Finite Group Databasebeta

← #11784970

Subgroups — #11784970

Order 1152.

1255 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
1C2×#64307C211yesmaximal, characteristic
2#10592705C211yesmaximal, characteristic, complemented
3C3×#5005C211yesmaximal, characteristic, complemented
4C3×#5001C211yesmaximal, characteristic, complemented
5#10591110C211yesmaximal, characteristic, complemented
6#10591086C211yesmaximal, characteristic, complemented
7#10591078C211yesmaximal, characteristic, complemented
8#70414—31nomaximal
9#70327—31nomaximal
10C2×#3828C2211yescharacteristic
11C2×C3×#235C2211yescharacteristic
12C22×C3×D12C2211yescharacteristic
13C2×D3×#67—22no—
14#64307—22no—
15#64307—22no—
16#64307—22no—
17#64307—22no—
18C2×#3802C412yescomplemented
19C2×D3×D12—21no—
20C2×#3796C412yescomplemented
21C2×#3795—21no—
22C32×#117C2211yescharacteristic, complemented
23#64135C2211yescharacteristic, complemented
24C2×C3×#208C2211yescharacteristic, complemented
25C2×C3×#208C2211yescharacteristic, complemented
26#63633—22no—
27C3×#809—22no—
28C3×#805—22no—
29#63568—22no—
30#63563—22no—
31#63561—21no—
32#63559—21no—
33C2×#1009—31no—
34C2×#1004—31no—
35C3×#421—31no—
36C3×#418—31no—
37#5236—31no—
38#5215—31no—
39#5139—31no—
40#5132—31no—
41#5132—31no—
42#5131—31no—
43#5006—31no—
44#5006—31no—
45#5006—61no—
46#5005D311yescharacteristic, complemented
47#5001D311yescharacteristic, complemented
48C23×C3×D3—21no—
49C22×D32—22no—
50C22×C32×C4C2311yescharacteristic
51C2×#638D412yes—
52#3828—21no—
53#3828—21no—
54#3828—21no—
55#3828—21no—
56C2×C4⋊(C3⋊D3)D411yescharacteristic
57C2×C4×C3⋊D3C2×C412yes—
58C2×#634D411yescharacteristic
59C2×C3×#67—21no—
60C3×#235—21no—
61C3×#235—21no—
62C3×#235D412yes—
63C3×#235D412yes—
64C2×C3×D12—21no—
65C2×C3×D12—21no—
66C2×C3×D12—21no—
67C2×C3×D12C2×C412yes—
68C2×C3×D12C2×C411yescharacteristic
69C2×C3×D12D412yes—
70C2×C3×D12D412yes—
71C2×C3×C4×D3—21no—
72C2×C3×Dic6C2×C411yescharacteristic
73D3×#67—28no—
74D3×#67—28no—
75C2×#626—22no—
76C2×#625—22no—
77C2×D3×Dic3—22no—
78#3802—24no—
79#3802—24no—
80D3×D12—24no—
81D3×D12—24no—
82#3796—24no—
83#3796—24no—
84#3795—24no—
85#3795—24no—
86C32×C4⋊C4—21no—
87C32×C22⋊C4—21no—
88C32×C42—21no—
89#3757—21no—
90C4⋊(C3⋊Dic3)—21no—
91C4×C3⋊Dic3—21no—
92C3×#208—21no—
93C3×#208—21no—
94C3×#208—21no—
95C3×#208—21no—
96C2×C3×C3⋊C8—21no—
97C2×C3×C3⋊C8—21no—
98#1795—91noSylow
99C2×C3×#41—31no—
100C22×C3×D4—31no—
101C22×#67—62no—
102#1009—62no—
103#1009—62no—
104#1009—62no—
105#1009—62no—
106C2×#239—32no—
107#1004—62no—
108#1004—62no—
109#1004—62no—
110#1004—62no—
111C2×#237—32no—
112C2×#237—61no—
113C2×D3×D4—32no—
114C2×D3×D4—32no—
115C2×D3×D4—61no—
116C2×D3×D4—62no—
117C2×#235—31no—
118C2×#235—31no—
119C2×#235—61no—
120C2×#235—61no—
121C2×#235C2×D311yescharacteristic
122C22×D12—61no—
