Finite Group Databasebeta

← C2×#85444 #11714757

Subgroups — C2×#85444 #11714757

Order 768.

1443 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 1443 conjugacy classes of subgroups — too many to draw. Only its 141 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×#5912C211yesmaximal, characteristic
2C22×A4⋊Q8C211yesmaximal, characteristic, complemented
3C2×A4×M16C211yesmaximal, characteristic, complemented
4#85444C218yesmaximal, direct product factor
5C2×#5401C212yesmaximal, complemented
6C2×#5398C212yesmaximal, complemented
7C2×#3056—31nomaximal, Sylow
8C22×C4×A4C2211yescharacteristic
9C2×GL(2,ℤ/4)—21no—
10C22×SL(2,ℤ/4)—21no—
11#5912—22no—
12#5912C2214yes—
13C2×C4⋊S4C2211yescharacteristic
14C2×C4×S4—21no—
15C2×A4⋊Q8—22no—
16C2×A4⋊Q8—21no—
17C2×A4⋊Q8C2212yes—
18C2×A4⋊Q8C2211yescharacteristic
19C2×A4⋊Q8C2214yescomplemented
20C2×#909—41nomaximal
21A4×M16C2214yescomplemented
22C2×C8×A4C2212yes—
23#5401C2218yescomplemented
24#5398C2218yescomplemented
25C22×#137—31no—
26C23×M16D311yescharacteristic, complemented
27C2×#525—31no—
28C22×C22⋊Q8—31no—
29#3056—38no—
30#3056—38no—
31C2×#470—32no—
32C2×#468—32no—
33C2×#467—32no—
34C2×#465—32no—
35C2×#428—31no—
36C2×#417—31no—
37C2×#419—31no—
38C23×A4D411yescharacteristic
39C22×S4—21no—
40C2×#235—41no—
41C22×Dic6—41no—
42C2×C4×A4D411yescharacteristic
43C2×C4×A4D412yes—
44C2×C4×A4C2311yescharacteristic
45C2×C4×A4C2312yes—
46GL(2,ℤ/4)—24no—
47C2×SL(2,ℤ/4)—22no—
48C2×SL(2,ℤ/4)—24no—
49C2×SL(2,ℤ/4)—21no—
50C4⋊S4—21no—
51C4⋊S4C2312yes—
52C4×S4—24no—
53A4⋊Q8—22no—
54A4⋊Q8—24no—
55A4⋊Q8—21no—
56A4⋊Q8C2314yes—
57A4⋊Q8C2312yes—
58C2×C3×M16—41no—
59#909—48no—
60C2×Dic12—42no—
61C2×#204—42no—
62C8×A4C2314yes—
63C22×#41—31no—
64C23×Q8—31no—
65C24×C4C2×D311yescharacteristic
66C2×#137—38no—
67C2×#137—61no—
68C2×#137—64no—
69C2×#137—62no—
70C2×#137—61no—
71C22×Q16—32no—
72C22×SD16—32no—
73C22×M16—31no—
74C22×M16—31no—
75C22×M16—32no—
76C22×M16—34no—
77C22×M16C2×D314yescomplemented
78C23×C8C2×D312yes—
79#525—34no—
80#525—34no—
81#525—62no—
82#525—62no—
83C2×#123—61no—
84C2×C22⋊Q8—32no—
85C2×C22⋊Q8—34no—
86C2×C22⋊Q8—31no—
87C2×C22⋊Q8—32no—
88C2×C22⋊Q8—31no—
89C2×C22⋊Q8—34no—
90C2×C22⋊Q8—61no—
91C2×C22⋊Q8—62no—
92C2×C22⋊Q8—61no—
93C2×C22⋊Q8—62no—
94C2×#121—31no—
95C2×#121—31no—
96C2×#120—61no—
97C2×C4×D4—61no—
98C2×C4×D4—61no—
99C2×#117—31no—
100C22×C4⋊C4—31no—
101C22×C4⋊C4—61no—
102C22×C22⋊C4—61no—
103#470—316no—
104#468—316no—
105#467—38no—
106#467—38no—
107#465—38no—
108#465—38no—
109#428—38no—
110C2×#107—32no—
111C2×#106—32no—
112#419—38no—
113#417—38no—
114C2×#103—32no—
115C2×#103—32no—
116C2×#103—32no—
117C2×#102—32no—
118C22×A4—22no—
119C22×A4C2×D412yes—
120C22×A4C2×D411yescharacteristic
121C2×S4—22no—
122C2×S4—42no—
