Finite Group Databasebeta

← #11714578

Subgroups — #11714578

Order 768.

1099 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 1099 conjugacy classes of subgroups — too many to draw. Only its 131 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×C4⋊Q8C211yesmaximal, characteristic
2Q8×SL(2,ℤ/4)C212yesmaximal
3#85502C212yesmaximal
4#85305C212yesmaximal
5#85304C214yesmaximal
6#85302C212yesmaximal
7C4×A4⋊Q8C211yesmaximal, characteristic
8C4×A4⋊Q8C211yesmaximal, characteristic
9#59144—31nomaximal, Sylow
10C2×Q8×A4C2212yes—
11C2×A4⋊Q8C2211yescharacteristic
12C2×A4⋊Q8C2211yescharacteristic
13#5721—41nomaximal
14A4×C4⋊C4C2212yes—
15A4×C4⋊C4C2212yes—
16C42×A4C2211yescharacteristic
17C4⋊SL(2,ℤ/4)C2211yescharacteristic
18C4⋊SL(2,ℤ/4)C2214yes—
19C4⋊SL(2,ℤ/4)C2211yescharacteristic
20SL(2,ℤ/4)⋊C4C2212yes—
21SL(2,ℤ/4)⋊C4C2212yes—
22SL(2,ℤ/4)⋊C4C2214yes—
23SL(2,ℤ/4)⋊C4C2214yes—
24C4×SL(2,ℤ/4)C2212yes—
25C4×SL(2,ℤ/4)C2214yes—
26C4×SL(2,ℤ/4)C2212yes—
27C2×#557—31no—
28C22×C4⋊Q8D311yescharacteristic
29#2713—34no—
30#2568—31no—
31#2550—32no—
32#2546—31no—
33#2546—32no—
34#2473—34no—
35#2451—32no—
36#2446—32no—
37#2442—32no—
38#2434—32no—
39#2365—31no—
40#2360—32no—
41Q8×C22⋊C4—32no—
42C4×C22⋊Q8—31no—
43C4×C22⋊Q8—31no—
44Q8×A4—24no—
45C2×C4×A4C2314yes—
46C2×C4×A4C2311yescharacteristic
47C2×C4×A4C2311yescharacteristic
48C2×C4×A4C2311yescharacteristic
49C2×SL(2,ℤ/4)C2312yes—
50C2×SL(2,ℤ/4)C2312yes—
51C2×SL(2,ℤ/4)C2312yes—
52C2×SL(2,ℤ/4)C2312yes—
53A4⋊Q8—22no—
54A4⋊Q8Q814yescomplemented
55C3×C4⋊Q8—41no—
56Q8×Dic3—42no—
57#944—42no—
58#890—42no—
59#889—44no—
60#887—42no—
61C4×Dic6—41no—
62C4×Dic6—41no—
63C23×Q8C2×D312yes—
64#557—62no—
65#557—62no—
66#557—62no—
67#557—61no—
68#557—61no—
69C2×C4⋊Q8—31no—
70C2×C4⋊Q8—31no—
71C2×C4⋊Q8—32no—
72C2×C4⋊Q8—32no—
73C2×C4⋊Q8—32no—
74C2×C4⋊Q8—32no—
75C2×C4⋊Q8—32no—
76C2×#125—32no—
77C2×#125—34no—
78C2×C22⋊Q8—31no—
79C2×C22⋊Q8—31no—
80C2×C4×Q8—31no—
81C2×C4×Q8—32no—
82C2×C4×Q8—32no—
83C2×C4×Q8—31no—
84C2×C4×Q8—32no—
85C22×C4⋊C4C2×D312yes—
86C22×C4⋊C4C2×D312yes—
87C22×C42C2×D311yescharacteristic
88#400—34no—
89#400—34no—
90#398—34no—
91#398—34no—
92#398—34no—
93#398—34no—
94#395—34no—
95#395—34no—
96#391—32no—
97#391—32no—
98#391—32no—
99#391—32no—
100#391—34no—
101#391—34no—
102#391—32no—
