Finite Group Databasebeta

← #11714037

Subgroups — #11714037

Order 768.

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

normal subgroups only

This group has 523 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
2#85480C211yesmaximal, characteristic
3#73141C211yesmaximal, characteristic
4#73052C211yesmaximal, characteristic, complemented
5#73052C211yesmaximal, characteristic, complemented
6#73051C211yesmaximal, characteristic, complemented
7#73050C211yesmaximal, characteristic, complemented
8#32985—31nomaximal, Sylow
9C22×C4×A4C2211yescharacteristic
10C2×A4⋊Q8C2211yescharacteristic
11A4×C4⋊C4C2211yescharacteristic
12A4×C4⋊C4C2211yescharacteristic
13A4×C22⋊C4—21no—
14C42×A4—21no—
15#5432—21no—
16C4⋊SL(2,ℤ/4)C2211yescharacteristic
17SL(2,ℤ/4)⋊C4—21no—
18C2×A4⋊C8C2211yescharacteristic
19C2×A4⋊C8C2211yescharacteristic
20#4996—41nomaximal
21C22×#117D311yescharacteristic
22C2×#485—31no—
23C2×#485—31no—
24#2848—31no—
25#1873—31no—
26#1873—31no—
27#1872—31no—
28#1872—31no—
29#1854—31no—
30#1854—31no—
31#1853—31no—
32#1852—31no—
33#1825—31no—
34#1825—31no—
35#1779—31no—
36C23×A4D411yescharacteristic
37C2×C4×A4—21no—
38C2×C4×A4—21no—
39C2×C4×A4—21no—
40C2×C4×A4—21no—
41C2×C4×A4D411yescharacteristic
42C2×C4×A4C2311yescharacteristic
43C2×SL(2,ℤ/4)—21no—
44C2×SL(2,ℤ/4)—21no—
45A4⋊Q8—21no—
46A4⋊Q8—21no—
47C3×#117—41no—
48#924—41no—
49A4⋊C8—21no—
50A4⋊C8—21no—
51#830—41no—
52#810—41no—
53#810—41no—
54#808—41no—
55#807—41no—
56C24×C4C2×D311yescharacteristic
57C2×C22⋊Q8—31no—
58C2×C22⋊Q8—31no—
59C2×C22⋊Q8—31no—
60C2×#117—31no—
61C2×#117—31no—
62C2×#117—31no—
63C2×#117—31no—
64C2×#117—61no—
65C2×#117—61no—
66C22×C4⋊C4C2×D311yescharacteristic
67C22×C4⋊C4C2×D311yescharacteristic
68C22×C22⋊C4—21no—
69C22×C42—21no—
70#485—62no—
71#485—62no—
72#485—61no—
73#485—61no—
74#485—61no—
75#485—61no—
76C2×#107—31no—
77C2×#107—31no—
78C2×#106—31no—
79C2×#106—31no—
80C2×#103—31no—
81C2×#103—31no—
82C2×#103—31no—
83C2×#103—31no—
84C2×C22⋊C8—31no—
85C2×C22⋊C8—31no—
86C2×C22⋊C8—31no—
87C2×C22⋊C8—31no—
88C2×C22⋊C8—31no—
89C2×C22⋊C8—31no—
90#385—61no—
91#385—61no—
92#385—61no—
93#380—31no—
94#380—31no—
95#380—31no—
96#379—61no—
97#340—31no—
98#340—31no—
99#340—31no—
100#340—31no—
101#340—31no—
102#340—31no—
103#340—31no—
104#340—31no—
105C22×A4—21no—
106C22×A4—21no—
107C22×A4—21no—
108C22×A4C2×D411yescharacteristic
109C22×C3×C4—41no—
110C2×Dic6—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—41no—
120C3×C22⋊C4—81no—
121C3×C42—81no—
122#217—81no—
123C4⋊Dic3—41no—
124Dic3⋊C4—81no—
125C2×C3⋊C8—41no—
126C2×C3⋊C8—41no—
127C25#6711yescharacteristic
128C22×Q8—31no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—31no—
134C23×C4—31no—
135C23×C4—31no—
136C23×C4—31no—
137C23×C4—61no—
138C23×C4—61no—
139C23×C4#6711yescharacteristic
140C23×C4C22×D311yescharacteristic
141C22×C8—31no—
142C22×C8—31no—
143C22×C8—31no—
144C22×C8—31no—
145C22⋊Q8—61no—
146C22⋊Q8—61no—
147C22⋊Q8—62no—
148C22⋊Q8—62no—
