Finite Group Databasebeta

← #11900865

Subgroups — #11900865

Order 1152.

3636 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#525C211yesmaximal, characteristic, complemented
2#10596303C211yesmaximal, characteristic, complemented
3#41×C3⋊Dic3C211yesmaximal, characteristic, complemented
4#10596262C211yesmaximal, characteristic, complemented
5C22×C4×C3⋊Dic3C211yesmaximal, characteristic, complemented
6#10596120C214yesmaximal, complemented
7#10596095C212yesmaximal, complemented
8#10596093C212yesmaximal, complemented
9#10596071C212yesmaximal, complemented
10#10596067C212yesmaximal, complemented
11#10596057C214yesmaximal, complemented
12#10596009C214yesmaximal, complemented
13C4×#3830C214yesmaximal, complemented
14#10595993C211yesmaximal, characteristic, complemented
15#10595988C211yesmaximal, characteristic, complemented
16#86018—34nomaximal
17C2×C32×#41C2211yescharacteristic
18C23×C32×C4C2211yescharacteristic
19C23×C3⋊Dic3C2212yes—
20C2×#3830C2214yes—
21C2×#3828C2211yescharacteristic
22C32×#123C2212yes—
23C32×C22⋊Q8C2212yes—
24C32×#121C2212yes—
25C32×#120C2212yes—
26C32×C4×D4C2214yes—
27C32×#117C2211yescharacteristic
28#64162C2212yes—
29#64156C2211yescharacteristic, complemented
30Q8×C3⋊Dic3C2212yescomplemented
31#64153C2211yescharacteristic, complemented
32#64149C2211yescharacteristic, complemented
33#64148C2214yes—
34#64148C2214yes—
35#64145C2211yescharacteristic, complemented
36#64144C2214yes—
37D4×C3⋊Dic3C2214yes—
38D4×C3⋊Dic3C2214yes—
39D4×C3⋊Dic3C2212yescomplemented
40#64139C2212yes—
41#64138C2212yes—
42C4×#638C2214yes—
43C4×#638C2214yescomplemented
44#64135C2211yescharacteristic
45#64135C2212yescomplemented
46#64133C2212yes—
47C2×C4×C3⋊Dic3C2212yes—
48C2×C4×C3⋊Dic3C2211yescharacteristic
49C2×C4×C3⋊Dic3C2211yescharacteristic
50C2×C4×C3⋊Dic3C2212yes—
51C2×C4×C3⋊Dic3C2212yes—
52C2×C4×C3⋊Dic3C2212yes—
53C2×C4×C3⋊Dic3C2212yes—
54C2×C4×C3⋊Dic3C2212yescomplemented
55#64107C2214yescomplemented
56#64102C2212yescomplemented
57#64101C2214yescomplemented
58#64099C2212yescomplemented
59#64098C2214yescomplemented
60#64097C2212yescomplemented
61#64096C2214yescomplemented
62#64096C2212yescomplemented
63#64095C2214yes—
64#64094C2214yescomplemented
65#64090C2214yes—
66#64090C2214yes—
67#64090C2214yescomplemented
68#64088C2214yescomplemented
69#64087C2214yes—
70#64087C2214yes—
71#64086C2214yes—
72#64086C2214yescomplemented
73#64081C2212yescomplemented
