Finite Group Databasebeta

← #11069326

Subgroups — #11069326

Order 768.

2958 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5727C211yesmaximal, characteristic, complemented
2C2×#5724C212yesmaximal, complemented
3C2×#5577C211yesmaximal, characteristic
4C3×#2546C211yesmaximal, characteristic, complemented
5#83255C212yesmaximal, complemented
6#83251C211yesmaximal, characteristic, complemented
7C4×#944C211yesmaximal, characteristic, complemented
8#77970C218yesmaximal, complemented
9#76640C212yesmaximal, complemented
10#76500C212yesmaximal, complemented
11#76010C214yesmaximal, complemented
12#76004C214yesmaximal, complemented
13#73712C211yesmaximal, characteristic, complemented
14#73617C211yesmaximal, characteristic, complemented
15#39363—31nomaximal, Sylow
16C22×#239C2212yes—
17C22×D3×Q8C2212yes—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×C4⋊Q8C2212yes—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×#947C2212yes—
22C2×#947C2212yes—
23C2×#946C2212yes—
24C2×#944C2211yescharacteristic
25#5727—28no—
26#5724—216no—
27#5577—28no—
28C2×#896C2214yes—
29C2×#896C2218yes—
30C2×#895C2212yes—
31C2×#895C2214yes—
32C2×#894C2218yes—
33C2×#893C2212yes—
34C2×#888C2212yes—
35C2×#875C2212yes—
36C2×#874C2211yescharacteristic
37C2×#874C2212yes—
38C2×C4×D12C2212yes—
39C2×C4×D12C2211yescharacteristic
40C2×#872C2212yes—
41C2×C4×Dic6C2211yescharacteristic
42C3×#398C2214yes—
43C3×#391C2212yes—
44C3×#389C2214yes—
45C3×C4×C4⋊C4C2211yescharacteristic
46#5229C2211yescharacteristic
47#4991C2218yes—
48#4989C2218yes—
49#4988C2214yes—
50#4988C2214yes—
51#4988C2212yes—
52#4988C2214yes—
53#4976C2214yes—
54#4975C2211yescharacteristic
55#4973C2211yescharacteristic
56#4972C2212yes—
57#4939C2212yes—
58C4×(C2×D3)⋊C4C2212yes—
59C4×Dic3⋊C4C2214yes—
60C42×Dic3C2211yescharacteristic
61#4676C2218yes—
62#4673C2218yes—
63#4672C2218yes—
64#4672C2214yes—
65#4669C2214yes—
66#4665C2218yes—
67C2×#550—31no—
68C2×D4×Q8—31no—
69C2×#536—32no—
70#2858—31no—
71#2681—38no—
72#2620—32no—
73#2548—31no—
74#2546D311yescharacteristic, complemented
75#2452—34no—
76#2445—34no—
77#2436—32no—
78#2432—32no—
79C4×C4⋊Q8—31no—
80#2293—31no—
81C22×C3×Q8C2312yes—
82C22×C3×Q8C2311yescharacteristic
83C2×#239—28no—
84C2×#239—28no—
85C2×#239—44no—
86C2×D3×Q8—28no—
87C2×D3×Q8—28no—
88C2×D3×Q8—44no—
89C22×D12C2312yes—
90C22×D12C2312yes—
91C22×D12C2314yes—
92C22×D12C2311yescharacteristic
93C22×D12C2314yes—
94C22×D12C2312yes—
95C22×C4×D3C2314yes—
96C22×C4×D3C2314yes—
97C22×C4×D3C2312yes—
98C22×Dic6C2314yes—
99C22×Dic6C2311yescharacteristic
100C3×C4⋊Q8—28no—
101C3×C4⋊Q8—24no—
