Finite Group Databasebeta

← #10822277

Subgroups — #10822277

Order 768.

2130 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
1C22×#942C211yesmaximal, characteristic, complemented
2C23×#215C211yesmaximal, characteristic, complemented
3C2×#5525C211yesmaximal, characteristic
4C3×#2310C211yesmaximal, characteristic, complemented
5C2×#5001C211yesmaximal, characteristic, complemented
6C2×#5002C211yesmaximal, characteristic, complemented
7#71579C211yesmaximal, characteristic, complemented
8#69689C214yesmaximal, complemented
9#69688C214yesmaximal, complemented
10#23423—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C23×D12C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C22×#216C2212yescomplemented
17C2×#942—24no—
18C2×#942—24no—
19C2×#942—24no—
20C22×#215—24no—
21C22×#215—24no—
22C22×#215—24no—
23C22×#215C2214yes—
24C22×#215C2212yescomplemented
25C22×#208C2211yescharacteristic
26C23×C3⋊C8C2211yescharacteristic
27#5525—24no—
28#5525—24no—
29C2×#895—22no—
30C2×#882—22no—
31C2×#875C2212yes—
32C2×#874C2212yes—
33C3×#391—22no—
34C3×#391C2214yescomplemented
35C3×#381—22no—
36#5002—24no—
37#5001—24no—
38C2×#810C2212yescomplemented
39C2×#809C2214yes—
40C2×#809C2212yescomplemented
41#4524C2214yescomplemented
42C22×#137—31no—
43C23×SD16—31no—
44C2×#535—31no—
45C2×#418—31no—
46C2×#416—31no—
47#2310D311yescharacteristic, complemented
48#2015—34no—
49#2014—34no—
50#1729—31no—
51C24×D3—21no—
52C22×C3×Q8—21no—
53C22×C3×Q8—22no—
54C22×C3×Q8—22no—
55C22×C3×Q8—22no—
56C22×C3×Q8—22no—
57C22×C3×Q8—22no—
58C22×C3×Q8D414yescomplemented
59C22×C3×Q8C2312yes—
60C23×C3×C4—21no—
61C23×C3×C4C2311yescharacteristic
62C22×#67—21no—
63C2×#235—22no—
64C2×#235—22no—
65C2×#235—22no—
66C2×#235—22no—
67C2×#235—22no—
68C2×#235D414yes—
69C22×D12—22no—
70C22×D12—22no—
71C22×D12—22no—
72C22×D12—22no—
73C22×D12—21no—
74C22×D12—22no—
75C22×D12—24no—
76C22×D12D414yes—
77C22×D12C2312yes—
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C22×Dic6C2311yescharacteristic
81C3×#117—22no—
82C3×#117—22no—
83C3×#117—22no—
84C3×#117—22no—
85C2×C3×C4⋊C4D412yes—
86C2×C3×C4⋊C4C2312yes—
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4D412yes—
89C2×C3×C42—21no—
90C2×C3×C42C2312yes—
91C2×#216—24no—
92C2×#216—24no—
93C2×#216—24no—
94#942—232no—
95C2×#215—232no—
96C2×#215—28no—
97C2×#215—24no—
98C2×#215—28no—
99C2×#215—24no—
100C2×#215—28no—
101C2×#215—24no—
102C2×(C2×D3)⋊C4—24no—
103C2×#208—22no—
104C2×#208—22no—
105C2×#208D414yescomplemented
106C22×C3⋊C8—22no—
107C22×C3⋊C8—22no—
108C22×C3⋊C8D414yescomplemented
109C22×C3⋊C8C2312yes—
110C22×C3⋊C8C2312yes—
111#895—216no—
112#882—216no—
113#875—24no—
114#875—24no—
115#874—24no—
116#874—24no—
117C3×#95—28no—
118#810—24no—
119#809—28no—
120#809—24no—
121C22×#41—31no—
122C23×Q8C2×D311yescharacteristic
123C23×D4—31no—
124C2×#137—64no—
125C2×#137—64no—
126C2×#137—64no—
127C22×Q16—32no—
128C22×SD16—34no—
129C22×SD16—32no—
130C22×SD16—64no—
131C22×SD16—64no—
132C22×SD16—64no—
133C22×M16—31no—
134C23×C8—31no—
135#535—64no—
