Finite Group Databasebeta

← #11864073

Subgroups — #11864073

Order 1152.

1681 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#235C211yesmaximal, characteristic, complemented
2#10596262C211yesmaximal, characteristic, complemented
3C2×#63833C211yesmaximal, characteristic, complemented
4C2×#63830C211yesmaximal, characteristic, complemented
5C3×#5207C211yesmaximal, characteristic, complemented
6#10591173C214yesmaximal, complemented
7#10591163C214yesmaximal, complemented
8#10591100C211yesmaximal, characteristic, complemented
9#10591086C211yesmaximal, characteristic, complemented
10#84081—31nomaximal
11#78638—31nomaximal
12C23×C32×C4C2211yescharacteristic
13C2×#3828C2211yescharacteristic
14C22×C3×#67—21no—
15C2×C3×#235—22no—
16C2×C3×#235—24no—
17C2×C3×#235—24no—
18C2×C3×#235C2214yes—
19C22×C3×D12C2211yescharacteristic
20C22×C3×C4×D3—21no—
21C22×C3×Dic6C2211yescharacteristic
22#64162—21no—
23#64139C2212yescomplemented
24#64138—21no—
25C4×#638—22no—
26#64135C2211yescharacteristic
27#64133C2212yescomplemented
28C2×C3×#208C2211yescharacteristic
29C2×C3×#208C2211yescharacteristic
30C2×#3719C2212yescomplemented
31#63833—24no—
32C2×#3717C2212yescomplemented
33#63830—24no—
34C2×#3716C2212yescomplemented
35C2×#3714C2212yescomplemented
36C3×#830C2214yescomplemented
37#63566C2214yescomplemented
38#63563C2214yescomplemented
39C22×C3×#41—31no—
40C22×#235D311yescharacteristic, complemented
41C2×#942—31no—
42C2×#932—31no—
43#5791—31no—
44#5791—31no—
45#5791—61no—
46C2×#909—31no—
47C2×#908—31no—
48C3×#407—31no—
49#5239—34no—
50#5238—34no—
51#5225—31no—
52#5215—31no—
53#5207D311yescharacteristic, complemented
54#5131—31no—
55#5120—31no—
56#4734—34no—
57#4733—34no—
58C24×C32D411yescharacteristic
59C23×C3×D3—21no—
60C22×C32×C4—22no—
61C22×C32×C4—22no—
62C22×C32×C4D411yescharacteristic
63C22×C32×C4D412yes—
64C22×C32×C4C2312yes—
65C22×C32×C4C2311yescharacteristic
66C2×#638—21no—
67C2×#638—22no—
68#3828—22no—
69#3828—22no—
70C2×C4⋊(C3⋊D3)C2311yescharacteristic
71C2×C4×C3⋊D3—21no—
72C2×#634C2311yescharacteristic
73C2×C3×#67—28no—
74C2×C3×#67—24no—
75C22×C3×Dic3—21no—
76C3×#235—216no—
77C3×#235—28no—
78C3×#235—28no—
79C2×C3×D12—21no—
80C2×C3×D12—22no—
81C2×C3×D12D414yescomplemented
82C2×C3×D12C2312yes—
83C2×C3×C4×D3—28no—
84C2×C3×C4×D3—24no—
85C2×C3×Dic6—21no—
86C2×C3×Dic6—22no—
87C2×C3×Dic6D414yescomplemented
88C2×C3×Dic6C2312yes—
89#3757—22no—
90#3757—21no—
91#3752—22no—
92C4⋊(C3⋊Dic3)C2312yes—
93#3750—22no—
94C4×C3⋊Dic3—21no—
95C3×#208—22no—
96C3×#208—22no—
97C2×C3×C3⋊C8C2312yes—
98C2×C3×C3⋊C8C2312yes—
99#3719—24no—
100#3717—24no—
101#3716—24no—
102#3714—24no—
103#3005—91noSylow
104C2×C3×#41—34no—
105C2×C3×#41—64no—
106C2×C3×#41—64no—
107C2×C3×#41—62no—
108C22×C3×Q8—31no—
109C22×C3×D4—31no—
