Finite Group Databasebeta

← #10822287

Subgroups — #10822287

Order 768.

2098 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#536C211yesmaximal, characteristic, complemented
2C22×#949C211yesmaximal, characteristic
3C23×#215C211yesmaximal, characteristic, complemented
4C2×#5233C211yesmaximal, characteristic, complemented
5C2×#5225C211yesmaximal, characteristic, complemented
6#82269C211yesmaximal, characteristic, complemented
7#71791C214yesmaximal, complemented
8#71728C214yesmaximal, complemented
9#71579C211yesmaximal, characteristic, complemented
10#23424—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C23×C3×Q8C2211yescharacteristic
13C23×D12C2211yescharacteristic
14C3×#536—24no—
15C3×#536—24no—
16C2×C3×C4⋊Q8C2212yescomplemented
17C2×C3×#124C2212yescomplemented
18C2×C3×#123—22no—
19C2×C3×C22⋊Q8—22no—
20C2×C3×#117C2211yescharacteristic
21C2×#949—24no—
22C2×#949—24no—
23C2×#949—24no—
24C22×#215—24no—
25C22×#215—24no—
26C22×#215—24no—
27C22×#215C2214yes—
28C22×#215C2212yes—
29C22×#213C2212yes—
30C2×#926C2211yescharacteristic
31C22×#208C2211yescharacteristic
32C23×C3⋊C8C2211yescharacteristic
33#5233—24no—
34#5225—24no—
35C2×#835C2214yescomplemented
36C2×#835C2212yescomplemented
37C2×#832C2212yescomplemented
38#4988—22no—
39#4955—22no—
40#4939C2214yes—
41#4524C2214yescomplemented
42C22×AGL(1,ℤ/8)—31no—
43C23×SD16—31no—
44C2×#536D311yescharacteristic, complemented
45C2×#419—31no—
46C2×#416—31no—
47#2309—31no—
48#2013—34no—
49#2012—34no—
50#1729—31no—
51C24×D3—21no—
52C2×C3×#41—22no—
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C2×C3×#41—22no—
56C2×C3×#41—22no—
57C2×C3×#41D414yes—
58C22×C3×Q8—21no—
59C22×C3×Q8—22no—
60C22×C3×Q8—22no—
61C22×C3×Q8—22no—
62C22×C3×Q8—22no—
63C22×C3×Q8—22no—
64C22×C3×Q8D414yes—
65C22×C3×Q8C2312yes—
66C22×C3×Q8C2311yescharacteristic
67C22×C3×D4—21no—
68C22×C3×D4C2311yescharacteristic
69C23×C3×C4—21no—
70C23×C3×C4—21no—
71C23×C3×C4C2311yescharacteristic
72C22×D12—22no—
73C22×D12—22no—
74C22×D12—22no—
75C22×D12—22no—
76C22×D12—21no—
77C22×D12—22no—
78C22×D12D414yes—
79C22×D12C2312yes—
80C3×C4⋊Q8—24no—
81C3×C4⋊Q8—24no—
82C3×#124—24no—
83C3×#124—24no—
84C3×#123—216no—
85C3×C22⋊Q8—216no—
86C3×#117—22no—
87C3×#117—22no—
88C2×C3×C4⋊C4—24no—
89C2×C3×C4⋊C4D412yes—
90C2×C3×C22⋊C4—24no—
91C2×C3×C22⋊C4D412yes—
92C2×C3×C42C2312yes—
93C2×#217—21no—
94#949—232no—
95C2×#215—232no—
96C2×#215—28no—
97C2×#215—24no—
98C2×#215—28no—
99C2×#215—24no—
100C2×#215—28no—
101C2×#215—24no—
102C2×#213—24no—
103C2×#213—24no—
104C2×#213—24no—
105C2×(C2×D3)⋊C4—28no—
106#926—22no—
107#926—22no—
108C2×C4⋊Dic3C2312yes—
109C2×C4×Dic3—21no—
110C2×#208—22no—
111C2×#208—22no—
112C2×#208D414yes—
113C22×C3⋊C8—22no—
114C22×C3⋊C8—22no—
115C22×C3⋊C8D414yes—
116C22×C3⋊C8C2312yes—
117C22×C3⋊C8C2312yes—
118#835—28no—
119#835—24no—
120#832—24no—
121C22×#41C2×D311yescharacteristic
122C23×Q8C2×D311yescharacteristic
123C23×D4—31no—
124C2×AGL(1,ℤ/8)—64no—
125C2×AGL(1,ℤ/8)—64no—
126C2×AGL(1,ℤ/8)—64no—
127C22×SD16—34no—
128C22×SD16—32no—
