Finite Group Databasebeta

← #10698053

Subgroups — #10698053

Order 768.

988 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 988 conjugacy classes of subgroups — too many to draw. Only its 105 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×#2848C211yesmaximal, characteristic, complemented
2#84032C211yesmaximal, characteristic
3#82736C211yesmaximal, characteristic, complemented
4#72126C211yesmaximal, characteristic, complemented
5#71690C211yesmaximal, characteristic, complemented
6#69509C211yesmaximal, characteristic, complemented
7#67625C211yesmaximal, characteristic, complemented
8#10322—31nomaximal, Sylow
9C24×C3×C4C2211yescharacteristic
10C2×C3×C22⋊Q8—21no—
11C2×C3×C22⋊Q8C2211yescharacteristic
12C2×#953—21no—
13C2×#930—21no—
14C2×#930C2211yescharacteristic
15C3×#385—21no—
16C3×#385—22no—
17C3×#380—21no—
18C3×#380C2211yescharacteristic
19C3×#379—21no—
20#5227—21no—
21#5213—21no—
22#5213—22no—
23#5212—21no—
24#5212C2211yescharacteristic
25C2×#830—22no—
26C2×#830C2211yescharacteristic
27C2×#830C2211yescharacteristic
28#5048—22no—
29#5046—21no—
30#5043—21no—
31#4968—21no—
32#4538—22no—
33#4451—22no—
34#3187—31no—
35#2848D311yescharacteristic, complemented
36#2847—31no—
37#1779—31no—
38#1520—31no—
39#1519—31no—
40#1289—31no—
41C25×C3D411yescharacteristic
42C22×C3×Q8D411yescharacteristic, complemented
43C23×C3×C4—21no—
44C23×C3×C4—21no—
45C23×C3×C4—21no—
46C23×C3×C4—21no—
47C23×C3×C4—21no—
48C23×C3×C4—21no—
49C23×C3×C4—44no—
50C23×C3×C4D411yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C22×#67—22no—
53C22×#67—21no—
54C22×D12D411yescharacteristic
55C3×C22⋊Q8—21no—
56C3×C22⋊Q8—21no—
57C3×C22⋊Q8—21no—
58C3×C22⋊Q8—21no—
59C3×C22⋊Q8—44no—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4D411yescharacteristic, complemented
64C2×C3×C22⋊C4—22no—
65C2×C3×C22⋊C4—22no—
66C2×C3×C22⋊C4—21no—
67C2×C3×C22⋊C4—21no—
68#953—42no—
69#953—42no—
70C2×#217—22no—
71C2×#217—21no—
72C2×#215—22no—
73#930—21no—
74#930—21no—
75#930—21no—
76#930—21no—
77#930—44no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—21no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3D411yescharacteristic
82C22×C3⋊C8—21no—
83C22×C3⋊C8D412yescomplemented
84C3×#95—22no—
85C3×#95D412yescomplemented
86#835—22no—
87C3⋊#95—22no—
88C3⋊#95D412yes—
89#830—21no—
90#830—21no—
91#830—21no—
92#830—21no—
93#830—44no—
94#809—22no—
95#808—22no—
96C24×C4C2×D311yescharacteristic
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8C2×D311yescharacteristic
99C2×#121—31no—
100C2×#121—61no—
101C2×#120—61no—
102#483—61no—
103#481—61no—
104#465—62no—
105#450—61no—
106#448—61no—
107C2×C22⋊C8—31no—
108C2×C22⋊C8—31no—
109C2×C22⋊C8—62no—
110#386—62no—
111#386—61no—
112#385—21no—
113#385—22no—
114#380—21no—
115#380—31no—
116#380—61no—
117#380C2×D311yescharacteristic
118#379—21no—
119#328—62no—
120#327—62no—
121C24×C3—21no—
122C24×C3—21no—
123C24×C3—21no—
124C24×C3—21no—
125C24×C3—21no—
126C24×C3—21no—
127C24×C3—42no—
128C24×C3—42no—
129C24×C3SD1612yescomplemented
130C24×C3C2×D411yescharacteristic
131C23×D3—21no—
132C2×C3×Q8—21no—
133C2×C3×Q8—21no—
134C2×C3×Q8—42no—
135C22×C3×C4—24no—
136C22×C3×C4—24no—
137C22×C3×C4—24no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—22no—
143C22×C3×C4—21no—
144C22×C3×C4—42no—
145C22×C3×C4—42no—
146C22×C3×C4—42no—
147C22×C3×C4—42no—
148C22×C3×C4—42no—
149C22×C3×C4—42no—
150C22×C3×C4—42no—
151C22×C3×C4—42no—
152C22×C3×C4—42no—
153C22×C3×C4—42no—
154C22×C3×C4—42no—
155C22×C3×C4—41no—
156C22×C3×C4—41no—
157C22×C3×C4—41no—
158C22×C3×C4—41no—
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4#4111yescharacteristic
165C2×#67—44no—
166C2×#67—44no—
167C2×#67—44no—
168C2×#67—41no—
169C2×#67—41no—
170C2×#67—41no—
171C2×#67—41no—
