Finite Group Databasebeta

← #13169378

Subgroups — #13169378

Order 1728.

1048 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723680C211yesmaximal, characteristic
2#11723406C211yesmaximal, characteristic
3#11723399C211yesmaximal, characteristic
4#11723399C211yesmaximal, characteristic
5C3×#63918C211yesmaximal, characteristic
6C3×C4×#627C211yesmaximal, characteristic
7C3×#63869C211yesmaximal, characteristic
8#11723032C212yesmaximal
9#11723016C212yesmaximal
10#11723016C212yesmaximal
11#11722941C212yesmaximal
12#10594318—31nomaximal
13#10593864—31nomaximal
14#10593852—31nomaximal
15C2×#6308C2211yescharacteristic
16C2×C3×#627C2211yescharacteristic
17C2×C3×#627C2212yes—
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C3×C4×C3⋊Dic3C2211yescharacteristic
21C32×C4⋊Dic3C2211yescharacteristic
22C32×C4×Dic3C2211yescharacteristic
23C32×C4×Dic3C2211yescharacteristic
24#88815C2212yes—
25#88815C2212yes—
26#88806C2212yes—
27#88806C2212yes—
28#88806C2212yes—
29Dic3×C3⋊Dic3C2212yes—
30Dic3×C3⋊Dic3C2212yes—
31Dic3×C3⋊Dic3C2212yes—
32C3×#3724C2211yescharacteristic
33C3×#3724C2212yes—
34C3×Dic3⋊Dic3C2211yescharacteristic
35C3×Dic3⋊Dic3C2212yes—
36C3×Dic3⋊Dic3C2212yes—
37C3×Dic3⋊Dic3C2211yescharacteristic
38#64096—31no—
39C4×#634—31no—
40C4×#634—31no—
41C3×#890—31no—
42C3×#888—31no—
43C3×#870—31no—
44C3×#869—31no—
45C3×C4×Dic6—31no—
46C3×C4×Dic6—31no—
47#63918D311yescharacteristic
48C4×#627D311yescharacteristic
49#63873—32no—
50#63872—31no—
51#63870—32no—
52#63870—32no—
53#63869—31no—
54#63869—31no—
55#63869D311yescharacteristic
56#63847—32no—
57#63847—32no—
58#63847—32no—
59#63847—32no—
60#63844—31no—
61#63844—31no—
62#63844—31no—
63#63844—31no—
64#63844—31no—
65#63844—31no—
66#63844—61no—
67#63844—61no—
68#63844—61no—
69#63843—32no—
70#63843—32no—
71#63839—32no—
72#63839—32no—
73#63838—32no—
74C2×C33×C4C2311yescharacteristic
75C2×C3⋊(C3⋊Dic3)C2312yes—
76#6308Q814yescomplemented
77C2×C3×C3⋊Dic3C2311yescharacteristic
78C2×C3×C3⋊Dic3C2311yescharacteristic
79C2×C3×C3⋊Dic3C2312yes—
80C2×C3×C3⋊Dic3C2312yes—
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#5685—91no—
90#5567—91no—
91C2×#634—31no—
92C2×#634—31no—
93C2×#634—31no—
94C2×#634—61no—
95C2×#634—61no—
96C2×#634—61no—
97C2×#634—121no—
98C2×C3×Dic6—32no—
99C2×C3×Dic6—31no—
100C2×C3×Dic6—32no—
101C2×C3×Dic6—31no—
102C2×#627C2×D312yes—
103C2×#627C2×D311yescharacteristic
104C32×C4⋊C4—31no—
105C32×C42—31no—
106C32×C42—31no—
107C4⋊(C3⋊Dic3)C2×D311yescharacteristic
108C4⋊(C3⋊Dic3)C2×D311yescharacteristic
109#3750—32no—
110#3750—32no—
111#3750—32no—
112C4×C3⋊Dic3—32no—
113C4×C3⋊Dic3—32no—
114C4×C3⋊Dic3—32no—
