Finite Group Databasebeta

← #13178211

Subgroups — #13178211

Order 1728.

1224 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64094C212yesmaximal, complemented
2C3×#64090C211yesmaximal, characteristic, complemented
3#11723725C212yesmaximal, complemented
4#11723717C211yesmaximal, characteristic, complemented
5#11723715C211yesmaximal, characteristic, complemented
6#11723695C211yesmaximal, characteristic, complemented
7#11723685C211yesmaximal, characteristic, complemented
8#11723683C212yesmaximal, complemented
9#11723678C211yesmaximal, characteristic, complemented
10#11723673C211yesmaximal, characteristic, complemented
11#11723670C212yesmaximal, complemented
12#10594201—32nomaximal
13#10594150—31nomaximal
14C2×C3×#638C2211yescharacteristic
15C2×C3×#638C2212yes—
16C22×C3×C3⋊Dic3C2211yescharacteristic
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C3×#634C2212yes—
19C2×#6295C2211yescharacteristic
20C2×#6295C2212yes—
21C2×#6294C2212yes—
22C33×C22⋊C4C2211yescharacteristic
23C3×#3757C2212yes—
24C3×#3752C2211yescharacteristic
25C3×#3752C2212yes—
26C3×#3752C2212yes—
27C3×#3750C2211yescharacteristic
28C3×C4×C3⋊Dic3C2211yescharacteristic
29C3×C4×C3⋊Dic3C2212yes—
30#88815C2211yescharacteristic
31#88815C2211yescharacteristic
32#88815C2212yes—
33#88815C2211yescharacteristic
34#88814C2212yes—
35#88814C2212yes—
36#88813C2212yes—
37#64094D312yescomplemented
38#64090D311yescharacteristic, complemented
39C3×#885—32no—
40C3×#885—32no—
41C3×#885—62no—
42C3×#67⋊C4—32no—
43C3×#67⋊C4—61no—
44#63980—31no—
45#63978—32no—
46#63971—32no—
47#63970—32no—
48#63958—32no—
49#63957—31no—
50#63955—32no—
51#63913—32no—
52#63912—31no—
53#63907—32no—
54#63907—32no—
55#63906—32no—
56#63902—31no—
57#63880—32no—
58#63878—32no—
59#63877—32no—
60#63874—31no—
61#63873—31no—
62#63866—32no—
63#63855—32no—
64#63836—32no—
65C23×C33C2311yescharacteristic
66C22×C3×C3⋊D3C2312yes—
67C22×C3×C3⋊D3C2311yescharacteristic
68C2×C33×C4C2311yescharacteristic
69C2×C33×C4C2311yescharacteristic
70C3×#638—22no—
71C3×#638—42no—
72C2×C3×C3⋊Dic3—22no—
73C2×C3×C3⋊Dic3C2312yes—
74C2×C3×C3⋊Dic3C2312yes—
75C2×C3×C3⋊Dic3C2311yescharacteristic
76C2×C3×C3⋊Dic3C2312yes—
77C2×C3×C3⋊Dic3C2311yescharacteristic
78C2×C3×C3⋊Dic3C2311yescharacteristic
79C3×C4×C3⋊D3—22no—
80C3×#634—42no—
81#6295—22no—
82#6295—42no—
83#6294—24no—
84#5670—92no—
85#5563—91no—
86C2×#638C2×D311yescharacteristic
87C2×#638C2×D312yes—
88C22×C3⋊Dic3C2×D311yescharacteristic