123C22×D12C2×D311yescharacteristic
124C2×C3×M16—31no—
125C2×C3×M16—31no—
126C3×#117—21no—
127C3×#117C2×D311yescharacteristic, complemented
128C3×#117C2×D311yescharacteristic, complemented
129#926—31no—
130#926—31no—
131#926—61no—
132C2×#208C2×D311yescharacteristic, complemented
133C2×#208C2×D311yescharacteristic, complemented
134C3×#104—62no—
135C3×#102—62no—
136#837—62no—
137#832—62no—
138#825—62no—
139#824—61no—
140#824—61no—
141#823—61no—
142#823—61no—
143#821—62no—
144#809—22no—
145#806—62no—
146#806—62no—
147#806—64no—
148#805—22no—
149C23×C32C2×D411yescharacteristic
150C22×C3⋊D3C22⋊C412yes—
151C22×C3×D3—21no—
152C22×C3×D3—22no—
153C22×C3×D3—22no—
154C22×C3×D3—22no—
155C22×C3×D3—82no—
156C2×D32—28no—
157C2×D32—84no—
158C2×C32×C4—41no—
159C2×C32×C4C22×C411yescharacteristic
160C2×C32×C4C2×D411yescharacteristic
161C2×C32×C4C2×D411yescharacteristic
162C2×C32×C4C2×D411yescharacteristic
163C2×C32×C4C2×D411yescharacteristic
164C2×C32×C4C2×D411yescharacteristic
165#638—22no—
166#638—22no—
167C2×C3⋊Dic3—41no—
168C2×C3⋊Dic3C22⋊C412yes—
169C4⋊(C3⋊D3)—21no—
170C4⋊(C3⋊D3)—21no—
171C4×C3⋊D3—22no—
172C4×C3⋊D3—22no—
173#634—21no—
174#634—21no—
175C3×#67—22no—
176C3×#67—22no—
177C2×C3×Dic3—21no—
178C3×D12—22no—
179C3×D12—21no—
180C3×D12—22no—
181C3×D12—22no—
182C3×D12C22⋊C414yes—
183C3×D12C22⋊C412yes—
184C3×C4×D3—22no—
185C3×C4×D3—22no—
186C3×Dic6—21no—
187C3×Dic6C22⋊C412yes—
188#626—28no—
189#625—28no—
190D3×Dic3—28no—
191C3×C3⋊C8—41no—
192C3×C3⋊C8—41no—
193C2×H(5,𝔽2)—91no—
194#421—31no—
195#421—91no—
196#418—31no—
197#418—91no—
198#413—91no—
199#410—91no—
200C24×C3—61no—
201C23×D3—21no—
202C23×D3—62no—
203C23×D3—62no—
204C3×#41—32no—
205C3×#41—32no—
206C3×#41—61no—
207C3×#41—61no—
208C2×C3×Q8—31no—
209C2×C3×D4—31no—
210C2×C3×D4—32no—
211C2×C3×D4—32no—
212C2×C3×D4—32no—
213C2×C3×D4—61no—
214C2×C3×D4—61no—
215C2×C3×D4—61no—
216C2×C3×D4—61no—
217C22×C3×C4—21no—
218C22×C3×C4—61no—
219C22×C3×C4C22×D311yescharacteristic
220C22×C3×C4C22×D311yescharacteristic
221C2×#67—21no—
222C2×#67—32no—
223C2×#67—32no—
224C2×#67—62no—
225C2×#67—62no—
226C2×#67—62no—
227C2×#67—62no—
228C2×#67—68no—
229C2×#67—68no—
230C22×Dic3—62no—
231#239—64no—
232#239—64no—
233#237—64no—
234#237—64no—
235#237—64no—
236#237—64no—
237D3×D4—64no—
238D3×D4—64no—
239D3×D4—64no—
240D3×D4—64no—
241D3×D4—68no—
242D3×D4—68no—
243D3×D4—64no—
244D3×D4—64no—
245#235—21no—
246#235—21no—
247#235—64no—
248#235—64no—
249#235—61no—
250#235—61no—
251#235—61no—
252#235—61no—
253#235—61no—
254#235—61no—
255#235—61no—
256#235—61no—
257#235—121no—
258#235—121no—
259#235—121no—
260#235—121no—
261#235D1212yes—
262#235#6712yes—
263C2×D12—21no—
264C2×D12—21no—
265C2×D12—21no—
266C2×D12—31no—
267C2×D12—31no—
268C2×D12—61no—
269C2×D12—64no—
270C2×D12—64no—
271C2×D12—62no—
