Finite Group Databasebeta

← #10732615

Subgroups — #10732615

Order 768.

3142 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#940C211yesmaximal, characteristic
2C22×#936C211yesmaximal, characteristic, complemented
3C22×#875C211yesmaximal, characteristic, complemented
4C2×C3×#390C211yesmaximal, characteristic, complemented
5C2×#4973C211yesmaximal, characteristic, complemented
6C2×#4988C211yesmaximal, characteristic, complemented
7#71550C218yesmaximal, complemented
8C4×C3⋊#95C211yesmaximal, characteristic, complemented
9#15227—31nomaximal, Sylow
10C23×C3×D4C2211yescharacteristic
11C23×#67C2214yes—
12C23×D12C2211yescharacteristic
13C23×Dic6C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C22⋊C4C2214yes—
16C22×C3×C42C2211yescharacteristic
17C22×#217C2214yes—
18C2×#940—24no—
19C2×#940—24no—
20C2×#940—24no—
21C2×#936—24no—
22C2×#936—24no—
23C2×#936—24no—
24C22×(C2×D3)⋊C4C2214yes—
25C22×(C2×D3)⋊C4C2214yes—
26C22×C4×Dic3C2211yescharacteristic
27C22×C4×Dic3C2211yescharacteristic
28C2×#875—24no—
29C2×#875—24no—
30C2×#875—24no—
31C3×#390—24no—
32C2×C3⋊#95C2214yes—
33C2×C3⋊#95C2214yes—
34#4988—24no—
35#4973—24no—
36C22×#127—31no—
37C22×#124—31no—
38C22×#124—31no—
39C2×#391—31no—
40C2×#390—31no—
41C2×#390D311yescharacteristic, complemented
42#1438—38no—
43C4×#95—31no—
44C25×C3C2312yes—
45C24×D3C2312yes—
46C22×C3×D4—22no—
47C22×C3×D4—22no—
48C22×C3×D4—22no—
49C22×C3×D4—22no—
50C22×C3×D4—22no—
51C22×C3×D4—22no—
52C22×C3×D4D414yes—
53C23×C3×C4C2312yes—
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C22×#67—28no—
57C22×#67—28no—
58C22×#67—28no—
59C22×#67—216no—
60C22×#67—48no—
61C23×Dic3C2312yes—
62C23×Dic3C2312yes—
63C23×Dic3C2312yes—
64C22×D12—22no—
65C22×D12—22no—
66C22×D12—22no—
67C22×D12—22no—
68C22×D12—22no—
69C22×D12—22no—
70C22×D12D414yes—
71C22×Dic6—22no—
72C22×Dic6—22no—
73C22×Dic6—22no—
74C22×Dic6—22no—
75C22×Dic6—22no—
76C22×Dic6—22no—
77C22×Dic6D414yescomplemented
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C22⋊C4—28no—
82C2×C3×C22⋊C4—48no—
83C2×C3×C42—22no—
84C2×C3×C42—22no—
85C2×C3×C42D414yescomplemented
86C2×#217—28no—
87C2×#217—28no—
88C2×#217—28no—
89#940—232no—
90#936—232no—
91C2×(C2×D3)⋊C4—28no—
92C2×(C2×D3)⋊C4—28no—
93C2×(C2×D3)⋊C4—28no—
94C2×(C2×D3)⋊C4—28no—
95C2×(C2×D3)⋊C4—48no—
96C2×C4×Dic3—22no—
97C2×C4×Dic3—22no—
98C2×C4×Dic3—22no—
99C2×C4×Dic3—22no—
100C2×C4×Dic3D414yes—
101C2×C4×Dic3D414yescomplemented
102#875—232no—
103C3⋊#95—28no—
104C3⋊#95—28no—
105C23×Q8—31no—
106C23×D4—34no—
107C23×D4—31no—
108C23×D4C2×D311yescharacteristic
109C2×#127—64no—
110C2×#127—64no—
111C2×#127—64no—
112C2×#124—64no—
113C2×#124—64no—
114C2×#124—64no—
115C2×#124—64no—
116C2×#124—64no—
117C2×#124—64no—
118C22×C4⋊C4C2×D311yescharacteristic
119C22×C22⋊C4—34no—
120C22×C22⋊C4—34no—
121C22×C22⋊C4—34no—
122C22×C22⋊C4C2×D314yes—
123C22×C42—31no—
124C22×C42—31no—
125C22×C42C2×D311yescharacteristic
126#391—64no—
127#390—24no—
128#390—64no—
129C2×#95—34no—
130C2×#95—34no—
131C24×C3—22no—
132C24×C3—22no—
133C24×C3—22no—
134C24×C3—22no—
135C24×C3—22no—
