Finite Group Databasebeta

← #10698052

Subgroups — #10698052

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
1C3×#3187C211yesmaximal, characteristic, complemented
2#82834C211yesmaximal, characteristic, complemented
3#82723C211yesmaximal, characteristic
4#71690C211yesmaximal, characteristic, complemented
5#71675C211yesmaximal, characteristic, complemented
6#69955C211yesmaximal, characteristic, complemented
7#67676C211yesmaximal, characteristic, complemented
8#10322—31nomaximal, Sylow
9C23×C4×D3C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C2×C3×#121C2211yescharacteristic
12C2×#946—21no—
13C2×#938—21no—
14C2×#937—21no—
15C3×#483—21no—
16C3×#481—21no—
17C3×#465—22no—
18C3×#450—21no—
19C3×#448—21no—
20C2×C3×C22⋊C8C2211yescharacteristic
21#5220—21no—
22#5212C2211yescharacteristic
23#5212C2211yescharacteristic
24C2×#830C2211yescharacteristic
25C2×(C2×D3)⋊C8—22no—
26#4989—22no—
27#4987—21no—
28#4987—21no—
29#4956—22no—
30#4955—21no—
31#4946—21no—
32#4538—22no—
33#4537—22no—
34#3187D311yescharacteristic, complemented
35#2848—31no—
36#2847—31no—
37#1779—31no—
38#1520—31no—
39#1519—31no—
40#1289—31no—
41C24×D3D411yescharacteristic
42C22×C3×Q8D411yescharacteristic, complemented
43C22×C3×D4—21no—
44C22×C3×D4D411yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C22×#67—22no—
47C23×Dic3D411yescharacteristic
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—
55C2×C3×SD16—22no—
56C22×C3×C8D412yescomplemented
57C3×C22⋊Q8—21no—
58C3×C22⋊Q8—21no—
59C3×C22⋊Q8—21no—
60C3×C22⋊Q8—21no—
61C3×#121—21no—
62C3×#121—21no—
63C3×#121—21no—
64C3×#121—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4D411yescharacteristic
67C2×C3×C4⋊C4D411yescharacteristic, complemented
68C2×C3×C22⋊C4—21no—
69C2×C3×C22⋊C4—21no—
70C2×#217—22no—
71C2×#217—21no—
72C2×#217—21no—
73#946—44no—
74#938—44no—
75#937—42no—
76#937—42no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—22no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3—21no—
82C2×Dic3⋊C4—22no—
83C22×C3⋊C8—21no—
84C3×#106—22no—
85C3×#103—22no—
86C3×#102—22no—
87C3×C22⋊C8—21no—
88C3×C22⋊C8—21no—
89C3⋊#95—22no—
90C3⋊#95—22no—
91C3⋊#95D412yes—
92C3⋊#95D412yescomplemented
93#830—21no—
94#830—21no—
95(C2×D3)⋊C8—44no—
96C24×C4—31no—
97C2×C22⋊Q8—61no—
98C2×C22⋊Q8C2×D311yescharacteristic
99C2×#121—61no—
100C2×#121C2×D311yescharacteristic
101C2×#120—61no—
102#483—21no—
103#481—21no—
104#465—22no—
105#450—21no—
106#448—21no—
107C2×C22⋊C8—31no—
108C2×C22⋊C8—62no—
109C2×C22⋊C8C2×D311yescharacteristic
110#386—62no—
111#386—61no—
112#385—62no—
113#385—61no—
114#380—31no—
115#380—31no—
116#380—61no—
117#380—61no—
118#379—61no—
119#328—62no—
120#327—62no—
121C24×C3—21no—
122C24×C3C2×D411yescharacteristic
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—
135C2×C3×D4—21no—
136C2×C3×D4—21no—
137C2×C3×D4—42no—
138C2×C3×D4—41no—
139C2×C3×D4—41no—
140C2×C3×D4—41no—
141C2×C3×D4—41no—
142C2×C3×D4—41no—
143C2×C3×D4—41no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—42no—
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4#4111yescharacteristic
154C2×#67—44no—
155C2×#67—44no—
156C2×#67—44no—
157C22×Dic3—21no—
158C22×Dic3—21no—
159C22×Dic3—21no—
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—42no—
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—
179C3×SD16—82no—
180C2×C3×C8—22no—
181C2×C3×C8—42no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—41no—
187C3×C4⋊C4—41no—
188C3×C22⋊C4—41no—
