Finite Group Databasebeta

← #11867803

Subgroups — #11867803

Order 1152.

1073 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64118C211yesmaximal, characteristic, complemented
2C2×C3×#907C211yesmaximal, characteristic, complemented
3#10595043C211yesmaximal, characteristic
4#10594856C211yesmaximal, characteristic, complemented
5C3×#5121C211yesmaximal, characteristic, complemented
6#10591105C211yesmaximal, characteristic, complemented
7C2×#63567C211yesmaximal, characteristic, complemented
8#10590853C212yesmaximal, complemented
9#10590852C212yesmaximal, complemented
10#10590838C212yesmaximal, complemented
11#10590830C212yesmaximal, complemented
12#83808—31nomaximal
13#79214—31nomaximal
14C2×#3828C2211yescharacteristic
15C22×#634C2211yescharacteristic
16C2×C3×#235C2211yescharacteristic
17C2×C32×M16C2211yescharacteristic
18#64118—24no—
19C2×#3746C2212yes—
20C2×#3744C2212yes—
21C3×#926C2211yescharacteristic
22C2×C3×C4⋊Dic3C2211yescharacteristic
23C22×C3×C3⋊C8C2211yescharacteristic
24C3×#907—24no—
25C2×C3×ΓL(1,𝔽25)C2212yes—
26C2×C3×D3×C8C2212yes—
27#63978—22no—
28#63962—22no—
29#63916—21no—
30#63909C2212yes—
31#63871—21no—
32#63866—21no—
33#63860—21no—
34#63857C2212yes—
35#63857C2212yes—
36#63857C2212yes—
37C3×#818C2212yes—
38C3×#817C2212yes—
39#63567C2212yes—
40#63567C2212yes—
41#63567C2211yescharacteristic
42#63567C2211yescharacteristic
43#63564C2211yescharacteristic
44#63564C2211yescharacteristic
45C2×C3×#131—31no—
46#5833—31no—
47#5825—31no—
48C2×#909—31no—
49C2×#909—31no—
50C2×#909—61no—
51C2×#907D311yescharacteristic, complemented
52#5526—31no—
53#5510—31no—
54C3×#428—31no—
55#5137—32no—
56#5134—32no—
57#5121D311yescharacteristic, complemented
58#5116—32no—
59#5114—32no—
60#5113—31no—
61C2×#816—31no—
62#5002—31no—
63C2×#810—31no—
64#4887—32no—
65#4886—32no—
66#4867—32no—
67#4864—32no—
68C22×C32×C4C2311yescharacteristic
69C2×#638—21no—
70C22×C3⋊Dic3—21no—
71#3828—22no—
72#3828—22no—
73C2×C4⋊(C3⋊D3)C2311yescharacteristic
74C2×C4×C3⋊D3—21no—
75C2×#634—22no—
76C2×#634—22no—
77C2×#634—21no—
78C2×#634C2312yes—
79C2×#634C2311yescharacteristic
80C2×C3×#67D412yes—
81C22×C3×Dic3—21no—
82C3×#235—22no—
83C3×#235—22no—
84C2×C3×D12D411yescharacteristic
85C2×C3×C4×D3C2312yes—
86C2×C3×Dic6D411yescharacteristic
87C2×#626—22no—
88C32×M16D414yescomplemented
89C2×C32×C8C2312yes—
90#3746—24no—
91#3744—24no—
92C3×#217—21no—
93C3×C4⋊Dic3—21no—
94C3×C4⋊Dic3C2312yes—
95C3×C4⋊Dic3C2311yescharacteristic
96C3×C4⋊Dic3C2311yescharacteristic
97C3×C4×Dic3—21no—
98C2×C3×C3⋊C8—22no—
99C2×C3×C3⋊C8C2311yescharacteristic
100C2×C3×C3⋊C8C2311yescharacteristic
101C3×ΓL(1,𝔽25)—24no—
102C3×D3×C8—24no—
103Dic3⋊Dic3—22no—
104Dic3⋊Dic3—22no—
