Finite Group Databasebeta

← #13155988

Subgroups — #13155988

Order 1728.

756 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723399C211yesmaximal, characteristic, complemented
2C2×#88796C211yesmaximal, characteristic, complemented
3C2×#88794C211yesmaximal, characteristic, complemented
4C3×#63911C211yesmaximal, characteristic
5#11722075C211yesmaximal, characteristic, complemented
6C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7C3×#63563C211yesmaximal, characteristic, complemented
8#10591439—31nomaximal
9#10590879—31nomaximal
10#10590377—31nomaximal
11C2×#6308C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C3×#626—21no—
15C2×C3×#626—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4×Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88806—21no—
21Dic3×C3⋊Dic3—21no—
22#88796—22no—
23#88794—22no—
24C2×#3754—31no—
25C2×#3744—31no—
26C4×#634—31no—
27C3×#940—31no—
28C3×#874—31no—
29#63911D311yescharacteristic
30#63844—31no—
31#63844—31no—
32#63844—61no—
33C2×#3716—31no—
34C2×#3716—31no—
35C2×#3716—31no—
36C2×#3716—31no—
37C2×#3716—61no—
38C2×#3716—61no—
39C3×#832—31no—
40C3×#821—31no—
41C3×Dic3⋊C8—31no—
42C3×C4⋊(C3⋊C8)—31no—
43Dic3⋊(C3⋊C8)D311yescharacteristic, complemented
44#63564—31no—
45#63563—31no—
46#63563D311yescharacteristic, complemented
47#63562—31no—
48C22×C3×C3⋊D3—21no—
49C22×C32×D3—21no—
50C2×C33×C4C2311yescharacteristic
51C2×C3⋊(C3⋊Dic3)—21no—
52#6308—21no—
53#6308D412yescomplemented
54C2×C3×C3⋊Dic3—21no—
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)—21no—
57C2×C32×Dic3D411yescharacteristic
58C2×C32×Dic3D411yescharacteristic
59C32×D12—21no—
60C32×D12—21no—
61C3×#626—42no—
62C3×#626—42no—
63C3×C3⋊(C3⋊C8)—21no—
64C32×C3⋊C8—21no—
65#5075—91no—
66#4848—91no—
67#4762—91no—
68C2×C32×D4—31no—
69C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
70C2×#634—31no—
71C2×#634—31no—
72C2×#634—31no—
73C2×#634—61no—
74C2×#634—61no—
75C2×#634—61no—
76C2×#634—121no—
77C2×C3×#67—61no—
78C2×C3×#67—61no—
79C2×C3×D12—21no—
80C2×C3×D12—31no—
81C2×C3×D12—31no—
82C2×C3×D12—61no—
83C2×C3×D12—61no—
84C2×C3×D12—61no—
85C2×C3×D12C2×D311yescharacteristic
86C2×C3×D12C2×D311yescharacteristic
87C2×#626—21no—
88C2×#626—21no—
89C2×C32×C8—31no—
90C32×C42—31no—
91#3754—62no—
92C4⋊(C3⋊Dic3)C2×D311yescharacteristic
93#3750—61no—
94C4×C3⋊Dic3—61no—
95C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
96#3744—62no—
97C3×C4⋊Dic3—31no—
98C3×C4⋊Dic3—31no—
99C3×C4⋊Dic3—61no—
100C3×C4×Dic3—21no—
101C3×C4×Dic3C2×D311yescharacteristic
102C3×C4×Dic3C2×D311yescharacteristic
103C2×C3×C3⋊C8—21no—
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—31no—
106C2×C3×C3⋊C8—61no—
107C2×C3×C3⋊C8C2×D311yescharacteristic
