Finite Group Databasebeta

← #11069491

Subgroups — #11069491

Order 768.

2078 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
1C2×#5682C212yesmaximal, complemented
2C3×C4×#124C211yesmaximal, characteristic, complemented
3#82499C211yesmaximal, characteristic, complemented
4#82460C211yesmaximal, characteristic, complemented
5#82416C211yesmaximal, characteristic
6#82333C211yesmaximal, characteristic, complemented
7#76622C212yesmaximal, complemented
8#76440C212yesmaximal, complemented
9#75690C214yesmaximal, complemented
10#75671C214yesmaximal, complemented
11#74194C212yesmaximal, complemented
12#73527C214yesmaximal, complemented
13#73504C214yesmaximal, complemented
14#73308C211yesmaximal, characteristic, complemented
15#73299C211yesmaximal, characteristic, complemented
16#39374—31nomaximal, Sylow
17C2×C3×#124C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×#946C2212yes—
21C2×Q8×Dic3C2212yes—
22C2×#938C2212yes—
23C2×D4×Dic3C2212yes—
24#5682—216no—
25C2×#886C2214yes—
26C2×#885C2214yes—
27C2×#879C2214yes—
28C2×#878C2214yes—
29C2×C4×D12C2211yescharacteristic
30C2×#872C2212yes—
31C2×#869C2212yes—
32C3×#391C2211yescharacteristic
33C3×#390C2211yescharacteristic
34C3×#388C2214yes—
35C3×#382C2211yescharacteristic
36C3×C4×C22⋊C4C2214yes—
37C3×C43C2211yescharacteristic
38#5213C2214yes—
39#5212C2212yes—
40C2×C3⋊#95C2212yes—
41#4991C2212yes—
42#4987C2212yes—
43#4987C2212yes—
44#4984C2214yes—
45#4984C2212yes—
46#4984C2212yes—
47#4981C2212yes—
48#4961C2214yes—
49#4960C2212yes—
50#4958C2214yes—
51#4953C2214yes—
52#4953C2212yes—
53#4950C2214yes—
54#4943C2214yes—
55#4940C2212yes—
56#4940C2214yes—
57#4940C2212yes—
58#4939C2212yes—
59C4×(C2×D3)⋊C4C2212yes—
60#4936C2212yes—
61#4936C2211yescharacteristic
62#4936C2211yescharacteristic
63#4936C2212yes—
64C4×C4⋊Dic3C2212yes—
65C4×C4⋊Dic3C2211yescharacteristic
66#4929C2214yes—
67#4929C2214yes—
68#4928C2212yes—
69#4677C2214yes—
70#4674C2214yes—
71#4673C2214yes—
72#4670C2214yes—
73#4659C2214yes—
74#4659C2214yes—
75#4656C2214yes—
76#4653C2214yes—
77C2×#542—32no—
78#2862—31no—
79#2710—34no—
80#2689—34no—
81#2628—32no—
82#2536—31no—
83#2535—31no—
84#2521—34no—
85#2488—34no—
86#2421—32no—
87#2420—32no—
88#2351—31no—
89#2349—31no—
90C4×#124D311yescharacteristic, complemented
91#2293—31no—
92C22×C3×Q8C2311yescharacteristic
93C22×C3×D4C2311yescharacteristic
94C23×C3×C4C2312yes—
95C22×#67C2314yes—
96C23×Dic3C2314yes—
97C22×D12C2311yescharacteristic
98C22×C4×D3C2312yes—
99C22×Dic6C2314yes—
100C3×#124D418yescomplemented
101C3×C4×Q8—24no—
102C3×C4×D4—24no—