136C24×C3—28no—
137C24×C3—22no—
138C24×C3—28no—
139C24×C3C2×D414yes—
140C24×C3C2411yescharacteristic
141C23×D3—22no—
142C23×D3—22no—
143C23×D3—22no—
144C23×D3—22no—
145C23×D3—22no—
146C23×D3—22no—
147C23×D3—28no—
148C23×D3—28no—
149C23×D3C2×D414yes—
150C2×C3×D4—28no—
151C2×C3×D4—28no—
152C2×C3×D4—28no—
153C2×C3×D4—44no—
154C2×C3×D4—44no—
155C2×C3×D4—44no—
156C2×C3×D4—44no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—24no—
160C22×C3×C4—24no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—28no—
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D414yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4#4112yes—
173C2×#67—232no—
174C2×#67—432no—
175C2×#67—432no—
176C2×#67—416no—
177C2×#67—416no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—24no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—24no—
189C22×Dic3—22no—
190C22×Dic3—24no—
191C22×Dic3C2×D414yes—
192C22×Dic3C2×D414yes—
193C22×Dic3C2×D414yes—
194C22×Dic3#4118yes—
195C2×D12—28no—
196C2×D12—28no—
197C2×D12—28no—
198C2×D12—44no—
199C2×D12—44no—
200C2×D12—44no—
201C2×D12—44no—
202C2×Dic6—28no—
203C2×Dic6—28no—
204C2×Dic6—28no—
205C2×Dic6—44no—
206C2×Dic6—44no—
207C2×Dic6—44no—
208C2×Dic6—44no—
209C3×C4⋊C4—44no—
210C3×C22⋊C4—416no—
211C3×C42—28no—
212#217—232no—
213(C2×D3)⋊C4—232no—
214(C2×D3)⋊C4—416no—
215C4×Dic3—28no—
216C4×Dic3—28no—
217C25—32no—
218C25C22×D312yes—
219C22×Q8—34no—
220C22×Q8—62no—
221C22×Q8—62no—
222C22×Q8—62no—
223C22×Q8—62no—
224C22×Q8—62no—
225C22×Q8—62no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4—34no—
233C22×D4—68no—
234C22×D4—616no—
235C22×D4—68no—
236C22×D4—68no—
237C22×D4—62no—
238C22×D4—62no—
239C22×D4—62no—
240C22×D4—62no—
241C22×D4—62no—
242C22×D4—62no—
243C22×D4—128no—
244C22×D4D1214yes—
245C23×C4—32no—
246C23×C4—32no—
247C23×C4—32no—
248C23×C4C22×D312yes—
249C23×C4C22×D312yes—
250C23×C4C22×D311yescharacteristic
251#127—632no—
252#124—632no—
253#124—632no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—48no—
259C2×C22⋊C4—68no—
260C2×C22⋊C4—68no—
261C2×C22⋊C4—68no—
262C2×C22⋊C4—68no—
263C2×C22⋊C4—68no—
264C2×C22⋊C4—68no—
265C2×C22⋊C4—68no—
266C2×C22⋊C4—128no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—34no—
270C2×C42—34no—
271C2×C42—62no—
272C2×C42—62no—
273C2×C42—62no—
274C2×C42—62no—
275C2×C42#6714yescomplemented
276#95—68no—
277#95—68no—
278C23×C3—24no—
279C23×C3—24no—
280C23×C3—24no—
281C23×C3—216no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3—44no—
286C23×C3—42no—
287C23×C3—44no—
288C23×C3—44no—
289C23×C3—44no—
290C23×C3—44no—
291C23×C3—44no—
292C23×C3—48no—
293C23×C3—48no—
294C23×C3—48no—
295C23×C3C22×D411yescharacteristic
296C23×C3C22×D411yescharacteristic
297C23×C3C22×D411yescharacteristic
298C23×C3C22×D411yescharacteristic
299C23×C3C22×D411yescharacteristic
300C23×C3C22×D411yescharacteristic
301C23×C3C2×#4111yescharacteristic
302C23×C3C2×#4114yes—
303C23×C3H(5,𝔽2)14yes—
304C22×D3—24no—
305C22×D3—24no—
306C22×D3—24no—
307C22×D3—216no—
308C22×D3—42no—
309C22×D3—42no—
310C22×D3—42no—
311C22×D3—44no—
312C22×D3—42no—