172C2×#67—41no—
173C2×#67—41no—
174C22×Dic3—21no—
175C22×Dic3—22no—
176C2×D12—21no—
177C2×D12—21no—
178C2×D12—42no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—42no—
182C3×C4⋊C4—44no—
183C3×C4⋊C4—41no—
184C3×C4⋊C4—41no—
185C3×C22⋊C4—44no—
186C3×C22⋊C4—44no—
187C3×C22⋊C4—41no—
188C3×C22⋊C4—41no—
189C3×C22⋊C4—41no—
190C3×C22⋊C4—41no—
191#217—44no—
192#217—41no—
193#217—41no—
194#215—82no—
195(C2×D3)⋊C4—44no—
196(C2×D3)⋊C4—41no—
197(C2×D3)⋊C4—41no—
198C4⋊Dic3—21no—
199C4⋊Dic3—21no—
200C4⋊Dic3—42no—
201C2×C3⋊C8—22no—
202C2×C3⋊C8—42no—
203C2×C3⋊C8—42no—
204C2×C3⋊C8—41no—
205C25#6711yescharacteristic
206C22×Q8#6711yescharacteristic, complemented
207C22×D4—31no—
208C22×D4—62no—
209C22×D4—61no—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—44no—
217C23×C4#6711yescharacteristic
218C23×C4C22×D311yescharacteristic
219C2×SD16—62no—
220C22×C8—32no—
221C22×C8—61no—
222C22⋊Q8—21no—
223C22⋊Q8—21no—
224C22⋊Q8—21no—
225C22⋊Q8—21no—
226C22⋊Q8—44no—
227#121—61no—
228#121—61no—
229#121—61no—
230#121—61no—
231#121—124no—
232#120—122no—
233#120—122no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—31no—
238C2×C4⋊C4—61no—
239C2×C4⋊C4#6711yescharacteristic, complemented
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—62no—
245C2×C22⋊C4—62no—
246C2×C22⋊C4—61no—
247C2×C22⋊C4—61no—
248#106—62no—
249#103—62no—
250#102—62no—
251C22⋊C8—61no—
252C22⋊C8—61no—
253C22⋊C8—61no—
254C22⋊C8—61no—
255C22⋊C8—124no—
256#95—22no—
257#95—32no—
258#95—62no—
259#95#6712yescomplemented
260C23×C3—22no—
261C23×C3—22no—
262C23×C3—22no—
263C23×C3—22no—
264C23×C3—22no—
265C23×C3—22no—
266C23×C3—22no—
267C23×C3—22no—
268C23×C3—22no—
269C23×C3—41no—
270C23×C3—44no—
271C23×C3—41no—
272C23×C3—41no—
273C23×C3—41no—
274C23×C3—41no—
275C23×C3—41no—
276C23×C3—41no—
277C23×C3—41no—
278C23×C3—44no—
279C23×C3—44no—
280C23×C3—44no—
281C23×C3—41no—
282C23×C3—41no—
283C23×C3—41no—
284C23×C3—41no—
285C23×C3#12011yescharacteristic
286C23×C3#12011yescharacteristic
287C23×C3#12111yescharacteristic
288C23×C3C2×SD1611yescharacteristic
289C23×C3#13511yescharacteristic
290C23×C3AGL(1,ℤ/8)11yescharacteristic
291C23×C3#13711yescharacteristic
292C22×D3—42no—
293C22×D3—42no—
294C22×D3—41no—
295C22×D3—41no—
296C22×D3—41no—
297C3×Q8—42no—
298C3×Q8—81no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—44no—
309C2×C3×C4—44no—
310C2×C3×C4—44no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—44no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—42no—
323C2×C3×C4#12011yescharacteristic
324C2×C3×C4#12111yescharacteristic
325C2×C3×C4#12111yescharacteristic
326C2×C3×C4#12711yescharacteristic
327#67—88no—
328#67—81no—
329#67—81no—
330#67—81no—
331#67—81no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3—41no—
335C2×Dic3—42no—
336C2×Dic3—41no—
337C2×Dic3—41no—
338D12—42no—
339D12—81no—
340C3⋊C8—42no—
341C3⋊C8—81no—
342C24—21no—
343C24—21no—
344C24—21no—
345C24—21no—
346C24—21no—
347C24—21no—
348C24—42no—
349C24—42no—
350C24—61no—
351C24#21512yescomplemented
352C24C2×#6711yescharacteristic
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—42no—
356C2×D4—61no—
357C2×D4—61no—
358C2×D4—122no—
359C2×D4—124no—
360C2×D4—124no—
361C2×D4—124no—
362C2×D4—121no—
363C2×D4—121no—
364C2×D4—121no—
365C2×D4—121no—
366C2×D4—121no—
367C2×D4—121no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—62no—
393C22×C4—61no—
394C22×C4D3×D411yescharacteristic
395C22×C4D3×D411yescharacteristic
396C22×C4D3×D411yescharacteristic
397C22×C4#23711yescharacteristic
398C22×C4C2×#6711yescharacteristic
399C22×C4C2×#6711yescharacteristic
400SD16—242no—
401C2×C8—62no—
402C2×C8—122no—
403C2×C8—122no—
404C2×C8—121no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—42no—