89C2×C4×C3⋊D3C2×D311yescharacteristic
90C2×#634C2×D312yes—
91C2×C3×#67—32no—
92C2×C3×#67—32no—
93C2×C3×#67—32no—
94C2×C3×#67—61no—
95C2×C3×#67—62no—
96C22×C3×Dic3—32no—
97C22×C3×Dic3—61no—
98C2×C3×C4×D3—32no—
99C2×C3×C4×D3—61no—
100C2×C3×Dic6—32no—
101C2×C3×Dic6—32no—
102C2×C3×Dic6—62no—
103C2×#626—32no—
104C2×#626—32no—
105C2×#626—32no—
106C2×#625—31no—
107C2×#625—32no—
108C2×#624—32no—
109C2×D3×Dic3—32no—
110C2×D3×Dic3—32no—
111C32×C22⋊C4—21no—
112C32×C22⋊C4—22no—
113C32×C22⋊C4—41no—
114C32×C22⋊C4C2×D312yes—
115C32×C22⋊C4C2×D311yescharacteristic
116#3757C2×D312yes—
117#3752C2×D311yescharacteristic
118#3752C2×D312yes—
119#3752C2×D312yes—
120#3750C2×D311yescharacteristic
121C4×C3⋊Dic3C2×D311yescharacteristic
122C4×C3⋊Dic3C2×D312yes—
123C3×#217—32no—
124C3×#217—32no—
125C3×#217—62no—
126C3×(C2×D3)⋊C4—32no—
127C3×(C2×D3)⋊C4—32no—
128C3×(C2×D3)⋊C4—32no—
129C3×(C2×D3)⋊C4—32no—
130C3×(C2×D3)⋊C4—32no—
131C3×(C2×D3)⋊C4—61no—
132C3×(C2×D3)⋊C4—62no—
133C3×(C2×D3)⋊C4—62no—
134C3×Dic3⋊C4—32no—
135C3×Dic3⋊C4—61no—
136C3×C4×Dic3—32no—
137C3×C4×Dic3—32no—
138C3×C4×Dic3—32no—
139C3×C4×Dic3—61no—
140C3×C4×Dic3—62no—
141#3724—31no—
142#3724—32no—
143#3724—31no—
144#3724—31no—
145Dic3⋊Dic3—32no—
146Dic3⋊Dic3—32no—
147Dic3⋊Dic3—32no—
148Dic3⋊Dic3—32no—
149Dic3⋊Dic3—32no—
150#3722—32no—
151#3722—32no—
152(C2×D3)⋊Dic3—32no—
153(C2×D3)⋊Dic3—32no—
154(C2×D3)⋊Dic3—32no—
155(C2×D3)⋊Dic3—32no—
156Dic32—32no—
157Dic32—32no—
158Dic32—32no—
159C22×C33—21no—
160C22×C33—21no—
161C22×C33—21no—
162C22×C33C2411yescharacteristic
163C2×C3×C3⋊D3—21no—
164C2×C3×C3⋊D3—22no—
165C2×C3×C3⋊D3—22no—
166C2×C3×C3⋊D3—21no—
167C2×C3×C3⋊D3—21no—
168C2×C3×C3⋊D3—22no—
169C33×C4—21no—
170C33×C4—21no—
171C3×C3⋊Dic3—22no—
172C3×C3⋊Dic3—21no—
173C3×C3⋊Dic3—22no—
174C3×C3⋊Dic3—22no—
175C3×C3⋊Dic3#4112yes—
176C3×C3⋊Dic3#4112yes—
177C3×#124—92no—
178C3×C4×D4—91no—
179#947—92no—
180#944—92no—
181#939—92no—
182#939—92no—
183#938—92no—
184#930—91no—
185#930—91no—
186#926—91no—
187#926—91no—
188#892—92no—
189#885—32no—
190#885—32no—
191#885—62no—
192#885—92no—
193#885—92no—
194#884—92no—
195#884—92no—
196#67⋊C4—32no—
197#67⋊C4—61no—
198#67⋊C4—92no—
199#877—92no—
200#877—92no—
201C4×D12—91no—
202#872—92no—
203#872—92no—
204#869—91no—
205C23×C32—21no—
206C23×C32—22no—
207C23×C32—41no—
208C23×C32C22×D311yescharacteristic
209C23×C32C22×D312yes—
210C22×C3⋊D3C22×D311yescharacteristic
211C22×C3⋊D3C22×D312yes—
212C22×C3×D3—32no—