272C2×D12C4×D312yes—
273C2×D12C4×D311yescharacteristic
274C2×D12D1212yes—
275C2×D12#6712yes—
276C2×C4×D3—21no—
277C2×C4×D3—32no—
278C2×C4×D3—32no—
279C2×C4×D3—62no—
280C2×C4×D3—62no—
281C2×Dic6—31no—
282C2×Dic6—31no—
283C2×Dic6—61no—
284C2×Dic6C4×D311yescharacteristic
285C3×M16—61no—
286C3×M16—61no—
287C3×M16—61no—
288C3×M16—61no—
289C2×C3×C8—61no—
290C2×C3×C8—61no—
291C3×C4⋊C4—21no—
292C3×C4⋊C4—21no—
293C3×C4⋊C4—41no—
294C3×C22⋊C4—22no—
295C3×C22⋊C4—21no—
296C3×C22⋊C4—21no—
297C3×C42—21no—
298C3×C42—21no—
299C3×C42—41no—
300#217—62no—
301#217—61no—
302#217—61no—
303C4⋊Dic3—61no—
304C4⋊Dic3—61no—
305C4⋊Dic3—121no—
306C4×Dic3—61no—
307C4×Dic3—61no—
308C4×Dic3—121no—
309#208—21no—
310#208—21no—
311#208—21no—
312#208—21no—
313C2×C3⋊C8—21no—
314C2×C3⋊C8—21no—
315C22×C32—41no—
316C22×C32C2×C22⋊C411yescharacteristic
317C22×C32#12011yescharacteristic
318C22×C32#12011yescharacteristic
319C2×C3⋊D3—22no—
320C2×C3⋊D3—42no—
321C2×C3×D3—24no—
322C2×C3×D3—22no—
323C2×C3×D3—42no—
324C2×C3×D3—44no—
325C2×C3×D3—82no—
326C2×C3×D3—82no—
327D32—88no—
328C32×C4—81no—
329C32×C4C2×C22⋊C411yescharacteristic
330C32×C4C2×C22⋊C411yescharacteristic
331C32×C4#12011yescharacteristic
332C32×C4#12011yescharacteristic
333C3⋊Dic3—22no—
334C3⋊Dic3—81no—
335C3×Dic3—22no—
336H(5,𝔽2)—182no—
337H(5,𝔽2)—182no—
338H(5,𝔽2)—182no—
339H(5,𝔽2)—182no—
340C2×#41—31no—
341C2×#41—91no—
342C2×#41—92no—
343C2×#41—181no—
344C22×D4—31no—
345C22×D4—92no—
346C22×D4—181no—
347C22×D4—182no—
348C2×M16—31no—
349C2×M16—31no—
350#117—91no—
351#117D3211yescharacteristic, complemented
352#104—62no—
353#104—182no—
354#102—62no—
355#102—182no—
356#101—181no—
357#100—181no—
358#99—182no—
359C23×C3—21no—
360C23×C3—62no—
361C23×C3—61no—
362C23×C3—62no—
363C23×C3—62no—
364C23×C3—242no—
365C23×C3D3×D411yescharacteristic
366C23×C3D3×D411yescharacteristic
367C22×D3—21no—
368C22×D3—22no—
369C22×D3—22no—
370C22×D3—22no—
371C22×D3—32no—
372C22×D3—32no—
373C22×D3—62no—
374C22×D3—68no—
375C22×D3—68no—
376C22×D3—82no—
377C22×D3—244no—
378C22×D3—244no—
379C3×Q8—32no—
380C3×Q8—61no—
381C3×D4—32no—
382C3×D4—34no—
383C3×D4—62no—
384C3×D4—62no—
385C3×D4—62no—
386C3×D4—61no—
387C3×D4—62no—
388C3×D4—62no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—21no—
395C2×C3×C4—42no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—61no—
399C2×C3×C4—62no—
400C2×C3×C4—62no—
401C2×C3×C4C2×C4×D311yescharacteristic
402C2×C3×C4C2×C4×D311yescharacteristic
403C2×C3×C4C2×D1211yescharacteristic
404C2×C3×C4C2×D1211yescharacteristic
405C2×C3×C4D3×D411yescharacteristic
406C2×C3×C4D3×D411yescharacteristic
407C2×C3×C4D3×D411yescharacteristic
408C2×C3×C4D3×D411yescharacteristic
409C2×C3×C4D3×D411yescharacteristic
410C2×C3×C4D3×D411yescharacteristic
411C2×C3×C4C2×#6711yescharacteristic
412C2×C3×C4C2×#6711yescharacteristic
