Finite Group Databasebeta

← #11865323

Subgroups — #11865323

Order 1152.

1611 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5964C211yesmaximal, characteristic, complemented
2C3×#5828C211yesmaximal, characteristic, complemented
3#10595022C212yesmaximal, complemented
4#10592939C211yesmaximal, characteristic, complemented
5#10591533C214yesmaximal, complemented
6#10591530C214yesmaximal, complemented
7#10591231C212yesmaximal, complemented
8#84125—31nomaximal
9#82936—31nomaximal
10C2×C32×#41C2211yescharacteristic
11C3×#1010—22no—
12C3×#1009—22no—
13C3×D3×#41—24no—
14C3×#1007—21no—
15C3×#1004—21no—
16C3×#1004C2214yes—
17C2×C3×#235—22no—
18C2×C3×#235C2212yes—
19C22×C3×C4×D3C2211yescharacteristic, complemented
20#64135C2212yescomplemented
21C3×#946—21no—
22C3×#938—21no—
23C3×#937C2212yescomplemented
24C3×#935C2212yescomplemented
25C3×C4×#67—22no—
26C3×#926C2211yescharacteristic, complemented
27#63976C2214yescomplemented
28#63961C2214yescomplemented
29#63908—22no—
30C4×#625—24no—
31#63856—22no—
32#63661C2214yescomplemented
33#63580C2218yescomplemented
34C3×#585—31no—
35#5964D311yescharacteristic, complemented
36C3×#525—31no—
37#5828—32no—
38#5828D311yescharacteristic, complemented
39#5791—32no—
40#5244—34no—
41#5243—34no—
42#5240—31no—
43#5240—31no—
44#5240—32no—
45#5240—61no—
46#5053—34no—
47#5032—34no—
48#5006—32no—
49C23×C3×D3C2311yescharacteristic
50C32×#41—22no—
51C32×#41—22no—
52C2×C32×Q8D411yescharacteristic, complemented
53C2×C32×D4D411yescharacteristic, complemented
54C2×C32×D4C2312yes—
55C22×C32×C4D412yescomplemented
56C22×C32×C4C2311yescharacteristic
57C2×C3×#67—24no—
58C2×C3×#67—22no—
59C2×C3×#67C2314yes—
60C22×C3×Dic3C2311yescharacteristic
61C3×#239—24no—
62C3×D3×Q8—24no—
63C3×#237—24no—
64C3×#237—28no—
65C3×D3×D4—24no—
66C3×D3×D4—28no—
67C3×#235—24no—
68C3×#235—24no—
69C3×#235—22no—
70C3×#235—24no—
71C3×#235—22no—
72C3×#235—22no—
73C3×#235—22no—
74C2×C3×D12—22no—
75C2×C3×D12D412yescomplemented
76C2×C3×C4×D3—24no—
77C2×C3×C4×D3—21no—
78C2×C3×C4×D3—21no—
79C2×C3×C4×D3—22no—
80C2×C3×C4×D3—22no—
81C2×C3×C4×D3D414yescomplemented
82C2×C3×Dic6—22no—
83C2×C3×Dic6D412yescomplemented
84C2×#625—24no—
85#3757C2314yes—
86C4⋊(C3⋊Dic3)—22no—
87C4×C3⋊Dic3—22no—
88C3×#217—22no—
89C3×#217C2312yes—
90C3×(C2×D3)⋊C4—22no—
91C3×C4⋊Dic3—21no—
92C3×Dic3⋊C4—22no—
93C3×C4×Dic3—21no—
94#3724—24no—
95(C2×D3)⋊Dic3—24no—
96(C2×D3)⋊Dic3—24no—
97#3025—91noSylow
98C3×2−1+4—62no—
99C3×2−1+4—61no—
100C3×H(5,𝔽2)—34no—
101C3×H(5,𝔽2)—62no—
102C3×H(5,𝔽2)—61no—
103C2×C3×#41—21no—
104C2×C3×#41—32no—
105C2×C3×#41—64no—
106C2×C3×#41—62no—
107C2×C3×#41C2×D311yescharacteristic
108C2×C3×#41C2×D311yescharacteristic
109C23×C3×C4—31no—
110#1010—22no—
111#1009—22no—
112D3×#41—24no—
113#1007—21no—
114#1004—21no—
