Finite Group Databasebeta

← #13178144

Subgroups — #13178144

Order 1728.

1174 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic
2C3×#64078C211yesmaximal, characteristic, complemented
3C3×C4×#634C211yesmaximal, characteristic, complemented
4#11723691C212yesmaximal
5#11723689C212yesmaximal, complemented
6#11723688C214yesmaximal, complemented
7#11723680C212yesmaximal, complemented
8#11723674C212yesmaximal, complemented
9#10593960—31nomaximal
10#10593958—31nomaximal
11#10593956—31nomaximal
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4×C3⋊D3C2212yes—
14C2×C3×#634C2211yescharacteristic
15C2×C3×#634C2212yes—
16C2×#6294C2214yes—
17C33×C42C2211yescharacteristic
18C3×#3752C2212yes—
19C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C3×C4⋊(C3⋊Dic3)C2212yes—
21C3×C4⋊(C3⋊Dic3)C2212yes—
22C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
23C3×#3750C2212yes—
24C3×C4×C3⋊Dic3C2212yes—
25#88815C2214yes—
26#88815C2214yes—
27#88814C2214yes—
28C4×C4⋊(C3⋊D3)D311yescharacteristic
29#64078D311yescharacteristic, complemented
30C4×#634D311yescharacteristic, complemented
31C3×C4×D12—31no—
32C3×C4×D12—31no—
33C3×C4×D12—61no—
34C3×#869—31no—
35C3×#869—31no—
36C3×#869—61no—
37C3×C4×Dic6—31no—
38C3×C4×Dic6—31no—
39C3×C4×Dic6—61no—
40#63900—32no—
41#63898—32no—
42#63896—34no—
43#63894—32no—
44Dic3×D12—32no—
45D3×C4⋊Dic3—32no—
46D3×C4⋊Dic3—32no—
47#63888—34no—
48#63887—34no—
49#63872—32no—
50#63871—32no—
51#63869—32no—
52#63869—32no—
53#63857—32no—
54#63856—32no—
55C22×C3×C3⋊D3C2312yes—
56C2×C33×C4C2311yescharacteristic
57C2×C33×C4C2311yescharacteristic
58C2×C33×C4C2311yescharacteristic
59C2×C3×C3⋊Dic3C2312yes—
60C2×C3×C3⋊Dic3C2312yes—
61C2×C3×C3⋊Dic3C2314yes—
62C2×C3×C3⋊Dic3C2312yes—
63C3×C4⋊(C3⋊D3)Q814yescomplemented
64C3×C4×C3⋊D3—24no—
65C3×#634—22no—
66C3×#634—42no—
67#6294—44no—
68#5730—91no—
69#5579—91no—
70#5543—91no—
71C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
72C2×C4×C3⋊D3C2×D312yes—
73C2×#634C2×D311yescharacteristic
74C2×#634C2×D312yes—
75C2×C3×D12—31no—
76C2×C3×D12—31no—
77C2×C3×D12—61no—
78C2×C3×C4×D3—32no—
79C2×C3×C4×D3—32no—
80C2×C3×C4×D3—62no—
81C2×C3×Dic6—31no—
82C2×C3×Dic6—31no—
83C2×C3×Dic6—32no—
84C2×C3×Dic6—32no—
85C2×C3×Dic6—61no—
86C2×C3×Dic6—62no—
87C2×#624—34no—
88C2×D3×Dic3—34no—
89C2×D3×Dic3—34no—
90C32×C42—21no—
91C32×C42—21no—
92C32×C42—21no—
93C32×C42—41no—
94C32×C42C2×D311yescharacteristic
95C32×C42C2×D311yescharacteristic
96C32×C42C2×D311yescharacteristic
97#3752C2×D312yes—
98C4⋊(C3⋊Dic3)C2×D311yescharacteristic