413#67—22no—
414#67—22no—
415#67—68no—
416#67—68no—
417#67—68no—
418#67—62no—
419#67—62no—
420#67—62no—
421#67—62no—
422#67—122no—
423#67—122no—
424C2×Dic3—21no—
425C2×Dic3—32no—
426C2×Dic3—32no—
427C2×Dic3—62no—
428C2×Dic3—68no—
429C2×Dic3—122no—
430C2×Dic3—121no—
431C2×Dic3—121no—
432D12—22no—
433D12—21no—
434D12—22no—
435D12—22no—
436D12—68no—
437D12—61no—
438D12—61no—
439D12—61no—
440D12—61no—
441D12—121no—
442D12—121no—
443D12(C2×D3)⋊C414yes—
444D12(C2×D3)⋊C412yes—
445C4×D3—22no—
446C4×D3—22no—
447C4×D3—68no—
448C4×D3—62no—
449C4×D3—62no—
450C4×D3—62no—
451C4×D3—62no—
452C4×D3—122no—
453C4×D3—122no—
454Dic6—21no—
455Dic6—61no—
456Dic6—61no—
457Dic6—61no—
458Dic6—61no—
459Dic6—121no—
460Dic6—121no—
461Dic6(C2×D3)⋊C412yes—
462C3×C8—121no—
463C3×C8—121no—
464C3⋊C8—41no—
465C3⋊C8—41no—
466C2×C32—21no—
467C2×C32—21no—
468C2×C32—21no—
469C2×C32#37911yescharacteristic
470C3⋊D3—42no—
471C3×D3—42no—
472C3×D3—44no—
473C24—61no—
474C24—182no—
475#41—32no—
476#41—32no—
477#41—61no—
478#41—61no—
479#41—184no—
480#41—184no—
481#41—184no—
482#41—184no—
483#41—181no—
484#41—181no—
485#41—181no—
486#41—181no—
487C2×Q8—31no—
488C2×Q8—91no—
489C2×D4—32no—
490C2×D4—31no—
491C2×D4—32no—
492C2×D4—32no—
493C2×D4—61no—
494C2×D4—61no—
495C2×D4—61no—
496C2×D4—61no—
497C2×D4—92no—
498C2×D4—91no—
499C2×D4—184no—
500C2×D4—184no—
501C2×D4—182no—
502C2×D4—182no—
503C2×D4—184no—
504C2×D4—184no—
505C2×D4—188no—
506C2×D4—188no—
507C22×C4—61no—
508C22×C4—92no—
509C22×C4—182no—
510C22×C4C2×D3211yescharacteristic
511M16—61no—
512M16—61no—
513M16—61no—
514M16—61no—
515C2×C8—61no—
516C2×C8—61no—
517C4⋊C4—21no—
518C4⋊C4—181no—
519C22⋊C4—21no—
520C22⋊C4—181no—
521C42—21no—
522C42—181no—
523C22×C3—21no—
524C22×C3—21no—
525C22×C3—21no—
526C22×C3—41no—
527C22×C3—41no—
528C22×C3—64no—
529C22×C3—62no—
530C22×C3—81no—
531C22×C3—122no—
532C22×C3—124no—
533C22×C3—242no—
534C22×C3—242no—
535C22×C3D3×C22⋊C411yescharacteristic
536C22×C3D3×C22⋊C411yescharacteristic
537C22×C3#88211yescharacteristic
538C22×C3#88211yescharacteristic
539C22×C3#93711yescharacteristic
540C22×C3#93711yescharacteristic
541C2×D3—22no—
542C2×D3—24no—
543C2×D3—44no—
544C2×D3—42no—
545C2×D3—62no—
546C2×D3—62no—
547C2×D3—82no—
548C2×D3—82no—
549C2×D3—122no—
550C2×D3—122no—
551C2×D3—122no—
552C2×D3—242no—
553C2×D3—248no—
554C2×D3—248no—
555C3×C4—21no—
556C3×C4—21no—
557C3×C4—21no—
558C3×C4—21no—
559C3×C4—62no—
560C3×C4—81no—
561C3×C4—81no—
562C3×C4—82no—
563C3×C4D3×C22⋊C411yescharacteristic
564C3×C4D3×C22⋊C411yescharacteristic
565C3×C4#88211yescharacteristic
566C3×C4#88211yescharacteristic
567C3×C4C2×(C2×D3)⋊C411yescharacteristic
568C3×C4C2×(C2×D3)⋊C411yescharacteristic
569C3×C4#93711yescharacteristic
570C3×C4#93711yescharacteristic
571Dic3—22no—
572Dic3—62no—
573Dic3—62no—
