Finite Group Databasebeta

← #11806898

Subgroups — #11806898

Order 1152.

1807 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×#67C212yesmaximal, complemented
2C2×#63979C211yesmaximal, characteristic
3C2×#63904C212yesmaximal, complemented
4#10592912C211yesmaximal, characteristic, complemented
5#10591722C211yesmaximal, characteristic, complemented
6#10591707C212yesmaximal, complemented
7#10591383C212yesmaximal, complemented
8#10591381C212yesmaximal, complemented
9Dic3×Dic3⋊C4C212yesmaximal, complemented
10#82343—32nomaximal
11C23×C32×C4C2211yescharacteristic
12C22×#638C2211yescharacteristic
13C22×C3×#67C2212yes—
14C22×C3×C4×D3C2212yes—
15C22×#626C2212yes—
16C22×#625C2211yescharacteristic
17C22×#624C2211yescharacteristic
18C2×#3757C2211yescharacteristic
19C2×#3752C2212yes—
20C2×C3×#217C2212yes—
21C3×C4×#67—28no—
22C2×C3×(C2×D3)⋊C4C2212yes—
23C2×C3×Dic3⋊C4C2212yes—
24C2×C3×C4×Dic3C2212yes—
25#63979C418yescomplemented
26C2×Dic3⋊Dic3C2212yes—
27C2×#3722C2211yescharacteristic
28C2×(C2×D3)⋊Dic3C2212yes—
29C2×(C2×D3)⋊Dic3C2212yes—
30C2×Dic32C2212yes—
31C2×Dic32C2211yescharacteristic
32#63904—28no—
33#63658C2212yes—
34#63579C2212yes—
35C2×C3×C4×D4—32no—
36C2×#940—32no—
37C2×C4×#67—32no—
38C2×C4×#67D312yescomplemented
39C2×#893—32no—
40#5221—32no—
41#5213—32no—
42#5213—32no—
43#5213—61no—
44Dic3×C4⋊C4—32no—
45#4957—32no—
46#4957—32no—
47#4956—32no—
48#4956—32no—
49#4939—32no—
50C4×Dic3⋊C4—32no—
51C24×C32C2311yescharacteristic
52C23×C3⋊D3C2311yescharacteristic
53C23×C3×D3C2312yes—
54C22×C32×C4—24no—
55C22×C32×C4—41no—
56C22×C32×C4C2312yes—
57C2×#638—41no—
58C2×#638—41no—
59C2×#638C2×C414yescomplemented
60C22×C3⋊Dic3C2312yes—
61C22×C3⋊Dic3C2311yescharacteristic
62C2×C3×#67—24no—
63C2×C3×#67—24no—
64C2×C3×#67C2×C418yescomplemented
65C22×C3×Dic3C2312yes—
66C22×C3×Dic3C2312yes—
67C22×C3×Dic3C2312yes—
68C2×C3×C4×D3—24no—
69C2×C3×C4×D3—24no—
70C2×C3×C4×D3—42no—
71C2×#626—24no—
72C2×#626—24no—
73C2×#626C2×C418yescomplemented
74C2×#625—41no—
75C2×#625—41no—
76C2×#625C2×C414yescomplemented
77C2×#624—24no—
78C2×#624—41no—
79#3757—41no—
80#3752—24no—
81C3×#217—24no—
82C3×(C2×D3)⋊C4—24no—
83C3×Dic3⋊C4D418yescomplemented
84C3×C4×Dic3—24no—
85Dic3⋊Dic3—24no—
86#3722—41no—
87(C2×D3)⋊Dic3—24no—
88(C2×D3)⋊Dic3—42no—
89Dic32—24no—
90Dic32C2×C414yescomplemented
91#2366—91noSylow
92C22×C3×D4—32no—
93C23×C3×C4—21no—
94C23×C3×C4—32no—
95C23×C3×C4C2×D312yes—
96C22×#67—32no—
97C22×#67—32no—
98C22×#67—32no—
99C22×#67—61no—
100C22×#67C2×D312yes—
101C22×D12—32no—
102C22×C4×D3—32no—
103C22×C4×D3C2×D312yes—
104C3×C4×D4—68no—
