Finite Group Databasebeta

← #11714414

Subgroups — #11714414

Order 768.

1201 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 1201 conjugacy classes of subgroups — too many to draw. Only its 137 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
1A4×C4⋊Q8C211yesmaximal, characteristic, complemented
2#85506C212yesmaximal
3Q8×SL(2,ℤ/4)C212yesmaximal, complemented
4#85312C212yesmaximal
5#85307C212yesmaximal
6#85305C212yesmaximal, complemented
7#85302C212yesmaximal, complemented
8C42×S4C211yesmaximal, characteristic
9#85283C211yesmaximal, characteristic, complemented
10#36129—31nomaximal, Sylow
11C2×Q8×A4C2212yes—
12C2×C4×S4C2211yescharacteristic
13C2×C4×S4C2211yescharacteristic
14C2×C4×S4C2211yescharacteristic
15C2×A4⋊Q8C2212yes—
16#5728—41nomaximal
17A4×C4⋊C4C2212yes—
18A4×C4⋊C4C2212yes—
19C42×A4C2211yescharacteristic
20(C2×S4)⋊C4C2214yes—
21C4⋊SL(2,ℤ/4)C2212yes—
22C4⋊SL(2,ℤ/4)C2214yes—
23C4⋊SL(2,ℤ/4)C2212yes—
24SL(2,ℤ/4)⋊C4C2214yes—
25C4×SL(2,ℤ/4)C2211yescharacteristic
26C4×SL(2,ℤ/4)C2214yes—
27C4×SL(2,ℤ/4)C2211yescharacteristic
28C4×SL(2,ℤ/4)C2211yescharacteristic
29C2×#533—31no—
30C22×C4⋊Q8D311yescharacteristic, complemented
31#2862—31no—
32#2857—31no—
33#2857—31no—
34#2550—32no—
35#2492—32no—
36#2451—32no—
37#2446—32no—
38#2442—32no—
39#2439—32no—
40#2429—32no—
41#2417—32no—
42#2409—32no—
43#2360—32no—
44#2352—32no—
45Q8×C22⋊C4—32no—
46C4×C4⋊Q8—31no—
47C42×D4—31no—
48C22×S4C2311yescharacteristic
49Q8×A4—24no—
50C2×C4×A4C2314yes—
51C2×C4×A4C2311yescharacteristic
52C2×C4×A4C2311yescharacteristic
53C2×C4×A4C2311yescharacteristic
54C2×SL(2,ℤ/4)C2314yes—
55C2×SL(2,ℤ/4)C2311yescharacteristic
56C2×SL(2,ℤ/4)C2311yescharacteristic
57C2×SL(2,ℤ/4)C2311yescharacteristic
58C4×S4—22no—
59C4×S4—22no—
60C4×S4Q814yescomplemented
61A4⋊Q8—24no—
62C3×C4⋊Q8—41no—
63#946—42no—
64Q8×Dic3—42no—
65#897—42no—
66#892—42no—
67#890—42no—
68#887—42no—
69C42×D3—41no—
70#869—41no—
71C23×Q8C2×D312yes—
72#533—62no—
73#533—62no—
74#533—62no—
75#533—61no—
76#533—61no—
77C2×C4⋊Q8—31no—
78C2×C4⋊Q8—31no—
79C2×C4⋊Q8—32no—
80C2×C4⋊Q8—32no—
81C2×C4⋊Q8—32no—
82C2×C4⋊Q8—31no—
83C2×#125—32no—
84C2×C22⋊Q8—32no—
85C2×C22⋊Q8—32no—
86C2×C22⋊Q8—32no—
87C2×C22⋊Q8—32no—
88C2×C4×Q8—32no—
89C2×C4×Q8—32no—
90C2×C4×Q8—32no—
91C2×C4×D4—31no—
92C2×C4×D4—31no—
93C2×C4×D4—31no—
94C2×#117—32no—
95C22×C4⋊C4C2×D312yes—
96C22×C4⋊C4C2×D312yes—
97C22×C42—31no—
98C22×C42C2×D311yescharacteristic
99#400—34no—
100#398—34no—
101#398—34no—
102#396—34no—
103#395—34no—
104#393—34no—
105#391—32no—
106#391—32no—
107#391—32no—
108#391—34no—
109#391—34no—
110#391—32no—
111#391—32no—
112#391—32no—
113#391—32no—
114#389—32no—
115#389—32no—
116#389—32no—
117#389—32no—
118#389—32no—
119#389—32no—
