Finite Group Databasebeta

← #13169375

Subgroups — #13169375

Order 1728.

1008 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723668C211yesmaximal, characteristic
2#11723454C211yesmaximal, characteristic
3#11723447C211yesmaximal, characteristic
4Dic3×#634C211yesmaximal, characteristic
5C3×#63918C211yesmaximal, characteristic
6C3×C4×#627C211yesmaximal, characteristic
7C3×#63844C211yesmaximal, characteristic
8#11723032C212yesmaximal
9#11723017C212yesmaximal
10#11723014C212yesmaximal
11#11722940C212yesmaximal
12#10594317—31nomaximal
13#10593864—31nomaximal
14#10593851—31nomaximal
15C2×C3×#634C2211yescharacteristic
16C2×C3×#627C2211yescharacteristic
17C2×C3×#627C2212yes—
18C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
19C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C3×C4×C3⋊Dic3C2211yescharacteristic
21C32×C4⋊Dic3C2211yescharacteristic
22C32×C4×Dic3C2211yescharacteristic
23C32×C4×Dic3C2211yescharacteristic
24#88815C2212yes—
25#88813C2212yes—
26#88807C2212yes—
27#88807C2212yes—
28#88806C2212yes—
29#88806C2212yes—
30Dic3⋊(C3⋊Dic3)C2212yes—
31Dic3×C3⋊Dic3C2212yes—
32C3×#3724C2211yescharacteristic
33C3×#3724C2212yes—
34C3×Dic3⋊Dic3C2212yes—
35C3×Dic3⋊Dic3C2211yescharacteristic
36C3×Dic3⋊Dic3C2211yescharacteristic
37C3×Dic32C2212yes—
38#64099—31no—
39#64079—31no—
40C4×#634—31no—
41C3×#888—31no—
42C3×#887—31no—
43C3×#869—31no—
44C3×C4×Dic6—31no—
45C3×C4×Dic6—31no—
46C3×C4×Dic6—31no—
47#63918D311yescharacteristic
48C4×#627D311yescharacteristic
49#63873—32no—
50#63869—31no—
51#63869—31no—
52#63869—31no—
53#63869—31no—
54#63869—61no—
55#63869—61no—
56#63867—32no—
57#63867—32no—
58#63847—32no—
59#63847—32no—
60#63846—32no—
61#63845—32no—
62#63844—31no—
63#63844D311yescharacteristic
64#63843—31no—
65#63843—32no—
66Dic3×Dic6—31no—
67Dic3×Dic6—31no—
68Dic3×Dic6—31no—
69Dic3×Dic6—32no—
70Dic3×Dic6—61no—
71#63839—32no—
72#63839—32no—
73#63837—32no—
74C2×C33×C4C2311yescharacteristic
75C2×C3⋊(C3⋊Dic3)C2312yes—
76C2×C3×C3⋊Dic3C2311yescharacteristic
77C2×C3×C3⋊Dic3C2311yescharacteristic
78C2×C3×C3⋊Dic3C2312yes—
79C2×C3×C3⋊Dic3C2312yes—
80C3×#634Q814yescomplemented
81C2×C32×Dic3C2312yes—
82C2×C32×Dic3C2311yescharacteristic
83C2×C32×Dic3C2311yescharacteristic
84C2×C32×Dic3C2311yescharacteristic
85C2×C32×Dic3C2311yescharacteristic
86C3×#627—22no—
87C3×#627—24no—
