Finite Group Databasebeta

← #11714034

Subgroups — #11714034

Order 768.

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

normal subgroups only

This group has 572 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×#117C211yesmaximal, characteristic, complemented
2#85486C211yesmaximal, characteristic
3#73141C211yesmaximal, characteristic, complemented
4#73053C211yesmaximal, characteristic, complemented
5#73053C211yesmaximal, characteristic, complemented
6#73051C211yesmaximal, characteristic, complemented
7#73050C211yesmaximal, characteristic, complemented
8#32981—31nomaximal, Sylow
9C22×C4×A4C2211yescharacteristic
10C2×GL(2,ℤ/4)—21no—
11C2×C4⋊S4C2211yescharacteristic
12A4×C4⋊C4C2211yescharacteristic
13A4×C4⋊C4C2211yescharacteristic
14A4×C22⋊C4—21no—
15C42×A4—21no—
16(C2×S4)⋊C4—21no—
17C4⋊SL(2,ℤ/4)C2211yescharacteristic
18C2×A4⋊C8C2211yescharacteristic
19C2×A4⋊C8C2211yescharacteristic
20#5003—41nomaximal
21C22×#117D311yescharacteristic, complemented
22C2×#482—31no—
23C2×#482—31no—
24#2847—31no—
25#1875—31no—
26#1875—31no—
27#1874—31no—
28#1874—31no—
29#1854—31no—
30#1854—31no—
31#1853—31no—
32#1852—31no—
33#1825—31no—
34#1825—31no—
35#1779—31no—
36C23×A4D411yescharacteristic
37C22×S4—21no—
38C2×C4×A4—21no—
39C2×C4×A4—21no—
40C2×C4×A4—21no—
41C2×C4×A4—21no—
42C2×C4×A4D411yescharacteristic
43C2×C4×A4C2311yescharacteristic
44GL(2,ℤ/4)—42no—
45C2×SL(2,ℤ/4)—21no—
46C4⋊S4—21no—
47C4⋊S4—21no—
48C3×#117—41no—
49#930—41no—
50A4⋊C8—21no—
51A4⋊C8—21no—
52#830—41no—
53#809—41no—
54#809—41no—
55#808—41no—
56#807—41no—
57C24×C4C2×D311yescharacteristic
58C2×#121—31no—
59C2×#121—31no—
60C2×#121—31no—
61C2×#120—61no—
62C2×#117—31no—
63C2×#117—31no—
64C2×#117—31no—
65C2×#117—31no—
66C2×#117—61no—
67C2×#117—61no—
68C22×C4⋊C4C2×D311yescharacteristic
69C22×C4⋊C4C2×D311yescharacteristic
70C22×C22⋊C4—21no—
71C22×C42—21no—
72#482—62no—
73#482—62no—
74#482—61no—
75#482—61no—
76#482—61no—
77#482—61no—
78C2×#107—31no—
79C2×#107—31no—
80C2×#106—31no—
81C2×#106—31no—
82C2×#102—31no—
83C2×#102—31no—
84C2×#102—31no—
85C2×#102—31no—
86C2×C22⋊C8—31no—
87C2×C22⋊C8—31no—
88C2×C22⋊C8—31no—
89C2×C22⋊C8—31no—
90C2×C22⋊C8—31no—
91C2×C22⋊C8—31no—
92#386—61no—
93#386—61no—
94#386—61no—
95#380—31no—
96#380—31no—
97#380—31no—
98#340—31no—
99#340—31no—
100#340—31no—
101#340—31no—
102#340—31no—
103#340—31no—
104#340—31no—
105#340—31no—
106C22×A4—21no—
107C22×A4—21no—
108C22×A4—21no—
109C22×A4C2×D411yescharacteristic
110C2×S4—41no—
111C2×S4—41no—
112C2×S4—41no—
113C22×C3×C4—41no—
114C2×#67—81no—
115C2×D12—41no—
116C4×A4—21no—
117C4×A4—41no—
118C4×A4—41no—
119C4×A4#4111yescharacteristic
120C4×A4#4111yescharacteristic
121SL(2,ℤ/4)—41no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C22⋊C4—81no—
125C3×C42—81no—
126(C2×D3)⋊C4—81no—
127C4⋊Dic3—41no—
128C2×C3⋊C8—41no—
129C2×C3⋊C8—41no—
130C25#6711yescharacteristic
131C22×D4—31no—
132C22×D4—61no—
133C22×D4—61no—
134C22×D4—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—
151#121—61no—
152#121—61no—
153#121—62no—
154#121—62no—
155#121—61no—
156#121—61no—
157#121—61no—
158#121—61no—
159#121—61no—
160#121—61no—
161#120—122no—
162#120—122no—
163#117—32no—
164#117—32no—