115C4×C3⋊Dic3C2×D311yescharacteristic
116C3×C4⋊Dic3—21no—
117C3×C4⋊Dic3—31no—
118C3×C4⋊Dic3—31no—
119C3×C4⋊Dic3—31no—
120C3×C4⋊Dic3—31no—
121C3×C4⋊Dic3—32no—
122C3×C4⋊Dic3—31no—
123C3×C4⋊Dic3—31no—
124C3×C4⋊Dic3—32no—
125C3×C4⋊Dic3—61no—
126C3×C4⋊Dic3—61no—
127C3×C4⋊Dic3C2×D311yescharacteristic
128C3×C4⋊Dic3C2×D311yescharacteristic
129C3×Dic3⋊C4—32no—
130C3×Dic3⋊C4—32no—
131C3×Dic3⋊C4—31no—
132C3×Dic3⋊C4—32no—
133C3×Dic3⋊C4—32no—
134C3×Dic3⋊C4—31no—
135C3×Dic3⋊C4—31no—
136C3×Dic3⋊C4—31no—
137C3×C4×Dic3—21no—
138C3×C4×Dic3—21no—
139C3×C4×Dic3—31no—
140C3×C4×Dic3—31no—
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—32no—
152#3724—32no—
153#3724—32no—
154#3724—62no—
155#3724—62no—
156#3724—62no—
157#3724C2×D311yescharacteristic
158#3724C2×D312yes—
159Dic3⋊Dic3—32no—
160Dic3⋊Dic3—32no—
161Dic3⋊Dic3—32no—
162Dic3⋊Dic3—32no—
163Dic3⋊Dic3C2×D311yescharacteristic
164Dic3⋊Dic3C2×D312yes—
165Dic3⋊Dic3C2×D312yes—
166Dic3⋊Dic3C2×D311yescharacteristic
167Dic32—32no—
168Dic32—32no—
169Dic32—32no—
170Dic32—32no—
171Dic32—32no—
172Dic32—32no—
173Dic32—62no—
174Dic32—62no—
175Dic32—62no—
176C22×C33C2411yescharacteristic
177C33×C4C2×Q812yes—
178C3⋊(C3⋊Dic3)C2×Q814yes—
179C3×C3⋊Dic3—22no—
180C3×C3⋊Dic3—21no—
181C3×C3⋊Dic3—21no—
182C3×C3⋊Dic3—22no—
183C32×Dic3—21no—
184C32×Dic3—22no—
185C32×Dic3—21no—
186C32×Dic3—21no—
187C32×Dic3#4112yes—
188C3×C4⋊Q8—91no—
189C3×#125—91no—
190C3×C4×Q8—91no—
191Q8×Dic3—92no—
192#944—92no—
193#944—92no—
194#890—31no—
195#890—91no—
196#890—91no—
197#889—92no—
198#889—92no—
199#889—92no—
200#889—92no—
201#889—92no—
202#889—92no—
203#888—31no—
204#888—92no—
205#887—91no—
206#887—91no—
207#887—92no—
208#887—92no—
209#887—91no—
210#887—91no—
211#887—181no—
212#870—31no—
213#870—91no—
214#869—31no—
215C4×Dic6—31no—
216C4×Dic6—31no—
217C4×Dic6—91no—
218C4×Dic6—91no—
219C4×Dic6—91no—
220C4×Dic6—91no—
221C4×Dic6—91no—
222C4×Dic6—181no—
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—31no—
230C2×C32×C4—32no—
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×D311yescharacteristic
245C2×C3⋊Dic3C22×D312yes—
246C2×C3⋊Dic3C22×D312yes—
247#634—34no—
248#634—34no—
249#634—34no—
250#634—64no—
251#634—64no—
252#634—64no—
253#634—124no—
254C2×C3×Dic3—21no—
255C2×C3×Dic3—21no—
256C2×C3×Dic3—22no—
257C2×C3×Dic3—21no—
258C2×C3×Dic3—21no—
259C2×C3×Dic3—31no—
260C2×C3×Dic3—31no—
261C2×C3×Dic3—32no—
262C2×C3×Dic3—32no—
263C2×C3×Dic3—32no—
264C2×C3×Dic3—32no—
265C2×C3×Dic3—31no—
266C2×C3×Dic3—31no—
267C2×C3×Dic3—61no—
268C2×C3×Dic3—62no—
269C2×C3×Dic3—62no—
270C2×C3×Dic3—61no—
271C2×C3×Dic3C22×D311yescharacteristic
272C2×C3×Dic3C22×D311yescharacteristic
273C2×C3×Dic3C22×D312yes—
