Finite Group Databasebeta

← C2×#73117 #11712708

Subgroups — C2×#73117 #11712708

Order 768.

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

normal subgroups only

This group has 931 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
1C2×#5912C211yesmaximal, characteristic
2C2×A4×M16C211yesmaximal, characteristic, complemented
3C2×C4×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
4#73117C214yesmaximal, direct product factor
5C2×#1871—31nomaximal, Sylow
6C22×C4×A4C2211yescharacteristic
7C2×GL(2,ℤ/4)—21no—
8C22×SL(2,ℤ/4)—21no—
9#5912—21no—
10#5912—21no—
11#5912C412yescomplemented
12#5912C2212yes—
13C2×C4⋊S4C411yescharacteristic, complemented
14C2×C4×S4—21no—
15C2×A4⋊Q8C411yescharacteristic, complemented
16A4×M16—21no—
17A4×M16C2212yescomplemented
18C2×C8×A4—21no—
19C4×SL(2,ℤ/4)—21no—
20C4×SL(2,ℤ/4)C2212yescomplemented
21C2×#825—41nomaximal
22C23×M16D311yescharacteristic, complemented
23C2×#525—31no—
24C22×#104—31no—
25C22×#104—31no—
26C2×C4×C22⋊C4—31no—
27#1871—34no—
28#1871—34no—
29C2×#339—31no—
30C2×#339—31no—
31C23×A4D411yescharacteristic
32C22×S4—21no—
33C2×#235—41no—
34C2×C4×A4C2×C411yescharacteristic
35C2×C4×A4C2×C411yescharacteristic
36C2×C4×A4D411yescharacteristic
37C2×C4×A4D411yescharacteristic
38C2×C4×A4D411yescharacteristic
39C2×C4×A4C2311yescharacteristic
40GL(2,ℤ/4)—22no—
41GL(2,ℤ/4)—22no—
42C2×SL(2,ℤ/4)—24no—
43C2×SL(2,ℤ/4)—21no—
44C2×SL(2,ℤ/4)—22no—
45C4⋊S4—21no—
46C4⋊S4C2×C412yescomplemented
47C4×S4—22no—
48C4×S4—22no—
49A4⋊Q8—21no—
50A4⋊Q8C2×C412yescomplemented
51C2×C3×M16—41no—
52C2×C4×Dic3—41no—
53C8×A4—22no—
54#825—44no—
55C22×#41—31no—
56C24×C4C2×D311yescharacteristic
57C22×M16—21no—
58C22×M16—31no—
59C22×M16—31no—
60C22×M16—32no—
61C22×M16—61no—
62C22×M16—61no—
63C22×M16C2×D312yescomplemented
64C23×C8—21no—
65#525—32no—
66#525—32no—
67#525—32no—
68#525—32no—
69#525—61no—
70#525—61no—
71#525—61no—
72#525—61no—
73C2×#123—61no—
74C2×C22⋊Q8—31no—
75C2×C22⋊Q8—31no—
76C2×#121—31no—
77C2×#121—31no—
78C2×#120—61no—
79C2×C4×D4—61no—
80C2×C4×D4—61no—
81C2×#117—31no—
82C22×C22⋊C4—61no—
83C22×C42—31no—
84C22×C42—31no—
85C2×#104—34no—
86C2×#104—34no—
87C2×#104—62no—
88C2×#104—62no—
89C2×#104—61no—
90C2×#104—62no—
91C2×#104—61no—
92C4×C22⋊C4—32no—
93C4×C22⋊C4—32no—
94C4×C22⋊C4—61no—
95C4×C22⋊C4—61no—
96C2×#95—61no—
97#339—34no—
98#339—34no—
99C22×A4—22no—
100C22×A4C22⋊C411yescharacteristic
101C22×A4C22⋊C411yescharacteristic
102C22×A4C2×D411yescharacteristic
103C2×S4—22no—
104C2×S4—42no—
105C22×C3×C4—41no—
106C2×#67—81no—
107C22×Dic3—81no—
108#235—42no—
109#235—42no—
110#235—81no—
111#235—81no—
112C2×D12—41no—
113C2×C4×D3—81no—
114C2×Dic6—41no—
115C4×A4C22⋊C411yescharacteristic
116C4×A4C22⋊C411yescharacteristic
117C4×A4C22×C411yescharacteristic
118C4×A4C2×D411yescharacteristic
119SL(2,ℤ/4)—24no—
120SL(2,ℤ/4)—22no—
121C3×M16—42no—
122C3×M16—81no—
123C2×C3×C8—81no—
124C4×Dic3—42no—
125C4×Dic3—81no—
126C25D1211yescharacteristic