123C22×C3×C4—41no—
124C2×#67—81no—
125C22×Dic3—81no—
126#235—44no—
127#235—82no—
128C2×D12—41no—
129C2×C4×D3—81no—
130C2×Dic6—42no—
131C2×Dic6—41no—
132C2×Dic6—44no—
133C2×Dic6—81no—
134C2×Dic6—82no—
135C4×A4C2×D411yescharacteristic
136C4×A4C2×D412yes—
137C4×A4C2411yescharacteristic
138SL(2,ℤ/4)—24no—
139SL(2,ℤ/4)—22no—
140C3×M16—44no—
141C2×C3×C8—42no—
142Dic12—48no—
143#204—48no—
144C25D1211yescharacteristic
145C2×#41—34no—
146C2×#41—62no—
147C2×#41—62no—
148C2×#41—61no—
149C2×#41—61no—
150C2×#41—62no—
151C2×#41—61no—
152C2×#41—61no—
153C22×Q8—32no—
154C22×Q8—31no—
155C22×Q8—34no—
156C22×Q8—61no—
157C22×Q8—62no—
158C22×Q8—62no—
159C22×Q8—61no—
160C22×Q8—61no—
161C22×Q8—62no—
162C22×Q8—61no—
163C22×Q8—61no—
164C22×D4—31no—
165C22×D4—61no—
166C22×D4—61no—
167C22×D4—61no—
168C23×C4—31no—
169C23×C4—31no—
170C23×C4—31no—
171C23×C4—31no—
172C23×C4—32no—
173C23×C4—32no—
174C23×C4—61no—
175C23×C4—61no—
176C23×C4—61no—
177C23×C4D1211yescharacteristic
178C23×C4D1212yes—
179C23×C4C22×D311yescharacteristic
180C23×C4C22×D312yes—
181#137—68no—
182#137—616no—
183#137—68no—
184C2×Q16—38no—
185C2×Q16—62no—
186C2×Q16—64no—
187C2×Q16—62no—
188C2×SD16—38no—
189C2×SD16—62no—
190C2×SD16—64no—
191C2×SD16—62no—
192C2×M16—31no—
193C2×M16—34no—
194C2×M16—34no—
195C2×M16—34no—
196C2×M16—38no—
197C2×M16—32no—
198C2×M16—61no—
199C2×M16—64no—
200C2×M16—62no—
201C2×M16S411yescharacteristic, complemented
202C22×C8—32no—
203C22×C8—32no—
204C22×C8—34no—
205C22×C8C22×D314yes—
206#123—64no—
207#123—64no—
208C22⋊Q8—34no—
209C22⋊Q8—38no—
210C22⋊Q8—32no—
211C22⋊Q8—34no—
212C22⋊Q8—34no—
213C22⋊Q8—32no—
214C22⋊Q8—64no—
215C22⋊Q8—64no—
216C22⋊Q8—64no—
217C22⋊Q8—64no—
218C22⋊Q8—62no—
219C22⋊Q8—62no—
220C22⋊Q8—61no—
221C22⋊Q8—64no—
222C22⋊Q8—62no—
223C22⋊Q8—61no—
224#121—32no—
225#121—34no—
226#121—32no—
227#121—62no—
228#121—61no—
229#121—61no—
230#120—64no—
231#120—64no—
232C4×D4—64no—
233C4×D4—64no—
234C4×D4—64no—
235C4×D4—64no—
236#117—34no—
237#117—62no—
238C2×C4⋊C4—32no—
239C2×C4⋊C4—34no—
240C2×C4⋊C4—31no—
241C2×C4⋊C4—31no—
242C2×C4⋊C4—61no—
243C2×C4⋊C4—62no—
244C2×C4⋊C4—62no—
245C2×C4⋊C4—62no—
246C2×C4⋊C4—64no—
247C2×C4⋊C4—64no—
248C2×C4⋊C4—61no—
249C2×C4⋊C4—61no—
250C2×C22⋊C4—61no—
251C2×C22⋊C4—61no—
252C2×C22⋊C4—62no—
253C2×C22⋊C4—64no—
254C2×C22⋊C4—61no—
255C2×C22⋊C4—61no—
256C2×C22⋊C4—64no—
257C2×C22⋊C4—62no—
258C2×C22⋊C4—61no—
259C2×C42—61no—
260#107—38no—
261#106—38no—
262#103—38no—
263#103—38no—
264#103—38no—
265#102—38no—
266C23×C3—41no—
267C22×D3—81no—
268C2×A4—22no—
269C2×A4#13712yescomplemented
270C2×A4C22×D411yescharacteristic
271S4—42no—
272C2×C3×C4—41no—
273C2×C3×C4—42no—
274C2×C3×C4—41no—
275C2×C3×C4—42no—
276#67—84no—
277C2×Dic3—84no—
278C2×Dic3—82no—
279C2×Dic3—81no—
280D12—42no—
281D12—81no—
