C2×C8—62no—
445C2×C8—62no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—122no—
454C4⋊C4—124no—
455C4⋊C4C2×#6712yes—
456C4⋊C4C2×#6712yes—
457C22⋊C4—24no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—124no—
462C22⋊C4—124no—
463C22⋊C4—121no—
464C22⋊C4—121no—
465C22⋊C4—122no—
466C42—24no—
467C42—24no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—61no—
474C42—61no—
475C42—61no—
476C42—61no—
477C42C2×#6712yes—
478C42C2×#6712yes—
479C22×C3—21no—
480C22×C3—21no—
481C22×C3—24no—
482C22×C3—42no—
483C22×C3—42no—
484C22×C3C2×#12011yescharacteristic
485C22×C3C2×#12111yescharacteristic
486C22×C3#52511yescharacteristic
487C2×D3—82no—
488C2×D3—82no—
489C2×D3—81no—
490C2×D3—81no—
491C2×D3—81no—
492C3×C4—24no—
493C3×C4—24no—
494C3×C4—22no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—42no—
498C3×C4—42no—
499C3×C4—42no—
500C3×C4C2×#12112yes—
501C3×C4#52512yes—
502Dic3—42no—
503Dic3—81no—
504Dic3—81no—
505C23—22no—
506C23—122no—
507C23—122no—
508C23—121no—
509C23—121no—
510C23—121no—
511C23C22×#6711yescharacteristic
512Q8—28no—
513Q8—24no—
514Q8—24no—
515Q8—22no—
516Q8—122no—
517Q8—122no—
518Q8#93014yes—
519D4—28no—
520D4—24no—
521D4—24no—
522D4—22no—
523D4—122no—
524D4—122no—
525D4#93014yes—
526C2×C4—216no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—62no—
540C2×C4—82no—
541C2×C4—84no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4#93014yes—
551C2×C4#95311yescharacteristic
552C2×C4#95311yescharacteristic
553C2×C4#95312yes—
554C2×C4C22×D1211yescharacteristic
555C2×C4C2×#23512yes—
556C2×C4C2×#23511yescharacteristic
557C2×C4C22×#6712yes—
558C8—122no—
559C8—122no—
560C2×C3—21no—
561C2×C3—21no—
562C2×C3—44no—
563C2×C3#284711yescharacteristic
564C2×C3#301811yescharacteristic
565C2×C3#302011yescharacteristic
566D3—161no—
567C22—21no—
568C22—21no—
569C22—24no—
570C22—42no—
571C22—42no—
572C22—242no—
573C22—242no—
574C22—241no—
575C22—241no—
576C22—241no—
577C22C2×#93011yescharacteristic
578C22#579111yescharacteristic
579C22C2×#95311yescharacteristic
580C4—24no—
581C4—24no—
582C4—21no—
583C4—21no—
584C4—22no—
585C4—42no—
586C4—42no—
587C4—42no—
588C4—122no—
589C4—241no—
590C4—241no—
591C4C2×#93012yes—
592C4#579112yes—
593C3#3292511yescharacteristic, complemented, Sylow
594C2—21no—
595C2—21no—
596C2—44no—
597C2—481no—
598C2#8403211yescharacteristic
599C2#8411611yescharacteristic
600C2#8412111yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×2−1+4C2
socleC22×C3C2×#120
prime radicalC2×#5245C2
hypercenterC4×2−1+4D3
hypocenterC3#32925
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×2−1+4

Complements

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

NH ≅ G/Ntype
1. C3×C4×2−1+4C2semidirect
2. C2×#5246C2semidirect
4. #82527C2semidirect
5. #82474C2semidirect
6. #71667C2semidirect
7. #71666C2semidirect
8. #71661C2semidirect
9. #71660C2semidirect
10. #71646C2semidirect
11. #71637C2semidirect
50. C4×2−1+4D3semidirect
593. C3#32925semidirect

Maximal subgroups

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

subgroupindex
C3×C4×2−1+42
C2×#52462
C2×#52452
#825272
#824742
#716672
#716662
#716612
#716602
#716462
#716372
#329253