Finite Group Databasebeta

← #10695548

Subgroups — #10695548

Order 768.

1008 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 1008 conjugacy classes of subgroups — too many to draw. Only its 85 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×#3006C211yesmaximal, characteristic, complemented
2#84096C211yesmaximal, characteristic, complemented
3#82708C211yesmaximal, characteristic, complemented
4#72138C211yesmaximal, characteristic, complemented
5#71674C211yesmaximal, characteristic, complemented
6#69967C211yesmaximal, characteristic, complemented
7#67682C211yesmaximal, characteristic, complemented
8#9938—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×C3×AGL(1,ℤ/8)—21no—
11C2×C3×#121C2211yescharacteristic
12C2×#937—21no—
13C2×#937—21no—
14C2×#935—21no—
15C22×(C2×D3)⋊C4C2211yescharacteristic
16C3×#449—22no—
17C3×#448—21no—
18C3×#447—21no—
19C3×#417—21no—
20C3×#407C2211yescharacteristic
21C2×#837—22no—
22#5231—21no—
23#5223—21no—
24#5223—21no—
25#5220—21no—
26#5213—22no—
27#5211C2211yescharacteristic
28C4×#217C2211yescharacteristic
29#5207C2211yescharacteristic
30#4993—21no—
31#4957—22no—
32#4537—22no—
33#4468—22no—
34#3006D311yescharacteristic, complemented
35#2539—31no—
36#2404—31no—
37#1789—31no—
38#1510—31no—
39#1462—31no—
40#1284—31no—
41C24×D3D411yescharacteristic
42C2×C3×#41—21no—
43C2×C3×#41—21no—
44C2×C3×#41—21no—
45C2×C3×#41—21no—
46C2×C3×#41—44no—
47C22×C3×Q8D411yescharacteristic, complemented
48C22×C3×D4—21no—
49C22×C3×D4—21no—
50C22×C3×D4D411yescharacteristic, complemented
51C22×C3×D4D411yescharacteristic, complemented
52C23×C3×C4—21no—
53C23×C3×C4C2311yescharacteristic
54C22×#67—22no—
55C22×#67—22no—
56C23×Dic3D411yescharacteristic
57C3×AGL(1,ℤ/8)—44no—
58C2×C3×SD16—22no—
59C2×C3×D8—22no—
60C2×C3×M16—21no—
61C3×#121—21no—
62C3×#121—21no—
63C3×#121—21no—
64C3×#121—21no—
65C2×C3×C4⋊C4D411yescharacteristic, complemented
66C2×C3×C22⋊C4—21no—
67C2×#217—21no—
68C2×#217—22no—
69C2×#217—21no—
70C2×#217—22no—
71#937—44no—
72#937—44no—
73#935—44no—
74C2×(C2×D3)⋊C4—22no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—21no—
77C2×(C2×D3)⋊C4—21no—
78C2×(C2×D3)⋊C4—21no—
79C2×(C2×D3)⋊C4—21no—
80C2×(C2×D3)⋊C4—21no—
81C2×(C2×D3)⋊C4—21no—
82C2×Dic3⋊C4—22no—
83C2×C4×Dic3D412yescomplemented
84C2×#208—21no—
85C3×#103—22no—
86C3×#102—22no—
87C3×C22⋊C8—21no—
88C3×C22⋊C8D412yescomplemented
89#837—44no—
90C3⋊#95—22no—
91C3⋊#95—21no—
92C3⋊#95—21no—
93C3⋊#95D412yescomplemented
94#830—21no—
95#830—21no—
96C22×#41C2×D311yescharacteristic
97C2×AGL(1,ℤ/8)—21no—
98C2×#123—61no—
99C2×#121C2×D311yescharacteristic
100C2×#120—61no—
101C2×#120—61no—
102C22×C22⋊C4—31no—
103#449—22no—
104#448—21no—
105#447—21no—
106C2×#104—62no—
107#417—21no—
108#407—31no—
109#407C2×D311yescharacteristic
110#399—61no—
111#393—61no—
112#392—61no—
113#392—61no—
114#390—62no—
115#386—62no—
116#386—61no—
117#381—31no—
118C4×C22⋊C4—31no—
119#327—62no—
120#323—62no—
121C24×C3—21no—
122C24×C3—21no—
123C24×C3C2×D411yescharacteristic
124C23×D3—21no—
125C23×D3—21no—
126C23×D3—21no—
127C23×D3—42no—
128C23×D3—42no—
129C23×D3—42no—
130C3×#41—24no—
131C3×#41—24no—
132C3×#41—42no—
133C3×#41—42no—
134C3×#41—42no—
135C3×#41—42no—
136C3×#41—42no—
137C3×#41—42no—
138C2×C3×Q8—21no—
139C2×C3×Q8—21no—
140C2×C3×Q8—42no—
141C2×C3×D4—24no—
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4—21no—
145C2×C3×D4—21no—
146C2×C3×D4—42no—
147C2×C3×D4—42no—
148C2×C3×D4—42no—
149C2×C3×D4—41no—
150C2×C3×D4—41no—
151C2×C3×D4—42no—
152C2×C3×D4—42no—
153C2×C3×D4—41no—
154C2×C3×D4—41no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—42no—
158C22×C3×C4—42no—
159C22×C3×C4—42no—
160C22×C3×C4—41no—