213C22×C3×D3—32no—
214C22×C3×D3—32no—
215C22×C3×D3—61no—
216C22×C3×D3—62no—
217C2×C32×C4—22no—
218C2×C32×C4—21no—
219C2×C32×C4—21no—
220C2×C32×C4—22no—
221C2×C32×C4—41no—
222C2×C32×C4—41no—
223C2×C32×C4C22×D311yescharacteristic
224C2×C32×C4C22×D312yes—
225C2×C32×C4C22×D312yes—
226C2×C32×C4C22×D311yescharacteristic
227#638—22no—
228#638—42no—
229C2×C3⋊Dic3—22no—
230C2×C3⋊Dic3C22×D311yescharacteristic
231C2×C3⋊Dic3C22×D312yes—
232C2×C3⋊Dic3C22×D312yes—
233C2×C3⋊Dic3C22×D312yes—
234C2×C3⋊Dic3C22×D311yescharacteristic
235C2×C3⋊Dic3C22×D311yescharacteristic
236C4×C3⋊D3—22no—
237#634—42no—
238C3×#67—64no—
239C3×#67—64no—
240C3×#67—122no—
241C3×#67—122no—
242C3×#67—124no—
243C2×C3×Dic3—32no—
244C2×C3×Dic3—32no—
245C2×C3×Dic3—32no—
246C2×C3×Dic3—32no—
247C2×C3×Dic3—32no—
248C2×C3×Dic3—32no—
249C2×C3×Dic3—32no—
250C2×C3×Dic3—32no—
251C2×C3×Dic3—32no—
252C2×C3×Dic3—61no—
253C2×C3×Dic3—62no—
254C2×C3×Dic3—61no—
255C2×C3×Dic3—62no—
256C2×C3×Dic3—61no—
257C2×C3×Dic3—62no—
258C2×C3×Dic3—64no—
259C2×C3×Dic3—122no—
260C3×C4×D3—64no—
261C3×C4×D3—122no—
262C3×Dic6—122no—
263C3×Dic6—122no—
264C3×Dic6—124no—
265#626—64no—
266#626—122no—
267#626—122no—
268#625—62no—
269#625—122no—
270#624—64no—
271D3×Dic3—64no—
272D3×Dic3—64no—
273#555—271noSylow
274C2×C33—41no—
275C2×C33C2×#4111yescharacteristic
276C2×C33C2×#4111yescharacteristic
277C2×C33H(5,𝔽2)11yescharacteristic
278C3×C3⋊D3—41no—
279C3×C3⋊D3—42no—
280C2×C3×Q8—92no—
281C2×C3×D4—91no—
282C2×C3×D4—92no—
283C22×C3×C4—91no—
284C22×C3×C4—91no—
285C2×#67—32no—
286C2×#67—32no—
287C2×#67—32no—
288C2×#67—61no—
289C2×#67—62no—
290C2×#67—92no—
291C2×#67—92no—
292C2×#67—92no—
293C22×Dic3—32no—
294C22×Dic3—61no—
295C22×Dic3—92no—
296C2×D12—91no—
297C2×D12—92no—
298C2×C4×D3—32no—
299C2×C4×D3—61no—
300C2×C4×D3—92no—
301C2×C4×D3—92no—
302C2×Dic6—32no—
303C2×Dic6—32no—
304C2×Dic6—62no—
305C3×C4⋊C4—91no—
306C3×C22⋊C4—21no—
307C3×C22⋊C4—22no—
308C3×C22⋊C4—41no—
309C3×C22⋊C4—92no—
310C3×C22⋊C4—92no—
311C3×C22⋊C4—92no—
312C3×C22⋊C4—91no—
313C3×C22⋊C4D3212yes—
314C3×C22⋊C4D3211yescharacteristic
315C3×C42—91no—
316C3×C42—92no—
317#217—32no—
318#217—32no—
319#217—62no—
320#217—92no—
321#217—92no—
322(C2×D3)⋊C4—32no—
323(C2×D3)⋊C4—32no—
324(C2×D3)⋊C4—32no—
325(C2×D3)⋊C4—32no—
326(C2×D3)⋊C4—32no—
327(C2×D3)⋊C4—61no—
328(C2×D3)⋊C4—62no—
329(C2×D3)⋊C4—62no—
330(C2×D3)⋊C4—92no—
331(C2×D3)⋊C4—92no—
332(C2×D3)⋊C4—92no—
333(C2×D3)⋊C4—92no—
334C4⋊Dic3—92no—
335C4⋊Dic3—91no—
