Finite Group Databasebeta

← #10830949

Subgroups — #10830949

Order 768.

1384 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D3×Q16C211yesmaximal, characteristic, complemented
2C2×C3×#470C211yesmaximal, characteristic, complemented
3C2×#5137C211yesmaximal, characteristic, complemented
4C2×#5127C211yesmaximal, characteristic
5#82751C212yesmaximal
6#72067C211yesmaximal, characteristic, complemented
7#71700C212yesmaximal, complemented
8#70577C211yesmaximal, characteristic, complemented
9#70575C211yesmaximal, characteristic, complemented
10#70285C212yesmaximal, complemented
11#70266C212yesmaximal, complemented
12#24273—31nomaximal, Sylow
13C22×D3×Q8C2212yes—
14C22×C3×Q16C2211yescharacteristic
15C22×C3×M16C2211yescharacteristic
16C2×C3×C22⋊Q8C2212yes—
17C22×#216C2212yes—
18C2×C4×#67C2211yescharacteristic
19C2×#924C2212yes—
20C2×D3×Q16—24no—
21C2×D3×Q16—24no—
22C2×D3×Q16—24no—
23C22×Dic12C2211yescharacteristic
24C22×D3×C8C2211yescharacteristic
25C3×#470—24no—
26C2×C3×#106C2211yescharacteristic
27C2×C3×#103C2212yes—
28#5229C2212yes—
29C2×#835C2212yes—
30C2×#830C2211yescharacteristic
31#5137—24no—
32#5127—24no—
33C2×(C2×D3)⋊C8C2211yescharacteristic
34C2×#817C2211yescharacteristic
35C2×#816C2212yes—
36C2×#815C2211yescharacteristic
37C2×Dic3⋊C8C2211yescharacteristic
38#4989—22no—
39#4987C2212yes—
40#4979—22no—
41#4973C2212yes—
42C2×#810C2212yes—
43#4956—22no—
44#4947—22no—
45#4529C2212yes—
46C23×Q16—31no—
47C2×#470—31no—
48C2×#470D311yescharacteristic, complemented
49C2×#435—31no—
50#2409—32no—
51#2056—32no—
52#2001—32no—
53#1971—31no—
54#1971—31no—
55#1807—31no—
56#1787—32no—
57C22×C3×Q8C2312yes—
58C23×C3×C4C2311yescharacteristic
59C22×#67D411yescharacteristic
60C2×D3×Q8—24no—
61C2×D3×Q8—24no—
62C2×D3×Q8—24no—
63C2×D3×Q8—24no—
64C2×D3×Q8—42no—
65C2×D3×Q8—42no—
66C22×C4×D3—22no—
67C22×C4×D3C2311yescharacteristic
68C22×Dic6—22no—
69C22×Dic6C2312yes—
70C2×C3×Q16—22no—
71C2×C3×Q16—22no—
72C2×C3×Q16D414yescomplemented
73C2×C3×M16—22no—
74C2×C3×M16—22no—
75C2×C3×M16—22no—
76C22×C3×C8C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C3×C22⋊Q8—24no—
79C3×C22⋊Q8—42no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C22⋊C4—22no—
83C2×#217—22no—
84C2×#217D411yescharacteristic
85C2×#216—24no—
86C2×#216—24no—
87C2×#216—24no—
88C4×#67—22no—
89C4×#67—22no—
90C2×(C2×D3)⋊C4—22no—
91C2×(C2×D3)⋊C4D411yescharacteristic
92C2×C4⋊Dic3C2312yes—
93#924—24no—
94#924—42no—
95C2×Dic3⋊C4—22no—
96C2×Dic3⋊C4D411yescharacteristic
97C2×C4×Dic3C2311yescharacteristic
98C22×C3⋊C8C2311yescharacteristic
99C22×C3⋊C8C2311yescharacteristic
100D3×Q16—232no—
101C2×Dic12—22no—
102C2×Dic12—22no—
103C2×Dic12D414yescomplemented
104C2×D3×C8—22no—
105C2×D3×C8—22no—
106C2×D3×C8D414yes—
107C3×#106—22no—
108C3×#103—24no—
109#835—42no—
110C3⋊#95—22no—
111C3⋊#95—22no—
112C3⋊#95—22no—
113C3⋊#95—22no—
114C3⋊#95—22no—
115#830—22no—
116(C2×D3)⋊C8—22no—
117#817—22no—
118#816—24no—