161C22×C3×C4—41no—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4#4111yescharacteristic
168C2×#67—44no—
169C2×#67—44no—
170C2×#67—44no—
171C2×#67—44no—
172C2×#67—82no—
173C2×#67—82no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—21no—
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—42no—
180C3×SD16—44no—
181C3×D8—44no—
182C3×M16—42no—
183C2×C3×C8—22no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4—21no—
186C3×C22⋊C4—41no—
187C3×C22⋊C4—41no—
188#217—24no—
189#217—44no—
190#217—42no—
191#217—82no—
192(C2×D3)⋊C4—44no—
193(C2×D3)⋊C4—44no—
194(C2×D3)⋊C4—41no—
195(C2×D3)⋊C4—41no—
196(C2×D3)⋊C4—41no—
197(C2×D3)⋊C4—41no—
198Dic3⋊C4—44no—
199C4×Dic3—42no—
200#208—42no—
201C2×C3⋊C8—41no—
202C25—31no—
203C2×#41—21no—
204C2×#41—21no—
205C2×#41—21no—
206C2×#41—21no—
207C2×#41—44no—
208C22×Q8#6711yescharacteristic, complemented
209C22×D4—21no—
210C22×D4—21no—
211C22×D4—62no—
212C22×D4—62no—
213C22×D4#6711yescharacteristic, complemented
214C22×D4#6711yescharacteristic, complemented
215C23×C4—21no—
216C23×C4—31no—
217C23×C4C22×D311yescharacteristic
218AGL(1,ℤ/8)—44no—
219C2×SD16—22no—
220C2×D8—22no—
221C2×M16—21no—
222C2×M16—61no—
223#123—124no—
224#121—21no—
225#121—21no—
226#121—21no—
227#121—21no—
228#120—124no—
229#120—124no—
230C2×C4⋊C4—62no—
231C2×C4⋊C4#6711yescharacteristic, complemented
232C2×C22⋊C4—21no—
233C2×C22⋊C4—61no—
234C2×C22⋊C4—62no—
235C2×C22⋊C4—62no—
236C2×C22⋊C4—62no—
237C2×C22⋊C4—61no—
238C2×C22⋊C4—61no—
239C2×C22⋊C4—61no—
240C2×C22⋊C4—61no—
241C2×C22⋊C4—61no—
242C2×C22⋊C4—61no—
243C2×C22⋊C4—61no—
244C2×C22⋊C4—62no—
245C2×C42—32no—
246#104—124no—
247#103—22no—
248#102—22no—
249C22⋊C8—21no—
250C22⋊C8—61no—
251C22⋊C8—61no—
252C22⋊C8#6712yescomplemented
253#95—32no—
254#95—62no—
255#95—61no—
256#95—61no—
257C23×C3—22no—
258C23×C3—22no—
259C23×C3—22no—
260C23×C3—41no—
261C23×C3—42no—
262C23×C3—41no—
263C23×C3—41no—
264C23×C3—41no—
265C23×C3—42no—
266C23×C3—41no—
267C23×C3—42no—
268C23×C3—41no—
269C23×C3—42no—
270C23×C3#12011yescharacteristic
271C23×C3#12011yescharacteristic
272C23×C3#12111yescharacteristic
273C22×D3—22no—
274C22×D3—22no—
275C22×D3—22no—
276C22×D3—42no—
277C22×D3—42no—
278C22×D3—41no—
279C22×D3—41no—
280C22×D3—41no—
281C22×D3—41no—
282C22×D3—42no—
283C22×D3—42no—
284C22×D3—88no—
285C22×D3—81no—
286C22×D3—81no—
287C3×Q8—22no—
288C3×Q8—22no—
289C3×Q8—41no—
290C3×Q8—41no—
291C3×D4—22no—
292C3×D4—22no—
293C3×D4—41no—
294C3×D4—41no—
295C3×D4—42no—
296C3×D4—44no—
297C3×D4—44no—
298C3×D4—81no—
299C3×D4—82no—
300C3×D4—82no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—44no—
309C2×C3×C4—44no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
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—
320#67—88no—
321C2×Dic3—24no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3—41no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3—81no—
331C3×C8—42no—
332C3⋊C8—81no—
333C24—21no—
334C24—21no—
335C24—61no—
336C24—61no—
337C24—61no—
338C24—122no—
339C24—122no—
340C24—122no—
341C24D3×D411yescharacteristic
342#41—24no—
343#41—24no—
344#41—42no—
345#41—42no—
346#41—42no—
347#41—42no—
348#41—42no—
349#41—42no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—42no—
353C2×D4—24no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—124no—
368C2×D4—124no—
369C2×D4—124no—
370C2×D4—124no—
371C2×D4—242no—
372C2×D4—242no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4—61no—
383C22×C4—61no—
384C22×C4—61no—
385C22×C4—122no—
386C22×C4D3×D411yescharacteristic
387C22×C4D3×D411yescharacteristic
388C22×C4#23711yescharacteristic
389C22×C4C2×#6711yescharacteristic
390C22×C4C2×#6711yescharacteristic
391C22×C4C2×#6711yescharacteristic
392SD16—44no—
393D8—44no—
394M16—42no—
395M16—122no—
396C2×C8—22no—
