Finite Group Databasebeta

← #13169328

Subgroups — #13169328

Order 1728.

1044 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723668C211yesmaximal, characteristic
2#11723455C211yesmaximal, characteristic
3#11723443C211yesmaximal, characteristic
4#11723399C211yesmaximal, characteristic
5C3×C4×#627C211yesmaximal, characteristic
6C3×#63866C211yesmaximal, characteristic
7C3×Dic3×Dic6C211yesmaximal, characteristic
8#11723017C212yesmaximal
9#11723013C212yesmaximal
10#11723012C212yesmaximal
11#11722940C212yesmaximal
12#10595420—31nomaximal
13#10593861—31nomaximal
14#10593860—31nomaximal
15C2×#6308C2211yescharacteristic
16C2×C3×#634C2211yescharacteristic
17C2×C32×Dic6C2211yescharacteristic
18C2×C3×#627C2211yescharacteristic
19C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C3×C4×C3⋊Dic3C2211yescharacteristic
21C32×C4×Dic3C2211yescharacteristic
22C32×C4×Dic3C2211yescharacteristic
23#88815C2212yes—
24#88813C2212yes—
25#88807C2212yes—
26#88807C2212yes—
27#88806C2212yes—
28Dic3⋊(C3⋊Dic3)C2212yes—
29Dic3⋊(C3⋊Dic3)C2212yes—
30Dic3×C3⋊Dic3C2212yes—
31C3×#3724C2211yescharacteristic
32C3×Dic3⋊Dic3C2211yescharacteristic
33C3×Dic3⋊Dic3C2212yes—
34C3×Dic3⋊Dic3C2211yescharacteristic
35C3×Dic3⋊Dic3C2212yes—
36C3×Dic3⋊Dic3C2212yes—
37C3×Dic32C2212yes—
38#64153—31no—
39#64078—31no—
40C4×#634—31no—
41C3×Q8×Dic3—31no—
42C3×#944—31no—
43C3×#869—31no—
44C3×C4×Dic6—31no—
45C3×C4×Dic6—31no—
46C3×C4×Dic6—31no—
47C4×#627D311yescharacteristic
48#63872—32no—
49#63869—32no—
50#63867—32no—
51#63866—31no—
52#63866—31no—
53#63866—31no—
54#63866—31no—
55#63866—61no—
56#63866—61no—
57#63866D311yescharacteristic
58#63847—32no—
59#63846—32no—
60#63846—32no—
61#63846—32no—
62#63845—32no—
63#63845—32no—
64#63844—31no—
65#63844—31no—
66#63844—31no—
67#63844—61no—
68#63843—31no—
69#63843—32no—
70Dic3×Dic6—31no—
71Dic3×Dic6—32no—
72Dic3×Dic6D311yescharacteristic
73#63837—32no—
74C2×C33×C4C2311yescharacteristic
75C2×C3⋊(C3⋊Dic3)C2312yes—
76#6308—22no—
77C2×C3×C3⋊Dic3C2311yescharacteristic
78C2×C3×C3⋊Dic3C2312yes—
79C2×C3×C3⋊Dic3C2312yes—
80C2×C3×C3⋊Dic3C2311yescharacteristic
81C3×#634—22no—
82C2×C32×Dic3C2312yes—
83C2×C32×Dic3C2311yescharacteristic
84C2×C32×Dic3C2311yescharacteristic
85C2×C32×Dic3C2311yescharacteristic
86C2×C32×Dic3C2311yescharacteristic
87C32×Dic6—22no—
88C3×#627Q814yescomplemented
89#5721—91no—
90#5570—91no—
91#5567—91no—
92C2×C32×Q8—31no—
93C2×#634—31no—
94C2×#634—31no—
95C2×#634—31no—
96C2×#634—61no—