336C4⋊Dic3—91no—
337C4⋊Dic3—91no—
338Dic3⋊C4—32no—
339Dic3⋊C4—61no—
340Dic3⋊C4—92no—
341Dic3⋊C4—92no—
342Dic3⋊C4—92no—
343Dic3⋊C4—92no—
344Dic3⋊C4—92no—
345C4×Dic3—32no—
346C4×Dic3—32no—
347C4×Dic3—32no—
348C4×Dic3—61no—
349C4×Dic3—62no—
350C4×Dic3—92no—
351C4×Dic3—92no—
352C4×Dic3—92no—
353C22×C32—21no—
354C22×C32—22no—
355C22×C32—21no—
356C22×C32—22no—
357C22×C32—22no—
358C22×C32—22no—
359C22×C32—21no—
360C22×C32—22no—
361C22×C32—21no—
362C22×C32—41no—
363C22×C32—42no—
364C22×C32—42no—
365C22×C32—41no—
366C22×C32—41no—
367C22×C32—42no—
368C22×C32—42no—
369C22×C32—81no—
370C22×C32—81no—
371C22×C32C23×D311yescharacteristic
372C22×C32C23×D312yes—
373C2×C3⋊D3—21no—
374C2×C3⋊D3—21no—
375C2×C3⋊D3—22no—
376C2×C3⋊D3—22no—
377C2×C3⋊D3—22no—
378C2×C3⋊D3—21no—
379C2×C3×D3—62no—
380C2×C3×D3—62no—
381C2×C3×D3—62no—
382C2×C3×D3—62no—
383C2×C3×D3—62no—
384C2×C3×D3—62no—
385C2×C3×D3—62no—
386C2×C3×D3—62no—
387C2×C3×D3—62no—
388C2×C3×D3—62no—
389C2×C3×D3—121no—
390C2×C3×D3—121no—
391C2×C3×D3—122no—
392C2×C3×D3—122no—
393C2×C3×D3—122no—
394C32×C4—21no—
395C32×C4—22no—
396C32×C4—22no—
397C32×C4—21no—
398C32×C4—42no—
399C32×C4—41no—
400C32×C4—42no—
401C32×C4—41no—
402C32×C4—81no—
403C32×C4—81no—
404C3⋊Dic3—22no—
405C3⋊Dic3—21no—
406C3⋊Dic3—22no—
407C3⋊Dic3—22no—
408C3⋊Dic3#23512yes—
409C3⋊Dic3#23712yes—
410C3×Dic3—34no—
411C3×Dic3—34no—
412C3×Dic3—62no—
413C3×Dic3—62no—
414C3×Dic3—64no—
415C3×Dic3—62no—
416C3×Dic3—62no—
417C3×Dic3—62no—
418C3×Dic3—62no—
419C3×Dic3—62no—
420C3×Dic3—62no—
421C3×Dic3—62no—
422C3×Dic3—122no—
423C3×Dic3—122no—
424C3×Dic3—121no—
425C4⋊Q8—271no—
426#124—92no—
427#124—272no—
428#121—271no—
429#121—272no—
430#121—271no—
431C4×D4—91no—
432C4×D4—271no—
433#117—271no—
434#117—272no—
435#117—271no—
436C33#55511yescharacteristic, complemented, Sylow
437C23×C3—21no—
438C23×C3—22no—
439C23×C3—41no—
440C23×C3—91no—
441C23×C3—92no—
442C23×C3C2×D3211yescharacteristic
443C23×C3C2×D3212yes—
444C22×D3—32no—
445C22×D3—32no—
446C22×D3—32no—
447C22×D3—61no—
448C22×D3—62no—
449C3×Q8—362no—
450C3×D4—182no—
451C3×D4—362no—
452C2×C3×C4—22no—
453C2×C3×C4—21no—
454C2×C3×C4—21no—
455C2×C3×C4—22no—
456C2×C3×C4—41no—
457C2×C3×C4—41no—
458C2×C3×C4—91no—
459C2×C3×C4—91no—
460C2×C3×C4—91no—
461C2×C3×C4—92no—
462C2×C3×C4—92no—
463C2×C3×C4—92no—
464C2×C3×C4—182no—
465C2×C3×C4—182no—
466C2×C3×C4C2×D3211yescharacteristic
467C2×C3×C4C2×D3212yes—
468C2×C3×C4C2×D3212yes—
469C2×C3×C4C2×D3211yescharacteristic
