Finite Group Databasebeta

← #11878728

Subgroups — #11878728

Order 1152.

935 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×C3×#944C211yesmaximal, characteristic, complemented
2#10594910C211yesmaximal, characteristic
3#10592926C211yesmaximal, characteristic, complemented
4C3×#5227C211yesmaximal, characteristic, complemented
5#10591623C212yesmaximal, complemented
6#10591618C212yesmaximal, complemented
7#10591565C212yesmaximal, complemented
8#10591564C212yesmaximal, complemented
9#10591174C211yesmaximal, characteristic, complemented
10#10591171C211yesmaximal, characteristic, complemented
11#10591053C211yesmaximal, characteristic, complemented
12#83213—31nomaximal
13#80034—31nomaximal
14C22×C32×Q8C2211yescharacteristic
15C22×C3×Dic6C2211yescharacteristic
16C2×#3804—21no—
17C2×#3755C2212yes—
18#64139C2211yescharacteristic
19C3×#944—21no—
20C3×#944—21no—
21C3×#944C2212yes—
22C3×#944C2212yes—
23C2×C3×#215C2212yes—
24C3×#930C2211yescharacteristic
25C2×C3×Dic3⋊C4—21no—
26C2×C3×Dic3⋊C4—21no—
27C2×C3×C4×Dic3C2211yescharacteristic
28#63978—21no—
29#63918C2211yescharacteristic
30#63911C2211yescharacteristic
31#63894C2211yescharacteristic
32Dic3×D12C2211yescharacteristic
33#63880—21no—
34#63852—21no—
35C2×#3717C2212yes—
36C2×#3715C2212yes—
37#63662C2212yes—
38#63657C2211yescharacteristic
39C3×#835C2212yes—
40C3×#830C2211yescharacteristic
41Dic3⋊(C3⋊C8)C2212yes—
42#63566C2212yes—
43#63565C2212yes—
44Dic3×C3⋊C8C2212yes—
45C2×C3×C4⋊Q8—31no—
46C2×#944D311yescharacteristic, complemented
47#5596—31no—
48#5477—31no—
49C3×#448—31no—
50#5227—31no—
51#5227—31no—
52#5227—31no—
53#5227—31no—
54#5227—61no—
55#5227D311yescharacteristic, complemented
56#5175—32no—
57#5162—32no—
58#5085—32no—
59#5083—32no—
60#5083—32no—
61#5079—32no—
62#5040—31no—
63#4925—31no—
64#4818—32no—
65#4792—32no—
66#4738—31no—
67#4718—31no—
68C2×C32×Q8—21no—
69C2×C32×Q8—42no—
70C2×C32×Q8C2312yes—
71C22×C32×C4—21no—
72C22×C32×C4C2311yescharacteristic
73C2×#638—21no—
74C2×C4⋊(C3⋊D3)C2311yescharacteristic
75C2×C3×#67—21no—
76C22×C3×Dic3—21no—
77C22×C3×Dic3D411yescharacteristic
78C22×C3×Dic3D411yescharacteristic
79C2×C3×D12C2311yescharacteristic
80C2×C3×Dic6—21no—
81C2×C3×Dic6—42no—
82C2×C3×Dic6C2312yes—
83C2×#627—21no—
84C2×#626—21no—
85C2×#626—21no—
86C2×#624—21no—
87#3804—44no—
88C2×D3×Dic3—21no—
89#3755—42no—
90#3752—21no—
91C4⋊(C3⋊Dic3)C2311yescharacteristic
92C2×C3⋊(C3⋊C8)C2312yes—
93C3×#215—42no—
94C3×(C2×D3)⋊C4—21no—
95C3×C4⋊Dic3C2311yescharacteristic
96C3×Dic3⋊C4—22no—
97C3×Dic3⋊C4—22no—
98C3×Dic3⋊C4—22no—
99C3×Dic3⋊C4—22no—
100C3×C4×Dic3D411yescharacteristic
101C3×C4×Dic3D411yescharacteristic
102C3×C4×Dic3C2311yescharacteristic