74C42×C3⋊D3C2212yescomplemented
75C4×#634C2214yescomplemented
76C3×#525D314yescomplemented
77#5832—34no—
78Dic3×#41—34no—
79#5791—34no—
80C22×C4×Dic3—34no—
81#5649—316no—
82#5624—38no—
83#5622—38no—
84#5600—38no—
85#5596—38no—
86#5586—316no—
87#5538—316no—
88C4×#237—316no—
89#5522—34no—
90#5517—34no—
91C24×C32C2311yescharacteristic
92C32×#41—24no—
93C2×C32×Q8C2311yescharacteristic
94C2×C32×D4C2314yes—
95C2×C32×D4C2312yes—
96C2×C32×D4C2311yescharacteristic
97C22×C32×C4—24no—
98C22×C32×C4C2312yes—
99C22×C32×C4C2312yes—
100C22×C32×C4C2311yescharacteristic
101C22×C32×C4C2312yes—
102C22×C32×C4C2311yescharacteristic
103C2×#638C2312yes—
104C2×#638C2314yes—
105C2×#638C2314yes—
106C22×C3⋊Dic3—28no—
107C22×C3⋊Dic3C2314yes—
108C22×C3⋊Dic3C2312yes—
109C22×C3⋊Dic3C2312yes—
110C22×C3⋊Dic3C2312yes—
111C22×C3⋊Dic3C2312yes—
112C22×C3⋊Dic3C2312yes—
113C22×C3⋊Dic3C2312yes—
114#3830—216no—
115#3828—24no—
116C2×C4⋊(C3⋊D3)C2311yescharacteristic
117C2×C4×C3⋊D3C2312yes—
118C2×C4×C3⋊D3C2314yes—
119C2×#634C2314yes—
120C2×#634C2311yescharacteristic
121C32×C4⋊C4C2312yes—
122C32×C4⋊C4C2314yes—
123C32×C22⋊C4C2314yes—
124C32×C22⋊C4C2314yes—
125C32×C22⋊C4C2312yes—
126C32×C42C2312yes—
127#3757C2314yes—
128#3757C2314yes—
129#3757C2312yes—
130#3757C2314yes—
131#3752C2314yes—
132#3752C2314yes—
133#3752C2314yes—
134C4⋊(C3⋊Dic3)C2314yes—
135C4⋊(C3⋊Dic3)C2314yes—
136C4⋊(C3⋊Dic3)C2312yes—
137#3750C2314yes—
138#3750C2314yes—
139#3750C2314yes—
140#3750C2314yes—
141#3750C2314yes—
142C4×C3⋊Dic3C2312yes—
143C4×C3⋊Dic3C2312yes—
144C4×C3⋊Dic3C2311yescharacteristic
145C4×C3⋊Dic3C2314yes—
146C4×C3⋊Dic3C2311yescharacteristic
147C4×C3⋊Dic3C2314yes—
148C4×C3⋊Dic3C2312yes—
149C4×C3⋊Dic3C2312yes—
150C4×C3⋊Dic3C2312yes—
151C4×C3⋊Dic3C2312yes—
152C4×C3⋊Dic3C2312yes—
153C4×C3⋊Dic3C2311yescharacteristic
154C4×C3⋊Dic3C2311yescharacteristic
155#3444—91noSylow
156C2×C3×#41C2×D314yes—
157C23×C3×C4C2×D314yes—
158C23×Dic3—38no—
159C2×#237—316no—
160C2×#235—34no—
161C3×#123C2×D318yes—
162C3×C22⋊Q8C2×D318yes—
163C3×#121C2×D318yes—
164C3×#120C2×D318yes—
165C3×C4×D4C2×D3116yes—
166C3×#117C2×D314yes—
167#953—38no—
168#947—34no—
169Q8×Dic3—38no—
170#944—34no—
171#940—34no—
172#939—316no—
173#939—316no—
174#936—34no—
175#935—316no—
176D4×Dic3—316no—
177D4×Dic3—316no—
178D4×Dic3—38no—
179#930—38no—
180#929—38no—
181C4×#67—316no—
182C4×#67—316no—
183#926—34no—
184#926—38no—
185#924—38no—
186C2×C4×Dic3—38no—
187C2×C4×Dic3—34no—