470#67—64no—
471#67—64no—
472#67—122no—
473#67—122no—
474#67—124no—
475#67—184no—
476#67—362no—
477#67—362no—
478C2×Dic3—32no—
479C2×Dic3—32no—
480C2×Dic3—32no—
481C2×Dic3—32no—
482C2×Dic3—32no—
483C2×Dic3—32no—
484C2×Dic3—32no—
485C2×Dic3—32no—
486C2×Dic3—32no—
487C2×Dic3—61no—
488C2×Dic3—62no—
489C2×Dic3—61no—
490C2×Dic3—62no—
491C2×Dic3—61no—
492C2×Dic3—62no—
493C2×Dic3—64no—
494C2×Dic3—122no—
495C2×Dic3—184no—
496D12—182no—
497D12—362no—
498C4×D3—64no—
499C4×D3—122no—
500C4×D3—184no—
501C4×D3—184no—
502Dic6—122no—
503Dic6—122no—
504Dic6—124no—
505C2×C32—22no—
506C2×C32—21no—
507C2×C32—22no—
508C2×C32—21no—
509C2×C32—21no—
510C2×C32—22no—
511C2×C32—41no—
512C2×C32—41no—
513C2×C32—41no—
514C2×C32—41no—
515C2×C32—42no—
516C2×C32—42no—
517C2×C32—82no—
518C2×C32—82no—
519C2×C32C2×#23511yescharacteristic
520C2×C32C2×#23711yescharacteristic
521C2×C32#100412yes—
522C2×C32D3×#4112yes—
523C2×C32D3×#4112yes—
524C2×C32#100911yescharacteristic
525C3⋊D3—41no—
526C3⋊D3—42no—
527C3×D3—122no—
528C3×D3—122no—
529C3×D3—122no—
530C3×D3—122no—
531C3×D3—242no—
532C2×Q8—92no—
533C2×D4—91no—
534C2×D4—92no—
535C2×D4—271no—
536C2×D4—272no—
537C22×C4—91no—
538C22×C4—91no—
539C22×C4—272no—
540C4⋊C4—91no—
541C4⋊C4—271no—
542C4⋊C4—272no—
543C4⋊C4—271no—
544C4⋊C4—271no—
545C22⋊C4—92no—
546C22⋊C4—92no—
547C22⋊C4—91no—
548C22⋊C4—92no—
549C22⋊C4—272no—
550C22⋊C4—272no—
551C22⋊C4#111311yescharacteristic
552C42—92no—
553C42—91no—
554C42—272no—
555C22×C3—22no—
556C22×C3—22no—
557C22×C3—21no—
558C22×C3—22no—
559C22×C3—21no—
560C22×C3—21no—
561C22×C3—22no—
562C22×C3—21no—
563C22×C3—22no—
564C22×C3—41no—
565C22×C3—41no—
566C22×C3—42no—
567C22×C3—42no—
568C22×C3—42no—
569C22×C3—42no—
570C22×C3—41no—
571C22×C3—81no—
572C22×C3—81no—
573C22×C3—182no—
574C22×C3—181no—
575C22×C3—182no—
576C22×C3—181no—
577C22×C3—181no—
578C22×C3—182no—
579C22×C3C22×D3212yes—
580C22×C3C22×D3211yescharacteristic
581C2×D3—62no—
582C2×D3—62no—
583C2×D3—62no—
584C2×D3—62no—
585C2×D3—62no—
586C2×D3—62no—
587C2×D3—62no—
588C2×D3—62no—
589C2×D3—62no—
590C2×D3—62no—
591C2×D3—121no—
592C2×D3—121no—
593C2×D3—122no—
594C2×D3—122no—
595C2×D3—122no—
596C3×C4—22no—
597C3×C4—21no—
598C3×C4—21no—
599C3×C4—22no—
600C3×C4—42no—
601C3×C4—41no—
602C3×C4—42no—
603C3×C4—41no—
604C3×C4—81no—
605C3×C4—81no—
606C3×C4—92no—
607C3×C4—92no—
608C3×C4—182no—
609C3×C4—181no—
610C3×C4—182no—
611C3×C4—182no—
612Dic3—34no—
613Dic3—34no—
614Dic3—62no—