274C2×C3×Dic3C22×D312yes—
275C2×C3×Dic3C22×D311yescharacteristic
276C2×C3×Dic3C22×D311yescharacteristic
277C2×C3×Dic3C22×D311yescharacteristic
278C2×C3×Dic3C22×D311yescharacteristic
279C2×C3×Dic3C22×D311yescharacteristic
280C2×C3×Dic3C22×D311yescharacteristic
281C3×Dic6—64no—
282C3×Dic6—62no—
283C3×Dic6—64no—
284C3×Dic6—62no—
285#627—22no—
286#627—24no—
287#557—271noSylow
288C2×C33C22×Q811yescharacteristic
289C2×C33C2×#4111yescharacteristic
290C2×C332−1+411yescharacteristic
291C2×C3×Q8—91no—
292C2×C3×Q8—92no—
293C2×Dic6—31no—
294C2×Dic6—31no—
295C2×Dic6—32no—
296C2×Dic6—32no—
297C2×Dic6—91no—
298C2×Dic6—91no—
299C2×Dic6—91no—
300C2×Dic6—181no—
301C2×Dic6—181no—
302C2×Dic6—181no—
303C2×Dic6—361no—
304C3×C4⋊C4—31no—
305C3×C4⋊C4—31no—
306C3×C4⋊C4—61no—
307C3×C4⋊C4—91no—
308C3×C4⋊C4—92no—
309C3×C4⋊C4—92no—
310C3×C4⋊C4—91no—
311C3×C4⋊C4—91no—
312C3×C4⋊C4—92no—
313C3×C4⋊C4—91no—
314C3×C4⋊C4—91no—
315C3×C42—31no—
316C3×C42—31no—
317C3×C42—31no—
318C3×C42—31no—
319C3×C42—61no—
320C3×C42—61no—
321C3×C42—91no—
322C4⋊Dic3—31no—
323C4⋊Dic3—31no—
324C4⋊Dic3—31no—
325C4⋊Dic3—31no—
326C4⋊Dic3—31no—
327C4⋊Dic3—31no—
328C4⋊Dic3—32no—
329C4⋊Dic3—32no—
330C4⋊Dic3—61no—
331C4⋊Dic3—61no—
332C4⋊Dic3—92no—
333C4⋊Dic3—92no—
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—32no—
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—92no—
356Dic3⋊C4—182no—
357Dic3⋊C4—182no—
358Dic3⋊C4—182no—
359C4×Dic3—31no—
360C4×Dic3—31no—
361C4×Dic3—61no—
362C4×Dic3—92no—
363C4×Dic3—92no—
364C4×Dic3—92no—
365C4×Dic3—92no—
366C4×Dic3—92no—
367C4×Dic3—92no—
368C4×Dic3—92no—
369C4×Dic3—92no—
370C4×Dic3—92no—
371C4×Dic3—182no—
372C4×Dic3—182no—
373C4×Dic3—182no—
374C4×Dic3D3211yescharacteristic
375C4×Dic3D3211yescharacteristic
376C22×C32—21no—
377C22×C32—21no—
378C22×C32—21no—
379C22×C32—41no—
380C22×C32C23×D311yescharacteristic
381C22×C32C23×D311yescharacteristic
382C22×C32C23×D311yescharacteristic
383C32×C4—22no—
384C32×C4—22no—
385C32×C4—22no—
386C32×C4—32no—
387C32×C4—42no—
388C32×C4—61no—
389C32×C4—62no—
390C32×C4—61no—
391C32×C4—61no—
392C32×C4D3×Q812yes—
393C32×C4D3×Q812yes—
394C32×C4D3×Q812yes—
395C3⋊Dic3—21no—
396C3⋊Dic3—21no—
397C3⋊Dic3—22no—
398C3⋊Dic3—22no—
399C3⋊Dic3—34no—
400C3⋊Dic3—34no—
401C3⋊Dic3—34no—
402C3⋊Dic3—64no—
403C3⋊Dic3—64no—
404C3⋊Dic3—64no—
405C3⋊Dic3—124no—
406C3×Dic3—22no—
407C3×Dic3—21no—
408C3×Dic3—22no—
409C3×Dic3—21no—
410C3×Dic3—21no—
411C3×Dic3—22no—
412C3×Dic3—21no—
413C3×Dic3—21no—
414C3×Dic3—21no—
415C3×Dic3—22no—
416C3×Dic3—41no—
417C3×Dic3—41no—
418C3×Dic3—42no—
419C3×Dic3—62no—
420C3×Dic3—62no—
421C3×Dic3—61no—
422C3×Dic3—61no—
423C3×Dic3—62no—