189C3×C22⋊C4—41no—
190C3×C22⋊C4—41no—
191C3×C22⋊C4—41no—
192#217—44no—
193#217—41no—
194#217—41no—
195#217—41no—
196#217—41no—
197(C2×D3)⋊C4—44no—
198(C2×D3)⋊C4—44no—
199(C2×D3)⋊C4—44no—
200C4⋊Dic3—42no—
201C4⋊Dic3—42no—
202Dic3⋊C4—44no—
203C2×C3⋊C8—42no—
204C2×C3⋊C8—41no—
205C25—31no—
206C22×Q8#6711yescharacteristic, complemented
207C22×D4—21no—
208C22×D4—62no—
209C22×D4#6711yescharacteristic
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—22no—
220C22×C8—61no—
221C22×C8#6712yescomplemented
222C22⋊Q8—21no—
223C22⋊Q8—21no—
224C22⋊Q8—21no—
225C22⋊Q8—21no—
226C22⋊Q8—124no—
227#121—21no—
228#121—21no—
229#121—21no—
230#121—21no—
231#121—124no—
232#120—122no—
233#120—122no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—62no—
236C2×C4⋊C4—61no—
237C2×C4⋊C4—61no—
238C2×C4⋊C4#6711yescharacteristic
239C2×C4⋊C4#6711yescharacteristic, 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—22no—
249#103—22no—
250#102—22no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253C22⋊C8—61no—
254C22⋊C8—61no—
255C22⋊C8—124no—
256#95—32no—
257#95—32no—
258#95—62no—
259#95—62no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3—42no—
263C23×C3—42no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3—41no—
267C23×C3#12011yescharacteristic
268C23×C3#12011yescharacteristic
269C23×C3#12111yescharacteristic
270C23×C3C2×SD1611yescharacteristic
271C23×C3#13511yescharacteristic
272C23×C3AGL(1,ℤ/8)11yescharacteristic
273C23×C3#13711yescharacteristic
274C22×D3—22no—
275C22×D3—22no—
276C22×D3—22no—
277C22×D3—22no—
278C22×D3—22no—
279C22×D3—22no—
280C22×D3—22no—
281C22×D3—22no—
282C22×D3—22no—
283C22×D3—44no—
284C22×D3—44no—
285C22×D3—44no—
286C22×D3—44no—
287C22×D3—41no—
288C22×D3—41no—
289C22×D3—41no—
290C22×D3—41no—
291C22×D3—41no—
292C22×D3—41no—
293C22×D3—41no—
294C22×D3—41no—
295C22×D3—41no—
296C22×D3—41no—
297C3×Q8—42no—
298C3×Q8—81no—
299C3×D4—42no—
300C3×D4—81no—
301C3×D4—81no—
302C3×D4—81no—
303C3×D4—81no—
304C3×D4—81no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4#12011yescharacteristic
319C2×C3×C4#12111yescharacteristic
320C2×C3×C4#12111yescharacteristic
321C2×C3×C4#12711yescharacteristic
322#67—88no—
323C2×Dic3—22no—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—41no—
327C2×Dic3—41no—
328C2×Dic3—41no—
329C2×Dic3—41no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3—42no—
336C4×D3—44no—
337C4×D3—44no—
338C4×D3—44no—
339C4×D3—44no—
340C3×C8—42no—
341C3⋊C8—81no—
342C24—21no—
343C24—32no—
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—21no—
357C2×D4—21no—
358C2×D4—42no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—124no—
366C2×D4—124no—
367C2×D4—124no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
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—62no—
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×C4D3×D411yescharacteristic
395C22×C4D3×D411yescharacteristic
396C22×C4#23711yescharacteristic
397C22×C4C2×#6711yescharacteristic
398C22×C4C2×#6711yescharacteristic
399C22×C4C2×#6711yescharacteristic
400SD16—82no—
401C2×C8—22no—
402C2×C8—42no—
403C2×C8—122no—
404C2×C8—121no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—122no—
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—82no—
437C22×C3—82no—
438C22×C3—81no—
439C22×C3—81no—
440C22×C3—81no—
441C22×C3#39211yescharacteristic
442C22×C3#44811yescharacteristic
443C22×C3#44911yescharacteristic
444C22×C3#45011yescharacteristic
