Finite Group Databasebeta

← #10698054

Subgroups — #10698054

Order 768.

988 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 988 conjugacy classes of subgroups — too many to draw. Only its 101 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
1#82834C211yesmaximal, characteristic, complemented
2#82736C211yesmaximal, characteristic, complemented
3#74048C211yesmaximal, characteristic
4C3×#1520C211yesmaximal, characteristic, complemented
5#69955C211yesmaximal, characteristic, complemented
6#69944C211yesmaximal, characteristic, complemented
7#67625C211yesmaximal, characteristic, complemented
8#10322—31nomaximal, Sylow
9C23×C4×D3C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C2×#946—21no—
12C2×#930C2211yescharacteristic
13C2×#895—21no—
14C2×#882—21no—
15C2×C3×C22⋊C8C2211yescharacteristic
16C3×#380C2211yescharacteristic
17#5227—21no—
18#5212C2211yescharacteristic
19C2×#830C2211yescharacteristic
20C2×(C2×D3)⋊C8—22no—
21#5048—22no—
22#5046—21no—
23#5043—21no—
24#4989—22no—
25#4987—21no—
26#4987—21no—
27#4968—21no—
28#4958—21no—
29#4955—21no—
30#4946—21no—
31#4665—22no—
32C3×#328—22no—
33#4470—22no—
34#3187—31no—
35#2848—31no—
36#2847—31no—
37#1779—31no—
38#1520D311yescharacteristic, complemented
39#1519—31no—
40#1289—31no—
41C24×D3D411yescharacteristic
42C22×C3×Q8D411yescharacteristic, complemented
43C23×C3×C4C2311yescharacteristic
44C22×#67—21no—
45C23×Dic3D411yescharacteristic
46C22×D12—22no—
47C22×D12D411yescharacteristic
48C22×C4×D3—21no—
49C22×C4×D3—21no—
50C22×C4×D3—21no—
51C22×C4×D3—21no—
52C22×C4×D3—21no—
53C22×C4×D3—21no—
54C22×C4×D3—44no—
55C22×C3×C8—21no—
56C3×C22⋊Q8—21no—
57C3×C22⋊Q8—21no—
58C3×C22⋊Q8—21no—
59C3×C22⋊Q8—21no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4D411yescharacteristic, complemented
63C2×C3×C22⋊C4—21no—
64C2×C3×C22⋊C4—21no—
65C2×#217—21no—
66#946—44no—
67C2×#215—22no—
68#930—21no—
69#930—21no—
70#930—21no—
71#930—21no—
72C2×(C2×D3)⋊C4—22no—
73C2×(C2×D3)⋊C4—22no—
74C2×(C2×D3)⋊C4—22no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—21no—
77C2×C4⋊Dic3—21no—
78C2×C4⋊Dic3D411yescharacteristic
79C2×Dic3⋊C4—22no—
80C22×C3⋊C8D412yescomplemented
81#895—44no—
82#882—42no—
83#882—42no—
84C3×C22⋊C8—21no—
85C3×C22⋊C8—21no—
86C3×#95D412yes—
87#835—22no—
88C3⋊#95—22no—
89C3⋊#95—22no—
90C3⋊#95D412yescomplemented
91#830—21no—
92#830—21no—
93(C2×D3)⋊C8—44no—
94#809—22no—
95#808—22no—
96C24×C4—31no—
97C2×C22⋊Q8—61no—
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—62no—
109C2×C22⋊C8C2×D311yescharacteristic
110#386—61no—
111#386—62no—
112#385—62no—
113#385—61no—
114#380—31no—
115#380—61no—
116#380—61no—
117#380C2×D311yescharacteristic
118#379—61no—
119#328—22no—
120#327—62no—
121C24×C3C2×D411yescharacteristic
122C23×D3—21no—
123C23×D3—21no—
124C23×D3—21no—
125C23×D3—21no—
126C23×D3—21no—
127C23×D3—21no—
128C23×D3—21no—
129C23×D3—42no—
130C23×D3—42no—
131C23×D3SD1612yescomplemented
132C2×C3×Q8—21no—
133C2×C3×Q8—21no—
134C2×C3×Q8—42no—
135C22×C3×C4—21no—
136C22×C3×C4—22no—
137C22×C3×C4—21no—
138C22×C3×C4—42no—
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4#4111yescharacteristic
145C2×#67—41no—
146C2×#67—41no—
147C2×#67—41no—
148C2×#67—41no—
149C2×#67—41no—
150C2×#67—41no—
151C22×Dic3—21no—
152C22×Dic3—21no—
153C22×Dic3—21no—
154C22×Dic3—21no—
155C22×Dic3—22no—
156C22×Dic3—42no—
157C2×D12—21no—
158C2×D12—21no—
159C2×D12—42no—
160C2×D12—44no—
161C2×D12—44no—
162C2×D12—44no—
163C2×C4×D3—24no—
164C2×C4×D3—24no—
165C2×C4×D3—24no—
166C2×C4×D3—42no—
167C2×C4×D3—42no—
168C2×C4×D3—42no—
169C2×C4×D3—42no—
170C2×C4×D3—42no—
171C2×C4×D3—42no—
172C2×C4×D3—42no—
173C2×C4×D3—42no—
174C2×C4×D3—42no—
175C2×C4×D3—41no—
176C2×C4×D3—41no—
177C2×C4×D3—41no—
178C2×C4×D3—41no—
179C2×C3×C8—42no—
180C2×C3×C8—41no—
