Finite Group Databasebeta

← #11711767

Subgroups — #11711767

Order 768.

1015 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 1015 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C22×A4⋊Q8C211yesmaximal, characteristic
2C2×C4⋊SL(2,ℤ/4)C212yesmaximal
3C2×SL(2,ℤ/4)⋊C4C211yesmaximal, characteristic
4A4×#95C211yesmaximal, characteristic
5A4⋊#95C212yesmaximal
6#14758—31nomaximal, Sylow
7C22×C4×A4C2211yescharacteristic
8C22×C4×A4C2212yes—
9C22×SL(2,ℤ/4)C2211yescharacteristic
10C22×SL(2,ℤ/4)C2212yes—
11C22×SL(2,ℤ/4)C2211yescharacteristic
12C2×A4⋊Q8—22no—
13C2×A4⋊Q8—24no—
14C4⋊SL(2,ℤ/4)—24no—
15SL(2,ℤ/4)⋊C4—22no—
16#4656—41nomaximal
17C22×C22⋊Q8—31no—
18C2×#395—31no—
19C2×#385—31no—
20C2×#380—32no—
21C22×#95D311yescharacteristic
22#1446—31no—
23#1444—32no—
24#1444—31no—
25#1441—31no—
26#1441—32no—
27C22⋊#95—32no—
28C23×A4C2311yescharacteristic
29C22×Dic6—41no—
30C2×C4×A4—22no—
31C2×C4×A4—22no—
32C2×C4×A4—22no—
33C2×C4×A4D412yes—
34C2×C4×A4Q814yescomplemented
35C2×SL(2,ℤ/4)—22no—
36C2×SL(2,ℤ/4)—22no—
37C2×SL(2,ℤ/4)—21no—
38C2×SL(2,ℤ/4)—21no—
39C2×SL(2,ℤ/4)—21no—
40C2×SL(2,ℤ/4)—22no—
41C2×SL(2,ℤ/4)D414yes—
42C2×SL(2,ℤ/4)Q812yescomplemented
43A4⋊Q8—28no—
44C2×C4⋊Dic3—42no—
45C2×Dic3⋊C4—41no—
46C3×#95—41no—
47C3⋊#95—42no—
48C23×Q8—31no—
49C24×C4C2×D312yes—
50C24×C4C2×D311yescharacteristic
51C2×C22⋊Q8—64no—
52C2×C22⋊Q8—64no—
53C2×C22⋊Q8—62no—
54C2×C22⋊Q8—62no—
55C22×C4⋊C4—31no—
56C22×C4⋊C4—31no—
57C22×C4⋊C4—32no—
58C22×C4⋊C4—32no—
59C22×C4⋊C4—32no—
60C22×C4⋊C4—31no—
61C22×C22⋊C4—32no—
62C22×C22⋊C4—31no—
63C22×C22⋊C4—31no—
64#395—61no—
65#395—62no—
66#395—61no—
67#385—62no—
68#385—62no—
69#380—64no—
70#380—64no—
71C2×#95—31no—
72C2×#95—32no—
73C2×#95—31no—
74C2×#95—32no—
75C2×#95—32no—
76C2×#95—31no—
77C2×#95—32no—
78C2×#95—32no—
79C2×#95—31no—
80C2×#95—32no—
81C2×#95—32no—
82C22×A4C2×D411yescharacteristic
83C22×A4C2×D412yes—
84C22×A4C2×Q812yes—
85C22×A4C2×Q811yescharacteristic
86C22×A4#4111yescharacteristic
87C22×C3×C4—42no—
88C22×C3×C4—41no—
89C22×Dic3—41no—
90C22×Dic3—42no—
91C22×Dic3—41no—
92C2×Dic6—84no—
93C2×Dic6—82no—
94C4×A4—22no—
95C4×A4—24no—
96SL(2,ℤ/4)—24no—
97SL(2,ℤ/4)—22no—
98SL(2,ℤ/4)—41no—
99C4⋊Dic3—84no—
100Dic3⋊C4—82no—
101C25C22×D311yescharacteristic
102C22×Q8—62no—
103C22×Q8—64no—
104C22×Q8—62no—
105C22×Q8—61no—
106C22×Q8—61no—
107C22×Q8—122no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—31no—
112C23×C4—32no—
113C23×C4—31no—
114C23×C4—32no—
115C23×C4—32no—
116C23×C4—34no—
117C23×C4—32no—
118C23×C4—32no—
119C23×C4—31no—
120C23×C4—31no—
121C23×C4—31no—
122C23×C4—31no—
123C23×C4—62no—
124C23×C4—62no—
125C23×C4—62no—
126C23×C4Dic614yescomplemented
127C23×C4D1212yes—
128C22⋊Q8—68no—
129C22⋊Q8—68no—
130C22⋊Q8—616no—
131C2×C4⋊C4—62no—
132C2×C4⋊C4—64no—
133C2×C4⋊C4—64no—
134C2×C4⋊C4—64no—
135C2×C4⋊C4—64no—
136C2×C4⋊C4—62no—
137C2×C4⋊C4—64no—
138C2×C4⋊C4—64no—
139C2×C4⋊C4—64no—
140C2×C4⋊C4—62no—
141C2×C4⋊C4—62no—
142C2×C4⋊C4—61no—
143C2×C4⋊C4—62no—
144C2×C4⋊C4—61no—
145C2×C4⋊C4—62no—