99C4⋊(C3⋊Dic3)C2×D312yes—
100C4⋊(C3⋊Dic3)C2×D312yes—
101C4⋊(C3⋊Dic3)C2×D311yescharacteristic
102#3750C2×D312yes—
103C4×C3⋊Dic3C2×D312yes—
104C3×(C2×D3)⋊C4—32no—
105C3×(C2×D3)⋊C4—32no—
106C3×(C2×D3)⋊C4—62no—
107C3×C4⋊Dic3—32no—
108C3×C4⋊Dic3—31no—
109C3×C4⋊Dic3—32no—
110C3×C4⋊Dic3—32no—
111C3×C4⋊Dic3—31no—
112C3×C4⋊Dic3—32no—
113C3×C4⋊Dic3—31no—
114C3×C4⋊Dic3—31no—
115C3×C4⋊Dic3—61no—
116C3×C4⋊Dic3—62no—
117C3×C4⋊Dic3—62no—
118C3×C4⋊Dic3—61no—
119C3×Dic3⋊C4—32no—
120C3×Dic3⋊C4—32no—
121C3×Dic3⋊C4—62no—
122C3×C4×Dic3—32no—
123C3×C4×Dic3—32no—
124C3×C4×Dic3—62no—
125#3724—34no—
126#3724—34no—
127Dic3⋊Dic3—34no—
128Dic3⋊Dic3—34no—
129Dic3⋊Dic3—34no—
130Dic3⋊Dic3—34no—
131#3722—34no—
132(C2×D3)⋊Dic3—34no—
133(C2×D3)⋊Dic3—34no—
134C22×C33C2411yescharacteristic
135C2×C3×C3⋊D3—22no—
136C2×C3×C3⋊D3—22no—
137C2×C3×C3⋊D3C2×Q814yes—
138C33×C4—21no—
139C33×C4C2×Q812yes—
140C33×C4#4112yes—
141C3×C3⋊Dic3—22no—
142C3×C3⋊Dic3—22no—
143C3×C3⋊Dic3—22no—
144C3×C3⋊Dic3—24no—
145C3×C4⋊Q8—91no—
146C3×C4×Q8—91no—
147C3×C4×D4—91no—
148#946—94no—
149#946—92no—
150D4×Dic3—92no—
151C2×C4⋊Dic3—92no—
152#924—92no—
153#897—92no—
154#896—94no—
155#893—92no—
156D3×C4⋊C4—92no—
157D3×C4⋊C4—92no—
158#890—92no—
159#890—92no—
160#878—94no—
161C4×D12—31no—
162C4×D12—31no—
163C4×D12—61no—
164C4×D12—92no—
165#870—92no—
166#869—31no—
167#869—31no—
168#869—61no—
169C4×Dic6—31no—
170C4×Dic6—31no—
171C4×Dic6—61no—
172C22×C3⋊D3C22×D312yes—
173C22×C3×D3—32no—
174C22×C3×D3—32no—
175C22×C3×D3—62no—
176C2×C32×C4—21no—
177C2×C32×C4—21no—
178C2×C32×C4—21no—
179C2×C32×C4—21no—
180C2×C32×C4—21no—
181C2×C32×C4—21no—
182C2×C32×C4—21no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4—41no—
186C2×C32×C4—41no—
187C2×C32×C4—41no—
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C22×D311yescharacteristic
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D311yescharacteristic
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C22×D311yescharacteristic
197C2×C3⋊Dic3C22×D312yes—
198C2×C3⋊Dic3C22×D314yes—
199C2×C3⋊Dic3C22×D312yes—
200C2×C3⋊Dic3C22×D312yes—
201C4⋊(C3⋊D3)Dic614yescomplemented
202C4×C3⋊D3—24no—
203#634—22no—
204#634—42no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—34no—
207C2×C3×Dic3—32no—
208C2×C3×Dic3—34no—
209C2×C3×Dic3—32no—
210C2×C3×Dic3—32no—
211C2×C3×Dic3—32no—
212C2×C3×Dic3—32no—
213C2×C3×Dic3—62no—
214C2×C3×Dic3—62no—
215C2×C3×Dic3—64no—
216C2×C3×Dic3—62no—
217C3×D12—34no—
218C3×D12—34no—
219C3×D12—64no—
220C3×C4×D3—64no—
221C3×C4×D3—64no—
222C3×C4×D3—124no—
223C3×Dic6—64no—
224C3×Dic6—62no—
