Finite Group Databasebeta

← #10698058

Subgroups — #10698058

Order 768.

988 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 988 conjugacy classes of subgroups — too many to draw. Only its 103 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#78612C211yesmaximal, characteristic, complemented
2#74049C211yesmaximal, characteristic, complemented
3#74048C211yesmaximal, characteristic
4#69955C211yesmaximal, characteristic, complemented
5#69510C211yesmaximal, characteristic, complemented
6#69509C211yesmaximal, characteristic, complemented
7C3×#1289C211yesmaximal, characteristic, complemented
8#10322—31nomaximal, Sylow
9C23×C4×D3C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C2×#924C2211yescharacteristic
12C2×#897—21no—
13C2×#895—21no—
14C2×#882—21no—
15C2×C3×C22⋊C8C2211yescharacteristic
16C3×#380C2211yescharacteristic
17C3×#380C2211yescharacteristic
18C2×#830C2211yescharacteristic
19#5114—22no—
20C2×(C2×D3)⋊C8—22no—
21#4987—21no—
22#4987—21no—
23#4958—21no—
24#4955—21no—
25#4950—21no—
26#4738—21no—
27#4735—21no—
28#4731—21no—
29#4721—21no—
30#4668—22no—
31#4665—22no—
32#4452—22no—
33#4451—22no—
34#3187—31no—
35#2848—31no—
36#2847—31no—
37#1779—31no—
38#1520—31no—
39#1519—31no—
40#1289D311yescharacteristic, complemented
41C24×D3D411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C22×#67—21no—
44C23×Dic3D411yescharacteristic
45C22×D12—22no—
46C22×D12D411yescharacteristic
47C22×C4×D3—21no—
48C22×C4×D3—21no—
49C22×C4×D3—21no—
50C22×C4×D3—21no—
51C22×C4×D3—21no—
52C22×C4×D3—21no—
53C22×C4×D3—44no—
54C22×Dic6D411yescharacteristic, complemented
55C22×C3×C8D412yescomplemented
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—21no—
60C2×#217—21no—
61#930—21no—
62#930—21no—
63#930—21no—
64#930—21no—
65C2×(C2×D3)⋊C4—22no—
66C2×(C2×D3)⋊C4—22no—
67C2×(C2×D3)⋊C4—22no—
68C2×(C2×D3)⋊C4—22no—
69C2×(C2×D3)⋊C4—21no—
70C2×C4⋊Dic3—22no—
71C2×C4⋊Dic3D411yescharacteristic
72C2×C4⋊Dic3D411yescharacteristic, complemented
73#924—21no—
74#924—21no—
75#924—21no—
76#924—21no—
77C2×Dic3⋊C4—21no—
78C22×C3⋊C8—21no—
79C2×#204—22no—
80#897—44no—
81#895—44no—
82#882—42no—
83#882—42no—
84C3×C22⋊C8—21no—
85C3×C22⋊C8—21no—
86C3×#95D412yes—
87C3×#95D412yescomplemented
88C3⋊#95—22no—
89C3⋊#95—22no—
90#830—21no—
91#830—21no—
92#821—22no—
93(C2×D3)⋊C8—44no—
94#817—22no—
95#816—22no—
96C24×C4—31no—
97C2×C22⋊Q8—31no—
98C2×C22⋊Q8—61no—
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—61no—
113#385—62no—
114#380—61no—
115#380—61no—
116#380C2×D311yescharacteristic
117#380C2×D311yescharacteristic
118#379—61no—
119#328—62no—
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
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—42no—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4#4111yescharacteristic
141C2×#67—41no—
142C2×#67—41no—
143C2×#67—41no—
144C2×#67—41no—
145C2×#67—41no—
146C2×#67—41no—
147C22×Dic3—21no—
148C22×Dic3—21no—
149C22×Dic3—21no—
150C22×Dic3—21no—
151C22×Dic3—21no—
152C22×Dic3—21no—
153C22×Dic3—42no—
154C2×D12—21no—
155C2×D12—21no—
156C2×D12—42no—
157C2×D12—44no—
158C2×D12—44no—
159C2×D12—44no—
160C2×C4×D3—24no—
161C2×C4×D3—24no—
162C2×C4×D3—24no—
163C2×C4×D3—42no—
164C2×C4×D3—42no—
165C2×C4×D3—42no—
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—41no—
173C2×C4×D3—41no—
174C2×C4×D3—41no—
175C2×C4×D3—41no—
176C2×Dic6—21no—
177C2×Dic6—21no—
178C2×Dic6—42no—
179C2×C3×C8—22no—
180C2×C3×C8—42no—
181C3×C4⋊C4—42no—
182C3×C4⋊C4—42no—
183C3×C22⋊C4—41no—
184C3×C22⋊C4—41no—
185C3×C22⋊C4—41no—
186C3×C22⋊C4—41no—
187#217—41no—
188#217—41no—
189(C2×D3)⋊C4—44no—
190(C2×D3)⋊C4—44no—
191(C2×D3)⋊C4—44no—
192(C2×D3)⋊C4—44no—
193(C2×D3)⋊C4—41no—
194(C2×D3)⋊C4—41no—
195C4⋊Dic3—21no—
196C4⋊Dic3—21no—
197C4⋊Dic3—21no—
198C4⋊Dic3—21no—
199C4⋊Dic3—44no—
200Dic3⋊C4—41no—
201Dic3⋊C4—41no—
202C2×C3⋊C8—42no—
203C2×C3⋊C8—41no—
204#204—82no—
205C25—31no—
206C22×Q8—31no—
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—61no—
221C22×C8D1212yescomplemented
222C22⋊Q8—61no—
223C22⋊Q8—61no—