136#535—64no—
137C2×#127—32no—
138C2×#124—32no—
139C2×#121—62no—
140C2×#120—62no—
141C2×#117C2×D311yescharacteristic
142C2×#117C2×D311yescharacteristic
143#418—64no—
144#416—64no—
145C2×#103—32no—
146C2×#102—32no—
147C2×#102—34no—
148#391—22no—
149#391C2×D314yescomplemented
150#381—22no—
151#336—34no—
152C24×C3C2×D411yescharacteristic
153C23×D3—22no—
154C23×D3—21no—
155C23×D3—24no—
156C23×D3—42no—
157C23×D3—42no—
158C23×D3—84no—
159C2×C3×Q8—24no—
160C2×C3×Q8—22no—
161C2×C3×Q8—28no—
162C2×C3×Q8—28no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—42no—
168C2×C3×Q8—44no—
169C2×C3×Q8—44no—
170C2×C3×Q8—42no—
171C2×C3×Q8C2×D414yes—
172C22×C3×C4—24no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—42no—
176C22×C3×C4—42no—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D414yes—
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4#4112yes—
185C22×C3×C4C2411yescharacteristic
186C2×#67—24no—
187C2×#67—42no—
188C2×#67—42no—
189C22×Dic3—21no—
190#235—28no—
191#235—28no—
192#235—44no—
193#235—44no—
194C2×D12—28no—
195C2×D12—28no—
196C2×D12—22no—
197C2×D12—21no—
198C2×D12—22no—
199C2×D12—22no—
200C2×D12—21no—
201C2×D12—21no—
202C2×D12—22no—
203C2×D12—21no—
204C2×D12—24no—
205C2×D12—21no—
206C2×D12—22no—
207C2×D12—216no—
208C2×D12—44no—
209C2×D12—44no—
210C2×D12—42no—
211C2×D12—42no—
212C2×D12—88no—
213C2×D12C2×D412yes—
214C2×D12C2×D414yes—
215C2×C4×D3—24no—
216C2×C4×D3—42no—
217C2×C4×D3—42no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6C2×D412yes—
224C3×C4⋊C4—24no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—22no—
227C3×C22⋊C4—24no—
228C3×C22⋊C4—42no—
229C3×C42—22no—
230C3×C42—22no—
231C3×C42—42no—
232#216—216no—
233#215—232no—
234#215—216no—
235(C2×D3)⋊C4—216no—
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—34no—
246C2×#41—62no—
247C2×#41—62no—
248C2×#41—62no—
249C2×#41—62no—
250C2×#41—62no—
251C22×Q8—22no—
252C22×Q8—21no—
253C22×Q8—22no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8—31no—
258C22×Q8D1214yescomplemented
259C22×Q8C22×D312yes—
260C22×D4—34no—
261C22×D4—32no—
262C22×D4—31no—
263C22×D4—61no—
264C22×D4—62no—
265C22×D4—62no—
266C22×D4—62no—
267C22×D4—62no—
268C22×D4—62no—
269C22×D4—61no—
270C22×D4—64no—
271C23×C4—21no—
272C23×C4—61no—
273C23×C4C22×D311yescharacteristic
274#137—632no—
275C2×Q16—64no—
276C2×Q16—64no—
277C2×Q16—64no—
278C2×SD16—68no—
279C2×SD16—64no—
280C2×SD16—68no—
281C2×SD16—64no—
282C2×SD16—632no—
283C2×SD16—68no—
284C2×SD16—64no—
285C2×M16—34no—
286C2×M16—62no—
287C2×M16—62no—
288C22×C8—34no—
289C22×C8—32no—
290C22×C8—32no—
291C22×C8—62no—
292C22×C8—62no—
293#127—64no—
294#127—64no—
295#124—64no—
296#124—64no—
297#121—616no—
298#120—616no—
299#117—22no—
300#117—22no—
301#117—22no—
302#117—22no—
303C2×C4⋊C4#6712yes—
304C2×C4⋊C4C22×D312yes—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—64no—
307C2×C22⋊C4#6712yes—
308C2×C42—21no—
309C2×C42C22×D312yes—
310#103—64no—
311#102—64no—
312#102—68no—
313#95—28no—
314C23×C3—24no—
315C23×C3#12012yes—