119#815—22no—
120Dic3⋊C8—22no—
121#810—42no—
122C23×Q8—32no—
123C22×Q16—32no—
124C22×Q16—31no—
125C22×Q16—64no—
126C22×Q16—64no—
127C22×Q16—64no—
128C22×Q16C2×D311yescharacteristic
129C22×M16C2×D311yescharacteristic
130C23×C8—31no—
131C2×C22⋊Q8—32no—
132C2×C22⋊Q8C2×D312yes—
133C2×C4×D4—31no—
134#470—24no—
135#470—64no—
136#435—64no—
137C2×#106—31no—
138C2×#106C2×D311yescharacteristic
139C2×C4⋊C8—31no—
140C2×#103—32no—
141C2×#103—32no—
142C2×#103—32no—
143C2×#103C2×D312yes—
144C2×C22⋊C8—31no—
145C2×C22⋊C8—31no—
146C2×#97—31no—
147#395—62no—
148#393—62no—
149#391—32no—
150#391—32no—
151#386—62no—
152#385—62no—
153#380—32no—
154#340—32no—
155C24×C3C2×D411yescharacteristic
156C23×D3C2×D411yescharacteristic
157C2×C3×Q8—22no—
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8C2×D414yes—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—22no—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2411yescharacteristic
173C2×#67—22no—
174C2×#67—24no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3C2×D411yescharacteristic
178C22×Dic3C2×D411yescharacteristic
179C22×Dic3C2×D411yescharacteristic
180D3×Q8—216no—
181D3×Q8—48no—
182D3×Q8—48no—
183D3×Q8—48no—
184C2×C4×D3—21no—
185C2×C4×D3—21no—
186C2×C4×D3—21no—
187C2×C4×D3—21no—
188C2×C4×D3—44no—
189C2×C4×D3—44no—
190C2×C4×D3—44no—
191C2×C4×D3C2×D412yes—
192C2×C4×D3#4112yes—
193C2×Dic6—22no—
194C2×Dic6—22no—
195C2×Dic6—22no—
196C2×Dic6—22no—
197C2×Dic6—22no—
198C2×Dic6—44no—
199C2×Dic6—44no—
200C2×Dic6—44no—
201C2×Dic6C2×D414yes—
202C3×Q16—28no—
203C3×M16—44no—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C2×C3×C8—21no—
207C2×C3×C8C2×D412yes—
208C2×C3×C8C2×D412yes—
209C2×C3×C8C2×D412yes—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—44no—
213C3×C22⋊C4—44no—
214#217—22no—
215#217—44no—
216#216—216no—
217(C2×D3)⋊C4—22no—
218(C2×D3)⋊C4—44no—
219C4⋊Dic3—22no—
220C4⋊Dic3—22no—
221Dic3⋊C4—22no—
222Dic3⋊C4—44no—
223C4×Dic3—21no—
224C4×Dic3—21no—
225C2×C3⋊C8—21no—
226C2×C3⋊C8—21no—
227C2×C3⋊C8—21no—
228C2×C3⋊C8—21no—
229C2×C3⋊C8—21no—
230C2×C3⋊C8C2×D412yes—
231Dic12—28no—
232D3×C8—28no—
233C22×Q8—32no—
234C22×Q8—62no—
235C22×Q8—64no—
236C22×Q8—64no—
237C22×Q8—64no—
238C22×Q8—64no—
239C22×Q8—122no—
240C22×Q8—122no—
241C22×Q8C22×D312yes—
242C22×D4—31no—
243C23×C4—31no—
244C23×C4—62no—
245C23×C4C22×D311yescharacteristic
246C2×Q16—22no—
247C2×Q16—22no—
248C2×Q16—34no—
249C2×Q16—64no—
250C2×Q16—632no—
251C2×Q16—64no—
252C2×Q16—64no—
253C2×Q16—62no—
254C2×Q16—62no—
255C2×Q16#6714yescomplemented
256C2×M16—22no—
257C2×M16—22no—
258C2×M16—22no—
259C22×C8—31no—
260C22×C8—34no—
261C22×C8—31no—
262C22×C8—62no—
263C22×C8—62no—
264C22×C8C22×D311yescharacteristic
265C22×C8C22×D311yescharacteristic
266C22⋊Q8—24no—
267C22⋊Q8—42no—
268C22⋊Q8—64no—
269C22⋊Q8—122no—
270C4×D4—62no—
271C4×D4—62no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—32no—
275C2×C4⋊C4—62no—
