Finite Group Databasebeta

← #11712606

Subgroups — #11712606

Order 768.

1149 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 1149 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
1C2×A4×#41C211yesmaximal, characteristic
2C2×C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
3C2×#5409C211yesmaximal, characteristic, complemented
4#73147C212yesmaximal, complemented
5#73144C212yesmaximal, complemented
6#20601—31nomaximal, Sylow
7A4×#41—22no—
8A4×#41C414yescomplemented
9C2×Q8×A4C2211yescharacteristic
10C2×D4×A4—21no—
11C2×D4×A4C2211yescharacteristic
12C22×C4×A4—21no—
13C22×C4×A4C2211yescharacteristic
14C22×SL(2,ℤ/4)—21no—
15C4⋊SL(2,ℤ/4)—21no—
16C4⋊SL(2,ℤ/4)C2212yes—
17#5409—22no—
18C2×A4⋊C8C2212yes—
19#5232—41nomaximal
20C23×#41D311yescharacteristic
21C2×#417—31no—
22C2×#417—31no—
23C2×#417—31no—
24C2×#417—31no—
25C2×#407—31no—
26C2×#380—31no—
27#1788—32no—
28#1788—32no—
29#1787—32no—
30#1786—32no—
31C23×A4—21no—
32C23×A4D411yescharacteristic
33C2×C3×#41—41no—
34Q8×A4—21no—
35Q8×A4C2×C412yescomplemented
36D4×A4—24no—
37D4×A4—21no—
38D4×A4C2×C412yescomplemented
39C2×C4×A4—21no—
40C2×C4×A4—24no—
41C2×C4×A4C2×C412yes—
42C2×C4×A4D411yescharacteristic
43C2×C4×A4D412yes—
44C2×C4×A4C2311yescharacteristic
45C2×SL(2,ℤ/4)—22no—
46C2×SL(2,ℤ/4)—42no—
47C2×C4⋊Dic3—41no—
48C2×#208—41no—
49A4⋊C8—22no—
50#835—42no—
51#832—42no—
52C22×#41—22no—
53C22×#41—31no—
54C22×#41—31no—
55C22×#41—34no—
56C22×#41—31no—
57C22×#41—31no—
58C22×#41—61no—
59C22×#41—61no—
60C22×#41—62no—
61C22×#41Dic314yescomplemented
62C23×Q8C2×D311yescharacteristic
63C23×D4—21no—
64C23×D4C2×D311yescharacteristic
65C24×C4—21no—
66C24×C4C2×D311yescharacteristic
67C22×M16—31no—
68C22×M16—31no—
69C22×C4⋊C4—31no—
70C22×C4⋊C4—31no—
71C22×C22⋊C4—61no—
72#417—62no—
73#417—64no—
74#417—62no—
75#417—64no—
76#417—61no—
77#417—61no—
78#417—61no—
79#417—61no—
80C2×#103—32no—
81C2×#103—32no—
82C2×#103—32no—
83C2×#103—32no—
84C2×#102—32no—
85C2×#102—32no—
86C2×#102—32no—
87C2×#102—32no—
88#407—62no—
89#407—62no—
90C2×C22⋊C8—32no—
91C2×C22⋊C8—32no—
92#380—32no—
93#380—32no—
94#380—61no—
95#380—61no—
96C2×#95—61no—
97C22×A4—21no—
98C22×A4—21no—
99C22×A4—22no—
100C22×A4—41no—
101C22×A4—41no—
102C22×A4C22⋊C412yes—
103C22×A4C2×D411yescharacteristic
104C3×#41—44no—
105C3×#41—82no—
106C2×C3×Q8—41no—
107C2×C3×D4—41no—
108C2×C3×D4—81no—
109C22×C3×C4—41no—
110C22×C3×C4—81no—
111C22×Dic3—81no—
112C4×A4—22no—
113C4×A4C22⋊C412yes—
114C4×A4C22×C411yescharacteristic
115C4×A4C2×D411yescharacteristic
116SL(2,ℤ/4)—42no—
117C4⋊Dic3—42no—
118C4⋊Dic3—81no—
119#208—82no—
120C2×C3⋊C8—42no—
121C25—21no—
122C25#6711yescharacteristic
123C2×#41—34no—
124C2×#41—32no—
125C2×#41—34no—
126C2×#41—32no—
127C2×#41—34no—
128C2×#41—34no—
129C2×#41—34no—
130C2×#41—34no—
131C2×#41—31no—
132C2×#41—31no—
133C2×#41—31no—
134C2×#41—61no—