225C3×Dic6—62no—
226C3×Dic6—122no—
227C3×Dic6—122no—
228C3×Dic6—124no—
229#624—124no—
230D3×Dic3—124no—
231D3×Dic3—124no—
232#554—271noSylow
233C2×C33C22×Q811yescharacteristic
234C2×C33C2×#4111yescharacteristic
235C2×C33H(5,𝔽2)11yescharacteristic
236C3×C3⋊D3—24no—
237C2×C3×Q8—91no—
238C2×C3×Q8—92no—
239C2×C3×D4—91no—
240C22×C3×C4—92no—
241C22×Dic3—94no—
242C2×D12—31no—
243C2×D12—31no—
244C2×D12—61no—
245C2×C4×D3—32no—
246C2×C4×D3—32no—
247C2×C4×D3—62no—
248C2×C4×D3—94no—
249C2×C4×D3—94no—
250C2×Dic6—31no—
251C2×Dic6—31no—
252C2×Dic6—32no—
253C2×Dic6—32no—
254C2×Dic6—61no—
255C2×Dic6—62no—
256C3×C4⋊C4—91no—
257C3×C4⋊C4—92no—
258C3×C4⋊C4—91no—
259C3×C4⋊C4—92no—
260C3×C4⋊C4—92no—
261C3×C22⋊C4—92no—
262C3×C42—21no—
263C3×C42—21no—
264C3×C42—21no—
265C3×C42—41no—
266C3×C42—92no—
267C3×C42D3211yescharacteristic
268C3×C42D3211yescharacteristic
269C3×C42D3211yescharacteristic
270#217—94no—
271(C2×D3)⋊C4—32no—
272(C2×D3)⋊C4—32no—
273(C2×D3)⋊C4—62no—
274(C2×D3)⋊C4—94no—
275(C2×D3)⋊C4—94no—
276C4⋊Dic3—31no—
277C4⋊Dic3—31no—
278C4⋊Dic3—32no—
279C4⋊Dic3—32no—
280C4⋊Dic3—32no—
281C4⋊Dic3—32no—
282C4⋊Dic3—31no—
283C4⋊Dic3—31no—
284C4⋊Dic3—61no—
285C4⋊Dic3—62no—
286C4⋊Dic3—62no—
287C4⋊Dic3—61no—
288C4⋊Dic3—94no—
289C4⋊Dic3—94no—
290Dic3⋊C4—32no—
291Dic3⋊C4—32no—
292Dic3⋊C4—62no—
293Dic3⋊C4—94no—
294Dic3⋊C4—94no—
295Dic3⋊C4—94no—
296Dic3⋊C4—94no—
297C4×Dic3—32no—
298C4×Dic3—32no—
299C4×Dic3—62no—
300C22×C32—21no—
301C22×C32—21no—
302C22×C32—21no—
303C22×C32—41no—
304C22×C32C23×D311yescharacteristic
305C22×C32C23×D311yescharacteristic
306C22×C32C23×D311yescharacteristic
307C2×C3⋊D3—22no—
308C2×C3⋊D3—22no—
309C2×C3⋊D3C2×Dic614yes—
310C2×C3×D3—34no—
311C2×C3×D3—34no—
312C2×C3×D3—64no—
313C2×C3×D3—62no—
314C2×C3×D3—62no—
315C2×C3×D3—62no—
316C2×C3×D3—62no—
317C2×C3×D3—122no—
318C2×C3×D3—122no—
319C32×C4—21no—
320C32×C4—22no—
321C32×C4—21no—
322C32×C4—21no—
323C32×C4—22no—
324C32×C4—22no—
325C32×C4—22no—
326C32×C4—22no—
327C32×C4—22no—
328C32×C4—22no—
329C32×C4—42no—
330C32×C4—42no—
331C32×C4—41no—
332C32×C4—41no—
333C32×C4—42no—
334C32×C4C2×Dic612yes—
335C32×C4#23512yes—
336C32×C4#23712yes—
337C32×C4D3×Q812yes—
338C32×C4D3×Q812yes—
339C32×C4#23912yes—
340C3⋊Dic3—22no—
341C3⋊Dic3—22no—
342C3⋊Dic3—24no—
343C3⋊Dic3—22no—
344C3×Dic3—64no—
345C3×Dic3—62no—
346C3×Dic3—62no—
347C3×Dic3—62no—
348C3×Dic3—62no—
349C3×Dic3—62no—
350C3×Dic3—64no—
351C3×Dic3—62no—
352C3×Dic3—64no—
353C3×Dic3—124no—
354C3×Dic3—122no—
355C3×Dic3—122no—
356C4⋊Q8—91no—
357#125—272no—
358C22⋊Q8—274no—
359C22⋊Q8—272no—