127C2×#41—32no—
128C2×#41—32no—
129C2×#41—61no—
130C2×#41—61no—
131C2×#41—61no—
132C2×#41—61no—
133C2×#41—61no—
134C2×#41—61no—
135C2×#41—61no—
136C2×#41—61no—
137C2×#41—61no—
138C2×#41—61no—
139C22×Q8—31no—
140C22×D4—31no—
141C22×D4—61no—
142C22×D4—61no—
143C22×D4—61no—
144C23×C4—31no—
145C23×C4—31no—
146C23×C4—31no—
147C23×C4—31no—
148C23×C4—31no—
149C23×C4—31no—
150C23×C4—31no—
151C23×C4—31no—
152C23×C4—61no—
153C23×C4—61no—
154C23×C4—61no—
155C23×C4C4×D311yescharacteristic
156C23×C4C4×D311yescharacteristic
157C23×C4D1211yescharacteristic
158C23×C4#6711yescharacteristic
159C23×C4#6711yescharacteristic
160C23×C4C22×D311yescharacteristic
161C2×M16—32no—
162C2×M16—32no—
163C2×M16—32no—
164C2×M16—31no—
165C2×M16—32no—
166C2×M16—32no—
167C2×M16—61no—
168C2×M16—61no—
169C2×M16—61no—
170C2×M16—61no—
171C2×M16—62no—
172C2×M16—64no—
173C2×M16—62no—
174C2×M16—61no—
175C2×M16S411yescharacteristic, complemented
176C22×C8—22no—
177C22×C8—61no—
178C22×C8—62no—
179C22×C8—61no—
180#123—62no—
181#123—62no—
182#123—62no—
183#123—62no—
184C22⋊Q8—32no—
185C22⋊Q8—32no—
186C22⋊Q8—32no—
187C22⋊Q8—32no—
188C22⋊Q8—61no—
189C22⋊Q8—61no—
190C22⋊Q8—61no—
191C22⋊Q8—61no—
192#121—32no—
193#121—32no—
194#121—32no—
195#121—32no—
196#121—61no—
197#121—61no—
198#121—61no—
199#121—61no—
200#120—62no—
201#120—62no—
202#120—62no—
203#120—62no—
204C4×D4—62no—
205C4×D4—62no—
206C4×D4—62no—
207C4×D4—62no—
208C4×D4—62no—
209C4×D4—62no—
210C4×D4—62no—
211C4×D4—62no—
212#117—32no—
213#117—32no—
214#117—61no—
215#117—61no—
216C2×C4⋊C4—31no—
217C2×C4⋊C4—31no—
218C2×C4⋊C4—61no—
219C2×C4⋊C4—61no—
220C2×C22⋊C4—61no—
221C2×C22⋊C4—61no—
222C2×C22⋊C4—64no—
223C2×C22⋊C4—62no—
224C2×C22⋊C4—61no—
225C2×C22⋊C4—61no—
226C2×C22⋊C4—64no—
227C2×C22⋊C4—61no—
228C2×C22⋊C4—62no—
229C2×C42—32no—
230C2×C42—32no—
231C2×C42—61no—
232C2×C42—61no—
233C2×C42—61no—
234C2×C42—61no—
235C2×C42—61no—
236C2×C42—61no—
237C2×C42—61no—
238C2×C42—61no—
239C2×C42—61no—
240C2×C42—61no—
241C2×C42—61no—
242#104—68no—
243#104—64no—
244#104—64no—
245#95—62no—
246#95—62no—
247C23×C3—41no—
248C22×D3—81no—
249C2×A4—21no—
250C2×A4—21no—
251C2×A4#10412yescomplemented
252C2×A4C2×C22⋊C411yescharacteristic
253S4—42no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260#67—82no—
261#67—82no—
262C2×Dic3—82no—
263C2×Dic3—81no—
264C2×Dic3—84no—
265D12—42no—
266D12—81no—
267C4×D3—82no—
268C4×D3—82no—
269Dic6—42no—
270Dic6—81no—
271C3×C8—82no—
272C24—22no—
273C24—31no—
274C24—31no—
275C24—31no—
276C24—31no—
277C24—62no—
278C24—61no—
279C24(C2×D3)⋊C411yescharacteristic
280C24(C2×D3)⋊C411yescharacteristic
281C24C2×D1211yescharacteristic
282#41—62no—
283#41—62no—
284#41—62no—
285#41—62no—
286#41—62no—
287#41—62no—
288#41—62no—
289#41—62no—
290#41—121no—
291#41—121no—
292#41—121no—
293#41—121no—
294#41—121no—
295#41—121no—
296#41—121no—
297#41—121no—
298C2×Q8—32no—
299C2×Q8—61no—
300C2×Q8—61no—
301C2×Q8—61no—
302C2×Q8—61no—
303C2×Q8—61no—
304C2×D4—32no—