120#389—34no—
121#388—34no—
122#387—34no—
123#387—34no—
124#386—34no—
125#385—34no—
126#384—31no—
127#382—32no—
128#382—32no—
129#380—32no—
130#380—32no—
131#380—32no—
132#380—34no—
133#380—32no—
134C4×C4⋊C4—32no—
135C4×C4⋊C4—32no—
136C4×C4⋊C4—31no—
137C4×C4⋊C4—31no—
138C4×C4⋊C4—31no—
139C4×C22⋊C4—31no—
140C4×C22⋊C4—31no—
141C4×C22⋊C4—31no—
142C4×C22⋊C4—31no—
143C4×C22⋊C4—34no—
144C4×C22⋊C4—31no—
145C4×C22⋊C4—31no—
146C43—31no—
147C22×A4C2411yescharacteristic
148C2×S4—21no—
149C2×S4—21no—
150C2×S4C2×Q812yes—
151C2×C3×Q8—42no—
152C2×C4×D3—41no—
153C2×C4×D3—41no—
154C2×C4×D3—41no—
155C2×Dic6—42no—
156C4×A4—24no—
157C4×A4C2×Q812yes—
158C4×A4#4112yes—
159C4×A4#4112yes—
160SL(2,ℤ/4)—24no—
161SL(2,ℤ/4)—21no—
162SL(2,ℤ/4)—21no—
163SL(2,ℤ/4)C2×Q812yes—
164C3×C4⋊C4—42no—
165C3×C4⋊C4—42no—
166C3×C42—41no—
167(C2×D3)⋊C4—44no—
168C4⋊Dic3—42no—
169C4⋊Dic3—42no—
170C4⋊Dic3—44no—
171Dic3⋊C4—44no—
172C4×Dic3—41no—
173C4×Dic3—41no—
174C4×Dic3—41no—
175C4×Dic3—44no—
176C22×Q8—24no—
177C22×Q8—32no—
178C22×Q8—32no—
179C22×Q8—34no—
180C22×Q8—32no—
181C22×Q8—64no—
182C22×D4—31no—
183C23×C4—31no—
184C23×C4—31no—
185C23×C4—31no—
186C23×C4C22×D314yes—
187C23×C4C22×D311yescharacteristic
188C23×C4C22×D311yescharacteristic
189C23×C4C22×D311yescharacteristic
190C4⋊Q8—32no—
191C4⋊Q8—32no—
192C4⋊Q8—32no—
193C4⋊Q8—31no—
194C4⋊Q8—61no—
195C4⋊Q8—61no—
196C4⋊Q8—61no—
197C4⋊Q8—61no—
198C4⋊Q8—61no—
199C4⋊Q8—61no—
200C4⋊Q8—64no—
201C4⋊Q8—61no—
202C4⋊Q8S411yescharacteristic, complemented
203#125—64no—
204#125—62no—
205#125—62no—
206C22⋊Q8—62no—
207C22⋊Q8—64no—
208C22⋊Q8—64no—
209C22⋊Q8—62no—
210C22⋊Q8—64no—
211C22⋊Q8—62no—
212C22⋊Q8—64no—
213C22⋊Q8—62no—
214C22⋊Q8—124no—
215C4×Q8—64no—
216C4×Q8—62no—
217C4×Q8—64no—
218C4×Q8—62no—
219C4×Q8—62no—
220C4×Q8—62no—
221C4×Q8—124no—
222C4×D4—34no—
223C4×D4—34no—
224C4×D4—62no—
225C4×D4—62no—
226C4×D4—62no—
227C4×D4—62no—
228#117—62no—
229#117—64no—
230#117—62no—
231C2×C4⋊C4—31no—
232C2×C4⋊C4—32no—
233C2×C4⋊C4—32no—
234C2×C4⋊C4—32no—
235C2×C4⋊C4—32no—
236C2×C4⋊C4—34no—
237C2×C4⋊C4—34no—
238C2×C4⋊C4—32no—
239C2×C4⋊C4—31no—
240C2×C4⋊C4—31no—
241C2×C4⋊C4—34no—
242C2×C4⋊C4—34no—
243C2×C4⋊C4—32no—
244C2×C4⋊C4—32no—
245C2×C4⋊C4—32no—
246C2×C4⋊C4—34no—
247C2×C4⋊C4—32no—
248C2×C4⋊C4—34no—
249C2×C4⋊C4—34no—
250C2×C22⋊C4—31no—
251C2×C22⋊C4—31no—
252C2×C22⋊C4—34no—
253C2×C22⋊C4—31no—
254C2×C22⋊C4—31no—
255C2×C22⋊C4—31no—
256C2×C22⋊C4—31no—
257C2×C22⋊C4—34no—
258C2×C22⋊C4—34no—
259C2×C42—31no—
260C2×C42—31no—
261C2×C42—31no—
262C2×C42—31no—
263C2×C42—31no—
264C2×C42—34no—
265C2×C42—31no—
266C2×C42—31no—
267C2×C42—31no—
268C2×C42—31no—
269C2×C42—31no—
270C2×C42—31no—
271C2×C42—31no—
272C2×C42—31no—
273C2×C42—34no—
274C2×C42—61no—
275C2×C42—61no—