360C4×Q8—91no—
361C4×D4—91no—
362C4×D4—272no—
363C2×C4⋊C4—272no—
364C33#55411yescharacteristic, complemented, Sylow
365C23×C3—92no—
366C22×D3—32no—
367C22×D3—32no—
368C22×D3—62no—
369C3×Q8—182no—
370C3×Q8—362no—
371C3×D4—94no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—92no—
385C2×C3×C4—92no—
386C2×C3×C4—92no—
387C2×C3×C4—94no—
388C2×C3×C4—184no—
389C2×C3×C4C2×D3211yescharacteristic
390C2×C3×C4C2×D3211yescharacteristic
391C2×C3×C4C2×D3211yescharacteristic
392C2×C3×C4C2×D3211yescharacteristic
393C2×C3×C4C2×D3211yescharacteristic
394C2×C3×C4C2×D3211yescharacteristic
395C2×C3×C4C2×D3211yescharacteristic
396C2×C3×C4C2×D3211yescharacteristic
397C2×C3×C4C2×D3211yescharacteristic
398C2×Dic3—32no—
399C2×Dic3—32no—
400C2×Dic3—32no—
401C2×Dic3—32no—
402C2×Dic3—32no—
403C2×Dic3—34no—
404C2×Dic3—32no—
405C2×Dic3—34no—
406C2×Dic3—62no—
407C2×Dic3—62no—
408C2×Dic3—64no—
409C2×Dic3—62no—
410C2×Dic3—364no—
411D12—34no—
412D12—34no—
413D12—64no—
414C4×D3—64no—
415C4×D3—64no—
416C4×D3—124no—
417C4×D3—364no—
418C4×D3—364no—
419Dic6—64no—
420Dic6—62no—
421Dic6—62no—
422Dic6—122no—
423Dic6—122no—
424Dic6—124no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—21no—
433C2×C32—21no—
434C2×C32—41no—
435C2×C32—41no—
436C2×C32—41no—
437C2×C32C22×Dic611yescharacteristic
438C2×C32C2×#23511yescharacteristic
439C2×C32C2×#23711yescharacteristic
440C2×C32#100411yescharacteristic
441C2×C32C2×D3×Q811yescharacteristic
442C2×C32C2×D3×Q811yescharacteristic
443C2×C32C2×#23911yescharacteristic
444C2×C32#100911yescharacteristic
445C2×C32#100911yescharacteristic
446C3⋊D3—24no—
447C3×D3—64no—
448C3×D3—64no—
449C3×D3—124no—
450C2×Q8—91no—
451C2×Q8—92no—
452C2×D4—91no—
453C22×C4—92no—
454C22×C4—274no—
455C4⋊C4—91no—
456C4⋊C4—92no—
457C4⋊C4—91no—
458C4⋊C4—92no—
459C4⋊C4—92no—
460C4⋊C4—274no—
461C4⋊C4—274no—
462C22⋊C4—92no—
463C22⋊C4—274no—
464C42—92no—
465C42#111311yescharacteristic
466C22×C3—21no—
467C22×C3—21no—
468C22×C3—21no—
469C22×C3—41no—
470C22×C3—94no—
471C22×C3—182no—
472C22×C3—182no—
473C22×C3C22×D3211yescharacteristic
474C22×C3C22×D3211yescharacteristic
475C22×C3C22×D3211yescharacteristic
476C2×D3—34no—
477C2×D3—34no—
478C2×D3—64no—
479C2×D3—62no—
480C2×D3—62no—
481C2×D3—62no—
482C2×D3—62no—
483C2×D3—122no—
484C2×D3—122no—
485C3×C4—22no—
486C3×C4—21no—
487C3×C4—21no—
488C3×C4—21no—
489C3×C4—22no—
490C3×C4—22no—
491C3×C4—22no—
492C3×C4—22no—
493C3×C4—22no—
494C3×C4—22no—
495C3×C4—41no—
496C3×C4—41no—
497C3×C4—42no—
498C3×C4—42no—
499C3×C4—42no—
500C3×C4—182no—
501C3×C4—182no—