108C2×C3×C3⋊C8C2×D311yescharacteristic
109#3724—62no—
110#3724—61no—
111#3724—61no—
112Dic32—62no—
113Dic32—61no—
114Dic32—61no—
115#3716—62no—
116#3716—62no—
117#3716—64no—
118#3716—62no—
119#3716—62no—
120#3716—64no—
121C22×C33C2×D411yescharacteristic
122C2×C3×C3⋊D3—41no—
123C2×C3×C3⋊D3—41no—
124C2×C3×C3⋊D3—41no—
125C2×C32×D3—41no—
126C2×C32×D3—41no—
127C2×C32×D3—41no—
128C33×C4C2×D411yescharacteristic
129C33×C4#4111yescharacteristic
130C3⋊(C3⋊Dic3)—22no—
131C3×C3⋊Dic3—41no—
132C32×Dic3—21no—
133C32×Dic3SD1612yescomplemented
134C3×#127—91no—
135#940—31no—
136C2×#214—91no—
137C2×#214—91no—
138C2×#214—91no—
139C2×#214—181no—
140C2×#204—91no—
141C2×#204—91no—
142C2×#204—91no—
143C2×#204—181no—
144#887—91no—
145#874—31no—
146C4×Dic6—91no—
147C4×Dic6—91no—
148C4×Dic6—181no—
149C3×C4⋊C8—91no—
150C3×#102—91no—
151#832—31no—
152#832—91no—
153#821—31no—
154#821—91no—
155Dic3⋊C8—31no—
156#809—91no—
157C4⋊(C3⋊C8)—31no—
158C23×C32—61no—
159C22×C3⋊D3—21no—
160C22×C3×D3—21no—
161C22×C3×D3—21no—
162C22×C3×D3—41no—
163C22×C3×D3—61no—
164C22×C3×D3—61no—
165C22×C3×D3—121no—
166C32×D4—61no—
167C32×D4—61no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—21no—
171C2×C32×C4—31no—
172C2×C32×C4—31no—
173C2×C32×C4—41no—
174C2×C32×C4C22×D311yescharacteristic
175C2×C32×C4C22×D311yescharacteristic
176C2×C32×C4C22×D311yescharacteristic
177C2×C3⋊Dic3—21no—
178C2×C3⋊Dic3—61no—
179C2×C3⋊Dic3—62no—
180C2×C3⋊Dic3—61no—
181C2×C3⋊Dic3—61no—
182C2×C3⋊Dic3—121no—
183C2×C3⋊Dic3—121no—
184C2×C3⋊Dic3—122no—
185C4⋊(C3⋊D3)—21no—
186C4⋊(C3⋊D3)—21no—
187#634—32no—
188#634—32no—
189#634—32no—
190#634—62no—
191#634—62no—
192#634—62no—
193#634—61no—
194#634—62no—
195#634—61no—
196#634—61no—
197#634—122no—
198#634—121no—
199#634—121no—
200#634—122no—
201C3×#67—122no—
202C3×#67—122no—
203C2×C3×Dic3—21no—
204C2×C3×Dic3—21no—
205C2×C3×Dic3—62no—
206C2×C3×Dic3—61no—
207C2×C3×Dic3—61no—
208C2×C3×Dic3D1211yescharacteristic
209C2×C3×Dic3D1211yescharacteristic
210C2×C3×Dic3#6711yescharacteristic
211C2×C3×Dic3#6711yescharacteristic
212C3×D12—22no—
213C3×D12—21no—
214C3×D12—21no—
215C3×D12—21no—
216C3×D12—21no—
217C3×D12—41no—
218C3×D12—62no—
219C3×D12—61no—
220C3×D12—61no—
221C3×D12—61no—
222C3×D12—61no—
223C3×D12—121no—
224C3×D12—122no—
225C3×D12—122no—
226#626—42no—
227#626—42no—
228C32×C8—61no—
229C3⋊(C3⋊C8)—21no—
230C3×C3⋊C8—22no—
231C3×C3⋊C8—21no—
232C3×C3⋊C8—21no—
233C3×C3⋊C8—62no—
234C3×C3⋊C8—61no—
235C3×C3⋊C8—61no—
236#460—271noSylow
237C2×C33#12111yescharacteristic
238C2×C33C2×SD1611yescharacteristic
239C2×C33AGL(1,ℤ/8)11yescharacteristic
240C3×C3⋊D3—81no—
241C32×D3—81no—
242C2×C3×D4—31no—
243C2×C3×D4—31no—
244C2×C3×D4—61no—
245C2×C3×D4—91no—
246C2×C3×D4—181no—
247C2×C3×D4—181no—