103C3×C4×Dic3C2311yescharacteristic
104C2×C3×C3⋊C8C2312yes—
105Dic3⋊Dic3—21no—
106#3722—21no—
107(C2×D3)⋊Dic3—21no—
108#3717—42no—
109#3715—42no—
110#3206—91noSylow
111C22×C3×Q8—21no—
112C22×C3×Q8—31no—
113C22×C3×Q8C2×D311yescharacteristic
114C22×C3×Q8C2×D311yescharacteristic
115C2×#237—61no—
116C2×#235—61no—
117C22×Dic6C2×D311yescharacteristic
118C2×C3×SD16—32no—
119C3×C4⋊Q8—32no—
120C3×C4⋊Q8—32no—
121C3×C4⋊Q8—61no—
122C3×C4⋊Q8—61no—
123C3×#121—31no—
124C2×C3×C4⋊C4—61no—
125C2×C3×C4⋊C4—61no—
126C2×C3×C42—31no—
127#944—21no—
128#944—21no—
129#944C2×D312yes—
130#944C2×D312yes—
131C2×#215—32no—
132C2×#215—32no—
133C2×#215—32no—
134C2×#215—32no—
135C2×#215—62no—
136C2×#215C2×D312yes—
137#940—31no—
138#939—61no—
139D4×Dic3—31no—
140C2×#214—32no—
141#930—31no—
142#930—31no—
143#930—61no—
144#930—61no—
145#930C2×D311yescharacteristic
146C2×Dic3⋊C4—21no—
147C2×Dic3⋊C4—21no—
148C2×C4×Dic3C2×D311yescharacteristic
149C2×#204—32no—
150#893—31no—
151#888—31no—
152#885—61no—
153#67⋊C4—61no—
154#875—61no—
155#874—31no—
156C4×D12—31no—
157C4×D12—31no—
158C4×D12—61no—
159#869—31no—
160C3×#103—32no—
161C3×C22⋊C8—31no—
162#835—32no—
163#835—32no—
164#835—32no—
165#835—32no—
166#835—62no—
167#835C2×D312yes—
168#830—31no—
169#830—31no—
170#830—61no—
171#830C2×D311yescharacteristic
172#816—32no—
173Dic3⋊C8—32no—
174C8×Dic3—32no—
175#810—32no—
176C4⋊(C3⋊C8)—32no—
177C4×C3⋊C8—32no—
178C23×C32C2×D411yescharacteristic
179C22×C3⋊D3—21no—
180C22×C3×D3—21no—
181C32×Q8—22no—
182C32×Q8—22no—
183C32×Q8—42no—
184C2×C32×C4—22no—
185C2×C32×C4—22no—
186C2×C32×C4—42no—
187C2×C32×C4C2×D411yescharacteristic
188C2×C32×C4C2411yescharacteristic
189#638—42no—
190C2×C3⋊Dic3—21no—
191C4⋊(C3⋊D3)—22no—
192C3×#67—42no—
193C2×C3×Dic3—22no—
194C2×C3×Dic3—21no—
195C2×C3×Dic3—22no—
196C2×C3×Dic3—42no—
197C2×C3×Dic3SD1614yescomplemented
198C2×C3×Dic3C2×D411yescharacteristic
199C2×C3×Dic3C2×D411yescharacteristic
200C2×C3×Dic3C2×D411yescharacteristic
201C2×C3×Dic3C2×D411yescharacteristic
202C3×D12—22no—
203C3×Dic6—22no—
204C3×Dic6—22no—
205C3×Dic6—42no—
206#627—42no—
207#626—42no—
208#626—42no—
209#624—42no—
210D3×Dic3—42no—
211C3⋊(C3⋊C8)—22no—
212C3×C3⋊C8—22no—
213#532—91no—
214C2×C4⋊Q8—31no—
215#492—92no—
216#487—92no—
217#461—92no—
218#461—92no—
219#448—31no—
220#448—91no—
221#448—91no—
222#448—91no—
223#431—91no—
224C2×C3×Q8—21no—
225C2×C3×Q8—21no—
226C2×C3×Q8—22no—
227C2×C3×Q8—32no—
228C2×C3×Q8—41no—
229C2×C3×Q8—42no—
230C2×C3×Q8—42no—
231C2×C3×Q8—44no—
232C2×C3×Q8—61no—
233C2×C3×Q8—122no—
234C2×C3×Q8C22×D312yes—
235C2×C3×Q8C22×D312yes—
236C2×C3×D4—31no—
237C2×C3×D4—61no—
238C22×C3×C4—21no—
239C22×C3×C4—21no—
240C22×C3×C4—21no—
241C22×C3×C4—31no—
242C22×C3×C4—31no—