97C2×#634—61no—
98C2×#634—61no—
99C2×#634—121no—
100C2×#634C2×D311yescharacteristic
101C2×C3×Dic6—21no—
102C2×C3×Dic6—31no—
103C2×C3×Dic6—31no—
104C2×C3×Dic6—31no—
105C2×C3×Dic6—31no—
106C2×C3×Dic6—61no—
107C2×C3×Dic6C2×D311yescharacteristic
108C2×C3×Dic6C2×D311yescharacteristic
109C2×#627C2×D311yescharacteristic
110C32×C42—31no—
111C32×C42—31no—
112C4⋊(C3⋊Dic3)—32no—
113C4⋊(C3⋊Dic3)—32no—
114C4⋊(C3⋊Dic3)C2×D311yescharacteristic
115#3750—32no—
116#3750—32no—
117#3750—32no—
118C4×C3⋊Dic3—32no—
119C4×C3⋊Dic3C2×D311yescharacteristic
120C3×C4⋊Dic3—31no—
121C3×C4⋊Dic3—31no—
122C3×C4⋊Dic3—32no—
123C3×C4⋊Dic3—32no—
124C3×C4⋊Dic3—32no—
125C3×C4⋊Dic3—31no—
126C3×C4⋊Dic3—31no—
127C3×C4⋊Dic3—61no—
128C3×Dic3⋊C4—32no—
129C3×Dic3⋊C4—32no—
130C3×Dic3⋊C4—32no—
131C3×Dic3⋊C4—31no—
132C3×Dic3⋊C4—31no—
133C3×Dic3⋊C4—31no—
134C3×Dic3⋊C4—31no—
135C3×C4×Dic3—21no—
136C3×C4×Dic3—21no—
137C3×C4×Dic3—31no—
138C3×C4×Dic3—31no—
139C3×C4×Dic3—32no—
140C3×C4×Dic3—32no—
141C3×C4×Dic3—61no—
142C3×C4×Dic3C2×D311yescharacteristic
143C3×C4×Dic3C2×D311yescharacteristic
144C3×C4×Dic3C2×D311yescharacteristic
145C3×C4×Dic3C2×D311yescharacteristic
146#3724—32no—
147#3724—32no—
148#3724—32no—
149#3724—62no—
150#3724C2×D311yescharacteristic
151Dic3⋊Dic3—32no—
152Dic3⋊Dic3—32no—
153Dic3⋊Dic3—32no—
154Dic3⋊Dic3—32no—
155Dic3⋊Dic3—32no—
156Dic3⋊Dic3—32no—
157Dic3⋊Dic3—32no—
158Dic3⋊Dic3—32no—
159Dic3⋊Dic3—32no—
160Dic3⋊Dic3—32no—
161Dic3⋊Dic3—62no—
162Dic3⋊Dic3—62no—
163Dic3⋊Dic3—62no—
164Dic3⋊Dic3—62no—
165Dic3⋊Dic3C2×D312yes—
166Dic3⋊Dic3C2×D312yes—
167Dic3⋊Dic3C2×D312yes—
168Dic3⋊Dic3C2×D311yescharacteristic
169Dic3⋊Dic3C2×D311yescharacteristic
170Dic32—32no—
171Dic32—32no—
172Dic32—32no—
173Dic32—32no—
174Dic32—32no—
175Dic32—62no—
176Dic32C2×D312yes—
177C22×C33C2411yescharacteristic
178C33×C4#4112yes—
179C3⋊(C3⋊Dic3)—22no—
180C3×C3⋊Dic3—22no—
181C3×C3⋊Dic3—21no—
182C3×C3⋊Dic3—22no—
183C3×C3⋊Dic3C2×Q812yes—
184C32×Dic3—22no—
185C32×Dic3—21no—
186C32×Dic3—21no—
187C32×Dic3C2×Q812yes—
188C32×Dic3C2×Q812yes—
189C3×C4⋊Q8—91no—
190C3×C4×Q8—91no—
191C3×C4×Q8—91no—
192Q8×Dic3—31no—
193Q8×Dic3—91no—
194Q8×Dic3—92no—
195#944—31no—
196#944—91no—
197#944—91no—
198#944—91no—
199#944—181no—
200#890—92no—
201#890—92no—
202#890—92no—
203#890—92no—
204#889—92no—
205#889—92no—
206#889—92no—
207#889—92no—
208#887—92no—
209#887—91no—
210#887—91no—
211#870—92no—
212#869—31no—
213#869—91no—
214#869—91no—
215#869—91no—
216#869—181no—
217C4×Dic6—31no—
