Finite Group Databasebeta

← #13178328

Subgroups — #13178328

Order 1728.

1064 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64107C212yesmaximal, complemented
2C3×#64096C211yesmaximal, characteristic, complemented
3C4⋊#6296C211yesmaximal, characteristic, complemented
4C4×#6296C211yesmaximal, characteristic, complemented
5#11723688C212yesmaximal, complemented
6#11723685C211yesmaximal, characteristic
7#11723683C212yesmaximal, complemented
8#11723675C212yesmaximal, complemented
9#11723672C211yesmaximal, characteristic
10#11723672C211yesmaximal, characteristic
11#11723665C211yesmaximal, characteristic, complemented
12#10594472—31nomaximal
13#10594421—32nomaximal
14C2×C3×#634C2211yescharacteristic
15C2×#6296C2211yescharacteristic
16C2×#6296C2211yescharacteristic
17C2×#6295C2211yescharacteristic
18C2×#6294C2212yes—
19C33×C4⋊C4C2211yescharacteristic
20C3×#3752C2212yes—
21C3×#3752C2212yes—
22C3×#3752C2212yes—
23C3×C4⋊(C3⋊Dic3)C2212yes—
24C3×#3750C2211yescharacteristic
25C3×#3750C2211yescharacteristic
26C3×#3750C2212yes—
27C3×C4×C3⋊Dic3C2211yescharacteristic
28C3×C4×C3⋊Dic3C2211yescharacteristic
29C3×C4×C3⋊Dic3C2211yescharacteristic
30C3×C4×C3⋊Dic3C2212yes—
31#88815C2211yescharacteristic
32#88815C2212yes—
33#88815C2212yes—
34#88814C2212yes—
35#88814C2212yes—
36#88813C2212yes—
37#64107D312yescomplemented
38#64096D311yescharacteristic, complemented
39C3×#898—32no—
40C3×#898—32no—
41C3×#898—62no—
42C3×#887—32no—
43C3×#887—61no—
44#63919—31no—
45#63918—32no—
46C4×#627—32no—
47C4×#627—31no—
48#63915—31no—
49#63915—31no—
50#63902—31no—
51#63896—32no—
52#63888—32no—
53#63887—32no—
54#63880—32no—
55#63878—32no—
56#63877—32no—
57#63861—32no—
58#63858—32no—
59#63852—32no—
60#63852—32no—
61#63848—32no—
62#63838—31no—
63#63837—32no—
64#63836—32no—
65C22×C3×C3⋊D3C2312yes—
66C2×C33×C4C2311yescharacteristic
67C2×C33×C4C2311yescharacteristic
68C2×C33×C4C2311yescharacteristic
69C2×C3×C3⋊Dic3C2311yescharacteristic
70C2×C3×C3⋊Dic3C2312yes—
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C2×C3×C3⋊Dic3C2312yes—
73C2×C3×C3⋊Dic3C2312yes—
74C2×C3×C3⋊Dic3C2311yescharacteristic
75C2×C3×C3⋊Dic3C2311yescharacteristic
76C3×#634—22no—
77#6296—22no—
78#6296—22no—
79#6295—22no—
80#6294—24no—
81#5702—92no—
82#5584—91no—
83C2×#634C2×D311yescharacteristic
84C2×C3×Dic6—32no—
85C2×C3×Dic6—61no—
86C2×#627—32no—
87C2×#627—31no—
88C2×#627—32no—
89C2×#627—31no—
90C2×#626—32no—
91C2×#625—31no—
92C2×#624—32no—
93C2×D3×Dic3—32no—
94C2×D3×Dic3—32no—
95C32×C4⋊C4—21no—
96C32×C4⋊C4—22no—
97C32×C4⋊C4—41no—
98C32×C4⋊C4C2×D312yes—
99C32×C4⋊C4C2×D311yescharacteristic
