Finite Group Databasebeta

← #11713222

Subgroups — #11713222

Order 768.

528 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 53 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×C22⋊Q8C211yesmaximal, characteristic
2#85482C211yesmaximal, characteristic
3#73147C211yesmaximal, characteristic, complemented
4#73147C211yesmaximal, characteristic, complemented
5#73141C211yesmaximal, characteristic
6#73051C211yesmaximal, characteristic, complemented
7#73050C211yesmaximal, characteristic, complemented
8#24010—31nomaximal, Sylow
9C2×Q8×A4C2211yescharacteristic
10C22×C4×A4C2211yescharacteristic
11A4×C4⋊C4—21no—
12A4×C4⋊C4C2211yescharacteristic
13A4×C22⋊C4—21no—
14#5432—21no—
15C4⋊SL(2,ℤ/4)C2211yescharacteristic
16C4⋊SL(2,ℤ/4)C2211yescharacteristic
17C4×SL(2,ℤ/4)—21no—
18C2×A4⋊C8C2211yescharacteristic
19C2×A4⋊C8C2211yescharacteristic
20#5045—41nomaximal
21C22×C22⋊Q8D311yescharacteristic
22C2×#485—31no—
23C2×#485—31no—
24#2280—31no—
25#2036—31no—
26#2036—31no—
27#2035—31no—
28#2035—31no—
29#1873—31no—
30#1873—31no—
31#1853—31no—
32#1852—31no—
33#1787—31no—
34#1787—31no—
35#1779—31no—
36C23×A4D411yescharacteristic
37Q8×A4—21no—
38Q8×A4—21no—
39C2×C4×A4—21no—
40C2×C4×A4—21no—
41C2×C4×A4—21no—
42C2×C4×A4—21no—
43C2×C4×A4D411yescharacteristic
44C2×C4×A4C2311yescharacteristic
45C2×SL(2,ℤ/4)—21no—
46C2×SL(2,ℤ/4)—21no—
47C3×C22⋊Q8—41no—
48#926—41no—
49A4⋊C8—21no—
50A4⋊C8—21no—
51#835—41no—
52#835—41no—
53#830—41no—
54#808—41no—
55#807—41no—
56C23×Q8C2×D311yescharacteristic
57C24×C4C2×D311yescharacteristic
58C2×C22⋊Q8—31no—
59C2×C22⋊Q8—31no—
60C2×C22⋊Q8—31no—
61C2×C22⋊Q8—31no—
62C2×C22⋊Q8—61no—
63C2×C22⋊Q8—61no—
64C2×#117—31no—
65C2×#117—31no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4C2×D311yescharacteristic
68C22×C22⋊C4—21no—
69#485—62no—
70#485—62no—
71#485—61no—
72#485—61no—
73#485—61no—
74#485—61no—
75C2×#107—31no—
76C2×#107—31no—
77C2×#106—31no—
78C2×#106—31no—
79C2×#103—31no—
80C2×#103—31no—
81C2×#103—31no—
82C2×#103—31no—
83C2×#103—31no—
84C2×#103—31no—
85C2×#103—31no—
86C2×#103—31no—
87C2×C22⋊C8—31no—
88C2×C22⋊C8—31no—
89C2×C22⋊C8—31no—
90C2×C22⋊C8—31no—
91C2×C22⋊C8—31no—
92C2×C22⋊C8—31no—
93#381—61no—
94#380—31no—
95#380—31no—
96#380—31no—
97#380—31no—
98#379—61no—
99C4×C22⋊C4—61no—
100C4×C22⋊C4—61no—
101#340—31no—
102#340—31no—
103#340—31no—
104#340—31no—
105C22×A4—21no—
106C22×A4—21no—
107C22×A4—21no—
108C22×A4C2×D411yescharacteristic
109C2×C3×Q8—41no—
110C22×C3×C4—41no—
111C4×A4—21no—
112C4×A4—41no—
113C4×A4—41no—
114C4×A4#4111yescharacteristic
115C4×A4#4111yescharacteristic
116SL(2,ℤ/4)—41no—
117SL(2,ℤ/4)—41no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—81no—
120C3×C22⋊C4—81no—
121#217—81no—
122C4⋊Dic3—41no—
123C4⋊Dic3—41no—
124C4×Dic3—81no—
125C2×C3⋊C8—41no—
126C2×C3⋊C8—41no—
127C25#6711yescharacteristic
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—31no—
131C22×Q8—31no—
132C22×Q8—61no—
133C22×Q8—61no—
134C22×Q8—61no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—31no—
140C23×C4—31no—
141C23×C4—31no—
142C23×C4—31no—
143C23×C4—61no—
144C23×C4—61no—
145C23×C4#6711yescharacteristic
146C23×C4C22×D311yescharacteristic
147C22×C8—31no—