243C22×C3×C4—41no—
244C22×C3×C4—61no—
245C22×C3×C4C22×D311yescharacteristic
246C22×C3×C4C22×D311yescharacteristic
247C2×#67—21no—
248C2×#67—61no—
249C2×#67—61no—
250C2×#67—61no—
251C2×#67—61no—
252C2×#67—121no—
253C22×Dic3—21no—
254C22×Dic3—61no—
255C22×Dic3#6711yescharacteristic
256C22×Dic3#6711yescharacteristic
257#237—124no—
258#235—124no—
259C2×D12—31no—
260C2×D12—31no—
261C2×D12—61no—
262C2×D12—61no—
263C2×D12—61no—
264C2×D12C22×D311yescharacteristic
265C2×C4×D3—61no—
266C2×C4×D3—61no—
267C2×C4×D3—61no—
268C2×Dic6—21no—
269C2×Dic6—42no—
270C2×Dic6—61no—
271C2×Dic6—61no—
272C2×Dic6C22×D312yes—
273C3×SD16—122no—
274C2×C3×C8—32no—
275C3×C4⋊C4—31no—
276C3×C4⋊C4—62no—
277C3×C4⋊C4—62no—
278C3×C4⋊C4—62no—
279C3×C4⋊C4—62no—
280C3×C22⋊C4—61no—
281C3×C42—31no—
282C3×C42—31no—
283C3×C42—31no—
284C3×C42—31no—
285#217—61no—
286#215—42no—
287#215—122no—
288#215—122no—
289#215—122no—
290#215—122no—
291#215—124no—
292#214—122no—
293(C2×D3)⋊C4—21no—
294(C2×D3)⋊C4—61no—
295(C2×D3)⋊C4—61no—
296(C2×D3)⋊C4—61no—
297(C2×D3)⋊C4—61no—
298(C2×D3)⋊C4—61no—
299(C2×D3)⋊C4—121no—
300C4⋊Dic3—31no—
301C4⋊Dic3—31no—
302C4⋊Dic3—61no—
303C4⋊Dic3—61no—
304C4⋊Dic3C22×D311yescharacteristic
305Dic3⋊C4—22no—
306Dic3⋊C4—22no—
307Dic3⋊C4—22no—
308Dic3⋊C4—22no—
309Dic3⋊C4—61no—
310C4×Dic3#6711yescharacteristic
311C4×Dic3#6711yescharacteristic
312C4×Dic3C22×D311yescharacteristic
313C4×Dic3C22×D311yescharacteristic
314C2×C3⋊C8—32no—
315C2×C3⋊C8—32no—
316C2×C3⋊C8—62no—
317C2×C3⋊C8C22×D312yes—
318#204—122no—
319C22×C32—21no—
320C22×C32—21no—
321C22×C32C2×SD1612yes—
322C22×C32C22×D411yescharacteristic
323C2×C3⋊D3—41no—
324C2×C3⋊D3—41no—
325C2×C3⋊D3—41no—
326C2×C3×D3—41no—
327C2×C3×D3—41no—
328C2×C3×D3—41no—
329C32×C4—21no—
330C32×C4—42no—
331C32×C4H(5,𝔽2)12yes—
332C3⋊Dic3—41no—
333C3×Dic3—21no—
334C3×Dic3—42no—
335C3×Dic3—41no—
336C3×Dic3C2×SD1612yes—
337C3×Dic3C2×SD1612yes—
338C3×Dic3#13712yes—
339C2×#41—181no—
340C22×Q8—31no—
341C22×Q8D3211yescharacteristic
342C2×SD16—32no—
343C2×SD16—92no—
344C2×SD16—92no—
345C2×SD16—92no—
346C4⋊Q8—32no—
347C4⋊Q8—32no—
348C4⋊Q8—61no—
349C4⋊Q8—61no—
350C4⋊Q8—91no—
351#127—91no—
352#124—181no—
353#121—31no—
354#121—91no—
355#121—181no—
356C4×D4—91no—
357C4×D4—91no—
358C4×D4—181no—
359C2×C4⋊C4—61no—
360C2×C4⋊C4—61no—
361C2×C42—31no—
362C4⋊C8—92no—
363#103—32no—
364#103—92no—
365#103—92no—
366#103—92no—
367C22⋊C8—31no—
368C22⋊C8—91no—
369C4×C8—92no—
370C23×C3—21no—
371C23×C3—61no—
372C23×C3D3×D411yescharacteristic
373C23×C3C2×#6711yescharacteristic
374C22×D3—21no—
375C22×D3—61no—
376C22×D3—61no—
377C22×D3—121no—
378C3×Q8—24no—
379C3×Q8—22no—
380C3×Q8—22no—
381C3×Q8—22no—
382C3×Q8—22no—
383C3×Q8—42no—