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C22⋊C4C2312yes—
107C2×C3×C22⋊C4C2314yes—
108C2×C3×C22⋊C4C2314yes—
109C2×C3×C42C2312yes—
110C2×C3×C42C2311yescharacteristic
111C2×C3×C42C2312yes—
112C2×C3×C42C2312yes—
113C2×C3×C42C2311yescharacteristic
114C2×C3×C42C2311yescharacteristic
115C2×#217C2314yes—
116C2×#217C2314yes—
117#946—28no—
118Q8×Dic3—28no—
119#938—28no—
120D4×Dic3—28no—
121C2×(C2×D3)⋊C4C2314yes—
122C2×(C2×D3)⋊C4C2312yes—
123C2×(C2×D3)⋊C4C2312yes—
124C2×(C2×D3)⋊C4C2312yes—
125C2×(C2×D3)⋊C4C2312yes—
126C2×(C2×D3)⋊C4C2314yes—
127C2×(C2×D3)⋊C4C2314yes—
128C2×(C2×D3)⋊C4C2312yes—
129C2×C4⋊Dic3C2314yes—
130C2×C4⋊Dic3C2314yes—
131C2×C4⋊Dic3C2311yescharacteristic
132C2×C4⋊Dic3C2312yes—
133C2×C4⋊Dic3C2312yes—
134C2×C4⋊Dic3C2314yes—
135C2×C4⋊Dic3C2314yes—
136C2×C4⋊Dic3C2312yes—
137C2×Dic3⋊C4C2314yes—
138C2×Dic3⋊C4C2312yes—
139C2×Dic3⋊C4C2312yes—
140C2×C4×Dic3C2314yes—
141C2×C4×Dic3C2312yes—
142#886—216no—
143#885—216no—
144#879—216no—
145#878—216no—
146C4×D12—24no—
147#872—28no—
148#869—28no—
149C3×#95C2314yes—
150C3×#95C2314yes—
151C3⋊#95C2312yes—
152C3⋊#95C2314yes—
153C3⋊#95C2314yes—
154C3⋊#95C2314yes—
155C3⋊#95C2312yes—
156C3⋊#95C2314yes—
157C3⋊#95C2314yes—
158C3⋊#95C2314yes—
159C3⋊#95C2312yes—
160C3⋊#95C2312yes—
161C3⋊#95C2312yes—
162C3⋊#95C2312yes—
163C3⋊#95C2311yescharacteristic
164C3⋊#95C2311yescharacteristic
165#542—616no—
166C2×C4⋊Q8—32no—
167C2×#126—34no—
168C2×#124—34no—
169C2×#124C2×D311yescharacteristic
170C2×#123—34no—
171C2×C22⋊Q8—32no—
172C2×C22⋊Q8—34no—
173C2×#121—32no—
174C2×C4×Q8—32no—
175C2×C4×Q8C2×D311yescharacteristic
176C2×C4×D4—32no—
177C2×C4×D4—31no—
178C2×C4×D4C2×D311yescharacteristic
179C2×#117—32no—
180#400—34no—
181#400—34no—
182#399—34no—
183#398—32no—
184#397—34no—
185#397—32no—
186#397—34no—
187#395—34no—
188#394—32no—
189#394—34no—
190#393—34no—
191#392—32no—
192#391—34no—
193#391—34no—
194#391C2×D311yescharacteristic
195#390—32no—
196#390C2×D311yescharacteristic
197#389—32no—
198#389—32no—
199#389—34no—
200#389—32no—
201#388—32no—
202#388—34no—
203#388—32no—
204#388—34no—
205#388C2×D314yes—
206#387—34no—
207#386—34no—
208#386—34no—
209#385—34no—
210#382—31no—
211#382—31no—
212#382—32no—
213#382—32no—
214#382C2×D311yescharacteristic
215#381—34no—
216#380—32no—
217#380—32no—
218#380—32no—
219C4×C4⋊C4—31no—
220C4×C4⋊C4—32no—
221C4×C22⋊C4—32no—
222C4×C22⋊C4C2×D314yes—
223C2×#95—32no—
224C43C2×D311yescharacteristic
225C24×C3C2412yes—
226C23×D3C2412yes—
227C2×C3×Q8—22no—
228C2×C3×Q8—22no—
229C2×C3×Q8C2×D414yes—
230C2×C3×D4—22no—
231C2×C3×D4—22no—
232C2×C3×D4C2×D414yes—
233C22×C3×C4—24no—