135C2×#41—61no—
136C2×#41—61no—
137C2×#41—62no—
138C2×#41—62no—
139C2×#41—64no—
140C2×#41—64no—
141C2×#41—62no—
142C2×#41—62no—
143C2×#41—62no—
144C2×#41—62no—
145C2×#41—64no—
146C2×#41—64no—
147C2×#41—62no—
148C2×#41—62no—
149C2×#41—62no—
150C2×#41—62no—
151C2×#41—62no—
152C2×#41—62no—
153C2×#41—68no—
154C2×#41—68no—
155C2×#41S411yescharacteristic
156C22×Q8—21no—
157C22×Q8—31no—
158C22×Q8—31no—
159C22×Q8—32no—
160C22×Q8—61no—
161C22×Q8—61no—
162C22×Q8C2×Dic312yescomplemented
163C22×D4—24no—
164C22×D4—21no—
165C22×D4—31no—
166C22×D4—31no—
167C22×D4—32no—
168C22×D4—61no—
169C22×D4—61no—
170C22×D4—61no—
171C22×D4—61no—
172C22×D4—64no—
173C22×D4—61no—
174C22×D4—61no—
175C22×D4C2×Dic312yescomplemented
176C23×C4—21no—
177C23×C4—24no—
178C23×C4—31no—
179C23×C4—32no—
180C23×C4—31no—
181C23×C4—31no—
182C23×C4—31no—
183C23×C4—32no—
184C23×C4—61no—
185C23×C4—61no—
186C23×C4—61no—
187C23×C4—61no—
188C23×C4—64no—
189C23×C4—61no—
190C23×C4—61no—
191C23×C4C2×Dic312yes—
192C23×C4#6711yescharacteristic
193C23×C4#6712yes—
194C23×C4C22×D311yescharacteristic
195C2×M16—62no—
196C2×M16—62no—
197C2×M16—61no—
198C2×M16—62no—
199C2×M16—61no—
200C2×M16—62no—
201C2×M16—61no—
202C2×M16—61no—
203C22×C8—32no—
204C22×C8—32no—
205C2×C4⋊C4—32no—
206C2×C4⋊C4—32no—
207C2×C4⋊C4—61no—
208C2×C4⋊C4—62no—
209C2×C4⋊C4—62no—
210C2×C4⋊C4—61no—
211C2×C4⋊C4—61no—
212C2×C4⋊C4—61no—
213C2×C4⋊C4—61no—
214C2×C4⋊C4—61no—
215C2×C22⋊C4—62no—
216C2×C22⋊C4—62no—
217C2×C22⋊C4—122no—
218C2×C22⋊C4—122no—
219#103—64no—
220#103—64no—
221#103—64no—
222#103—62no—
223#103—62no—
224#102—64no—
225#102—64no—
226#102—64no—
227#102—62no—
228#102—62no—
229C22⋊C8—64no—
230C22⋊C8—62no—
231C22⋊C8—62no—
232#95—62no—
233#95—62no—
234C23×C3—41no—
235C23×C3—81no—
236C2×A4—22no—
237C2×A4—42no—
238C2×A4C2×C22⋊C411yescharacteristic
239C2×A4AGL(1,ℤ/8)11yescharacteristic
240C2×A4#13711yescharacteristic
241C3×Q8—42no—
242C3×Q8—81no—
243C3×D4—42no—
244C3×D4—81no—
245C3×D4—84no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—84no—
251C2×C3×C4—81no—
252C2×Dic3—82no—
253C2×Dic3—162no—
254C3⋊C8—82no—
255C24—21no—
256C24—21no—
257C24—22no—
258C24—31no—
259C24—31no—
260C24—32no—
261C24—41no—
262C24—41no—
263C24—61no—
264C24—61no—
265C24—61no—
266C24—61no—
267C24—62no—
268C24—121no—
269C24—121no—
270C24#21712yes—
271C24C2×#6711yescharacteristic
272#41—22no—
273#41—38no—
274#41—38no—
275#41—34no—
276#41—34no—
277#41—34no—
278#41—64no—
279#41—64no—
280#41—64no—
281#41—64no—
282#41—62no—
283#41—64no—
284#41—62no—
285#41—64no—
286#41—64no—
287#41—64no—
288#41—68no—
289#41—68no—
290#41—68no—
291#41—68no—
292#41—62no—
293#41SL(2,ℤ/4)14yescomplemented
294C2×Q8—32no—
295C2×Q8—32no—
296C2×Q8—32no—
297C2×Q8—32no—
298C2×Q8—32no—
299C2×Q8—31no—
300C2×Q8—61no—
301C2×Q8—61no—
302C2×Q8—61no—
303C2×Q8—62no—
304C2×Q8—61no—
305C2×Q8—61no—
