Finite Group Databasebeta

← #13177999

Subgroups — #13177999

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#11724579C211yesmaximal, characteristic, complemented
3C3×#64159C211yesmaximal, characteristic, complemented
4C32×#949C211yesmaximal, characteristic, complemented
5#11723587C212yesmaximal, complemented
6#11723572C212yesmaximal, complemented
7#11723564C212yesmaximal, complemented
8#11723547C212yesmaximal, complemented
9#11723371C211yesmaximal, characteristic, complemented
10C2×#88790C211yesmaximal, characteristic, complemented
11#11723357C211yesmaximal, characteristic, complemented
12#10596245—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
22C3×#3828C2211yescharacteristic, complemented
23C2×C32×D12C2211yescharacteristic, complemented
24D3×#638—22no—
25#89026C2212yescomplemented
26C3⋊D3×D12—21no—
27#89015C2212yescomplemented
28#89014—21no—
29C3×#3756C2211yescharacteristic, complemented
30C3×#3755C2211yescharacteristic, complemented
31C3×#3754C2211yescharacteristic, complemented
32C3×#3753C2211yescharacteristic, complemented
33C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
34C32×#215C2212yescomplemented
35C32×#213C2212yescomplemented
36C32×#208C2211yescharacteristic, complemented
37#88797C2212yescomplemented
38#88791C2212yescomplemented
39#88790C2212yescomplemented
40#88790C2212yescomplemented
41#88788C2212yescomplemented
42#88785C2212yescomplemented
43#64366—31no—
44#64366—34no—
45#64366—64no—
46#64361—31no—
47#64307—34no—
48C32×AGL(1,ℤ/8)—31no—
49#64159D311yescharacteristic, complemented
50C2×#3753—31no—
51#64125—32no—
52#64123—32no—
53#64120—32no—
54D8×C3⋊D3—32no—
55#64117—31no—
56#64116—31no—
57C3×#950—34no—
58C3×#949D314yescomplemented
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×C32×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×#638—21no—
85C3×C4⋊(C3⋊D3)C2311yescharacteristic
86C3×C4×C3⋊D3—21no—
87C3×#634C2311yescharacteristic
88C32×D12—21no—
89C32×D12C2312yes—
90#6291—22no—
91#6290—22no—
92D3×C3⋊Dic3—22no—
93C3×C3⋊(C3⋊C8)C2311yescharacteristic
94C3×C3⋊(C3⋊C8)C2311yescharacteristic
95C32×C3⋊C8C2312yes—
96#5774—94no—
97#5769—91no—
98C32×#41—24no—
99C32×#41C2×D314yescomplemented
100C32×#41C2×D311yescharacteristic, complemented
101C2×C32×D4—31no—
102C2×#638—62no—
103#3832—32no—
104#3832—38no—
105#3832—68no—
106#3830—32no—
107#3830—61no—
108D4×C3⋊D3—32no—
109D4×C3⋊D3—38no—
110D4×C3⋊D3—32no—
111D4×C3⋊D3—61no—
112D4×C3⋊D3—62no—
113D4×C3⋊D3—68no—
114D4×C3⋊D3—68no—
115D4×C3⋊D3—616no—
116#3828—31no—
117#3828—34no—
118#3828—61no—
119#3828—64no—
120#3828—64no—
121#3828—68no—
122#3828C2×D311yescharacteristic, complemented
123C2×C4⋊(C3⋊D3)—31no—
124C2×C4⋊(C3⋊D3)—34no—
125C2×C4⋊(C3⋊D3)—61no—
126C2×C4⋊(C3⋊D3)—64no—
127C2×C4⋊(C3⋊D3)—64no—
128C2×C4⋊(C3⋊D3)—68no—
129C3×#235—34no—
130C2×C3×D12C2×D314yescomplemented
131D3×#67—68no—
132#3802—38no—
133D3×D12—64no—
134#3796—38no—
135#3795—64no—
136C32×SD16—32no—
137C32×D8—32no—
138C32×M16—31no—
139#3756C2×D311yescharacteristic, complemented
140#3755C2×D311yescharacteristic, complemented
141#3754C2×D311yescharacteristic, complemented
142#3753—32no—
143#3753—32no—
144#3753C2×D311yescharacteristic, complemented
145C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
146#3745—32no—