574Dic3—122no—
575Dic3—241no—
576Dic3—241no—
577Dic3—242no—
578C32#179511yescharacteristic, complemented, Sylow
579C23—61no—
580C23—62no—
581C23—62no—
582C23—62no—
583C23—92no—
584C23—188no—
585C23—242no—
586C23—724no—
587C23#381111yescharacteristic
588Q8—32no—
589Q8—61no—
590Q8—181no—
591Q8—181no—
592D4—34no—
593D4—32no—
594D4—62no—
595D4—62no—
596D4—62no—
597D4—61no—
598D4—62no—
599D4—62no—
600D4—188no—
601D4—188no—
602D4—182no—
603D4—182no—
604D4—181no—
605D4—181no—
606C2×C4—41no—
607C2×C4—61no—
608C2×C4—62no—
609C2×C4—62no—
610C2×C4—92no—
611C2×C4—188no—
612C2×C4—182no—
613C2×C4—182no—
614C2×C4—361no—
615C2×C4C4×D3211yescharacteristic
616C2×C4D3×D1211yescharacteristic
617C2×C4D3×D1211yescharacteristic
618C2×C4#380211yescharacteristic
619C2×C4D3×#6711yescharacteristic
620C2×C4D3×#6711yescharacteristic
621C8—121no—
622C8—121no—
623C2×C3—21no—
624C2×C3—21no—
625C2×C3—21no—
626C2×C3—21no—
627C2×C3—21no—
628C2×C3—21no—
629C2×C3—21no—
630C2×C3—41no—
631C2×C3—41no—
632C2×C3—41no—
633C2×C3—124no—
634C2×C3—122no—
635C2×C3#495511yescharacteristic
636C2×C3#495511yescharacteristic
637D3—42no—
638D3—44no—
639D3—122no—
640D3—122no—
641D3—242no—
642C22—41no—
643C22—62no—
644C22—64no—
645C22—124no—
646C22—122no—
647C22—182no—
648C22—242no—
649C22—242no—
650C22—362no—
651C22—728no—
652C22#6392111yescharacteristic
653C22#6392311yescharacteristic
654C22#6398311yescharacteristic
655C4—62no—
656C4—81no—
657C4—182no—
658C4—721no—
659C4D3×(C2×D3)⋊C411yescharacteristic
660C4D3×(C2×D3)⋊C411yescharacteristic
661C4#6392211yescharacteristic
662C4#6392211yescharacteristic
663C3—21no—
664C3#7032711yescharacteristic, complemented
665C3#7041411yescharacteristic, complemented
666C2—21no—
667C2—21no—
668C2—21no—
669C2—124no—
670C2—122no—
671C2—362no—
672C2#1059139911yescharacteristic
subgroupG/H
centerC2#10591399
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC32×#117C22
socleC2×C32#379
prime radicalC2×#64307C2
cosocleC22×C32×C4C23
hypercenter#117D32
hypocenterC32#1795
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. #10592705C2semidirect
3. C3×#5005C2semidirect
4. C3×#5001C2semidirect
5. #10591110C2semidirect
6. #10591086C2semidirect
7. #10591078C2semidirect
18. C2×#3802C4semidirect
20. C2×#3796C4semidirect
22. C32×#117C22semidirect
23. #64135C22semidirect
24. C2×C3×#208C22semidirect
25. C2×C3×#208C22semidirect
46. #5005D3semidirect
47. #5001D3semidirect
127. C3×#117C2×D3semidirect
128. C3×#117C2×D3semidirect
132. C2×#208C2×D3semidirect
133. C2×#208C2×D3semidirect
351. #117D32semidirect
578. C32#1795semidirect
664. C3#70327semidirect
665. C3#70414semidirect

Maximal subgroups

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

subgroupindex
C2×#643072
#105927052
C3×#50052
C3×#50012
#105911102
#105910862
#105910782
#704143
#703273