149C22⋊Q8—61no—
150C22⋊Q8—61no—
151C22⋊Q8—61no—
152C22⋊Q8—61no—
153C22⋊Q8—61no—
154C22⋊Q8—61no—
155#117—32no—
156#117—32no—
157#117—31no—
158#117—31no—
159#117—31no—
160#117—61no—
161#117—61no—
162#117—61no—
163#117—62no—
164#117—62no—
165#117S411yescharacteristic
166C2×C4⋊C4—31no—
167C2×C4⋊C4—31no—
168C2×C4⋊C4—31no—
169C2×C4⋊C4—31no—
170C2×C4⋊C4—31no—
171C2×C4⋊C4—31no—
172C2×C4⋊C4—31no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C4⋊C4—61no—
178C2×C22⋊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×C42—61no—
189C2×C42—61no—
190C2×C42—61no—
191C2×C42—61no—
192#107—62no—
193#107—61no—
194#107—61no—
195#106—62no—
196#106—61no—
197#106—61no—
198#103—62no—
199#103—62no—
200#103—61no—
201#103—61no—
202#103—61no—
203#103—61no—
204C22⋊C8—61no—
205C22⋊C8—61no—
206C22⋊C8—62no—
207C22⋊C8—62no—
208C22⋊C8—61no—
209C22⋊C8—61no—
210C22⋊C8—61no—
211C22⋊C8—61no—
212C22⋊C8—61no—
213C22⋊C8—61no—
214#95—61no—
215#95—61no—
216#95—61no—
217C23×C3—41no—
218C2×A4—41no—
219C2×A4#12311yescharacteristic
220C2×A4#13511yescharacteristic
221C2×A4#13711yescharacteristic
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—81no—
225C2×C3×C4—81no—
226C2×C3×C4—81no—
227C2×C3×C4—81no—
228C2×Dic3—81no—
229C2×Dic3—81no—
230Dic6—81no—
231Dic6—81no—
232C3⋊C8—81no—
233C3⋊C8—81no—
234C24—21no—
235C24—21no—
236C24—21no—
237C24—31no—
238C24—31no—
239C24—61no—
240C24—61no—
241C24—61no—
242C24C2×#6711yescharacteristic
243C2×Q8—61no—
244C2×Q8—61no—
245C2×Q8—61no—
246C2×Q8—61no—
247C2×Q8—61no—
248C2×Q8—61no—
249C22×C4—21no—
250C22×C4—31no—
251C22×C4—31no—
252C22×C4—31no—
253C22×C4—31no—
254C22×C4—31no—
255C22×C4—32no—
256C22×C4—32no—
257C22×C4—31no—
258C22×C4—31no—
259C22×C4—31no—
260C22×C4—31no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—61no—
264C22×C4—61no—
265C22×C4—61no—
266C22×C4—61no—
267C22×C4—61no—
268C22×C4—61no—
269C22×C4—61no—
270C22×C4—61no—
271C22×C4—62no—
272C22×C4—61no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4—61no—
277C22×C4—61no—
278C22×C4—62no—
279C22×C4—61no—
280C22×C4—61no—
281C22×C4—61no—
282C22×C4—61no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—121no—
286C22×C4—121no—
287C22×C4#23511yescharacteristic
288C22×C4#23511yescharacteristic
289C22×C4C2×S411yescharacteristic
290C2×C8—61no—
291C2×C8—61no—
292C2×C8—61no—
293C2×C8—61no—
294C2×C8—61no—
295C2×C8—61no—
296C2×C8—61no—
297C2×C8—61no—
298C2×C8—61no—
299C2×C8—61no—
300C2×C8—61no—
301C2×C8—61no—
302C4⋊C4—32no—
303C4⋊C4—32no—
304C4⋊C4—31no—
305C4⋊C4—31no—
306C4⋊C4—61no—
307C4⋊C4—61no—
308C4⋊C4—61no—
309C4⋊C4—61no—
310C4⋊C4—61no—
311C4⋊C4—61no—
312C4⋊C4—61no—
313C4⋊C4—61no—
314C4⋊C4—121no—
315C4⋊C4—121no—
316C4⋊C4—122no—
317C4⋊C4—121no—
318C4⋊C4—121no—
319C4⋊C4C2×S411yescharacteristic
320C4⋊C4C2×S411yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—61no—
323C22⋊C4—61no—
324C22⋊C4—62no—
325C22⋊C4—61no—
326C22⋊C4—121no—
327C22⋊C4—121no—
328C22⋊C4—122no—
329C22⋊C4—121no—
330C22⋊C4—122no—