316C23×C3#12112yes—
317C23×C3#12712yes—
318C23×C3C22×D411yescharacteristic
319C22×D3—24no—
320C22×D3—22no—
321C22×D3—41no—
322C22×D3—42no—
323C22×D3—41no—
324C22×D3—42no—
325C22×D3—44no—
326C22×D3—48no—
327C22×D3—48no—
328C22×D3—88no—
329C22×D3—88no—
330C22×D3—88no—
331C3×Q8—24no—
332C3×Q8—24no—
333C3×Q8—42no—
334C3×Q8—42no—
335C3×Q8—48no—
336C2×C3×C4—24no—
337C2×C3×C4—24no—
338C2×C3×C4—28no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—48no—
342C2×C3×C4—42no—
343C2×C3×C4—44no—
344C2×C3×C4#12014yes—
345C2×C3×C4#12014yes—
346C2×C3×C4#12012yes—
347C2×C3×C4#12114yes—
348C2×C3×C4#12114yes—
349C2×C3×C4#12112yes—
350C2×C3×C4#12712yes—
351C2×C3×C4C22×D411yescharacteristic
352C2×C3×C4C22×D411yescharacteristic
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4C22×D411yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357#67—48no—
358C2×Dic3—22no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361D12—24no—
362D12—22no—
363D12—24no—
364D12—22no—
365D12—42no—
366D12—41no—
367D12—42no—
368D12—41no—
369D12—48no—
370D12—816no—
371C4×D3—48no—
372Dic6—22no—
373Dic6—22no—
374Dic6—41no—
375Dic6—41no—
376C3⋊C8—24no—
377C3⋊C8—24no—
378C24—62no—
379C24—61no—
380C24—64no—
381C24—122no—
382C24—122no—
383C24—244no—
384C24C2×#6711yescharacteristic
385#41—68no—
386#41—68no—
387#41—124no—
388#41—124no—
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—28no—
392C2×Q8—28no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—32no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—61no—
403C2×Q8—61no—
404C2×Q8—61no—
405C2×Q8—61no—
406C2×Q8—61no—
407C2×Q8C2×D1214yes—
408C2×D4—34no—
409C2×D4—32no—
410C2×D4—616no—
411C2×D4—64no—
412C2×D4—68no—
413C2×D4—68no—
414C2×D4—64no—
415C2×D4—62no—
416C2×D4—61no—
417C2×D4—62no—
418C2×D4—61no—
419C2×D4—62no—
420C2×D4—61no—
421C2×D4—62no—
422C2×D4—61no—
423C2×D4—62no—
424C2×D4—61no—
425C2×D4—122no—
426C2×D4—122no—
427C2×D4—124no—
428C2×D4—124no—
429C2×D4—122no—
430C2×D4—122no—
431C2×D4—248no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—61no—
438C22×C4—64no—
439C22×C4—122no—
440C22×C4—122no—
441C22×C4C2×D1212yes—
442C22×C4#23512yes—
443C22×C4D3×D412yes—
444C22×C4D3×D412yes—
445C22×C4D3×D412yes—
446C22×C4D3×D412yes—
447C22×C4C2×#6714yes—
448C22×C4C2×#6711yescharacteristic
449C22×C4C23×D311yescharacteristic
450Q16—616no—
451SD16—632no—
452SD16—616no—
453M16—68no—
454C2×C8—34no—
455C2×C8—34no—
456C2×C8—68no—
457C2×C8—62no—
458C2×C8—62no—
459C2×C8—62no—
460C2×C8—62no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C22⋊C4—24no—
465C22⋊C4—42no—
466C22⋊C4—616no—
467C42—22no—
468C42—22no—
469C42—42no—
470C22×C3—24no—
471C22×C3—24no—
472C22×C3—24no—
473C22×C3#39214yes—
474C22×C3C2×#12011yescharacteristic
475C22×C3C2×#12111yescharacteristic
476C22×C3C2×#12711yescharacteristic
477C22×C3#57714yes—
478C2×D3—42no—
479C2×D3—44no—
480C2×D3—44no—
481C2×D3—48no—
482C2×D3—816no—
483C2×D3—82no—
484C2×D3—84no—
485C2×D3—82no—
486C2×D3—84no—
487C3×C4—44no—
488C3×C4—44no—
489C3×C4—82no—
490C3×C4#39214yes—
