Finite Group Databasebeta

← #11778025

Subgroups — #11778025

Order 1152.

726 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5634C211yesmaximal, characteristic, complemented
2#10592745C211yesmaximal, characteristic, complemented
3#10591395C211yesmaximal, characteristic
4#10591183C211yesmaximal, characteristic, complemented
5C3×#4538C211yesmaximal, characteristic, complemented
6#10589153C211yesmaximal, characteristic, complemented
7#10589145C211yesmaximal, characteristic, complemented
8#71892—31nomaximal
9#68316—31nomaximal
10C22×C4⋊(C3⋊D3)C2211yescharacteristic
11C22×#626—21no—
12C32×C22⋊Q8C2211yescharacteristic
13C2×#3755—21no—
14C3×#946—21no—
15C2×C3×(C2×D3)⋊C4C2211yescharacteristic
16C3×#924C2211yescharacteristic
17C3×#897—21no—
18C3×#896—21no—
19C3×#896—21no—
20C3×D3×C22⋊C4—21no—
21C3×#878—21no—
22C2×#3722—21no—
23C2×#3717—21no—
24#63657C2211yescharacteristic
25#63636—21no—
26C3×C3⋊#95C2211yescharacteristic
27C3×#830C2211yescharacteristic
28#63564—21no—
29C3×#543—31no—
30#5634D311yescharacteristic, complemented
31#5046—31no—
32#5046—31no—
33#5046—31no—
34#5046—61no—
35#4960—31no—
36#4731—31no—
37#4671—31no—
38C3×#328—31no—
39#4538—31no—
40#4538D311yescharacteristic, complemented
41#4525—31no—
42#4468—31no—
43#4452—31no—
44C23×C3⋊D3—21no—
45C23×C3×D3D411yescharacteristic
46C2×C32×Q8D411yescharacteristic, complemented
47C22×C32×C4C2311yescharacteristic
48C2×C4⋊(C3⋊D3)—21no—
49C2×C4⋊(C3⋊D3)—21no—
50C2×C4⋊(C3⋊D3)—42no—
51C22×C3×Dic3—21no—
52C22×C3×Dic3D411yescharacteristic
53C2×C3×C4×D3—21no—
54C2×C3×C4×D3—21no—
55C2×C3×Dic6—21no—
56C2×C3×Dic6D411yescharacteristic, complemented
57C2×#626—42no—
58C2×#626—42no—
59C2×#626—42no—
60C32×C4⋊C4—21no—
61C32×C4⋊C4D411yescharacteristic, complemented
62C32×C22⋊C4—21no—
63#3755—42no—
64C2×C3⋊(C3⋊C8)—21no—
65C3×#217—21no—
66C3×(C2×D3)⋊C4—21no—
67C3×(C2×D3)⋊C4—21no—
68C3×(C2×D3)⋊C4—21no—
69C3×(C2×D3)⋊C4—21no—
70C3×C4⋊Dic3—21no—
71C3×C4⋊Dic3D411yescharacteristic, complemented
72C3×Dic3⋊C4—21no—
73C3×Dic3⋊C4—21no—
74C3×Dic3⋊C4—21no—
75C2×C3×C3⋊C8—21no—
76#3722—42no—
77#3717—42no—
78#1596—91noSylow
79C22×#67—61no—
80C22×D12—31no—
81C22×D12—31no—
82C22×D12—61no—
83C22×D12—61no—
84C3×C22⋊Q8—21no—
85C3×C22⋊Q8—31no—
86C3×C22⋊Q8—61no—
87C3×C22⋊Q8—61no—
88C3×C22⋊Q8—61no—
89C3×C22⋊Q8—61no—
90C3×C22⋊Q8—61no—
91C3×C22⋊Q8C2×D311yescharacteristic
92C3×C22⋊Q8C2×D311yescharacteristic
93C2×C3×C22⋊C4—31no—
94C2×C3×C22⋊C4—61no—
95#946—21no—
96C2×#215—61no—
97C2×#215—62no—
98C2×#215—61no—
99C2×#215—61no—
100C2×(C2×D3)⋊C4—61no—
101C2×(C2×D3)⋊C4—61no—
102C2×(C2×D3)⋊C4C2×D311yescharacteristic
103#924C2×D311yescharacteristic
104C2×#204—61no—
105#897—21no—
106#896—21no—
107#896—21no—
108D3×C22⋊C4—21no—
109#878—21no—
110C3×C22⋊C8—31no—
111C3×#95—31no—
112C3⋊#95C2×D311yescharacteristic
113#830—31no—
114#830—31no—
115#830—61no—
116#830C2×D311yescharacteristic
117#821—61no—
118#809—61no—
119#809—62no—
120#809—61no—
121#809—61no—
122C23×C32C2×D411yescharacteristic
123C22×C3⋊D3—22no—
124C22×C3⋊D3—41no—
125C22×C3⋊D3—41no—
126C22×C3⋊D3—41no—
127C22×C3⋊D3—41no—
128C22×C3⋊D3—41no—
129C22×C3⋊D3—41no—
130C22×C3×D3—21no—