102C3×C4×Q8—24no—
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C4⋊C4C2312yes—
106C2×C3×C4⋊C4C2312yes—
107C2×C3×C4⋊C4C2312yes—
108C2×C3×C4⋊C4C2318yes—
109C2×C3×C42C2312yes—
110C2×C3×C42C2312yes—
111C2×C3×C42C2311yescharacteristic
112#947—28no—
113#947—44no—
114#946—28no—
115#944D418yescomplemented
116C2×(C2×D3)⋊C4C2318yes—
117C2×(C2×D3)⋊C4C2314yes—
118C2×(C2×D3)⋊C4C2314yes—
119C2×(C2×D3)⋊C4C2318yes—
120C2×(C2×D3)⋊C4C2314yes—
121C2×(C2×D3)⋊C4C2314yes—
122C2×(C2×D3)⋊C4C2318yes—
123C2×(C2×D3)⋊C4C2318yes—
124C2×(C2×D3)⋊C4C2312yes—
125C2×C4⋊Dic3C2311yescharacteristic
126C2×C4⋊Dic3C2311yescharacteristic
127C2×Dic3⋊C4C2314yes—
128C2×Dic3⋊C4C2312yes—
129C2×Dic3⋊C4C2314yes—
130C2×Dic3⋊C4C2314yes—
131C2×Dic3⋊C4C2314yes—
132C2×C4×Dic3C2312yes—
133C2×C4×Dic3C2311yescharacteristic
134C2×C4×Dic3C2312yes—
135C2×C4×Dic3C2312yes—
136C2×C4×Dic3C2311yescharacteristic
137#896—216no—
138#896—232no—
139#895—28no—
140#895—216no—
141#894—232no—
142#893—28no—
143#888—28no—
144#875—28no—
145#874—28no—
146#874—24no—
147C4×D12—28no—
148C4×D12D418yes—
149#872—28no—
150C4×Dic6—24no—
151C3×#95C2318yes—
152C3×#95C2312yes—
153C3⋊#95C2314yes—
154C3⋊#95C2314yes—
155C3⋊#95C2314yes—
156C22×#41—32no—
157C23×Q8—32no—
158#550—68no—
159D4×Q8—68no—
160#536—616no—
161C2×C4⋊Q8—31no—
162C2×C4⋊Q8—32no—
163C2×C4⋊Q8C2×D311yescharacteristic
164C2×C4⋊Q8C2×D312yes—
165C2×#127—31no—
166C2×#127—32no—
167C2×#124—32no—
168C2×#124—32no—
169C2×#124—32no—
170C2×#123—38no—
171C2×C22⋊Q8—32no—
172C2×C22⋊Q8—38no—
173C2×C22⋊Q8—34no—
174C2×#121—32no—
175C2×#121—34no—
176C2×C4×Q8—31no—
177C2×C4×Q8C2×D311yescharacteristic
178C2×C4×D4—32no—
179C2×C4×D4—31no—
180C2×C4×D4—32no—
181C2×#117—32no—
182#398C2×D314yes—
183#396—38no—
184#394—38no—
185#394—38no—
186#394—34no—
187#393—38no—
188#391—31no—
189#391—31no—
190#391C2×D312yes—
191#390—34no—
192#390—34no—
193#390—32no—
194#390—34no—
195#390—32no—
196#389—34no—
197#389—32no—
198#389C2×D314yes—
199#388—34no—
200#386—38no—
201#385—38no—
202#384—31no—
203C4×C4⋊C4—34no—
204C4×C4⋊C4C2×D311yescharacteristic
205C4×C22⋊C4—32no—
206C43—31no—
207C23×D3C2414yes—
208C23×D3C2412yes—
209C2×C3×Q8—22no—
210C2×C3×Q8—24no—
211C2×C3×Q8—24no—
212C2×C3×Q8—22no—
213C2×C3×Q8—24no—
214C2×C3×Q8C2×D414yes—
215C22×C3×C4C2412yes—
216C22×C3×C4C2414yes—
217C22×C3×C4C2414yes—
218C22×C3×C4C2412yes—
219C22×C3×C4C2411yescharacteristic
220C22×C3×C4C2411yescharacteristic
221C22×C3×C4C2411yescharacteristic
222C22×Dic3C2412yes—
223C22×Dic3C2412yes—
224C22×Dic3C2412yes—
225C22×Dic3C2412yes—
226C22×Dic3C2412yes—
227#239—432no—
228D3×Q8—432no—
229C2×D12—24no—