306C2×Q8—62no—
307C2×Q8—64no—
308C2×Q8C2×S411yescharacteristic
309C2×D4—21no—
310C2×D4—32no—
311C2×D4—32no—
312C2×D4—32no—
313C2×D4—32no—
314C2×D4—32no—
315C2×D4—31no—
316C2×D4—61no—
317C2×D4—61no—
318C2×D4—61no—
319C2×D4—61no—
320C2×D4—62no—
321C2×D4—61no—
322C2×D4—61no—
323C2×D4—62no—
324C2×D4—64no—
325C2×D4—64no—
326C2×D4—64no—
327C2×D4—61no—
328C2×D4—64no—
329C2×D4—62no—
330C2×D4—64no—
331C2×D4—64no—
332C2×D4—61no—
333C2×D4—64no—
334C2×D4—64no—
335C2×D4—64no—
336C2×D4C2×S411yescharacteristic
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—31no—
340C22×C4—32no—
341C22×C4—32no—
342C22×C4—31no—
343C22×C4—31no—
344C22×C4—32no—
345C22×C4—32no—
346C22×C4—31no—
347C22×C4—31no—
348C22×C4—31no—
349C22×C4—31no—
350C22×C4—32no—
351C22×C4—31no—
352C22×C4—32no—
353C22×C4—32no—
354C22×C4—32no—
355C22×C4—32no—
356C22×C4—32no—
357C22×C4—32no—
358C22×C4—31no—
359C22×C4—64no—
360C22×C4—61no—
361C22×C4—64no—
362C22×C4—61no—
363C22×C4—61no—
364C22×C4—61no—
365C22×C4—62no—
366C22×C4—61no—
367C22×C4—62no—
368C22×C4—61no—
369C22×C4—62no—
370C22×C4—64no—
371C22×C4—64no—
372C22×C4—64no—
373C22×C4—64no—
374C22×C4—64no—
375C22×C4—64no—
376C22×C4—61no—
377C22×C4—62no—
378C22×C4—61no—
379C22×C4—62no—
380C22×C4—62no—
381C22×C4—122no—
382C22×C4—121no—
383C22×C4—122no—
384C22×C4—122no—
385C22×C4—122no—
386C22×C4—122no—
387C22×C4—121no—
388C22×C4#21712yes—
389C22×C4C22×Dic311yescharacteristic
390C22×C4C2×#6711yescharacteristic
391C22×C4C2×S411yescharacteristic
392M16—122no—
393M16—122no—
394M16—122no—
395M16—122no—
396C2×C8—62no—
397C2×C8—62no—
398C2×C8—62no—
399C2×C8—62no—
400C2×C8—62no—
401C2×C8—62no—
402C4⋊C4—62no—
403C4⋊C4—62no—
404C4⋊C4—62no—
405C4⋊C4—62no—
406C4⋊C4—121no—
407C4⋊C4—122no—
408C4⋊C4—121no—
409C22⋊C4—122no—
410C22⋊C4—124no—
411C22⋊C4—122no—
412C22×C3—41no—
413C22×C3—42no—
414C22×C3—81no—
415C22×C3—81no—
416C22×C3—82no—
417C22×C3—161no—
418C22×C3—161no—
419A4#41711yescharacteristic, complemented
420C3×C4—41no—
421C3×C4—41no—
422C3×C4—42no—
423C3×C4—82no—
424Dic3—162no—
425C23—21no—
426C23—22no—
427C23—31no—
428C23—31no—
429C23—31no—
430C23—31no—
431C23—32no—
432C23—31no—
433C23—32no—
434C23—32no—
435C23—42no—
436C23—61no—
437C23—61no—
438C23—61no—
439C23—61no—
440C23—62no—
441C23—62no—
442C23—61no—
443C23—62no—
444C23—61no—
445C23—62no—
446C23—62no—
447C23—61no—
448C23—61no—
449C23—61no—
450C23—62no—
451C23—62no—
452C23—61no—
453C23—121no—
454C23—121no—
455C23—121no—
456C23—121no—
457C23—122no—
458C23—122no—
459C23—122no—
460C23—121no—
461C23—121no—
462C23—121no—
463C23—121no—
464C23—122no—
465C23—124no—
466C23#94911yescharacteristic
467C23#95111yescharacteristic
468C23C2×#21711yescharacteristic
469C23GL(2,ℤ/4)11yescharacteristic
470Q8—21no—
471Q8—34no—
472Q8—32no—
473Q8—62no—
474Q8—62no—
475Q8—62no—
476Q8—61no—
477Q8C2×SL(2,ℤ/4)12yescomplemented