181C3×C4⋊C4—21no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—42no—
184C3×C4⋊C4—41no—
185C3×C4⋊C4—41no—
186C3×C22⋊C4—41no—
187C3×C22⋊C4—41no—
188C3×C22⋊C4—41no—
189C3×C22⋊C4—41no—
190#217—41no—
191#217—41no—
192#215—82no—
193(C2×D3)⋊C4—44no—
194(C2×D3)⋊C4—44no—
195(C2×D3)⋊C4—44no—
196(C2×D3)⋊C4—44no—
197(C2×D3)⋊C4—41no—
198(C2×D3)⋊C4—41no—
199C4⋊Dic3—21no—
200C4⋊Dic3—21no—
201C4⋊Dic3—42no—
202Dic3⋊C4—44no—
203C2×C3⋊C8—22no—
204C2×C3⋊C8—42no—
205C25—31no—
206C22×Q8D1211yescharacteristic, complemented
207C22×D4—31no—
208C22×D4—62no—
209C22×D4—61no—
210C23×C4—31no—
211C23×C4—61no—
212C23×C4—61no—
213C23×C4—61no—
214C23×C4—61no—
215C23×C4—61no—
216C23×C4—61no—
217C23×C4—124no—
218C23×C4C22×D311yescharacteristic
219C2×SD16—62no—
220C22×C8—21no—
221C22×C8—32no—
222C22⋊Q8—21no—
223C22⋊Q8—21no—
224C22⋊Q8—21no—
225C22⋊Q8—21no—
226C22⋊Q8—124no—
227#121—61no—
228#121—61no—
229#121—61no—
230#121—61no—
231#121—124no—
232#120—122no—
233#120—122no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—31no—
237C2×C4⋊C4—62no—
238C2×C4⋊C4—61no—
239C2×C4⋊C4D1211yescharacteristic, complemented
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—62no—
243C2×C22⋊C4—62no—
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—21no—
252C22⋊C8—21no—
253C22⋊C8—61no—
254C22⋊C8—61no—
255C22⋊C8—124no—
256#95—32no—
257#95—62no—
258#95—62no—
259#95#6712yes—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3#12011yescharacteristic
263C23×C3#12011yescharacteristic
264C23×C3#12111yescharacteristic
265C23×C3C2×SD1611yescharacteristic
266C23×C3#13511yescharacteristic
267C23×C3AGL(1,ℤ/8)11yescharacteristic
268C23×C3#13711yescharacteristic
269C22×D3—22no—
270C22×D3—22no—
271C22×D3—22no—
272C22×D3—22no—
273C22×D3—22no—
274C22×D3—22no—
275C22×D3—22no—
276C22×D3—22no—
277C22×D3—22no—
278C22×D3—44no—
279C22×D3—44no—
280C22×D3—44no—
281C22×D3—44no—
282C22×D3—41no—
283C22×D3—41no—
284C22×D3—41no—
285C22×D3—41no—
286C22×D3—41no—
287C22×D3—41no—
288C22×D3—41no—
289C22×D3—41no—
290C22×D3—41no—
291C22×D3—41no—
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—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—42no—
312C2×C3×C4#12011yescharacteristic
313C2×C3×C4#12111yescharacteristic
314C2×C3×C4#12111yescharacteristic
315C2×C3×C4#12711yescharacteristic
316#67—81no—
317#67—81no—
318#67—81no—
319#67—81no—
320C2×Dic3—22no—
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—41no—
326C2×Dic3—41no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—41no—
330C2×Dic3—42no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333D12—42no—
334D12—81no—
335D12—88no—
336C4×D3—44no—
337C4×D3—44no—
338C4×D3—44no—
339C4×D3—44no—
340C3×C8—81no—
341C3⋊C8—42no—
342C24—32no—
343C24—61no—
344C24—61no—
345C24—61no—
346C24—61no—
347C24—61no—
348C24—61no—
349C24—61no—
350C24—122no—
351C24—122no—
352C24D3×D411yescharacteristic
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—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—42no—
372C22×C4—64no—
373C22×C4—64no—
374C22×C4—64no—
375C22×C4—61no—
376C22×C4—61no—
377C22×C4—61no—
378C22×C4—62no—
379C22×C4—61no—
380C22×C4—122no—
381C22×C4—122no—
382C22×C4—122no—
383C22×C4—122no—
384C22×C4—122no—
385C22×C4—122no—
386C22×C4—122no—
387C22×C4—122no—
388C22×C4—122no—
389C22×C4—122no—
390C22×C4—121no—
391C22×C4—121no—
392C22×C4—121no—
393C22×C4—121no—
394C22×C4C2×D1211yescharacteristic
395C22×C4#23511yescharacteristic
396C22×C4D3×D411yescharacteristic
397C22×C4D3×D411yescharacteristic
398C22×C4D3×D411yescharacteristic
399C22×C4C2×#6711yescharacteristic
400SD16—242no—
401C2×C8—42no—
402C2×C8—41no—
403C2×C8—62no—
404C2×C8—122no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—42no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—61no—
411C4⋊C4—61no—
412C4⋊C4—122no—
413C4⋊C4—124no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C22⋊C4—124no—