230C2×D12—28no—
231C2×D12—24no—
232C2×D12—24no—
233C2×D12—24no—
234C2×D12—28no—
235C2×D12—24no—
236C2×D12—28no—
237C2×D12—28no—
238C2×D12—24no—
239C2×D12—44no—
240C2×D12—44no—
241C2×D12—48no—
242C2×D12C2×D414yes—
243C2×C4×D3—24no—
244C2×C4×D3—24no—
245C2×C4×D3—28no—
246C2×C4×D3—28no—
247C2×C4×D3—28no—
248C2×C4×D3—28no—
249C2×C4×D3—44no—
250C2×C4×D3—44no—
251C2×C4×D3C2×D418yes—
252C2×Dic6—22no—
253C2×Dic6—22no—
254C2×Dic6—28no—
255C2×Dic6—28no—
256C2×Dic6—44no—
257C2×Dic6C2×D414yes—
258C3×C4⋊C4—24no—
259C3×C4⋊C4—216no—
260C3×C4⋊C4—24no—
261C3×C4⋊C4—22no—
262C3×C4⋊C4—24no—
263C3×C4⋊C4—24no—
264C3×C42—24no—
265C3×C42—24no—
266C3×C42C2×D414yes—
267(C2×D3)⋊C4—28no—
268(C2×D3)⋊C4—28no—
269(C2×D3)⋊C4—28no—
270(C2×D3)⋊C4—216no—
271(C2×D3)⋊C4—216no—
272(C2×D3)⋊C4—44no—
273(C2×D3)⋊C4—48no—
274(C2×D3)⋊C4—48no—
275(C2×D3)⋊C4C2×D418yes—
276C4⋊Dic3—22no—
277C4⋊Dic3C2×D414yes—
278Dic3⋊C4—24no—
279Dic3⋊C4—28no—
280Dic3⋊C4—28no—
281Dic3⋊C4—44no—
282Dic3⋊C4C2×D4116yes—
283C4×Dic3—24no—
284C4×Dic3—22no—
285C4×Dic3—24no—
286C4×Dic3—42no—
287C4×Dic3C2×D414yes—
288C2×#41—68no—
289C2×#41—68no—
290C2×#41—124no—
291C22×Q8—31no—
292C22×Q8—34no—
293C22×Q8—68no—
294C22×Q8—68no—
295C22×Q8—124no—
296C22×Q8C22×D312yes—
297C22×Q8C22×D311yescharacteristic
298C22×D4—31no—
299C22×D4—32no—
300C22×D4—32no—
301C22×D4—34no—
302C22×D4—34no—
303C22×D4—32no—
304C23×C4—32no—
305C23×C4—34no—
306C23×C4—34no—
307C4⋊Q8—28no—
308C4⋊Q8—24no—
309C4⋊Q8—38no—
310C4⋊Q8—68no—
311#127—68no—
312#127—64no—
313#124—68no—
314#124—68no—
315#124—124no—
316#123—632no—
317C22⋊Q8—616no—
318C22⋊Q8—632no—
319C22⋊Q8—68no—
320#121—616no—
321#121—68no—
322C4×Q8—24no—
323C4×Q8—64no—
324C4×D4—38no—
325C4×D4—68no—
326C4×D4—68no—
327#117—68no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—34no—
330C2×C4⋊C4—34no—
331C2×C4⋊C4—31no—
332C2×C4⋊C4—32no—
333C2×C4⋊C4—34no—
334C2×C4⋊C4—34no—
335C2×C4⋊C4C22×D312yes—
336C2×C4⋊C4C22×D311yescharacteristic
337C2×C4⋊C4C22×D312yes—
338C2×C4⋊C4C22×D312yes—
339C2×C4⋊C4C22×D318yes—
340C2×C4⋊C4C22×D312yes—
341C2×C22⋊C4—32no—
342C2×C22⋊C4—34no—
343C2×C22⋊C4—34no—
344C2×C22⋊C4—34no—
345C2×C22⋊C4—38no—
346C2×C22⋊C4—38no—
347C2×C22⋊C4—38no—
348C2×C22⋊C4—34no—
349C2×C22⋊C4—38no—
350C2×C42—31no—
351C2×C42—32no—
352C2×C42—31no—
353C2×C42—32no—
354C2×C42—32no—
355C2×C42C22×D312yes—
356C2×C42C22×D312yes—
357C2×C42C22×D311yescharacteristic
358#95—34no—
359#95—34no—
360#95—34no—
361#95C22×D312yes—
362#95C22×D318yes—
363C23×C3C2511yescharacteristic
364C22×D3—22no—
365C22×D3—22no—
366C22×D3—24no—
367C22×D3—24no—
368C22×D3—24no—
369C22×D3—24no—
370C22×D3—24no—