146C2×C4⋊C4—122no—
147C2×C4⋊C4—122no—
148C2×C4⋊C4—122no—
149C2×C22⋊C4—32no—
150C2×C22⋊C4—34no—
151C2×C22⋊C4—34no—
152C2×C22⋊C4—32no—
153C2×C22⋊C4—61no—
154C2×C22⋊C4—62no—
155C2×C22⋊C4—62no—
156C2×C22⋊C4—61no—
157C2×C22⋊C4—62no—
158C2×C22⋊C4—61no—
159C2×C22⋊C4—62no—
160C2×C22⋊C4—61no—
161C2×C22⋊C4—62no—
162C2×C22⋊C4—61no—
163C2×C22⋊C4—61no—
164C2×C22⋊C4—62no—
165#95—31no—
166#95—32no—
167#95—61no—
168#95—64no—
169#95—64no—
170#95—64no—
171#95—62no—
172#95—62no—
173#95—62no—
174#95—62no—
175#95—122no—
176#95—122no—
177#95S411yescharacteristic
178C23×C3—41no—
179C2×A4#12011yescharacteristic
180C2×A4C22⋊Q812yes—
181C2×A4C22⋊Q811yescharacteristic
182C2×A4C4⋊Q811yescharacteristic
183C2×A4C4⋊Q812yes—
184C2×C3×C4—44no—
185C2×C3×C4—42no—
186C2×C3×C4—82no—
187C2×C3×C4—82no—
188C2×C3×C4—82no—
189C2×Dic3—44no—
190C2×Dic3—42no—
191C2×Dic3—81no—
192C2×Dic3—82no—
193C2×Dic3—82no—
194C2×Dic3—81no—
195C2×Dic3—81no—
196C2×Dic3—82no—
197Dic6—88no—
198C24—31no—
199C24—32no—
200C24—31no—
201C24—31no—
202C24—31no—
203C24—32no—
204C24C2×Dic612yes—
205C24C2×D1211yescharacteristic
206C24D3×D412yes—
207C24#23711yescharacteristic
208C24D3×Q811yescharacteristic
209C2×Q8—68no—
210C2×Q8—124no—
211C2×Q8—122no—
212C2×Q8—128no—
213C2×Q8—124no—
214C2×Q8—124no—
215C2×Q8—122no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—32no—
219C22×C4—31no—
220C22×C4—34no—
221C22×C4—34no—
222C22×C4—32no—
223C22×C4—32no—
224C22×C4—32no—
225C22×C4—34no—
226C22×C4—34no—
227C22×C4—34no—
228C22×C4—32no—
229C22×C4—32no—
230C22×C4—62no—
231C22×C4—62no—
232C22×C4—62no—
233C22×C4—62no—
234C22×C4—62no—
235C22×C4—62no—
236C22×C4—62no—
237C22×C4—61no—
238C22×C4—62no—
239C22×C4—64no—
240C22×C4—62no—
241C22×C4—62no—
242C22×C4—62no—
243C22×C4—62no—
244C22×C4—62no—
245C22×C4—64no—
246C22×C4—64no—
247C22×C4—64no—
248C22×C4—62no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4—62no—
252C22×C4—62no—
253C22×C4—61no—
254C22×C4—62no—
255C22×C4—62no—
256C22×C4—62no—
257C22×C4—62no—
258C22×C4—62no—
259C22×C4—62no—
260C22×C4—62no—
261C22×C4—61no—
262C22×C4—61no—
263C22×C4—62no—
264C22×C4—61no—
265C22×C4—62no—
266C22×C4—62no—
267C22×C4—61no—
268C22×C4—62no—
269C22×C4—61no—
270C22×C4—62no—
271C22×C4—61no—
272C22×C4—62no—
273C22×C4—62no—
274C22×C4—61no—
275C22×C4—62no—
276C22×C4—61no—
277C22×C4—61no—
278C22×C4—62no—
279C22×C4—61no—
280C22×C4—61no—
281C22×C4—124no—
282C22×C4—122no—
283C22×C4C2×S412yes—
284C22×C4C2×S411yescharacteristic
285C4⋊C4—68no—
286C4⋊C4—616no—
287C4⋊C4—124no—
288C4⋊C4—122no—
289C4⋊C4—128no—
290C4⋊C4—124no—
291C4⋊C4—124no—
292C4⋊C4—122no—
293C22⋊C4—62no—
294C22⋊C4—62no—
295C22⋊C4—62no—
296C22⋊C4—64no—
297C22⋊C4—64no—
298C22⋊C4—64no—
299C22⋊C4—64no—
300C22⋊C4—62no—
301C22⋊C4—121no—
302C22⋊C4—122no—
303C22⋊C4—121no—
304C22×C3—42no—
305C22×C3—41no—
306C22×C3—42no—
307C22×C3—41no—
308C22×C3—41no—
309A4#39511yescharacteristic, complemented
310C3×C4—82no—
311C3×C4—84no—
312Dic3—84no—
313Dic3—82no—
314Dic3—161no—
315C23—32no—
316C23—32no—
317C23—31no—
318C23—31no—