419C22⋊C4—124no—
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—41no—
433C22×C3—41no—
434C22×C3—41no—
435C22×C3—41no—
436C22×C3#39211yescharacteristic
437C22×C3#44811yescharacteristic
438C22×C3#44911yescharacteristic
439C22×C3#45011yescharacteristic
440C22×C3#45211yescharacteristic
441C22×C3#46511yescharacteristic
442C22×C3#46811yescharacteristic
443C2×D3—24no—
444C2×D3—24no—
445C2×D3—24no—
446C2×D3—42no—
447C2×D3—42no—
448C2×D3—42no—
449C2×D3—42no—
450C2×D3—42no—
451C2×D3—42no—
452C2×D3—42no—
453C2×D3—42no—
454C2×D3—42no—
455C2×D3—42no—
456C2×D3—42no—
457C2×D3—42no—
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—82no—
467C3×C4—81no—
468C3×C4—81no—
469C3×C4B(4,𝔽2)12yes—
470Dic3—42no—
471Dic3—82no—
472Dic3—81no—
473C23—41no—
474C23—41no—
475C23—62no—
476C23—62no—
477C23—62no—
478C23—62no—
479C23—62no—
480C23—62no—
481C23—62no—
482C23—62no—
483C23—62no—
484C23—121no—
485C23—121no—
486C23—121no—
487C23—121no—
488C23—121no—
489C23—121no—
490C23—122no—
491C23—122no—
492C23—124no—
493C23—124no—
494C23—124no—
495C23—124no—
496C23—121no—
497C23—121no—
498C23—121no—
499C23—121no—
500C23—121no—
501C23—121no—
502C23—121no—
503C23#88211yescharacteristic
504C23#88411yescharacteristic
505C23D3×SD1611yescharacteristic
506C23#91411yescharacteristic
507C23#91811yescharacteristic
508C23#91911yescharacteristic
509C23#93711yescharacteristic
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—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—62no—
533C2×C4—62no—
534C2×C4—62no—
535C2×C4—124no—
536C2×C4—124no—
537C2×C4—124no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—124no—
543C2×C4—122no—
544C2×C4—122no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—122no—
549C2×C4#88211yescharacteristic
550C2×C4#88411yescharacteristic
551C2×C4#93011yescharacteristic
552C2×C4#94011yescharacteristic
553C8—81no—
554C8—122no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3#200011yescharacteristic
562C2×C3#201011yescharacteristic
563C2×C3#202111yescharacteristic
564D3—44no—
565D3—161no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—64no—
577C22—64no—
578C22—64no—
579C22—122no—
580C22—122no—
581C22—122no—
582C22—122no—
583C22—122no—
584C22—122no—
585C22—122no—
586C22—122no—
587C22—122no—
588C22—122no—
589C22—122no—
590C22—122no—
591C22—242no—
592C22—242no—
593C22—241no—
594C22—241no—
595C22—241no—
596C22#480611yescharacteristic
597C22#480711yescharacteristic
598C22#481011yescharacteristic
599C22#481211yescharacteristic
600C22#496011yescharacteristic
601C22#516911yescharacteristic
602C22#518711yescharacteristic
603C4—21no—
604C4—21no—
605C4—21no—
606C4—82no—
607C4—81no—
608C4—81no—
609C4—122no—
610C4—242no—
611C4—241no—
612C4#474112yes—
613C3#1032211yescharacteristic, complemented, Sylow
614C2—21no—
615C2—21no—
616C2—21no—
617C2—21no—
618C2—21no—
619C2—21no—
620C2—124no—
621C2—481no—
622C2#7003711yescharacteristic
623C2#7027211yescharacteristic
624C2#7040611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1520C2
socleC22×C3#392
prime radical#74048C2
hypercenter#1520D3
hypocenterC3#10322
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1520

Complements

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

NH ≅ G/Ntype
1. #82834C2semidirect
2. #82736C2semidirect
4. C3×#1520C2semidirect
5. #69955C2semidirect
6. #69944C2semidirect
7. #67625C2semidirect
38. #1520D3semidirect
42. C22×C3×Q8D4semidirect
62. C2×C3×C4⋊C4D4semidirect
80. C22×C3⋊C8D4semidirect
90. C3⋊#95D4semidirect
131. C23×D3SD16semidirect
206. C22×Q8D12semidirect
239. C2×C4⋊C4D12semidirect
613. C3#10322semidirect

Maximal subgroups

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

subgroupindex
#828342
#827362
#740482
C3×#15202
#699552
#699442
#676252
#103223