Finite Group Databasebeta

← #13177997

Subgroups — #13177997

Order 1728.

2272 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11724684C211yesmaximal, characteristic, complemented
2#11724580C211yesmaximal, characteristic, complemented
3C3×#64158C211yesmaximal, characteristic, complemented
4C32×#950C211yesmaximal, characteristic, complemented
5#11723589C212yesmaximal, complemented
6#11723572C212yesmaximal, complemented
7#11723566C212yesmaximal, complemented
8#11723547C212yesmaximal, complemented
9C2×#88791C211yesmaximal, characteristic, complemented
10#11723367C211yesmaximal, characteristic, complemented
11#11723356C211yesmaximal, characteristic, complemented
12#10596240—31nomaximal
13#10595461—34nomaximal
14C33×#41C2211yescharacteristic, complemented
15#89080C2212yescomplemented
16D4×C3⋊(C3⋊D3)—22no—
17D4×C3⋊(C3⋊D3)C2212yescomplemented
18#89076—21no—
19#89076C2211yescharacteristic
20C2×C4⋊(C3⋊(C3⋊D3))—21no—
21C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
22C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
23C32×#235C2211yescharacteristic, complemented
24C3⋊D3×#67—22no—
25#89026C2212yescomplemented
26D3×C4⋊(C3⋊D3)—21no—
27#89021—21no—
28#89017C2212yescomplemented
29C3×#3755C2212yescomplemented
30C3×#3753C2212yescomplemented
31C3×#3748C2211yescharacteristic, complemented
32C32×#216C2211yescharacteristic, complemented
33C32×#215C2211yescharacteristic, complemented
34C32×#214C2211yescharacteristic, complemented
35C32×#213C2211yescharacteristic, complemented
36C2×C32×C3⋊C8C2211yescharacteristic, complemented
37#88795C2212yescomplemented
38#88791C2212yescomplemented
39#88791C2212yescomplemented
40#88790C2212yescomplemented
41#88788C2212yescomplemented
42#88785C2212yescomplemented
43#64366—31no—
44#64366—31no—
45#64366—34no—
46#64366—64no—
47#64307—34no—
48C32×#135—31no—
49#64158—31no—
50#64158D311yescharacteristic, complemented
51#64128—32no—
52#64123—32no—
53#64123—32no—
54D8×C3⋊D3—32no—
55C2×#3745—31no—
56#64110—31no—
57C3×#950D314yescomplemented
58C3×#949—34no—
59#63950—38no—
60#63941—38no—
61#63937—38no—
62#63926—38no—
63#63825—34no—
64C2×#3714—34no—
65#63817—34no—
66C22×C3⋊(C3⋊D3)—21no—
67C22×C3⋊(C3⋊D3)—22no—
68C22×C3×C3⋊D3—21no—
69C2×D3×C3⋊D3—22no—
70C33×Q8C2311yescharacteristic
71C33×D4—21no—
72C33×D4C2311yescharacteristic
73C2×C33×C4—21no—
74C2×C33×C4C2311yescharacteristic
75#6312—22no—
76#6312D412yes—
77C4⋊(C3⋊(C3⋊D3))—22no—
78C4⋊(C3⋊(C3⋊D3))—21no—
79C4⋊(C3⋊(C3⋊D3))D411yescharacteristic
80C4⋊(C3⋊(C3⋊D3))C2312yes—
81C4×C3⋊(C3⋊D3)—22no—
82C4×C3⋊(C3⋊D3)C2312yes—
