Finite Group Databasebeta

← #11713940

Subgroups — #11713940

Order 768.

1055 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 1055 conjugacy classes of subgroups — too many to draw. Only its 71 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×Q8×A4C211yesmaximal, characteristic
2C2×#5416C212yesmaximal, complemented
3#85480C211yesmaximal, characteristic
4#73147C212yesmaximal, complemented
5#73141C211yesmaximal, characteristic
6#32882—31nomaximal, Sylow
7C2×Q8×A4—24no—
8C2×Q8×A4—21no—
9C2×Q8×A4C2212yes—
10C22×C4×A4—21no—
11C22×C4×A4C2211yescharacteristic
12C2×A4⋊Q8C2211yescharacteristic
13#5432—21no—
14#5416—24no—
15C4⋊SL(2,ℤ/4)C2211yescharacteristic
16SL(2,ℤ/4)⋊C4—21no—
17C2×A4⋊C8C2212yes—
18#5228—41nomaximal
19C24×Q8D311yescharacteristic
20C2×#451—32no—
21C2×#451—31no—
22C2×#451—31no—
23C2×#451—32no—
24#2848—31no—
25#1787—32no—
26#1787—31no—
27#1787—31no—
28#1787—32no—
29#1779—31no—
30C23×A4D411yescharacteristic
31C22×C3×Q8—41no—
32Q8×A4—24no—
33Q8×A4—22no—
34Q8×A4D414yes—
35C2×C4×A4—24no—
36C2×C4×A4—22no—
37C2×C4×A4—22no—
38C2×C4×A4D411yescharacteristic
39C2×C4×A4C2311yescharacteristic
40C2×SL(2,ℤ/4)—21no—
41C2×SL(2,ℤ/4)—21no—
42A4⋊Q8—22no—
43C2×#216—42no—
44#924—41no—
45A4⋊C8—22no—
46#835—42no—
47#830—41no—
48C23×Q8—24no—
49C23×Q8—21no—
50C23×Q8—32no—
51C23×Q8—31no—
52C23×Q8—31no—
53C23×Q8—61no—
54C23×Q8—61no—
55C23×Q8—64no—
56C23×Q8C2×D312yes—
57C24×C4—21no—
58C24×C4C2×D311yescharacteristic
59C22×Q16—32no—
60C22×Q16—32no—
61C2×C22⋊Q8—31no—
62C2×C22⋊Q8—31no—
63C2×C22⋊Q8—31no—
64#451—64no—
65#451—68no—
66#451—62no—
67#451—62no—
68#451—62no—
69#451—61no—
70#451—64no—
71#451—61no—
72C2×#103—32no—
73C2×#103—32no—
74C2×#103—32no—
75C2×#103—32no—
76C2×#103—32no—
77C2×#103—32no—
78C2×C22⋊C8—32no—
79C2×C22⋊C8—31no—
80C2×C22⋊C8—31no—
81C2×C22⋊C8—32no—
82#385—61no—
83#385—61no—
84#385—61no—
85#380—31no—
86#380—31no—
87#380—31no—
88#379—61no—
89C22×A4—21no—
90C22×A4—21no—
91C22×A4Q1612yescomplemented
92C22×A4C2×D411yescharacteristic
93C2×C3×Q8—42no—
94C2×C3×Q8—81no—
95C2×C3×Q8—84no—
96C22×C3×C4—41no—
97C22×C3×C4—81no—
98C2×Dic6—41no—
99C4×A4—24no—
100C4×A4—21no—
101C4×A4C2×D412yes—
102SL(2,ℤ/4)—41no—
103SL(2,ℤ/4)—41no—
104#217—81no—
105#216—84no—
106C4⋊Dic3—41no—
107Dic3⋊C4—81no—
108C2×C3⋊C8—42no—
109C25#6711yescharacteristic
110C22×Q8—24no—
111C22×Q8—22no—
112C22×Q8—32no—
113C22×Q8—32no—
114C22×Q8—34no—
115C22×Q8—38no—
116C22×Q8—32no—
117C22×Q8—32no—
118C22×Q8—32no—
119C22×Q8—31no—
120C22×Q8—31no—
121C22×Q8—61no—
122C22×Q8—61no—
123C22×Q8—61no—
124C22×Q8—62no—
125C22×Q8—62no—
126C22×Q8—61no—
127C22×Q8—62no—
128C22×Q8—61no—
129C22×Q8—64no—
130C22×Q8—64no—
131C22×Q8—64no—
132C22×Q8—64no—
133C22×Q8—64no—
134C22×Q8—64no—
135C22×Q8—64no—
136C22×Q8—64no—
137C22×Q8—68no—
138C22×Q8—64no—
139C22×Q8—68no—
140C22×Q8—68no—
141C22×Q8—68no—
142C22×Q8#6714yes—
143C22×Q8S411yescharacteristic