188C2×C4×Dic3—34no—
189C2×C4×Dic3—38no—
190C2×C4×Dic3—38no—
191C2×C4×Dic3—38no—
192C2×C4×Dic3—38no—
193C2×C4×Dic3—38no—
194#898—316no—
195#893—38no—
196#892—316no—
197#890—38no—
198#889—316no—
199#888—38no—
200#887—316no—
201#887—38no—
202#886—316no—
203#885—316no—
204#67⋊C4—316no—
205#67⋊C4—316no—
206#67⋊C4—316no—
207#879—316no—
208#878—316no—
209#878—316no—
210#877—316no—
211#877—316no—
212#872—38no—
213C42×D3—38no—
214C4×Dic6—316no—
215C23×C32—24no—
216C23×C32—22no—
217C23×C32C2412yes—
218C23×C32C2411yescharacteristic
219C23×C32C2412yes—
220C22×C3⋊D3C2412yes—
221C32×Q8—22no—
222C32×D4—28no—
223C32×D4—24no—
224C32×D4—22no—
225C2×C32×C4—22no—
226C2×C32×C4—24no—
227C2×C32×C4—24no—
228C2×C32×C4—24no—
229C2×C32×C4#4114yescomplemented
230C2×C32×C4C2412yes—
231C2×C32×C4C2412yes—
232C2×C32×C4C2414yes—
233C2×C32×C4C2411yescharacteristic
234C2×C32×C4C2411yescharacteristic
235#638—24no—
236#638—28no—
237#638—28no—
238C2×C3⋊Dic3—24no—
239C2×C3⋊Dic3—28no—
240C2×C3⋊Dic3—24no—
241C2×C3⋊Dic3—24no—
242C2×C3⋊Dic3—24no—
243C2×C3⋊Dic3—24no—
244C2×C3⋊Dic3#4118yescomplemented
245C2×C3⋊Dic3C2412yes—
246C2×C3⋊Dic3C2412yes—
247C2×C3⋊Dic3C2412yes—
248C2×C3⋊Dic3C2414yes—
249C2×C3⋊Dic3C2412yes—
250C2×C3⋊Dic3C2412yes—
251C4⋊(C3⋊D3)—22no—
252C4×C3⋊D3—24no—
253C4×C3⋊D3—28no—
254#634—22no—
255#634—28no—
256#533—92no—
257#533—91no—
258#532—91no—
259#532—92no—
260#529—94no—
261#529—92no—
262#529—92no—
263#529—94no—
264#525—91no—
265#525—94no—
266#525C3⋊D311yescharacteristic, complemented
267C4×#41—91no—
268C4×#41—94no—
269C4×#41—91no—
270C22×C42—91no—
271C24×C3C22×D314yes—
272C3×#41—216no—
273C2×C3×Q8C22×D314yes—
274C2×C3×D4C22×D314yes—
275C2×C3×D4C22×D318yes—
276C2×C3×D4C22×D3116yes—
277C22×C3×C4—216no—
278C22×C3×C4C22×D318yes—
279C22×C3×C4C22×D318yes—
280C22×C3×C4C22×D314yes—
281C22×C3×C4C22×D318yes—
282C22×C3×C4C22×D314yes—
283C2×#67—38no—
284C2×#67—316no—
285C2×#67—316no—
286C22×Dic3—38no—
287C22×Dic3—316no—
288C22×Dic3—38no—
289C22×Dic3—38no—
290C22×Dic3—38no—
291C22×Dic3—38no—
292C22×Dic3—38no—
293C22×Dic3—632no—
294#237—664no—
295#235—616no—
296C2×D12—34no—
297C2×C4×D3—38no—
298C2×C4×D3—316no—
299C2×Dic6—316no—
300C2×Dic6—34no—
301C3×C4⋊C4C22×D318yes—
302C3×C4⋊C4C22×D3116yes—
303C3×C22⋊C4C22×D3116yes—
304C3×C22⋊C4C22×D3116yes—
305C3×C22⋊C4C22×D318yes—
306C3×C42C22×D318yes—
307#217—316no—
308#217—316no—
309#217—38no—
310#217—316no—
311(C2×D3)⋊C4—316no—
312(C2×D3)⋊C4—316no—