100#3752C2×D312yes—
101#3752C2×D312yes—
102#3752C2×D312yes—
103C4⋊(C3⋊Dic3)C2×D312yes—
104#3750C2×D312yes—
105#3750C2×D311yescharacteristic
106#3750C2×D311yescharacteristic
107C4×C3⋊Dic3C2×D311yescharacteristic
108C4×C3⋊Dic3C2×D311yescharacteristic
109C4×C3⋊Dic3C2×D311yescharacteristic
110C4×C3⋊Dic3C2×D312yes—
111C3×(C2×D3)⋊C4—32no—
112C3×(C2×D3)⋊C4—32no—
113C3×(C2×D3)⋊C4—32no—
114C3×(C2×D3)⋊C4—32no—
115C3×(C2×D3)⋊C4—32no—
116C3×(C2×D3)⋊C4—32no—
117C3×(C2×D3)⋊C4—62no—
118C3×(C2×D3)⋊C4—62no—
119C3×(C2×D3)⋊C4—62no—
120C3×C4⋊Dic3—32no—
121C3×C4⋊Dic3—32no—
122C3×C4⋊Dic3—62no—
123C3×Dic3⋊C4—32no—
124C3×Dic3⋊C4—32no—
125C3×Dic3⋊C4—32no—
126C3×Dic3⋊C4—32no—
127C3×Dic3⋊C4—61no—
128C3×Dic3⋊C4—61no—
129C3×Dic3⋊C4—62no—
130C3×C4×Dic3—32no—
131C3×C4×Dic3—32no—
132C3×C4×Dic3—32no—
133C3×C4×Dic3—32no—
134C3×C4×Dic3—32no—
135C3×C4×Dic3—61no—
136C3×C4×Dic3—61no—
137C3×C4×Dic3—61no—
138C3×C4×Dic3—62no—
139#3724—31no—
140#3724—32no—
141#3724—32no—
142Dic3⋊Dic3—32no—
143Dic3⋊Dic3—32no—
144Dic3⋊Dic3—32no—
145Dic3⋊Dic3—32no—
146Dic3⋊Dic3—32no—
147#3722—32no—
148#3722—32no—
149(C2×D3)⋊Dic3—32no—
150(C2×D3)⋊Dic3—32no—
151(C2×D3)⋊Dic3—32no—
152(C2×D3)⋊Dic3—32no—
153Dic32—32no—
154Dic32—32no—
155Dic32—32no—
156C22×C33C2411yescharacteristic
157C2×C3×C3⋊D3—22no—
158C2×C3×C3⋊D3—22no—
159C2×C3×C3⋊D3—22no—
160C33×C4—21no—
161C33×C4—21no—
162C33×C4—21no—
163C3×C3⋊Dic3—21no—
164C3×C3⋊Dic3—22no—
165C3×C3⋊Dic3—22no—
166C3×C3⋊Dic3—22no—
167C3×C3⋊Dic3—21no—
168C3×C3⋊Dic3#4112yes—
169C3×C3⋊Dic3#4112yes—
170C3×#126—92no—
171C3×C4×Q8—91no—
172#946—92no—
173#898—32no—
174#898—32no—
175#898—62no—
176#898—92no—
177#898—92no—
178#896—92no—
179#892—92no—
180#888—92no—
181#887—32no—
182#887—61no—
183#887—92no—
184#885—92no—
185#885—92no—
186#67⋊C4—92no—
187#879—92no—
188#878—92no—
189#877—92no—
190#875—91no—
191#875—91no—
192C4×D12—91no—
193#872—92no—
194#869—91no—
195C4×Dic6—92no—
196C4×Dic6—91no—
197C4×Dic6—91no—
198C22×C3⋊D3C22×D312yes—
199C22×C3×D3—32no—
200C22×C3×D3—32no—
201C22×C3×D3—62no—
202C2×C32×C4—21no—
203C2×C32×C4—21no—
204C2×C32×C4—21no—
205C2×C32×C4—22no—
206C2×C32×C4—22no—
207C2×C32×C4—22no—
208C2×C32×C4—41no—
209C2×C32×C4—41no—
210C2×C32×C4—41no—
211C2×C32×C4C22×D312yes—
212C2×C32×C4C22×D311yescharacteristic
213C2×C32×C4C22×D312yes—
214C2×C32×C4C22×D312yes—
215C2×C32×C4C22×D311yescharacteristic
216C2×C32×C4C22×D311yescharacteristic
217C2×C3⋊Dic3C22×D311yescharacteristic
218C2×C3⋊Dic3C22×D312yes—
219C2×C3⋊Dic3C22×D312yes—
220C2×C3⋊Dic3C22×D311yescharacteristic
221C2×C3⋊Dic3C22×D312yes—
222C2×C3⋊Dic3C22×D311yescharacteristic
223C2×C3⋊Dic3C22×D311yescharacteristic