305C2×D4—61no—
306C2×D4—61no—
307C2×D4—61no—
308C2×D4—61no—
309C2×D4—62no—
310C2×D4—62no—
311C2×D4—62no—
312C2×D4—62no—
313C2×D4—62no—
314C2×D4—62no—
315C2×D4—61no—
316C2×D4—122no—
317C2×D4—121no—
318C2×D4—121no—
319C2×D4—121no—
320C2×D4—124no—
321C2×D4—122no—
322C2×D4—121no—
323C22×C4—31no—
324C22×C4—31no—
325C22×C4—31no—
326C22×C4—31no—
327C22×C4—31no—
328C22×C4—31no—
329C22×C4—31no—
330C22×C4—31no—
331C22×C4—31no—
332C22×C4—31no—
333C22×C4—31no—
334C22×C4—31no—
335C22×C4—31no—
336C22×C4—31no—
337C22×C4—31no—
338C22×C4—31no—
339C22×C4—31no—
340C22×C4—31no—
341C22×C4—31no—
342C22×C4—31no—
343C22×C4—31no—
344C22×C4—31no—
345C22×C4—31no—
346C22×C4—31no—
347C22×C4—31no—
348C22×C4—31no—
349C22×C4—31no—
350C22×C4—31no—
351C22×C4—31no—
352C22×C4—31no—
353C22×C4—31no—
354C22×C4—61no—
355C22×C4—61no—
356C22×C4—61no—
357C22×C4—61no—
358C22×C4—61no—
359C22×C4—61no—
360C22×C4—61no—
361C22×C4—62no—
362C22×C4—62no—
363C22×C4—62no—
364C22×C4—62no—
365C22×C4—61no—
366C22×C4—61no—
367C22×C4—61no—
368C22×C4—64no—
369C22×C4—64no—
370C22×C4—121no—
371C22×C4—121no—
372C22×C4—121no—
373C22×C4—121no—
374C22×C4—121no—
375C22×C4—121no—
376C22×C4—122no—
377C22×C4—122no—
378C22×C4—121no—
379C22×C4—121no—
380C22×C4(C2×D3)⋊C411yescharacteristic
381C22×C4(C2×D3)⋊C411yescharacteristic
382C22×C4C2×C4×D311yescharacteristic
383C22×C4C2×#6711yescharacteristic
384C22×C4C2×S411yescharacteristic
385M16—21no—
386M16—34no—
387M16—32no—
388M16—62no—
389M16—62no—
390M16—62no—
391M16—61no—
392M16C2×S412yescomplemented
393C2×C8—21no—
394C2×C8—62no—
395C2×C8—62no—
396C2×C8—62no—
397C2×C8—61no—
398C2×C8—62no—
399C4⋊C4—32no—
400C4⋊C4—32no—
401C4⋊C4—61no—
402C4⋊C4—61no—
403C4⋊C4—62no—
404C4⋊C4—62no—
405C4⋊C4—62no—
406C4⋊C4—62no—
407C22⋊C4—62no—
408C22⋊C4—62no—
409C22⋊C4—62no—
410C22⋊C4—62no—
411C22⋊C4—64no—
412C22⋊C4—62no—
413C22⋊C4—62no—
414C22⋊C4—62no—
415C22⋊C4—62no—
416C22⋊C4—62no—
417C22⋊C4—64no—
418C22⋊C4—68no—
419C22⋊C4—62no—
420C42—62no—
421C42—62no—
422C42—62no—
423C42—62no—
424C42—62no—
425C42—62no—
426C42—121no—
427C42—121no—
428C42—121no—
429C42—121no—
430C22×C3—41no—
431C22×C3—41no—
432C22×C3—41no—
433C22×C3—82no—
434C2×D3—82no—
435C2×D3—162no—
436A4C2×#10411yescharacteristic, complemented
437C3×C4—41no—
438C3×C4—41no—
439C3×C4—41no—
440C3×C4—41no—
441Dic3—82no—
442Dic3—84no—
443C23—21no—
444C23—21no—
445C23—31no—
446C23—31no—
447C23—32no—
448C23—31no—
449C23—31no—
450C23—31no—
451C23—31no—
452C23—31no—
453C23—31no—
454C23—31no—
455C23—62no—
456C23—62no—
457C23—61no—
458C23—61no—
459C23—61no—
460C23—62no—
461C23—62no—
462C23—62no—
463C23—62no—
464C23—62no—
465C23—121no—
466C23—122no—
467C23—122no—
468C23—121no—
469C23—121no—
470C23—121no—
471C23#82512yescomplemented
472C23C2×(C2×D3)⋊C411yescharacteristic
473C23C4⋊S411yescharacteristic
474Q8—62no—
475Q8—62no—
476Q8—121no—
477Q8—121no—
478D4—62no—
479D4—62no—
480D4—121no—
481D4—124no—
482D4—122no—
483D4—122no—
484D4—122no—
485D4—124no—
486D4—124no—