105C2×C3×C4⋊C4—32no—
106C2×C3×C22⋊C4—32no—
107C2×C3×C22⋊C4—32no—
108C2×C3×C42—32no—
109C2×#217—32no—
110C2×#217—32no—
111C2×#217—32no—
112C2×#217—61no—
113C2×#217C2×D312yes—
114#940—316no—
115C4×#67—28no—
116C4×#67—68no—
117C2×(C2×D3)⋊C4—32no—
118C2×(C2×D3)⋊C4—32no—
119C2×(C2×D3)⋊C4—32no—
120C2×(C2×D3)⋊C4—32no—
121C2×(C2×D3)⋊C4—32no—
122C2×(C2×D3)⋊C4—62no—
123C2×(C2×D3)⋊C4C2×D312yes—
124C2×C4⋊Dic3—32no—
125C2×Dic3⋊C4—32no—
126C2×Dic3⋊C4C2×D312yes—
127C2×C4×Dic3—32no—
128C2×C4×Dic3—32no—
129C2×C4×Dic3—32no—
130C2×C4×Dic3C2×D312yes—
131#893—68no—
132C3⋊#95—32no—
133C3⋊#95—32no—
134C3⋊#95—32no—
135C3⋊#95—32no—
136C3⋊#95—62no—
137C23×C32—21no—
138C23×C32—21no—
139C23×C32—22no—
140C23×C32—22no—
141C23×C32C22×C412yes—
142C23×C32C2411yescharacteristic
143C22×C3⋊D3—22no—
144C22×C3⋊D3—22no—
145C22×C3⋊D3—21no—
146C22×C3⋊D3—21no—
147C22×C3⋊D3C22×C412yes—
148C22×C3×D3—22no—
149C22×C3×D3—22no—
150C22×C3×D3—22no—
151C22×C3×D3—22no—
152C22×C3×D3—22no—
153C22×C3×D3—22no—
154C22×C3×D3C22×C414yes—
155C2×C32×C4—22no—
156C2×C32×C4—22no—
157C2×C32×C4—44no—
158C2×C32×C4C2×D414yes—
159#638—44no—
160C2×C3⋊Dic3—21no—
161C2×C3⋊Dic3—21no—
162C2×C3⋊Dic3—22no—
163C2×C3⋊Dic3—22no—
164C2×C3⋊Dic3—22no—
165C2×C3⋊Dic3C22×C412yes—
166C3×#67—216no—
167C2×C3×Dic3—22no—
168C2×C3×Dic3—22no—
169C2×C3×Dic3—22no—
170C2×C3×Dic3—22no—
171C2×C3×Dic3—22no—
172C2×C3×Dic3C22×C414yes—
173C2×C3×Dic3C2×D414yes—
174C2×C3×Dic3C2×D414yes—
175C2×C3×Dic3#4114yes—
176C3×C4×D3—48no—
177#626—216no—
178#625—44no—
179#624—44no—
180C2×#127—91no—
181C2×C4×D4—32no—
182C2×C4×D4—92no—
183#390—92no—
184#390—92no—
185#386—91no—
186#386—92no—
187#386—91no—
188C4×C4⋊C4—92no—
189C24×C3—21no—
190C24×C3—32no—
191C24×C3C22×D312yes—
192C23×D3—32no—
193C23×D3—61no—
194C23×D3C22×D312yes—
195C2×C3×D4—38no—
196C2×C3×D4—64no—
197C2×C3×D4—64no—
198C22×C3×C4—24no—
199C22×C3×C4—24no—
200C22×C3×C4—22no—
201C22×C3×C4—32no—
202C22×C3×C4—32no—
203C22×C3×C4—32no—
204C22×C3×C4—44no—
205C22×C3×C4—42no—
206C22×C3×C4—42no—
207C22×C3×C4—64no—
208C22×C3×C4—64no—
209C22×C3×C4—122no—
210C22×C3×C4C22×D312yes—
211C22×C3×C4C22×D312yes—
212C2×#67—24no—
213C2×#67—24no—
214C2×#67—38no—
215C2×#67—38no—
216C2×#67—38no—
217C2×#67—64no—
218C2×#67—64no—
219C2×#67—64no—
220C2×#67—122no—
221C2×#67—122no—
222C2×#67—122no—
223C2×#67—122no—
224C2×#67—122no—
225C2×#67—122no—
226C2×#67C4×D318yescomplemented
227C22×Dic3—32no—
228C22×Dic3—32no—
229C22×Dic3—32no—
230C22×Dic3—61no—
231C22×Dic3—62no—
232C22×Dic3C22×D312yes—
233C22×Dic3C22×D312yes—
234C22×Dic3C22×D312yes—
235C2×D12—38no—
236C2×D12—64no—
237C2×D12—64no—
238C2×C4×D3—24no—