147#3744—32no—
148#3743—32no—
149C8×C3⋊D3—32no—
150C3×#216—34no—
151C3×#215—34no—
152C3×#215C2×D318yescomplemented
153C3×#214—34no—
154C3×#213—34no—
155C3×#213C2×D318yescomplemented
156C3×#208C2×D314yescomplemented
157C2×C3×C3⋊C8—34no—
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—21no—
171C2×C32×D3—22no—
172C2×C32×D3—81no—
173D3×C3⋊D3—82no—
174C33×C4—21no—
175C33×C4C2×D411yescharacteristic
176C33×C4C2411yescharacteristic
177C3⋊(C3⋊Dic3)C2×D412yes—
178C3×C3⋊Dic3—21no—
179#1009—94no—
180#1009—91no—
181#1009—91no—
182#1009—184no—
183#1004—94no—
184C3×AGL(1,ℤ/8)—34no—
185C3×#135—91no—
186#950—34no—
187#949—91no—
188#949C3⋊D311yescharacteristic, complemented
189C2×#213—94no—
190#919—92no—
191#916—98no—
192#914—98no—
193#914—92no—
194#914—92no—
195#911—98no—
196D3×D8—98no—
197D3×D8—92no—
198#908—94no—
199#907—94no—
200C2×D24—91no—
201#901—91no—
202C23×C32—61no—
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—24no—
211C2×D32—68no—
212C32×Q8—24no—
213C32×Q8C22×D314yes—
214C32×Q8C22×D311yescharacteristic
215C32×D4—28no—
216C32×D4—24no—
217C32×D4—21no—
218C32×D4—24no—
219C32×D4—32no—
220C32×D4—61no—
221C32×D4C22×D314yes—
222C32×D4C22×D311yescharacteristic
223C2×C32×C4—28no—
224C2×C32×C4—24no—
225C2×C32×C4—21no—
226C2×C32×C4—24no—
227C2×C32×C4C22×D314yes—
228C2×C32×C4C22×D311yescharacteristic
229#638—21no—
230#638—32no—
231#638—38no—
232#638—68no—
233#638—62no—
234#638—62no—
235#638—62no—
236#638—68no—
237#638—616no—
238C2×C3⋊Dic3—62no—
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)—64no—
250C4⋊(C3⋊D3)—61no—
251C4⋊(C3⋊D3)C22×D311yescharacteristic
252C4×C3⋊D3—21no—
253C4×C3⋊D3—32no—
254C4×C3⋊D3—38no—
255C4×C3⋊D3—68no—
256C4×C3⋊D3—62no—
257C4×C3⋊D3—68no—
258C4×C3⋊D3—616no—
259#634—31no—
260#634—34no—
261#634—64no—
262#634C22×D311yescharacteristic
263C3×#67—64no—
264C3×D12—24no—
265C3×D12—34no—
266C3×D12C22×D318yes—
267C3×C4×D3—64no—
268C3×Dic6—34no—
269#626—68no—
270#625—68no—
271D3×Dic3—68no—
272C32×C8—32no—
273C3⋊(C3⋊C8)C22×D311yescharacteristic
274C3⋊(C3⋊C8)C22×D311yescharacteristic
275C3×C3⋊C8—34no—
276C3×C3⋊C8—34no—
277C3×C3⋊C8C22×D318yes—
278#576—271noSylow
279C2×C33—21no—
280C2×C33—41no—
281C2×C33C22×D411yescharacteristic
282C3⋊(C3⋊D3)—22no—
283C3⋊(C3⋊D3)—42no—
284C3×C3⋊D3—41no—
285C32×D3—42no—
286C3×#41—24no—
287C3×#41—91no—
288C3×#41D3214yescomplemented
289C3×#41C2×C3⋊D311yescharacteristic, complemented
290C2×C3×D4—34no—
291C2×#67—188no—
292#239—98no—
293#239—92no—
294#239—92no—
295#239—188no—
296#237—98no—
297#237—184no—
298D3×D4—98no—
299D3×D4—98no—
300D3×D4—92no—
301D3×D4—92no—
302D3×D4—188no—
303D3×D4—184no—
304D3×D4—188no—
305D3×D4—182no—
306D3×D4—182no—
307D3×D4—1816no—
308#235—34no—
309#235—94no—
310#235—91no—
311#235—184no—
312#235—181no—
313#235—184no—
314#235—181no—
315#235—188no—
316C2×D12—94no—
317C2×D12—91no—
318C2×D12—184no—
319C2×D12—181no—
320C2×D12—184no—
321C2×D12—181no—
322C2×D12—188no—
323C2×D12C2×C3⋊D311yescharacteristic, complemented
324C3×Q16—91no—
325C3×SD16—38no—
326C3×SD16—91no—
327C3×SD16—91no—
328C3×D8—38no—
329C3×D8—91no—
330C3×M16—34no—
331C2×C3×C8—91no—
332#216—34no—
333#215—34no—
334#215—92no—
335#215C2×C3⋊D312yescomplemented
336#214—34no—
337#213—34no—
338#213—98no—
339#213—98no—
340#213—92no—
