Finite Group Databasebeta

← #11784910

Subgroups — #11784910

Order 1152.

1189 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64297C211yesmaximal, characteristic
2C2×#63632C211yesmaximal, characteristic, complemented
3C3×#4929C211yesmaximal, characteristic, complemented
4C2×C3×#805C211yesmaximal, characteristic, complemented
5#10591106C211yesmaximal, characteristic, complemented
6#10591087C211yesmaximal, characteristic, complemented
7C2×#63560C211yesmaximal, characteristic, complemented
8#70387—31nomaximal
9#69692—31nomaximal
10C2×#3828C2211yescharacteristic
11C2×C3×#235C2211yescharacteristic
12C22×C3×Dic6C2211yescharacteristic
13C2×#3804—22no—
14C2×#3799—21no—
15C2×#3797—21no—
16#64297—22no—
17#64297—22no—
18#64297—22no—
19#64297C414yescomplemented
20C2×#3796—21no—
21C2×D3×Dic6—21no—
22C2×C32×C42C2211yescharacteristic
23C2×#3748C2211yescharacteristic
24C2×C3×C4⋊Dic3C2211yescharacteristic
25C2×C3×#208C2211yescharacteristic
26#63632—22no—
27C3×C3⋊#95—22no—
28C3×#805—22no—
29#63567—21no—
30#63565—21no—
31#63564—21no—
32#63562—21no—
33#63560—22no—
34C2×#1010—31no—
35C2×#1007—31no—
36C2×C3×#104—31no—
37C3×#391—31no—
38#5232—31no—
39C2×#836—31no—
40C2×#824—31no—
41#5133—31no—
42#4966—31no—
43#4966—31no—
44#4929D311yescharacteristic, complemented
45C2×#805—31no—
46C2×#805—31no—
47C2×#805—61no—
48C2×#805D311yescharacteristic, complemented
49C22×C32×C4—21no—
50C22×C32×C4C2311yescharacteristic
51C2×#638—21no—
52#3828—21no—
53#3828—21no—
54#3828C2×C412yes—
55#3828D412yes—
56C2×C4⋊(C3⋊D3)D411yescharacteristic
57C2×C4×C3⋊D3—21no—
58C2×#634D411yescharacteristic
59C2×C3×#67—21no—
60C22×C3×Dic3—21no—
61C22×C3×Dic3D412yes—
62C3×#235—21no—
63C3×#235—21no—
64C3×#235C2×C412yes—
65C3×#235D412yes—
66C2×C3×D12D411yescharacteristic
67C2×C3×C4×D3—21no—
68C2×C3×Dic6—21no—
69C2×C3×Dic6—21no—
70C2×C3×Dic6—21no—
71C2×C3×Dic6—21no—
72C2×C3×Dic6C2×C412yes—
73C2×C3×Dic6D412yes—
74C2×C3×Dic6D411yescharacteristic
75C2×#627—22no—
76C2×#626—22no—
77C2×#624—22no—
78#3804—28no—
79#3804—28no—
80C2×D3×Dic3—22no—
81#3799—24no—
82#3799—24no—
83#3797—24no—
84#3797—24no—
85#3796—24no—
86#3796—24no—
87D3×Dic6—24no—
88D3×Dic6—24no—
89C32×C42—21no—
90C32×C42—21no—
91#3748—21no—
92#3748—21no—
93C2×C3⋊(C3⋊C8)—21no—
94C3×C4⋊Dic3—21no—
95C3×C4⋊Dic3—21no—
96C3×#208—21no—
97C3×#208—21no—
98C2×C3×C3⋊C8—21no—
99#1793—91noSylow
100C2×C3×#41—31no—
101C22×C3×Q8—31no—
102#1010—34no—
103#1010—62no—
104#1010—62no—
105#1010—62no—
106#1007—34no—
107#1007—62no—
108#1007—62no—
109#1007—62no—
110C2×#239—61no—
111C2×#239—61no—
112C2×D3×Q8—61no—
113C2×D3×Q8—61no—
114C2×D3×Q8—61no—
115C2×#237—61no—
116C2×#237—62no—
117C2×#235—31no—
118C2×#235—31no—
119C2×#235—61no—
120C2×#235—61no—
121C2×#235—62no—
122C2×#235C2×D311yescharacteristic
123C22×Dic6—61no—