445C22×C3#45211yescharacteristic
446C22×C3#46511yescharacteristic
447C22×C3#46811yescharacteristic
448C2×D3—24no—
449C2×D3—24no—
450C2×D3—24no—
451C2×D3—42no—
452C2×D3—42no—
453C2×D3—42no—
454C2×D3—42no—
455C2×D3—42no—
456C2×D3—42no—
457C2×D3—42no—
458C2×D3—42no—
459C2×D3—42no—
460C2×D3—42no—
461C2×D3—42no—
462C2×D3—42no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—81no—
467C3×C4—81no—
468C3×C4—81no—
469C3×C4B(4,𝔽2)12yes—
470Dic3—42no—
471Dic3—82no—
472Dic3—82no—
473C23—41no—
474C23—41no—
475C23—42no—
476C23—42no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—62no—
481C23—62no—
482C23—62no—
483C23—62no—
484C23—62no—
485C23—62no—
486C23—62no—
487C23—62no—
488C23—62no—
489C23—121no—
490C23—121no—
491C23—121no—
492C23—121no—
493C23—121no—
494C23—121no—
495C23—124no—
496C23—124no—
497C23—124no—
498C23—124no—
499C23—121no—
500C23—121no—
501C23—121no—
502C23—121no—
503C23D3×SD1611yescharacteristic
504C23#91411yescharacteristic
505C23#91511yescharacteristic
506C23#91611yescharacteristic
507C23#93711yescharacteristic
508C23#93711yescharacteristic
509C23#93811yescharacteristic
510Q8—42no—
511Q8—81no—
512D4—42no—
513D4—81no—
514D4—81no—
515D4—81no—
516D4—81no—
517D4—81no—
518D4—248no—
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—41no—
528C2×C4—41no—
529C2×C4—41no—
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—122no—
546C2×C4—122no—
547C2×C4—122no—
548C2×C4—122no—
549C2×C4#93911yescharacteristic
550C2×C4#93911yescharacteristic
551C2×C4#94011yescharacteristic
552C2×C4#95311yescharacteristic
553C8—42no—
554C8—241no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—161no—
562C2×C3#200011yescharacteristic
563C2×C3#201011yescharacteristic
564C2×C3#202111yescharacteristic
565D3—44no—
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—82no—
580C22—82no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—122no—
585C22—122no—
586C22—122no—
587C22—122no—
588C22—122no—
589C22—122no—
590C22—122no—
591C22—122no—
592C22—122no—
593C22—122no—
594C22—122no—
595C22—122no—
596C22#516911yescharacteristic
597C22#517011yescharacteristic
598C22#517111yescharacteristic
599C22#517211yescharacteristic
600C22#517311yescharacteristic
601C22#517411yescharacteristic
602C22#522311yescharacteristic
603C4—21no—
604C4—21no—
605C4—21no—
606C4—81no—
607C4—81no—
608C4—81no—
609C4—122no—
610C4—242no—
611C4—242no—
612C4#524412yes—
613C3#1032211yescharacteristic, complemented, Sylow
614C2—21no—
615C2—21no—
616C2—21no—
617C2—21no—
618C2—21no—
619C2—21no—
620C2—124no—
621C2—161no—
622C2#7185111yescharacteristic
623C2#7205411yescharacteristic
624C2#7211811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#3187C2
socleC22×C3#392
prime radical#82723C2
hypercenter#3187D3
hypocenterC3#10322
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3187

Complements

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

NH ≅ G/Ntype
1. C3×#3187C2semidirect
2. #82834C2semidirect
4. #71690C2semidirect
5. #71675C2semidirect
6. #69955C2semidirect
7. #67676C2semidirect
34. #3187D3semidirect
42. C22×C3×Q8D4semidirect
56. C22×C3×C8D4semidirect
67. C2×C3×C4⋊C4D4semidirect
92. C3⋊#95D4semidirect
131. C23×D3SD16semidirect
206. C22×Q8#67semidirect
221. C22×C8#67semidirect
239. C2×C4⋊C4#67semidirect
613. C3#10322semidirect

Maximal subgroups

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

subgroupindex
C3×#31872
#828342
#827232
#716902
#716752
#699552
#676762
#103223