83#6308D411yescharacteristic
84C3×C4⋊(C3⋊D3)—21no—
85C3×C4⋊(C3⋊D3)C2312yes—
86C32×#67—21no—
87C32×D12C2311yescharacteristic
88C32×C4×D3—21no—
89C32×Dic6C2311yescharacteristic
90#6292—22no—
91#6290—22no—
92Dic3×C3⋊D3—22no—
93C3×C3⋊(C3⋊C8)C2312yes—
94C32×C3⋊C8C2311yescharacteristic
95C32×C3⋊C8C2311yescharacteristic
96#5774—91no—
97#5769—94no—
98C32×#41—24no—
99C32×#41—31no—
100C32×#41C2×D314yescomplemented
101C32×#41C2×D311yescharacteristic, complemented
102#3832—32no—
103#3832—38no—
104#3832—32no—
105#3832—68no—
106D4×C3⋊D3—32no—
107D4×C3⋊D3—38no—
108D4×C3⋊D3—32no—
109D4×C3⋊D3—62no—
110D4×C3⋊D3—68no—
111D4×C3⋊D3—62no—
112D4×C3⋊D3—68no—
113D4×C3⋊D3—616no—
114#3828—31no—
115#3828—34no—
116#3828—61no—
117#3828—61no—
118#3828—64no—
119#3828—64no—
120#3828—68no—
121C2×C4⋊(C3⋊D3)—31no—
122C2×C4⋊(C3⋊D3)—34no—
123C2×C4⋊(C3⋊D3)—61no—
124C2×C4⋊(C3⋊D3)—61no—
125C2×C4⋊(C3⋊D3)—64no—
126C2×C4⋊(C3⋊D3)—64no—
127C2×C4⋊(C3⋊D3)—68no—
128C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
129C3×#235C2×D314yescomplemented
130C2×C3×D12—34no—
131D3×#67—68no—
132#3802—38no—
133D3×D12—64no—
134#3796—38no—
135#3795—64no—
136C32×Q16—31no—
137C32×SD16—31no—
138C32×SD16—31no—
139C32×D8—31no—
140C2×C32×C8—31no—
141#3755—32no—
142#3755C2×D312yescomplemented
143#3753—32no—
144#3753C2×D312yescomplemented
145#3748C2×D311yescharacteristic, complemented
146#3745—32no—
147#3745—32no—
148#3743—32no—
149C8×C3⋊D3—32no—
150C3×#216C2×D314yescomplemented
151C3×#215—38no—
152C3×#215C2×D314yescomplemented
153C3×#214C2×D314yescomplemented
154C3×#213—38no—
155C3×#213C2×D314yescomplemented
156C3×#208—34no—
157C2×C3×C3⋊C8C2×D314yescomplemented
158#3717—38no—
159#3714—38no—
160#3714—38no—
161#3714—38no—
162#3712—38no—
163#3710—38no—
164C22×C33—21no—
165C22×C33C2×D411yescharacteristic
166C2×C3⋊(C3⋊D3)—22no—
167C2×C3⋊(C3⋊D3)—82no—
168C2×C3⋊(C3⋊D3)—81no—
169C2×C3⋊(C3⋊D3)C2×D412yes—
170C2×C3×C3⋊D3—22no—
171C2×C3×C3⋊D3—81no—
172C2×C32×D3—21no—
173D3×C3⋊D3—82no—
174C33×C4—21no—
175C33×C4C2×D411yescharacteristic
176C33×C4C2411yescharacteristic
177C3⋊(C3⋊Dic3)C2×D412yes—
178C32×Dic3—21no—
179#1009—94no—
180#1009—94no—
181#1009—91no—
182#1009—184no—
183#1004—91no—
184C3×AGL(1,ℤ/8)—91no—
185C3×#135—34no—
186#950C3⋊D311yescharacteristic, complemented
187#949—34no—
188#949—94no—
189C2×#213—91no—
190#919—98no—
191#916—92no—
192#914—98no—
193#914—98no—
194#914—92no—
195#911—92no—
196D3×D8—98no—
197D3×D8—92no—
198#908—91no—
199#907—91no—
200C2×D24—94no—
201#901—94no—
202C22×C3⋊D3—21no—
203C22×C3⋊D3—62no—
204C22×C3⋊D3—62no—