224C22⋊Q8—61no—
225C22⋊Q8—61no—
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—31no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—61no—
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—62no—
257#95—62no—
258#95D1212yes—
259#95D1212yescomplemented
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—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4#12011yescharacteristic
310C2×C3×C4#12111yescharacteristic
311C2×C3×C4#12111yescharacteristic
312C2×C3×C4#12711yescharacteristic
313#67—81no—
314#67—81no—
315#67—81no—
316#67—81no—
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—22no—
320C2×Dic3—41no—
321C2×Dic3—41no—
322C2×Dic3—41no—
323C2×Dic3—41no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326C2×Dic3—41no—
327C2×Dic3—41no—
328C2×Dic3—41no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331D12—42no—
332D12—81no—
333D12—88no—
334C4×D3—44no—
335C4×D3—44no—
336C4×D3—44no—
337C4×D3—44no—
338Dic6—42no—
339Dic6—81no—
340C3×C8—42no—
341C3⋊C8—81no—
342C24—32no—
343C24—61no—
344C24—61no—
345C24—61no—
346C24—61no—
347C24—61no—
348C24—61no—
349C24—61no—
350C24—122no—
351C24—122no—
352C24D3×D411yescharacteristic
353C2×Q8—61no—
354C2×Q8—61no—
355C2×Q8—122no—
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—22no—
369C22×C4—22no—
370C22×C4—42no—
371C22×C4—64no—
372C22×C4—64no—
373C22×C4—64no—
374C22×C4—61no—
375C22×C4—61no—
376C22×C4—61no—
377C22×C4—61no—
378C22×C4—61no—
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×C4C2×D1211yescharacteristic
396C22×C4C2×D1211yescharacteristic
397C22×C4D3×D411yescharacteristic
398C22×C4D3×D411yescharacteristic
399C22×C4#23911yescharacteristic
400SD16—242no—
401C2×C8—22no—
402C2×C8—42no—
403C2×C8—122no—
404C2×C8—121no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—61no—
408C4⋊C4—61no—
409C4⋊C4—61no—
410C4⋊C4—61no—
411C4⋊C4—124no—
412C4⋊C4—121no—
413C4⋊C4—121no—
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—82no—
468C3×C4B(4,𝔽2)12yes—
469Dic3—42no—
470Dic3—81no—
471Dic3—81no—
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#88211yescharacteristic
505C23#89511yescharacteristic
506C23#91111yescharacteristic
507C23D3×SD1611yescharacteristic
508C23#91611yescharacteristic
509C23#91811yescharacteristic
510Q8—122no—
511Q8—241no—
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—42no—
531C2×C4—62no—
532C2×C4—62no—
533C2×C4—62no—
534C2×C4—124no—
535C2×C4—124no—
536C2×C4—124no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—124no—
542C2×C4—122no—
543C2×C4—122no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4#87411yescharacteristic
550C2×C4#88211yescharacteristic
551C2×C4#89511yescharacteristic
552C2×C4#89511yescharacteristic
553C8—42no—
554C8—241no—
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#467111yescharacteristic
597C22#477611yescharacteristic
598C22#478111yescharacteristic
599C22#480611yescharacteristic
600C22#480811yescharacteristic
601C22#486411yescharacteristic
602C22#488611yescharacteristic
603C4—21no—
604C4—21no—
605C4—21no—
606C4—82no—
607C4—82no—
608C4—122no—
609C4—241no—
610C4—241no—
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#6870311yescharacteristic
623C2#6888711yescharacteristic
624C2#6896511yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1289C2
socleC22×C3#392
prime radical#74048C2
hypercenter#1289D3
hypocenterC3#10322
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1289

Complements

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

NH ≅ G/Ntype
1. #78612C2semidirect
2. #74049C2semidirect
4. #69955C2semidirect
5. #69510C2semidirect
6. #69509C2semidirect
7. C3×#1289C2semidirect
40. #1289D3semidirect
54. C22×Dic6D4semidirect
55. C22×C3×C8D4semidirect
72. C2×C4⋊Dic3D4semidirect
87. C3×#95D4semidirect
131. C23×D3SD16semidirect
221. C22×C8D12semidirect
259. #95D12semidirect
613. C3#10322semidirect

Maximal subgroups

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

subgroupindex
#786122
#740492
#740482
#699552
#695102
#695092
C3×#12892
#103223