148C22×C8—31no—
149C22×C8—31no—
150C22×C8—31no—
151C22⋊Q8—32no—
152C22⋊Q8—32no—
153C22⋊Q8—31no—
154C22⋊Q8—31no—
155C22⋊Q8—31no—
156C22⋊Q8—61no—
157C22⋊Q8—61no—
158C22⋊Q8—61no—
159C22⋊Q8—62no—
160C22⋊Q8—62no—
161C22⋊Q8S411yescharacteristic
162#117—61no—
163#117—61no—
164#117—61no—
165#117—61no—
166#117—121no—
167#117—121no—
168C2×C4⋊C4—31no—
169C2×C4⋊C4—31no—
170C2×C4⋊C4—31no—
171C2×C4⋊C4—31no—
172C2×C4⋊C4—31no—
173C2×C4⋊C4—31no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—61no—
179C2×C22⋊C4—61no—
180C2×C22⋊C4—61no—
181C2×C22⋊C4—61no—
182C2×C22⋊C4—61no—
183C2×C22⋊C4—61no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—61no—
186C2×C22⋊C4—61no—
187C2×C22⋊C4—61no—
188C2×C22⋊C4—61no—
189C2×C42—61no—
190C2×C42—61no—
191#107—62no—
192#107—61no—
193#107—61no—
194#106—62no—
195#106—61no—
196#106—61no—
197#103—62no—
198#103—62no—
199#103—61no—
200#103—61no—
201#103—61no—
202#103—61no—
203#103—121no—
204#103—121no—
205#103—121no—
206#103—121no—
207C22⋊C8—61no—
208C22⋊C8—61no—
209C22⋊C8—61no—
210C22⋊C8—61no—
211C22⋊C8—61no—
212C22⋊C8—61no—
213C22⋊C8—61no—
214C22⋊C8—61no—
215C22⋊C8—121no—
216C22⋊C8—121no—
217#95—61no—
218#95—61no—
219#95—61no—
220#95—61no—
221C23×C3—41no—
222C2×A4—41no—
223C2×A4#12311yescharacteristic
224C2×A4#13511yescharacteristic
225C2×A4#13711yescharacteristic
226C3×Q8—81no—
227C3×Q8—81no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—81no—
231C2×C3×C4—81no—
232C2×C3×C4—81no—
233C2×C3×C4—81no—
234C2×Dic3—81no—
235C2×Dic3—81no—
236C3⋊C8—81no—
237C3⋊C8—81no—
238C24—21no—
239C24—21no—
240C24—21no—
241C24—31no—
242C24—31no—
243C24—61no—
244C24—61no—
245C24—61no—
246C24C2×#6711yescharacteristic
247C2×Q8—32no—
248C2×Q8—31no—
249C2×Q8—61no—
250C2×Q8—61no—
251C2×Q8—61no—
252C2×Q8—61no—
253C2×Q8—61no—
254C2×Q8—61no—
255C2×Q8—61no—
256C2×Q8—61no—
257C2×Q8—61no—
258C2×Q8—121no—
259C2×Q8—121no—
260C2×Q8—122no—
261C2×Q8C2×S411yescharacteristic
262C22×C4—21no—
263C22×C4—31no—
264C22×C4—31no—
265C22×C4—31no—
266C22×C4—31no—
267C22×C4—31no—
268C22×C4—31no—
269C22×C4—31no—
270C22×C4—31no—
271C22×C4—31no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4—61no—
277C22×C4—61no—
278C22×C4—61no—
279C22×C4—61no—
280C22×C4—61no—
281C22×C4—61no—
282C22×C4—61no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4—61no—
287C22×C4—61no—
288C22×C4—61no—
289C22×C4—61no—
290C22×C4—61no—
291C22×C4—61no—
292C22×C4—61no—
293C22×C4—61no—
294C22×C4—61no—
295C22×C4—61no—
296C22×C4—121no—
297C22×C4—121no—
298C22×C4—121no—
299C22×C4—121no—
300C22×C4#23711yescharacteristic
301C22×C4#23711yescharacteristic
302C22×C4C2×S411yescharacteristic
303C2×C8—61no—
304C2×C8—61no—
305C2×C8—61no—
306C2×C8—61no—
307C2×C8—61no—
308C2×C8—61no—
309C2×C8—61no—
310C2×C8—61no—
311C2×C8—61no—
312C2×C8—61no—
313C2×C8—61no—
314C2×C8—61no—
315C4⋊C4—21no—
316C4⋊C4—32no—
317C4⋊C4—31no—
318C4⋊C4—61no—
319C4⋊C4—61no—
320C4⋊C4—61no—
321C4⋊C4—62no—
322C4⋊C4—61no—
323C4⋊C4—61no—
324C4⋊C4—61no—
325C4⋊C4—61no—
326C4⋊C4—61no—
327C4⋊C4—61no—
328C4⋊C4—61no—
329C4⋊C4—61no—
330C4⋊C4—61no—
331C4⋊C4C2×S411yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—61no—
334C22⋊C4—61no—
335C22⋊C4—62no—
336C22⋊C4—61no—
