Finite Group Databasebeta

← #10698056

Subgroups — #10698056

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#82723C211yesmaximal, characteristic
2#82630C211yesmaximal, characteristic, complemented
3#74049C211yesmaximal, characteristic, complemented
4C3×#1519C211yesmaximal, characteristic, complemented
5#69955C211yesmaximal, characteristic, complemented
6#69946C211yesmaximal, characteristic, complemented
7#67625C211yesmaximal, characteristic, complemented
8#10322—31nomaximal, Sylow
9C23×C4×D3C2211yescharacteristic
10C2×C3×#121C2211yescharacteristic
11C2×#938—21no—
12C2×#937—21no—
13C2×#924C2211yescharacteristic
14C2×#897—21no—
15C2×C3×C22⋊C8C2211yescharacteristic
16C3×#380C2211yescharacteristic
17#5220—21no—
18#5218—21no—
19#5212C2211yescharacteristic
20C2×#830C2211yescharacteristic
21C2×(C2×D3)⋊C8—22no—
22#5041—22no—
23#5040—21no—
24#5034—21no—
25#4987—21no—
26#4987—21no—
27#4971—21no—
28#4956—22no—
29#4955—21no—
30#4950—21no—
31#4668—22no—
32C3×#327—22no—
33#4467—22no—
34#3187—31no—
35#2848—31no—
36#2847—31no—
37#1779—31no—
38#1520—31no—
39#1519D311yescharacteristic, complemented
40#1289—31no—
41C24×D3D411yescharacteristic
42C22×C3×D4—21no—
43C22×C3×D4D411yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C22×#67—22no—
46C23×Dic3D411yescharacteristic
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×C8—21no—
56C3×#121—21no—
57C3×#121—21no—
58C3×#121—21no—
59C3×#121—21no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4D411yescharacteristic
62C2×C3×C22⋊C4—21no—
63C2×C3×C22⋊C4—21no—
64C2×#217—22no—
65C2×#217—21no—
66C2×#217—21no—
67#938—44no—
68#937—42no—
69#937—42no—
70C2×#214—22no—
71C2×(C2×D3)⋊C4—22no—
72C2×(C2×D3)⋊C4—22no—
73C2×(C2×D3)⋊C4—22no—
74C2×C4⋊Dic3—22no—
75C2×C4⋊Dic3—21no—
76C2×C4⋊Dic3D411yescharacteristic, complemented
77#924—21no—
78#924—21no—
79#924—21no—
80#924—21no—
81C2×Dic3⋊C4—21no—
82C22×C3⋊C8D412yescomplemented
83#897—44no—
84C3×C22⋊C8—21no—
85C3×C22⋊C8—21no—
86C3×#95D412yescomplemented
87#832—22no—
88C3⋊#95—22no—
89C3⋊#95—22no—
90C3⋊#95D412yes—
91#830—21no—
92#830—21no—
93(C2×D3)⋊C8—44no—
94#810—22no—
95#808—22no—
96C24×C4—31no—
97C2×C22⋊Q8—31no—
98C2×C22⋊Q8—61no—
99C2×#121—61no—
100C2×#121C2×D311yescharacteristic
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—62no—
111#386—61no—
112#385—61no—
113#385—62no—
114#380—31no—
115#380—61no—
116#380—61no—
117#380C2×D311yescharacteristic
118#379—61no—
119#328—62no—
120#327—22no—
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×D4—21no—
133C2×C3×D4—21no—
134C2×C3×D4—42no—
135C2×C3×D4—41no—
136C2×C3×D4—41no—
137C2×C3×D4—41no—
138C2×C3×D4—41no—
139C2×C3×D4—41no—
140C2×C3×D4—41no—
141C22×C3×C4—21no—
142C22×C3×C4—22no—
143C22×C3×C4—42no—
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4#4111yescharacteristic
150C2×#67—44no—
151C2×#67—44no—
152C2×#67—44no—
153C22×Dic3—21no—
154C22×Dic3—21no—
155C22×Dic3—21no—
156C22×Dic3—21no—
157C22×Dic3—22no—
158C22×Dic3—21no—
159C22×Dic3—42no—
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—42no—
180C2×C3×C8—41no—
181C3×C4⋊C4—21no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—42no—
184C3×C22⋊C4—41no—
185C3×C22⋊C4—41no—
186C3×C22⋊C4—41no—
187C3×C22⋊C4—41no—
188#217—44no—
189#217—41no—
190#217—41no—
191#217—41no—
192#217—41no—
193#214—82no—
194(C2×D3)⋊C4—44no—
195(C2×D3)⋊C4—44no—
196(C2×D3)⋊C4—44no—
197C4⋊Dic3—21no—
198C4⋊Dic3—21no—
199C4⋊Dic3—44no—
200C4⋊Dic3—42no—
201Dic3⋊C4—41no—
202Dic3⋊C4—41no—
203C2×C3⋊C8—22no—
204C2×C3⋊C8—42no—
205C25—31no—
206C22×Q8—31no—