248C2×#67—61no—
249C2×#67—61no—
250C2×D12—31no—
251C2×D12—31no—
252C2×D12—61no—
253C2×D12—61no—
254C2×D12—61no—
255C2×D12D3211yescharacteristic
256C2×Dic6—91no—
257C2×Dic6—91no—
258C2×Dic6—91no—
259C2×Dic6—181no—
260C2×Dic6—181no—
261C2×Dic6—181no—
262C2×Dic6—361no—
263C2×C3×C8—31no—
264C2×C3×C8—31no—
265C2×C3×C8—61no—
266C2×C3×C8—91no—
267C3×C4⋊C4—91no—
268C3×C42—31no—
269C3×C42—31no—
270C3×C42—61no—
271#214—182no—
272#214—182no—
273#214—182no—
274#214—184no—
275C4⋊Dic3—31no—
276C4⋊Dic3—31no—
277C4⋊Dic3—61no—
278Dic3⋊C4—181no—
279Dic3⋊C4—181no—
280Dic3⋊C4—182no—
281Dic3⋊C4—181no—
282C4×Dic3—181no—
283C4×Dic3—181no—
284C4×Dic3—182no—
285C4×Dic3—181no—
286C4×Dic3D3211yescharacteristic
287C2×C3⋊C8—31no—
288C2×C3⋊C8—31no—
289C2×C3⋊C8—61no—
290C2×C3⋊C8D3211yescharacteristic
291#204—182no—
292#204—182no—
293#204—184no—
294#204—182no—
295C22×C32—21no—
296C22×C32—21no—
297C22×C32—21no—
298C22×C32—41no—
299C22×C32—121no—
300C22×C32—121no—
301C22×C32—121no—
302C22×C32C2×D1211yescharacteristic
303C22×C32D3×D411yescharacteristic
304C22×C32C2×#6711yescharacteristic
305C2×C3⋊D3—41no—
306C2×C3⋊D3—41no—
307C2×C3⋊D3—41no—
308C2×C3×D3—42no—
309C2×C3×D3—41no—
310C2×C3×D3—41no—
311C2×C3×D3—41no—
312C2×C3×D3—41no—
313C2×C3×D3—41no—
314C2×C3×D3—41no—
315C2×C3×D3—81no—
316C2×C3×D3—81no—
317C2×C3×D3—122no—
318C2×C3×D3—121no—
319C2×C3×D3—121no—
320C2×C3×D3—121no—
321C2×C3×D3—121no—
322C2×C3×D3—121no—
323C2×C3×D3—121no—
324C2×C3×D3—241no—
325C2×C3×D3—241no—
326C32×C4—21no—
327C32×C4—21no—
328C32×C4—21no—
329C32×C4—21no—
330C32×C4—21no—
331C32×C4—21no—
332C32×C4—32no—
333C32×C4—41no—
334C32×C4—41no—
335C32×C4—61no—
336C32×C4#23511yescharacteristic
337C32×C4D3×D411yescharacteristic
338C32×C4D3×D411yescharacteristic
339C32×C4D3×D411yescharacteristic
340C32×C4#23711yescharacteristic
341C32×C4#23911yescharacteristic
342C3⋊Dic3—41no—
343C3⋊Dic3—62no—
344C3⋊Dic3—64no—
345C3⋊Dic3—62no—
346C3⋊Dic3—62no—
347C3⋊Dic3—122no—
348C3⋊Dic3—122no—
349C3⋊Dic3—124no—
350C3×Dic3—21no—
351C3×Dic3—21no—
352C3×Dic3—22no—
353C3×Dic3—41no—
354C3×Dic3—122no—
355C3×Dic3—121no—
356C3×Dic3—121no—
357C3×Dic3#20412yescomplemented
358C3×Dic3#21412yescomplemented
359C2×SD16—271no—
360C2×SD16—271no—
361#127—91no—
362C4×Q8—271no—
363C4⋊C8—91no—
364#102—91no—
365#102—271no—
366C33#46011yescharacteristic, complemented, Sylow
367C23×C3—61no—
368C23×C3—61no—
369C23×C3—121no—
370C23×C3—181no—
371C22×D3—21no—
372C22×D3—61no—
373C22×D3—61no—
374C22×D3—121no—
375C3×D4—62no—
376C3×D4—61no—
377C3×D4—61no—
378C3×D4—61no—
379C3×D4—61no—
380C3×D4—121no—
381C3×D4—181no—
382C3×D4—181no—
383C3×D4—362no—
384C3×D4—362no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—31no—
389C2×C3×C4—31no—
390C2×C3×C4—31no—