88#5721—91no—
89#5569—91no—
90#5567—91no—
91C2×#634C2×D311yescharacteristic
92C2×C3×Dic6—31no—
93C2×C3×Dic6—31no—
94C2×C3×Dic6—32no—
95C2×C3×Dic6—31no—
96C2×C3×Dic6—32no—
97C2×C3×Dic6—31no—
98C2×C3×Dic6—61no—
99C2×#627C2×D312yes—
100C2×#627C2×D311yescharacteristic
101C32×C4⋊C4—31no—
102C32×C42—31no—
103C32×C42—31no—
104C4⋊(C3⋊Dic3)—31no—
105C4⋊(C3⋊Dic3)—31no—
106C4⋊(C3⋊Dic3)—31no—
107C4⋊(C3⋊Dic3)—32no—
108C4⋊(C3⋊Dic3)—61no—
109C4⋊(C3⋊Dic3)—61no—
110C4⋊(C3⋊Dic3)—61no—
111C4⋊(C3⋊Dic3)—121no—
112C4⋊(C3⋊Dic3)C2×D311yescharacteristic
113#3750—32no—
114#3750—32no—
115#3750—32no—
116#3750—32no—
117C4×C3⋊Dic3—32no—
118C4×C3⋊Dic3C2×D311yescharacteristic
119C3×C4⋊Dic3—21no—
120C3×C4⋊Dic3—31no—
121C3×C4⋊Dic3—31no—
122C3×C4⋊Dic3—31no—
123C3×C4⋊Dic3—31no—
124C3×C4⋊Dic3—32no—
125C3×C4⋊Dic3—61no—
126C3×C4⋊Dic3C2×D311yescharacteristic
127C3×C4⋊Dic3C2×D311yescharacteristic
128C3×Dic3⋊C4—32no—
129C3×Dic3⋊C4—32no—
130C3×Dic3⋊C4—31no—
131C3×Dic3⋊C4—32no—
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—32no—
150#3724—32no—
151#3724—62no—
152#3724—62no—
153#3724C2×D312yes—
154#3724C2×D311yescharacteristic
155Dic3⋊Dic3—32no—
156Dic3⋊Dic3—32no—
157Dic3⋊Dic3—32no—
158Dic3⋊Dic3—32no—
159Dic3⋊Dic3—32no—
160Dic3⋊Dic3—32no—
161Dic3⋊Dic3—32no—
162Dic3⋊Dic3—32no—
163Dic3⋊Dic3—62no—
164Dic3⋊Dic3—62no—
165Dic3⋊Dic3—62no—
166Dic3⋊Dic3C2×D311yescharacteristic
167Dic3⋊Dic3C2×D311yescharacteristic
168Dic3⋊Dic3C2×D312yes—
169Dic32—32no—
170Dic32—32no—
171Dic32—32no—
172Dic32—32no—
173Dic32—32no—
174Dic32—62no—
175Dic32C2×D312yes—
176C22×C33C2411yescharacteristic
177C33×C4C2×Q812yes—
178C3⋊(C3⋊Dic3)—22no—
179C3×C3⋊Dic3—22no—
180C3×C3⋊Dic3—21no—
181C3×C3⋊Dic3—21no—
182C3×C3⋊Dic3C2×Q814yes—
183C32×Dic3—21no—
184C32×Dic3—22no—
185C32×Dic3—21no—
186C32×Dic3—21no—
187C32×Dic3#4112yes—
188C3×C4⋊Q8—91no—
189C3×C4×Q8—91no—
190C3×C4×Q8—91no—
191Q8×Dic3—92no—
192Q8×Dic3—91no—
193Q8×Dic3—91no—
194#944—92no—
195#944—92no—
196#890—91no—
197#890—91no—
198#890—92no—
199#890—91no—
200#890—181no—
201#889—92no—
202#889—92no—
203#889—92no—
204#889—92no—
205#888—31no—
206#888—92no—
207#887—31no—
208#887—91no—
209#887—92no—
210#870—92no—
211#870—91no—
212#870—91no—
213#870—91no—
214#870—181no—
215#869—31no—
216C4×Dic6—31no—
217C4×Dic6—31no—
218C4×Dic6—31no—
219C4×Dic6—92no—
220C4×Dic6—91no—
221C4×Dic6—91no—
222C4×Dic6—91no—