165#117—31no—
166#117—31no—
167#117—31no—
168#117—61no—
169#117—61no—
170#117—61no—
171#117—62no—
172#117—62no—
173#117S411yescharacteristic, complemented
174C2×C4⋊C4—31no—
175C2×C4⋊C4—31no—
176C2×C4⋊C4—31no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—31no—
180C2×C4⋊C4—31no—
181C2×C4⋊C4—61no—
182C2×C4⋊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×C22⋊C4—61no—
190C2×C22⋊C4—61no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4—61no—
193C2×C42—61no—
194C2×C42—61no—
195C2×C42—61no—
196C2×C42—61no—
197#107—62no—
198#107—61no—
199#107—61no—
200#106—62no—
201#106—61no—
202#106—61no—
203#102—62no—
204#102—62no—
205#102—61no—
206#102—61no—
207#102—61no—
208#102—61no—
209C22⋊C8—61no—
210C22⋊C8—61no—
211C22⋊C8—62no—
212C22⋊C8—62no—
213C22⋊C8—61no—
214C22⋊C8—61no—
215C22⋊C8—61no—
216C22⋊C8—61no—
217C22⋊C8—61no—
218C22⋊C8—61no—
219#95—61no—
220#95—61no—
221#95—61no—
222C23×C3—41no—
223C22×D3—81no—
224C2×A4—41no—
225C2×A4#12311yescharacteristic
226C2×A4#13511yescharacteristic
227C2×A4AGL(1,ℤ/8)11yescharacteristic
228S4—81no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—81no—
232C2×C3×C4—81no—
233C2×C3×C4—81no—
234C2×C3×C4—81no—
235#67—162no—
236C2×Dic3—81no—
237D12—81no—
238D12—81no—
239C3⋊C8—81no—
240C3⋊C8—81no—
241C24—21no—
242C24—21no—
243C24—21no—
244C24—31no—
245C24—31no—
246C24—61no—
247C24—61no—
248C24—61no—
249C24—61no—
250C24C2×#6711yescharacteristic
251C2×D4—61no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—61no—
257C2×D4—121no—
258C2×D4—121no—
259C2×D4—122no—
260C2×D4—121no—
261C2×D4—121no—
262C2×D4—122no—
263C2×D4—122no—
264C2×D4—122no—
265C2×D4—122no—
266C2×D4—121no—
267C2×D4—121no—
268C2×D4—121no—
269C2×D4—121no—
270C22×C4—21no—
271C22×C4—31no—
272C22×C4—31no—
273C22×C4—31no—
274C22×C4—31no—
275C22×C4—31no—
276C22×C4—32no—
277C22×C4—32no—
278C22×C4—31no—
279C22×C4—31no—
280C22×C4—31no—
281C22×C4—31no—
282C22×C4—41no—
283C22×C4—41no—
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—62no—
293C22×C4—61no—
294C22×C4—61no—
295C22×C4—61no—
296C22×C4—61no—
297C22×C4—61no—
298C22×C4—61no—
299C22×C4—62no—
300C22×C4—61no—
301C22×C4—61no—
302C22×C4—61no—
303C22×C4—61no—
304C22×C4—61no—
305C22×C4—121no—
306C22×C4—121no—
307C22×C4#23511yescharacteristic
308C22×C4#23511yescharacteristic
309C22×C4C2×S411yescharacteristic
310C2×C8—61no—
311C2×C8—61no—
312C2×C8—61no—
313C2×C8—61no—
314C2×C8—61no—
315C2×C8—61no—
316C2×C8—61no—
317C2×C8—61no—
318C2×C8—61no—
319C2×C8—61no—
320C2×C8—61no—
321C2×C8—61no—
322C4⋊C4—32no—
323C4⋊C4—32no—
324C4⋊C4—31no—
325C4⋊C4—31no—
326C4⋊C4—61no—
327C4⋊C4—61no—
328C4⋊C4—61no—
329C4⋊C4—61no—
330C4⋊C4—61no—
331C4⋊C4—61no—
332C4⋊C4—61no—
333C4⋊C4—61no—
334C4⋊C4C2×S411yescharacteristic
335C4⋊C4C2×S411yescharacteristic
336C22⋊C4—21no—
337C22⋊C4—61no—
338C22⋊C4—61no—
339C22⋊C4—62no—
340C22⋊C4—61no—
341C22⋊C4—121no—
342C22⋊C4—121no—
343C22⋊C4—122no—
344C22⋊C4—121no—
345C22⋊C4—121no—
346C22⋊C4—122no—
347C22⋊C4—122no—
348C22⋊C4—121no—
349C22⋊C4—121no—
350C42—21no—
351C42—61no—
352C42—61no—
353C42—62no—
354C42—61no—
355C22×C3—41no—
356C22×C3—81no—
357C22×C3—81no—
358C22×C3—81no—
359C2×D3—161no—
360C2×D3—161no—
361C2×D3—161no—