239C2×C4×D3—24no—
240C2×C4×D3—42no—
241C2×C4×D3—64no—
242C2×C4×D3—64no—
243C2×C4×D3—122no—
244C3×C4⋊C4—38no—
245C3×C22⋊C4—64no—
246C3×C22⋊C4—64no—
247C3×C42—64no—
248#217—24no—
249#217—64no—
250#217—122no—
251#217—122no—
252#217—122no—
253(C2×D3)⋊C4—24no—
254(C2×D3)⋊C4—64no—
255(C2×D3)⋊C4—64no—
256(C2×D3)⋊C4—64no—
257(C2×D3)⋊C4—124no—
258(C2×D3)⋊C4—122no—
259(C2×D3)⋊C4—122no—
260C4⋊Dic3—64no—
261Dic3⋊C4—64no—
262Dic3⋊C4#6718yescomplemented
263C4×Dic3—24no—
264C4×Dic3—38no—
265C4×Dic3—64no—
266C4×Dic3—64no—
267C22×C32—22no—
268C22×C32—24no—
269C22×C32—41no—
270C22×C32—42no—
271C22×C32—41no—
272C22×C32C23×C411yescharacteristic
273C22×C32C22×D412yes—
274C22×C32C2×#4112yes—
275C22×C32H(5,𝔽2)11yescharacteristic
276C22×C32H(5,𝔽2)11yescharacteristic
277C2×C3⋊D3—24no—
278C2×C3⋊D3—22no—
279C2×C3⋊D3—41no—
280C2×C3⋊D3—42no—
281C2×C3⋊D3—41no—
282C2×C3×D3—24no—
283C2×C3×D3—24no—
284C2×C3×D3—24no—
285C2×C3×D3—42no—
286C2×C3×D3—42no—
287C2×C3×D3—42no—
288C2×C3×D3—42no—
289C32×C4—24no—
290C3⋊Dic3—22no—
291C3⋊Dic3—42no—
292C3×Dic3—24no—
293C3×Dic3—42no—
294C3×Dic3C4×D418yescomplemented
295C22×D4—32no—
296C22×D4—92no—
297C22×D4—91no—
298C22×D4—91no—
299C23×C4—32no—
300C23×C4—91no—
301C23×C4D3211yescharacteristic
302#127—98no—
303C4×D4—68no—
304C4×D4—188no—
305C2×C4⋊C4—32no—
306C2×C4⋊C4—92no—
307C2×C22⋊C4—32no—
308C2×C22⋊C4—32no—
309C2×C22⋊C4—92no—
310C2×C22⋊C4—92no—
311C2×C22⋊C4—91no—
312C2×C22⋊C4—92no—
313C2×C22⋊C4—91no—
314C2×C42—32no—
315C2×C42—91no—
316C2×C42—92no—
317#95—92no—
318#95—92no—
319C23×C3—22no—
320C23×C3—22no—
321C23×C3—22no—
322C23×C3—22no—
323C23×C3—22no—
324C23×C3—22no—
325C23×C3—22no—
326C23×C3—22no—
327C23×C3—21no—
328C23×C3—22no—
329C23×C3—34no—
330C23×C3—42no—
331C23×C3—42no—
332C23×C3—62no—
333C23×C3—62no—
334C23×C3—62no—
335C23×C3—62no—
336C23×C3—62no—
337C23×C3—62no—
338C23×C3C2×C4×D314yes—
339C23×C3C23×D312yes—
340C22×D3—22no—
341C22×D3—22no—
342C22×D3—22no—
343C22×D3—22no—
344C22×D3—22no—
345C22×D3—22no—
346C22×D3—34no—
347C22×D3—62no—
348C22×D3—62no—
349C22×D3—62no—
350C22×D3—62no—
351C22×D3—62no—
352C22×D3—62no—
353C22×D3—62no—
354C22×D3—122no—
355C22×D3—121no—
356C22×D3—121no—
357C22×D3—122no—
358C22×D3C2×C4×D314yes—
359C3×D4—616no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—24no—
365C2×C3×C4—34no—
366C2×C3×C4—34no—
367C2×C3×C4—34no—
368C2×C3×C4—34no—
369C2×C3×C4—42no—
370C2×C3×C4—42no—
371C2×C3×C4—48no—
372C2×C3×C4—48no—
373C2×C3×C4—62no—
374C2×C3×C4—62no—
375C2×C3×C4—62no—
376C2×C3×C4—62no—
377C2×C3×C4—62no—
378C2×C3×C4—128no—
379C2×C3×C4D3×D414yes—
380C2×C3×C4C2×#6714yes—