124C22×Dic6C2×D311yescharacteristic
125C2×C3×M16—31no—
126C2×C3×C4⋊C4—31no—
127C2×C3×C42—21no—
128C2×C3×C42C2×D311yescharacteristic
129C2×C3×C42C2×D311yescharacteristic
130C2×C4⋊Dic3C2×D311yescharacteristic
131C2×#208—31no—
132C2×#208—31no—
133C2×#208—61no—
134C2×#208C2×D311yescharacteristic
135C3×#104—62no—
136C3×#95—62no—
137#836—62no—
138#835—61no—
139#832—61no—
140C3⋊#95—22no—
141#824—62no—
142#821—61no—
143#816—61no—
144#810—61no—
145#810—61no—
146#809—61no—
147#809—61no—
148#805—22no—
149#805—62no—
150#805—62no—
151#805—64no—
152C23×C32C2×D411yescharacteristic
153C22×C3⋊D3—21no—
154C22×C3×D3—21no—
155C2×C32×C4—22no—
156C2×C32×C4—41no—
157C2×C32×C4—41no—
158C2×C32×C4C22×C411yescharacteristic
159C2×C32×C4C2×D411yescharacteristic
160C2×C32×C4C2×D411yescharacteristic
161C2×C32×C4C2×D411yescharacteristic
162C2×C32×C4C2×D411yescharacteristic
163C2×C32×C4C2×D411yescharacteristic
164#638—22no—
165#638—22no—
166C2×C3⋊Dic3—21no—
167C4⋊(C3⋊D3)—21no—
168C4⋊(C3⋊D3)C22⋊C412yes—
169C4×C3⋊D3—22no—
170C4×C3⋊D3—22no—
171#634—21no—
172#634C22⋊C412yes—
173C3×#67—22no—
174C3×#67—22no—
175C2×C3×Dic3—22no—
176C2×C3×Dic3—22no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—42no—
179C2×C3×Dic3—41no—
180C2×C3×Dic3C22⋊C414yes—
181C3×D12—21no—
182C3×D12C22⋊C412yes—
183C3×C4×D3—22no—
184C3×C4×D3—22no—
185C3×Dic6—22no—
186C3×Dic6—22no—
187C3×Dic6—22no—
188C3×Dic6—22no—
189C3×Dic6—21no—
190C3×Dic6C22⋊C412yes—
191#627—28no—
192#626—28no—
193#624—28no—
194D3×Dic3—28no—
195C3⋊(C3⋊C8)—41no—
196C3×C3⋊C8—41no—
197C2×2−1+4—91no—
198C2×#104—31no—
199C2×#104—91no—
200#417—91no—
201#417—91no—
202C2×#101—91no—
203#391—31no—
204C3×#41—32no—
205C3×#41—32no—
206C3×#41—61no—
207C3×#41—61no—
208C2×C3×Q8—32no—
209C2×C3×Q8—32no—
210C2×C3×Q8—31no—
211C2×C3×Q8—61no—
212C2×C3×Q8—61no—
213C2×C3×Q8—61no—
214C2×C3×Q8—61no—
215C2×C3×D4—31no—
216C2×C3×D4—61no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—32no—
221C22×C3×C4—41no—
222C22×C3×C4—61no—
223C22×C3×C4—61no—
224C22×C3×C4C22×D311yescharacteristic
225C22×C3×C4C22×D311yescharacteristic
226C2×#67—21no—
227C2×#67—62no—
228C2×#67—61no—
229C2×#67—61no—
230C2×#67—121no—
231C22×Dic3—21no—
232C22×Dic3—62no—
233C22×Dic3#6712yes—
234#239—64no—
235#239—64no—
236#239—64no—
237#239—64no—
238D3×Q8—64no—
239D3×Q8—64no—
240D3×Q8—64no—
241D3×Q8—64no—
242D3×Q8—64no—
243D3×Q8—64no—
244#237—64no—
245#237—64no—
246#237—68no—
247#237—68no—
248#235—21no—
249#235—21no—
250#235—32no—
251#235—32no—
252#235—32no—
253#235—32no—
254#235—62no—
255#235—62no—
256#235—64no—
257#235—64no—
258#235—68no—
259#235—68no—
260#235—61no—
261#235—61no—
262#235—61no—
263#235—61no—
264#235—121no—
265#235—121no—
266#235C4×D312yes—
267#235#6712yes—
268C2×D12—31no—
269C2×D12—31no—
270C2×D12—61no—
271C2×D12—62no—
272C2×D12D1211yescharacteristic