362A4#48211yescharacteristic, complemented
363C3×C4—41no—
364C3×C4—41no—
365C3×C4—81no—
366C3×C4—161no—
367C3×C4—161no—
368Dic3—161no—
369C23—31no—
370C23—31no—
371C23—31no—
372C23—31no—
373C23—32no—
374C23—31no—
375C23—41no—
376C23—61no—
377C23—61no—
378C23—61no—
379C23—61no—
380C23—61no—
381C23—61no—
382C23—61no—
383C23—61no—
384C23—61no—
385C23—61no—
386C23—61no—
387C23—121no—
388C23—121no—
389C23—121no—
390C23—121no—
391C23—121no—
392C23—121no—
393C23—121no—
394C23—121no—
395C23—121no—
396C23—122no—
397C23—121no—
398C23#92911yescharacteristic
399C23#94911yescharacteristic
400C23#95011yescharacteristic
401C23GL(2,ℤ/4)11yescharacteristic
402D4—121no—
403D4—121no—
404D4—121no—
405D4—121no—
406D4—242no—
407D4—241no—
408D4—241no—
409D4—241no—
410D4—244no—
411D4—242no—
412D4—241no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—31no—
418C2×C4—31no—
419C2×C4—31no—
420C2×C4—31no—
421C2×C4—31no—
422C2×C4—31no—
423C2×C4—31no—
424C2×C4—32no—
425C2×C4—31no—
426C2×C4—61no—
427C2×C4—61no—
428C2×C4—61no—
429C2×C4—61no—
430C2×C4—61no—
431C2×C4—61no—
432C2×C4—61no—
433C2×C4—62no—
434C2×C4—62no—
435C2×C4—61no—
436C2×C4—61no—
437C2×C4—61no—
438C2×C4—61no—
439C2×C4—61no—
440C2×C4—121no—
441C2×C4—121no—
442C2×C4—121no—
443C2×C4—121no—
444C2×C4—121no—
445C2×C4—121no—
446C2×C4—121no—
447C2×C4—121no—
448C2×C4—121no—
449C2×C4—121no—
450C2×C4—121no—
451C2×C4—121no—
452C2×C4—121no—
453C2×C4—121no—
454C2×C4—121no—
455C2×C4—121no—
456C2×C4—121no—
457C2×C4GL(2,ℤ/4)11yescharacteristic
458C2×C4C22×S411yescharacteristic
459C8—121no—
460C8—121no—
461C8—121no—
462C8—121no—
463C2×C3—41no—
464C2×C3—41no—
465C2×C3—41no—
466C2×C3—161no—
467D3—321no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—31no—
472C22—31no—
473C22—31no—
474C22—31no—
475C22—31no—
476C22—31no—
477C22—31no—
478C22—31no—
479C22—31no—
480C22—61no—
481C22—62no—
482C22—62no—
483C22—61no—
484C22—61no—
485C22—61no—
486C22—121no—
487C22—121no—
488C22—121no—
489C22—121no—
490C22—121no—
491C22—121no—
492C22—241no—
493C22—241no—
494C22—241no—
495C22—241no—
496C22—241no—
497C22—241no—
498C22—241no—
499C22#500311yescharacteristic, complemented
500C22C2×GL(2,ℤ/4)11yescharacteristic
501C4—21no—
502C4—31no—
503C4—31no—
504C4—41no—
505C4—41no—
506C4—61no—
507C4—121no—
508C4—121no—
509C4—241no—
510C4—241no—
511C4—241no—
512C4#591211yescharacteristic
513C4#591211yescharacteristic
514C3—41noSylow
515C2—31no—
516C2—31no—
517C2—31no—
518C2—31no—
519C2—41no—
520C2—121no—
521C2—481no—
522C2#8548511yescharacteristic
523C2#8551311yescharacteristic
524C2#8551411yescharacteristic
subgroupG/H
centerC22C2×GL(2,ℤ/4)
commutator, cosocleC2×C4×A4C23
FrattiniC2×C4C22×S4
FittingC22×#117D3
socleC24C2×#67
prime radical#85486C2
hypercenter#117S4
hypocenterA4#482
LCSG ▷ C2×C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. A4×#117C2semidirect
3. #73141C2semidirect
4. #73053C2semidirect
5. #73053C2semidirect
6. #73051C2semidirect
7. #73050C2semidirect
21. C22×#117D3semidirect
173. #117S4semidirect
362. A4#482semidirect
499. C22#5003semidirect

Maximal subgroups

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

subgroupindex
A4×#1172
#854862
#731412
#730532
#730512
#730502
#329813
#50034