381#67—216no—
382#67—616no—
383#67—128no—
384#67—128no—
385#67—128no—
386C2×Dic3—22no—
387C2×Dic3—22no—
388C2×Dic3—22no—
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—34no—
392C2×Dic3—62no—
393C2×Dic3—62no—
394C2×Dic3—62no—
395C2×Dic3—62no—
396C2×Dic3—62no—
397C2×Dic3—62no—
398C2×Dic3—62no—
399C2×Dic3—62no—
400C2×Dic3—62no—
401C2×Dic3—121no—
402C2×Dic3—121no—
403C2×Dic3—122no—
404C2×Dic3—122no—
405C2×Dic3—122no—
406C2×Dic3C2×C4×D314yes—
407C2×Dic3#23514yes—
408C2×Dic3C2×#6714yes—
409C2×Dic3C2×#6714yes—
410D12—616no—
411C4×D3—48no—
412C4×D3—128no—
413C2×C32—42no—
414C2×C32C2×C4×D412yes—
415C2×C32#51811yescharacteristic
416C2×C32D4211yescharacteristic
417C2×C32#55012yes—
418C2×C32#55911yescharacteristic
419C3⋊D3—42no—
420C3×D3—44no—
421C24—32no—
422C24—91no—
423C24C2×D3211yescharacteristic
424C2×D4—38no—
425C2×D4—64no—
426C2×D4—64no—
427C2×D4—98no—
428C2×D4—94no—
429C2×D4—94no—
430C2×D4—184no—
431C2×D4—184no—
432C2×D4—361no—
433C2×D4—361no—
434C2×D4—361no—
435C2×D4—361no—
436C22×C4—24no—
437C22×C4—32no—
438C22×C4—32no—
439C22×C4—32no—
440C22×C4—41no—
441C22×C4—64no—
442C22×C4—64no—
443C22×C4—91no—
444C22×C4—92no—
445C22×C4—122no—
446C22×C4—184no—
447C22×C4—361no—
448C22×C4C2×D3212yes—
449C4⋊C4—38no—
450C4⋊C4—184no—
451C22⋊C4—64no—
452C22⋊C4—64no—
453C22⋊C4—184no—
454C22⋊C4—184no—
455C22⋊C4—361no—
456C22⋊C4—362no—
457C22⋊C4—361no—
458C42—64no—
459C42—94no—
460C42—184no—
461C22×C3—24no—
462C22×C3—24no—
463C22×C3—24no—
464C22×C3—24no—
465C22×C3—21no—
466C22×C3—22no—
467C22×C3—21no—
468C22×C3—22no—
469C22×C3—21no—
470C22×C3—44no—
471C22×C3—42no—
472C22×C3—44no—
473C22×C3—42no—
474C22×C3—42no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3—42no—
478C22×C3—64no—
479C22×C3—64no—
480C22×C3—64no—
481C22×C3—122no—
482C22×C3—122no—
483C22×C3—122no—
484C22×C3—122no—
485C22×C3C22×C4×D312yes—
486C22×C3C2×#23512yes—
487C22×C3C2×D3×D412yes—
488C22×C3#100412yes—
489C22×C3D3×#4112yes—
490C22×C3#100912yes—
491C22×C3C22×#6712yes—
492C2×D3—24no—
493C2×D3—24no—
494C2×D3—24no—
495C2×D3—42no—
496C2×D3—42no—
497C2×D3—42no—
498C2×D3—42no—
499C2×D3—64no—
500C2×D3—64no—
501C2×D3—64no—
502C2×D3—122no—
503C2×D3—124no—
504C2×D3—122no—
505C2×D3—122no—
506C2×D3—122no—
507C2×D3—122no—
508C2×D3—242no—
509C2×D3—241no—
510C2×D3—241no—
511C3×C4—24no—
512C3×C4—24no—
513C3×C4—38no—
514C3×C4—44no—
515C3×C4—64no—
516C3×C4—122no—
517Dic3—24no—
518Dic3—42no—
519Dic3—64no—
520Dic3—122no—
521Dic3—122no—
522Dic3—122no—
523Dic3—242no—
524Dic3C4×#6718yescomplemented
525C32#236611yescharacteristic, complemented, Sylow
526C23—22no—
527C23—21no—
