Finite Group Databasebeta

← #11069488

Subgroups — #11069488

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×#5561C212yesmaximal, complemented
2C3×#2349C211yesmaximal, characteristic, complemented
3#82416C211yesmaximal, characteristic
4#82405C212yesmaximal, complemented
5C4×#936C211yesmaximal, characteristic, complemented
6#76250C212yesmaximal, complemented
7#76214C212yesmaximal, complemented
8#74526C214yesmaximal, complemented
9#74513C214yesmaximal, complemented
10#74240C211yesmaximal, characteristic, complemented
11#74221C211yesmaximal, characteristic, complemented
12#73929C214yesmaximal, complemented
13#73637C211yesmaximal, characteristic, complemented
14#73617C211yesmaximal, characteristic, complemented
15#73352C214yesmaximal, complemented
16#39374—31nomaximal, Sylow
17C2×C3×C4×D4C2212yes—
18C2×C3×C4×D4C2211yescharacteristic
19C2×#938C2212yes—
20C2×#936C2211yescharacteristic
21C2×#929C2214yes—
22C2×#924C2214yes—
23#5561—216no—
24C2×#897C2212yes—
25C2×#888C2212yes—
26C2×#885C2214yes—
27C2×#879C2214yes—
28C2×C4×D12C2211yescharacteristic
29C2×#872C2212yes—
30C2×C4×Dic6C2211yescharacteristic
31C2×C4×Dic6C2212yes—
32C3×#389C2212yes—
33C3×#385C2214yes—
34C3×#382C2211yescharacteristic
35C3×#380C2212yes—
36C3×C4×C4⋊C4C2211yescharacteristic
37C2×C3×#95C2212yes—
38#5221C2211yescharacteristic
39#5213C2214yes—
40#5213C2214yes—
41#5211C2214yes—
42C4×#217C2214yes—
43#4993C2214yes—
44#4990C2214yes—
45#4988C2212yes—
46#4987C2212yes—
47#4987C2212yes—
48#4980C2214yes—
49#4979C2214yes—
50#4977C2211yescharacteristic
51#4977C2212yes—
52#4976C2214yes—
53#4973C2211yescharacteristic
54#4972C2212yes—
55#4960C2212yes—
56#4959C2214yes—
57#4953C2212yes—
58#4952C2214yes—
59#4945C2214yes—
60C4×(C2×D3)⋊C4C2212yes—
61#4936C2211yescharacteristic
62C42×Dic3C2211yescharacteristic
63#4928C2212yes—
64#4675C2214yes—
65#4673C2214yes—
66#4672C2212yes—
67#4670C2214yes—
68#4669C2212yes—
69#4669C2212yes—
70#4661C2212yes—
71#4660C2214yes—
72#4658C2214yes—
73#4650C2212yes—
74#4649C2212yes—
75#4646C2214yes—
76#4646C2214yes—
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#2349D311yescharacteristic, complemented
90C4×#124—31no—
91#2293—31no—
92C22×C3×D4C2311yescharacteristic
93C23×C3×C4C2312yes—
94C23×C3×C4C2314yes—
95C22×#67C2314yes—
96C22×D12C2311yescharacteristic
97C22×C4×D3C2312yes—
98C22×Dic6C2311yescharacteristic
99C22×Dic6C2314yes—
100C3×C4×D4—28no—
101C3×C4×D4—24no—
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C4⋊C4C2312yes—
105C2×C3×C4⋊C4C2314yes—
106C2×C3×C22⋊C4C2312yes—
107C2×C3×C22⋊C4C2314yes—
108C2×C3×C42C2312yes—
109C2×C3×C42C2311yescharacteristic
110C2×#217C2314yes—
111C2×#217C2314yes—
112C2×#217C2314yes—
113#938—28no—
114#936D418yescomplemented
115#929—216no—
116C2×(C2×D3)⋊C4C2312yes—
117C2×(C2×D3)⋊C4C2312yes—
118C2×(C2×D3)⋊C4C2312yes—
119C2×(C2×D3)⋊C4C2314yes—
120C2×(C2×D3)⋊C4C2312yes—
121C2×(C2×D3)⋊C4C2312yes—
122C2×(C2×D3)⋊C4C2314yes—
123C2×(C2×D3)⋊C4C2314yes—
124C2×C4⋊Dic3C2314yes—