397C2×C8—121no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—124no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—64no—
404C22⋊C4—124no—
405C22⋊C4—124no—
406C22⋊C4—124no—
407C22⋊C4—122no—
408C22⋊C4—121no—
409C22⋊C4—121no—
410C22⋊C4—121no—
411C22⋊C4—121no—
412C22⋊C4—242no—
413C42—122no—
414C22×C3—21no—
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—44no—
422C22×C3—44no—
423C22×C3—42no—
424C22×C3—82no—
425C22×C3—82no—
426C22×C3—82no—
427C22×C3—81no—
428C22×C3—81no—
429C22×C3—81no—
430C22×C3—81no—
431C22×C3—81no—
432C22×C3#39211yescharacteristic
433C22×C3#45312yes—
434C22×C3B(4,𝔽2)12yes—
435C2×D3—44no—
436C2×D3—44no—
437C2×D3—42no—
438C2×D3—82no—
439C2×D3—82no—
440C2×D3—82no—
441C2×D3—81no—
442C2×D3—81no—
443C2×D3—82no—
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—21no—
447C3×C4—21no—
448C3×C4—42no—
449C3×C4—81no—
450Dic3—44no—
451Dic3—82no—
452Dic3—81no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—41no—
457C23—41no—
458C23—42no—
459C23—41no—
460C23—41no—
461C23—42no—
462C23—42no—
463C23—41no—
464C23—41no—
465C23—42no—
466C23—62no—
467C23—62no—
468C23—62no—
469C23—122no—
470C23—122no—
471C23—121no—
472C23—121no—
473C23—121no—
474C23—122no—
475C23—121no—
476C23—122no—
477C23—241no—
478C23—241no—
479C23—248no—
480C23#93711yescharacteristic
481C23#93711yescharacteristic
482C23#93811yescharacteristic
483Q8—22no—
484Q8—22no—
485Q8—41no—
486Q8—41no—
487D4—22no—
488D4—22no—
489D4—41no—
490D4—41no—
491D4—42no—
492D4—44no—
493D4—44no—
494D4—81no—
495D4—82no—
496D4—82no—
497D4—248no—
498D4—248no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—64no—
514C2×C4—122no—
515C2×C4—122no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4—122no—
519C2×C4—122no—
520C2×C4—122no—
521C2×C4—121no—
522C2×C4—241no—
523C2×C4#93911yescharacteristic
524C2×C4#93911yescharacteristic
525C2×C4#94011yescharacteristic
526C2×C4#95311yescharacteristic
527C8—42no—
528C8—241no—
529C2×C3—22no—
530C2×C3—22no—
531C2×C3—41no—
532C2×C3—82no—
533C2×C3—161no—
534C2×C3#200211yescharacteristic
535C2×C3#201011yescharacteristic
536C2×C3#202111yescharacteristic
537D3—82no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—44no—
546C22—44no—
547C22—42no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—124no—
557C22—124no—
558C22—122no—
559C22—242no—
560C22—242no—
561C22—242no—
562C22—242no—
563C22—241no—
564C22—241no—
565C22#503212yes—
566C22#519812yes—
567C22#522311yescharacteristic
568C4—21no—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—42no—
573C4—81no—
574C4—124no—
575C4—242no—
576C4—241no—
577C3#993811yescharacteristic, complemented, Sylow
578C2—22no—
579C2—22no—
580C2—41no—
581C2—82no—
582C2—161no—
583C2—242no—
584C2#7167111yescharacteristic
585C2#7209411yescharacteristic
586C2#7211811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#3006C2
socleC22×C3#392
hypercenter#3006D3
hypocenterC3#9938
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3006

Complements

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

NH ≅ G/Ntype
1. C3×#3006C2semidirect
2. #84096C2semidirect
3. #82708C2semidirect
4. #72138C2semidirect
5. #71674C2semidirect
6. #69967C2semidirect
7. #67682C2semidirect
34. #3006D3semidirect
47. C22×C3×Q8D4semidirect
50. C22×C3×D4D4semidirect
51. C22×C3×D4D4semidirect
65. C2×C3×C4⋊C4D4semidirect
83. C2×C4×Dic3D4semidirect
88. C3×C22⋊C8D4semidirect
93. C3⋊#95D4semidirect
208. C22×Q8#67semidirect
213. C22×D4#67semidirect
214. C22×D4#67semidirect
231. C2×C4⋊C4#67semidirect
252. C22⋊C8#67semidirect
577. C3#9938semidirect

Maximal subgroups

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

subgroupindex
C3×#30062
#840962
#827082
#721382
#716742
#699672
#676822
#99383