273C2×C4×D3—21no—
274C2×C4×D3—62no—
275C2×C4×D3—62no—
276C2×C4×D3—62no—
277C2×C4×D3—61no—
278C2×C4×D3—61no—
279C2×C4×D3—121no—
280C2×Dic6—21no—
281C2×Dic6—21no—
282C2×Dic6—21no—
283C2×Dic6—21no—
284C2×Dic6—31no—
285C2×Dic6—31no—
286C2×Dic6—61no—
287C2×Dic6—64no—
288C2×Dic6—64no—
289C2×Dic6—62no—
290C2×Dic6—62no—
291C2×Dic6C4×D312yes—
292C2×Dic6D1212yes—
293C2×Dic6D1211yescharacteristic
294C3×M16—61no—
295C3×M16—61no—
296C2×C3×C8—61no—
297C3×C4⋊C4—61no—
298C3×C4⋊C4—61no—
299C3×C42—22no—
300C3×C42—21no—
301C3×C42—21no—
302C3×C42—21no—
303C3×C42—21no—
304C3×C42—41no—
305C4⋊Dic3—21no—
306C4⋊Dic3—21no—
307#208—21no—
308#208—21no—
309#208—62no—
310#208—61no—
311#208—61no—
312#208—61no—
313#208—61no—
314#208—121no—
315C2×C3⋊C8—21no—
316C2×C3⋊C8—61no—
317C2×C3⋊C8—61no—
318C2×C3⋊C8—121no—
319C22×C32—21no—
320C22×C32—21no—
321C22×C32C2×C22⋊C411yescharacteristic
322C22×C32#12011yescharacteristic
323C22×C32#12011yescharacteristic
324C2×C3⋊D3—22no—
325C2×C3⋊D3—41no—
326C2×C3⋊D3—41no—
327C2×C3×D3—22no—
328C2×C3×D3—41no—
329C2×C3×D3—41no—
330C32×C4—42no—
331C32×C4C2×C22⋊C411yescharacteristic
332C32×C4C2×C22⋊C411yescharacteristic
333C32×C4#12011yescharacteristic
334C32×C4#12011yescharacteristic
335C3⋊Dic3—22no—
336C3×Dic3—24no—
337C3×Dic3—22no—
338C3×Dic3—81no—
3392−1+4—94no—
3402−1+4—182no—
3412−1+4—182no—
3422−1+4—182no—
343C2×#41—31no—
344C2×#41—91no—
345C2×#41—182no—
346C2×#41—181no—
347C2×#41—181no—
348C22×Q8—31no—
349C22×Q8—181no—
350C22×Q8—181no—
351C2×M16—31no—
352C2×M16—91no—
353C2×C4⋊C4—31no—
354C2×C42D3211yescharacteristic
355#104—62no—
356#104—182no—
357#103—181no—
358#103—181no—
359#102—181no—
360#102—181no—
361#101—182no—
362#95—62no—
363C23×C3—21no—
364C23×C3—61no—
365C23×C3D3×D411yescharacteristic
366C23×C3C2×#6711yescharacteristic
367C22×D3—21no—
368C22×D3—61no—
369C22×D3—61no—
370C22×D3—121no—
371C3×Q8—32no—
372C3×Q8—62no—
373C3×Q8—61no—
374C3×Q8—62no—
375C3×Q8—62no—
376C3×Q8—62no—
377C3×D4—32no—
378C3×D4—62no—
379C3×D4—62no—
380C3×D4—61no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—34no—
390C2×C3×C4—42no—
391C2×C3×C4—42no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—42no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—62no—
398C2×C3×C4—61no—
399C2×C3×C4—62no—
400C2×C3×C4—62no—
401C2×C3×C4—62no—
402C2×C3×C4—122no—
403C2×C3×C4—121no—
404C2×C3×C4C2×C4×D311yescharacteristic
405C2×C3×C4C2×C4×D311yescharacteristic
406C2×C3×C4C2×D1211yescharacteristic
407C2×C3×C4C2×D1211yescharacteristic
408C2×C3×C4D3×D411yescharacteristic
409C2×C3×C4D3×D411yescharacteristic
410C2×C3×C4D3×D411yescharacteristic
411C2×C3×C4D3×D411yescharacteristic
412C2×C3×C4D3×D411yescharacteristic
413C2×C3×C4D3×D411yescharacteristic
414C2×C3×C4D3×D411yescharacteristic
415C2×C3×C4C2×#6711yescharacteristic
416#67—22no—