218C4×Dic6—31no—
219C4×Dic6—31no—
220C4×Dic6—92no—
221C4×Dic6—91no—
222C4×Dic6—91no—
223C4×Dic6—91no—
224C4×Dic6—181no—
225C32×Q8—62no—
226C2×C32×C4—21no—
227C2×C32×C4—21no—
228C2×C32×C4—21no—
229C2×C32×C4—31no—
230C2×C32×C4—31no—
231C2×C32×C4—32no—
232C2×C32×C4—31no—
233C2×C32×C4—31no—
234C2×C32×C4—41no—
235C2×C32×C4C22×D311yescharacteristic
236C2×C32×C4C22×D311yescharacteristic
237C2×C32×C4C22×D311yescharacteristic
238C2×C3⋊Dic3—32no—
239C2×C3⋊Dic3—32no—
240C2×C3⋊Dic3—32no—
241C2×C3⋊Dic3—62no—
242C2×C3⋊Dic3—62no—
243C2×C3⋊Dic3—62no—
244C2×C3⋊Dic3—122no—
245C2×C3⋊Dic3C22×D312yes—
246C2×C3⋊Dic3C22×D312yes—
247C2×C3⋊Dic3C22×D311yescharacteristic
248C2×C3⋊Dic3C22×D311yescharacteristic
249#634—22no—
250#634—62no—
251#634—64no—
252#634—62no—
253#634—62no—
254#634—122no—
255#634—122no—
256#634—124no—
257C2×C3×Dic3—21no—
258C2×C3×Dic3—21no—
259C2×C3×Dic3—21no—
260C2×C3×Dic3—22no—
261C2×C3×Dic3—21no—
262C2×C3×Dic3—31no—
263C2×C3×Dic3—31no—
264C2×C3×Dic3—32no—
265C2×C3×Dic3—32no—
266C2×C3×Dic3—32no—
267C2×C3×Dic3—32no—
268C2×C3×Dic3—31no—
269C2×C3×Dic3—31no—
270C2×C3×Dic3—61no—
271C2×C3×Dic3—61no—
272C2×C3×Dic3—62no—
273C2×C3×Dic3—62no—
274C2×C3×Dic3C22×D312yes—
275C2×C3×Dic3C22×D312yes—
276C2×C3×Dic3C22×D311yescharacteristic
277C2×C3×Dic3C22×D311yescharacteristic
278C2×C3×Dic3C22×D311yescharacteristic
279C2×C3×Dic3C22×D311yescharacteristic
280C2×C3×Dic3C22×D311yescharacteristic
281C2×C3×Dic3C22×D311yescharacteristic
282C2×C3×Dic3C22×D311yescharacteristic
283C2×C3×Dic3C22×D311yescharacteristic
284C3×Dic6—22no—
285C3×Dic6—22no—
286C3×Dic6—34no—
287C3×Dic6—34no—
288C3×Dic6—42no—
289C3×Dic6—62no—
290C3×Dic6—62no—
291C3×Dic6—122no—
292#627Dic614yescomplemented
293#557—271noSylow
294C2×C33C22×Q811yescharacteristic
295C2×C33C2×#4111yescharacteristic
296C2×C332−1+411yescharacteristic
297C2×C3×Q8—31no—
298C2×C3×Q8—31no—
299C2×C3×Q8—61no—
300C2×C3×Q8—91no—
301C2×C3×Q8—91no—
302C2×Dic6—31no—
303C2×Dic6—31no—
304C2×Dic6—31no—
305C2×Dic6—31no—
306C2×Dic6—61no—
307C2×Dic6—91no—
308C2×Dic6—91no—
309C2×Dic6—91no—
310C2×Dic6—181no—
311C2×Dic6—181no—
312C2×Dic6—181no—
313C2×Dic6—361no—
314C2×Dic6D3211yescharacteristic
315C3×C4⋊C4—92no—
316C3×C4⋊C4—92no—
317C3×C4⋊C4—92no—
318C3×C4⋊C4—91no—
319C3×C4⋊C4—91no—
320C3×C4⋊C4—91no—
321C3×C4⋊C4—91no—
322C3×C42—31no—
323C3×C42—31no—
324C3×C42—31no—
325C3×C42—31no—
326C3×C42—61no—
327C3×C42—61no—
328C3×C42—92no—
329C3×C42—91no—
330C4⋊Dic3—31no—
331C4⋊Dic3—31no—