391C2×C3×C4—31no—
392C2×C3×C4—41no—
393C2×C3×C4—61no—
394C2×C3×C4—61no—
395C2×C3×C4—181no—
396C2×C3×C4C2×D3211yescharacteristic
397C2×C3×C4C2×D3211yescharacteristic
398C2×C3×C4C2×D3211yescharacteristic
399#67—122no—
400#67—122no—
401C2×Dic3—62no—
402C2×Dic3—61no—
403C2×Dic3—61no—
404C2×Dic3—182no—
405C2×Dic3—181no—
406C2×Dic3—181no—
407C2×Dic3—181no—
408C2×Dic3—361no—
409C2×Dic3—361no—
410C2×Dic3—362no—
411C2×Dic3#62611yescharacteristic
412C2×Dic3#62611yescharacteristic
413D12—21no—
414D12—21no—
415D12—62no—
416D12—61no—
417D12—61no—
418D12—61no—
419D12—61no—
420D12—121no—
421D12—122no—
422D12—122no—
423Dic6—92no—
424Dic6—92no—
425Dic6—92no—
426Dic6—182no—
427Dic6—182no—
428Dic6—182no—
429Dic6—182no—
430Dic6—181no—
431Dic6—181no—
432Dic6—181no—
433Dic6—362no—
434Dic6—361no—
435Dic6—361no—
436Dic6—362no—
437C3×C8—62no—
438C3×C8—61no—
439C3×C8—61no—
440C3×C8—181no—
441C3⋊C8—21no—
442C3⋊C8—62no—
443C3⋊C8—61no—
444C3⋊C8—61no—
445C2×C32—21no—
446C2×C32—21no—
447C2×C32—21no—
448C2×C32—21no—
449C2×C32—21no—
450C2×C32—21no—
451C2×C32—21no—
452C2×C32—21no—
453C2×C32—21no—
454C2×C32—41no—
455C2×C32—41no—
456C2×C32—41no—
457C2×C32—241no—
458C2×C32#88411yescharacteristic
459C2×C32#89511yescharacteristic
460C2×C32C2×#20411yescharacteristic
461C2×C32#90811yescharacteristic
462C2×C32#91111yescharacteristic
463C2×C32D3×SD1611yescharacteristic
464C2×C32#93211yescharacteristic
465C2×C32C2×#21411yescharacteristic
466C2×C32#93811yescharacteristic
467C3⋊D3—81no—
468C3×D3—82no—
469C3×D3—81no—
470C3×D3—81no—
471C3×D3—242no—
472C3×D3—241no—
473C3×D3—241no—
474C2×Q8—271no—
475C2×D4—31no—
476C2×D4—91no—
477C2×D4—181no—
478C2×D4—181no—
479SD16—542no—
480SD16—542no—
481C2×C8—31no—
482C2×C8—91no—
483C4⋊C4—91no—
484C4⋊C4—541no—
485C42—31no—
486C42—541no—
487C22×C3—21no—
488C22×C3—21no—
489C22×C3—21no—
490C22×C3—41no—
491C22×C3—122no—
492C22×C3—121no—
493C22×C3—121no—
494C22×C3—121no—
495C22×C3—121no—
496C22×C3—121no—
497C22×C3—121no—
498C22×C3—241no—
499C22×C3—241no—
500C22×C3—361no—
501C22×C3—361no—
502C22×C3—361no—
503C22×C3D3×D1211yescharacteristic
504C22×C3C2×#62611yescharacteristic
505C22×C3D3×#6711yescharacteristic
506C2×D3—41no—
507C2×D3—41no—
508C2×D3—41no—
509C2×D3—122no—
510C2×D3—121no—
511C2×D3—121no—
512C2×D3—121no—
513C2×D3—121no—
514C2×D3—121no—
515C2×D3—121no—
516C2×D3—241no—
517C2×D3—241no—
518C3×C4—21no—
519C3×C4—21no—
520C3×C4—21no—
521C3×C4—21no—
522C3×C4—21no—
523C3×C4—21no—
524C3×C4—32no—
525C3×C4—32no—
526C3×C4—41no—
527C3×C4—41no—
528C3×C4—62no—
529C3×C4—61no—
530C3×C4—61no—
531C3×C4—121no—
532C3×C4—361no—
533C3×C4#379611yescharacteristic
534C3×C4#379911yescharacteristic
535C3×C4#380211yescharacteristic
536C3×C4#380411yescharacteristic