103#391—31no—
104#391—31no—
105#389—34no—
106#389—32no—
107#389—32no—
108#389—32no—
109#389—31no—
110#389—34no—
111#389—31no—
112#387—34no—
113#387—34no—
114#387—34no—
115#387—32no—
116#387—32no—
117#385—34no—
118#385—34no—
119#385—32no—
120#385—32no—
121#382—32no—
122#380—31no—
123#380—34no—
124#380—31no—
125C4×C4⋊C4—31no—
126C4×C4⋊C4—32no—
127C4×C4⋊C4—31no—
128C4×C4⋊C4—32no—
129C4×C4⋊C4—32no—
130C4×C22⋊C4—34no—
131C4×C22⋊C4—32no—
132C4×C22⋊C4—32no—
133C22×A4C2411yescharacteristic
134C2×C3×Q8—42no—
135C2×Dic6—41no—
136C2×Dic6—41no—
137C4×A4—24no—
138C4×A4—21no—
139C4×A4C2×Q812yes—
140C4×A4#4112yes—
141SL(2,ℤ/4)—22no—
142SL(2,ℤ/4)—22no—
143SL(2,ℤ/4)—22no—
144SL(2,ℤ/4)C2×Q814yes—
145C3×C4⋊C4—42no—
146C3×C4⋊C4—42no—
147C3×C42—41no—
148C4⋊Dic3—41no—
149C4⋊Dic3—41no—
150C4⋊Dic3—44no—
151Dic3⋊C4—42no—
152Dic3⋊C4—42no—
153Dic3⋊C4—44no—
154Dic3⋊C4—44no—
155C4×Dic3—42no—
156C4×Dic3—42no—
157C4×Dic3—44no—
158C22×Q8—24no—
159C22×Q8—31no—
160C22×Q8—32no—
161C22×Q8—32no—
162C22×Q8—34no—
163C22×Q8—31no—
164C22×Q8—64no—
165C23×C4C22×D314yes—
166C23×C4C22×D311yescharacteristic
167C23×C4C22×D311yescharacteristic
168C23×C4C22×D311yescharacteristic
169C4⋊Q8—32no—
170C4⋊Q8—32no—
171C4⋊Q8—32no—
172C4⋊Q8—31no—
173C4⋊Q8—61no—
174C4⋊Q8—61no—
175C4⋊Q8—61no—
176C4⋊Q8—64no—
177C4⋊Q8—64no—
178C4⋊Q8—62no—
179C4⋊Q8—62no—
180C4⋊Q8—124no—
181C4⋊Q8S411yescharacteristic
182#125—64no—
183#125—64no—
184#125—64no—
185#125—62no—
186#125—64no—
187#125—64no—
188#125—62no—
189C22⋊Q8—34no—
190C22⋊Q8—34no—
191C22⋊Q8—62no—
192C22⋊Q8—62no—
193C4×Q8—61no—
194C4×Q8—61no—
195C4×Q8—61no—
196C4×Q8—61no—
197C4×Q8—61no—
198C4×Q8—61no—
199C4×Q8—62no—
200C4×Q8—64no—
201C4×Q8—62no—
202C4×Q8—64no—
203C4×Q8—62no—
204C4×Q8—62no—
205C4×Q8—61no—
206C4×Q8—61no—
207C4×Q8—124no—
208C2×C4⋊C4—31no—
209C2×C4⋊C4—32no—
210C2×C4⋊C4—32no—
211C2×C4⋊C4—32no—
212C2×C4⋊C4—32no—
213C2×C4⋊C4—34no—
214C2×C4⋊C4—34no—
215C2×C4⋊C4—32no—
216C2×C4⋊C4—31no—
217C2×C4⋊C4—31no—
218C2×C4⋊C4—31no—
219C2×C4⋊C4—34no—
220C2×C4⋊C4—32no—
221C2×C4⋊C4—32no—
222C2×C4⋊C4—34no—
223C2×C4⋊C4—32no—
224C2×C4⋊C4—31no—
225C2×C4⋊C4—34no—
226C2×C4⋊C4—31no—
227C2×C4⋊C4—32no—
228C2×C4⋊C4—32no—
229C2×C4⋊C4—34no—
230C2×C4⋊C4—34no—
231C2×C4⋊C4—34no—
232C2×C22⋊C4—32no—
233C2×C22⋊C4—32no—
234C2×C22⋊C4—32no—
235C2×C22⋊C4—32no—