110C22×C3×D4—61no—
111C23×C3×C4—21no—
112C23×C3×C4—61no—
113C23×C3×C4C2×D311yescharacteristic
114C23×C3×C4C2×D311yescharacteristic
115C22×#67—21no—
116C2×#235—22no—
117C2×#235—24no—
118C2×#235—24no—
119C2×#235—31no—
120C2×#235—31no—
121C2×#235—61no—
122C2×#235C2×D314yes—
123C22×D12C2×D311yescharacteristic
124C22×C4×D3—21no—
125C22×Dic6C2×D311yescharacteristic
126C2×C3×M16—31no—
127C2×C3×M16—31no—
128#953—62no—
129#953—61no—
130#953—61no—
131C2×#216—32no—
132#942—64no—
133C2×#215—32no—
134C2×#214—32no—
135#932—64no—
136C2×#213—32no—
137#930—32no—
138#930—32no—
139#930—62no—
140#929—62no—
141#929—61no—
142#929—61no—
143C4×#67—64no—
144C4×#67—62no—
145C4×#67—62no—
146#926—31no—
147#926—31no—
148#926—61no—
149#924—32no—
150#924—32no—
151#924—62no—
152C2×#208C2×D311yescharacteristic
153C2×#208C2×D311yescharacteristic
154#909—64no—
155#908—64no—
156C2×Dic12—32no—
157C2×D24—32no—
158C2×#204—32no—
159C2×#204—32no—
160C3×C22⋊C8—34no—
161#835—34no—
162#832—34no—
163#830C2×D314yescomplemented
164#821—34no—
165#816—34no—
166C23×C32—22no—
167C23×C32—42no—
168C23×C32C2×D412yes—
169C23×C32C2×D411yescharacteristic
170C22×C3⋊D3—21no—
171C22×C3×D3—24no—
172C22×C3×D3—22no—
173C22×C3×D3—44no—
174C2×C32×C4—22no—
175C2×C32×C4—24no—
176C2×C32×C4—22no—
177C2×C32×C4—22no—
178C2×C32×C4C2×D412yes—
179C2×C32×C4C2×D411yescharacteristic
180C2×C32×C4C2×D412yes—
181C2×C32×C4C2×D412yes—
182C2×C32×C4C2411yescharacteristic
183#638—44no—
184#638—42no—
185C2×C3⋊Dic3—22no—
186C2×C3⋊Dic3—21no—
187C4⋊(C3⋊D3)—21no—
188C4⋊(C3⋊D3)—21no—
189C4×C3⋊D3—42no—
190#634—21no—
191#634—21no—
192C3×#67—216no—
193C2×C3×Dic3—24no—
194C2×C3×Dic3—22no—
195C3×D12—24no—
196C3×D12—22no—
197C3×D12C2×D414yes—
198C3×C4×D3—216no—
199C3×Dic6—24no—
200C3×Dic6—22no—
201C3×Dic6C2×D414yes—
202C3×C3⋊C8—22no—
203C3×C3⋊C8—22no—
204C22×#41—31no—
205C2×#137—91no—
206C2×AGL(1,ℤ/8)—91no—
207#525—91no—
208#452—94no—
209#449—94no—
210#419—91no—
211#418—91no—
212#407—31no—
213C24×C3—21no—
214C24×C3—61no—
215C24×C3D1211yescharacteristic
216C24×C3#6711yescharacteristic
217C23×D3—21no—
218C3×#41—68no—
219C3×#41—68no—
220C3×#41—616no—
221C2×C3×Q8—34no—
222C2×C3×Q8—32no—
223C2×C3×Q8—61no—
224C2×C3×Q8—62no—
225C2×C3×D4—34no—
226C2×C3×D4—32no—
227C2×C3×D4—68no—
228C2×C3×D4—64no—
229C2×C3×D4—61no—
230C2×C3×D4—62no—
231C22×C3×C4—22no—
232C22×C3×C4—22no—
233C22×C3×C4—22no—
234C22×C3×C4—22no—
235C22×C3×C4—22no—
236C22×C3×C4—21no—
237C22×C3×C4—21no—
238C22×C3×C4—22no—
239C22×C3×C4—42no—
240C22×C3×C4—42no—
241C22×C3×C4—68no—
242C22×C3×C4—61no—
243C22×C3×C4—64no—
244C22×C3×C4D1212yes—
245C22×C3×C4D1211yescharacteristic
246C22×C3×C4#6712yes—
247C22×C3×C4#6711yescharacteristic
248C22×C3×C4C22×D312yes—
249C22×C3×C4C22×D312yes—
250C22×C3×C4C22×D311yescharacteristic