487D4—122no—
488D4—122no—
489D4—122no—
490D4—121no—
491C2×C4—31no—
492C2×C4—31no—
493C2×C4—31no—
494C2×C4—31no—
495C2×C4—31no—
496C2×C4—31no—
497C2×C4—31no—
498C2×C4—31no—
499C2×C4—31no—
500C2×C4—31no—
501C2×C4—31no—
502C2×C4—31no—
503C2×C4—31no—
504C2×C4—31no—
505C2×C4—31no—
506C2×C4—31no—
507C2×C4—31no—
508C2×C4—31no—
509C2×C4—31no—
510C2×C4—31no—
511C2×C4—31no—
512C2×C4—31no—
513C2×C4—31no—
514C2×C4—31no—
515C2×C4—31no—
516C2×C4—31no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—62no—
524C2×C4—62no—
525C2×C4—62no—
526C2×C4—64no—
527C2×C4—64no—
528C2×C4—122no—
529C2×C4—122no—
530C2×C4—124no—
531C2×C4—122no—
532C2×C4—121no—
533C2×C4—122no—
534C2×C4—122no—
535C2×C4—122no—
536C2×C4—124no—
537C2×C4—122no—
538C2×C4—124no—
539C2×C4—124no—
540C2×C4—122no—
541C2×C4—121no—
542C2×C4—122no—
543C2×C4—122no—
544C2×C4C4×S411yescharacteristic
545C2×C4C4×S411yescharacteristic
546C2×C4C4⋊S411yescharacteristic
547C2×C4GL(2,ℤ/4)11yescharacteristic
548C2×C4GL(2,ℤ/4)11yescharacteristic
549C2×C4C22×S411yescharacteristic
550C8—22no—
551C8—62no—
552C2×C3—42no—
553C2×C3—41no—
554C2×C3—81no—
555C2×C3—81no—
556D3—162no—
557C22—22no—
558C22—32no—
559C22—31no—
560C22—32no—
561C22—32no—
562C22—31no—
563C22—31no—
564C22—31no—
565C22—31no—
566C22—61no—
567C22—61no—
568C22—62no—
569C22—62no—
570C22—61no—
571C22—61no—
572C22—61no—
573C22—61no—
574C22—61no—
575C22—61no—
576C22—61no—
577C22—62no—
578C22—122no—
579C22—122no—
580C22—122no—
581C22—122no—
582C22—242no—
583C22—242no—
584C22C2×#82511yescharacteristic, complemented
585C22(C2×S4)⋊C411yescharacteristic
586C22(C2×S4)⋊C411yescharacteristic
587C22C2×C4⋊S411yescharacteristic
588C4—31no—
589C4—31no—
590C4—31no—
591C4—31no—
592C4—124no—
593C4—122no—
594C4—122no—
595C4—124no—
596C4—122no—
597C4(C2×S4)⋊C411yescharacteristic
598C4(C2×S4)⋊C411yescharacteristic
599C4C2×C4×S411yescharacteristic
600C4C2×GL(2,ℤ/4)11yescharacteristic
601C3—41noSylow
602C2—21no—
603C2—21no—
604C2—31no—
605C2—31no—
606C2—32no—
607C2—61no—
608C2—61no—
609C2—242no—
610C2#7311712yesdirect product factor
611C2C2×(C2×S4)⋊C411yescharacteristic
subgroupG/H
centerC2×C4GL(2,ℤ/4)
commutatorC4×A4C22×C4
FrattiniC2×C4C22×S4
FittingC23×M16D3
socleC24(C2×D3)⋊C4
prime radicalC2×#5912C2
cosocleC2×C4×A4C23
hypercenterC2×M16S4
hypocenterA4C2×#104
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. C2×A4×M16C2semidirect
3. C2×C4×SL(2,ℤ/4)C2semidirect
4. #73117C2direct (N × H)
11. #5912C4semidirect
13. C2×C4⋊S4C4semidirect
15. C2×A4⋊Q8C4semidirect
17. A4×M16C22semidirect
20. C4×SL(2,ℤ/4)C22semidirect
22. C23×M16D3semidirect
46. C4⋊S4C2×C4semidirect
50. A4⋊Q8C2×C4semidirect
63. C22×M16C2×D3semidirect
175. C2×M16S4semidirect
251. C2×A4#104semidirect
392. M16C2×S4semidirect
436. A4C2×#104semidirect
471. C23#825semidirect
584. C22C2×#825semidirect
610. C2#73117direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#59122
C2×A4×M162
C2×C4×SL(2,ℤ/4)2
#731172
C2×#18713
C2×#8254