131C22×C3×D3—21no—
132C22×C3×D3—41no—
133C22×C3×D3—41no—
134C22×C3×D3SD1612yescomplemented
135C32×Q8—41no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—41no—
139C2×C32×C4C2×D411yescharacteristic
140C2×C32×C4C2×D411yescharacteristic
141C4⋊(C3⋊D3)—22no—
142C4⋊(C3⋊D3)—22no—
143C4⋊(C3⋊D3)—41no—
144C4⋊(C3⋊D3)—41no—
145C2×C3×Dic3—22no—
146C2×C3×Dic3—21no—
147C2×C3×Dic3—21no—
148C2×C3×Dic3—21no—
149C2×C3×Dic3—41no—
150C2×C3×Dic3—41no—
151C2×C3×Dic3—41no—
152C3×C4×D3—81no—
153C3×C4×D3—81no—
154C3×Dic6—41no—
155C3×Dic6—81no—
156#626—48no—
157C3⋊(C3⋊C8)—41no—
158C3×C3⋊C8—41no—
159#543—31no—
160#450—91no—
161#450—91no—
162#392—91no—
163#328—31no—
164#328—91no—
165#323—91no—
166C24×C3—31no—
167C23×D3—61no—
168C23×D3—61no—
169C23×D3—121no—
170C23×D3#6711yescharacteristic
171C2×C3×Q8—21no—
172C2×C3×Q8—31no—
173C2×C3×Q8—61no—
174C2×C3×Q8D1211yescharacteristic, complemented
175C2×C3×Q8#6711yescharacteristic, complemented
176C22×C3×C4—21no—
177C22×C3×C4—31no—
178C22×C3×C4—61no—
179C22×C3×C4—61no—
180C22×C3×C4—61no—
181C22×C3×C4C22×D311yescharacteristic
182C22×C3×C4C22×D311yescharacteristic
183C2×#67—122no—
184C2×#67—122no—
185C2×#67—122no—
186C22×Dic3—21no—
187C22×Dic3#6711yescharacteristic
188C2×D12—62no—
189C2×D12—61no—
190C2×D12—61no—
191C2×D12—61no—
192C2×D12—61no—
193C2×D12—121no—
194C2×D12—122no—
195C2×D12—122no—
196C2×D12—124no—
197C2×D12—122no—
198C2×D12—122no—
199C2×D12—122no—
200C2×C4×D3—21no—
201C2×C4×D3—21no—
202C2×Dic6—21no—
203C2×Dic6#6711yescharacteristic, complemented
204C2×C3×C8—61no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—31no—
209C3×C4⋊C4—41no—
210C3×C4⋊C4—61no—
211C3×C4⋊C4—61no—
212C3×C4⋊C4—61no—
213C3×C4⋊C4—61no—
214C3×C4⋊C4D1211yescharacteristic, complemented
215C3×C4⋊C4#6711yescharacteristic, complemented
216C3×C22⋊C4—21no—
217C3×C22⋊C4—21no—
218C3×C22⋊C4—41no—
219C3×C22⋊C4—61no—
220C3×C22⋊C4—61no—
221C3×C22⋊C4—61no—
222C3×C22⋊C4—61no—
223C3×C22⋊C4—61no—
224#217—21no—
225#215—122no—
226#215—122no—
227#215—122no—
228#215—124no—
229(C2×D3)⋊C4—21no—
230(C2×D3)⋊C4—21no—
231(C2×D3)⋊C4—21no—
232(C2×D3)⋊C4—21no—
233(C2×D3)⋊C4—122no—
234(C2×D3)⋊C4—122no—
235C4⋊Dic3—21no—
236C4⋊Dic3#6711yescharacteristic, complemented
237Dic3⋊C4—21no—
238Dic3⋊C4—21no—
239Dic3⋊C4—21no—
240C2×C3⋊C8—21no—
241C2×C3⋊C8—62no—
242C2×C3⋊C8—61no—
243C2×C3⋊C8—61no—
244#204—122no—
245C22×C32—21no—
246C22×C32—21no—
247C22×C32#12011yescharacteristic
248C22×C32C2×SD1611yescharacteristic
249C22×C32#13711yescharacteristic
250C2×C3⋊D3—42no—
251C2×C3⋊D3—42no—
252C2×C3⋊D3—42no—
253C2×C3⋊D3—81no—
254C2×C3⋊D3—81no—
255C2×C3⋊D3—81no—
256C2×C3⋊D3—81no—
257C2×C3×D3—22no—
258C2×C3×D3—41no—
259C2×C3×D3—41no—
260C2×C3×D3—42no—
261C2×C3×D3—81no—
262C32×C4—21no—
263C32×C4—21no—
264C32×C4—41no—
265C32×C4—41no—
266C3×Dic3—42no—
267C3×Dic3—41no—
268C3×Dic3—41no—
269C3×Dic3—41no—
270C22×D4—91no—
271C22×D4—181no—
272C2×SD16—181no—
273C2×SD16—181no—
274C22⋊Q8—31no—
275C22⋊Q8—61no—
276C22⋊Q8—61no—
277C22⋊Q8—61no—
278C22⋊Q8—61no—
279C22⋊Q8—61no—
280C22⋊Q8D3211yescharacteristic
281C2×C22⋊C4—31no—
282C2×C22⋊C4—61no—
283C2×C22⋊C4—181no—
284#102—181no—