224#634—22no—
225C2×C3×Dic3—32no—
226C2×C3×Dic3—32no—
227C2×C3×Dic3—32no—
228C2×C3×Dic3—32no—
229C2×C3×Dic3—32no—
230C2×C3×Dic3—32no—
231C2×C3×Dic3—32no—
232C2×C3×Dic3—32no—
233C2×C3×Dic3—32no—
234C2×C3×Dic3—32no—
235C2×C3×Dic3—61no—
236C2×C3×Dic3—61no—
237C2×C3×Dic3—61no—
238C2×C3×Dic3—61no—
239C2×C3×Dic3—62no—
240C2×C3×Dic3—62no—
241C2×C3×Dic3—62no—
242C3×Dic6—64no—
243C3×Dic6—64no—
244#627—64no—
245#627—64no—
246#627—62no—
247#627—62no—
248#626—64no—
249#625—62no—
250#624—64no—
251D3×Dic3—64no—
252D3×Dic3—64no—
253#556—271noSylow
254C2×C33C2×#4111yescharacteristic
255C2×C33C2×#4111yescharacteristic
256C2×C332−1+411yescharacteristic
257C3×C3⋊D3—42no—
258C2×C3×Q8—91no—
259C2×#67—92no—
260C22×Dic3—92no—
261C2×D12—91no—
262C2×C4×D3—92no—
263C2×C4×D3—92no—
264C2×Dic6—32no—
265C2×Dic6—61no—
266C2×Dic6—91no—
267C2×Dic6—91no—
268C2×Dic6—92no—
269C2×Dic6—92no—
270C3×C4⋊C4—21no—
271C3×C4⋊C4—22no—
272C3×C4⋊C4—41no—
273C3×C4⋊C4—92no—
274C3×C4⋊C4—92no—
275C3×C4⋊C4—91no—
276C3×C4⋊C4—91no—
277C3×C4⋊C4D3212yes—
278C3×C4⋊C4D3211yescharacteristic
279C3×C22⋊C4—92no—
280C3×C22⋊C4—92no—
281C3×C22⋊C4—92no—
282C3×C42—92no—
283C3×C42—91no—
284C3×C42—91no—
285C3×C42—91no—
286#217—92no—
287#217—92no—
288(C2×D3)⋊C4—32no—
289(C2×D3)⋊C4—32no—
290(C2×D3)⋊C4—32no—
291(C2×D3)⋊C4—32no—
292(C2×D3)⋊C4—32no—
293(C2×D3)⋊C4—32no—
294(C2×D3)⋊C4—62no—
295(C2×D3)⋊C4—62no—
296(C2×D3)⋊C4—62no—
297(C2×D3)⋊C4—92no—
298(C2×D3)⋊C4—92no—
299(C2×D3)⋊C4—92no—
300(C2×D3)⋊C4—92no—
301C4⋊Dic3—32no—
302C4⋊Dic3—32no—
303C4⋊Dic3—62no—
304C4⋊Dic3—91no—
305C4⋊Dic3—92no—
306C4⋊Dic3—92no—
307Dic3⋊C4—32no—
308Dic3⋊C4—32no—
309Dic3⋊C4—32no—
310Dic3⋊C4—32no—
311Dic3⋊C4—61no—
312Dic3⋊C4—61no—
313Dic3⋊C4—62no—
314Dic3⋊C4—92no—
315Dic3⋊C4—92no—
316Dic3⋊C4—92no—
317Dic3⋊C4—92no—
318Dic3⋊C4—92no—
319C4×Dic3—32no—
320C4×Dic3—32no—
321C4×Dic3—32no—
322C4×Dic3—32no—
323C4×Dic3—32no—
324C4×Dic3—61no—
325C4×Dic3—61no—
326C4×Dic3—61no—
327C4×Dic3—62no—
328C4×Dic3—92no—
329C4×Dic3—92no—
330C4×Dic3—92no—
331C22×C32—21no—
332C22×C32—22no—
333C22×C32—41no—
334C22×C32C23×D311yescharacteristic
335C22×C32C23×D312yes—
336C2×C3⋊D3—22no—
337C2×C3⋊D3—22no—
338C2×C3⋊D3—22no—
339C2×C3×D3—62no—
340C2×C3×D3—62no—
341C2×C3×D3—62no—
342C2×C3×D3—62no—
343C2×C3×D3—62no—
344C2×C3×D3—62no—
345C2×C3×D3—122no—
346C2×C3×D3—122no—
347C2×C3×D3—122no—
348C32×C4—21no—
349C32×C4—22no—
350C32×C4—22no—
351C32×C4—22no—
352C32×C4—22no—
353C32×C4—21no—
354C32×C4—21no—
355C32×C4—42no—
356C32×C4—42no—
357C32×C4—41no—
358C32×C4—41no—
359C32×C4—42no—
360C32×C4—42no—
361C32×C4—81no—
362C32×C4—81no—
363C3⋊Dic3—21no—
364C3⋊Dic3—22no—