332C4⋊Dic3—31no—
333C4⋊Dic3—31no—
334C4⋊Dic3—32no—
335C4⋊Dic3—32no—
336C4⋊Dic3—32no—
337C4⋊Dic3—61no—
338C4⋊Dic3—92no—
339C4⋊Dic3—92no—
340C4⋊Dic3—92no—
341C4⋊Dic3—92no—
342C4⋊Dic3—92no—
343C4⋊Dic3—92no—
344C4⋊Dic3—92no—
345C4⋊Dic3—182no—
346C4⋊Dic3—182no—
347Dic3⋊C4—31no—
348Dic3⋊C4—31no—
349Dic3⋊C4—31no—
350Dic3⋊C4—31no—
351Dic3⋊C4—32no—
352Dic3⋊C4—32no—
353Dic3⋊C4—32no—
354Dic3⋊C4—92no—
355Dic3⋊C4—92no—
356Dic3⋊C4—92no—
357Dic3⋊C4—92no—
358Dic3⋊C4—92no—
359Dic3⋊C4—92no—
360Dic3⋊C4—92no—
361Dic3⋊C4—92no—
362Dic3⋊C4—92no—
363Dic3⋊C4—92no—
364Dic3⋊C4—92no—
365Dic3⋊C4—182no—
366Dic3⋊C4—182no—
367Dic3⋊C4—182no—
368C4×Dic3—31no—
369C4×Dic3—31no—
370C4×Dic3—32no—
371C4×Dic3—32no—
372C4×Dic3—61no—
373C4×Dic3—92no—
374C4×Dic3—92no—
375C4×Dic3—92no—
376C4×Dic3—92no—
377C4×Dic3—92no—
378C4×Dic3—92no—
379C4×Dic3—182no—
380C4×Dic3D3211yescharacteristic
381C4×Dic3D3211yescharacteristic
382C22×C32—21no—
383C22×C32—21no—
384C22×C32—21no—
385C22×C32—41no—
386C22×C32C23×D311yescharacteristic
387C22×C32C23×D311yescharacteristic
388C22×C32C23×D311yescharacteristic
389C32×C4—22no—
390C32×C4—22no—
391C32×C4—22no—
392C32×C4—32no—
393C32×C4—32no—
394C32×C4—42no—
395C32×C4—61no—
396C32×C4—62no—
397C32×C4—61no—
398C32×C4#23512yes—
399C32×C4#23712yes—
400C32×C4#23912yes—
401C3⋊Dic3—21no—
402C3⋊Dic3—22no—
403C3⋊Dic3—22no—
404C3⋊Dic3—62no—
405C3⋊Dic3—64no—
406C3⋊Dic3—62no—
407C3⋊Dic3—62no—
408C3⋊Dic3—122no—
409C3⋊Dic3—122no—
410C3⋊Dic3—124no—
411C3⋊Dic3C2×Dic612yes—
412C3×Dic3—22no—
413C3×Dic3—22no—
414C3×Dic3—22no—
415C3×Dic3—21no—
416C3×Dic3—21no—
417C3×Dic3—21no—
418C3×Dic3—21no—
419C3×Dic3—22no—
420C3×Dic3—22no—
421C3×Dic3—32no—
422C3×Dic3—32no—
423C3×Dic3—42no—
424C3×Dic3—41no—
425C3×Dic3—62no—
426C3×Dic3—62no—
427C3×Dic3—62no—
428C3×Dic3—62no—
429C3×Dic3—61no—
430C3×Dic3—62no—
431C3×Dic3—62no—
432C3×Dic3—61no—
433C3×Dic3—122no—
434C3×Dic3—122no—
435C3×Dic3C2×Dic612yes—
436C3×Dic3C2×Dic612yes—
437C3×Dic3D3×Q812yes—
438C3×Dic3D3×Q812yes—
439C4⋊Q8—91no—
440C4⋊Q8—271no—
441C4⋊Q8—271no—
442#125—272no—
443#125—272no—
444#125—272no—
445C4×Q8—91no—
446C4×Q8—91no—
447C4×Q8—271no—
448C4×Q8—272no—
449C4×Q8—271no—
450C33#55711yescharacteristic, complemented, Sylow
451C3×Q8—62no—
452C3×Q8—62no—
453C3×Q8—94no—
454C3×Q8—122no—
455C3×Q8—182no—
456C2×C3×C4—21no—
457C2×C3×C4—21no—
458C2×C3×C4—21no—
459C2×C3×C4—31no—
460C2×C3×C4—31no—
461C2×C3×C4—31no—
462C2×C3×C4—31no—
463C2×C3×C4—32no—
464C2×C3×C4—31no—
465C2×C3×C4—32no—