537C3×C4#381111yescharacteristic
538C3×C4#381111yescharacteristic
539Dic3—21no—
540Dic3—122no—
541Dic3—121no—
542Dic3—121no—
543Dic3—184no—
544Dic3—182no—
545Dic3—182no—
546Dic3—182no—
547Dic3—362no—
548Dic3—362no—
549Dic3—364no—
550Dic3#371612yescomplemented
551C32—21no—
552C32—21no—
553C32—21no—
554C32—41no—
555C32#476211yescharacteristic, complemented
556C32#484811yescharacteristic, complemented
557C32#507511yescharacteristic, complemented
558C23—61no—
559C23—181no—
560Q8—272no—
561Q8—541no—
562D4—61no—
563D4—61no—
564D4—181no—
565D4—181no—
566D4—362no—
567D4—362no—
568C2×C4—31no—
569C2×C4—31no—
570C2×C4—181no—
571C2×C4—541no—
572C2×C4D3311yescharacteristic
573C8—61no—
574C8—181no—
575C2×C3—21no—
576C2×C3—21no—
577C2×C3—21no—
578C2×C3—21no—
579C2×C3—21no—
580C2×C3—21no—
581C2×C3—21no—
582C2×C3—21no—
583C2×C3—21no—
584C2×C3—41no—
585C2×C3—41no—
586C2×C3—41no—
587C2×C3—242no—
588C2×C3—241no—
589C2×C3—241no—
590C2×C3—721no—
591C2×C3D3×#20411yescharacteristic
592C2×C3#6379511yescharacteristic
593C2×C3C2×#371611yescharacteristic
594C2×C3#6383011yescharacteristic
595C2×C3#6391011yescharacteristic
596C2×C3#6391611yescharacteristic
597C2×C3D3×#21411yescharacteristic
598C2×C3#6393211yescharacteristic
599C2×C3#6395811yescharacteristic
600D3—81no—
601D3—242no—
602D3—241no—
603D3—241no—
604C22—121no—
605C22—121no—
606C22—121no—
607C22—361no—
608C22—361no—
609C22—361no—
610C22D3×#62611yescharacteristic
611C4—32no—
612C4—61no—
613C4—361no—
614C4—542no—
615C4#8895411yescharacteristic
616C4#8896211yescharacteristic
617C3—21no—
618C3—21no—
619C3—21no—
620C3—41no—
621C3#1059037711yescharacteristic, complemented
622C3#1059087911yescharacteristic, complemented
623C3#1059143911yescharacteristic, complemented
624C2—241no—
625C2—721no—
626C2D3×#371611yescharacteristic
627C2#1172286911yescharacteristic
628C2#1172296311yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC3×#63911C2
hypocenterC33#460
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723399C2semidirect
2. C2×#88796C2semidirect
3. C2×#88794C2semidirect
5. #11722075C2semidirect
6. C3×Dic3⋊(C3⋊C8)C2semidirect
7. C3×#63563C2semidirect
43. Dic3⋊(C3⋊C8)D3semidirect
46. #63563D3semidirect
53. #6308D4semidirect
133. C32×Dic3SD16semidirect
357. C3×Dic3#204semidirect
358. C3×Dic3#214semidirect
366. C33#460semidirect
550. Dic3#3716semidirect
555. C32#4762semidirect
556. C32#4848semidirect
557. C32#5075semidirect
621. C3#10590377semidirect
622. C3#10590879semidirect
623. C3#10591439semidirect

Maximal subgroups

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

subgroupindex
#117233992
C2×#887962
C2×#887942
C3×#639112
#117220752
C3×Dic3⋊(C3⋊C8)2
C3×#635632
#105914393
#105908793
#105903773