285#102—181no—
286C22⋊C8—31no—
287C22⋊C8—91no—
288#95—31no—
289C23×C3—21no—
290C23×C3—32no—
291C23×C3—61no—
292C23×C3—61no—
293C23×C3—121no—
294C23×C3—121no—
295C23×C3D3×D411yescharacteristic
296C23×C3C2×#6711yescharacteristic
297C22×D3—21no—
298C22×D3—21no—
299C22×D3—41no—
300C22×D3—41no—
301C22×D3—62no—
302C22×D3—62no—
303C22×D3—122no—
304C22×D3—121no—
305C22×D3—121no—
306C22×D3—122no—
307C22×D3—121no—
308C22×D3—121no—
309C22×D3—121no—
310C22×D3—121no—
311C22×D3—122no—
312C22×D3—121no—
313C22×D3—121no—
314C22×D3—121no—
315C22×D3—121no—
316C22×D3—121no—
317C22×D3—121no—
318C22×D3—241no—
319C22×D3—241no—
320C22×D3—241no—
321C22×D3—241no—
322C22×D3#21512yescomplemented
323C3×Q8—41no—
324C3×Q8—41no—
325C3×Q8—42no—
326C3×Q8—121no—
327C3×Q8—241no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—42no—
339C2×C3×C4—62no—
340C2×C3×C4—61no—
341C2×C3×C4—61no—
342C2×C3×C4—61no—
343C2×C3×C4—121no—
344C2×C3×C4—121no—
345C2×C3×C4—121no—
346C2×C3×C4—241no—
347C2×C3×C4—241no—
348C2×C3×C4C2×D1211yescharacteristic
349C2×C3×C4D3×D411yescharacteristic
350C2×C3×C4C2×#6711yescharacteristic
351C2×C3×C4C2×#6711yescharacteristic
352#67—128no—
353C2×Dic3—22no—
354C2×Dic3—21no—
355C2×Dic3—21no—
356C2×Dic3—21no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360D12—62no—
361D12—64no—
362D12—62no—
363D12—62no—
364D12—62no—
365D12—122no—
366D12—122no—
367D12—121no—
368D12—121no—
369D12—121no—
370D12—121no—
371D12—122no—
372D12—128no—
373C4×D3—81no—
374C4×D3—81no—
375Dic6—41no—
376Dic6—81no—
377C3×C8—121no—
378C3⋊C8—41no—
379C3⋊C8—122no—
380C3⋊C8—121no—
381C3⋊C8—121no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32#44811yescharacteristic
385C2×C32#45311yescharacteristic
386C2×C32B(4,𝔽2)11yescharacteristic
387C3⋊D3—82no—
388C3×D3—42no—
389C24—31no—
390C24—181no—
391C2×Q8—31no—
392C2×Q8—61no—
393C2×Q8#62611yescharacteristic, complemented
394C2×D4—181no—
395C2×D4—181no—
396C2×D4—362no—
397C2×D4—362no—
398C2×D4—362no—
399C2×D4—362no—
400C22×C4—31no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4C2×D3211yescharacteristic
405SD16—362no—
406SD16—362no—
407C2×C8—61no—
408C2×C8—181no—
409C4⋊C4—21no—
410C4⋊C4—31no—
411C4⋊C4—61no—
412C4⋊C4—61no—
413C4⋊C4—61no—
414C4⋊C4—61no—
415C4⋊C4#62611yescharacteristic, complemented
416C22⋊C4—21no—
417C22⋊C4—61no—
418C22⋊C4—61no—
419C22⋊C4—61no—
420C22⋊C4—61no—
421C22⋊C4—61no—
422C22⋊C4—362no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—21no—
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—41no—
431C22×C3—41no—
432C22×C3—62no—
433C22×C3—122no—
434C22×C3—121no—
435C22×C3—121no—
436C22×C3—241no—
437C22×C3#88211yescharacteristic
438C22×C3D3×SD1611yescharacteristic
439C22×C3#91811yescharacteristic
440C22×C3C2×#21511yescharacteristic
441C22×C3#94211yescharacteristic
442C22×C3#95311yescharacteristic
443C2×D3—22no—
444C2×D3—41no—
445C2×D3—41no—
446C2×D3—42no—
447C2×D3—81no—
448C2×D3—124no—
449C2×D3—122no—
450C2×D3—122no—
451C2×D3—122no—
452C2×D3—122no—
453C2×D3—122no—
454C2×D3—122no—
455C2×D3—242no—
456C2×D3—242no—
457C2×D3—241no—
458C2×D3—241no—
459C2×D3—241no—
460C2×D3—241no—
461C2×D3—242no—