223C4×Dic6—181no—
224C2×C32×C4—21no—
225C2×C32×C4—21no—
226C2×C32×C4—21no—
227C2×C32×C4—31no—
228C2×C32×C4—31no—
229C2×C32×C4—32no—
230C2×C32×C4—31no—
231C2×C32×C4—31no—
232C2×C32×C4—41no—
233C2×C32×C4C22×D311yescharacteristic
234C2×C32×C4C22×D311yescharacteristic
235C2×C32×C4C22×D311yescharacteristic
236C2×C3⋊Dic3—32no—
237C2×C3⋊Dic3—32no—
238C2×C3⋊Dic3—32no—
239C2×C3⋊Dic3—62no—
240C2×C3⋊Dic3—62no—
241C2×C3⋊Dic3—62no—
242C2×C3⋊Dic3—122no—
243C2×C3⋊Dic3C22×D311yescharacteristic
244C2×C3⋊Dic3C22×D312yes—
245C2×C3⋊Dic3C22×D311yescharacteristic
246C2×C3⋊Dic3C22×D312yes—
247#634Dic614yescomplemented
248C2×C3×Dic3—21no—
249C2×C3×Dic3—21no—
250C2×C3×Dic3—21no—
251C2×C3×Dic3—22no—
252C2×C3×Dic3—21no—
253C2×C3×Dic3—32no—
254C2×C3×Dic3—32no—
255C2×C3×Dic3—31no—
256C2×C3×Dic3—31no—
257C2×C3×Dic3—32no—
258C2×C3×Dic3—32no—
259C2×C3×Dic3—31no—
260C2×C3×Dic3—31no—
261C2×C3×Dic3—62no—
262C2×C3×Dic3—61no—
263C2×C3×Dic3—62no—
264C2×C3×Dic3—61no—
265C2×C3×Dic3C22×D311yescharacteristic
266C2×C3×Dic3C22×D312yes—
267C2×C3×Dic3C22×D311yescharacteristic
268C2×C3×Dic3C22×D311yescharacteristic
269C2×C3×Dic3C22×D311yescharacteristic
270C2×C3×Dic3C22×D312yes—
271C2×C3×Dic3C22×D311yescharacteristic
272C2×C3×Dic3C22×D311yescharacteristic
273C2×C3×Dic3C22×D311yescharacteristic
274C2×C3×Dic3C22×D311yescharacteristic
275C3×Dic6—34no—
276C3×Dic6—34no—
277C3×Dic6—64no—
278C3×Dic6—64no—
279C3×Dic6—62no—
280C3×Dic6—64no—
281C3×Dic6—62no—
282#627—22no—
283#627—24no—
284#557—271noSylow
285C2×C33C22×Q811yescharacteristic
286C2×C33C2×#4111yescharacteristic
287C2×C332−1+411yescharacteristic
288C2×C3×Q8—91no—
289C2×C3×Q8—92no—
290C2×C3×Q8—91no—
291C2×Dic6—31no—
292C2×Dic6—31no—
293C2×Dic6—31no—
294C2×Dic6—31no—
295C2×Dic6—32no—
296C2×Dic6—32no—
297C2×Dic6—61no—
298C3×C4⋊C4—31no—
299C3×C4⋊C4—31no—
300C3×C4⋊C4—61no—
301C3×C4⋊C4—91no—
302C3×C4⋊C4—92no—
303C3×C4⋊C4—91no—
304C3×C4⋊C4—91no—
305C3×C4⋊C4—92no—
306C3×C4⋊C4—91no—
307C3×C42—31no—
308C3×C42—31no—
309C3×C42—31no—
310C3×C42—31no—
311C3×C42—61no—
312C3×C42—61no—
313C3×C42—92no—
314C3×C42—91no—
315C4⋊Dic3—31no—
316C4⋊Dic3—31no—
317C4⋊Dic3—31no—
318C4⋊Dic3—31no—
319C4⋊Dic3—32no—
320C4⋊Dic3—61no—
321C4⋊Dic3—92no—
322C4⋊Dic3—92no—
323C4⋊Dic3—92no—
324C4⋊Dic3—91no—
325C4⋊Dic3—91no—
326C4⋊Dic3—91no—
327C4⋊Dic3—92no—
328C4⋊Dic3—92no—
329C4⋊Dic3—182no—