205C22×C3⋊D3—68no—
206C22×C3⋊D3—616no—
207C22×C3⋊D3—68no—
208C22×C3⋊D3—64no—
209C22×C3⋊D3—61no—
210C22×C3×D3—64no—
211C2×D32—68no—
212C32×Q8—24no—
213C32×Q8—31no—
214C32×Q8C22×D314yes—
215C32×Q8C22×D311yescharacteristic
216C32×D4—28no—
217C32×D4—24no—
218C32×D4—21no—
219C32×D4—24no—
220C32×D4—31no—
221C32×D4—61no—
222C32×D4C22×D314yes—
223C32×D4C22×D311yescharacteristic
224C2×C32×C4—28no—
225C2×C32×C4—24no—
226C2×C32×C4—21no—
227C2×C32×C4—24no—
228C2×C32×C4—61no—
229C2×C32×C4C22×D314yes—
230C2×C32×C4C22×D311yescharacteristic
231#638—32no—
232#638—38no—
233#638—68no—
234#638—62no—
235#638—62no—
236#638—68no—
237#638—616no—
238C4⋊(C3⋊D3)—21no—
239C4⋊(C3⋊D3)—31no—
240C4⋊(C3⋊D3)—32no—
241C4⋊(C3⋊D3)—34no—
242C4⋊(C3⋊D3)—38no—
243C4⋊(C3⋊D3)—64no—
244C4⋊(C3⋊D3)—68no—
245C4⋊(C3⋊D3)—62no—
246C4⋊(C3⋊D3)—68no—
247C4⋊(C3⋊D3)—68no—
248C4⋊(C3⋊D3)—616no—
249C4⋊(C3⋊D3)—62no—
250C4⋊(C3⋊D3)—64no—
251C4⋊(C3⋊D3)—61no—
252C4⋊(C3⋊D3)C22×D312yes—
253C4×C3⋊D3—32no—
254C4×C3⋊D3—38no—
255C4×C3⋊D3—68no—
256C4×C3⋊D3—62no—
257C4×C3⋊D3—68no—
258C4×C3⋊D3—62no—
259C4×C3⋊D3—616no—
260#634—31no—
261#634—34no—
262#634—64no—
263C3×#67—24no—
264C3×D12—38no—
265C3×D12—64no—
266C3×D12C22×D314yes—
267C3×C4×D3—24no—
268C3×Dic6C22×D314yes—
269#626—68no—
270#625—68no—
271D3×Dic3—68no—
272C32×C8—31no—
273C32×C8—31no—
274C3⋊(C3⋊C8)C22×D312yes—
275C3×C3⋊C8—38no—
276C3×C3⋊C8C22×D314yes—
277C3×C3⋊C8C22×D314yes—
278#576—271noSylow
279C2×C33—21no—
280C2×C33—41no—
281C2×C33C22×D411yescharacteristic
282C3⋊(C3⋊D3)—22no—
283C3⋊(C3⋊D3)—42no—
284C3×C3⋊D3—42no—
285C32×D3—41no—
286C3×#41—24no—
287C3×#41—34no—
288C3×#41D3214yescomplemented
289C3×#41C2×C3⋊D311yescharacteristic, complemented
290C2×C3×D4—91no—
291C2×#67—182no—
292#239—98no—
293#239—98no—
294#239—92no—
295#239—188no—
296#237—92no—
297#237—181no—
298D3×D4—98no—
299D3×D4—98no—
300D3×D4—92no—
301D3×D4—92no—
302D3×D4—188no—
303D3×D4—181no—
304D3×D4—188no—
305D3×D4—182no—
306D3×D4—188no—
307D3×D4—1816no—
308#235—94no—
309#235—91no—
310#235—184no—
311#235—184no—
312#235—184no—
313#235—181no—
314#235—188no—
315#235C2×C3⋊D311yescharacteristic, complemented
316C2×D12—34no—
317C2×D12—94no—
318C2×D12—91no—
319C2×D12—184no—
320C2×D12—184no—
321C2×D12—184no—
322C2×D12—181no—
323C2×D12—188no—
324C3×Q16—34no—
325C3×SD16—34no—
326C3×SD16—34no—
327C3×SD16—92no—
328C3×D8—34no—
329C3×D8—92no—
330C3×M16—91no—
331C2×C3×C8—34no—
332#216C2×C3⋊D311yescharacteristic, complemented
333#215—38no—
334#215—98no—
335#215C2×C3⋊D311yescharacteristic, complemented
336#214C2×C3⋊D311yescharacteristic, complemented