502C3×C4—182no—
503C3×C4—184no—
504C3×C4D3×Dic612yes—
505C3×C4D3×Dic612yes—
506C3×C4#379512yes—
507C3×C4#379612yes—
508C3×C4#379712yes—
509C3×C4#379912yes—
510Dic3—64no—
511Dic3—62no—
512Dic3—62no—
513Dic3—62no—
514Dic3—62no—
515Dic3—64no—
516Dic3—64no—
517Dic3—62no—
518Dic3—62no—
519Dic3—124no—
520Dic3—122no—
521Dic3—122no—
522C32—21no—
523C32—21no—
524C32—21no—
525C32—41no—
526C32#554311yescharacteristic, complemented
527C32#557911yescharacteristic, complemented
528C32#573011yescharacteristic, complemented
529C23—92no—
530Q8—182no—
531Q8—362no—
532D4—94no—
533C2×C4—92no—
534C2×C4—92no—
535C2×C4—92no—
536C2×C4—94no—
537C2×C4—184no—
538C2×C4—1084no—
539C2×C4C2×#111311yescharacteristic
540C2×C4C2×#111311yescharacteristic
541C2×C4C2×#111311yescharacteristic
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—21no—
547C2×C3—21no—
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—41no—
552C2×C3—41no—
553C2×C3—41no—
554C2×C3—184no—
555C2×C3C2×D3×Dic611yescharacteristic
556C2×C3C2×D3×Dic611yescharacteristic
557C2×C3C2×#379511yescharacteristic
558C2×C3C2×#379611yescharacteristic
559C2×C3C2×#379711yescharacteristic
560C2×C3C2×#379911yescharacteristic
561C2×C3#6430711yescharacteristic
562C2×C3#6430711yescharacteristic
563C2×C3#6430811yescharacteristic
564D3—64no—
565D3—64no—
566D3—124no—
567C22—94no—
568C22—182no—
569C22—182no—
570C22C22×#111311yescharacteristic
571C4—21no—
572C4—182no—
573C4—182no—
574C4—182no—
575C4—184no—
576C4#8904012yes—
577C4#8904112yes—
578C3—21no—
579C3—21no—
580C3—21no—
581C3—41no—
582C3#1059395611yescharacteristic, complemented
583C3#1059395811yescharacteristic, complemented
584C3#1059396011yescharacteristic, complemented
585C2—184no—
586C2C2×#8904011yescharacteristic
587C2C2×#8904111yescharacteristic
588C2#1172461911yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC42#1113
hypocenterC33#554
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C3×#64078C2semidirect
3. C3×C4×#634C2semidirect
5. #11723689C2semidirect
6. #11723688C2semidirect
7. #11723680C2semidirect
8. #11723674C2semidirect
29. #64078D3semidirect
30. C4×#634D3semidirect
63. C3×C4⋊(C3⋊D3)Q8semidirect
201. C4⋊(C3⋊D3)Dic6semidirect
364. C33#554semidirect
526. C32#5543semidirect
527. C32#5579semidirect
528. C32#5730semidirect
582. C3#10593956semidirect
583. C3#10593958semidirect
584. C3#10593960semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C4⋊(C3⋊D3)2
C3×#640782
C3×C4×#6342
#117236912
#117236892
#117236882
#117236802
#117236742
#105939603
#105939583
#105939563