115#1004C2×D314yes—
116C2×#235—22no—
117C2×#235C2×D312yes—
118C22×C4×D3C2×D311yescharacteristic, complemented
119C3×#123—32no—
120C3×C22⋊Q8—61no—
121C3×#121—61no—
122C3×#120—32no—
123C3×C4×D4—62no—
124C3×#117—31no—
125#953—34no—
126#946—21no—
127#946—62no—
128#938—21no—
129#938—62no—
130#937—34no—
131#937C2×D312yescomplemented
132#935—34no—
133#935C2×D312yescomplemented
134#930—62no—
135#929—34no—
136C4×#67—22no—
137C4×#67—64no—
138C4×#67—64no—
139#926—32no—
140#926—32no—
141#926—62no—
142#926C2×D311yescharacteristic, complemented
143#924—62no—
144#834—34no—
145#834—34no—
146#834—38no—
147#834—64no—
148#806—38no—
149C23×C32C2×D412yes—
150C23×C32C2411yescharacteristic
151C22×C3×D3—22no—
152C22×C3×D3—21no—
153C22×C3×D3—24no—
154C22×C3×D3—42no—
155C22×C3×D3C2×D414yes—
156C32×Q8—22no—
157C32×D4—24no—
158C32×D4—22no—
159C2×C32×C4—22no—
160C2×C32×C4—22no—
161C2×C32×C4C2×D412yes—
162C2×C32×C4C2×D412yes—
163C2×C32×C4C2×D412yes—
164C2×C32×C4C2×D411yescharacteristic
165C2×C32×C4C2×D411yescharacteristic
166C2×C3⋊Dic3—24no—
167C3×#67—28no—
168C3×#67—24no—
169C3×#67—24no—
170C3×#67—44no—
171C2×C3×Dic3—24no—
172C2×C3×Dic3—22no—
173C2×C3×Dic3—22no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3C2×D414yes—
176C3×D12—24no—
177C3×D12—22no—
178C3×D12—22no—
179C3×C4×D3—28no—
180C3×C4×D3—24no—
181C3×C4×D3—24no—
182C3×C4×D3—41no—
183C3×C4×D3—42no—
184C3×C4×D3—41no—
185C3×Dic6—24no—
186C3×Dic6—22no—
187C3×Dic6—22no—
188#625—48no—
189#585—31no—
190#525—31no—
191#525—92no—
192#458—94no—
193B(4,𝔽2)—94no—
194#410—92no—
195#410—91no—
196C24×C3—31no—
197C23×D3C22×D311yescharacteristic
198C3×#41—22no—
199C3×#41—22no—
200C3×#41—22no—
201C3×#41—22no—
202C3×#41—42no—
203C3×#41—42no—
204C3×#41—64no—
205C3×#41—64no—
206C3×#41—68no—
207C3×#41—64no—
208C3×#41—64no—
209C3×#41—62no—
210C3×#41—62no—
211C3×#41—62no—
212C3×#41—62no—
213C3×#41—64no—
214C2×C3×Q8—21no—
215C2×C3×Q8—32no—
216C2×C3×Q8—62no—
217C2×C3×Q8—64no—
218C2×C3×Q8#6711yescharacteristic, complemented
219C2×C3×Q8#6711yescharacteristic, complemented
220C2×C3×D4—21no—
221C2×C3×D4—22no—
222C2×C3×D4—34no—
223C2×C3×D4—32no—
224C2×C3×D4—62no—
225C2×C3×D4—68no—
226C2×C3×D4—64no—
227C2×C3×D4—64no—
228C2×C3×D4—62no—
229C2×C3×D4#6711yescharacteristic, complemented
230C2×C3×D4#6711yescharacteristic, complemented
231C2×C3×D4C22×D312yes—
232C2×C3×D4C22×D312yes—
233C22×C3×C4—22no—
234C22×C3×C4—21no—
235C22×C3×C4—34no—
236C22×C3×C4—31no—
237C22×C3×C4—64no—
238C22×C3×C4—62no—
239C22×C3×C4—62no—
240C22×C3×C4—61no—
241C22×C3×C4—61no—
242C22×C3×C4#6712yescomplemented
243C22×C3×C4#6712yescomplemented
244C22×C3×C4C22×D311yescharacteristic
245C22×C3×C4C22×D311yescharacteristic
246C2×#67—24no—
247C2×#67—22no—
248C2×#67—64no—
249C2×#67—64no—
250C2×#67C22×D314yes—