337#213—38no—
338#213—98no—
339#213—92no—
340#213—92no—
341#213C2×C3⋊D311yescharacteristic, complemented
342#208—34no—
343C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
344D24—98no—
345D24—98no—
346D24—92no—
347#204—92no—
348ΓL(1,𝔽25)—98no—
349ΓL(1,𝔽25)—92no—
350D3×C8—98no—
351D3×C8—92no—
352C22×C32—28no—
353C22×C32—24no—
354C22×C32—21no—
355C22×C32—24no—
356C22×C32—61no—
357C22×C32D3×D414yes—
358C22×C32D3×D411yescharacteristic
359C2×C3⋊D3—22no—
360C2×C3⋊D3—32no—
361C2×C3⋊D3—38no—
362C2×C3⋊D3—68no—
363C2×C3⋊D3—616no—
364C2×C3⋊D3—68no—
365C2×C3⋊D3—62no—
366C2×C3⋊D3—81no—
367C2×C3⋊D3—242no—
368C2×C3⋊D3—242no—
369C2×C3⋊D3—248no—
370C2×C3⋊D3—241no—
371C2×C3⋊D3—244no—
372C2×C3⋊D3—2416no—
373C2×C3⋊D3—248no—
374C2×C3×D3—24no—
375C2×C3×D3—68no—
376C2×C3×D3—244no—
377D32—248no—
378C32×C4—28no—
379C32×C4—24no—
380C32×C4—21no—
381C32×C4—24no—
382C32×C4—24no—
383C32×C4—61no—
384C32×C4D3×D414yes—
385C32×C4D3×D411yescharacteristic
386C32×C4C23×D314yes—
387C32×C4C23×D311yescharacteristic
388C3⋊Dic3—32no—
389C3⋊Dic3—38no—
390C3⋊Dic3—68no—
391C3×Dic3—24no—
392H(5,𝔽2)—271no—
393H(5,𝔽2)—271no—
394AGL(1,ℤ/8)—91no—
395AGL(1,ℤ/8)—271no—
396AGL(1,ℤ/8)—272no—
397AGL(1,ℤ/8)—272no—
398#135—31no—
399#135—272no—
400C2×D8—271no—
401C2×D8—272no—
402#131—271no—
403C33#57611yescharacteristic, complemented, Sylow
404C23×C3—181no—
405C22×D3—64no—
406C22×D3—182no—
407C22×D3—188no—
408C22×D3—188no—
409C22×D3—182no—
410C22×D3—1816no—
411C22×D3—188no—
412C22×D3—184no—
413C22×D3—181no—
414C3×Q8—24no—
415C3×Q8—34no—
416C3×Q8C2×D3214yes—
417C3×Q8C22×C3⋊D311yescharacteristic
418C3×D4—28no—
419C3×D4—24no—
420C3×D4—21no—
421C3×D4—24no—
422C3×D4—34no—
423C3×D4—64no—
424C3×D4—92no—
425C3×D4—181no—
426C3×D4C2×D3214yes—
427C3×D4C22×C3⋊D311yescharacteristic
428C2×C3×C4—28no—
429C2×C3×C4—24no—
430C2×C3×C4—21no—
431C2×C3×C4—24no—
432C2×C3×C4—64no—
433C2×C3×C4C2×D3214yes—
434C2×C3×C4C22×C3⋊D311yescharacteristic
435#67—21no—
436#67—98no—
437#67—92no—
438#67—188no—
439#67—182no—
440#67—182no—
441#67—188no—
442#67—188no—
443#67—182no—
444#67—1816no—
445C2×Dic3—182no—
446D12—38no—
447D12—64no—
448D12—98no—
449D12—92no—
450D12—94no—
451D12—91no—
452D12—184no—
453D12—188no—
454D12—188no—
455D12—182no—
456D12—188no—
457D12—1816no—
458D12—188no—
459D12—184no—
460D12—181no—
461D12C22×C3⋊D311yescharacteristic
462C4×D3—21no—
463C4×D3—98no—
464C4×D3—92no—
465C4×D3—188no—
466C4×D3—188no—
467C4×D3—182no—
468C4×D3—188no—
469C4×D3—1816no—
470Dic6—94no—
471Dic6—91no—
472Dic6—184no—
473Dic6C22×C3⋊D311yescharacteristic
474C3×C8—34no—