408C4⋊C4—44no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—61no—
412C4⋊C4—61no—
413C4⋊C4—122no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C22⋊C4—41no—
419C22⋊C4—41no—
420C22⋊C4—124no—
421C22⋊C4—124no—
422C22⋊C4—121no—
423C22⋊C4—121no—
424C22⋊C4—121no—
425C22⋊C4—121no—
426C22×C3—21no—
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—24no—
433C22×C3—24no—
434C22×C3—24no—
435C22×C3—41no—
436C22×C3—41no—
437C22×C3—41no—
438C22×C3—41no—
439C22×C3—42no—
440C22×C3—42no—
441C22×C3—42no—
442C22×C3—42no—
443C22×C3—42no—
444C22×C3—42no—
445C22×C3—42no—
446C22×C3—42no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—42no—
450C22×C3—42no—
451C22×C3#39211yescharacteristic
452C22×C3#44811yescharacteristic
453C22×C3#44911yescharacteristic
454C22×C3#45011yescharacteristic
455C22×C3#45211yescharacteristic
456C22×C3#46511yescharacteristic
457C22×C3#46811yescharacteristic
458C2×D3—82no—
459C2×D3—82no—
460C2×D3—81no—
461C2×D3—81no—
462C2×D3—81no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—42no—
467C3×C4—82no—
468C3×C4—81no—
469C3×C4—81no—
470C3×C4B(4,𝔽2)12yes—
471Dic3—82no—
472Dic3—81no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—41no—
483C23—44no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—122no—
499C23—122no—
500C23—121no—
501C23—121no—
502C23—121no—
503C23#93711yescharacteristic
504C23#93911yescharacteristic
505C23C2×#21511yescharacteristic
506C23#94211yescharacteristic
507C23#94911yescharacteristic
508C23#95011yescharacteristic
509C23#95311yescharacteristic
510Q8—42no—
511Q8—81no—
512D4—122no—
513D4—241no—
514D4—248no—
515D4—241no—
516D4—241no—
517D4—241no—
518D4—241no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—122no—
544C2×C4—122no—
545C2×C4—121no—
546C2×C4—122no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4#93711yescharacteristic
550C2×C4#93811yescharacteristic
551C2×C4#93911yescharacteristic
552C2×C4#94011yescharacteristic
553C8—122no—
554C8—241no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—44no—
562C2×C3#200011yescharacteristic
563C2×C3#201011yescharacteristic
564C2×C3#202111yescharacteristic
565D3—161no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—24no—
573C22—24no—
574C22—24no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—242no—
592C22—242no—
593C22—241no—
594C22—241no—
595C22—241no—
596C22#504611yescharacteristic
597C22#504711yescharacteristic
598C22#504811yescharacteristic
599C22#504911yescharacteristic
600C22#522311yescharacteristic
601C22#522711yescharacteristic
602C22#523811yescharacteristic
603C4—21no—
604C4—21no—
605C4—21no—
606C4—42no—
607C4—82no—
608C4—81no—
609C4—81no—
610C4—242no—
611C4—241no—
612C4#503212yes—
613C3#1032211yescharacteristic, complemented, Sylow
614C2—21no—
615C2—21no—
616C2—21no—
617C2—21no—
618C2—21no—
619C2—21no—
620C2—44no—
621C2—481no—
622C2#7167111yescharacteristic
623C2#7169911yescharacteristic
624C2#7171511yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2848C2
socleC22×C3#392
prime radical#84032C2
hypercenter#2848D3
hypocenterC3#10322
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2848

Complements

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

NH ≅ G/Ntype
1. C3×#2848C2semidirect
3. #82736C2semidirect
4. #72126C2semidirect
5. #71690C2semidirect
6. #69509C2semidirect
7. #67625C2semidirect
35. #2848D3semidirect
42. C22×C3×Q8D4semidirect
63. C2×C3×C4⋊C4D4semidirect
83. C22×C3⋊C8D4semidirect
85. C3×#95D4semidirect
129. C24×C3SD16semidirect
206. C22×Q8#67semidirect
239. C2×C4⋊C4#67semidirect
259. #95#67semidirect
351. C24#215semidirect
613. C3#10322semidirect

Maximal subgroups

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

subgroupindex
C3×#28482
#840322
#827362
#721262
#716902
#695092
#676252
#103223