236C2×C42—31no—
237C2×C42—31no—
238C2×C42—31no—
239C2×C42—31no—
240C2×C42—32no—
241C2×C42—34no—
242C2×C42—32no—
243C2×C42—32no—
244C2×C42—32no—
245C2×C42—34no—
246#95—32no—
247#95—32no—
248#95—32no—
249#95—34no—
250#95—32no—
251#95—34no—
252#95—32no—
253#95—32no—
254#95—32no—
255#95—32no—
256#95—34no—
257#95—32no—
258#95—32no—
259#95—34no—
260#95—34no—
261#95—34no—
262#95—34no—
263#95—34no—
264C2×A4C22×Q811yescharacteristic
265C2×A4C2×#4111yescharacteristic
266C2×A42−1+411yescharacteristic
267C3×Q8—84no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—44no—
271C2×C3×C4—41no—
272C2×Dic3—42no—
273C2×Dic3—42no—
274C2×Dic3—42no—
275C2×Dic3—42no—
276Dic6—44no—
277Dic6—82no—
278C24C23×D311yescharacteristic
279C2×Q8—34no—
280C2×Q8—34no—
281C2×Q8—32no—
282C2×Q8—62no—
283C2×Q8—64no—
284C2×Q8—64no—
285C2×Q8—61no—
286C2×Q8—62no—
287C2×Q8—64no—
288C2×Q8—64no—
289C2×Q8—68no—
290C2×Q8—61no—
291C2×Q8—62no—
292C2×Q8—62no—
293C2×Q8—61no—
294C2×Q8—61no—
295C2×Q8—124no—
296C2×Q8C2×S412yes—
297C22×C4—24no—
298C22×C4—21no—
299C22×C4—31no—
300C22×C4—31no—
301C22×C4—31no—
302C22×C4—31no—
303C22×C4—31no—
304C22×C4—32no—
305C22×C4—32no—
306C22×C4—32no—
307C22×C4—32no—
308C22×C4—31no—
309C22×C4—34no—
310C22×C4—34no—
311C22×C4—32no—
312C22×C4—32no—
313C22×C4—32no—
314C22×C4—32no—
315C22×C4—32no—
316C22×C4—32no—
317C22×C4—61no—
318C22×C4—64no—
319C22×C4#23712yes—
320C22×C4D3×Q812yes—
321C4⋊C4—34no—
322C4⋊C4—38no—
323C4⋊C4—34no—
324C4⋊C4—32no—
325C4⋊C4—34no—
326C4⋊C4—32no—
327C4⋊C4—62no—
328C4⋊C4—62no—
329C4⋊C4—62no—
330C4⋊C4—64no—
331C4⋊C4—64no—
332C4⋊C4—61no—
333C4⋊C4—61no—
334C4⋊C4—62no—
335C4⋊C4—62no—
336C4⋊C4—64no—
337C4⋊C4—64no—
338C4⋊C4—62no—
339C4⋊C4—62no—
340C4⋊C4—64no—
341C4⋊C4—64no—
342C4⋊C4—61no—
343C4⋊C4—62no—
344C4⋊C4—62no—
345C4⋊C4—64no—
346C4⋊C4—61no—
347C4⋊C4—64no—
348C4⋊C4—64no—
349C4⋊C4—124no—
350C4⋊C4—124no—
351C4⋊C4—124no—
352C4⋊C4C2×S412yes—
353C4⋊C4C2×S412yes—
354C22⋊C4—34no—
355C22⋊C4—34no—
356C22⋊C4—62no—
357C22⋊C4—62no—
358C22⋊C4—62no—
359C22⋊C4—62no—
360C22⋊C4—62no—
361C22⋊C4—62no—
362C42—31no—
363C42—31no—
364C42—31no—
365C42—61no—
366C42—64no—
367C42—62no—
368C42—64no—
369C42—62no—
370C42—64no—
371C42—64no—
372C42—62no—
373C42—62no—
374C42—124no—
375C42C2×S411yescharacteristic
376C22×C3—41no—