475C3×C8—34no—
476C3×C8—92no—
477C3⋊C8—38no—
478C3⋊C8C22×C3⋊D311yescharacteristic
479C3⋊C8C22×C3⋊D311yescharacteristic
480C2×C32—21no—
481C2×C32—24no—
482C2×C32—24no—
483C2×C32—44no—
484C2×C32—48no—
485C2×C32—44no—
486C2×C32—41no—
487C2×C32—121no—
488C2×C32C2×D3×D414yes—
489C2×C32C2×D3×D411yescharacteristic
490C3⋊D3—42no—
491C3⋊D3—62no—
492C3⋊D3—68no—
493C3⋊D3—128no—
494C3⋊D3—1216no—
495C3⋊D3—128no—
496C3⋊D3—122no—
497C3×D3—44no—
498C3×D3—128no—
499#41—31no—
500#41—271no—
501#41—272no—
502#41—272no—
503#41—541no—
504#41—541no—
505#41D3×C3⋊D311yescharacteristic, complemented
506C2×D4—91no—
507C2×D4—272no—
508C2×D4—272no—
509C2×D4—271no—
510C2×D4—541no—
511C2×D4—541no—
512C2×D4—542no—
513C2×D4—542no—
514Q16—31no—
515SD16—31no—
516SD16—31no—
517SD16—92no—
518SD16—272no—
519D8—31no—
520D8—92no—
521D8—272no—
522D8—272no—
523D8—272no—
524M16—91no—
525M16—272no—
526C2×C8—31no—
527C2×C8—272no—
528C22×C3—28no—
529C22×C3—24no—
530C22×C3—21no—
531C22×C3—24no—
532C22×C3—64no—
533C22×C3—182no—
534C22×C3—721no—
535C22×C3#381114yes—
536C22×C3D4×C3⋊D311yescharacteristic
537C2×D3—21no—
538C2×D3—68no—
539C2×D3—98no—
540C2×D3—92no—
541C2×D3—188no—
542C2×D3—1816no—
543C2×D3—188no—
544C2×D3—182no—
545C2×D3—244no—
546C2×D3—722no—
547C2×D3—728no—
548C2×D3—728no—
549C2×D3—722no—
550C2×D3—724no—
551C2×D3—721no—
552C2×D3—728no—
553C2×D3—7216no—
554C3×C4—28no—
555C3×C4—24no—
556C3×C4—21no—
557C3×C4—24no—
558C3×C4—24no—
559C3×C4—64no—
560C3×C4#381114yes—
561C3×C4D4×C3⋊D311yescharacteristic
562C3×C4C22×D3214yes—
563C3×C4C23×C3⋊D311yescharacteristic
564Dic3—21no—
565Dic3—98no—
566Dic3—92no—
567Dic3—188no—
568C32—24no—
569C32#576914yescomplemented
570C32#577411yescharacteristic, complemented
571C23—181no—
572C23—542no—
573C23—542no—
574C23—541no—
575Q8—31no—
576Q8—271no—
577Q8C2×D3×C3⋊D311yescharacteristic
578D4—21no—
579D4—31no—
580D4—61no—
581D4—92no—
582D4—181no—
583D4—272no—
584D4—272no—
585D4—271no—
586D4—542no—
587D4—542no—
588D4—542no—
589D4—542no—
590D4—541no—
591D4C2×D3×C3⋊D311yescharacteristic
592C2×C4—21no—
593C2×C4—61no—
594C2×C4—272no—
595C2×C4—542no—
596C2×C4—542no—
597C2×C4C2×D3×C3⋊D311yescharacteristic
598C8—31no—
599C8—31no—
600C8—92no—
601C2×C3—24no—
602C2×C3—21no—
603C2×C3—24no—
604C2×C3—44no—
605C2×C3—48no—
606C2×C3—44no—
607C2×C3—41no—
608C2×C3—124no—
609C2×C3—362no—
610C2×C3C2×#381114yes—
611C2×C3C2×D4×C3⋊D311yescharacteristic
612D3—41no—
613D3—128no—
614D3—188no—
615D3—182no—
616D3—368no—
617D3—3616no—
618D3—368no—
619D3—362no—
620C22—21no—
621C22—61no—
622C22—182no—
623C22—272no—