105Dic3⋊Dic3—22no—
106#3722—22no—
107(C2×D3)⋊Dic3—22no—
108(C2×D3)⋊Dic3—22no—
109(C2×D3)⋊Dic3—22no—
110#3063—91noSylow
111C2×C3×#41—31no—
112C2×#235—31no—
113C2×#235—31no—
114C2×#235—61no—
115C2×#235C2×D311yescharacteristic
116C22×Dic6—31no—
117C22×Dic6—31no—
118C22×Dic6—61no—
119C3×#131—64no—
120C2×C3×M16—21no—
121C2×C3×M16—32no—
122C2×C3×M16C2×D311yescharacteristic
123C2×C3×M16C2×D311yescharacteristic
124C22×C3×C8—31no—
125C22×C3×C8—32no—
126C3×#117—31no—
127C2×C3×C4⋊C4—31no—
128#946—61no—
129#944—61no—
130#939—62no—
131#938—61no—
132#936—61no—
133#935—62no—
134#930—32no—
135#930—62no—
136#926C2×D311yescharacteristic
137C2×C4⋊Dic3C2×D311yescharacteristic
138#924—32no—
139#924—32no—
140#924—32no—
141C22×C3⋊C8C2×D311yescharacteristic
142#909—68no—
143#909—64no—
144#909—64no—
145#907—24no—
146C2×Dic12—32no—
147C2×Dic12—32no—
148C2×Dic12—62no—
149C2×#204—32no—
150C2×#204—32no—
151C2×#204—62no—
152C2×ΓL(1,𝔽25)C2×D312yes—
153C2×D3×C8C2×D312yes—
154#897—32no—
155#897—32no—
156#897—32no—
157#897—61no—
158#895—32no—
159#895—61no—
160#886—62no—
161#875—61no—
162#869—61no—
163C3×#107—32no—
164C3×#106—32no—
165#821—31no—
166#821—31no—
167#818C2×D312yes—
168#817C2×D312yes—
169#816—32no—
170#816—31no—
171#816—32no—
172#816—31no—
173#810—32no—
174#810—31no—
175#810—32no—
176#810—31no—
177#809—31no—
178#809—31no—
179C23×C32C2×D411yescharacteristic
180C22×C3⋊D3—21no—
181C22×C3×D3C2×D412yes—
182C2×C32×C4C2×D411yescharacteristic
183C2×C32×C4C2×D412yes—
184C2×C32×C4#4112yes—
185C2×C32×C4C2411yescharacteristic
186#638—42no—
187C2×C3⋊Dic3—22no—
188C2×C3⋊Dic3—21no—
189C2×C3⋊Dic3—42no—
190C4⋊(C3⋊D3)—21no—
191C4⋊(C3⋊D3)—21no—
192C4×C3⋊D3—42no—
193#634—22no—
194#634—21no—
195#634—22no—
196#634—21no—
197#634—42no—
198C3×#67—24no—
199C2×C3×Dic3—21no—
200C2×C3×Dic3—21no—
201C2×C3×Dic3—22no—
202C2×C3×Dic3—81no—
203C2×C3×Dic3C2×D412yes—
204C3×D12—22no—
205C3×C4×D3—22no—
206C3×C4×D3—22no—
207C3×Dic6—22no—
208#626—82no—
209C32×C8C2×D414yes—
210C3×C3⋊C8—21no—
211C3×C3⋊C8—21no—
212C2×#137—91no—
213C2×#131—31no—
214#537—91no—
215#536—91no—
216#470—92no—
217#468—92no—
218#467—92no—
219#465—92no—
220#428—31no—
221#416—91no—
222C2×#103—91no—
223C3×#41—62no—
224C3×#41—62no—
225C2×C3×Q8—31no—
226C2×C3×D4—32no—
227C2×C3×D4—31no—
228C22×C3×C4—21no—
229C22×C3×C4—32no—
230C22×C3×C4—61no—
231C22×C3×C4C22×D311yescharacteristic
232C22×C3×C4C22×D311yescharacteristic
233C2×#67—62no—
234C2×#67—61no—
235C2×#67—61no—
236C2×#67—121no—
237C2×#67D1212yes—
238C22×Dic3—21no—
239C22×Dic3—61no—
240C22×Dic3—61no—
241C22×Dic3—121no—
242#235—22no—
243#235—22no—
244#235—64no—
245#235—62no—
246#235—62no—
247#235—62no—
248#235—62no—
249#235—122no—
250C2×D12—31no—