371C22×D3—28no—
372C22×D3—28no—
373C22×D3C22×D414yes—
374C3×Q8—28no—
375C3×Q8—48no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—24no—
380C2×C3×C4—24no—
381C2×C3×C4—22no—
382C2×C3×C4—24no—
383C2×C3×C4—24no—
384C2×C3×C4—24no—
385C2×C3×C4—24no—
386C2×C3×C4—24no—
387C2×C3×C4C22×D412yes—
388C2×C3×C4C22×D414yes—
389C2×C3×C4C22×D412yes—
390C2×C3×C4C22×D412yes—
391C2×C3×C4C22×D412yes—
392C2×C3×C4H(5,𝔽2)12yes—
393C2×C3×C42−1+414yes—
394C2×Dic3—22no—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3—22no—
399C2×Dic3—22no—
400C2×Dic3—22no—
401C2×Dic3—22no—
402C2×Dic3—22no—
403C2×Dic3—22no—
404C2×Dic3—24no—
405C2×Dic3C22×D414yes—
406C2×Dic3C22×D414yes—
407C2×Dic3C22×D414yes—
408D12—28no—
409D12—416no—
410D12—48no—
411D12—48no—
412D12—84no—
413D12—88no—
414C4×D3—216no—
415C4×D3—416no—
416C4×D3—416no—
417Dic6—28no—
418Dic6—416no—
419C24—32no—
420C24—34no—
421#41—1232no—
422C2×Q8—22no—
423C2×Q8—24no—
424C2×Q8—24no—
425C2×Q8—22no—
426C2×Q8—24no—
427C2×Q8—34no—
428C2×Q8—68no—
429C2×Q8—68no—
430C2×Q8—62no—
431C2×Q8—62no—
432C2×Q8—124no—
433C2×Q8—1232no—
434C2×Q8D3×D414yes—
435C2×D4—34no—
436C2×D4—68no—
437C2×D4—68no—
438C2×D4—64no—
439C2×D4—64no—
440C2×D4—68no—
441C2×D4—64no—
442C2×D4—64no—
443C2×D4—64no—
444C2×D4—68no—
445C2×D4—64no—
446C2×D4—124no—
447C2×D4—124no—
448C2×D4—128no—
449C22×C4—32no—
450C22×C4—32no—
451C22×C4—32no—
452C22×C4—38no—
453C22×C4—32no—
454C22×C4—32no—
455C22×C4—68no—
456C22×C4—68no—
457C22×C4—68no—
458C22×C4—64no—
459C22×C4—64no—
460C22×C4—68no—
461C22×C4—124no—
462C22×C4—124no—
463C22×C4C23×D312yes—
464C22×C4C23×D314yes—
465C22×C4C23×D314yes—
466C22×C4C23×D312yes—
467C22×C4C23×D311yescharacteristic
468C22×C4C23×D311yescharacteristic
469C22×C4C23×D311yescharacteristic
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—216no—
475C4⋊C4—24no—
476C4⋊C4—34no—
477C4⋊C4—316no—
478C4⋊C4—68no—
479C4⋊C4—68no—
480C4⋊C4—64no—
481C4⋊C4—62no—
482C4⋊C4—124no—
483C22⋊C4—38no—
484C22⋊C4—68no—
485C22⋊C4—68no—
486C22⋊C4—616no—
487C22⋊C4—616no—
488C22⋊C4—68no—
489C22⋊C4—128no—
490C22⋊C4—124no—
491C22⋊C4—128no—
492C42—24no—
493C42—24no—
494C42—34no—
495C42—64no—
496C42—64no—
497C42—62no—
498C42—122no—
499C42D3×D414yes—
500C22×C3C23×D411yescharacteristic
501C22×C3C23×D411yescharacteristic
502C22×C3C2×H(5,𝔽2)11yescharacteristic
503C22×C3C2×2−1+412yes—
504C22×C3#58512yes—
505C2×D3—24no—
506C2×D3—24no—
507C2×D3—24no—
508C2×D3—44no—
509C2×D3—44no—
510C2×D3—44no—
511C2×D3—44no—
512C2×D3—48no—
513C2×D3—44no—
514C2×D3—48no—
515C3×C4—24no—
516C3×C4—22no—
517C3×C4—44no—
518C3×C4—44no—
519C3×C4—42no—
520C3×C4#53614yescomplemented