417#67—22no—
418#67—68no—
419#67—62no—
420#67—62no—
421#67—62no—
422#67—62no—
423#67—122no—
424#67—122no—
425C2×Dic3—21no—
426C2×Dic3—22no—
427C2×Dic3—22no—
428C2×Dic3—42no—
429C2×Dic3—41no—
430C2×Dic3—68no—
431C2×Dic3—61no—
432C2×Dic3—61no—
433C2×Dic3—121no—
434C2×Dic3(C2×D3)⋊C414yes—
435D12—21no—
436D12—32no—
437D12—32no—
438D12—62no—
439D12—68no—
440D12—61no—
441D12—61no—
442D12—121no—
443D12(C2×D3)⋊C412yes—
444C4×D3—22no—
445C4×D3—22no—
446C4×D3—68no—
447C4×D3—68no—
448C4×D3—68no—
449C4×D3—62no—
450C4×D3—62no—
451C4×D3—62no—
452C4×D3—62no—
453C4×D3—122no—
454C4×D3—122no—
455Dic6—22no—
456Dic6—22no—
457Dic6—22no—
458Dic6—22no—
459Dic6—21no—
460Dic6—32no—
461Dic6—32no—
462Dic6—62no—
463Dic6—68no—
464Dic6—68no—
465Dic6—61no—
466Dic6—61no—
467Dic6—121no—
468Dic6(C2×D3)⋊C412yes—
469C3×C8—121no—
470C3⋊C8—41no—
471C3⋊C8—122no—
472C3⋊C8—121no—
473C3⋊C8—121no—
474C2×C32—21no—
475C2×C32—21no—
476C2×C32#37911yescharacteristic
477C2×C32#45411yescharacteristic
478C2×C32#45611yescharacteristic
479C3⋊D3—42no—
480C3×D3—42no—
481#41—32no—
482#41—32no—
483#41—61no—
484#41—61no—
485#41—92no—
486#41—92no—
487#41—188no—
488#41—188no—
489#41—184no—
490#41—184no—
491#41—184no—
492#41—184no—
493#41—181no—
494#41—181no—
495C2×Q8—32no—
496C2×Q8—32no—
497C2×Q8—31no—
498C2×Q8—61no—
499C2×Q8—61no—
500C2×Q8—61no—
501C2×Q8—61no—
502C2×Q8—91no—
503C2×Q8—182no—
504C2×Q8—184no—
505C2×Q8—184no—
506C2×Q8—184no—
507C2×Q8—184no—
508C2×D4—31no—
509C2×D4—61no—
510C2×D4—91no—
511C2×D4—182no—
512C2×D4—181no—
513C22×C4—21no—
514C22×C4—32no—
515C22×C4—61no—
516C22×C4—61no—
517C22×C4—182no—
518C22×C4—182no—
519C22×C4—181no—
520C22×C4C2×D3211yescharacteristic
521M16—61no—
522M16—61no—
523M16—181no—
524M16—181no—
525C2×C8—61no—
526C2×C8—181no—
527C4⋊C4—61no—
528C4⋊C4—61no—
529C42—21no—
530C42—21no—
531C22×C3—21no—
532C22×C3—21no—
533C22×C3—21no—
534C22×C3—21no—
535C22×C3—21no—
536C22×C3—21no—
537C22×C3—21no—
538C22×C3—41no—
539C22×C3—41no—
540C22×C3—62no—
541C22×C3—121no—
542C22×C3—121no—
543C22×C3D3×C22⋊C411yescharacteristic
544C22×C3#88211yescharacteristic
545C22×C3C2×(C2×D3)⋊C411yescharacteristic
546C22×C3#93711yescharacteristic
547C22×C3#93711yescharacteristic
548C22×C3#93711yescharacteristic
549C2×D3—22no—
550C2×D3—41no—
551C2×D3—41no—
552C2×D3—62no—
553C2×D3—62no—
554C2×D3—122no—
555C2×D3—121no—
556C2×D3—121no—
557C2×D3—121no—
558C2×D3—121no—
559C2×D3—241no—
560C2×D3—241no—
561C3×C4—21no—
562C3×C4—21no—
563C3×C4—21no—
564C3×C4—21no—
565C3×C4—42no—
566C3×C4—42no—
567C3×C4—44no—
568C3×C4—62no—
569C3×C4—64no—
570C3×C4—241no—
571C3×C4D3×C22⋊C411yescharacteristic
572C3×C4D3×C22⋊C411yescharacteristic
573C3×C4D3×C22⋊C411yescharacteristic
574C3×C4#88211yescharacteristic
575C3×C4#88211yescharacteristic
576C3×C4#88211yescharacteristic