125C2×C4⋊Dic3C2311yescharacteristic
126C2×C4⋊Dic3C2311yescharacteristic
127C2×C4⋊Dic3C2312yes—
128#924—216no—
129C2×Dic3⋊C4C2314yes—
130C2×Dic3⋊C4C2314yes—
131C2×Dic3⋊C4C2312yes—
132C2×Dic3⋊C4C2312yes—
133C2×Dic3⋊C4C2312yes—
134C2×Dic3⋊C4C2314yes—
135C2×C4×Dic3C2312yes—
136C2×C4×Dic3C2314yes—
137C2×C4×Dic3C2311yescharacteristic
138C2×C4×Dic3C2312yes—
139C2×C4×Dic3C2312yes—
140C2×C4×Dic3C2311yescharacteristic
141#897—28no—
142#888—28no—
143#885—216no—
144#879—216no—
145C4×D12—24no—
146#872—28no—
147C4×Dic6—28no—
148C4×Dic6—24no—
149C3×#95C2312yes—
150C3×#95C2312yes—
151C3×#95C2311yescharacteristic
152C3×#95C2311yescharacteristic
153C3×#95C2314yes—
154C3⋊#95C2312yes—
155C3⋊#95C2314yes—
156C3⋊#95C2314yes—
157C3⋊#95C2312yes—
158C3⋊#95C2314yes—
159C3⋊#95C2314yes—
160C3⋊#95C2314yes—
161C3⋊#95C2314yes—
162C3⋊#95C2312yes—
163C3⋊#95C2314yes—
164C3⋊#95C2312yes—
165#542—616no—
166C2×C4⋊Q8—32no—
167C2×#126—34no—
168C2×#124—31no—
169C2×#124—34no—
170C2×#123—34no—
171C2×C22⋊Q8—34no—
172C2×C22⋊Q8—32no—
173C2×#121—32no—
174C2×C4×Q8—32no—
175C2×C4×Q8—31no—
176C2×C4×D4—31no—
177C2×C4×D4C2×D312yes—
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—34no—
189#394—32no—
190#393—34no—
191#392—32no—
192#391—31no—
193#391—34no—
194#391—34no—
195#390—31no—
196#390—32no—
197#389—34no—
198#389—32no—
199#389—32no—
200#389C2×D312yes—
201#388—34no—
202#388—34no—
203#388—32no—
204#388—32no—
205#388—34no—
206#387—34no—
207#386—34no—
208#386—34no—
209#385C2×D314yes—
210#382—32no—
211#382—31no—
212#382—31no—
213#382—32no—
214#382C2×D311yescharacteristic
215#381—34no—
216#380—32no—
217#380—32no—
218#380C2×D312yes—
219C4×C4⋊C4—32no—
220C4×C4⋊C4C2×D311yescharacteristic
221C4×C22⋊C4—34no—
222C4×C22⋊C4—32no—
223C2×#95C2×D312yes—
224C43—31no—
225C24×C3C2412yes—
226C23×D3C2412yes—
227C2×C3×D4—22no—
228C2×C3×D4—22no—
229C2×C3×D4C2×D414yes—
230C22×C3×C4—28no—
231C22×C3×C4—24no—
232C22×C3×C4—44no—
233C22×C3×C4—44no—
234C22×C3×C4—44no—
235C22×C3×C4C2412yes—
236C22×C3×C4C2412yes—
237C22×C3×C4C2414yes—
238C22×C3×C4C2412yes—
239C22×C3×C4C2411yescharacteristic
240C22×C3×C4C2411yescharacteristic
241C22×C3×C4C2411yescharacteristic
242C2×#67—28no—
243C2×#67—44no—
244C2×#67—44no—
245C22×Dic3C2412yes—
246C22×Dic3C2412yes—
247C22×Dic3C2414yes—
248C22×Dic3C2412yes—
249C22×Dic3C2412yes—
250C22×Dic3C2412yes—
251C2×D12—22no—
252C2×D12—42no—
253C2×C4×D3—24no—
254C2×C4×D3—24no—
255C2×C4×D3—24no—
256C2×Dic6—22no—
257C2×Dic6—22no—
258C2×Dic6—28no—
259C2×Dic6—44no—
260C2×Dic6—44no—
261C2×Dic6C2×D414yes—
262C3×C4⋊C4—24no—
263C3×C4⋊C4—22no—
264C3×C4⋊C4—42no—
265C3×C4⋊C4—44no—
266C3×C22⋊C4—28no—
267C3×C22⋊C4—24no—
268C3×C42—24no—
269C3×C42#4114yes—
270#217—28no—
271#217—44no—
272#217C2×D4116yes—
273(C2×D3)⋊C4—24no—
274(C2×D3)⋊C4—24no—
275(C2×D3)⋊C4—24no—
276(C2×D3)⋊C4—28no—
277(C2×D3)⋊C4—42no—
278(C2×D3)⋊C4—42no—
279(C2×D3)⋊C4—44no—
280(C2×D3)⋊C4—44no—
281C4⋊Dic3—24no—
282C4⋊Dic3—22no—
283C4⋊Dic3—22no—
284C4⋊Dic3—28no—
285Dic3⋊C4—24no—
286Dic3⋊C4—28no—