144C23×C4—24no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—31no—
148C23×C4—31no—
149C23×C4—31no—
150C23×C4—31no—
151C23×C4—61no—
152C23×C4—62no—
153C23×C4—62no—
154C23×C4—61no—
155C23×C4—64no—
156C23×C4#6711yescharacteristic
157C23×C4C22×D311yescharacteristic
158C2×Q16—64no—
159C2×Q16—64no—
160C2×Q16—64no—
161C2×Q16—64no—
162C2×Q16—64no—
163C2×Q16—62no—
164C2×Q16—62no—
165C22×C8—32no—
166C22×C8—32no—
167C22⋊Q8—62no—
168C22⋊Q8—64no—
169C22⋊Q8—61no—
170C22⋊Q8—61no—
171C22⋊Q8—61no—
172C22⋊Q8—62no—
173C22⋊Q8—61no—
174C2×C4⋊C4—31no—
175C2×C4⋊C4—31no—
176C2×C4⋊C4—31no—
177C2×C4⋊C4—61no—
178C2×C4⋊C4—61no—
179C2×C4⋊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—
186#103—64no—
187#103—64no—
188#103—64no—
189#103—64no—
190#103—64no—
191#103—62no—
192#103—62no—
193C22⋊C8—62no—
194C22⋊C8—64no—
195C22⋊C8—61no—
196C22⋊C8—61no—
197C22⋊C8—61no—
198C22⋊C8—62no—
199C22⋊C8—61no—
200#95—61no—
201#95—61no—
202#95—61no—
203C23×C3—41no—
204C2×A4—22no—
205C2×A4#12011yescharacteristic
206C2×A4C2×Q1611yescharacteristic
207C2×A4#13711yescharacteristic
208C3×Q8—44no—
209C3×Q8—82no—
210C3×Q8—84no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—82no—
214C2×C3×C4—82no—
215C2×C3×C4—84no—
216C2×Dic3—81no—
217C2×Dic3—81no—
218Dic6—82no—
219C3⋊C8—82no—
220C24—21no—
221C24—21no—
222C24—31no—
223C24—31no—
224C24—32no—
225C24—61no—
226C24—61no—
227C24#21612yescomplemented
228C24C2×#6711yescharacteristic
229C2×Q8—24no—
230C2×Q8—21no—
231C2×Q8—34no—
232C2×Q8—34no—
233C2×Q8—34no—
234C2×Q8—34no—
235C2×Q8—34no—
236C2×Q8—32no—
237C2×Q8—62no—
238C2×Q8—62no—
239C2×Q8—64no—
240C2×Q8—68no—
241C2×Q8—62no—
242C2×Q8—62no—
243C2×Q8—64no—
244C2×Q8—62no—
245C2×Q8—62no—
246C2×Q8—64no—
247C2×Q8—68no—
248C2×Q8—62no—
249C2×Q8—61no—
250C2×Q8—62no—
251C2×Q8—62no—
252C2×Q8—64no—
253C2×Q8—64no—
254C2×Q8—68no—
255C2×Q8—64no—
256C2×Q8—68no—
257C2×Q8—68no—
258C2×Q8—616no—
259C2×Q8—68no—
260C2×Q8—68no—
261C2×Q8—64no—
262C2×Q8—68no—
263C2×Q8—64no—
264C2×Q8—616no—
265C2×Q8—64no—
266C2×Q8—61no—
267C2×Q8—61no—
268C2×Q8C2×S412yes—
269C22×C4—21no—
270C22×C4—24no—
271C22×C4—21no—
272C22×C4—31no—
273C22×C4—31no—
274C22×C4—31no—
275C22×C4—32no—
276C22×C4—34no—
277C22×C4—31no—
278C22×C4—31no—
279C22×C4—31no—
280C22×C4—31no—
281C22×C4—62no—
282C22×C4—62no—
283C22×C4—62no—
284C22×C4—62no—
285C22×C4—64no—
286C22×C4—64no—
287C22×C4—61no—
288C22×C4—61no—
289C22×C4—62no—
290C22×C4—62no—
291C22×C4—62no—
292C22×C4—62no—
293C22×C4—62no—
294C22×C4—68no—
295C22×C4—62no—
296C22×C4—64no—
297C22×C4—61no—
298C22×C4—64no—
299C22×C4—62no—
300C22×C4—68no—
301C22×C4—64no—
302C22×C4—61no—
303C22×C4—61no—
304C22×C4—61no—
305C22×C4—61no—
306C22×C4C2×#6712yes—
307C22×C4C2×S411yescharacteristic
308Q16—128no—
309Q16—124no—
310Q16—124no—
311C2×C8—62no—
312C2×C8—62no—
313C2×C8—62no—
314C2×C8—62no—
315C2×C8—62no—
316C2×C8—62no—
317C4⋊C4—61no—
318C4⋊C4—62no—