276C2×C4⋊C4C22×D312yes—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—31no—
279C2×C22⋊C4—31no—
280C2×C22⋊C4—62no—
281C2×C22⋊C4—62no—
282C2×C42—31no—
283#106—22no—
284#106—62no—
285C4⋊C8—62no—
286#103—24no—
287#103—64no—
288#103—122no—
289#103—122no—
290C22⋊C8—62no—
291C22⋊C8—62no—
292#97—62no—
293#95—62no—
294#95—62no—
295#95—62no—
296#95—62no—
297#95—62no—
298C23×C3—21no—
299C23×C3—21no—
300C23×C3—21no—
301C23×C3—22no—
302C23×C3—22no—
303C23×C3C22×D411yescharacteristic
304C22×D3—21no—
305C22×D3—21no—
306C22×D3—22no—
307C22×D3#12112yes—
308C22×D3#13714yes—
309C3×Q8—24no—
310C3×Q8—24no—
311C3×Q8—42no—
312C3×Q8—42no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—24no—
316C2×C3×C4—21no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4C22×D411yescharacteristic
327C2×C3×C4C22×D411yescharacteristic
328C2×C3×C4C22×D411yescharacteristic
329C2×C3×C4C2×#4111yescharacteristic
330C2×C3×C4H(5,𝔽2)11yescharacteristic
331C2×C3×C4H(5,𝔽2)11yescharacteristic
332#67—44no—
333C2×Dic3—21no—
334C2×Dic3—21no—
335C2×Dic3—21no—
336C2×Dic3—21no—
337C2×Dic3—24no—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—42no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3#12112yes—
346C4×D3—22no—
347C4×D3—22no—
348C4×D3—22no—
349C4×D3—22no—
350C4×D3—416no—
351Dic6—24no—
352Dic6—24no—
353Dic6—416no—
354Dic6—42no—
355Dic6—42no—
356C3×C8—41no—
357C3×C8#12014yes—
358C3⋊C8—22no—
359C3⋊C8—41no—
360C24—31no—
361C24D3×D411yescharacteristic
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—34no—
368C2×Q8—62no—
369C2×Q8—616no—
370C2×Q8—62no—
371C2×Q8—62no—
372C2×Q8—62no—
373C2×Q8—62no—
374C2×Q8—124no—
375C2×Q8—124no—
376C2×Q8—124no—
377C2×Q8—128no—
378C2×Q8—128no—
379C2×Q8—128no—
380C2×Q8C2×#6714yes—
381C2×D4—62no—
382C2×D4—64no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—31no—
392C22×C4—32no—
393C22×C4—32no—
394C22×C4—31no—
395C22×C4—31no—
396C22×C4—61no—
397C22×C4—61no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—62no—
401C22×C4—62no—
402C22×C4—124no—
403C22×C4—124no—
404C22×C4—124no—
405C22×C4D3×D411yescharacteristic
406C22×C4C23×D311yescharacteristic
407Q16—28no—
408Q16—616no—
409Q16—68no—
410M16—44no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—32no—
415C2×C8—61no—
416C2×C8—61no—
417C2×C8—68no—
418C2×C8—61no—
419C2×C8—61no—
420C2×C8—61no—
421C2×C8D3×D412yes—
422C2×C8D3×D412yes—
423C2×C8C2×#6712yes—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—44no—
427C4⋊C4—62no—
428C4⋊C4—62no—
429C4⋊C4—62no—
430C4⋊C4—124no—
431C22⋊C4—44no—
432C22⋊C4—62no—
433C22⋊C4—62no—
434C22⋊C4—124no—
435C22⋊C4—124no—
436C42—61no—
437C42—61no—
438C22×C3—41no—
439C22×C3—42no—
440C22×C3—42no—
441C22×C3—41no—
442C22×C3—41no—
443C22×C3C2×#12111yescharacteristic
444C22×C3D4211yescharacteristic
445C22×C3D4211yescharacteristic
446C22×C3C2×#13712yes—
447C22×C3#57812yes—
448C2×D3—22no—
449C2×D3—24no—
450C2×D3—24no—
451C2×D3—22no—
452C2×D3#47014yescomplemented
453C3×C4—44no—