234C22×C3×C4—44no—
235C22×C3×C4C2412yes—
236C22×C3×C4C2412yes—
237C22×C3×C4C2412yes—
238C22×C3×C4C2412yes—
239C22×C3×C4C2412yes—
240C22×C3×C4C2411yescharacteristic
241C22×C3×C4C2411yescharacteristic
242C22×C3×C4C2411yescharacteristic
243C2×#67—28no—
244C2×#67—44no—
245C2×#67—44no—
246C22×Dic3—28no—
247C22×Dic3—44no—
248C22×Dic3—44no—
249C22×Dic3C2412yes—
250C22×Dic3C2414yes—
251C22×Dic3C2412yes—
252C22×Dic3C2414yes—
253C22×Dic3C2412yes—
254C2×D12—22no—
255C2×D12—42no—
256C2×C4×D3—24no—
257C2×C4×D3—24no—
258C2×C4×D3—24no—
259C2×Dic6—28no—
260C2×Dic6—44no—
261C2×Dic6—44no—
262C3×C4⋊C4—22no—
263C3×C4⋊C4—24no—
264C3×C4⋊C4—22no—
265C3×C22⋊C4—24no—
266C3×C22⋊C4—44no—
267C3×C22⋊C4C2×D4116yes—
268C3×C42—24no—
269C3×C42—22no—
270C3×C42—42no—
271C3×C42—42no—
272C3×C42C2×D414yes—
273C3×C42#4114yes—
274#217—28no—
275#217—28no—
276(C2×D3)⋊C4—24no—
277(C2×D3)⋊C4—24no—
278(C2×D3)⋊C4—28no—
279(C2×D3)⋊C4—24no—
280(C2×D3)⋊C4—42no—
281(C2×D3)⋊C4—44no—
282(C2×D3)⋊C4—44no—
283(C2×D3)⋊C4—42no—
284C4⋊Dic3—28no—
285C4⋊Dic3—24no—
286C4⋊Dic3—24no—
287C4⋊Dic3—28no—
288C4⋊Dic3—22no—
289C4⋊Dic3—44no—
290C4⋊Dic3—42no—
291C4⋊Dic3—44no—
292Dic3⋊C4—24no—
293Dic3⋊C4—24no—
294Dic3⋊C4—28no—
295C4×Dic3—24no—
296C4×Dic3—28no—
297C22×Q8—34no—
298C22×Q8C22×D311yescharacteristic
299C22×D4—34no—
300C22×D4—31no—
301C22×D4C22×D311yescharacteristic
302C23×C4—34no—
303C23×C4—32no—
304C23×C4C22×D312yes—
305C4⋊Q8—68no—
306#126—616no—
307#124—616no—
308#124D1218yescomplemented
309#123—616no—
310C22⋊Q8—68no—
311C22⋊Q8—616no—
312#121—68no—
313C4×Q8—24no—
314C4×Q8—68no—
315C4×D4—24no—
316C4×D4—68no—
317C4×D4—64no—
318#117—68no—
319C2×C4⋊C4—32no—
320C2×C4⋊C4—32no—
321C2×C4⋊C4—32no—
322C2×C4⋊C4—31no—
323C2×C4⋊C4—32no—
324C2×C4⋊C4—34no—
325C2×C4⋊C4—34no—
326C2×C4⋊C4—34no—
327C2×C4⋊C4—32no—
328C2×C4⋊C4—34no—
329C2×C4⋊C4—34no—
330C2×C4⋊C4C22×D311yescharacteristic
331C2×C4⋊C4C22×D311yescharacteristic
332C2×C4⋊C4C22×D312yes—
333C2×C22⋊C4—34no—
334C2×C22⋊C4—34no—
335C2×C22⋊C4—32no—
336C2×C22⋊C4—32no—
337C2×C22⋊C4—32no—
338C2×C22⋊C4—34no—
339C2×C22⋊C4—32no—
340C2×C22⋊C4—32no—
341C2×C22⋊C4—34no—
342C2×C22⋊C4—34no—
343C2×C22⋊C4C22×D312yes—
344C2×C22⋊C4C22×D314yes—
345C2×C22⋊C4C22×D314yes—
346C2×C42—32no—
347C2×C42—34no—
348C2×C42C22×D312yes—
349C2×C42C22×D311yescharacteristic
350C2×C42C22×D312yes—
351C2×C42C22×D312yes—
352C2×C42C22×D311yescharacteristic
353C2×C42C22×D311yescharacteristic
354#95—31no—
355#95—32no—
356#95—34no—
357#95—32no—
358#95—32no—
359#95—31no—
360#95—32no—
361#95—34no—
362#95—34no—
363#95—34no—
364#95—34no—
365#95—32no—
366#95—34no—