384C3×Q8—42no—
385C3×Q8—42no—
386C3×Q8—44no—
387C3×Q8—62no—
388C3×Q8—62no—
389C3×Q8—122no—
390C3×D4—62no—
391C3×D4—122no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—22no—
395C2×C3×C4—22no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—34no—
399C2×C3×C4—31no—
400C2×C3×C4—31no—
401C2×C3×C4—31no—
402C2×C3×C4—31no—
403C2×C3×C4—42no—
404C2×C3×C4—42no—
405C2×C3×C4—42no—
406C2×C3×C4—42no—
407C2×C3×C4—44no—
408C2×C3×C4—62no—
409C2×C3×C4—62no—
410C2×C3×C4—61no—
411C2×C3×C4—122no—
412C2×C3×C4D3×D411yescharacteristic
413C2×C3×C4C2×#6711yescharacteristic
414C2×C3×C4C23×D311yescharacteristic
415C2×C3×C4C23×D311yescharacteristic
416#67—42no—
417#67—122no—
418#67—122no—
419#67—122no—
420#67—122no—
421#67—242no—
422C2×Dic3—22no—
423C2×Dic3—21no—
424C2×Dic3—22no—
425C2×Dic3—42no—
426C2×Dic3—61no—
427C2×Dic3—61no—
428C2×Dic3—121no—
429C2×Dic3—122no—
430C2×Dic3#21514yescomplemented
431C2×Dic3C2×#6711yescharacteristic
432C2×Dic3C2×#6711yescharacteristic
433C2×Dic3C2×#6711yescharacteristic
434C2×Dic3C2×#6711yescharacteristic
435D12—22no—
436D12—62no—
437D12—62no—
438D12—122no—
439D12—122no—
440D12—122no—
441C4×D3—122no—
442C4×D3—122no—
443C4×D3—122no—
444Dic6—22no—
445Dic6—22no—
446Dic6—42no—
447Dic6—122no—
448Dic6—122no—
449C3×C8—62no—
450C3⋊C8—22no—
451C3⋊C8—62no—
452C3⋊C8—62no—
453C3⋊C8—122no—
454C2×C32—22no—
455C2×C32#53511yescharacteristic
456C2×C32C22×SD1611yescharacteristic
457C2×C32C2×#13711yescharacteristic
458C3⋊D3—81no—
459C3×D3—81no—
460#41—364no—
461C2×Q8—21no—
462C2×Q8—32no—
463C2×Q8—42no—
464C2×Q8—61no—
465C2×Q8—122no—
466C2×Q8—181no—
467C2×Q8C2×D3212yes—
468C2×D4—31no—
469C2×D4—61no—
470C2×D4—91no—
471C2×D4—181no—
472C2×D4—181no—
473C2×D4—181no—
474C22×C4—21no—
475C22×C4—31no—
476C22×C4—31no—
477C22×C4—61no—
478C22×C4—181no—
479C22×C4—181no—
480C22×C4C2×D3211yescharacteristic
481SD16—122no—
482SD16—362no—
483SD16—362no—
484SD16—362no—
485C2×C8—32no—
486C2×C8—92no—
487C4⋊C4—31no—
488C4⋊C4—62no—
489C4⋊C4—62no—
490C4⋊C4—62no—
491C4⋊C4—62no—
492C4⋊C4—91no—
493C4⋊C4—181no—
494C22⋊C4—61no—
495C22⋊C4—181no—
496C22⋊C4—181no—
497C22⋊C4—181no—
498C42—31no—
499C42—31no—
500C42—31no—
501C42—31no—
502C22×C3—21no—
503C22×C3—21no—
504C22×C3—21no—
505C22×C3—21no—
506C22×C3—21no—
507C22×C3—22no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3—121no—
511C22×C3—121no—
512C22×C3—121no—
513C22×C3D3×SD1612yes—
514C22×C3C2×#21512yes—
515C22×C3C2×D3×D411yescharacteristic
516C22×C3C22×#6711yescharacteristic
517C2×D3—41no—
518C2×D3—41no—
519C2×D3—41no—
520C2×D3—121no—
521C2×D3—121no—
522C2×D3—121no—
523C2×D3—121no—
524C2×D3—121no—
525C2×D3—121no—
526C2×D3—241no—
527C2×D3—241no—
528C2×D3—241no—
529C3×C4—21no—
530C3×C4—21no—