276C2×C42—61no—
277C2×C42—61no—
278#95—31no—
279#95—31no—
280#95—31no—
281#95—31no—
282#95—31no—
283#95—31no—
284#95—34no—
285#95—34no—
286#95—34no—
287#95—34no—
288#95—34no—
289#95—34no—
290#95—34no—
291#95—34no—
292#95—34no—
293#95—34no—
294C22×D3—41no—
295C2×A4C22×Q811yescharacteristic
296C2×A4C2×#4111yescharacteristic
297C2×A4C2×#4111yescharacteristic
298S4—22no—
299C3×Q8—84no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—44no—
304C2×Dic3—41no—
305C2×Dic3—41no—
306C2×Dic3—41no—
307C2×Dic3—44no—
308C4×D3—44no—
309C4×D3—82no—
310C4×D3—82no—
311Dic6—84no—
312C24—31no—
313C24C23×D311yescharacteristic
314C2×Q8—34no—
315C2×Q8—32no—
316C2×Q8—62no—
317C2×Q8—64no—
318C2×Q8—64no—
319C2×Q8—64no—
320C2×Q8—64no—
321C2×Q8—68no—
322C2×Q8—62no—
323C2×Q8—64no—
324C2×Q8—64no—
325C2×Q8—62no—
326C2×Q8—124no—
327C2×Q8C2×S412yes—
328C2×D4—32no—
329C2×D4—32no—
330C2×D4—61no—
331C2×D4—61no—
332C2×D4—61no—
333C2×D4—61no—
334C22×C4—24no—
335C22×C4—31no—
336C22×C4—31no—
337C22×C4—31no—
338C22×C4—31no—
339C22×C4—31no—
340C22×C4—31no—
341C22×C4—32no—
342C22×C4—32no—
343C22×C4—32no—
344C22×C4—31no—
345C22×C4—31no—
346C22×C4—31no—
347C22×C4—34no—
348C22×C4—34no—
349C22×C4—34no—
350C22×C4—31no—
351C22×C4—31no—
352C22×C4—31no—
353C22×C4—34no—
354C22×C4—34no—
355C22×C4—31no—
356C22×C4—61no—
357C22×C4—62no—
358C22×C4—61no—
359C22×C4—61no—
360C22×C4—62no—
361C22×C4—62no—
362C22×C4—62no—
363C22×C4—62no—
364C22×C4—61no—
365C22×C4—64no—
366C22×C4—61no—
367C22×C4—61no—
368C22×C4#23712yes—
369C22×C4D3×Q812yes—
370C22×C4#23912yes—
371C4⋊C4—34no—
372C4⋊C4—34no—
373C4⋊C4—32no—
374C4⋊C4—34no—
375C4⋊C4—32no—
376C4⋊C4—62no—
377C4⋊C4—62no—
378C4⋊C4—64no—
379C4⋊C4—62no—
380C4⋊C4—62no—
381C4⋊C4—62no—
382C4⋊C4—64no—
383C4⋊C4—64no—
384C4⋊C4—62no—
385C4⋊C4—64no—
386C4⋊C4—64no—
387C4⋊C4—68no—
388C4⋊C4—62no—
389C4⋊C4—64no—
390C4⋊C4—64no—
391C4⋊C4—62no—
392C4⋊C4—124no—
393C4⋊C4—124no—
394C4⋊C4C2×S412yes—
395C4⋊C4C2×S412yes—
396C22⋊C4—34no—
397C22⋊C4—32no—
398C22⋊C4—32no—
399C22⋊C4—64no—
400C22⋊C4—62no—
401C22⋊C4—62no—
402C22⋊C4—61no—
403C22⋊C4—61no—
404C22⋊C4—64no—
405C22⋊C4—61no—
406C22⋊C4—61no—
407C22⋊C4—64no—
408C22⋊C4—64no—
409C22⋊C4—124no—
410C42—34no—
411C42—31no—
412C42—31no—
413C42—31no—
414C42—61no—
415C42—62no—
416C42—62no—
417C42—61no—
418C42—61no—
419C42—61no—
420C42—64no—
421C42—62no—
422C42—62no—
423C42—62no—
424C42—61no—
425C42—61no—
426C42—61no—
427C42—64no—
428C42—124no—
429C42C2×S411yescharacteristic
430C22×C3—41no—
431C2×D3—42no—
432C2×D3—81no—
433C2×D3—81no—
434A4#53311yescharacteristic, complemented
435C3×C4—42no—
436C3×C4—42no—
437C3×C4—42no—
438C3×C4—84no—
439Dic3—42no—
440Dic3—84no—
441Dic3—81no—
442Dic3—81no—
443C23—31no—
444C23—31no—
445C23—31no—