331C22⋊C4—122no—
332C22⋊C4—121no—
333C22⋊C4—121no—
334C22⋊C4—121no—
335C42—21no—
336C42—61no—
337C42—61no—
338C42—62no—
339C42—61no—
340C22×C3—41no—
341C22×C3—81no—
342C22×C3—81no—
343C22×C3—81no—
344A4#48511yescharacteristic, complemented
345C3×C4—41no—
346C3×C4—41no—
347C3×C4—81no—
348C3×C4—161no—
349C3×C4—161no—
350Dic3—161no—
351Dic3—161no—
352C23—31no—
353C23—31no—
354C23—31no—
355C23—31no—
356C23—32no—
357C23—31no—
358C23—41no—
359C23—61no—
360C23—61no—
361C23—61no—
362C23—61no—
363C23—61no—
364C23—61no—
365C23—61no—
366C23—61no—
367C23—61no—
368C23—61no—
369C23—61no—
370C23—121no—
371C23—121no—
372C23—121no—
373C23—122no—
374C23#92911yescharacteristic
375C23#95011yescharacteristic
376C23#95111yescharacteristic
377C23GL(2,ℤ/4)11yescharacteristic
378Q8—121no—
379Q8—121no—
380Q8—121no—
381Q8—121no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—31no—
387C2×C4—31no—
388C2×C4—31no—
389C2×C4—31no—
390C2×C4—31no—
391C2×C4—31no—
392C2×C4—31no—
393C2×C4—32no—
394C2×C4—31no—
395C2×C4—61no—
396C2×C4—61no—
397C2×C4—61no—
398C2×C4—61no—
399C2×C4—61no—
400C2×C4—61no—
401C2×C4—61no—
402C2×C4—62no—
403C2×C4—62no—
404C2×C4—61no—
405C2×C4—61no—
406C2×C4—61no—
407C2×C4—61no—
408C2×C4—61no—
409C2×C4—121no—
410C2×C4—121no—
411C2×C4—121no—
412C2×C4—121no—
413C2×C4—121no—
414C2×C4—121no—
415C2×C4—121no—
416C2×C4—121no—
417C2×C4—121no—
418C2×C4—121no—
419C2×C4—121no—
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×C4GL(2,ℤ/4)11yescharacteristic
430C2×C4C22×S411yescharacteristic
431C8—121no—
432C8—121no—
433C8—121no—
434C8—121no—
435C2×C3—41no—
436C2×C3—41no—
437C2×C3—41no—
438C2×C3—161no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—31no—
443C22—31no—
444C22—31no—
445C22—31no—
446C22—31no—
447C22—31no—
448C22—31no—
449C22—31no—
450C22—31no—
451C22—61no—
452C22—62no—
453C22—62no—
454C22—61no—
455C22—61no—
456C22—61no—
457C22—121no—
458C22—121no—
459C22—121no—
460C22—121no—
461C22—121no—
462C22—121no—
463C22#499611yescharacteristic, complemented
464C22C2×GL(2,ℤ/4)11yescharacteristic
465C4—21no—
466C4—31no—
467C4—31no—
468C4—41no—
469C4—41no—
470C4—61no—
471C4—121no—
472C4—121no—
473C4—241no—
474C4—241no—
475C4—241no—
476C4—241no—
477C4#591211yescharacteristic
478C4#591211yescharacteristic
479C3—41noSylow
480C2—31no—
481C2—31no—
482C2—31no—
483C2—31no—
484C2—41no—
485C2—121no—
486C2#8548511yescharacteristic
487C2#8551211yescharacteristic
488C2#8551411yescharacteristic
subgroupG/H
centerC22C2×GL(2,ℤ/4)
commutator, cosocleC2×C4×A4C23
FrattiniC2×C4C22×S4
FittingC22×#117D3
socleC24C2×#67
prime radicalC23×A4D4
hypercenter#117S4
hypocenterA4#485
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
4. #73052C2semidirect
5. #73052C2semidirect
6. #73051C2semidirect
7. #73050C2semidirect
344. A4#485semidirect
463. C22#4996semidirect

Maximal subgroups

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

subgroupindex
A4×#1172
#854802
#731412
#730522
#730512
#730502
#329853
#49964