Finite Group Databasebeta

← #11878676

Subgroups — #11878676

Order 1152.

1134 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5548C211yesmaximal, characteristic, complemented
2D3×#874C211yesmaximal, characteristic, complemented
3#10592714C211yesmaximal, characteristic, complemented
4C3×#5015C211yesmaximal, characteristic, complemented
5#10591515C212yesmaximal, complemented
6#10591482C212yesmaximal, complemented
7#10590595C212yesmaximal, complemented
8#10590548C212yesmaximal, complemented
9#10590349C211yesmaximal, characteristic, complemented
10#10590347C211yesmaximal, characteristic, complemented
11#10590248C211yesmaximal, characteristic, complemented
12#83302—31nomaximal
13#80336—31nomaximal
14C2×C3×#235—21no—
15C2×D3×D12—21no—
16C2×D3×D12—21no—
17C2×D3×D12C2212yes—
18C32×C4×D4C2211yescharacteristic, complemented
19C2×#3753C2212yescomplemented
20#64083C2211yescharacteristic
21C3×#940C2211yescharacteristic, complemented
22C3×#938C2211yescharacteristic
23C2×C3×#213C2212yescomplemented
24C3×C4×#67—21no—
25C3×#895C2211yescharacteristic
26C3×#888C2211yescharacteristic, complemented
27C3×#885—21no—
28C3×#884—21no—
29C3×#874C2211yescharacteristic
30C3×C4×D12C2211yescharacteristic, complemented
31C3×C42×D3C2211yescharacteristic
32#63911—21no—
33#63911—21no—
34#63911C2212yes—
35C2×#3714C2212yescomplemented
36C2×#3713C2212yescomplemented
37#63636C2212yescomplemented
38C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
39C3×#809C2212yescomplemented
40C3×C4⋊(C3⋊C8)C2211yescharacteristic, complemented
41#63564C2212yescomplemented
42#63562C2212yescomplemented
43(C2×D3)⋊(C3⋊C8)C2212yes—
44Dic3×C3⋊C8C2212yescomplemented
45C3×#532—31no—
46D3×#127—31no—
47#5548D311yescharacteristic, complemented
48C2×#874—31no—
49C3×#459—31no—
50#5156—32no—
51#5151—32no—
52#5088—32no—
53#5073—32no—
54#5072—31no—
55#5036—32no—
56#5036—32no—
57#5015—31no—
58#5015—31no—
59#5015—31no—
60#5015—31no—
61#5015—61no—
62#5015D311yescharacteristic, complemented
63#4925—31no—
64#4843—31no—
65#4833—31no—
66#4773—32no—
67#4760—32no—
68C22×D32—22no—
69C2×C32×D4C2311yescharacteristic
70C22×C32×C4—21no—
71C2×C4⋊(C3⋊D3)—21no—
72C2×C4⋊(C3⋊D3)—21no—
73C2×C4⋊(C3⋊D3)C2312yes—
74C2×C3×#67—21no—
75C2×C3×#67—21no—
76C2×C3×#67—21no—
77C3×#235—44no—
78C2×C3×D12—21no—
79C2×C3×D12—21no—
80C2×C3×D12—21no—
81C2×C3×D12C2311yescharacteristic
82C2×C3×D12C2312yes—
83C2×C3×C4×D3—21no—
84C2×C3×C4×D3D411yescharacteristic
85C2×C3×C4×D3D411yescharacteristic
86C2×C3×C4×D3C2311yescharacteristic
87C2×C3×Dic6—21no—
88C2×#626—22no—
89C2×#626—22no—
90C2×#626—22no—
91D3×D12—42no—
92D3×D12—42no—
93D3×D12—44no—
94D3×D12—44no—