531C3×C4—22no—
532C3×C4—32no—
533C3×C4—32no—
534C3×C4—32no—
535C3×C4—41no—
536C3×C4—44no—
537C3×C4—42no—
538C3×C4—42no—
539C3×C4—61no—
540C3×C4—121no—
541C3×C4—122no—
542C3×C4#100412yes—
543C3×C4#100412yes—
544Dic3—21no—
545Dic3—41no—
546Dic3—42no—
547Dic3—121no—
548Dic3—121no—
549Dic3—241no—
550Dic3C2×#21512yes—
551Dic3C2×#21512yes—
552Dic3#94212yes—
553C32#320611yescharacteristic, complemented, Sylow
554C23—61no—
555C23—181no—
556C23D3×#6711yescharacteristic
557Q8—22no—
558Q8—22no—
559Q8—42no—
560Q8—62no—
561Q8—62no—
562Q8—122no—
563Q8—362no—
564D4—62no—
565D4—122no—
566D4—182no—
567D4—362no—
568D4—362no—
569D4—362no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—31no—
573C2×C4—31no—
574C2×C4—31no—
575C2×C4—34no—
576C2×C4—31no—
577C2×C4—42no—
578C2×C4—62no—
579C2×C4—61no—
580C2×C4—62no—
581C2×C4—122no—
582C2×C4—181no—
583C2×C4—362no—
584C2×C4—362no—
585C2×C4D3×#6711yescharacteristic
586C2×C4C22×D3211yescharacteristic
587C8—62no—
588C8—182no—
589C2×C3—22no—
590C2×C3—22no—
591C2×C3—21no—
592C2×C3—21no—
593C2×C3—21no—
594C2×C3—42no—
595C2×C3—241no—
596C2×C3#559211yescharacteristic
597C2×C3C2×D3×SD1611yescharacteristic
598C2×C3D3×#13711yescharacteristic
599C2×C3#580511yescharacteristic
600C2×C3C22×#21511yescharacteristic
601C2×C3C2×#94211yescharacteristic
602D3—81no—
603D3—241no—
604D3—241no—
605D3—481no—
606C22—21no—
607C22—21no—
608C22—121no—
609C22—121no—
610C22—121no—
611C22—361no—
612C22—361no—
613C22—361no—
614C22D3×#21512yes—
615C22C2×D3×#6711yescharacteristic
616C4—21no—
617C4—32no—
618C4—32no—
619C4—32no—
620C4—42no—
621C4—61no—
622C4—122no—
623C4—121no—
624C4—361no—
625C4#6433012yes—
626C3—21no—
627C3#8003411yescharacteristic, complemented
628C3#8321311yescharacteristic, complemented
629C2—22no—
630C2—241no—
631C2—721no—
632C2C2×D3×#21511yescharacteristic
633C2D3×#94211yescharacteristic
634C2#1059551311yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC22×C32×Q8C22
socleC22×C32C22×D4
prime radical#10594910C2
hypercenterC22×Q8D32
hypocenterC32#3206
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#944C2semidirect
3. #10592926C2semidirect
4. C3×#5227C2semidirect
5. #10591623C2semidirect
6. #10591618C2semidirect
7. #10591565C2semidirect
8. #10591564C2semidirect
9. #10591174C2semidirect
10. #10591171C2semidirect
11. #10591053C2semidirect
46. C2×#944D3semidirect
55. #5227D3semidirect
197. C2×C3×Dic3SD16semidirect
430. C2×Dic3#215semidirect
553. C32#3206semidirect
627. C3#80034semidirect
628. C3#83213semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9442
#105949102
#105929262
C3×#52272
#105916232
#105916182
#105915652
#105915642
#105911742
#105911712
#105910532
#832133
#800343