251C22×Dic3C22×D311yescharacteristic
252#239—24no—
253D3×Q8—24no—
254#237—24no—
255#237—28no—
256D3×D4—24no—
257D3×D4—28no—
258#235—24no—
259#235—24no—
260#235—22no—
261#235—24no—
262#235—22no—
263#235—22no—
264#235—22no—
265C2×D12—22no—
266C2×D12#6712yescomplemented
267C2×C4×D3—24no—
268C2×C4×D3—22no—
269C2×C4×D3—22no—
270C2×C4×D3—21no—
271C2×C4×D3—21no—
272C2×C4×D3#6714yescomplemented
273C2×Dic6—22no—
274C2×Dic6#6712yescomplemented
275C3×C4⋊C4—62no—
276C3×C4⋊C4—61no—
277C3×C22⋊C4—32no—
278C3×C22⋊C4—62no—
279C3×C22⋊C4—62no—
280C3×C42—61no—
281#217—22no—
282#217—34no—
283#217—34no—
284#217—64no—
285#217—64no—
286#217—64no—
287#217C22×D312yes—
288(C2×D3)⋊C4—22no—
289(C2×D3)⋊C4—64no—
290(C2×D3)⋊C4—64no—
291C4⋊Dic3—21no—
292C4⋊Dic3—62no—
293C4⋊Dic3—62no—
294C4⋊Dic3—122no—
295Dic3⋊C4—22no—
296Dic3⋊C4—64no—
297Dic3⋊C4—64no—
298C4×Dic3—21no—
299C4×Dic3—62no—
300C4×Dic3—62no—
301C4×Dic3—122no—
302C22×C32—22no—
303C22×C32—41no—
304C22×C32—42no—
305C22×C32C22×D412yes—
306C22×C32C22×D411yescharacteristic
307C2×C3×D3—24no—
308C2×C3×D3—22no—
309C2×C3×D3—24no—
310C2×C3×D3—41no—
311C2×C3×D3—42no—
312C2×C3×D3—41no—
313C2×C3×D3—44no—
314C2×C3×D3—84no—
315C32×C4—22no—
316C32×C4#12011yescharacteristic
317C32×C4#12012yes—
318C32×C4#12011yescharacteristic
319C3⋊Dic3—44no—
320C3×Dic3—24no—
321C3×Dic3—42no—
322C3×Dic3—41no—
3232−1+4—62no—
3242−1+4—61no—
325H(5,𝔽2)—34no—
326H(5,𝔽2)—62no—
327H(5,𝔽2)—61no—
328C2×#41—32no—
329C2×#41—64no—
330C2×#41—62no—
331C2×#41D3211yescharacteristic
332C23×C4—31no—
333#123—32no—
334#123—94no—
335C22⋊Q8—61no—
336C22⋊Q8—182no—
337#121—61no—
338#121—182no—
339#120—32no—
340#120—94no—
341C4×D4—62no—
342C4×D4—184no—
343#117—31no—
344#117—92no—
345#99—98no—
346#99—94no—
347C23×C3—21no—
348C23×C3—22no—
349C23×C3—34no—
350C23×C3—64no—
351C23×C3—62no—
352C23×C3—61no—
353C23×C3—122no—
354C23×C3C2×#6712yes—
355C23×C3C2×#6712yes—
356C23×C3C23×D311yescharacteristic
357C23×C3C23×D311yescharacteristic
358C22×D3—22no—
359C22×D3—21no—
360C22×D3—24no—
361C22×D3—42no—
362C22×D3C2×#6714yes—
363C3×Q8—22no—
364C3×Q8—22no—
365C3×Q8—42no—
366C3×Q8—64no—
367C3×Q8—62no—
368C3×Q8—62no—
369C3×D4—24no—
370C3×D4—24no—
371C3×D4—22no—
372C3×D4—22no—
373C3×D4—42no—
374C3×D4—44no—
375C3×D4—64no—
376C3×D4—68no—
377C3×D4—64no—
378C3×D4—64no—
379C3×D4—62no—
380C3×D4—62no—
381C3×D4—124no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—22no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—34no—
392C2×C3×C4—42no—
393C2×C3×C4—42no—
394C2×C3×C4—68no—
395C2×C3×C4—64no—
396C2×C3×C4—64no—
397C2×C3×C4—64no—
398C2×C3×C4—62no—
399C2×C3×C4—62no—
400C2×C3×C4—61no—
401C2×C3×C4—121no—
402C2×C3×C4—122no—
403C2×C3×C4—121no—
404C2×C3×C4D3×D412yes—
405C2×C3×C4D3×D412yes—
406C2×C3×C4C2×#6712yes—
407C2×C3×C4C2×#6712yes—