367#95—32no—
368#95C22×D314yes—
369#95C22×D314yes—
370C23×C3—22no—
371C23×C3—22no—
372C23×C3—24no—
373C23×C3—24no—
374C23×C3C22×D414yes—
375C23×C3C2511yescharacteristic
376C22×D3—22no—
377C22×D3—22no—
378C22×D3—22no—
379C22×D3—22no—
380C22×D3—22no—
381C22×D3—22no—
382C22×D3—22no—
383C3×Q8—28no—
384C3×D4—28no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—24no—
395C2×C3×C4—24no—
396C2×C3×C4—22no—
397C2×C3×C4—22no—
398C2×C3×C4—84no—
399C2×C3×C4C22×D414yes—
400C2×C3×C4C22×D414yes—
401C2×C3×C4C22×D412yes—
402C2×C3×C4C2×#4112yes—
403C2×C3×C4C2×#4112yes—
404C2×C3×C4C2×#4112yes—
405C2×C3×C42−1+412yes—
406#67—88no—
407C2×Dic3—22no—
408C2×Dic3—22no—
409C2×Dic3—24no—
410C2×Dic3—22no—
411C2×Dic3—24no—
412C2×Dic3—22no—
413C2×Dic3—22no—
414C2×Dic3—22no—
415C2×Dic3—22no—
416C2×Dic3—24no—
417C2×Dic3—24no—
418C2×Dic3—22no—
419C2×Dic3—24no—
420C2×Dic3—24no—
421C2×Dic3—22no—
422C2×Dic3—88no—
423D12—44no—
424C4×D3—48no—
425Dic6—88no—
426C24—32no—
427C24C23×D312yes—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—68no—
431C2×Q8—124no—
432C2×Q8—124no—
433C2×Q8C2×D1214yes—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—68no—
437C2×D4—62no—
438C2×D4—122no—
439C2×D4—124no—
440C2×D4—124no—
441C2×D4C2×D1214yes—
442C22×C4—24no—
443C22×C4—32no—
444C22×C4—34no—
445C22×C4—32no—
446C22×C4—34no—
447C22×C4—32no—
448C22×C4—44no—
449C22×C4—68no—
450C22×C4—64no—
451C22×C4—64no—
452C22×C4—64no—
453C22×C4—124no—
454C22×C4—124no—
455C22×C4C23×D312yes—
456C22×C4C23×D312yes—
457C22×C4C23×D312yes—
458C22×C4C23×D312yes—
459C22×C4C23×D312yes—
460C22×C4C23×D311yescharacteristic
461C22×C4C23×D311yescharacteristic
462C22×C4C23×D311yescharacteristic
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—64no—
467C4⋊C4—64no—
468C4⋊C4—64no—
469C4⋊C4—64no—
470C4⋊C4—68no—
471C4⋊C4—68no—
472C4⋊C4—68no—
473C4⋊C4—62no—
474C4⋊C4—122no—
475C4⋊C4—124no—
476C4⋊C4—124no—
477C22⋊C4—24no—
478C22⋊C4—44no—
479C22⋊C4—68no—
480C22⋊C4—68no—
481C22⋊C4—64no—
482C22⋊C4—64no—
483C22⋊C4—68no—
484C22⋊C4—64no—
485C22⋊C4—122no—
486C22⋊C4—122no—
487C22⋊C4—124no—
488C22⋊C4—124no—
489C22⋊C4C2×D12116yes—
490C42—24no—
491C42—22no—
492C42—42no—
493C42—42no—
494C42—64no—
495C42—68no—
496C42C2×D1214yes—
497C42#23714yes—
498C22×C3—24no—
499C22×C3—28no—
500C22×C3—42no—
501C22×C3—42no—
502C22×C3—44no—
503C22×C3C23×D411yescharacteristic
504C22×C3C22×#4111yescharacteristic
505C22×C3C2×2−1+411yescharacteristic
506C22×C3#58512yes—
507C22×C3#58512yes—
508C2×D3—42no—
509C2×D3—42no—
510C2×D3—42no—
511C2×D3—42no—
512C2×D3—42no—
513C2×D3—42no—
514C2×D3—42no—
515C3×C4—22no—