129C22×SD16—64no—
130C22×SD16—64no—
131C22×SD16—64no—
132C22×D8—32no—
133C22×M16—31no—
134C23×C8—31no—
135#536—24no—
136#536—24no—
137C2×C4⋊Q8C2×D312yescomplemented
138C2×#124C2×D312yescomplemented
139C2×#123—22no—
140C2×C22⋊Q8—22no—
141C2×#117—31no—
142C2×#117C2×D311yescharacteristic
143#419—64no—
144#416—64no—
145C2×#103—32no—
146C2×#103—34no—
147C2×#102—32no—
148#390—34no—
149#390—62no—
150#379—62no—
151#336—34no—
152C24×C3—21no—
153C24×C3C2×D411yescharacteristic
154C23×D3—22no—
155C23×D3—24no—
156C23×D3—42no—
157C23×D3—42no—
158C23×D3—84no—
159C3×#41—28no—
160C3×#41—28no—
161C3×#41—44no—
162C3×#41—44no—
163C2×C3×Q8—24no—
164C2×C3×Q8—22no—
165C2×C3×Q8—21no—
166C2×C3×Q8—28no—
167C2×C3×Q8—28no—
168C2×C3×Q8—22no—
169C2×C3×Q8—21no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8—22no—
175C2×C3×Q8—21no—
176C2×C3×Q8—44no—
177C2×C3×Q8—42no—
178C2×C3×Q8—44no—
179C2×C3×Q8—42no—
180C2×C3×Q8C2×D414yes—
181C2×C3×Q8C2×D412yes—
182C2×C3×D4—24no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—42no—
189C2×C3×D4—42no—
190C2×C3×D4C2×D412yes—
191C22×C3×C4—24no—
192C22×C3×C4—22no—
193C22×C3×C4—21no—
194C22×C3×C4—24no—
195C22×C3×C4—42no—
196C22×C3×C4—42no—
197C22×C3×C4—42no—
198C22×C3×C4—42no—
199C22×C3×C4C2×D412yes—
200C22×C3×C4C2×D412yes—
201C22×C3×C4C2×D412yes—
202C22×C3×C4C2×D414yes—
203C22×C3×C4C2×D412yes—
204C22×C3×C4C2×D412yes—
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4#4112yes—
207C22×C3×C4C2411yescharacteristic
208C22×Dic3—22no—
209C2×D12—28no—
210C2×D12—28no—
211C2×D12—22no—
212C2×D12—22no—
213C2×D12—22no—
214C2×D12—22no—
215C2×D12—24no—
216C2×D12—22no—
217C2×D12—42no—
218C2×D12—44no—
219C2×D12—44no—
220C2×D12—42no—
221C2×D12C2×D414yes—
222C3×C4⋊C4—216no—
223C3×C4⋊C4—24no—
224C3×C22⋊C4—216no—
225C3×C22⋊C4—24no—
226C3×C42—22no—
227C3×C42—22no—
228#217—42no—
229#215—232no—
230#215—216no—
231#213—216no—
232(C2×D3)⋊C4—88no—
233C4⋊Dic3—22no—
234C4⋊Dic3—22no—
235C4×Dic3—42no—
236#208—28no—
237C2×C3⋊C8—28no—
238C2×C3⋊C8—22no—
239C2×C3⋊C8—22no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8C2×D414yes—
243C2×C3⋊C8C2×D414yes—
244C25—61no—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—22no—
250C2×#41#6714yes—
251C22×Q8—21no—
252C22×Q8—22no—
253C22×Q8—22no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8#6714yes—
258C22×Q8C22×D312yes—
259C22×Q8C22×D311yescharacteristic
260C22×D4—21no—
261C22×D4—34no—
262C22×D4—32no—
263C22×D4—62no—
264C22×D4—62no—
265C22×D4—62no—
266C22×D4—62no—
267C22×D4—62no—
268C22×D4—61no—
269C22×D4C22×D311yescharacteristic
270C23×C4—21no—
271C23×C4—21no—
272C23×C4C22×D311yescharacteristic
273AGL(1,ℤ/8)—632no—
274C2×SD16—68no—
275C2×SD16—64no—
276C2×SD16—68no—
277C2×SD16—64no—
278C2×SD16—632no—
279C2×SD16—68no—
280C2×SD16—64no—
281C2×D8—64no—
282C2×D8—64no—
283C2×D8—64no—
284C2×M16—34no—
285C2×M16—62no—
286C2×M16—62no—
287C22×C8—34no—
288C22×C8—32no—
289C22×C8—32no—
290C22×C8—62no—
291C22×C8—62no—
292C4⋊Q8—24no—
293C4⋊Q8—24no—
294#124—24no—
295#124—24no—
296#123—216no—
297C22⋊Q8—216no—
298#117—22no—