466C2×C3×C4—31no—
467C2×C3×C4—31no—
468C2×C3×C4—31no—
469C2×C3×C4—41no—
470C2×C3×C4—61no—
471C2×C3×C4—61no—
472C2×C3×C4—61no—
473C2×C3×C4—61no—
474C2×C3×C4—62no—
475C2×C3×C4—91no—
476C2×C3×C4—92no—
477C2×C3×C4—92no—
478C2×C3×C4—91no—
479C2×C3×C4C2×D3211yescharacteristic
480C2×C3×C4C2×D3211yescharacteristic
481C2×C3×C4C2×D3211yescharacteristic
482C2×Dic3—31no—
483C2×Dic3—31no—
484C2×Dic3—32no—
485C2×Dic3—32no—
486C2×Dic3—31no—
487C2×Dic3—31no—
488C2×Dic3—32no—
489C2×Dic3—32no—
490C2×Dic3—61no—
491C2×Dic3—61no—
492C2×Dic3—62no—
493C2×Dic3—62no—
494C2×Dic3—92no—
495C2×Dic3—92no—
496C2×Dic3—92no—
497C2×Dic3—182no—
498C2×Dic3—182no—
499C2×Dic3—182no—
500C2×Dic3—362no—
501C2×Dic3C2×D3212yes—
502C2×Dic3C2×D3211yescharacteristic
503C2×Dic3C2×D3211yescharacteristic
504C2×Dic3C2×D3211yescharacteristic
505C2×Dic3C2×D3211yescharacteristic
506Dic6—22no—
507Dic6—34no—
508Dic6—34no—
509Dic6—62no—
510Dic6—62no—
511Dic6—122no—
512Dic6—184no—
513Dic6—182no—
514Dic6—182no—
515Dic6—182no—
516Dic6—362no—
517Dic6—362no—
518Dic6—364no—
519C2×C32—21no—
520C2×C32—21no—
521C2×C32—21no—
522C2×C32—21no—
523C2×C32—21no—
524C2×C32—21no—
525C2×C32—21no—
526C2×C32—21no—
527C2×C32—21no—
528C2×C32—41no—
529C2×C32—41no—
530C2×C32—41no—
531C2×C32C22×Dic611yescharacteristic
532C2×C32C2×#23511yescharacteristic
533C2×C32C2×#23711yescharacteristic
534C2×C32C2×D3×Q811yescharacteristic
535C2×C32C2×D3×Q811yescharacteristic
536C2×C32C2×#23911yescharacteristic
537C2×C32#100711yescharacteristic
538C2×C32#101011yescharacteristic
539C2×C32#101011yescharacteristic
540C2×Q8—31no—
541C2×Q8—91no—
542C2×Q8—91no—
543C2×Q8—271no—
544C4⋊C4—91no—
545C4⋊C4—91no—
546C4⋊C4—91no—
547C4⋊C4—92no—
548C4⋊C4—92no—
549C4⋊C4—92no—
550C4⋊C4—91no—
551C4⋊C4—272no—
552C4⋊C4—272no—
553C4⋊C4—272no—
554C4⋊C4—272no—
555C4⋊C4—272no—
556C4⋊C4—272no—
557C42—31no—
558C42—31no—
559C42—92no—
560C42—91no—
561C42—272no—
562C42—272no—
563C22×C3—21no—
564C22×C3—21no—
565C22×C3—21no—
566C22×C3—41no—
567C22×C3C22×D3211yescharacteristic
568C22×C3C22×D3211yescharacteristic
569C22×C3C22×D3211yescharacteristic
570C3×C4—22no—
571C3×C4—22no—
572C3×C4—22no—
573C3×C4—32no—
574C3×C4—32no—
575C3×C4—32no—
576C3×C4—32no—
577C3×C4—42no—
578C3×C4—62no—
579C3×C4—62no—
580C3×C4—62no—
581C3×C4—62no—
582C3×C4—61no—
583C3×C4—61no—
584C3×C4—62no—
585C3×C4—61no—
586C3×C4—61no—
587C3×C4—92no—
588C3×C4—122no—
589C3×C4—121no—
590C3×C4—182no—
591C3×C4—182no—
592C3×C4—181no—
593C3×C4#379612yes—