95C32×C4⋊C4C2311yescharacteristic
96C32×C22⋊C4—21no—
97C32×C42C2311yescharacteristic
98#3753—42no—
99C2×C3⋊(C3⋊C8)C2312yes—
100C3×#217—21no—
101C3×#213—42no—
102C3×(C2×D3)⋊C4—21no—
103C3×(C2×D3)⋊C4—21no—
104C3×C4⋊Dic3C2311yescharacteristic
105C3×Dic3⋊C4—21no—
106C3×C4×Dic3D411yescharacteristic
107C3×C4×Dic3D411yescharacteristic
108C3×C4×Dic3C2311yescharacteristic
109C2×C3×C3⋊C8C2312yes—
110#3714—42no—
111#3713—42no—
112#3205—91noSylow
113C2×C3×#41—61no—
114C2×D3×D4—32no—
115C2×D3×D4—61no—
116C2×D3×D4—61no—
117C2×#235—21no—
118C22×D12—32no—
119C22×D12—61no—
120C22×D12—61no—
121C2×C3×D8—32no—
122C3×C4⋊Q8—31no—
123C3×#127—31no—
124C3×#127—31no—
125C3×#124—61no—
126C3×#121—31no—
127C3×#121—31no—
128C3×#121—61no—
129C3×C4×D4—21no—
130C3×C4×D4—31no—
131C3×C4×D4—61no—
132C3×C4×D4C2×D311yescharacteristic, complemented
133C3×C4×D4C2×D311yescharacteristic, complemented
134C2×C3×C42—31no—
135#940—32no—
136#940—61no—
137#940—61no—
138#940C2×D311yescharacteristic, complemented
139#938C2×D311yescharacteristic
140C2×#213—32no—
141C2×#213—32no—
142C2×#213—32no—
143C2×#213—32no—
144C2×#213—32no—
145C2×#213—62no—
146C2×#213C2×D312yescomplemented
147C4×#67—21no—
148C2×D24—32no—
149#895C2×D311yescharacteristic
150#888C2×D311yescharacteristic, complemented
151#885—21no—
152#884—21no—
153#874—32no—
154#874—31no—
155#874—31no—
156#874—61no—
157#874—61no—
158#874—61no—
159#874C2×D311yescharacteristic
160C4×D12C2×D311yescharacteristic, complemented
161C42×D3C2×D311yescharacteristic
162C3×C4⋊C8—31no—
163C3×#102—32no—
164#832—32no—
165#830—32no—
166#821—32no—
167(C2×D3)⋊C8—32no—
168C8×Dic3—32no—
169#809—32no—
170#809—32no—
171#809—32no—
172#809—32no—
173#809—62no—
174#809C2×D312yescomplemented
175C4⋊(C3⋊C8)—31no—
176C4⋊(C3⋊C8)—31no—
177C4⋊(C3⋊C8)—61no—
178C4⋊(C3⋊C8)C2×D311yescharacteristic, complemented
179C4×C3⋊C8—32no—
180C23×C32—21no—
181C22×C3⋊D3—22no—
182C22×C3×D3—22no—
183C22×C3×D3—21no—
184C22×C3×D3C2×D411yescharacteristic
185C2×D32—44no—
186C2×D32—44no—
187C2×D32—44no—
188C32×D4—21no—
189C32×D4—21no—
190C2×C32×C4—21no—
191C2×C32×C4—42no—
192C2×C32×C4C2×D411yescharacteristic
193C2×C32×C4C2×D411yescharacteristic
194C2×C32×C4C2411yescharacteristic
195C4⋊(C3⋊D3)—22no—
196C4⋊(C3⋊D3)—22no—
197C4⋊(C3⋊D3)—42no—
198C4⋊(C3⋊D3)—42no—
199C3×#67—42no—
200C3×#67—42no—
201C3×#67—42no—
202C2×C3×Dic3—21no—
203C2×C3×Dic3C2×D411yescharacteristic
204C2×C3×Dic3C2×D411yescharacteristic
205C2×C3×Dic3C2×D411yescharacteristic
206C3×D12—21no—
207C3×D12—22no—
208C3×D12—21no—
209C3×D12—22no—
210C3×D12—42no—
211C3×D12—42no—
212C3×D12—42no—
213C3×C4×D3—21no—
214C3×C4×D3—21no—
215C3×C4×D3—22no—