299#117—22no—
300#117—62no—
301#117—62no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—32no—
304C2×C4⋊C4#6712yes—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—68no—
308C2×C22⋊C4#6712yes—
309C2×C42—61no—
310C2×C42C22×D312yes—
311#103—64no—
312#103—68no—
313#102—64no—
314C23×C3—24no—
315C23×C3—22no—
316C23×C3—41no—
317C23×C3—41no—
318C23×C3—44no—
319C23×C3#12012yes—
320C23×C3#12112yes—
321C23×C3#12712yes—
322C23×C3C22×D411yescharacteristic
323C22×D3—24no—
324C22×D3—42no—
325C22×D3—42no—
326C22×D3—48no—
327C22×D3—48no—
328C22×D3—88no—
329C22×D3—88no—
330C22×D3—88no—
331C3×Q8—24no—
332C3×Q8—22no—
333C3×Q8—24no—
334C3×Q8—22no—
335C3×Q8—42no—
336C3×Q8—41no—
337C3×Q8—42no—
338C3×Q8—41no—
339C3×Q8—48no—
340C3×D4—22no—
341C3×D4—22no—
342C3×D4—48no—
343C3×D4—41no—
344C3×D4—41no—
345C2×C3×C4—24no—
346C2×C3×C4—24no—
347C2×C3×C4—22no—
348C2×C3×C4—28no—
349C2×C3×C4—41no—
350C2×C3×C4—42no—
351C2×C3×C4—41no—
352C2×C3×C4—42no—
353C2×C3×C4—48no—
354C2×C3×C4—48no—
355C2×C3×C4#12014yes—
356C2×C3×C4#12014yes—
357C2×C3×C4#12012yes—
358C2×C3×C4#12112yes—
359C2×C3×C4#12114yes—
360C2×C3×C4#12114yes—
361C2×C3×C4#12712yes—
362C2×C3×C4C22×D411yescharacteristic
363C2×C3×C4C22×D411yescharacteristic
364C2×C3×C4C22×D411yescharacteristic
365C2×C3×C4C22×D411yescharacteristic
366C2×C3×C4C22×D411yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×Dic3—42no—
369C2×Dic3—44no—
370D12—24no—
371D12—24no—
372D12—42no—
373D12—42no—
374D12—48no—
375C3⋊C8—24no—
376C3⋊C8—24no—
377C24—21no—
378C24—62no—
379C24—64no—
380C24—122no—
381C24—122no—
382C24—244no—
383C24C2×#6711yescharacteristic
384#41—28no—
385#41—28no—
386#41—44no—
387#41—44no—
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—21no—
391C2×Q8—28no—
392C2×Q8—28no—
393C2×Q8—22no—
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—21no—
401C2×Q8—44no—
402C2×Q8—42no—
403C2×Q8—44no—
404C2×Q8—42no—
405C2×Q8C2×#6714yes—
406C2×Q8C2×#6712yes—
407C2×D4—24no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—34no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—68no—
417C2×D4—68no—
418C2×D4—64no—
419C2×D4—62no—
420C2×D4—62no—
421C2×D4—62no—
422C2×D4—62no—
423C2×D4—62no—
424C2×D4—124no—
425C2×D4—124no—
426C2×D4—122no—
427C2×D4—122no—
428C2×D4C2×#6712yes—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—62no—
438C22×C4D3×D412yes—
439C22×C4D3×D412yes—
440C22×C4D3×D412yes—
441C22×C4#23712yes—
442C22×C4C2×#6714yes—
443C22×C4C2×#6712yes—
444C22×C4C2×#6712yes—
445C22×C4C2×#6711yescharacteristic
446C22×C4C23×D311yescharacteristic
447SD16—632no—
448SD16—616no—
449D8—616no—
450M16—68no—
451C2×C8—34no—
452C2×C8—34no—
453C2×C8—68no—
454C2×C8—62no—
455C2×C8—62no—
456C2×C8—62no—
457C2×C8—62no—
458C4⋊C4—216no—
459C4⋊C4—24no—
460C4⋊C4—62no—
461C4⋊C4—62no—
462C22⋊C4—216no—
463C22⋊C4—24no—
464C22⋊C4—122no—
465C22⋊C4—248no—
466C42—22no—
467C42—22no—
468C42—122no—
469C22×C3—24no—
470C22×C3—24no—
471C22×C3—24no—
472C22×C3—42no—
473C22×C3—44no—
474C22×C3—82no—
475C22×C3—82no—
476C22×C3#39214yes—