615Dic3—62no—
616Dic3—64no—
617Dic3—62no—
618Dic3—62no—
619Dic3—62no—
620Dic3—62no—
621Dic3—62no—
622Dic3—62no—
623Dic3—62no—
624Dic3—122no—
625Dic3—121no—
626Dic3—122no—
627C32—21no—
628C32—22no—
629C32—41no—
630C32#556311yescharacteristic, complemented
631C32#567012yescomplemented
632C23—92no—
633C23—91no—
634C23C2×#111311yescharacteristic
635Q8—362no—
636D4—182no—
637D4—362no—
638D4—542no—
639D4—1082no—
640C2×C4—91no—
641C2×C4—91no—
642C2×C4—92no—
643C2×C4—92no—
644C2×C4—91no—
645C2×C4—92no—
646C2×C4—182no—
647C2×C4—182no—
648C2×C4—544no—
649C2×C4C2×#111311yescharacteristic
650C2×C4C2×#111311yescharacteristic
651C2×C3—21no—
652C2×C3—22no—
653C2×C3—21no—
654C2×C3—22no—
655C2×C3—22no—
656C2×C3—21no—
657C2×C3—42no—
658C2×C3—41no—
659C2×C3—42no—
660C2×C3—41no—
661C2×C3—41no—
662C2×C3—41no—
663C2×C3—82no—
664C2×C3—82no—
665C2×C3—361no—
666C2×C3—362no—
667C2×C3D3×#23512yes—
668C2×C3#6430611yescharacteristic
669C2×C3D3×#23712yes—
670C2×C3#6431211yescharacteristic
671C2×C3#6431311yescharacteristic
672C2×C3#6431412yes—
673D3—122no—
674D3—122no—
675D3—122no—
676D3—122no—
677D3—242no—
678C22—21no—
679C22—21no—
680C22—21no—
681C22—182no—
682C22—181no—
683C22—181no—
684C22—182no—
685C22—182no—
686C22—181no—
687C22C22×#111311yescharacteristic
688C4—21no—
689C4—21no—
690C4—92no—
691C4—92no—
692C4—182no—
693C4—181no—
694C4—182no—
695C4—182no—
696C3—21no—
697C3—22no—
698C3—41no—
699C3#1059415011yescharacteristic, complemented
700C3#1059420112yescomplemented
701C2—41no—
702C2—361no—
703C2—362no—
704C2#1172461811yescharacteristic
705C2#1172462211yescharacteristic
706C2#1172462311yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊C4C22
hypercenterC22⋊C4#1113
hypocenterC33#555
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#64094C2semidirect
2. C3×#64090C2semidirect
3. #11723725C2semidirect
4. #11723717C2semidirect
5. #11723715C2semidirect
6. #11723695C2semidirect
7. #11723685C2semidirect
8. #11723683C2semidirect
9. #11723678C2semidirect
10. #11723673C2semidirect
11. #11723670C2semidirect
37. #64094D3semidirect
38. #64090D3semidirect
436. C33#555semidirect
630. C32#5563semidirect
631. C32#5670semidirect
699. C3#10594150semidirect
700. C3#10594201semidirect

Maximal subgroups

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

subgroupindex
C3×#640942
C3×#640902
#117237252
#117237172
#117237152
#117236952
#117236852
#117236832
#117236782
#117236732
#117236702
#105942013
#105941503