577C3×C4C2×(C2×D3)⋊C411yescharacteristic
578C3×C4#93711yescharacteristic
579Dic3—24no—
580Dic3—22no—
581Dic3—62no—
582Dic3—62no—
583Dic3—81no—
584Dic3—122no—
585C32#179311yescharacteristic, complemented, Sylow
586C23—61no—
587C23—181no—
588C23D3×#6711yescharacteristic
589Q8—32no—
590Q8—62no—
591Q8—62no—
592Q8—62no—
593Q8—62no—
594Q8—61no—
595Q8—92no—
596Q8—188no—
597Q8—181no—
598D4—32no—
599D4—62no—
600D4—62no—
601D4—61no—
602D4—92no—
603D4—188no—
604D4—182no—
605D4—182no—
606D4—181no—
607C2×C4—22no—
608C2×C4—34no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—62no—
612C2×C4—61no—
613C2×C4—62no—
614C2×C4—62no—
615C2×C4—62no—
616C2×C4—122no—
617C2×C4—121no—
618C2×C4—188no—
619C2×C4—188no—
620C2×C4—181no—
621C2×C4—182no—
622C2×C4—182no—
623C2×C4C4×D3211yescharacteristic
624C2×C4D3×D1211yescharacteristic
625C2×C4D3×D1211yescharacteristic
626C2×C4#380211yescharacteristic
627C2×C4D3×#6711yescharacteristic
628C2×C4#381111yescharacteristic
629C8—121no—
630C8—361no—
631C2×C3—21no—
632C2×C3—21no—
633C2×C3—21no—
634C2×C3—21no—
635C2×C3—21no—
636C2×C3—21no—
637C2×C3—21no—
638C2×C3—41no—
639C2×C3—41no—
640C2×C3—122no—
641C2×C3#482611yescharacteristic
642C2×C3#482711yescharacteristic
643C2×C3#495511yescharacteristic
644C2×C3#495511yescharacteristic
645C2×C3#506211yescharacteristic
646C2×C3#509511yescharacteristic
647D3—42no—
648D3—122no—
649D3—122no—
650D3—242no—
651C22—21no—
652C22—21no—
653C22—62no—
654C22—121no—
655C22—121no—
656C22—182no—
657C22—361no—
658C22—361no—
659C22D3×(C2×D3)⋊C411yescharacteristic
660C22#6392211yescharacteristic
661C22#6398311yescharacteristic
662C4—42no—
663C4—64no—
664C4—62no—
665C4—182no—
666C4—241no—
667C4D3×(C2×D3)⋊C411yescharacteristic
668C4#6392111yescharacteristic
669C4#6392211yescharacteristic
670C4#6392311yescharacteristic
671C3—21no—
672C3#6969211yescharacteristic, complemented
673C3#7038711yescharacteristic, complemented
674C2—21no—
675C2—21no—
676C2—122no—
677C2—362no—
678C2#1059039911yescharacteristic
679C2#1059040011yescharacteristic
680C2#1059139911yescharacteristic
subgroupG/H
centerC22#63922
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×C42C22
socleC22×C32#120
prime radicalC2×#64297C2
cosocleC22×C32×C4C23
hypercenterC2×C42D32
hypocenterC32#1793
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
2. C2×#63632C2semidirect
3. C3×#4929C2semidirect
4. C2×C3×#805C2semidirect
5. #10591106C2semidirect
6. #10591087C2semidirect
7. C2×#63560C2semidirect
19. #64297C4semidirect
44. #4929D3semidirect
48. C2×#805D3semidirect
585. C32#1793semidirect
672. C3#69692semidirect
673. C3#70387semidirect

Maximal subgroups

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

subgroupindex
C2×#642972
C2×#636322
C3×#49292
C2×C3×#8052
#105911062
#105910872
C2×#635602
#703873
#696923