341#213C2×C3⋊D312yescomplemented
342#208C2×C3⋊D311yescharacteristic, complemented
343C2×C3⋊C8—34no—
344D24—98no—
345D24—92no—
346D24—92no—
347#204—98no—
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—62no—
357C22×C32—241no—
358C22×C32D3×D414yes—
359C22×C32D3×D411yescharacteristic
360C2×C3⋊D3—21no—
361C2×C3⋊D3—32no—
362C2×C3⋊D3—38no—
363C2×C3⋊D3—68no—
364C2×C3⋊D3—616no—
365C2×C3⋊D3—68no—
366C2×C3⋊D3—62no—
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—28no—
375C2×C3×D3—64no—
376C2×C3×D3—84no—
377D32—248no—
378C32×C4—28no—
379C32×C4—24no—
380C32×C4—21no—
381C32×C4—24no—
382C32×C4—24no—
383C32×C4D3×D414yes—
384C32×C4D3×D411yescharacteristic
385C32×C4C23×D314yes—
386C32×C4C23×D311yescharacteristic
387C3⋊Dic3—21no—
388C3⋊Dic3—32no—
389C3⋊Dic3—38no—
390C3⋊Dic3—68no—
391C3×Dic3—64no—
392H(5,𝔽2)—271no—
393H(5,𝔽2)—271no—
394AGL(1,ℤ/8)—31no—
395AGL(1,ℤ/8)—271no—
396AGL(1,ℤ/8)—272no—
397AGL(1,ℤ/8)—272no—
398#135—91no—
399#135—272no—
400C2×D8—271no—
401C2×D8—272no—
402#131—271no—
403C33#57611yescharacteristic, complemented, Sylow
404C23×C3—64no—
405C22×D3—21no—
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—91no—
416C3×Q8C2×D3214yes—
417C3×Q8C22×C3⋊D311yescharacteristic
418C3×D4—28no—
419C3×D4—24no—
420C3×D4—21no—
421C3×D4—24no—
422C3×D4—38no—
423C3×D4—64no—
424C3×D4—91no—
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—181no—
433C2×C3×C4C2×D3214yes—
434C2×C3×C4C22×C3⋊D311yescharacteristic
435#67—64no—
436#67—98no—
437#67—92no—
438#67—188no—
439#67—188no—
440#67—182no—
441#67—188no—
442#67—188no—
443#67—182no—
444#67—1816no—
445C2×Dic3—188no—
446D12—21no—
447D12—34no—
448D12—98no—
449D12—92no—
450D12—94no—
451D12—91no—
452D12—184no—
453D12—188no—
454D12—188no—
455D12—182no—
456D12—188no—
457D12—1816no—
458D12—182no—
459D12—184no—
460D12—181no—
461D12C22×C3⋊D312yes—
462C4×D3—64no—
463C4×D3—98no—
464C4×D3—92no—
465C4×D3—188no—
466C4×D3—188no—
467C4×D3—182no—
468C4×D3—182no—
469C4×D3—1816no—
470Dic6—34no—
471Dic6—94no—
472Dic6—91no—
473Dic6—184no—
474C3×C8—38no—
475C3×C8—91no—
476C3×C8—91no—
477C3⋊C8—34no—
478C3⋊C8—34no—
479C3⋊C8C22×C3⋊D312yes—
480C2×C32—21no—
481C2×C32—24no—
482C2×C32—24no—
483C2×C32—44no—
484C2×C32—48no—
485C2×C32—41no—
486C2×C32—44no—
487C2×C32—122no—
488C2×C32C2×D3×D414yes—
489C2×C32C2×D3×D411yescharacteristic
490C3⋊D3—41no—
491C3⋊D3—62no—
492C3⋊D3—68no—
493C3⋊D3—128no—
494C3⋊D3—1216no—
495C3⋊D3—122no—
496C3⋊D3—128no—
497C3×D3—48no—
498C3×D3—124no—
499#41—91no—
500#41—271no—
501#41—272no—
502#41—272no—
503#41—541no—
504#41—541no—
505#41D3×C3⋊D311yescharacteristic, complemented
506C2×D4—31no—
507C2×D4—272no—
508C2×D4—272no—
509C2×D4—271no—
510C2×D4—541no—
511C2×D4—541no—
512C2×D4—542no—
513C2×D4—542no—
514Q16—91no—
515SD16—32no—
516SD16—91no—
517SD16—91no—
518SD16—272no—
519D8—32no—
520D8—91no—
521D8—272no—
522D8—272no—
523D8—272no—
524M16—31no—
525M16—272no—
526C2×C8—91no—
527C2×C8—272no—
528C22×C3—28no—
529C22×C3—24no—
530C22×C3—21no—
531C22×C3—24no—
532C22×C3—68no—
533C22×C3—181no—
534C22×C3—244no—
535C22×C3#381114yes—
536C22×C3D4×C3⋊D311yescharacteristic
537C2×D3—22no—
538C2×D3—64no—
539C2×D3—81no—
540C2×D3—98no—
541C2×D3—92no—
542C2×D3—188no—
543C2×D3—1816no—
544C2×D3—188no—
545C2×D3—182no—