528C23—22no—
529C23—21no—
530C23—34no—
531C23—62no—
532C23—62no—
533C23—62no—
534C23—62no—
535C23—62no—
536C23—62no—
537C23—92no—
538C23—182no—
539C23—181no—
540C23—182no—
541C23—181no—
542C23C4×D3212yes—
543C23C22×D3211yescharacteristic
544D4—616no—
545D4—1816no—
546D4—364no—
547D4—364no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—34no—
551C2×C4—34no—
552C2×C4—34no—
553C2×C4—34no—
554C2×C4—44no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—62no—
558C2×C4—62no—
559C2×C4—62no—
560C2×C4—92no—
561C2×C4—128no—
562C2×C4—181no—
563C2×C4—181no—
564C2×C4—182no—
565C2×C4—182no—
566C2×C4—182no—
567C2×C4—364no—
568C2×C4D3×#6714yes—
569C2×C3—22no—
570C2×C3—21no—
571C2×C3—21no—
572C2×C3—22no—
573C2×C3—21no—
574C2×C3—44no—
575C2×C3—44no—
576C2×C3—124no—
577C2×C3C4×D3×D412yes—
578C2×C3#554512yes—
579C2×C3#555312yes—
580C2×C3#555912yes—
581C2×C3C2×C4×#6712yes—
582C2×C3D4×#6712yes—
583C2×C3#583412yes—
584D3—44no—
585D3—124no—
586D3—242no—
587C22—24no—
588C22—22no—
589C22—42no—
590C22—41no—
591C22—41no—
592C22—64no—
593C22—64no—
594C22—64no—
595C22—122no—
596C22—122no—
597C22—122no—
598C22—122no—
599C22—184no—
600C22—182no—
601C22—362no—
602C22—361no—
603C22—361no—
604C22C2×C4×D3211yescharacteristic
605C22D3×#23512yes—
606C22#6430811yescharacteristic
607C22C2×D3×#6712yes—
608C22#6433011yescharacteristic
609C4—24no—
610C4—38no—
611C4—64no—
612C4—122no—
613C4—182no—
614C4—362no—
615C3—21no—
616C3#8234312yescomplemented
617C2—42no—
618C2—124no—
619C2—362no—
620C2#1059500712yes—
621C2#1059503011yescharacteristic
622C2C4×D3×#6712yes—
623C2#1059505811yescharacteristic
624C2#67211yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutatorC22×C32C23×C4
FittingC23×C32×C4C22
socle, cosocleC23×C32C24
prime radicalC2×#63979C2
hypercenterC23×C4D32
hypocenterC32#2366
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×#67C2semidirect
3. C2×#63904C2semidirect
4. #10592912C2semidirect
5. #10591722C2semidirect
6. #10591707C2semidirect
7. #10591383C2semidirect
8. #10591381C2semidirect
9. Dic3×Dic3⋊C4C2semidirect
25. #63979C4semidirect
38. C2×C4×#67D3semidirect
59. C2×#638C2×C4semidirect
64. C2×C3×#67C2×C4semidirect
73. C2×#626C2×C4semidirect
76. C2×#625C2×C4semidirect
83. C3×Dic3⋊C4D4semidirect
90. Dic32C2×C4semidirect
226. C2×#67C4×D3semidirect
262. Dic3⋊C4#67semidirect
294. C3×Dic3C4×D4semidirect
524. Dic3C4×#67semidirect
525. C32#2366semidirect
616. C3#82343semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×#672
C2×#639792
C2×#639042
#105929122
#105917222
#105917072
#105913832
#105913812
Dic3×Dic3⋊C42
#823433