282C4×D3—84no—
283Dic6—44no—
284Dic6—42no—
285Dic6—82no—
286Dic6—84no—
287Dic6—81no—
288C3×C8—44no—
289C24—22no—
290C24—31no—
291C24—31no—
292C24—32no—
293C24—62no—
294C24—61no—
295C24C2×D1212yes—
296C24C2×D1211yescharacteristic
297#41—64no—
298#41—64no—
299#41—64no—
300#41—64no—
301#41—122no—
302#41—122no—
303#41—122no—
304#41—122no—
305C2×Q8—34no—
306C2×Q8—32no—
307C2×Q8—64no—
308C2×Q8—62no—
309C2×Q8—61no—
310C2×Q8—62no—
311C2×Q8—61no—
312C2×Q8—64no—
313C2×Q8—64no—
314C2×Q8—62no—
315C2×Q8—62no—
316C2×Q8—64no—
317C2×Q8—61no—
318C2×Q8—61no—
319C2×Q8—62no—
320C2×Q8—61no—
321C2×Q8—64no—
322C2×Q8—122no—
323C2×Q8—121no—
324C2×Q8—122no—
325C2×Q8—122no—
326C2×Q8—122no—
327C2×Q8—122no—
328C2×Q8—121no—
329C2×D4—32no—
330C2×D4—61no—
331C2×D4—61no—
332C2×D4—61no—
333C2×D4—61no—
334C2×D4—62no—
335C2×D4—64no—
336C2×D4—64no—
337C2×D4—62no—
338C2×D4—61no—
339C2×D4—122no—
340C2×D4—122no—
341C2×D4—121no—
342C2×D4—124no—
343C2×D4—122no—
344C2×D4—121no—
345C22×C4—31no—
346C22×C4—31no—
347C22×C4—32no—
348C22×C4—32no—
349C22×C4—31no—
350C22×C4—31no—
351C22×C4—32no—
352C22×C4—32no—
353C22×C4—31no—
354C22×C4—31no—
355C22×C4—31no—
356C22×C4—32no—
357C22×C4—32no—
358C22×C4—32no—
359C22×C4—32no—
360C22×C4—32no—
361C22×C4—31no—
362C22×C4—32no—
363C22×C4—32no—
364C22×C4—31no—
365C22×C4—61no—
366C22×C4—62no—
367C22×C4—61no—
368C22×C4—62no—
369C22×C4—61no—
370C22×C4—64no—
371C22×C4—61no—
372C22×C4—61no—
373C22×C4—62no—
374C22×C4—64no—
375C22×C4—61no—
376C22×C4—64no—
377C22×C4—62no—
378C22×C4—121no—
379C22×C4—122no—
380C22×C4—122no—
381C22×C4—121no—
382C22×C4—124no—
383C22×C4—121no—
384C22×C4—121no—
385C22×C4C2×D1212yes—
386C22×C4C2×D1211yescharacteristic
387C22×C4C2×S411yescharacteristic
388C22×C4C23×D311yescharacteristic
389Q16—68no—
390Q16—68no—
391SD16—68no—
392SD16—68no—
393M16—34no—
394M16—38no—
395M16—64no—
396M16C2×S414yescomplemented
397C2×C8—32no—
398C2×C8—34no—
399C2×C8—34no—
400C2×C8—34no—
401C2×C8—62no—
402C2×C8C2×S412yes—
403C4⋊C4—34no—
404C4⋊C4—32no—
405C4⋊C4—32no—
406C4⋊C4—64no—
407C4⋊C4—62no—
408C4⋊C4—61no—
409C4⋊C4—61no—
410C4⋊C4—64no—
411C4⋊C4—68no—
412C4⋊C4—64no—
413C4⋊C4—62no—
414C4⋊C4—64no—
415C4⋊C4—62no—
416C22⋊C4—62no—
417C22⋊C4—64no—
418C22⋊C4—62no—
419C22⋊C4—64no—
420C22⋊C4—68no—
421C22⋊C4—64no—
422C22⋊C4—62no—
423C22⋊C4—64no—
424C22⋊C4—64no—
425C22⋊C4—62no—
426C42—64no—
427C22×C3—42no—
428C22×C3—41no—
429C22×C3—82no—
430C2×D3—82no—
431C2×D3—162no—
432A4C2×#13711yescharacteristic, complemented
433C3×C4—41no—
434C3×C4—42no—
435C3×C4—41no—
436Dic3—84no—
437Dic3—82no—
438C23—22no—
439C23—31no—
440C23—31no—
441C23—32no—
442C23—32no—
443C23—32no—
444C23—32no—
445C23—31no—
446C23—62no—
447C23—62no—
448C23—61no—
449C23—62no—
450C23—62no—
451C23—64no—
452C23—62no—
453C23—62no—