313(C2×D3)⋊C4—316no—
314C4⋊Dic3—316no—
315C4⋊Dic3—316no—
316C4⋊Dic3—38no—
317Dic3⋊C4—316no—
318Dic3⋊C4—316no—
319Dic3⋊C4—316no—
320Dic3⋊C4—316no—
321Dic3⋊C4—316no—
322C4×Dic3—38no—
323C4×Dic3—38no—
324C4×Dic3—34no—
325C4×Dic3—316no—
326C4×Dic3—34no—
327C4×Dic3—38no—
328C4×Dic3—316no—
329C4×Dic3—38no—
330C4×Dic3—38no—
331C4×Dic3—38no—
332C4×Dic3—38no—
333C4×Dic3—34no—
334C4×Dic3—34no—
335C22×C32—28no—
336C22×C32—22no—
337C22×C32—22no—
338C22×C32—22no—
339C22×C32—24no—
340C22×C32—22no—
341C22×C32C2×#4112yes—
342C22×C32C2×#4114yes—
343C22×C32C2511yescharacteristic
344C2×C3⋊D3—22no—
345C2×C3⋊D3—24no—
346C32×C4—24no—
347C32×C4—22no—
348C32×C4—22no—
349C32×C4C2×#4112yes—
350C32×C4C2×#4112yes—
351C3⋊Dic3—22no—
352C3⋊Dic3—24no—
353C3⋊Dic3—22no—
354C3⋊Dic3—22no—
355C3⋊Dic3C2×#4114yes—
356C3⋊Dic3C2×#4114yes—
357C2×#41—94no—
358C2×#41—91no—
359C2×#41C2×C3⋊D311yescharacteristic
360C23×C4—92no—
361C23×C4C2×C3⋊D311yescharacteristic
362C4⋊Q8—91no—
363C4⋊Q8—92no—
364#127—91no—
365#126—94no—
366#126—94no—
367#125—94no—
368#125—92no—
369#124—91no—
370#124—91no—
371#124—94no—
372#123—94no—
373#123—92no—
374#123—94no—
375#123C2×C3⋊D312yes—
376C22⋊Q8—92no—
377C22⋊Q8—94no—
378C22⋊Q8—94no—
379C22⋊Q8C2×C3⋊D312yes—
380#121—94no—
381#121—94no—
382#121—92no—
383#121C2×C3⋊D312yes—
384#120—92no—
385#120C2×C3⋊D312yes—
386C4×Q8—92no—
387C4×Q8—94no—
388C4×Q8—92no—
389C4×Q8—94no—
390C4×D4—94no—
391C4×D4—94no—
392C4×D4—92no—
393C4×D4—94no—
394C4×D4—94no—
395C4×D4—92no—
396C4×D4—94no—
397C4×D4—94no—
398C4×D4—94no—
399C4×D4C2×C3⋊D314yes—
400#117—91no—
401#117—92no—
402#117—94no—
403#117—94no—
404#117—94no—
405#117—92no—
406#117C2×C3⋊D311yescharacteristic
407C2×C42—92no—
408C2×C42—92no—
409C2×C42—91no—
410C2×C42—92no—
411C2×C42—92no—
412C2×C42—92no—
413C2×C42—92no—
414C2×C42—91no—
415C2×C42—92no—
416C23×C3—28no—
417C23×C3—216no—
418C23×C3C23×D318yes—
419C23×C3C23×D314yes—
420C23×C3C23×D318yes—
421C22×D3—38no—
422C3×Q8—28no—
423C3×D4—216no—
424C3×D4—232no—
425C3×D4—28no—
426C2×C3×C4—28no—
427C2×C3×C4—216no—
428C2×C3×C4—216no—
429C2×C3×C4—216no—
430C2×C3×C4#237116yescomplemented
431C2×C3×C4C23×D318yes—
432C2×C3×C4C23×D318yes—
433C2×C3×C4C23×D3116yes—
434C2×C3×C4C23×D314yes—
435C2×C3×C4C23×D314yes—
436#67—632no—
437#67—616no—
438#67—632no—
439C2×Dic3—332no—
440C2×Dic3—38no—
441C2×Dic3—38no—
442C2×Dic3—38no—
443C2×Dic3—38no—
444C2×Dic3—38no—
445C2×Dic3—316no—
446C2×Dic3—632no—
447C2×Dic3—616no—