287Dic3⋊C4—24no—
288Dic3⋊C4—24no—
289Dic3⋊C4—28no—
290Dic3⋊C4—44no—
291C4×Dic3—24no—
292C4×Dic3—22no—
293C4×Dic3—28no—
294C4×Dic3—42no—
295C4×Dic3—42no—
296C4×Dic3C2×D414yes—
297C22×Q8—31no—
298C22×Q8—34no—
299C22×D4—34no—
300C22×D4—31no—
301C22×D4C22×D311yescharacteristic
302C23×C4—32no—
303C23×C4C22×D314yes—
304C23×C4C22×D312yes—
305C4⋊Q8—68no—
306#126—616no—
307#124—38no—
308#124—616no—
309#123—616no—
310C22⋊Q8—616no—
311C22⋊Q8—68no—
312#121—68no—
313C4×Q8—68no—
314C4×Q8—64no—
315C4×D4—28no—
316C4×D4—24no—
317C4×D4—64no—
318#117—68no—
319C2×C4⋊C4—32no—
320C2×C4⋊C4—32no—
321C2×C4⋊C4—31no—
322C2×C4⋊C4—32no—
323C2×C4⋊C4—34no—
324C2×C4⋊C4—31no—
325C2×C4⋊C4—32no—
326C2×C4⋊C4—34no—
327C2×C4⋊C4—34no—
328C2×C4⋊C4—34no—
329C2×C4⋊C4C22×D312yes—
330C2×C4⋊C4C22×D311yescharacteristic
331C2×C4⋊C4C22×D312yes—
332C2×C4⋊C4C22×D314yes—
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—34no—
340C2×C22⋊C4—32no—
341C2×C22⋊C4—32no—
342C2×C22⋊C4—34no—
343C2×C22⋊C4—34no—
344C2×C22⋊C4C22×D314yes—
345C2×C22⋊C4C22×D312yes—
346C2×C42—31no—
347C2×C42—34no—
348C2×C42—32no—
349C2×C42—31no—
350C2×C42—32no—
351C2×C42—32no—
352C2×C42C22×D312yes—
353C2×C42C22×D311yescharacteristic
354#95—32no—
355#95—34no—
356#95—32no—
357#95—34no—
358#95—32no—
359#95—34no—
360#95—34no—
361#95—34no—
362#95—34no—
363#95—34no—
364#95—32no—
365#95C22×D311yescharacteristic
366#95C22×D312yes—
367#95C22×D314yes—
368#95C22×D311yescharacteristic
369#95C22×D312yes—
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×D4—28no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—24no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—24no—
393C2×C3×C4—24no—
394C2×C3×C4—22no—
395C2×C3×C4—24no—
396C2×C3×C4—22no—
397C2×C3×C4—84no—
398C2×C3×C4—88no—
399C2×C3×C4C22×D412yes—
400C2×C3×C4C2×#4112yes—
401C2×C3×C4C2×#4112yes—
402C2×C3×C4C2×#4112yes—
403C2×C3×C42−1+412yes—
404#67—88no—
405C2×Dic3—22no—
406C2×Dic3—22no—
407C2×Dic3—22no—
408C2×Dic3—22no—
409C2×Dic3—24no—
410C2×Dic3—22no—
411C2×Dic3—22no—
412C2×Dic3—22no—
413C2×Dic3—22no—
414C2×Dic3—22no—
415C2×Dic3—22no—
416C2×Dic3—22no—
417C2×Dic3—24no—
418C2×Dic3—24no—
419C2×Dic3—24no—
420C2×Dic3C22×D414yes—
421C2×Dic3C22×D414yes—
422D12—44no—
423C4×D3—48no—
424Dic6—28no—
425Dic6—88no—
426C24—32no—
427C24C23×D312yes—
428C2×Q8—34no—
429C2×Q8—68no—
430C2×Q8—62no—
431C2×Q8—62no—
432C2×Q8—124no—
433C2×Q8—124no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—68no—
437C2×D4—62no—
438C2×D4—122no—
439C2×D4—124no—
440C2×D4—124no—
441C2×D4D3×D414yes—
442C22×C4—28no—
443C22×C4—24no—
444C22×C4—32no—
445C22×C4—32no—
446C22×C4—32no—
447C22×C4—32no—
448C22×C4—34no—
449C22×C4—32no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—64no—
454C22×C4—64no—
455C22×C4—64no—
456C22×C4C23×D312yes—
457C22×C4C23×D312yes—
458C22×C4C23×D314yes—
459C22×C4C23×D312yes—
460C22×C4C23×D311yescharacteristic
461C22×C4C23×D311yescharacteristic
462C22×C4C23×D311yescharacteristic
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—64no—