330C4⋊Dic3—181no—
331C4⋊Dic3—181no—
332C4⋊Dic3—181no—
333C4⋊Dic3—361no—
334C4⋊Dic3D3211yescharacteristic
335Dic3⋊C4—31no—
336Dic3⋊C4—31no—
337Dic3⋊C4—31no—
338Dic3⋊C4—31no—
339Dic3⋊C4—32no—
340Dic3⋊C4—32no—
341Dic3⋊C4—32no—
342Dic3⋊C4—92no—
343Dic3⋊C4—92no—
344Dic3⋊C4—92no—
345Dic3⋊C4—92no—
346Dic3⋊C4—92no—
347Dic3⋊C4—92no—
348Dic3⋊C4—92no—
349Dic3⋊C4—92no—
350Dic3⋊C4—92no—
351Dic3⋊C4—92no—
352Dic3⋊C4—92no—
353Dic3⋊C4—92no—
354Dic3⋊C4—92no—
355Dic3⋊C4—182no—
356Dic3⋊C4—182no—
357Dic3⋊C4—182no—
358Dic3⋊C4—182no—
359C4×Dic3—31no—
360C4×Dic3—31no—
361C4×Dic3—32no—
362C4×Dic3—32no—
363C4×Dic3—61no—
364C4×Dic3—92no—
365C4×Dic3—92no—
366C4×Dic3—92no—
367C4×Dic3—92no—
368C4×Dic3—92no—
369C4×Dic3—92no—
370C4×Dic3—182no—
371C4×Dic3D3211yescharacteristic
372C4×Dic3D3211yescharacteristic
373C22×C32—21no—
374C22×C32—21no—
375C22×C32—21no—
376C22×C32—41no—
377C22×C32C23×D311yescharacteristic
378C22×C32C23×D311yescharacteristic
379C22×C32C23×D311yescharacteristic
380C32×C4—22no—
381C32×C4—22no—
382C32×C4—22no—
383C32×C4—32no—
384C32×C4—42no—
385C32×C4—62no—
386C32×C4—61no—
387C32×C4—61no—
388C32×C4—61no—
389C32×C4C2×Dic612yes—
390C32×C4D3×Q812yes—
391C32×C4D3×Q812yes—
392C3⋊Dic3—21no—
393C3⋊Dic3—22no—
394C3⋊Dic3—21no—
395C3⋊Dic3—62no—
396C3⋊Dic3—62no—
397C3⋊Dic3—62no—
398C3⋊Dic3—122no—
399C3⋊Dic3—122no—
400C3⋊Dic3—122no—
401C3⋊Dic3—242no—
402C3⋊Dic3C2×Dic614yes—
403C3×Dic3—21no—
404C3×Dic3—22no—
405C3×Dic3—21no—
406C3×Dic3—21no—
407C3×Dic3—21no—
408C3×Dic3—22no—
409C3×Dic3—21no—
410C3×Dic3—21no—
411C3×Dic3—22no—
412C3×Dic3—34no—
413C3×Dic3—34no—
414C3×Dic3—41no—
415C3×Dic3—41no—
416C3×Dic3—41no—
417C3×Dic3—42no—
418C3×Dic3—64no—
419C3×Dic3—62no—
420C3×Dic3—61no—
421C3×Dic3—61no—
422C3×Dic3—61no—
423C3×Dic3—61no—
424C3×Dic3—62no—
425C3×Dic3—122no—
426C3×Dic3—121no—
427C3×Dic3—121no—
428C3×Dic3#23512yes—
429C3×Dic3#23712yes—
430C4⋊Q8—91no—
431C4⋊Q8—272no—
432#125—271no—
433#125—271no—
434#125—272no—
435#125—272no—
436C4×Q8—91no—
437C4×Q8—91no—
438C4×Q8—271no—
439C4×Q8—272no—
440C4×Q8—271no—
441C33#55711yescharacteristic, complemented, Sylow
442C3×Q8—94no—
443C3×Q8—184no—
444C3×Q8—182no—
445C2×C3×C4—21no—
446C2×C3×C4—21no—
447C2×C3×C4—21no—
448C2×C3×C4—31no—
449C2×C3×C4—31no—
450C2×C3×C4—31no—
451C2×C3×C4—32no—
452C2×C3×C4—31no—
453C2×C3×C4—31no—
454C2×C3×C4—31no—
455C2×C3×C4—32no—