448C2×Dic3—616no—
449C2×Dic3—616no—
450C2×Dic3—616no—
451C2×Dic3—616no—
452D12—68no—
453C4×D3—632no—
454C4×D3—616no—
455Dic6—632no—
456Dic6—68no—
457C2×C32—22no—
458C2×C32—24no—
459C2×C32—42no—
460C2×C32C22×#4111yescharacteristic
461C2×C32C22×#4112yes—
462C3⋊D3—42no—
463C24C22×C3⋊D311yescharacteristic
464#41—24no—
465#41—1816no—
466#41—184no—
467C2×Q8—94no—
468C2×Q8—91no—
469C2×Q8C22×C3⋊D311yescharacteristic
470C2×D4—92no—
471C2×D4—94no—
472C2×D4—94no—
473C2×D4—91no—
474C2×D4C22×C3⋊D312yes—
475C2×D4C22×C3⋊D311yescharacteristic
476C2×D4C22×C3⋊D314yes—
477C22×C4—24no—
478C22×C4—92no—
479C22×C4—92no—
480C22×C4—92no—
481C22×C4—92no—
482C22×C4—92no—
483C22×C4—92no—
484C22×C4—94no—
485C22×C4—94no—
486C22×C4—92no—
487C22×C4—188no—
488C22×C4C22×C3⋊D312yes—
489C22×C4C22×C3⋊D311yescharacteristic
490C22×C4C22×C3⋊D312yes—
491C22×C4C22×C3⋊D312yes—
492C22×C4C22×C3⋊D311yescharacteristic
493C4⋊C4—94no—
494C4⋊C4—94no—
495C4⋊C4—92no—
496C4⋊C4—94no—
497C4⋊C4—94no—
498C4⋊C4—94no—
499C4⋊C4—94no—
500C4⋊C4—94no—
501C4⋊C4C22×C3⋊D314yes—
502C4⋊C4C22×C3⋊D312yes—
503C22⋊C4—92no—
504C22⋊C4—94no—
505C22⋊C4—94no—
506C22⋊C4—94no—
507C22⋊C4—94no—
508C22⋊C4—94no—
509C22⋊C4—94no—
510C22⋊C4C22×C3⋊D314yes—
511C22⋊C4C22×C3⋊D314yes—
512C22⋊C4C22×C3⋊D312yes—
513C42—92no—
514C42—92no—
515C42—92no—
516C42—92no—
517C42—91no—
518C42—92no—
519C42—91no—
520C42—94no—
521C42—94no—
522C42—91no—
523C42—91no—
524C42—92no—
525C42—92no—
526C42C22×C3⋊D312yes—
527C22×C3—232no—
528C22×C3—28no—
529C22×C3—28no—
530C22×C3—28no—
531C22×C3—28no—
532C22×C3—216no—
533C22×C3C2×#23718yes—
534C22×C3D3×#41116yes—
535C22×C3C24×D314yes—
536C2×D3—68no—
537C2×D3—616no—
538C3×C4—28no—
539C3×C4—28no—
540C3×C4—216no—
541C3×C4C2×#23718yes—
542C3×C4C2×#23718yes—
543Dic3—316no—
544Dic3—316no—
545Dic3—68no—
546Dic3—616no—
547Dic3—68no—
548Dic3—68no—
549C32#344411yescharacteristic, complemented, Sylow
550C23—22no—
551C23—24no—
552C23—92no—
553C23C23×C3⋊D312yes—
554C23C23×C3⋊D311yescharacteristic
555C23C23×C3⋊D312yes—
556Q8—22no—
557Q8—188no—
558Q8—182no—
559D4—24no—
560D4—28no—
561D4—22no—
562D4—188no—
563D4—184no—
564D4—188no—
565D4—182no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—98no—
571C2×C4—92no—
572C2×C4—92no—
573C2×C4—92no—
574C2×C4—92no—
575C2×C4—92no—
576C2×C4—94no—
577C2×C4—188no—
578C2×C4—184no—
579C2×C4—184no—
580C2×C4—184no—
581C2×C4—184no—
582C2×C4—188no—
583C2×C4—184no—
584C2×C4—184no—