491C3×C4C2×#12011yescharacteristic
492C3×C4C2×#12011yescharacteristic
493C3×C4C2×#12111yescharacteristic
494C3×C4C2×#12111yescharacteristic
495Dic3—42no—
496C23—24no—
497C23—64no—
498C23—62no—
499C23—128no—
500C23—121no—
501C23—122no—
502C23—121no—
503C23—122no—
504C23—124no—
505C23—128no—
506C23—248no—
507C23—248no—
508C23—248no—
509C23#93012yes—
510C23#93712yes—
511C23#94012yes—
512C23C22×#6711yescharacteristic
513Q8—24no—
514Q8—24no—
515Q8—48no—
516Q8—42no—
517Q8—42no—
518Q8—62no—
519Q8—62no—
520Q8—121no—
521Q8—121no—
522D4—64no—
523D4—62no—
524D4—64no—
525D4—62no—
526D4—128no—
527D4—128no—
528D4—122no—
529D4—121no—
530D4—122no—
531D4—121no—
532D4—2416no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—48no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—62no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—128no—
545C2×C4#88214yes—
546C2×C4#88214yes—
547C2×C4#88414yes—
548C2×C4#88414yes—
549C2×C4#93012yes—
550C2×C4#93712yes—
551C2×C4#94012yes—
552C2×C4C22×D1211yescharacteristic
553C2×C4C2×#23511yescharacteristic
554C2×C4C2×D3×D411yescharacteristic
555C2×C4C2×D3×D411yescharacteristic
556C2×C4C2×D3×D411yescharacteristic
557C2×C4C2×D3×D411yescharacteristic
558C8—64no—
559C8—64no—
560C2×C3—24no—
561C2×C3C2×#39211yescharacteristic
562C2×C3#300912yes—
563C2×C3#304312yes—
564C2×C3#315412yes—
565D3—82no—
566D3—84no—
567C22—24no—
568C22—24no—
569C22—24no—
570C22—122no—
571C22—124no—
572C22—124no—
573C22—128no—
574C22—2416no—
575C22—244no—
576C22—242no—
577C22—244no—
578C22—242no—
579C22#496014yes—
580C22C2×#93011yescharacteristic
581C22C2×#93711yescharacteristic
582C22C2×#94011yescharacteristic
583C22#583714yes—
584C4—44no—
585C4—44no—
586C4—82no—
587C4—122no—
588C4#496014yes—
589C4C2×#88211yescharacteristic
590C4C2×#88211yescharacteristic
591C4C2×#88411yescharacteristic
592C4C2×#88411yescharacteristic
593C3#2342311yescharacteristic, complemented, Sylow
594C2—24no—
595C2—244no—
596C2—242no—
597C2C2×#496011yescharacteristic
598C2#8255112yes—
599C2#8340212yes—
600C2#8340412yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2310C2
socleC23×C3C22×D4
prime radicalC2×#5525C2
hypercenter#2310D3
hypocenterC3#23423
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2310

Complements

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

NH ≅ G/Ntype
1. C22×#942C2semidirect
2. C23×#215C2semidirect
4. C3×#2310C2semidirect
5. C2×#5001C2semidirect
6. C2×#5002C2semidirect
7. #71579C2semidirect
8. #69689C2semidirect
9. #69688C2semidirect
16. C22×#216C22semidirect
24. C22×#215C22semidirect
34. C3×#391C22semidirect
38. C2×#810C22semidirect
40. C2×#809C22semidirect
41. #4524C22semidirect
47. #2310D3semidirect
58. C22×C3×Q8D4semidirect
105. C2×#208D4semidirect
108. C22×C3⋊C8D4semidirect
149. #391C2×D3semidirect
258. C22×Q8D12semidirect
593. C3#23423semidirect

Maximal subgroups

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

subgroupindex
C22×#9422
C23×#2152
C2×#55252
C3×#23102
C2×#50012
C2×#50022
#715792
#696892
#696882
#234233