251C22×C3×C4C22×D311yescharacteristic
252C2×#67—28no—
253C2×#67—24no—
254C2×#67—64no—
255C2×#67—62no—
256C2×#67—62no—
257C2×#67—61no—
258C2×#67—62no—
259C2×#67—61no—
260C22×Dic3—21no—
261#235—216no—
262#235—28no—
263#235—28no—
264#235—62no—
265#235—62no—
266#235—62no—
267#235—62no—
268#235—122no—
269#235—122no—
270C2×D12—21no—
271C2×D12—22no—
272C2×D12—31no—
273C2×D12—31no—
274C2×D12—61no—
275C2×D12#6714yescomplemented
276C2×D12C22×D312yes—
277C2×C4×D3—28no—
278C2×C4×D3—24no—
279C2×C4×D3—62no—
280C2×C4×D3—61no—
281C2×C4×D3—61no—
282C2×Dic6—21no—
283C2×Dic6—22no—
284C2×Dic6—31no—
285C2×Dic6—31no—
286C2×Dic6—61no—
287C2×Dic6#6714yescomplemented
288C2×Dic6C22×D312yes—
289C3×M16—62no—
290C3×M16—62no—
291C2×C3×C8—32no—
292C2×C3×C8—32no—
293#217—64no—
294#217—62no—
295#217—62no—
296#217—61no—
297#217—62no—
298#217—61no—
299#216—64no—
300#215—64no—
301#214—64no—
302#213—64no—
303(C2×D3)⋊C4—64no—
304(C2×D3)⋊C4—62no—
305(C2×D3)⋊C4—62no—
306C4⋊Dic3—32no—
307C4⋊Dic3—32no—
308C4⋊Dic3—62no—
309Dic3⋊C4—64no—
310Dic3⋊C4—62no—
311Dic3⋊C4—62no—
312C4×Dic3—62no—
313C4×Dic3—61no—
314C4×Dic3—61no—
315#208—22no—
316#208—22no—
317C2×C3⋊C8C22×D312yes—
318C2×C3⋊C8C22×D312yes—
319Dic12—64no—
320D24—64no—
321#204—64no—
322#204—64no—
323C22×C32—22no—
324C22×C32—22no—
325C22×C32—22no—
326C22×C32—22no—
327C22×C32—44no—
328C22×C32#12012yes—
329C22×C32C22×D411yescharacteristic
330C2×C3⋊D3—41no—
331C2×C3⋊D3—42no—
332C2×C3×D3—24no—
333C2×C3×D3—48no—
334C2×C3×D3—44no—
335C32×C4—22no—
336C32×C4#12012yes—
337C32×C4C22×D411yescharacteristic
338C32×C4C22×D411yescharacteristic
339C3⋊Dic3—41no—
340C3⋊Dic3—42no—
341C3×Dic3—24no—
342C2×#41—34no—
343C2×#41—64no—
344C2×#41—64no—
345C2×#41—62no—
346C2×#41—91no—
347C22×Q8—31no—
348C22×D4—31no—
349C22×D4—61no—
350C23×C4—61no—
351C23×C4D3211yescharacteristic
352#137—184no—
353AGL(1,ℤ/8)—184no—
354C2×Q16—92no—
355C2×SD16—92no—
356C2×SD16—92no—
357C2×D8—92no—
358C2×M16—31no—
359C2×M16—31no—
360#123—181no—
361C22⋊Q8—92no—
362#121—92no—
363#120—181no—
364C4×D4—182no—
365#117—91no—
366#103—94no—
367#102—94no—
368C22⋊C8—34no—
369C23×C3—22no—
370C23×C3—22no—
371C23×C3—22no—
372C23×C3—21no—
373C23×C3—42no—
374C23×C3—42no—
375C23×C3—42no—
376C23×C3—44no—
377C23×C3—64no—
378C23×C3—62no—
379C23×C3—124no—
380C23×C3C2×D1212yes—
381C23×C3C2×D1211yescharacteristic
382C23×C3C2×#6712yes—
383C23×C3C2×#6711yescharacteristic
384C22×D3—24no—
385C22×D3—22no—
386C22×D3—44no—
387C22×D3—62no—
388C22×D3—61no—
389C22×D3—61no—
390C3×Q8—34no—
391C3×Q8—64no—
392C3×Q8—62no—
393C3×D4—34no—
394C3×D4—616no—
395C3×D4—64no—
396C3×D4—62no—
397C2×C3×C4—22no—
398C2×C3×C4—22no—
399C2×C3×C4—24no—
400C2×C3×C4—24no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4—22no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—22no—