454C23—121no—
455C23—122no—
456C23—122no—
457C23—122no—
458C23—121no—
459C23#90912yescomplemented
460C23C4⋊S411yescharacteristic
461C23C22×D1211yescharacteristic
462Q8—64no—
463Q8—62no—
464Q8—64no—
465Q8—62no—
466Q8—124no—
467Q8—122no—
468Q8—121no—
469Q8—124no—
470Q8—122no—
471Q8—121no—
472D4—62no—
473D4—62no—
474D4—121no—
475D4—124no—
476D4—122no—
477D4—124no—
478D4—128no—
479D4—124no—
480D4—122no—
481D4—121no—
482C2×C4—32no—
483C2×C4—32no—
484C2×C4—31no—
485C2×C4—31no—
486C2×C4—32no—
487C2×C4—31no—
488C2×C4—32no—
489C2×C4—31no—
490C2×C4—32no—
491C2×C4—32no—
492C2×C4—32no—
493C2×C4—31no—
494C2×C4—31no—
495C2×C4—32no—
496C2×C4—32no—
497C2×C4—31no—
498C2×C4—31no—
499C2×C4—61no—
500C2×C4—62no—
501C2×C4—62no—
502C2×C4—61no—
503C2×C4—62no—
504C2×C4—62no—
505C2×C4—64no—
506C2×C4—62no—
507C2×C4—64no—
508C2×C4—124no—
509C2×C4—124no—
510C2×C4—121no—
511C2×C4—122no—
512C2×C4—124no—
513C2×C4—122no—
514C2×C4—122no—
515C2×C4—128no—
516C2×C4—124no—
517C2×C4—121no—
518C2×C4—122no—
519C2×C4—124no—
520C2×C4C4⋊S412yes—
521C2×C4C4⋊S411yescharacteristic
522C2×C4C22×S412yes—
523C2×C4C22×S411yescharacteristic
524C8—34no—
525C8C22×S414yes—
526C2×C3—41no—
527C2×C3—42no—
528C2×C3—82no—
529D3—162no—
530C22—22no—
531C22—32no—
532C22—31no—
533C22—31no—
534C22—31no—
535C22—32no—
536C22—32no—
537C22—32no—
538C22—62no—
539C22—62no—
540C22—64no—
541C22—61no—
542C22—62no—
543C22—62no—
544C22—62no—
545C22—122no—
546C22—122no—
547C22—122no—
548C22—122no—
549C22—242no—
550C22—242no—
551C22C2×#90911yescharacteristic, complemented
552C22C2×C4⋊S412yes—
553C22C2×C4⋊S411yescharacteristic
554C4—32no—
555C4—31no—
556C4—31no—
557C4—124no—
558C4—122no—
559C4—122no—
560C4—122no—
561C4—124no—
562C4C2×C4⋊S412yes—
563C4C2×C4⋊S411yescharacteristic
564C4C23×S411yescharacteristic
565C3—41noSylow
566C2—22no—
567C2—31no—
568C2—31no—
569C2—32no—
570C2—62no—
571C2—242no—
572C2#8544412yesdirect product factor
573C2C22×C4⋊S411yescharacteristic
subgroupG/H
centerC22C2×C4⋊S4
commutator, cosocleC4×A4C24
FrattiniC4C23×S4
FittingC23×M16D3
socleC24C2×D12
prime radicalC2×#5912C2
hypercenterC2×M16S4
hypocenterA4C2×#137
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. C22×A4⋊Q8C2semidirect
3. C2×A4×M16C2semidirect
4. #85444C2direct (N × H)
5. C2×#5401C2semidirect
6. C2×#5398C2semidirect
19. C2×A4⋊Q8C22semidirect
21. A4×M16C22semidirect
23. #5401C22semidirect
24. #5398C22semidirect
26. C23×M16D3semidirect
77. C22×M16C2×D3semidirect
201. C2×M16S4semidirect
269. C2×A4#137semidirect
396. M16C2×S4semidirect
432. A4C2×#137semidirect
459. C23#909semidirect
551. C22C2×#909semidirect
572. C2#85444direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#59122
C22×A4⋊Q82
C2×A4×M162
#854442
C2×#54012
C2×#53982
C2×#30563
C2×#9094