585C2×C4#383014yescomplemented
586C2×C4C23×C3⋊D312yes—
587C2×C4C23×C3⋊D312yes—
588C2×C4C23×C3⋊D314yes—
589C2×C4C23×C3⋊D311yescharacteristic
590C2×C4C23×C3⋊D311yescharacteristic
591C2×C3—28no—
592C2×C3—216no—
593C2×C3—48no—
594C2×C3C22×#23714yes—
595C2×C3C2×D3×#4118yes—
596D3—128no—
597C22—28no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—24no—
603C22—182no—
604C22—184no—
605C22C2×#383012yes—
606C22#41×C3⋊D314yes—
607C22C24×C3⋊D311yescharacteristic
608C4—22no—
609C4—22no—
610C4—24no—
611C4—94no—
612C4—94no—
613C4—182no—
614C4—182no—
615C4—182no—
616C4—184no—
617C4C2×#383012yes—
618C4C2×#383012yes—
619C3#8601814yescomplemented
620C2—22no—
621C2—24no—
622C2—42no—
623C2—362no—
624C2C22×#383011yescharacteristic
625C2C2×#41×C3⋊D312yes—
subgroupG/H
centerC2×C4C23×C3⋊D3
commutator, socle, cosocleC22×C32C25
FrattiniC22C24×C3⋊D3
FittingC32×#525C2
hypercenter#525C3⋊D3
hypocenterC32#3444
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C32×#525C2semidirect
2. #10596303C2semidirect
3. #41×C3⋊Dic3C2semidirect
4. #10596262C2semidirect
5. C22×C4×C3⋊Dic3C2semidirect
6. #10596120C2semidirect
7. #10596095C2semidirect
8. #10596093C2semidirect
9. #10596071C2semidirect
10. #10596067C2semidirect
11. #10596057C2semidirect
12. #10596009C2semidirect
13. C4×#3830C2semidirect
14. #10595993C2semidirect
15. #10595988C2semidirect
29. #64156C22semidirect
30. Q8×C3⋊Dic3C22semidirect
31. #64153C22semidirect
32. #64149C22semidirect
35. #64145C22semidirect
39. D4×C3⋊Dic3C22semidirect
43. C4×#638C22semidirect
45. #64135C22semidirect
54. C2×C4×C3⋊Dic3C22semidirect
55. #64107C22semidirect
56. #64102C22semidirect
57. #64101C22semidirect
58. #64099C22semidirect
59. #64098C22semidirect
60. #64097C22semidirect
61. #64096C22semidirect
62. #64096C22semidirect
64. #64094C22semidirect
67. #64090C22semidirect
68. #64088C22semidirect
72. #64086C22semidirect
73. #64081C22semidirect
74. C42×C3⋊D3C22semidirect
75. C4×#634C22semidirect
76. C3×#525D3semidirect
229. C2×C32×C4#41semidirect
244. C2×C3⋊Dic3#41semidirect
266. #525C3⋊D3semidirect
430. C2×C3×C4#237semidirect
549. C32#3444semidirect
585. C2×C4#3830semidirect
619. C3#86018semidirect

Maximal subgroups

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

subgroupindex
C32×#5252
#105963032
#41×C3⋊Dic32
#105962622
C22×C4×C3⋊Dic32
#105961202
#105960952
#105960932
#105960712
#105960672
#105960572
#105960092
C4×#38302
#105959932
#105959882
#860183