377A4#55711yescharacteristic, complemented
378C3×C4—42no—
379C3×C4—42no—
380C3×C4—84no—
381C3×C4—81no—
382Dic3—44no—
383Dic3—82no—
384Dic3—82no—
385Dic3—82no—
386C23—31no—
387C23—31no—
388C23—31no—
389C23—31no—
390C23C2×#23711yescharacteristic
391C23C2×D3×Q811yescharacteristic
392C23#100711yescharacteristic
393Q8—24no—
394Q8—68no—
395Q8—64no—
396Q8—64no—
397Q8—64no—
398Q8—124no—
399Q8—122no—
400Q8—122no—
401C2×C4—31no—
402C2×C4—31no—
403C2×C4—32no—
404C2×C4—32no—
405C2×C4—32no—
406C2×C4—32no—
407C2×C4—32no—
408C2×C4—32no—
409C2×C4—34no—
410C2×C4—34no—
411C2×C4—32no—
412C2×C4—31no—
413C2×C4—34no—
414C2×C4—64no—
415C2×C4—61no—
416C2×C4—64no—
417C2×C4—61no—
418C2×C4—61no—
419C2×C4—61no—
420C2×C4—62no—
421C2×C4—62no—
422C2×C4—62no—
423C2×C4—62no—
424C2×C4—64no—
425C2×C4—61no—
426C2×C4—64no—
427C2×C4—62no—
428C2×C4—62no—
429C2×C4—62no—
430C2×C4—62no—
431C2×C4—62no—
432C2×C4—62no—
433C2×C4—62no—
434C2×C4—62no—
435C2×C4—62no—
436C2×C4—62no—
437C2×C4—62no—
438C2×C4—62no—
439C2×C4—62no—
440C2×C4—62no—
441C2×C4—62no—
442C2×C4—62no—
443C2×C4—62no—
444C2×C4—62no—
445C2×C4C22×S411yescharacteristic
446C2×C4C22×S414yes—
447C2×C4C22×S411yescharacteristic
448C2×C4C22×S411yescharacteristic
449C2×C3—41no—
450C2×C3—41no—
451C2×C3—41no—
452C22—31no—
453C22—31no—
454C22—31no—
455C22—31no—
456C22—31no—
457C22—31no—
458C22—31no—
459C22—31no—
460C22—31no—
461C22—61no—
462C22#572111yescharacteristic, complemented
463C22C23×S411yescharacteristic
464C4—24no—
465C4—21no—
466C4—32no—
467C4—32no—
468C4—64no—
469C4—64no—
470C4—64no—
471C4—61no—
472C4—122no—
473C4—122no—
474C4—122no—
475C4—122no—
476C4—122no—
477C4—122no—
478C4#591412yes—
479C4Q8×S412yes—
480C3—41noSylow
481C2—31no—
482C2—31no—
483C2—31no—
484C2—31no—
485C2C2×#591411yescharacteristic
486C2C2×Q8×S411yescharacteristic
487C2#8748311yescharacteristic
subgroupG/H
center, FrattiniC22C23×S4
commutator, prime radical, cosocleC22×A4C24
FittingC22×C4⋊Q8D3
socleC24C23×D3
hypercenterC4⋊Q8S4
hypocenterA4#557
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
54. A4⋊Q8Q8semidirect
377. A4#557semidirect
462. C22#5721semidirect

Maximal subgroups

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

subgroupindex
A4×C4⋊Q82
Q8×SL(2,ℤ/4)2
#855022
#853052
#853042
#853022
C4×A4⋊Q82
#591443
#57214