456C2×C3×C4—31no—
457C2×C3×C4—31no—
458C2×C3×C4—41no—
459C2×C3×C4—61no—
460C2×C3×C4—61no—
461C2×C3×C4—61no—
462C2×C3×C4—61no—
463C2×C3×C4—62no—
464C2×C3×C4—92no—
465C2×C3×C4—91no—
466C2×C3×C4—92no—
467C2×C3×C4—91no—
468C2×C3×C4C2×D3211yescharacteristic
469C2×C3×C4C2×D3211yescharacteristic
470C2×C3×C4C2×D3211yescharacteristic
471C2×Dic3—32no—
472C2×Dic3—32no—
473C2×Dic3—31no—
474C2×Dic3—31no—
475C2×Dic3—31no—
476C2×Dic3—31no—
477C2×Dic3—32no—
478C2×Dic3—32no—
479C2×Dic3—62no—
480C2×Dic3—61no—
481C2×Dic3—61no—
482C2×Dic3—62no—
483C2×Dic3—92no—
484C2×Dic3—92no—
485C2×Dic3—92no—
486C2×Dic3—182no—
487C2×Dic3—182no—
488C2×Dic3—182no—
489C2×Dic3—362no—
490C2×Dic3C2×D3211yescharacteristic
491C2×Dic3C2×D3212yes—
492C2×Dic3C2×D3211yescharacteristic
493C2×Dic3C2×D3211yescharacteristic
494C2×Dic3C2×D3211yescharacteristic
495Dic6—34no—
496Dic6—34no—
497Dic6—64no—
498Dic6—62no—
499Dic6—64no—
500Dic6—62no—
501Dic6—64no—
502C2×C32—21no—
503C2×C32—21no—
504C2×C32—21no—
505C2×C32—21no—
506C2×C32—21no—
507C2×C32—21no—
508C2×C32—21no—
509C2×C32—21no—
510C2×C32—21no—
511C2×C32—41no—
512C2×C32—41no—
513C2×C32—41no—
514C2×C32C22×Dic611yescharacteristic
515C2×C32C2×#23511yescharacteristic
516C2×C32C2×#23711yescharacteristic
517C2×C32C2×D3×Q811yescharacteristic
518C2×C32C2×D3×Q811yescharacteristic
519C2×C32#100711yescharacteristic
520C2×C32#100711yescharacteristic
521C2×C32D3×#4111yescharacteristic
522C2×C32#101011yescharacteristic
523C2×Q8—91no—
524C2×Q8—92no—
525C2×Q8—91no—
526C4⋊C4—31no—
527C4⋊C4—91no—
528C4⋊C4—91no—
529C4⋊C4—91no—
530C4⋊C4—92no—
531C4⋊C4—92no—
532C4⋊C4—91no—
533C4⋊C4—272no—
534C4⋊C4—272no—
535C4⋊C4—272no—
536C4⋊C4—272no—
537C4⋊C4—272no—
538C4⋊C4—272no—
539C4⋊C4—271no—
540C42—31no—
541C42—31no—
542C42—92no—
543C42—91no—
544C42—272no—
545C42—272no—
546C22×C3—21no—
547C22×C3—21no—
548C22×C3—21no—
549C22×C3—41no—
550C22×C3C22×D3211yescharacteristic
551C22×C3C22×D3211yescharacteristic
552C22×C3C22×D3211yescharacteristic
553C3×C4—22no—
554C3×C4—22no—
555C3×C4—22no—
556C3×C4—32no—
557C3×C4—32no—
558C3×C4—42no—
559C3×C4—62no—
560C3×C4—62no—
561C3×C4—61no—
562C3×C4—61no—
563C3×C4—61no—
564C3×C4—62no—
565C3×C4—61no—
566C3×C4—61no—
567C3×C4—61no—
568C3×C4—94no—
569C3×C4—121no—
570C3×C4—122no—
571C3×C4—121no—
572C3×C4—121no—
573C3×C4—182no—
574C3×C4—181no—
575C3×C4—181no—