408C2×C3×C4—22no—
409C2×C3×C4—22no—
410C2×C3×C4—42no—
411C2×C3×C4—42no—
412C2×C3×C4—44no—
413C2×C3×C4—42no—
414C2×C3×C4—616no—
415C2×C3×C4—64no—
416C2×C3×C4—62no—
417C2×C3×C4C2×D1212yes—
418C2×C3×C4C2×D1211yescharacteristic
419C2×C3×C4D3×D412yes—
420C2×C3×C4D3×D412yes—
421C2×C3×C4C2×#6712yes—
422C2×C3×C4C2×#6711yescharacteristic
423C2×C3×C4C2×#6712yes—
424C2×C3×C4C2×#6712yes—
425C2×C3×C4C23×D311yescharacteristic
426C2×C3×C4C23×D311yescharacteristic
427#67—216no—
428#67—124no—
429#67—128no—
430#67—124no—
431#67—124no—
432#67—122no—
433#67—122no—
434C2×Dic3—24no—
435C2×Dic3—22no—
436C2×Dic3—64no—
437C2×Dic3—62no—
438C2×Dic3—62no—
439C2×Dic3—61no—
440C2×Dic3—62no—
441C2×Dic3—61no—
442D12—24no—
443D12—22no—
444D12—61no—
445D12—61no—
446D12—61no—
447D12—61no—
448D12—121no—
449D12—121no—
450D12C2×#6714yes—
451C4×D3—216no—
452C4×D3—124no—
453C4×D3—122no—
454C4×D3—122no—
455Dic6—24no—
456Dic6—22no—
457Dic6—61no—
458Dic6—61no—
459Dic6—61no—
460Dic6—61no—
461Dic6—121no—
462Dic6—121no—
463Dic6C2×#6714yes—
464C3×C8—62no—
465C3×C8—62no—
466C3⋊C8—22no—
467C3⋊C8—22no—
468C2×C32—22no—
469C2×C32—42no—
470C2×C32C2×#12011yescharacteristic
471C2×C32#57512yes—
472C3⋊D3—81no—
473C3×D3—44no—
474C24—61no—
475C24#62611yescharacteristic
476#41—68no—
477#41—68no—
478#41—616no—
479#41—182no—
480#41—182no—
481C2×Q8—32no—
482C2×Q8—34no—
483C2×Q8—61no—
484C2×Q8—62no—
485C2×Q8—91no—
486C2×D4—32no—
487C2×D4—34no—
488C2×D4—68no—
489C2×D4—64no—
490C2×D4—61no—
491C2×D4—62no—
492C2×D4—91no—
493C2×D4—182no—
494C2×D4—181no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—68no—
498C22×C4—61no—
499C22×C4—64no—
500C22×C4—181no—
501C22×C4#62612yes—
502C22×C4#62611yescharacteristic
503C22×C4C2×D3212yes—
504C22×C4C2×D3211yescharacteristic
505Q16—184no—
506SD16—184no—
507SD16—184no—
508D8—184no—
509M16—62no—
510M16—62no—
511C2×C8—32no—
512C2×C8—32no—
513C4⋊C4—92no—
514C4⋊C4—182no—
515C22⋊C4—181no—
516C22⋊C4—182no—
517C22⋊C4—182no—
518C42—181no—
519C22×C3—22no—
520C22×C3—22no—
521C22×C3—24no—
522C22×C3—22no—
523C22×C3—22no—
524C22×C3—22no—
525C22×C3—22no—
526C22×C3—22no—
527C22×C3—22no—
528C22×C3—21no—
529C22×C3—22no—
530C22×C3—42no—
531C22×C3—42no—
532C22×C3—42no—
533C22×C3—44no—
534C22×C3—44no—
535C22×C3—48no—
536C22×C3—64no—
537C22×C3—128no—
538C22×C3—124no—
539C22×C3#88212yes—
540C22×C3#95312yes—
541C22×C3C22×D1211yescharacteristic
542C22×C3C22×#6711yescharacteristic
543C2×D3—24no—
544C2×D3—48no—
545C2×D3—44no—
546C2×D3—122no—
547C2×D3—124no—
548C2×D3—121no—
549C2×D3—122no—
550C2×D3—122no—
551C2×D3—121no—
552C3×C4—22no—
553C3×C4—22no—
554C3×C4—21no—
555C3×C4—21no—
556C3×C4—22no—
557C3×C4—42no—
558C3×C4—64no—
559C3×C4#88212yes—
560C3×C4#95312yes—
561C3×C4C2×D3×D411yescharacteristic