516C3×C4—24no—
517C3×C4—24no—
518C3×C4—22no—
519C3×C4—42no—
520C3×C4—42no—
521C3×C4—42no—
522C3×C4#54714yes—
523Dic3—42no—
524Dic3—44no—
525Dic3—44no—
526Dic3—42no—
527Dic3—42no—
528C23—22no—
529C23—22no—
530C23—24no—
531C23—24no—
532C23—62no—
533C23—62no—
534C23—62no—
535C23—62no—
536C23—62no—
537C23—62no—
538C23—62no—
539C23C22×D1214yes—
540C23C24×D311yescharacteristic
541Q8—28no—
542Q8—248no—
543D4—28no—
544D4—124no—
545D4—248no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—62no—
560C2×C4—62no—
561C2×C4—64no—
562C2×C4—62no—
563C2×C4—64no—
564C2×C4—62no—
565C2×C4—62no—
566C2×C4—62no—
567C2×C4—62no—
568C2×C4—64no—
569C2×C4—64no—
570C2×C4—62no—
571C2×C4—62no—
572C2×C4—64no—
573C2×C4—64no—
574C2×C4—84no—
575C2×C4—128no—
576C2×C4—248no—
577C2×C4C22×D1214yes—
578C2×C4C22×D1214yes—
579C2×C4C22×D1212yes—
580C2×C4C2×#23712yes—
581C2×C4C2×#23712yes—
582C2×C4C2×#23712yes—
583C2×C4#101012yes—
584C2×C3—44no—
585C2×C3#345511yescharacteristic
586C2×C3C2×#54711yescharacteristic
587C2×C3#350012yes—
588C2×C3#351011yescharacteristic
589C2×C3#356512yes—
590D3—82no—
591C22—28no—
592C22—24no—
593C22—42no—
594C22—44no—
595C22—42no—
596C22—122no—
597C22—122no—
598C22—122no—
599C22—122no—
600C22—122no—
601C22—122no—
602C22—122no—
603C22C23×D1211yescharacteristic
604C22C22×#23711yescharacteristic
605C22C2×#101011yescharacteristic
606C22#596412yes—
607C22#596412yes—
608C4—22no—
609C4—24no—
610C4—22no—
611C4—24no—
612C4—42no—
613C4—42no—
614C4—42no—
615C4—122no—
616C4—124no—
617C4—124no—
618C4—122no—
619C4—122no—
620C4#555514yes—
621C3#3937411yescharacteristic, complemented, Sylow
622C2—44no—
623C2—242no—
624C2C2×#555511yescharacteristic
625C2#8584712yes—
626C2#8606511yescharacteristic
627C2#8618212yes—
628C2#8623611yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×C4×#124C2
prime radical#82416C2
hypercenterC4×#124D3
hypocenterC3#39374
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
1. C2×#5682C2semidirect
2. C3×C4×#124C2semidirect
3. #82499C2semidirect
4. #82460C2semidirect
6. #82333C2semidirect
7. #76622C2semidirect
8. #76440C2semidirect
9. #75690C2semidirect
10. #75671C2semidirect
11. #74194C2semidirect
12. #73527C2semidirect
13. #73504C2semidirect
14. #73308C2semidirect
15. #73299C2semidirect
90. C4×#124D3semidirect
100. C3×#124D4semidirect
308. #124D12semidirect
621. C3#39374semidirect

Maximal subgroups

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

subgroupindex
C2×#56822
C3×C4×#1242
#824992
#824602
#824162
#823332
#766222
#764402
#756902
#756712
#741942
#735272
#735042
#733082
#732992
#393743