454C3×C4—41no—
455C3×C4—82no—
456C3×C4C2×#12011yescharacteristic
457C3×C4#53511yescharacteristic
458C3×C4#54611yescharacteristic
459C3×C4#54611yescharacteristic
460Dic3—22no—
461Dic3—44no—
462Dic3—41no—
463Dic3—41no—
464Dic3—82no—
465C23—22no—
466C23—22no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—34no—
471C23—32no—
472C23—62no—
473C23—61no—
474C23—61no—
475C23C2×D3×D411yescharacteristic
476Q8—24no—
477Q8—24no—
478Q8—42no—
479Q8—42no—
480Q8—64no—
481Q8—64no—
482Q8—1216no—
483Q8—122no—
484Q8—122no—
485D4—124no—
486C2×C4—24no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—32no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—61no—
501C2×C4—61no—
502C2×C4—62no—
503C2×C4—62no—
504C2×C4—62no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—62no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—64no—
511C2×C4—1216no—
512C2×C4—122no—
513C2×C4—122no—
514C2×C4—122no—
515C2×C4—122no—
516C2×C4—122no—
517C2×C4C2×D3×D411yescharacteristic
518C2×C4C2×D3×D411yescharacteristic
519C2×C4#100411yescharacteristic
520C2×C4D3×#4111yescharacteristic
521C2×C4#100911yescharacteristic
522C2×C4C22×#6711yescharacteristic
523C8—41no—
524C8—62no—
525C8—121no—
526C8#93714yes—
527C2×C3—81no—
528C2×C3#240811yescharacteristic
529C2×C3C2×#47011yescharacteristic
530C2×C3#305911yescharacteristic
531C2×C3#327012yes—
532C2×C3#327012yes—
533D3—24no—
534C22—34no—
535C22—41no—
536C22—41no—
537C22—42no—
538C22—42no—
539C22—41no—
540C22—62no—
541C22—64no—
542C22—64no—
543C22—62no—
544C22D3×#12111yescharacteristic
545C22#560911yescharacteristic
546C22D3×#13712yes—
547C22#577912yes—
548C22D4×#6711yescharacteristic
549C4—41no—
550C4—44no—
551C4—62no—
552C4—82no—
553C4—121no—
554C4—121no—
555C4—124no—
556C4—242no—
557C4#560711yescharacteristic
558C4#561211yescharacteristic
559C4C2×#93711yescharacteristic
560C4#582911yescharacteristic
561C3#2427311yescharacteristic, complemented, Sylow
562C2—64no—
563C2—81no—
564C2D3×#47011yescharacteristic
565C2#8090511yescharacteristic
566C2#8092612yes—
567C2#8264011yescharacteristic
568C2#8398912yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#470C2
socleC23×C3C22×D4
prime radicalC22×#67D4
hypercenterC2×#470D3
hypocenterC3#24273
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#470

Complements

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

NH ≅ G/Ntype
1. C22×D3×Q16C2semidirect
2. C2×C3×#470C2semidirect
3. C2×#5137C2semidirect
6. #72067C2semidirect
7. #71700C2semidirect
8. #70577C2semidirect
9. #70575C2semidirect
10. #70285C2semidirect
11. #70266C2semidirect
48. C2×#470D3semidirect
72. C2×C3×Q16D4semidirect
103. C2×Dic12D4semidirect
255. C2×Q16#67semidirect
452. C2×D3#470semidirect
561. C3#24273semidirect

Maximal subgroups

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

subgroupindex
C22×D3×Q162
C2×C3×#4702
C2×#51372
C2×#51272
#827512
#720672
#717002
#705772
#705752
#702852
#702662
#242733