468C4⋊C4—64no—
469C4⋊C4—68no—
470C4⋊C4—68no—
471C4⋊C4—68no—
472C4⋊C4—62no—
473C4⋊C4—64no—
474C4⋊C4—62no—
475C4⋊C4—64no—
476C4⋊C4—124no—
477C22⋊C4—28no—
478C22⋊C4—24no—
479C22⋊C4—316no—
480C22⋊C4—64no—
481C22⋊C4—64no—
482C22⋊C4—68no—
483C22⋊C4—68no—
484C22⋊C4—64no—
485C22⋊C4—124no—
486C22⋊C4—122no—
487C22⋊C4—124no—
488C22⋊C4—122no—
489C22⋊C4—124no—
490C42—24no—
491C42—34no—
492C42—68no—
493C42—64no—
494C42—62no—
495C42—122no—
496C42—122no—
497C42#23514yes—
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—22no—
517C3×C4—44no—
518C3×C4—42no—
519C3×C4—42no—
520C3×C4—42no—
521C3×C4#54714yes—
522Dic3—24no—
523Dic3—24no—
524Dic3—42no—
525Dic3—42no—
526Dic3—44no—
527Dic3—42no—
528C23—22no—
529C23—22no—
530C23—24no—
531C23—24no—
532C23—62no—
533C23—62no—
534C23—62no—
535C23—62no—
536C23—62no—
537C23—62no—
538C23—62no—
539C23C2×D3×D414yes—
540C23C24×D311yescharacteristic
541Q8—68no—
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—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—34no—
560C2×C4—34no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—62no—
566C2×C4—62no—
567C2×C4—64no—
568C2×C4—62no—
569C2×C4—62no—
570C2×C4—62no—
571C2×C4—62no—
572C2×C4—64no—
573C2×C4—64no—
574C2×C4—62no—
575C2×C4—64no—
576C2×C4—84no—
577C2×C4—88no—
578C2×C4—128no—
579C2×C4C2×#23512yes—
580C2×C4C2×#23512yes—
581C2×C4C2×#23512yes—
582C2×C4C2×D3×D412yes—
583C2×C4#100712yes—
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—42no—
595C22—44no—
596C22—122no—
597C22—122no—
598C22—122no—
599C22—122no—
600C22—122no—
601C22—122no—
602C22—122no—
603C22C22×#23511yescharacteristic
604C22C22×D3×D411yescharacteristic
605C22C2×#100711yescharacteristic
606C22#596412yes—
607C22#596612yes—
608C4—22no—
609C4—22no—
610C4—42no—
611C4—42no—
612C4—44no—
613C4—42no—
614C4—64no—
615C4—64no—
616C4—122no—
617C4—124no—
618C4—122no—
619C4—122no—
620C4#550714yes—
621C3#3937411yescharacteristic, complemented, Sylow
622C2—44no—
623C2—242no—
624C2C2×#550711yescharacteristic
625C2#8583711yescharacteristic
626C2#8586311yescharacteristic
627C2#8663212yes—
628C2#8694512yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2349C2
prime radical#82416C2
hypercenter#2349D3
hypocenterC3#39374
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2349

Complements

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

NH ≅ G/Ntype
1. C2×#5561C2semidirect
2. C3×#2349C2semidirect
4. #82405C2semidirect
5. C4×#936C2semidirect
6. #76250C2semidirect
7. #76214C2semidirect
8. #74526C2semidirect
9. #74513C2semidirect
10. #74240C2semidirect
11. #74221C2semidirect
12. #73929C2semidirect
13. #73637C2semidirect
14. #73617C2semidirect
15. #73352C2semidirect
89. #2349D3semidirect
114. #936D4semidirect
621. C3#39374semidirect

Maximal subgroups

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

subgroupindex
C2×#55612
C3×#23492
#824162
#824052
C4×#9362
#762502
#762142
#745262
#745132
#742402
#742212
#739292
#736372
#736172
#733522
#393743