216C3×C4×D3—42no—
217C3×C4×D3D814yescomplemented
218C3×Dic6—42no—
219#626—44no—
220#626—44no—
221#626—44no—
222C3⋊(C3⋊C8)—22no—
223C3×C3⋊C8—22no—
224#532—31no—
225C2×#127—91no—
226#493—92no—
227#486—92no—
228#459—31no—
229#459—91no—
230#459—91no—
231#459—91no—
232#447—92no—
233#447—92no—
234#431—91no—
235C23×D3—62no—
236C23×D3—62no—
237C3×#41—124no—
238C2×C3×Q8—61no—
239C2×C3×D4—21no—
240C2×C3×D4—32no—
241C2×C3×D4—31no—
242C2×C3×D4—61no—
243C2×C3×D4—61no—
244C2×C3×D4—61no—
245C2×C3×D4—61no—
246C2×C3×D4—61no—
247C2×C3×D4—61no—
248C2×C3×D4C22×D311yescharacteristic
249C2×C3×D4C22×D311yescharacteristic
250C22×C3×C4—21no—
251C22×C3×C4—21no—
252C22×C3×C4—31no—
253C22×C3×C4—31no—
254C22×C3×C4—31no—
255C22×C3×C4—41no—
256C22×C3×C4—61no—
257C2×#67—21no—
258C2×#67—21no—
259C2×#67—21no—
260C2×#67—62no—
261C2×#67—62no—
262C2×#67—62no—
263D3×D4—122no—
264D3×D4—122no—
265D3×D4—124no—
266D3×D4—124no—
267#235—44no—
268C2×D12—21no—
269C2×D12—21no—
270C2×D12—21no—
271C2×D12—32no—
272C2×D12—32no—
273C2×D12—62no—
274C2×D12—62no—
275C2×D12—62no—
276C2×D12—62no—
277C2×D12—61no—
278C2×D12—61no—
279C2×D12—61no—
280C2×D12—61no—
281C2×D12—121no—
282C2×D12—121no—
283C2×D12—122no—
284C2×D12—122no—
285C2×D12—124no—
286C2×D12—124no—
287C2×D12C22×D312yes—
288C2×D12C22×D311yescharacteristic
289C2×C4×D3—21no—
290C2×C4×D3#6711yescharacteristic
291C2×C4×D3#6711yescharacteristic
292C2×C4×D3C22×D311yescharacteristic
293C2×Dic6—21no—
294C3×D8—122no—
295C2×C3×C8—32no—
296C3×C4⋊C4—21no—
297C3×C4⋊C4—31no—
298C3×C4⋊C4—61no—
299C3×C4⋊C4C22×D311yescharacteristic
300C3×C4⋊C4C22×D311yescharacteristic
301C3×C22⋊C4—21no—
302C3×C22⋊C4—21no—
303C3×C22⋊C4—41no—
304C3×C22⋊C4—61no—
305C3×C22⋊C4—61no—
306C3×C22⋊C4—61no—
307C3×C42—21no—
308C3×C42—31no—
309C3×C42—31no—
310C3×C42—31no—
311C3×C42C22×D311yescharacteristic
312C3×C42C22×D311yescharacteristic
313#217—21no—
314#213—42no—
315#213—122no—
316#213—122no—
317#213—122no—
318#213—122no—
319#213—122no—
320#213—124no—
321(C2×D3)⋊C4—21no—
322(C2×D3)⋊C4—21no—
323C4⋊Dic3C22×D311yescharacteristic
324Dic3⋊C4—21no—
325C4×Dic3#6711yescharacteristic
326C4×Dic3#6711yescharacteristic
327C4×Dic3C22×D311yescharacteristic
328C2×C3⋊C8—32no—
329C2×C3⋊C8—32no—
330C2×C3⋊C8—62no—
331C2×C3⋊C8C22×D312yes—
332D24—122no—
333C22×C32—41no—
334C22×C32—41no—
335C22×C32—41no—
336C22×C32C22×D411yescharacteristic
337C2×C3⋊D3—42no—
338C2×C3⋊D3—42no—
339C2×C3⋊D3—42no—
340C2×C3×D3—21no—
341C2×C3×D3—21no—
342C2×C3×D3—42no—
343C2×C3×D3—41no—
344C2×C3×D3—42no—
345C2×C3×D3—41no—
346C2×C3×D3—42no—
347C2×C3×D3—41no—
348C2×C3×D3C2×D812yes—
349D32—88no—
350C32×C4—21no—
351C32×C4—41no—