408C2×C3×C4C2×#6711yescharacteristic
409C2×C3×C4C2×#6711yescharacteristic
410C2×C3×C4C2×#6711yescharacteristic
411C2×C3×C4C2×#6711yescharacteristic
412C2×C3×C4C2×#6712yes—
413C2×C3×C4C2×#6712yes—
414#67—28no—
415#67—24no—
416#67—24no—
417#67—44no—
418#67—128no—
419#67—128no—
420C2×Dic3—24no—
421C2×Dic3—22no—
422C2×Dic3—22no—
423C2×Dic3—21no—
424C2×Dic3—64no—
425C2×Dic3—64no—
426C2×Dic3—124no—
427C2×Dic3C2×#6714yes—
428D12—24no—
429D12—22no—
430D12—22no—
431C4×D3—28no—
432C4×D3—24no—
433C4×D3—24no—
434C4×D3—41no—
435C4×D3—42no—
436C4×D3—41no—
437Dic6—24no—
438Dic6—22no—
439Dic6—22no—
440C2×C32—21no—
441C2×C32—22no—
442C2×C32—42no—
443C2×C32C2×#12011yescharacteristic
444C3×D3—42no—
445C3×D3—44no—
446C24—31no—
447#41—22no—
448#41—22no—
449#41—64no—
450#41—64no—
451#41—68no—
452#41—64no—
453#41—64no—
454#41—62no—
455#41—62no—
456#41—64no—
457#41—62no—
458#41—62no—
459C2×Q8—32no—
460C2×Q8—62no—
461C2×Q8—64no—
462C2×Q8#62511yescharacteristic, complemented
463C2×D4—34no—
464C2×D4—32no—
465C2×D4—62no—
466C2×D4—68no—
467C2×D4—64no—
468C2×D4—64no—
469C2×D4—62no—
470C2×D4—184no—
471C2×D4#62511yescharacteristic, complemented
472C2×D4C2×D3212yes—
473C22×C4—34no—
474C22×C4—31no—
475C22×C4—64no—
476C22×C4—61no—
477C22×C4—61no—
478C22×C4—62no—
479C22×C4—62no—
480C22×C4#62512yescomplemented
481C22×C4C2×D3211yescharacteristic
482C4⋊C4—61no—
483C4⋊C4—62no—
484C4⋊C4—184no—
485C4⋊C4—182no—
486C22⋊C4—32no—
487C22⋊C4—62no—
488C22⋊C4—62no—
489C22⋊C4—94no—
490C22⋊C4—184no—
491C22⋊C4—184no—
492C42—61no—
493C42—182no—
494C22×C3—22no—
495C22×C3—22no—
496C22×C3—21no—
497C22×C3—22no—
498C22×C3—42no—
499C22×C3—41no—
500C22×C3—41no—
501C22×C3—42no—
502C22×C3—42no—
503C22×C3—64no—
504C22×C3—62no—
505C22×C3—64no—
506C22×C3—81no—
507C22×C3—82no—
508C22×C3—124no—
509C22×C3—122no—
510C22×C3—121no—
511C22×C3—121no—
512C22×C3—244no—
513C22×C3C2×D3×D412yes—
514C22×C3C22×#6711yescharacteristic
515C22×C3C22×#6711yescharacteristic
516C22×C3C22×#6712yes—
517C2×D3—22no—
518C2×D3—24no—
519C2×D3—24no—
520C2×D3—41no—
521C2×D3—42no—
522C2×D3—41no—
523C2×D3—44no—
524C2×D3—84no—
525C3×C4—22no—
526C3×C4—22no—
527C3×C4—21no—
528C3×C4—22no—
529C3×C4—21no—
530C3×C4—42no—
531C3×C4—64no—
532C3×C4—122no—
533C3×C4—121no—
534C3×C4#93711yescharacteristic
535C3×C4#93712yes—
536C3×C4#93711yescharacteristic
537C3×C4#95311yescharacteristic
538C3×C4#95312yes—
539C3×C4#95311yescharacteristic
540Dic3—24no—
541Dic3—42no—
542Dic3—41no—
543Dic3—124no—
544Dic3—124no—
545Dic3—244no—
546C32#302511yescharacteristic, complemented, Sylow
547C23—34no—
548C23—64no—
549C23—61no—
550C23—62no—
551C23—122no—
552C23C2×#62512yes—
553C23C22×D3211yescharacteristic
554Q8—22no—
555Q8—64no—
556Q8—62no—
557Q8—62no—
558D4—24no—
559D4—22no—
560D4—64no—
561D4—68no—
562D4—64no—