207C22×D4—21no—
208C22×D4—62no—
209C22×D4D1211yescharacteristic
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—61no—
223C22⋊Q8—61no—
224C22⋊Q8—61no—
225C22⋊Q8—61no—
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—31no—
236C2×C4⋊C4—62no—
237C2×C4⋊C4—61no—
238C2×C4⋊C4—61no—
239C2×C4⋊C4D1211yescharacteristic
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#6712yescomplemented
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×D4—42no—
298C3×D4—81no—
299C3×D4—81no—
300C3×D4—81no—
301C3×D4—81no—
302C3×D4—81no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—41no—
312C2×C3×C4—42no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4#12011yescharacteristic
316C2×C3×C4#12111yescharacteristic
317C2×C3×C4#12111yescharacteristic
318C2×C3×C4#12711yescharacteristic
319#67—88no—
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—42no—
330C2×Dic3—42no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333C2×Dic3—42no—
334C4×D3—44no—
335C4×D3—44no—
336C4×D3—44no—
337C4×D3—44no—
338Dic6—42no—
339Dic6—81no—
340C3×C8—81no—
341C3⋊C8—42no—
342C24—21no—
343C24—32no—
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—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—22no—
370C22×C4—42no—
371C22×C4—64no—
372C22×C4—64no—
373C22×C4—64no—
374C22×C4—61no—
375C22×C4—61no—
376C22×C4—61no—
377C22×C4—62no—
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×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—61no—
409C4⋊C4—61no—
410C4⋊C4—122no—
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—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—82no—
467C3×C4—81no—
468C3×C4B(4,𝔽2)12yes—
469Dic3—42no—
470Dic3—82no—
471Dic3—81no—
472Dic3—81no—
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—
503C23#88211yescharacteristic
504C23#88411yescharacteristic
505C23#91111yescharacteristic
506C23#91211yescharacteristic
507C23D3×SD1611yescharacteristic
508C23#91511yescharacteristic
509C23#93711yescharacteristic
510Q8—122no—
511Q8—241no—
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—42no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
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—122no—
545C2×C4—122no—
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—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#477611yescharacteristic
597C22#477811yescharacteristic
598C22#478011yescharacteristic
599C22#478311yescharacteristic
600C22#496011yescharacteristic
601C22#515811yescharacteristic
602C22#517011yescharacteristic
603C4—21no—
604C4—21no—
605C4—21no—
606C4—82no—
607C4—81no—
608C4—122no—
609C4—242no—
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—161no—
622C2#7003711yescharacteristic
623C2#7020511yescharacteristic
624C2#7040611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1519C2
socleC22×C3#392
prime radical#82723C2
hypercenter#1519D3
hypocenterC3#10322
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1519

Complements

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

NH ≅ G/Ntype
2. #82630C2semidirect
3. #74049C2semidirect
4. C3×#1519C2semidirect
5. #69955C2semidirect
6. #69946C2semidirect
7. #67625C2semidirect
39. #1519D3semidirect
54. C22×Dic6D4semidirect
76. C2×C4⋊Dic3D4semidirect
82. C22×C3⋊C8D4semidirect
86. C3×#95D4semidirect
131. C23×D3SD16semidirect
259. #95#67semidirect
613. C3#10322semidirect

Maximal subgroups

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

subgroupindex
#827232
#826302
#740492
C3×#15192
#699552
#699462
#676252
#103223