478D4—24no—
479D4—21no—
480D4—34no—
481D4—32no—
482D4—64no—
483D4—62no—
484D4—62no—
485D4—62no—
486D4—64no—
487D4—68no—
488D4—64no—
489D4—61no—
490D4C2×SL(2,ℤ/4)12yescomplemented
491C2×C4—21no—
492C2×C4—24no—
493C2×C4—31no—
494C2×C4—31no—
495C2×C4—32no—
496C2×C4—32no—
497C2×C4—31no—
498C2×C4—31no—
499C2×C4—32no—
500C2×C4—31no—
501C2×C4—31no—
502C2×C4—32no—
503C2×C4—32no—
504C2×C4—32no—
505C2×C4—32no—
506C2×C4—32no—
507C2×C4—32no—
508C2×C4—31no—
509C2×C4—31no—
510C2×C4—64no—
511C2×C4—62no—
512C2×C4—62no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—62no—
516C2×C4—62no—
517C2×C4—64no—
518C2×C4—68no—
519C2×C4—62no—
520C2×C4—61no—
521C2×C4—64no—
522C2×C4—62no—
523C2×C4—122no—
524C2×C4—122no—
525C2×C4—122no—
526C2×C4—122no—
527C2×C4—122no—
528C2×C4—122no—
529C2×C4—242no—
530C2×C4—244no—
531C2×C4—242no—
532C2×C4C2×SL(2,ℤ/4)12yes—
533C2×C4GL(2,ℤ/4)12yes—
534C2×C4GL(2,ℤ/4)11yescharacteristic
535C2×C4C22×S411yescharacteristic
536C8—122no—
537C8—122no—
538C2×C3—41no—
539C2×C3—41no—
540C2×C3—41no—
541C2×C3—82no—
542C2×C3—162no—
543C22—21no—
544C22—21no—
545C22—22no—
546C22—31no—
547C22—31no—
548C22—31no—
549C22—31no—
550C22—31no—
551C22—31no—
552C22—31no—
553C22—31no—
554C22—31no—
555C22—32no—
556C22—41no—
557C22—41no—
558C22—62no—
559C22—62no—
560C22—62no—
561C22—62no—
562C22—61no—
563C22—62no—
564C22—62no—
565C22—61no—
566C22—61no—
567C22—62no—
568C22—122no—
569C22—122no—
570C22—122no—
571C22—122no—
572C22—121no—
573C22—121no—
574C22—122no—
575C22—122no—
576C22—122no—
577C22—122no—
578C22—121no—
579C22—121no—
580C22#523211yescharacteristic, complemented
581C22#543212yes—
582C22C2×GL(2,ℤ/4)11yescharacteristic
583C4—22no—
584C4—31no—
585C4—32no—
586C4—31no—
587C4—62no—
588C4—242no—
589C4—242no—
590C4#543212yes—
591C4C22×SL(2,ℤ/4)11yescharacteristic
592C4C2×GL(2,ℤ/4)11yescharacteristic
593C3—41noSylow
594C2—22no—
595C2—31no—
596C2—31no—
597C2—31no—
598C2—31no—
599C2—42no—
600C2—62no—
601C2—122no—
602C2#8551211yescharacteristic
603C2#8551311yescharacteristic
604C2C2×#543211yescharacteristic
subgroupG/H
centerC22C2×GL(2,ℤ/4)
commutatorC4×A4C22×C4
FrattiniC2×C4C22×S4
FittingC23×#41D3
socleC24C2×#67
prime radicalC2×A4×#41C2
cosocleC2×C4×A4C23
hypercenterC2×#41S4
hypocenterA4#417
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊SL(2,ℤ/4)C2semidirect
3. C2×#5409C2semidirect
4. #73147C2semidirect
5. #73144C2semidirect
8. A4×#41C4semidirect
35. Q8×A4C2×C4semidirect
38. D4×A4C2×C4semidirect
61. C22×#41Dic3semidirect
162. C22×Q8C2×Dic3semidirect
175. C22×D4C2×Dic3semidirect
293. #41SL(2,ℤ/4)semidirect
419. A4#417semidirect
477. Q8C2×SL(2,ℤ/4)semidirect
490. D4C2×SL(2,ℤ/4)semidirect
580. C22#5232semidirect

Maximal subgroups

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

subgroupindex
C2×A4×#412
C2×C4⋊SL(2,ℤ/4)2
C2×#54092
#731472
#731442
#206013
#52324