424C3×Dic3—61no—
425C3×Dic3—62no—
426C3×Dic3—62no—
427C3×Dic3—61no—
428C3×Dic3—122no—
429C3×Dic3—122no—
430C3×Dic3—121no—
431C3×Dic3#23512yes—
432C3×Dic3#23712yes—
433C4⋊Q8—91no—
434C4⋊Q8—272no—
435#125—91no—
436#125—271no—
437#125—272no—
438#125—272no—
439C4×Q8—91no—
440C4×Q8—271no—
441C4×Q8—271no—
442C4×Q8—271no—
443C4×Q8—272no—
444C33#55711yescharacteristic, complemented, Sylow
445C3×Q8—184no—
446C3×Q8—182no—
447C2×C3×C4—21no—
448C2×C3×C4—21no—
449C2×C3×C4—21no—
450C2×C3×C4—31no—
451C2×C3×C4—31no—
452C2×C3×C4—31no—
453C2×C3×C4—31no—
454C2×C3×C4—32no—
455C2×C3×C4—32no—
456C2×C3×C4—31no—
457C2×C3×C4—31no—
458C2×C3×C4—31no—
459C2×C3×C4—31no—
460C2×C3×C4—41no—
461C2×C3×C4—61no—
462C2×C3×C4—61no—
463C2×C3×C4—62no—
464C2×C3×C4—61no—
465C2×C3×C4—61no—
466C2×C3×C4—91no—
467C2×C3×C4—92no—
468C2×C3×C4—92no—
469C2×C3×C4—91no—
470C2×C3×C4C2×D3211yescharacteristic
471C2×C3×C4C2×D3211yescharacteristic
472C2×C3×C4C2×D3211yescharacteristic
473C2×Dic3—31no—
474C2×Dic3—31no—
475C2×Dic3—31no—
476C2×Dic3—31no—
477C2×Dic3—32no—
478C2×Dic3—32no—
479C2×Dic3—32no—
480C2×Dic3—32no—
481C2×Dic3—61no—
482C2×Dic3—61no—
483C2×Dic3—62no—
484C2×Dic3—62no—
485C2×Dic3—92no—
486C2×Dic3—92no—
487C2×Dic3—92no—
488C2×Dic3—182no—
489C2×Dic3—182no—
490C2×Dic3—182no—
491C2×Dic3—362no—
492C2×Dic3C2×D3211yescharacteristic
493C2×Dic3C2×D3212yes—
494C2×Dic3C2×D3211yescharacteristic
495C2×Dic3C2×D3211yescharacteristic
496C2×Dic3C2×D3211yescharacteristic
497Dic6—62no—
498Dic6—64no—
499Dic6—62no—
500Dic6—64no—
501Dic6—94no—
502Dic6—94no—
503Dic6—94no—
504Dic6—184no—
505Dic6—184no—
506Dic6—184no—
507Dic6—364no—
508C2×C32—21no—
509C2×C32—21no—
510C2×C32—21no—
511C2×C32—21no—
512C2×C32—21no—
513C2×C32—21no—
514C2×C32—21no—
515C2×C32—21no—
516C2×C32—21no—
517C2×C32—41no—
518C2×C32—41no—
519C2×C32—41no—
520C2×C32C2×#23511yescharacteristic
521C2×C32C2×#23711yescharacteristic
522C2×C32C2×D3×Q811yescharacteristic
523C2×C32C2×D3×Q811yescharacteristic
524C2×C32C2×D3×Q811yescharacteristic
525C2×C32#100711yescharacteristic
526C2×C32D3×#4111yescharacteristic
527C2×C32#101011yescharacteristic
528C2×C32#101011yescharacteristic
529C2×Q8—91no—
530C2×Q8—92no—
531C2×Q8—271no—
532C4⋊C4—31no—
533C4⋊C4—91no—
534C4⋊C4—91no—
535C4⋊C4—91no—
536C4⋊C4—92no—
537C4⋊C4—92no—
538C4⋊C4—92no—
539C4⋊C4—91no—
540C4⋊C4—91no—
541C4⋊C4—272no—
542C4⋊C4—272no—
543C4⋊C4—272no—
544C4⋊C4—272no—
545C4⋊C4—272no—
546C42—31no—
547C42—31no—
548C42—91no—
549C42—272no—
550C42—272no—
551C42—272no—
552C22×C3—21no—
553C22×C3—21no—
554C22×C3—21no—
555C22×C3—41no—
556C22×C3C22×D3211yescharacteristic
557C22×C3C22×D3211yescharacteristic
558C22×C3C22×D3211yescharacteristic