462C2×D3—242no—
463C2×D3—242no—
464C2×D3—242no—
465C2×D3—241no—
466C2×D3—241no—
467C2×D3—241no—
468C2×D3—241no—
469C3×C4—22no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—41no—
475C3×C4—41no—
476C3×C4—41no—
477C3×C4—41no—
478C3×C4—41no—
479C3×C4—81no—
480C3×C4—81no—
481C3×C4—122no—
482C3×C4—121no—
483C3×C4—121no—
484C3×C4—121no—
485Dic3—42no—
486Dic3—41no—
487Dic3—41no—
488Dic3—41no—
489C32#159611yescharacteristic, complemented, Sylow
490C23—32no—
491C23—61no—
492C23—61no—
493C23—121no—
494C23—121no—
495C23—182no—
496C23—361no—
497C23—361no—
498C23—361no—
499C23—361no—
500C23—361no—
501C23—361no—
502C23D3×#6711yescharacteristic
503Q8—41no—
504Q8—121no—
505Q8—241no—
506D4—182no—
507D4—182no—
508D4—361no—
509D4—361no—
510D4—368no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—41no—
514C2×C4—62no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—61no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—121no—
521C2×C4—241no—
522C2×C4—241no—
523C2×C4C2×#62611yescharacteristic
524C2×C4D3×#6711yescharacteristic
525C8—121no—
526C8—361no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—41no—
535C2×C3—41no—
536C2×C3—122no—
537C2×C3#474111yescharacteristic
538C2×C3#480611yescharacteristic
539C2×C3#482211yescharacteristic
540C2×C3#522711yescharacteristic
541C2×C3#524111yescharacteristic
542C2×C3#524411yescharacteristic
543D3—42no—
544D3—244no—
545D3—242no—
546D3—242no—
547C22—21no—
548C22—21no—
549C22—62no—
550C22—122no—
551C22—121no—
552C22—121no—
553C22—241no—
554C22—362no—
555C22—362no—
556C22—362no—
557C22—721no—
558C22—721no—
559C22—721no—
560C22—721no—
561C22D3×#21511yescharacteristic
562C22#6394811yescharacteristic
563C22#6397711yescharacteristic
564C4—21no—
565C4—21no—
566C4—41no—
567C4—41no—
568C4—122no—
569C4—121no—
570C4—121no—
571C4—121no—
572C3—21no—
573C3#6831611yescharacteristic, complemented
574C3#7189211yescharacteristic, complemented
575C2—21no—
576C2—21no—
577C2—122no—
578C2—722no—
579C2#1059153111yescharacteristic
580C2#1059158411yescharacteristic
581C2#1059166611yescharacteristic
subgroupG/H
centerC22#63977
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC32×C22⋊Q8C22
socleC22×C32#120
prime radical#10591395C2
hypercenterC22⋊Q8D32
hypocenterC32#1596
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×#5634C2semidirect
2. #10592745C2semidirect
4. #10591183C2semidirect
5. C3×#4538C2semidirect
6. #10589153C2semidirect
7. #10589145C2semidirect
30. #5634D3semidirect
40. #4538D3semidirect
46. C2×C32×Q8D4semidirect
56. C2×C3×Dic6D4semidirect
61. C32×C4⋊C4D4semidirect
71. C3×C4⋊Dic3D4semidirect
134. C22×C3×D3SD16semidirect
174. C2×C3×Q8D12semidirect
175. C2×C3×Q8#67semidirect
203. C2×Dic6#67semidirect
214. C3×C4⋊C4D12semidirect
215. C3×C4⋊C4#67semidirect
236. C4⋊Dic3#67semidirect
322. C22×D3#215semidirect
393. C2×Q8#626semidirect
415. C4⋊C4#626semidirect
489. C32#1596semidirect
573. C3#68316semidirect
574. C3#71892semidirect

Maximal subgroups

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

subgroupindex
C3×#56342
#105927452
#105913952
#105911832
C3×#45382
#105891532
#105891452
#718923
#683163