352C32×C4C2×D812yes—
353C32×C4H(5,𝔽2)11yescharacteristic
354C32×C4H(5,𝔽2)11yescharacteristic
355C3×Dic3—21no—
356C3×Dic3—41no—
357C3×Dic3C2×D812yes—
358C3×Dic3AGL(1,ℤ/8)12yes—
359C2×#41—61no—
360C22×D4—92no—
361C22×D4—181no—
362C22×D4—181no—
363C2×D8—32no—
364C2×D8—92no—
365C2×D8—92no—
366C2×D8—92no—
367C4⋊Q8—31no—
368#127—31no—
369#127—31no—
370#127—92no—
371#127—91no—
372#127—181no—
373#127—181no—
374#124—61no—
375#121—31no—
376#121—31no—
377#121—61no—
378C4×D4—31no—
379C4×D4—61no—
380C4×D4D3211yescharacteristic, complemented
381C2×C42—31no—
382C4⋊C8—31no—
383C4⋊C8—91no—
384#102—32no—
385#102—92no—
386#102—92no—
387#102—92no—
388C22⋊C8—92no—
389C4×C8—92no—
390C23×C3—21no—
391C23×C3—21no—
392C23×C3—31no—
393C23×C3—41no—
394C23×C3—61no—
395C23×C3—62no—
396C22×D3—22no—
397C22×D3—21no—
398C22×D3—62no—
399C22×D3—62no—
400C22×D3—122no—
401C22×D3—124no—
402C22×D3—124no—
403C22×D3—124no—
404C22×D3—124no—
405C22×D3—124no—
406C22×D3—124no—
407C22×D3C2×#6711yescharacteristic
408C3×Q8—122no—
409C3×D4—22no—
410C3×D4—21no—
411C3×D4—21no—
412C3×D4—21no—
413C3×D4—21no—
414C3×D4—41no—
415C3×D4—61no—
416C3×D4—61no—
417C3×D4—62no—
418C3×D4—62no—
419C3×D4—122no—
420C3×D4—122no—
421C3×D4—122no—
422C3×D4—122no—
423C3×D4—122no—
424C3×D4—122no—
425C2×C3×C4—21no—
426C2×C3×C4—21no—
427C2×C3×C4—21no—
428C2×C3×C4—21no—
429C2×C3×C4—21no—
430C2×C3×C4—31no—
431C2×C3×C4—31no—
432C2×C3×C4—34no—
433C2×C3×C4—31no—
434C2×C3×C4—41no—
435C2×C3×C4—42no—
436C2×C3×C4—42no—
437C2×C3×C4—62no—
438C2×C3×C4—61no—
439C2×C3×C4—61no—
440C2×C3×C4—61no—
441C2×C3×C4—82no—
442C2×C3×C4—122no—
443C2×C3×C4D3×D411yescharacteristic
444C2×C3×C4D3×D411yescharacteristic
445C2×C3×C4C2×#6711yescharacteristic
446C2×C3×C4C2×#6711yescharacteristic
447C2×C3×C4C23×D311yescharacteristic
448C2×C3×C4C23×D311yescharacteristic
449#67—42no—
450#67—42no—
451#67—42no—
452#67—124no—
453#67—124no—
454#67—124no—
455C2×Dic3—21no—
456C2×Dic3C2×#6711yescharacteristic
457C2×Dic3C2×#6711yescharacteristic
458C2×Dic3C2×#6711yescharacteristic
459D12—22no—
460D12—21no—
461D12—21no—
462D12—22no—
463D12—42no—
464D12—42no—
465D12—42no—
466D12—62no—
467D12—62no—
468D12—62no—
469D12—62no—
470D12—122no—
471D12—122no—
472D12—122no—
473D12—122no—
474D12—124no—
475D12—124no—
476D12—124no—
477D12—122no—
478D12—122no—
479D12—242no—
480D12—242no—
481C4×D3—21no—
482C4×D3—21no—
483C4×D3—22no—
484C4×D3—42no—
485C4×D3#21314yescomplemented
486Dic6—42no—
487C3×C8—62no—
488C3⋊C8—22no—
489C3⋊C8—62no—
490C3⋊C8—62no—
491C3⋊C8—122no—
492C2×C32—81no—
493C2×C32#53511yescharacteristic
494C2×C32C22×D811yescharacteristic
495C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