251C2×D12—31no—
252C2×D12—61no—
253C2×D12—62no—
254C2×D12D1211yescharacteristic
255C2×C4×D3—61no—
256C2×C4×D3—61no—
257C2×C4×D3—121no—
258C2×C4×D3C22×D312yes—
259C2×Dic6—32no—
260C2×Dic6—31no—
261C2×Dic6—32no—
262C2×Dic6—31no—
263C2×Dic6—62no—
264C2×Dic6—61no—
265C2×Dic6—64no—
266C2×Dic6—62no—
267C2×Dic6—62no—
268C2×Dic6—62no—
269C2×Dic6—62no—
270C2×Dic6—61no—
271C2×Dic6—61no—
272C2×Dic6—121no—
273C2×Dic6—122no—
274C2×Dic6D1211yescharacteristic
275C3×M16—24no—
276C3×M16—64no—
277C3×M16D1214yescomplemented
278C3×M16#6714yescomplemented
279C2×C3×C8—22no—
280C2×C3×C8—31no—
281C2×C3×C8—31no—
282C2×C3×C8—62no—
283C2×C3×C8—64no—
284C2×C3×C8C22×D312yes—
285C2×C3×C8C22×D312yes—
286C3×C4⋊C4—31no—
287C3×C4⋊C4—31no—
288C3×C4⋊C4—32no—
289C3×C4⋊C4—61no—
290C3×C22⋊C4—61no—
291C3×C42—61no—
292#217—21no—
293#217—62no—
294#217—62no—
295#217—62no—
296(C2×D3)⋊C4—62no—
297(C2×D3)⋊C4—62no—
298(C2×D3)⋊C4—62no—
299(C2×D3)⋊C4—62no—
300(C2×D3)⋊C4—62no—
301C4⋊Dic3—21no—
302C4⋊Dic3—62no—
303C4⋊Dic3—62no—
304C4⋊Dic3—62no—
305C4⋊Dic3C22×D312yes—
306C4⋊Dic3C22×D311yescharacteristic
307C4⋊Dic3C22×D311yescharacteristic
308Dic3⋊C4—62no—
309Dic3⋊C4—62no—
310Dic3⋊C4—62no—
311C4×Dic3—21no—
312C2×C3⋊C8—22no—
313C2×C3⋊C8C22×D311yescharacteristic
314C2×C3⋊C8C22×D311yescharacteristic
315Dic12—68no—
316Dic12—64no—
317Dic12—64no—
318#204—68no—
319#204—64no—
320#204—64no—
321ΓL(1,𝔽25)—24no—
322D3×C8—24no—
323C22×C32—21no—
324C22×C32—21no—
325C22×C32#12112yes—
326C22×C32C22×D411yescharacteristic
327C2×C3⋊D3—41no—
328C2×C3⋊D3—41no—
329C2×C3⋊D3—41no—
330C2×C3×D3—22no—
331C2×C3×D3—22no—
332C2×C3×D3—22no—
333C32×C4#12112yes—
334C32×C4C22×D411yescharacteristic
335C32×C4C2×#4111yescharacteristic
336C3⋊Dic3—42no—
337C3⋊Dic3—41no—
338C3×Dic3—22no—
339C3×Dic3—42no—
340C3×Dic3—41no—
341C3×Dic3—41no—
342C2×#41—31no—
343C2×#41—91no—
344C22×Q8—91no—
345#137—184no—
346C2×Q16—92no—
347C2×SD16—92no—
348#131—64no—
349C2×M16—32no—
350C2×M16D3211yescharacteristic
351C22×C8—31no—
352C22×C8—32no—
353C4⋊Q8—181no—
354#124—181no—
355#123—182no—
356C22⋊Q8—92no—
357C22⋊Q8—92no—
358C22⋊Q8—92no—
359C22⋊Q8—181no—
360#121—92no—
361#121—181no—
362#121—182no—
363#117—31no—
364C2×C4⋊C4—31no—
365#107—32no—
366#106—32no—
367#103—92no—
368#103—92no—
369#103—91no—
370#103—91no—
371#102—91no—
372#102—91no—
373C23×C3—21no—
374C23×C3—32no—
375C23×C3C2×D1211yescharacteristic
376C23×C3D3×D411yescharacteristic
377C22×D3—61no—
378C22×D3—61no—
379C22×D3—121no—
380C22×D3C2×D1212yes—
381C3×Q8—62no—
382C3×D4—64no—
383C3×D4—62no—
384C2×C3×C4—21no—
385C2×C3×C4—22no—
386C2×C3×C4—21no—
387C2×C3×C4—22no—
388C2×C3×C4—32no—
389C2×C3×C4—62no—
390C2×C3×C4—62no—
391C2×C3×C4—61no—
392C2×C3×C4—61no—
393C2×C3×C4—62no—