313C22×D3—44no—
314C22×D3—44no—
315C22×D3—44no—
316C22×D3—44no—
317C22×D3—44no—
318C22×D3—48no—
319C22×D3—48no—
320C22×D3—48no—
321C3×D4—416no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—44no—
326C2×C3×C4—44no—
327C2×C3×C4—42no—
328C2×C3×C4—44no—
329C2×C3×C4—44no—
330C2×C3×C4—44no—
331C2×C3×C4#12014yes—
332C2×C3×C4#12014yes—
333C2×C3×C4#12014yes—
334C2×C3×C4#12114yes—
335C2×C3×C4#12114yes—
336C2×C3×C4#12114yes—
337C2×C3×C4#12714yes—
338#67—464no—
339C2×Dic3—24no—
340C2×Dic3—24no—
341C2×Dic3—28no—
342C2×Dic3—28no—
343C2×Dic3—24no—
344C2×Dic3—24no—
345C2×Dic3—24no—
346C2×Dic3—24no—
347C2×Dic3—24no—
348C2×Dic3—24no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—42no—
352C2×Dic3—42no—
353C2×Dic3—44no—
354C2×Dic3—44no—
355C2×Dic3#12418yescomplemented
356D12—416no—
357Dic6—416no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—28no—
364C24—28no—
365C24—22no—
366C24—34no—
367C24—62no—
368C24—62no—
369C24—62no—
370C24—62no—
371C24—62no—
372C24—68no—
373C24—68no—
374C24—62no—
375C24C2×D1214yes—
376C24C23×D311yescharacteristic
377C2×Q8—68no—
378C2×Q8—68no—
379C2×Q8—68no—
380C2×Q8—124no—
381C2×Q8—124no—
382C2×Q8—124no—
383C2×Q8—124no—
384C2×D4—28no—
385C2×D4—28no—
386C2×D4—28no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—632no—
392C2×D4—68no—
393C2×D4—68no—
394C2×D4—68no—
395C2×D4—1216no—
396C2×D4—1232no—
397C2×D4—1232no—
398C2×D4—1216no—
399C2×D4—124no—
400C2×D4—124no—
401C2×D4—124no—
402C2×D4—124no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—28no—
410C22×C4—22no—
411C22×C4—34no—
412C22×C4—38no—
413C22×C4—34no—
414C22×C4—34no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—64no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4—62no—
424C22×C4—62no—
425C22×C4—64no—
426C22×C4—64no—
427C22×C4—62no—
428C22×C4C2×D1212yes—
429C22×C4#23512yes—
430C22×C4D3×D412yes—
431C22×C4D3×D412yes—
432C22×C4D3×D412yes—
433C22×C4D3×D412yes—
434C22×C4C2×#6714yes—
435C22×C4C2×#6712yes—
436C4⋊C4—44no—
437C22⋊C4—416no—
438C22⋊C4—632no—
439C22⋊C4—632no—
440C22⋊C4—1216no—
441C42—28no—
442C42—68no—
443C42—68no—
444C22×C3—44no—
445C22×C3—48no—
446C22×C3—48no—
447C22×C3—48no—
448C22×C3—82no—
449C22×C3—82no—
450C22×C3—84no—
451C22×C3—84no—
452C22×C3—84no—
453C22×C3C2×#12011yescharacteristic
454C22×C3C2×#12011yescharacteristic
455C22×C3C2×#12011yescharacteristic
456C22×C3C2×#12111yescharacteristic
457C22×C3C2×#12111yescharacteristic
458C22×C3C2×#12111yescharacteristic
459C22×C3C2×#12412yes—
460C22×C3C2×#12711yescharacteristic
461C22×C3#53212yes—
462C22×C3#53212yes—
463C22×C3#53512yes—
464C22×C3#53512yes—
465C22×C3#53512yes—
466C22×C3#54614yes—
467C22×C3#54614yes—
468C22×C3#55014yes—
469C22×C3#55914yes—
470C2×D3—44no—
471C2×D3—48no—
472C2×D3—48no—
473C2×D3—48no—
474C2×D3—82no—
475C2×D3—82no—
476C2×D3—84no—
477C2×D3—84no—
478C2×D3—84no—
479C3×C4—44no—
480C3×C4—82no—
481C3×C4#39218yes—
482Dic3—28no—
483Dic3—44no—
484Dic3—44no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—216no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—44no—
498C23—48no—
499C23—44no—