319C4⋊C4—62no—
320C4⋊C4—61no—
321C4⋊C4—122no—
322C4⋊C4—122no—
323C4⋊C4—121no—
324C4⋊C4—121no—
325C22⋊C4—121no—
326C22⋊C4—122no—
327C22⋊C4—122no—
328C22⋊C4—121no—
329C22⋊C4—122no—
330C22⋊C4—121no—
331C22⋊C4—121no—
332C22⋊C4—121no—
333C22⋊C4—121no—
334C22×C3—42no—
335C22×C3—41no—
336C22×C3—81no—
337C22×C3—81no—
338A4#45111yescharacteristic, complemented
339C3×C4—42no—
340C3×C4—81no—
341C3×C4—84no—
342Dic3—161no—
343Dic3—161no—
344C23—22no—
345C23—31no—
346C23—31no—
347C23—31no—
348C23—32no—
349C23—31no—
350C23—32no—
351C23—32no—
352C23—32no—
353C23—32no—
354C23—31no—
355C23—61no—
356C23—61no—
357C23—61no—
358C23—61no—
359C23—62no—
360C23—61no—
361C23—61no—
362C23—62no—
363C23—62no—
364C23—61no—
365C23—64no—
366C23#94211yescharacteristic
367C23C2×#21611yescharacteristic
368C23#95311yescharacteristic
369C23GL(2,ℤ/4)11yescharacteristic
370Q8—24no—
371Q8—22no—
372Q8—38no—
373Q8—34no—
374Q8—64no—
375Q8—64no—
376Q8—68no—
377Q8—64no—
378Q8—64no—
379Q8—68no—
380Q8—62no—
381Q8—122no—
382Q8—122no—
383Q8GL(2,ℤ/4)14yes—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—31no—
388C2×C4—32no—
389C2×C4—32no—
390C2×C4—32no—
391C2×C4—32no—
392C2×C4—31no—
393C2×C4—62no—
394C2×C4—61no—
395C2×C4—64no—
396C2×C4—61no—
397C2×C4—64no—
398C2×C4—68no—
399C2×C4—61no—
400C2×C4—62no—
401C2×C4—64no—
402C2×C4—64no—
403C2×C4—64no—
404C2×C4—68no—
405C2×C4—62no—
406C2×C4—62no—
407C2×C4—61no—
408C2×C4—64no—
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×C4GL(2,ℤ/4)11yescharacteristic
422C2×C4C22×S411yescharacteristic
423C8—122no—
424C8—122no—
425C2×C3—41no—
426C2×C3—41no—
427C2×C3—41no—
428C2×C3—82no—
429C22—21no—
430C22—21no—
431C22—31no—
432C22—31no—
433C22—31no—
434C22—31no—
435C22—31no—
436C22—31no—
437C22—31no—
438C22—31no—
439C22—31no—
440C22—34no—
441C22—32no—
442C22—62no—
443C22—62no—
444C22—62no—
445C22—62no—
446C22—61no—
447C22—62no—
448C22—62no—
449C22—62no—
450C22—61no—
451C22—61no—
452C22#522811yescharacteristic, complemented
453C22#541612yescomplemented
454C22C2×GL(2,ℤ/4)11yescharacteristic
455C4—24no—
456C4—21no—
457C4—32no—
458C4—61no—
459C4—64no—
460C4—241no—
461C4—241no—
462C4—241no—
463C4—241no—
464C4C2×GL(2,ℤ/4)12yes—
465C3—41noSylow
466C2—22no—
467C2—31no—
468C2—31no—
469C2—31no—
470C2—31no—
471C2—62no—
472C2C2×#541611yescharacteristic
473C2#8549911yescharacteristic
474C2#8554611yescharacteristic
subgroupG/H
centerC22C2×GL(2,ℤ/4)
commutator, cosocleC2×C4×A4C23
FrattiniC2×C4C22×S4
FittingC24×Q8D3
socleC24C2×#67
prime radicalC23×A4D4
hypercenterC22×Q8S4
hypocenterA4#451
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
2. C2×#5416C2semidirect
4. #73147C2semidirect
91. C22×A4Q16semidirect
227. C24#216semidirect
338. A4#451semidirect
452. C22#5228semidirect
453. C22#5416semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×A42
C2×#54162
#854802
#731472
#731412
#328823
#52284