496C3⋊D3—82no—
497C3×D3—22no—
498C3×D3—82no—
499C3×D3—81no—
500C24—182no—
501#41—124no—
502C2×Q8—61no—
503C2×D4—32no—
504C2×D4—31no—
505C2×D4—61no—
506C2×D4—61no—
507C2×D4—61no—
508C2×D4—61no—
509C2×D4—61no—
510C2×D4—61no—
511C2×D4—92no—
512C2×D4—182no—
513C2×D4—182no—
514C2×D4—182no—
515C2×D4—181no—
516C2×D4—181no—
517C2×D4—364no—
518C2×D4—362no—
519C2×D4—362no—
520C2×D4—364no—
521C2×D4C2×D3211yescharacteristic
522C22×C4—21no—
523C22×C4—31no—
524C22×C4—31no—
525C22×C4—31no—
526C22×C4—61no—
527D8—122no—
528D8—362no—
529D8—362no—
530D8—362no—
531C2×C8—32no—
532C2×C8—92no—
533C4⋊C4—31no—
534C4⋊C4—61no—
535C4⋊C4C2×D3211yescharacteristic
536C22⋊C4—21no—
537C22⋊C4—61no—
538C22⋊C4—61no—
539C22⋊C4—61no—
540C42—31no—
541C42—31no—
542C42—31no—
543C42C2×D3211yescharacteristic
544C22×C3—21no—
545C22×C3—32no—
546C22×C3—42no—
547C22×C3—41no—
548C22×C3—41no—
549C22×C3—41no—
550C22×C3—41no—
551C22×C3—41no—
552C22×C3—41no—
553C22×C3—61no—
554C22×C3—61no—
555C22×C3—81no—
556C22×C3—81no—
557C22×C3—121no—
558C22×C3—121no—
559C22×C3—122no—
560C22×C3—122no—
561C22×C3—122no—
562C22×C3—121no—
563C22×C3C2×D3×D411yescharacteristic
564C22×C3C22×#6711yescharacteristic
565C2×D3—21no—
566C2×D3—21no—
567C2×D3—42no—
568C2×D3—42no—
569C2×D3—41no—
570C2×D3—41no—
571C2×D3—41no—
572C2×D3—42no—
573C2×D3—122no—
574C2×D3—122no—
575C2×D3—122no—
576C2×D3—122no—
577C2×D3—122no—
578C2×D3—122no—
579C2×D3—242no—
580C2×D3—242no—
581C2×D3—242no—
582C2×D3—248no—
583C2×D3—248no—
584C2×D3C2×#21312yes—
585C3×C4—21no—
586C3×C4—21no—
587C3×C4—21no—
588C3×C4—21no—
589C3×C4—22no—
590C3×C4—32no—
591C3×C4—32no—
592C3×C4—41no—
593C3×C4—41no—
594C3×C4—41no—
595C3×C4—61no—
596C3×C4—81no—
597C3×C4—121no—
598C3×C4D3×D812yes—
599C3×C4C2×#21312yes—
600C3×C4#100411yescharacteristic
601C3×C4#100911yescharacteristic
602C3×C4#100911yescharacteristic
603C3×C4#100911yescharacteristic
604Dic3—21no—
605Dic3—41no—
606Dic3C2×#21312yes—
607Dic3#94912yes—
608C32#320511yescharacteristic, complemented, Sylow
609C23—21no—
610C23—31no—
611C23—61no—
612C23—62no—
613C23—182no—
614C23—364no—
615C23—364no—
616C23—364no—
617Q8—122no—
618D4—21no—
619D4—21no—
620D4—62no—
621D4—61no—
622D4—61no—
623D4—62no—
624D4—122no—
625D4—122no—
626D4—122no—
627D4—122no—
628D4—122no—
629D4—122no—
630D4—182no—
631D4—182no—
632D4—362no—
633D4—364no—
634D4—364no—
635D4—364no—
636D4—362no—
637C2×C4—21no—
638C2×C4—31no—
639C2×C4—31no—
640C2×C4—34no—
641C2×C4—31no—
642C2×C4—42no—
643C2×C4—62no—
644C2×C4—61no—
645C2×C4—61no—
646C2×C4—61no—
647C2×C4—122no—
648C2×C4D3×#6711yescharacteristic
649C2×C4D3×#6711yescharacteristic
650C2×C4C22×D3211yescharacteristic