500C23—48no—
501C23—48no—
502C23—64no—
503C23—64no—
504C23—64no—
505C23—616no—
506C23—124no—
507C23—124no—
508C23—124no—
509C23—124no—
510C23—122no—
511C23—122no—
512C23—122no—
513C23—122no—
514C23—124no—
515C23—128no—
516C23—128no—
517C23—124no—
518C23—128no—
519C23C22×D1211yescharacteristic
520C23C2×#23511yescharacteristic
521C23C2×D3×D411yescharacteristic
522C23C2×D3×D411yescharacteristic
523C23C2×D3×D411yescharacteristic
524C23C2×D3×D411yescharacteristic
525C23#100414yes—
526C23D3×#4114yes—
527C23C22×#6711yescharacteristic
528Q8—1216no—
529D4—416no—
530D4—1264no—
531D4—1216no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—38no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—64no—
543C2×C4—64no—
544C2×C4—68no—
545C2×C4—68no—
546C2×C4—64no—
547C2×C4—64no—
548C2×C4—64no—
549C2×C4—64no—
550C2×C4—64no—
551C2×C4—64no—
552C2×C4—122no—
553C2×C4—122no—
554C2×C4—122no—
555C2×C4—122no—
556C2×C4—124no—
557C2×C4—124no—
558C2×C4#88214yes—
559C2×C4#88214yes—
560C2×C4#88414yes—
561C2×C4#88414yes—
562C2×C4#93014yes—
563C2×C4#93714yes—
564C2×C4#94014yes—
565C2×C3—84no—
566C2×C3C2×#39211yescharacteristic
567C2×C3#243112yes—
568C2×C3#243112yes—
569C2×C3#243512yes—
570C2×C3#254412yes—
571C2×C3#254412yes—
572C2×C3#254812yes—
573C2×C3#265711yescharacteristic
574C2×C3#285811yescharacteristic
575D3—84no—
576C22—48no—
577C22—48no—
578C22—44no—
579C22—48no—
580C22—82no—
581C22—84no—
582C22—84no—
583C22—84no—
584C22—82no—
585C22—128no—
586C22—128no—
587C22—124no—
588C22—128no—
589C22—244no—
590C22—242no—
591C22—244no—
592C22—244no—
593C22—242no—
594C22C2×#88211yescharacteristic
595C22C2×#88211yescharacteristic
596C22C2×#88411yescharacteristic
597C22C2×#88411yescharacteristic
598C22#555414yes—
599C22#555914yes—
600C22#559914yes—
601C22#561214yes—
602C22#566812yes—
603C22D3×#12412yes—
604C22#571512yes—
605C22#571512yes—
606C22C2×#93011yescharacteristic
607C22C2×#93711yescharacteristic
608C22C2×#94011yescharacteristic
609C22#580512yes—
610C22#583212yes—
611C4—44no—
612C4—68no—
613C4—82no—
614C4—124no—
615C4—124no—
616C4#496018yes—
617C3#1522711yescharacteristic, complemented, Sylow
618C2—84no—
619C2—244no—
620C2#7631312yes—
621C2#7631512yes—
622C2#7635012yes—
623C2#7635212yes—
624C2C2×#496011yescharacteristic
625C2#8238112yes—
626C2#8265212yes—
627C2#8299311yescharacteristic
628C2#8318111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#390C2
prime radicalC22×#940C2
hypercenterC2×#390D3
hypocenterC3#15227
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C22×#936C2semidirect
3. C22×#875C2semidirect
4. C2×C3×#390C2semidirect
5. C2×#4973C2semidirect
6. C2×#4988C2semidirect
7. #71550C2semidirect
8. C4×C3⋊#95C2semidirect
41. C2×#390D3semidirect
77. C22×Dic6D4semidirect
85. C2×C3×C42D4semidirect
101. C2×C4×Dic3D4semidirect
275. C2×C42#67semidirect
355. C2×Dic3#124semidirect
617. C3#15227semidirect

Maximal subgroups

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

subgroupindex
C22×#9402
C22×#9362
C22×#8752
C2×C3×#3902
C2×#49732
C2×#49882
#715502
C4×C3⋊#952
#152273