594C3×C4#379912yes—
595C3×C4#380412yes—
596Dic3—21no—
597Dic3—21no—
598Dic3—22no—
599Dic3—32no—
600Dic3—32no—
601Dic3—62no—
602Dic3—62no—
603Dic3—62no—
604Dic3—61no—
605Dic3—61no—
606Dic3—62no—
607Dic3—62no—
608Dic3—62no—
609Dic3—122no—
610Dic3—122no—
611Dic3—184no—
612Dic3—182no—
613Dic3—182no—
614Dic3—182no—
615Dic3—362no—
616Dic3—362no—
617Dic3—364no—
618Dic3D3×Dic612yes—
619Dic3D3×Dic612yes—
620C32—21no—
621C32—21no—
622C32—21no—
623C32—41no—
624C32#556711yescharacteristic, complemented
625C32#557011yescharacteristic, complemented
626C32#572111yescharacteristic, complemented
627Q8—62no—
628Q8—94no—
629Q8—182no—
630Q8—542no—
631C2×C4—31no—
632C2×C4—31no—
633C2×C4—31no—
634C2×C4—31no—
635C2×C4—32no—
636C2×C4—91no—
637C2×C4—92no—
638C2×C4—91no—
639C2×C4—92no—
640C2×C4—272no—
641C2×C4D3311yescharacteristic
642C2×C3—21no—
643C2×C3—21no—
644C2×C3—21no—
645C2×C3—21no—
646C2×C3—21no—
647C2×C3—21no—
648C2×C3—21no—
649C2×C3—21no—
650C2×C3—21no—
651C2×C3—41no—
652C2×C3—41no—
653C2×C3—41no—
654C2×C3C2×D3×Dic611yescharacteristic
655C2×C3C2×D3×Dic611yescharacteristic
656C2×C3C2×#379611yescharacteristic
657C2×C3#6429711yescharacteristic
658C2×C3#6429811yescharacteristic
659C2×C3C2×#379911yescharacteristic
660C2×C3#6431511yescharacteristic
661C2×C3D32×Q811yescharacteristic
662C2×C3C2×#380411yescharacteristic
663C22C2×D3311yescharacteristic
664C4—32no—
665C4—32no—
666C4—61no—
667C4—61no—
668C4—62no—
669C4—92no—
670C4—181no—
671C4—182no—
672C4—182no—
673C4—542no—
674C4#8896212yes—
675C3—21no—
676C3—21no—
677C3—21no—
678C3—41no—
679C3#1059386011yescharacteristic, complemented
680C3#1059386111yescharacteristic, complemented
681C3#1059542011yescharacteristic, complemented
682C2D32×Dic611yescharacteristic
683C2#1172439611yescharacteristic
684C2C2×#8896211yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, prime radical, cosocleC22×C33C24
FittingC2×C33×C4C23
hypercenterC2×C4D33
hypocenterC33#557
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
88. C3×#627Q8semidirect
292. #627Dic6semidirect
450. C33#557semidirect
624. C32#5567semidirect
625. C32#5570semidirect
626. C32#5721semidirect
679. C3#10593860semidirect
680. C3#10593861semidirect
681. C3#10595420semidirect

Maximal subgroups

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

subgroupindex
#117236682
#117234552
#117234432
#117233992
C3×C4×#6272
C3×#638662
C3×Dic3×Dic62
#117230172
#117230132
#117230122
#117229402
#105954203
#105938613
#105938603