576C3×C4D3×Dic612yes—
577C3×C4D3×Dic612yes—
578C3×C4#379712yes—
579Dic3—21no—
580Dic3—21no—
581Dic3—21no—
582Dic3—22no—
583Dic3—34no—
584Dic3—34no—
585Dic3—64no—
586Dic3—61no—
587Dic3—62no—
588Dic3—61no—
589Dic3—61no—
590Dic3—61no—
591Dic3—62no—
592Dic3—122no—
593Dic3—121no—
594Dic3—121no—
595Dic3—182no—
596Dic3—182no—
597Dic3—182no—
598Dic3—362no—
599Dic3—362no—
600Dic3—362no—
601Dic3—722no—
602Dic3#379512yes—
603C32—21no—
604C32—21no—
605C32—21no—
606C32—41no—
607C32#556711yescharacteristic, complemented
608C32#556911yescharacteristic, complemented
609C32#572111yescharacteristic, complemented
610Q8—94no—
611Q8—182no—
612Q8—184no—
613C2×C4—31no—
614C2×C4—31no—
615C2×C4—31no—
616C2×C4—31no—
617C2×C4—32no—
618C2×C4—92no—
619C2×C4—91no—
620C2×C4—91no—
621C2×C4—92no—
622C2×C4—272no—
623C2×C4D3311yescharacteristic
624C2×C3—21no—
625C2×C3—21no—
626C2×C3—21no—
627C2×C3—21no—
628C2×C3—21no—
629C2×C3—21no—
630C2×C3—21no—
631C2×C3—21no—
632C2×C3—21no—
633C2×C3—41no—
634C2×C3—41no—
635C2×C3—41no—
636C2×C3C2×D3×Dic611yescharacteristic
637C2×C3C2×D3×Dic611yescharacteristic
638C2×C3C2×#379511yescharacteristic
639C2×C3#6429711yescharacteristic
640C2×C3#6429711yescharacteristic
641C2×C3C2×#379711yescharacteristic
642C2×C3D3×#23511yescharacteristic
643C2×C3D3×#23711yescharacteristic
644C2×C3#6431611yescharacteristic
645C22C2×D3311yescharacteristic
646C4—32no—
647C4—61no—
648C4—62no—
649C4—61no—
650C4—61no—
651C4—94no—
652C4—181no—
653C4—182no—
654C4—181no—
655C4—542no—
656C4#8895712yes—
657C3—21no—
658C3—21no—
659C3—21no—
660C3—41no—
661C3#1059385111yescharacteristic, complemented
662C3#1059386411yescharacteristic, complemented
663C3#1059431711yescharacteristic, complemented
664C2D3×#379511yescharacteristic
665C2#1172439511yescharacteristic
666C2C2×#8895711yescharacteristic
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
80. C3×#634Q8semidirect
247. #634Dic6semidirect
441. C33#557semidirect
607. C32#5567semidirect
608. C32#5569semidirect
609. C32#5721semidirect
661. C3#10593851semidirect
662. C3#10593864semidirect
663. C3#10594317semidirect

Maximal subgroups

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

subgroupindex
#117236682
#117234542
#117234472
Dic3×#6342
C3×#639182
C3×C4×#6272
C3×#638442
#117230322
#117230172
#117230142
#117229402
#105943173
#105938643
#105938513