559C3×C4—22no—
560C3×C4—22no—
561C3×C4—22no—
562C3×C4—32no—
563C3×C4—32no—
564C3×C4—42no—
565C3×C4—62no—
566C3×C4—62no—
567C3×C4—62no—
568C3×C4—61no—
569C3×C4—61no—
570C3×C4—61no—
571C3×C4—62no—
572C3×C4—61no—
573C3×C4—61no—
574C3×C4—61no—
575C3×C4—121no—
576C3×C4—122no—
577C3×C4—121no—
578C3×C4—182no—
579C3×C4—182no—
580C3×C4—181no—
581C3×C4—181no—
582C3×C4#379712yes—
583C3×C4#379712yes—
584C3×C4#379712yes—
585Dic3—21no—
586Dic3—21no—
587Dic3—21no—
588Dic3—22no—
589Dic3—62no—
590Dic3—61no—
591Dic3—62no—
592Dic3—61no—
593Dic3—61no—
594Dic3—62no—
595Dic3—61no—
596Dic3—62no—
597Dic3—62no—
598Dic3—94no—
599Dic3—94no—
600Dic3—94no—
601Dic3—122no—
602Dic3—121no—
603Dic3—122no—
604Dic3—184no—
605Dic3—184no—
606Dic3—184no—
607Dic3—364no—
608Dic3#379512yes—
609C32—21no—
610C32—21no—
611C32—21no—
612C32—41no—
613C32#556711yescharacteristic, complemented
614C32#568511yescharacteristic, complemented
615C32#572111yescharacteristic, complemented
616Q8—182no—
617Q8—184no—
618Q8—274no—
619C2×C4—31no—
620C2×C4—31no—
621C2×C4—31no—
622C2×C4—31no—
623C2×C4—32no—
624C2×C4—91no—
625C2×C4—91no—
626C2×C4—92no—
627C2×C4—92no—
628C2×C4—272no—
629C2×C4D3311yescharacteristic
630C2×C3—21no—
631C2×C3—21no—
632C2×C3—21no—
633C2×C3—21no—
634C2×C3—21no—
635C2×C3—21no—
636C2×C3—21no—
637C2×C3—21no—
638C2×C3—21no—
639C2×C3—41no—
640C2×C3—41no—
641C2×C3—41no—
642C2×C3C2×#379511yescharacteristic
643C2×C3#6429711yescharacteristic
644C2×C3#6429811yescharacteristic
645C2×C3C2×#379711yescharacteristic
646C2×C3C2×#379711yescharacteristic
647C2×C3C2×#379711yescharacteristic
648C2×C3D3×#23511yescharacteristic
649C2×C3D3×#23711yescharacteristic
650C2×C3#6431511yescharacteristic
651C22C2×D3311yescharacteristic
652C4—32no—
653C4—61no—
654C4—62no—
655C4—61no—
656C4—61no—
657C4—181no—
658C4—182no—
659C4—181no—
660C4—182no—
661C4—274no—
662C4#8895812yes—
663C3—21no—
664C3—21no—
665C3—21no—
666C3—41no—
667C3#1059385211yescharacteristic, complemented
668C3#1059386411yescharacteristic, complemented
669C3#1059431811yescharacteristic, complemented
670C2D3×#379511yescharacteristic
671C2#1172439611yescharacteristic
672C2C2×#8895811yescharacteristic
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
76. #6308Q8semidirect
444. C33#557semidirect
613. C32#5567semidirect
614. C32#5685semidirect
615. C32#5721semidirect
667. C3#10593852semidirect
668. C3#10593864semidirect
669. C3#10594318semidirect

Maximal subgroups

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

subgroupindex
#117236802
#117234062
#117233992
C3×#639182
C3×C4×#6272
C3×#638692
#117230322
#117230162
#117229412
#105943183
#105938643
#105938523