651C8—62no—
652C8—182no—
653C2×C3—21no—
654C2×C3—21no—
655C2×C3—21no—
656C2×C3—62no—
657C2×C3—82no—
658C2×C3—81no—
659C2×C3—81no—
660C2×C3—242no—
661C2×C3—241no—
662C2×C3#560711yescharacteristic
663C2×C3C2×D3×D811yescharacteristic
664C2×C3D3×AGL(1,ℤ/8)11yescharacteristic
665C2×C3C22×#21311yescharacteristic
666C2×C3C2×#94911yescharacteristic
667C2×C3#583011yescharacteristic
668D3—22no—
669D3—81no—
670D3—82no—
671D3—242no—
672D3—242no—
673D3—482no—
674C22—32no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22—61no—
679C22—61no—
680C22—121no—
681C22—122no—
682C22—122no—
683C22—121no—
684C22—122no—
685C22—121no—
686C22—362no—
687C22—362no—
688C22—362no—
689C22—728no—
690C22C2×D3×#6711yescharacteristic
691C4—21no—
692C4—32no—
693C4—32no—
694C4—41no—
695C4—61no—
696C4—121no—
697C4D3×#21312yes—
698C4#6430711yescharacteristic
699C4#6430811yescharacteristic
700C3—21no—
701C3#8033611yescharacteristic, complemented
702C3#8330211yescharacteristic, complemented
703C2—62no—
704C2—81no—
705C2—242no—
706C2—241no—
707C2—722no—
708C2#1059506411yescharacteristic
709C2C2×D3×#21311yescharacteristic
710C2D3×#94911yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4×D4C22
socleC22×C32C22×D4
hypercenterC4×D4D32
hypocenterC32#3205
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C3×#5548C2semidirect
2. D3×#874C2semidirect
3. #10592714C2semidirect
4. C3×#5015C2semidirect
5. #10591515C2semidirect
6. #10591482C2semidirect
7. #10590595C2semidirect
8. #10590548C2semidirect
9. #10590349C2semidirect
10. #10590347C2semidirect
11. #10590248C2semidirect
18. C32×C4×D4C22semidirect
19. C2×#3753C22semidirect
21. C3×#940C22semidirect
23. C2×C3×#213C22semidirect
26. C3×#888C22semidirect
30. C3×C4×D12C22semidirect
35. C2×#3714C22semidirect
36. C2×#3713C22semidirect
37. #63636C22semidirect
38. C4⋊(C3⋊(C3⋊C8))C22semidirect
39. C3×#809C22semidirect
40. C3×C4⋊(C3⋊C8)C22semidirect
41. #63564C22semidirect
42. #63562C22semidirect
44. Dic3×C3⋊C8C22semidirect
47. #5548D3semidirect
62. #5015D3semidirect
132. C3×C4×D4C2×D3semidirect
133. C3×C4×D4C2×D3semidirect
138. #940C2×D3semidirect
146. C2×#213C2×D3semidirect
150. #888C2×D3semidirect
160. C4×D12C2×D3semidirect
174. #809C2×D3semidirect
178. C4⋊(C3⋊C8)C2×D3semidirect
217. C3×C4×D3D8semidirect
380. C4×D4D32semidirect
485. C4×D3#213semidirect
608. C32#3205semidirect
701. C3#80336semidirect
702. C3#83302semidirect

Maximal subgroups

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

subgroupindex
C3×#55482
D3×#8742
#105927142
C3×#50152
#105915152
#105914822
#105905952
#105905482
#105903492
#105903472
#105902482
#833023
#803363