562C3×C4C2×D3×D411yescharacteristic
563C3×C4C22×#6711yescharacteristic
564C3×C4C22×#6711yescharacteristic
565Dic3—24no—
566Dic3—122no—
567Dic3—124no—
568Dic3—121no—
569Dic3—122no—
570Dic3—122no—
571Dic3—121no—
572C32#300511yescharacteristic, complemented, Sylow
573C23—22no—
574C23—42no—
575C23—62no—
576C23—64no—
577C23—124no—
578C23—181no—
579C23C2×#62612yes—
580C23C2×#62611yescharacteristic
581Q8—34no—
582Q8—62no—
583Q8—64no—
584Q8—181no—
585Q8—181no—
586D4—34no—
587D4—616no—
588D4—62no—
589D4—64no—
590D4—181no—
591D4—181no—
592D4—364no—
593D4—362no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—616no—
599C2×C4—62no—
600C2×C4—64no—
601C2×C4—182no—
602C2×C4—181no—
603C2×C4—362no—
604C2×C4C2×#62612yes—
605C2×C4C2×#62611yescharacteristic
606C2×C4D3×#6712yes—
607C2×C4D3×#6712yes—
608C2×C4C22×D3211yescharacteristic
609C8—62no—
610C8—62no—
611C2×C3—22no—
612C2×C3—22no—
613C2×C3—21no—
614C2×C3—22no—
615C2×C3—42no—
616C2×C3—44no—
617C2×C3—42no—
618C2×C3—42no—
619C2×C3—124no—
620C2×C3C2×#88211yescharacteristic
621C2×C3#574812yes—
622C2×C3#582212yes—
623C2×C3C2×#95311yescharacteristic
624D3—44no—
625D3—242no—
626D3—241no—
627D3—241no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—44no—
633C22—64no—
634C22—128no—
635C22—124no—
636C22—361no—
637C22—362no—
638C22#6397712yes—
639C22C22×#62611yescharacteristic
640C4—22no—
641C4—64no—
642C4—361no—
643C4—362no—
644C4#6397712yes—
645C4C2×D3×#6711yescharacteristic
646C4C2×D3×#6711yescharacteristic
647C3—21no—
648C3#7863811yescharacteristic, complemented
649C3#8408111yescharacteristic, complemented
650C2—22no—
651C2—42no—
652C2—124no—
653C2—721no—
654C2#1059479512yes—
655C2C2×#6397711yescharacteristic
subgroupG/H
centerC22C22×#626
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC23×C32×C4C22
socleC22×C32C22×D4
hypercenterC23×C4D32
hypocenterC32#3005
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×#235C2semidirect
2. #10596262C2semidirect
3. C2×#63833C2semidirect
4. C2×#63830C2semidirect
5. C3×#5207C2semidirect
6. #10591173C2semidirect
7. #10591163C2semidirect
8. #10591100C2semidirect
9. #10591086C2semidirect
23. #64139C22semidirect
27. #64133C22semidirect
30. C2×#3719C22semidirect
32. C2×#3717C22semidirect
34. C2×#3716C22semidirect
35. C2×#3714C22semidirect
36. C3×#830C22semidirect
37. #63566C22semidirect
38. #63563C22semidirect
40. C22×#235D3semidirect
53. #5207D3semidirect
81. C2×C3×D12D4semidirect
87. C2×C3×Dic6D4semidirect
163. #830C2×D3semidirect
275. C2×D12#67semidirect
287. C2×Dic6#67semidirect
572. C32#3005semidirect
648. C3#78638semidirect
649. C3#84081semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#2352
#105962622
C2×#638332
C2×#638302
C3×#52072
#105911732
#105911632
#105911002
#105910862
#840813
#786383