521C3×C4D4214yes—
522Dic3—24no—
523Dic3—24no—
524Dic3—24no—
525Dic3—42no—
526Dic3—42no—
527C23—34no—
528C23—62no—
529C23—64no—
530C23—62no—
531C23—64no—
532C23—64no—
533C23—64no—
534C23—68no—
535C23—64no—
536C23—68no—
537C23C24×D311yescharacteristic
538Q8—28no—
539Q8—48no—
540Q8—68no—
541Q8—1216no—
542D4—68no—
543D4—1216no—
544D4—128no—
545D4—128no—
546D4—244no—
547D4—248no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—34no—
560C2×C4—34no—
561C2×C4—34no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—62no—
566C2×C4—62no—
567C2×C4—62no—
568C2×C4—616no—
569C2×C4—62no—
570C2×C4—62no—
571C2×C4—62no—
572C2×C4—62no—
573C2×C4—64no—
574C2×C4—1216no—
575C2×C4—1216no—
576C2×C4C2×D3×D412yes—
577C2×C4C2×D3×D414yes—
578C2×C4C2×D3×D412yes—
579C2×C4C2×D3×D412yes—
580C2×C4C2×D3×D412yes—
581C2×C4#100714yes—
582C2×C4#100912yes—
583C2×C3C2×#53611yescharacteristic
584C2×C3C2×D4211yescharacteristic
585C2×C3#349712yes—
586C2×C3#350011yescharacteristic
587C2×C3#355412yes—
588D3—44no—
589D3—84no—
590C22—64no—
591C22—64no—
592C22—64no—
593C22—124no—
594C22—124no—
595C22—124no—
596C22—124no—
597C22—124no—
598C22—128no—
599C22—128no—
600C22C22×D3×D411yescharacteristic
601C22C22×D3×D411yescharacteristic
602C22C2×#100712yes—
603C22C2×#100911yescharacteristic
604C22#596812yes—
605C4—22no—
606C4—24no—
607C4—42no—
608C4—44no—
609C4—44no—
610C4—64no—
611C4—64no—
612C4—64no—
613C4—122no—
614C4—122no—
615C4#560914yes—
616C4#572414yescomplemented
617C3#3936311yescharacteristic, complemented, Sylow
618C2—124no—
619C2—244no—
620C2C2×#560911yescharacteristic
621C2C2×#572411yescharacteristic
622C2#8634111yescharacteristic
623C2#8676012yes—
624C2#8703512yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2546C2
prime radicalC2×#5577C2
hypercenter#2546D3
hypocenterC3#39363
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2546

Complements

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

NH ≅ G/Ntype
1. C2×#5727C2semidirect
2. C2×#5724C2semidirect
4. C3×#2546C2semidirect
5. #83255C2semidirect
6. #83251C2semidirect
7. C4×#944C2semidirect
8. #77970C2semidirect
9. #76640C2semidirect
10. #76500C2semidirect
11. #76010C2semidirect
12. #76004C2semidirect
13. #73712C2semidirect
14. #73617C2semidirect
74. #2546D3semidirect
115. #944D4semidirect
520. C3×C4#536semidirect
616. C4#5724semidirect
617. C3#39363semidirect

Maximal subgroups

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

subgroupindex
C2×#57272
C2×#57242
C2×#55772
C3×#25462
#832552
#832512
C4×#9442
#779702
#766402
#765002
#760102
#760042
#737122
#736172
#393633