394C2×C3×C4—241no—
395C2×C3×C4C2×D1211yescharacteristic
396C2×C3×C4C2×D1212yes—
397C2×C3×C4#23512yes—
398C2×C3×C4D3×D411yescharacteristic
399C2×C3×C4#23912yes—
400C2×C3×C4C2×#6712yes—
401C2×C3×C4C23×D311yescharacteristic
402C2×C3×C4C23×D311yescharacteristic
403#67—24no—
404#67—124no—
405#67—122no—
406#67—122no—
407#67—242no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—22no—
411C2×Dic3—62no—
412C2×Dic3—61no—
413C2×Dic3—62no—
414C2×Dic3—61no—
415C2×Dic3—81no—
416C2×Dic3—122no—
417C2×Dic3—121no—
418C2×Dic3—124no—
419C2×Dic3—122no—
420C2×Dic3—122no—
421C2×Dic3C2×D1212yes—
422D12—22no—
423D12—62no—
424D12—61no—
425D12—61no—
426D12—61no—
427D12—61no—
428D12—121no—
429D12—242no—
430C4×D3—22no—
431C4×D3—22no—
432C4×D3—124no—
433C4×D3—122no—
434C4×D3—122no—
435Dic6—22no—
436Dic6—64no—
437Dic6—62no—
438Dic6—62no—
439Dic6—61no—
440Dic6—62no—
441Dic6—61no—
442Dic6—62no—
443Dic6—61no—
444Dic6—62no—
445Dic6—61no—
446Dic6—122no—
447Dic6—121no—
448Dic6—124no—
449Dic6—122no—
450Dic6—122no—
451C3×C8—24no—
452C3×C8—61no—
453C3×C8—61no—
454C3×C8C2×D1214yes—
455C3×C8C2×#6714yes—
456C3⋊C8—21no—
457C3⋊C8—21no—
458C2×C32—22no—
459C2×C32C2×#12111yescharacteristic
460C2×C32#57711yescharacteristic
461C2×C32#57811yescharacteristic
462C3⋊D3—81no—
463C3×D3—42no—
464#41—62no—
465#41—62no—
466#41—182no—
467#41—182no—
468C2×Q8—31no—
469C2×Q8—92no—
470C2×Q8—91no—
471C2×Q8—181no—
472C2×Q8—182no—
473C2×Q8—182no—
474C2×D4—32no—
475C2×D4—31no—
476C2×D4—91no—
477C2×D4—182no—
478C2×D4—181no—
479C22×C4—32no—
480C22×C4—61no—
481C22×C4—181no—
482C22×C4—181no—
483C22×C4C2×D3211yescharacteristic
484Q16—184no—
485SD16—184no—
486M16—64no—
487M16#62614yescomplemented
488C2×C8—31no—
489C2×C8—31no—
490C2×C8—62no—
491C2×C8—64no—
492C2×C8C2×D3212yes—
493C4⋊C4—31no—
494C4⋊C4—31no—
495C4⋊C4—32no—
496C4⋊C4—61no—
497C4⋊C4—182no—
498C4⋊C4—182no—
499C4⋊C4—182no—
500C22⋊C4—61no—
501C22⋊C4—182no—
502C22⋊C4—182no—
503C22⋊C4—182no—
504C22⋊C4—182no—
505C42—61no—
506C22×C3—21no—
507C22×C3—21no—
508C22×C3—21no—
509C22×C3—21no—
510C22×C3—21no—
511C22×C3—22no—
512C22×C3—41no—
513C22×C3—41no—
514C22×C3—62no—
515C22×C3—62no—
516C22×C3—62no—
517C22×C3#89512yes—
518C22×C3#93012yes—
519C22×C3C22×D1211yescharacteristic
520C22×C3C2×D3×D411yescharacteristic
521C2×D3—22no—
522C2×D3—22no—
523C2×D3—22no—
524C2×D3—122no—
525C2×D3—121no—
526C2×D3—121no—
527C2×D3—121no—
528C2×D3—121no—
529C2×D3—121no—
530C2×D3—121no—
531C2×D3—241no—
532C2×D3—241no—
533C3×C4—21no—
534C3×C4—21no—
535C3×C4—22no—
536C3×C4—62no—
537C3×C4—122no—
538C3×C4—121no—
539C3×C4—121no—
540C3×C4#89512yes—
541C3×C4#93012yes—
542C3×C4C22×D1211yescharacteristic
543C3×C4C2×#23511yescharacteristic