365C3⋊Dic3—22no—
366C3⋊Dic3—22no—
367C3⋊Dic3—21no—
368C3⋊Dic3#23512yes—
369C3⋊Dic3#23912yes—
370C3×Dic3—34no—
371C3×Dic3—34no—
372C3×Dic3—62no—
373C3×Dic3—62no—
374C3×Dic3—62no—
375C3×Dic3—62no—
376C3×Dic3—62no—
377C3×Dic3—62no—
378C3×Dic3—62no—
379C3×Dic3—62no—
380C3×Dic3—62no—
381C3×Dic3—62no—
382C3×Dic3—62no—
383C3×Dic3—122no—
384C3×Dic3—121no—
385C3×Dic3—122no—
386C3×Dic3—122no—
387C4⋊Q8—271no—
388#126—92no—
389#126—272no—
390#124—271no—
391#124—271no—
392C22⋊Q8—272no—
393C4×Q8—91no—
394C4×Q8—271no—
395C4×Q8—271no—
396C4×D4—271no—
397#117—272no—
398C33#55611yescharacteristic, complemented, Sylow
399C23×C3—92no—
400C22×D3—32no—
401C22×D3—32no—
402C22×D3—62no—
403C3×Q8—182no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—22no—
408C2×C3×C4—22no—
409C2×C3×C4—22no—
410C2×C3×C4—41no—
411C2×C3×C4—41no—
412C2×C3×C4—41no—
413C2×C3×C4—91no—
414C2×C3×C4—91no—
415C2×C3×C4—91no—
416C2×C3×C4—92no—
417C2×C3×C4—92no—
418C2×C3×C4—91no—
419C2×C3×C4—92no—
420C2×C3×C4C2×D3212yes—
421C2×C3×C4C2×D3211yescharacteristic
422C2×C3×C4C2×D3212yes—
423C2×C3×C4C2×D3212yes—
424C2×C3×C4C2×D3211yescharacteristic
425C2×C3×C4C2×D3211yescharacteristic
426#67—184no—
427C2×Dic3—32no—
428C2×Dic3—32no—
429C2×Dic3—32no—
430C2×Dic3—32no—
431C2×Dic3—32no—
432C2×Dic3—32no—
433C2×Dic3—32no—
434C2×Dic3—32no—
435C2×Dic3—32no—
436C2×Dic3—32no—
437C2×Dic3—61no—
438C2×Dic3—61no—
439C2×Dic3—61no—
440C2×Dic3—61no—
441C2×Dic3—62no—
442C2×Dic3—62no—
443C2×Dic3—62no—
444C2×Dic3—184no—
445D12—182no—
446C4×D3—184no—
447C4×D3—184no—
448Dic6—64no—
449Dic6—64no—
450Dic6—184no—
451Dic6—184no—
452Dic6—182no—
453Dic6—182no—
454C2×C32—22no—
455C2×C32—21no—
456C2×C32—21no—
457C2×C32—22no—
458C2×C32—22no—
459C2×C32—21no—
460C2×C32—41no—
461C2×C32—41no—
462C2×C32—41no—
463C2×C32C2×#23511yescharacteristic
464C2×C32C2×#23911yescharacteristic
465C2×C32D3×#4112yes—
466C2×C32D3×#4112yes—
467C2×C32#101012yes—
468C2×C32#101011yescharacteristic
469C3⋊D3—42no—
470C3×D3—122no—
471C3×D3—122no—
472C3×D3—242no—
473C2×Q8—91no—
474C2×Q8—271no—
475C2×Q8—271no—
476C2×D4—271no—
477C22×C4—272no—
478C4⋊C4—91no—
479C4⋊C4—91no—
480C4⋊C4—92no—
481C4⋊C4—92no—
482C4⋊C4—272no—
483C4⋊C4—271no—
484C4⋊C4—272no—
485C4⋊C4#111311yescharacteristic
486C22⋊C4—92no—
487C22⋊C4—92no—
488C22⋊C4—92no—
489C22⋊C4—272no—
490C22⋊C4—272no—
491C42—92no—
492C42—91no—
493C42—91no—
494C42—91no—
495C42—272no—
496C22×C3—21no—
497C22×C3—22no—
498C22×C3—41no—
499C22×C3—182no—
500C22×C3—182no—
501C22×C3—182no—
502C22×C3C22×D3212yes—
503C22×C3C22×D3211yescharacteristic
504C2×D3—62no—
505C2×D3—62no—
506C2×D3—62no—
507C2×D3—62no—
508C2×D3—62no—