624C22—542no—
625C22—721no—
626C22—2161no—
627C22—2162no—
628C22—2162no—
629C22#8903911yescharacteristic
630C4—21no—
631C4—61no—
632C4—272no—
633C4#8903911yescharacteristic
634C4C22×D3×C3⋊D311yescharacteristic
635C3—24no—
636C3#1059546114yescomplemented
637C3#1059624011yescharacteristic, complemented
638C2—21no—
639C2—41no—
640C2—121no—
641C2—362no—
642C2—542no—
643C2—1082no—
644C2C2×#8903911yescharacteristic
subgroupG/H
centerC2C2×#89039
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC33×#41C22
socleC2×C33C22×D4
hypercenter#41D3×C3⋊D3
hypocenterC33#576
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. #11724684C2semidirect
2. #11724580C2semidirect
3. C3×#64158C2semidirect
4. C32×#950C2semidirect
5. #11723589C2semidirect
6. #11723572C2semidirect
7. #11723566C2semidirect
8. #11723547C2semidirect
9. C2×#88791C2semidirect
10. #11723367C2semidirect
11. #11723356C2semidirect
14. C33×#41C22semidirect
15. #89080C22semidirect
17. D4×C3⋊(C3⋊D3)C22semidirect
21. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
22. C2×C3×C4⋊(C3⋊D3)C22semidirect
23. C32×#235C22semidirect
25. #89026C22semidirect
28. #89017C22semidirect
29. C3×#3755C22semidirect
30. C3×#3753C22semidirect
31. C3×#3748C22semidirect
32. C32×#216C22semidirect
33. C32×#215C22semidirect
34. C32×#214C22semidirect
35. C32×#213C22semidirect
36. C2×C32×C3⋊C8C22semidirect
37. #88795C22semidirect
38. #88791C22semidirect
39. #88791C22semidirect
40. #88790C22semidirect
41. #88788C22semidirect
42. #88785C22semidirect
50. #64158D3semidirect
57. C3×#950D3semidirect
100. C32×#41C2×D3semidirect
101. C32×#41C2×D3semidirect
128. C2×C4⋊(C3⋊D3)C2×D3semidirect
129. C3×#235C2×D3semidirect
142. #3755C2×D3semidirect
144. #3753C2×D3semidirect
145. #3748C2×D3semidirect
150. C3×#216C2×D3semidirect
152. C3×#215C2×D3semidirect
153. C3×#214C2×D3semidirect
155. C3×#213C2×D3semidirect
157. C2×C3×C3⋊C8C2×D3semidirect
186. #950C3⋊D3semidirect
288. C3×#41D32semidirect
289. C3×#41C2×C3⋊D3semidirect
315. #235C2×C3⋊D3semidirect
332. #216C2×C3⋊D3semidirect
335. #215C2×C3⋊D3semidirect
336. #214C2×C3⋊D3semidirect
341. #213C2×C3⋊D3semidirect
343. C2×C3⋊C8C2×C3⋊D3semidirect
403. C33#576semidirect
505. #41D3×C3⋊D3semidirect
569. C32#5769semidirect
570. C32#5774semidirect
636. C3#10595461semidirect
637. C3#10596240semidirect

Maximal subgroups

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

subgroupindex
#117246842
#117245802
C3×#641582
C32×#9502
#117235892
#117235722
#117235662
#117235472
C2×#887912
#117233672
#117233562
#105962403
#105954613