544C3×C4C2×#23911yescharacteristic
545C3×C4C22×#6711yescharacteristic
546Dic3—22no—
547Dic3—42no—
548Dic3—41no—
549Dic3—41no—
550Dic3—124no—
551Dic3—122no—
552Dic3—122no—
553Dic3—121no—
554Dic3—122no—
555Dic3—121no—
556C32#306311yescharacteristic, complemented, Sylow
557C23—32no—
558C23—181no—
559C23D3×D1211yescharacteristic
560Q8—62no—
561Q8—182no—
562Q8—181no—
563Q8—182no—
564Q8—181no—
565Q8—362no—
566D4—64no—
567D4—62no—
568D4—181no—
569D4—181no—
570D4—362no—
571D4—722no—
572C2×C4—32no—
573C2×C4—62no—
574C2×C4—62no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—62no—
578C2×C4—181no—
579C2×C4—182no—
580C2×C4—241no—
581C2×C4—362no—
582C2×C4—362no—
583C2×C4#379912yes—
584C2×C4D3×D1211yescharacteristic
585C2×C4C2×#62612yes—
586C2×C4C22×D3211yescharacteristic
587C8—61no—
588C8—61no—
589C8C2×#62614yes—
590C2×C3—22no—
591C2×C3—22no—
592C2×C3—21no—
593C2×C3—21no—
594C2×C3—21no—
595C2×C3—42no—
596C2×C3—122no—
597C2×C3C2×#89511yescharacteristic
598C2×C3#575111yescharacteristic
599C2×C3#575311yescharacteristic
600C2×C3#577511yescharacteristic
601C2×C3#577911yescharacteristic
602C2×C3C2×#93011yescharacteristic
603D3—42no—
604D3—242no—
605D3—241no—
606D3—241no—
607C22—21no—
608C22—21no—
609C22—62no—
610C22—62no—
611C22—62no—
612C22—361no—
613C22—361no—
614C22—361no—
615C22#6390912yes—
616C22C2×D3×D1211yescharacteristic
617C4—62no—
618C4—122no—
619C4—121no—
620C4—121no—
621C4—361no—
622C4—362no—
623C4#6390912yes—
624C4C2×#379911yescharacteristic
625C4C22×#62611yescharacteristic
626C3—21no—
627C3#7921411yescharacteristic, complemented
628C3#8380811yescharacteristic, complemented
629C2—22no—
630C2—122no—
631C2—721no—
632C2#1059471311yescharacteristic
633C2#1059472111yescharacteristic
634C2C2×#6390911yescharacteristic
subgroupG/H
centerC22C2×D3×D12
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC2×C32×M16C22
socleC22×C32C22×D4
prime radical#10595043C2
hypercenterC2×M16D32
hypocenterC32#3063
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. C2×#64118C2semidirect
2. C2×C3×#907C2semidirect
4. #10594856C2semidirect
5. C3×#5121C2semidirect
6. #10591105C2semidirect
7. C2×#63567C2semidirect
8. #10590853C2semidirect
9. #10590852C2semidirect
10. #10590838C2semidirect
11. #10590830C2semidirect
51. C2×#907D3semidirect
57. #5121D3semidirect
88. C32×M16D4semidirect
277. C3×M16D12semidirect
278. C3×M16#67semidirect
487. M16#626semidirect
556. C32#3063semidirect
627. C3#79214semidirect
628. C3#83808semidirect

Maximal subgroups

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

subgroupindex
C2×#641182
C2×C3×#9072
#105950432
#105948562
C3×#51212
#105911052
C2×#635672
#105908532
#105908522
#105908382
#105908302
#838083
#792143