319C23—32no—
320C23—32no—
321C23—31no—
322C23—31no—
323C23—31no—
324C23—31no—
325C23—31no—
326C23—32no—
327C23—31no—
328C23—31no—
329C23—31no—
330C23—32no—
331C23—32no—
332C23—32no—
333C23—32no—
334C23—32no—
335C23—31no—
336C23—32no—
337C23—32no—
338C23—62no—
339C23—61no—
340C23—61no—
341C23—62no—
342C23—61no—
343C23#86911yescharacteristic
344C23#87812yes—
345C23#88211yescharacteristic
346C23#88812yes—
347C23#89711yescharacteristic
348C23C22×S411yescharacteristic
349Q8—128no—
350Q8—128no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—34no—
355C2×C4—32no—
356C2×C4—62no—
357C2×C4—64no—
358C2×C4—64no—
359C2×C4—64no—
360C2×C4—64no—
361C2×C4—62no—
362C2×C4—64no—
363C2×C4—62no—
364C2×C4—62no—
365C2×C4—62no—
366C2×C4—62no—
367C2×C4—62no—
368C2×C4—64no—
369C2×C4—64no—
370C2×C4—64no—
371C2×C4—64no—
372C2×C4—62no—
373C2×C4—62no—
374C2×C4—64no—
375C2×C4—64no—
376C2×C4—64no—
377C2×C4—62no—
378C2×C4—64no—
379C2×C4—64no—
380C2×C4—62no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4—122no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4—122no—
387C2×C4—122no—
388C2×C4—122no—
389C2×C4—122no—
390C2×C4—122no—
391C2×C4—121no—
392C2×C4—121no—
393C2×C4—122no—
394C2×C4—122no—
395C2×C4—121no—
396C2×C4—122no—
397C2×C4—122no—
398C2×C4—121no—
399C2×C4—121no—
400C2×C4—121no—
401C2×C4—122no—
402C2×C4—121no—
403C2×C4—121no—
404C2×C4—122no—
405C2×C4—121no—
406C2×C4—122no—
407C2×C4—121no—
408C2×C4—122no—
409C2×C4A4⋊Q814yescomplemented
410C2×C4C4⋊S412yes—
411C2×C3—41no—
412C2×C3—42no—
413C2×C3—41no—
414C2×C3—42no—
415C2×C3—41no—
416C22—31no—
417C22—32no—
418C22—31no—
419C22—32no—
420C22—31no—
421C22—31no—
422C22—31no—
423C22—32no—
424C22—32no—
425C22—32no—
426C22—32no—
427C22—32no—
428C22—31no—
429C22—31no—
430C22—32no—
431C22—32no—
432C22—32no—
433C22—32no—
434C22—32no—
435C22—31no—
436C22—31no—
437C22—31no—
438C22—31no—
439C22—61no—
440C22—62no—
441C22—61no—
442C22—61no—
443C22—62no—
444C22#465611yescharacteristic, complemented
445C22C2×A4⋊Q812yes—
446C22C2×C4⋊S411yescharacteristic
447C22D4×S412yes—
448C22#591411yescharacteristic
449C22Q8×S411yescharacteristic
450C4—24no—
451C4—22no—
452C4—64no—
453C4—62no—
454C4—124no—
455C4—122no—
456C4—124no—
457C4—122no—
458C4—241no—
459C4—241no—
460C3—41noSylow
461C2—31no—
462C2—31no—
463C2—31no—
464C2—31no—
465C2—32no—
466C2—32no—
467C2#8528311yescharacteristic
468C2#8529312yes—
469C2#8529711yescharacteristic
470C2#8530312yes—
471C2#8531211yescharacteristic
subgroupG/H
center, FrattiniC23C22×S4
commutator, prime radical, cosocleC23×A4C23
FittingC22×#95D3
socleC25C22×D3
hypercenter#95S4
hypocenterA4#395
LCSG ▷ C23×A4 ▷ A4
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
34. C2×C4×A4Q8semidirect
42. C2×SL(2,ℤ/4)Q8semidirect
126. C23×C4Dic6semidirect
309. A4#395semidirect
409. C2×C4A4⋊Q8semidirect
444. C22#4656semidirect

Maximal subgroups

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

subgroupindex
C22×A4⋊Q82
C2×C4⋊SL(2,ℤ/4)2
C2×SL(2,ℤ/4)⋊C42
A4×#952
A4⋊#952
#147583
#46564