337C22⋊C4—121no—
338C22⋊C4—121no—
339C22⋊C4—122no—
340C22⋊C4—121no—
341C22⋊C4—122no—
342C22⋊C4—122no—
343C22⋊C4—121no—
344C22⋊C4—121no—
345C22⋊C4—121no—
346C42—121no—
347C42—122no—
348C42—121no—
349C22×C3—41no—
350C22×C3—81no—
351C22×C3—81no—
352C22×C3—81no—
353A4#48511yescharacteristic, complemented
354C3×C4—41no—
355C3×C4—41no—
356C3×C4—81no—
357C3×C4—161no—
358C3×C4—161no—
359Dic3—161no—
360Dic3—161no—
361C23—31no—
362C23—31no—
363C23—31no—
364C23—31no—
365C23—32no—
366C23—31no—
367C23—41no—
368C23—61no—
369C23—61no—
370C23—61no—
371C23—61no—
372C23—61no—
373C23—61no—
374C23—61no—
375C23—61no—
376C23—61no—
377C23—61no—
378C23—61no—
379C23—121no—
380C23—121no—
381C23—121no—
382C23—122no—
383C23#93511yescharacteristic
384C23#94211yescharacteristic
385C23#95011yescharacteristic
386C23GL(2,ℤ/4)11yescharacteristic
387Q8—21no—
388Q8—21no—
389Q8—62no—
390Q8—62no—
391Q8—61no—
392Q8—61no—
393Q8—121no—
394Q8—121no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—31no—
400C2×C4—31no—
401C2×C4—31no—
402C2×C4—31no—
403C2×C4—31no—
404C2×C4—31no—
405C2×C4—31no—
406C2×C4—31no—
407C2×C4—61no—
408C2×C4—61no—
409C2×C4—61no—
410C2×C4—61no—
411C2×C4—61no—
412C2×C4—61no—
413C2×C4—61no—
414C2×C4—61no—
415C2×C4—61no—
416C2×C4—61no—
417C2×C4—61no—
418C2×C4—61no—
419C2×C4—61no—
420C2×C4—121no—
421C2×C4—121no—
422C2×C4—121no—
423C2×C4—121no—
424C2×C4—121no—
425C2×C4—121no—
426C2×C4—121no—
427C2×C4—121no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—121no—
434C2×C4—121no—
435C2×C4—121no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4—121no—
439C2×C4—121no—
440C2×C4—121no—
441C2×C4—121no—
442C2×C4GL(2,ℤ/4)11yescharacteristic
443C2×C4C22×S411yescharacteristic
444C8—121no—
445C8—121no—
446C8—121no—
447C8—121no—
448C2×C3—41no—
449C2×C3—41no—
450C2×C3—41no—
451C2×C3—161no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—31no—
456C22—31no—
457C22—31no—
458C22—31no—
459C22—31no—
460C22—31no—
461C22—31no—
462C22—31no—
463C22—31no—
464C22—61no—
465C22—62no—
466C22—62no—
467C22—61no—
468C22—61no—
469C22—61no—
470C22—121no—
471C22—121no—
472C22—121no—
473C22—121no—
474C22—121no—
475C22—121no—
476C22#504511yescharacteristic, complemented
477C22C2×GL(2,ℤ/4)11yescharacteristic
478C4—21no—
479C4—31no—
480C4—31no—
481C4—41no—
482C4—41no—
483C4—61no—
484C4—121no—
485C4—121no—
486C4—241no—
487C4—241no—
488C4—241no—
489C4—241no—
490C4#591411yescharacteristic
491C4#591411yescharacteristic
492C3—41noSylow
493C2—31no—
494C2—31no—
495C2—31no—
496C2—31no—
497C2—41no—
498C2—121no—
499C2#8549111yescharacteristic
500C2#8549911yescharacteristic
501C2#8551411yescharacteristic
subgroupG/H
centerC22C2×GL(2,ℤ/4)
commutator, cosocleC2×C4×A4C23
FrattiniC2×C4C22×S4
FittingC22×C22⋊Q8D3
socleC24C2×#67
prime radicalC23×A4D4
hypercenterC22⋊Q8S4
hypocenterA4#485
LCSG ▷ C2×C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #73147C2semidirect
4. #73147C2semidirect
6. #73051C2semidirect
7. #73050C2semidirect
353. A4#485semidirect
476. C22#5045semidirect

Maximal subgroups

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

subgroupindex
A4×C22⋊Q82
#854822
#731472
#731412
#730512
#730502
#240103
#50454