563D4—64no—
564D4—62no—
565D4—62no—
566D4—124no—
567D4—368no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—34no—
571C2×C4—68no—
572C2×C4—64no—
573C2×C4—64no—
574C2×C4—64no—
575C2×C4—62no—
576C2×C4—61no—
577C2×C4—62no—
578C2×C4—121no—
579C2×C4—122no—
580C2×C4—121no—
581C2×C4—184no—
582C2×C4C2×#62512yes—
583C2×C4C2×#62511yescharacteristic
584C2×C4C2×#62511yescharacteristic
585C2×C4D3×#6712yes—
586C2×C4D3×#6712yes—
587C2×C3—22no—
588C2×C3—21no—
589C2×C3—21no—
590C2×C3—22no—
591C2×C3—21no—
592C2×C3—42no—
593C2×C3—41no—
594C2×C3—42no—
595C2×C3—42no—
596C2×C3—82no—
597C2×C3—122no—
598C2×C3—124no—
599C2×C3C2×#93711yescharacteristic
600C2×C3C2×#95311yescharacteristic
601D3—44no—
602D3—42no—
603C22—22no—
604C22—41no—
605C22—42no—
606C22—62no—
607C22—64no—
608C22—64no—
609C22—124no—
610C22—121no—
611C22—122no—
612C22—121no—
613C22—244no—
614C22C22×#62511yescharacteristic
615C22C2×D3×#6712yes—
616C4—22no—
617C4—64no—
618C4—122no—
619C4—121no—
620C4—364no—
621C4#6397611yescharacteristic
622C4#6397612yes—
623C4#6397611yescharacteristic
624C3—21no—
625C3#8293611yescharacteristic, complemented
626C3#8412511yescharacteristic, complemented
627C2—21no—
628C2—22no—
629C2—42no—
630C2—122no—
631C2—124no—
632C2C2×#6397611yescharacteristic
subgroupG/H
centerC4#63976
commutator, cosocleC23×C32C24
FrattiniC23C22×D32
FittingC2×C32×#41C22
socleC2×C32C2×#120
hypercenterC2×#41D32
hypocenterC32#3025
LCSG ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C3×#5964C2semidirect
2. C3×#5828C2semidirect
3. #10595022C2semidirect
4. #10592939C2semidirect
5. #10591533C2semidirect
6. #10591530C2semidirect
7. #10591231C2semidirect
19. C22×C3×C4×D3C22semidirect
20. #64135C22semidirect
23. C3×#937C22semidirect
24. C3×#935C22semidirect
26. C3×#926C22semidirect
27. #63976C22semidirect
28. #63961C22semidirect
32. #63661C22semidirect
33. #63580C22semidirect
35. #5964D3semidirect
38. #5828D3semidirect
52. C2×C32×Q8D4semidirect
53. C2×C32×D4D4semidirect
55. C22×C32×C4D4semidirect
75. C2×C3×D12D4semidirect
81. C2×C3×C4×D3D4semidirect
83. C2×C3×Dic6D4semidirect
118. C22×C4×D3C2×D3semidirect
131. #937C2×D3semidirect
133. #935C2×D3semidirect
142. #926C2×D3semidirect
218. C2×C3×Q8#67semidirect
219. C2×C3×Q8#67semidirect
229. C2×C3×D4#67semidirect
230. C2×C3×D4#67semidirect
242. C22×C3×C4#67semidirect
243. C22×C3×C4#67semidirect
266. C2×D12#67semidirect
272. C2×C4×D3#67semidirect
274. C2×Dic6#67semidirect
462. C2×Q8#625semidirect
471. C2×D4#625semidirect
480. C22×C4#625semidirect
546. C32#3025semidirect
625. C3#82936semidirect
626. C3#84125semidirect

Maximal subgroups

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

subgroupindex
C3×#59642
C3×#58282
#105950222
#105929392
#105915332
#105915302
#105912312
#841253
#829363