509C2×D3—62no—
510C2×D3—122no—
511C2×D3—122no—
512C2×D3—122no—
513C3×C4—22no—
514C3×C4—22no—
515C3×C4—21no—
516C3×C4—22no—
517C3×C4—22no—
518C3×C4—21no—
519C3×C4—21no—
520C3×C4—42no—
521C3×C4—41no—
522C3×C4—41no—
523C3×C4—42no—
524C3×C4—42no—
525C3×C4—42no—
526C3×C4—81no—
527C3×C4—81no—
528C3×C4—92no—
529C3×C4—92no—
530C3×C4—182no—
531C3×C4—181no—
532C3×C4—181no—
533C3×C4—182no—
534C3×C4—182no—
535Dic3—34no—
536Dic3—34no—
537Dic3—62no—
538Dic3—62no—
539Dic3—62no—
540Dic3—62no—
541Dic3—62no—
542Dic3—62no—
543Dic3—62no—
544Dic3—62no—
545Dic3—62no—
546Dic3—62no—
547Dic3—62no—
548Dic3—122no—
549Dic3—121no—
550Dic3—122no—
551Dic3—122no—
552C32—22no—
553C32—21no—
554C32—41no—
555C32#558411yescharacteristic, complemented
556C32#570212yescomplemented
557C23—92no—
558Q8—182no—
559Q8—542no—
560Q8—542no—
561D4—542no—
562C2×C4—91no—
563C2×C4—91no—
564C2×C4—91no—
565C2×C4—92no—
566C2×C4—92no—
567C2×C4—91no—
568C2×C4—92no—
569C2×C4—544no—
570C2×C4C2×#111311yescharacteristic
571C2×C4C2×#111311yescharacteristic
572C2×C4C2×#111311yescharacteristic
573C2×C3—21no—
574C2×C3—22no—
575C2×C3—21no—
576C2×C3—22no—
577C2×C3—21no—
578C2×C3—22no—
579C2×C3—41no—
580C2×C3—41no—
581C2×C3—41no—
582C2×C3—362no—
583C2×C3D3×#23512yes—
584C2×C3#6430611yescharacteristic
585C2×C3#6430912yes—
586C2×C3#6430911yescharacteristic
587C2×C3D3×#23912yes—
588C2×C3#6431911yescharacteristic
589D3—122no—
590D3—122no—
591D3—242no—
592C22—182no—
593C22—182no—
594C22—182no—
595C22C22×#111311yescharacteristic
596C4—21no—
597C4—21no—
598C4—21no—
599C4—92no—
600C4—92no—
601C4—182no—
602C4—181no—
603C4—181no—
604C4—182no—
605C4—182no—
606C3—22no—
607C3—21no—
608C3—41no—
609C3#1059442112yescomplemented
610C3#1059447211yescharacteristic, complemented
611C2—362no—
612C2#1172461811yescharacteristic
613C2#1172462011yescharacteristic
614C2#1172462611yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
prime radicalC2×#6295C22
hypercenterC4⋊C4#1113
hypocenterC33#556
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#64107C2semidirect
2. C3×#64096C2semidirect
3. C4⋊#6296C2semidirect
4. C4×#6296C2semidirect
5. #11723688C2semidirect
7. #11723683C2semidirect
8. #11723675C2semidirect
11. #11723665C2semidirect
37. #64107D3semidirect
38. #64096D3semidirect
398. C33#556semidirect
555. C32#5584semidirect
556. C32#5702semidirect
609. C3#10594421semidirect
610. C3#10594472semidirect

Maximal subgroups

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

subgroupindex
C3×#641072
C3×#640962
C4⋊#62962
C4×#62962
#117236882
#117236852
#117236832
#117236752
#117236722
#117236652
#105944723
#105944213