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—181no—
560C3×C4#381114yes—
561C3×C4D4×C3⋊D311yescharacteristic
562C3×C4C22×D3214yes—
563C3×C4C23×C3⋊D311yescharacteristic
564Dic3—64no—
565Dic3—98no—
566Dic3—92no—
567Dic3—188no—
568C32—24no—
569C32#576911yescharacteristic, complemented
570C32#577414yescomplemented
571C23—61no—
572C23—542no—
573C23—542no—
574C23—541no—
575Q8—91no—
576Q8—271no—
577Q8C2×D3×C3⋊D311yescharacteristic
578D4—21no—
579D4—32no—
580D4—61no—
581D4—91no—
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—181no—
594C2×C4—272no—
595C2×C4—542no—
596C2×C4—542no—
597C2×C4C2×D3×C3⋊D311yescharacteristic
598C8—32no—
599C8—91no—
600C8—91no—
601C2×C3—24no—
602C2×C3—21no—
603C2×C3—24no—
604C2×C3—44no—
605C2×C3—48no—
606C2×C3—44no—
607C2×C3—41no—
608C2×C3—128no—
609C2×C3—361no—
610C2×C3C2×#381114yes—
611C2×C3C2×D4×C3⋊D311yescharacteristic
612D3—42no—
613D3—124no—
614D3—188no—
615D3—182no—
616D3—368no—
617D3—3616no—
618D3—368no—
619D3—362no—
620C22—21no—
621C22—62no—
622C22—181no—
623C22—241no—
624C22—272no—
625C22—542no—
626C22—2161no—
627C22—2162no—
628C22—2162no—
629C22#8903911yescharacteristic
630C4—21no—
631C4—181no—
632C4—272no—
633C4#8903911yescharacteristic
634C4C22×D3×C3⋊D311yescharacteristic
635C3—24no—
636C3#1059546114yescomplemented
637C3#1059624511yescharacteristic, complemented
638C2—21no—
639C2—41no—
640C2—122no—
641C2—361no—
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. #11724579C2semidirect
3. C3×#64159C2semidirect
4. C32×#949C2semidirect
5. #11723587C2semidirect
6. #11723572C2semidirect
7. #11723564C2semidirect
8. #11723547C2semidirect
9. #11723371C2semidirect
10. C2×#88790C2semidirect
11. #11723357C2semidirect
14. C33×#41C22semidirect
15. #89080C22semidirect
17. D4×C3⋊(C3⋊D3)C22semidirect
21. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
22. C3×#3828C22semidirect
23. C2×C32×D12C22semidirect
25. #89026C22semidirect
27. #89015C22semidirect
29. C3×#3756C22semidirect
30. C3×#3755C22semidirect
31. C3×#3754C22semidirect
32. C3×#3753C22semidirect
33. C2×C3×C3⋊(C3⋊C8)C22semidirect
34. C32×#215C22semidirect
35. C32×#213C22semidirect
36. C32×#208C22semidirect
37. #88797C22semidirect
38. #88791C22semidirect
39. #88790C22semidirect
40. #88790C22semidirect
41. #88788C22semidirect
42. #88785C22semidirect
49. #64159D3semidirect
58. C3×#949D3semidirect
99. C32×#41C2×D3semidirect
100. C32×#41C2×D3semidirect
122. #3828C2×D3semidirect
130. C2×C3×D12C2×D3semidirect
139. #3756C2×D3semidirect
140. #3755C2×D3semidirect
141. #3754C2×D3semidirect
144. #3753C2×D3semidirect
145. C2×C3⋊(C3⋊C8)C2×D3semidirect
152. C3×#215C2×D3semidirect
155. C3×#213C2×D3semidirect
156. C3×#208C2×D3semidirect
188. #949C3⋊D3semidirect
288. C3×#41D32semidirect
289. C3×#41C2×C3⋊D3semidirect
323. C2×D12C2×C3⋊D3semidirect
335. #215C2×C3⋊D3semidirect
341. #213C2×C3⋊D3semidirect
342. #208C2×C3⋊D3semidirect
403. C33#576semidirect
505. #41D3×C3⋊D3semidirect
569. C32#5769semidirect
570. C32#5774semidirect
636. C3#10595461semidirect
637. C3#10596245semidirect

Maximal subgroups

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

subgroupindex
#117246842
#117245792
C3×#641592
C32×#9492
#117235872
#117235722
#117235642
#117235472
#117233712
C2×#887902
#117233572
#105962453
#105954613