446C23—31no—
447C23—32no—
448C23—61no—
449C23—61no—
450C23—61no—
451C23—61no—
452C23—61no—
453C23—61no—
454C23C2×#23711yescharacteristic
455C23C2×D3×Q811yescharacteristic
456C23C2×#23911yescharacteristic
457Q8—24no—
458Q8—68no—
459Q8—64no—
460Q8—124no—
461Q8—124no—
462Q8—124no—
463D4—62no—
464D4—62no—
465D4—62no—
466D4—62no—
467C2×C4—31no—
468C2×C4—31no—
469C2×C4—31no—
470C2×C4—32no—
471C2×C4—32no—
472C2×C4—32no—
473C2×C4—32no—
474C2×C4—32no—
475C2×C4—32no—
476C2×C4—32no—
477C2×C4—32no—
478C2×C4—32no—
479C2×C4—32no—
480C2×C4—32no—
481C2×C4—32no—
482C2×C4—34no—
483C2×C4—64no—
484C2×C4—64no—
485C2×C4—61no—
486C2×C4—61no—
487C2×C4—61no—
488C2×C4—64no—
489C2×C4—61no—
490C2×C4—61no—
491C2×C4—61no—
492C2×C4—61no—
493C2×C4—61no—
494C2×C4—61no—
495C2×C4—64no—
496C2×C4—64no—
497C2×C4—64no—
498C2×C4—61no—
499C2×C4—61no—
500C2×C4—61no—
501C2×C4—61no—
502C2×C4—61no—
503C2×C4—61no—
504C2×C4—64no—
505C2×C4—64no—
506C2×C4—61no—
507C2×C4—61no—
508C2×C4—64no—
509C2×C4—64no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—64no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—64no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4—122no—
519C2×C4—122no—
520C2×C4C22×S414yes—
521C2×C4C22×S411yescharacteristic
522C2×C4C22×S411yescharacteristic
523C2×C4C22×S411yescharacteristic
524C2×C3—41no—
525C2×C3—41no—
526C2×C3—41no—
527D3—82no—
528C22—31no—
529C22—31no—
530C22—31no—
531C22—31no—
532C22—31no—
533C22—31no—
534C22—31no—
535C22—31no—
536C22—31no—
537C22—61no—
538C22—62no—
539C22—62no—
540C22—62no—
541C22—121no—
542C22—121no—
543C22—121no—
544C22—121no—
545C22#572811yescharacteristic, complemented
546C22C23×S411yescharacteristic
547C4—24no—
548C4—32no—
549C4—32no—
550C4—32no—
551C4—62no—
552C4—62no—
553C4—64no—
554C4—62no—
555C4—121no—
556C4—124no—
557C4—121no—
558C4—121no—
559C4—121no—
560C4—124no—
561C4#591412yes—
562C4Q8×S412yes—
563C4#591912yes—
564C3—41noSylow
565C2—31no—
566C2—31no—
567C2—31no—
568C2—31no—
569C2—122no—
570C2C2×#591411yescharacteristic
571C2C2×Q8×S411yescharacteristic
572C2C2×#591911yescharacteristic
subgroupG/H
center, FrattiniC22C23×S4
commutator, cosocleC22×A4C24
FittingC22×C4⋊Q8D3
socleC24C23×D3
prime radicalC22×S4C23
hypercenterC4⋊Q8S4
hypocenterA4#533
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. A4×C4⋊Q8C2semidirect
3. Q8×SL(2,ℤ/4)C2semidirect
6. #85305C2semidirect
7. #85302C2semidirect
9. #85283C2semidirect
30. C22×C4⋊Q8D3semidirect
60. C4×S4Q8semidirect
202. C4⋊Q8S4semidirect
434. A4#533semidirect
545. C22#5728semidirect

Maximal subgroups

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

subgroupindex
A4×C4⋊Q82
#855062
Q8×SL(2,ℤ/4)2
#853122
#853072
#853052
#853022
C42×S42
#852832
#361293
#57284