477C22×C3C2×#12011yescharacteristic
478C22×C3C2×#12111yescharacteristic
479C22×C3C2×#12711yescharacteristic
480C22×C3#57714yes—
481C2×D3—44no—
482C2×D3—48no—
483C2×D3—816no—
484C2×D3—84no—
485C2×D3—84no—
486C3×C4—42no—
487C3×C4—44no—
488C3×C4—44no—
489C3×C4#39214yes—
490C3×C4C2×#12011yescharacteristic
491C3×C4C2×#12011yescharacteristic
492C3×C4C2×#12111yescharacteristic
493C3×C4C2×#12111yescharacteristic
494Dic3—82no—
495C23—24no—
496C23—22no—
497C23—41no—
498C23—41no—
499C23—44no—
500C23—64no—
501C23—128no—
502C23—122no—
503C23—128no—
504C23—122no—
505C23—248no—
506C23—248no—
507C23—248no—
508C23#93812yes—
509C23#94012yes—
510C23#95312yes—
511C23C22×#6711yescharacteristic
512Q8—24no—
513Q8—22no—
514Q8—24no—
515Q8—22no—
516Q8—48no—
517Q8—42no—
518Q8—41no—
519Q8—42no—
520Q8—41no—
521D4—22no—
522D4—22no—
523D4—48no—
524D4—41no—
525D4—41no—
526D4—64no—
527D4—64no—
528D4—128no—
529D4—122no—
530D4—122no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—48no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—48no—
541C2×C4—122no—
542C2×C4—124no—
543C2×C4#93714yes—
544C2×C4#93714yes—
545C2×C4#93812yes—
546C2×C4#93914yes—
547C2×C4#93914yes—
548C2×C4#94012yes—
549C2×C4#95312yes—
550C2×C4C2×D3×D411yescharacteristic
551C2×C4C2×D3×D411yescharacteristic
552C2×C4C2×D3×D411yescharacteristic
553C2×C4C2×#23711yescharacteristic
554C2×C4C22×#6711yescharacteristic
555C2×C4C22×#6711yescharacteristic
556C8—64no—
557C8—64no—
558C2×C3—24no—
559C2×C3—82no—
560C2×C3C2×#39211yescharacteristic
561C2×C3#301012yes—
562C2×C3#304212yes—
563C2×C3#315412yes—
564D3—84no—
565C22—24no—
566C22—24no—
567C22—24no—
568C22—42no—
569C22—44no—
570C22—82no—
571C22—82no—
572C22—124no—
573C22—128no—
574C22—2416no—
575C22—244no—
576C22—244no—
577C22#522314yes—
578C22C2×#93811yescharacteristic
579C22C2×#94011yescharacteristic
580C22#583714yes—
581C22C2×#95311yescharacteristic
582C4—42no—
583C4—44no—
584C4—44no—
585C4—242no—
586C4#522314yes—
587C4C2×#93711yescharacteristic
588C4C2×#93711yescharacteristic
589C4C2×#93911yescharacteristic
590C4C2×#93911yescharacteristic
591C3#2342411yescharacteristic, complemented, Sylow
592C2—24no—
593C2—82no—
594C2—244no—
595C2#8340212yes—
596C2#8340512yes—
597C2C2×#522311yescharacteristic
598C2#8411312yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#536C2
socleC23×C3C22×D4
prime radicalC22×#949C2
hypercenterC2×#536D3
hypocenterC3#23424
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
1. C2×C3×#536C2semidirect
3. C23×#215C2semidirect
4. C2×#5233C2semidirect
5. C2×#5225C2semidirect
6. #82269C2semidirect
7. #71791C2semidirect
8. #71728C2semidirect
9. #71579C2semidirect
16. C2×C3×C4⋊Q8C22semidirect
17. C2×C3×#124C22semidirect
35. C2×#835C22semidirect
36. C2×#835C22semidirect
37. C2×#832C22semidirect
41. #4524C22semidirect
44. C2×#536D3semidirect
137. C2×C4⋊Q8C2×D3semidirect
138. C2×#124C2×D3semidirect
591. C3#23424semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5362
C22×#9492
C23×#2152
C2×#52332
C2×#52252
#822692
#717912
#717282
#715792
#234243