Finite Group Databasebeta

← #11714470

Subgroups — #11714470

Order 768.

1313 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 1313 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C2×C42×A4C211yesmaximal, characteristic
2#85482C212yesmaximal
3#85480C214yesmaximal
4#85284C212yesmaximal, complemented
5#85283C212yesmaximal, complemented
6C4×A4⋊Q8C214yesmaximal, complemented
7#36625—31nomaximal, Sylow
8C22×C4×A4C2212yes—
9C22×C4×A4C2211yescharacteristic
10C2×A4⋊Q8C2214yes—
11#5470—41nomaximal
12C42×A4C2212yes—
13C42×A4C2211yescharacteristic
14C42×A4C2211yescharacteristic
15#5432C2214yes—
16C4⋊SL(2,ℤ/4)C2214yes—
17C4⋊SL(2,ℤ/4)C2214yes—
18SL(2,ℤ/4)⋊C4C2218yes—
19C4×SL(2,ℤ/4)C2214yes—
20C2×#533—31no—
21C23×C42D311yescharacteristic
22#2857—32no—
23#2857—32no—
24#2855—34no—
25#2851—32no—
26#2851—32no—
27#2848—34no—
28C4×C22⋊Q8—34no—
29C4×C22⋊Q8—34no—
30#2296—32no—
31#2285—31no—
32#2280—32no—
33C23×A4C2311yescharacteristic
34C2×C4×A4—24no—
35C2×C4×A4Q814yescomplemented
36C2×C4×A4C2312yes—
37C2×C4×A4C2311yescharacteristic
38C2×C4×A4C2311yescharacteristic
39C2×C4×A4C2312yes—
40C2×SL(2,ℤ/4)C2318yes—
41A4⋊Q8—28no—
42C2×C3×C42—41no—
43#926—42no—
44#924—44no—
45#870—42no—
46#869—42no—
47C4×Dic6—44no—
48C24×C4C2×D312yes—
49C24×C4C2×D311yescharacteristic
50#533—62no—
51#533—61no—
52#533—61no—
53#533—61no—
54#533—62no—
55#533—61no—
56C2×C4⋊Q8—32no—
57C2×#125—32no—
58C2×C22⋊Q8—34no—
59C2×C22⋊Q8—34no—
60C2×C22⋊Q8—34no—
61C2×C4×Q8—34no—
62C2×#117—32no—
63C2×#117—32no—
64C22×C42—31no—
65C22×C42—31no—
66C22×C42—32no—
67C22×C42—31no—
68C22×C42—31no—
69C22×C42—32no—
70C22×C42C2×D311yescharacteristic
71C22×C42C2×D311yescharacteristic
72C22×C42C2×D312yes—
73#391—34no—
74#391—34no—
75#389—34no—
76#389—34no—
77#389—34no—
78#389—34no—
79#387—38no—
80#387—38no—
81#385—38no—
82#385—38no—
83#385—38no—
84#384—32no—
85#382—32no—
86#381—34no—
87#380—34no—
88#380—34no—
89#380—34no—
90#380—34no—
91#380—34no—
92#379—34no—
93C4×C4⋊C4—34no—
94C4×C4⋊C4—38no—
95C4×C22⋊C4—34no—
96C4×C22⋊C4—34no—
97C22×A4—22no—
98C22×A4C2×Q812yes—
99C22×A4C2411yescharacteristic
100C22×C3×C4—41no—
101C22×C3×C4—42no—
102C2×Dic6—44no—
103C4×A4—22no—
104C4×A4C2×Q812yes—
105C4×A4C2×Q812yes—
106C4×A4#4114yes—
107SL(2,ℤ/4)—28no—
108C3×C42—42no—
109C3×C42—41no—
110C3×C42—41no—
111#217—44no—
112C4⋊Dic3—44no—
113C4⋊Dic3—44no—
114Dic3⋊C4—48no—
115C4×Dic3—44no—
116C25C22×D311yescharacteristic
117C22×Q8—34no—
118C23×C4—24no—
119C23×C4—31no—
120C23×C4—32no—
121C23×C4—34no—
122C23×C4—31no—
123C23×C4—31no—
124C23×C4—31no—
125C23×C4—32no—
126C23×C4—32no—
127C23×C4—32no—
128C23×C4—64no—
129C23×C4Dic614yescomplemented
130C23×C4C22×D312yes—
131C23×C4C22×D311yescharacteristic
132C23×C4C22×D311yescharacteristic
133C23×C4C22×D312yes—
134C4⋊Q8—62no—
135C4⋊Q8—62no—
136C4⋊Q8—62no—
137C4⋊Q8—62no—
138#125—64no—
139#125—62no—
140#125—62no—
141C22⋊Q8—68no—
142C22⋊Q8—616no—
143C22⋊Q8—64no—
144C22⋊Q8—64no—
145C22⋊Q8—64no—
146C22⋊Q8—68no—
147C22⋊Q8—64no—
148C4×Q8—64no—
149C4×Q8—64no—
150C4×Q8—64no—
151C4×Q8—64no—
152#117—68no—
153#117—62no—
154#117—62no—
155#117—62no—
156#117—62no—
157C2×C4⋊C4—34no—
158C2×C4⋊C4—34no—
159C2×C4⋊C4—38no—
160C2×C4⋊C4—34no—
161C2×C4⋊C4—34no—
162C2×C4⋊C4—34no—
163C2×C4⋊C4—38no—
164C2×C4⋊C4—38no—
165C2×C22⋊C4—34no—
166C2×C22⋊C4—34no—
167C2×C22⋊C4—38no—
168C2×C22⋊C4—38no—
169C2×C42—32no—
170C2×C42—31no—
171C2×C42—32no—
172C2×C42—31no—
173C2×C42—31no—
174C2×C42—31no—
175C2×C42—31no—
176C2×C42—32no—
177C2×C42—32no—
178C2×C42—32no—
179C2×C42—32no—
180C2×C42—31no—
181C2×C42—31no—
182C2×C42—32no—
183C2×C42—32no—
184C2×C42—34no—
185C2×C42—34no—
186C2×C42—62no—
187C2×C42—62no—
188C2×C42—61no—
189C2×C42—61no—
190C2×C42—61no—
191C2×C42S411yescharacteristic
192#95—38no—
193#95—38no—
194#95—38no—
195#95—38no—
196#95—38no—
197C23×C3—41no—
198C2×A4—22no—
199C2×A4C22×Q811yescharacteristic
200C2×A4C2×#4112yes—
201C2×C3×C4—44no—
202C2×C3×C4—41no—
203C2×C3×C4—42no—
204C2×C3×C4—42no—
205C2×C3×C4—41no—
206C2×C3×C4—84no—
207C2×Dic3—48no—
208Dic6—88no—
209C24—22no—
210C24—31no—
211C24—31no—
212C24—32no—
213C24—62no—
214C24C2×Dic612yes—
215C24C23×D311yescharacteristic
216C2×Q8—64no—
217C2×Q8—68no—
218C2×Q8—68no—
219C2×Q8—64no—
220C22×C4—22no—
221C22×C4—31no—
222C22×C4—34no—
223C22×C4—34no—
224C22×C4—31no—
225C22×C4—32no—
226C22×C4—31no—
227C22×C4—31no—
228C22×C4—31no—
229C22×C4—31no—
230C22×C4—31no—
231C22×C4—32no—
232C22×C4—32no—
233C22×C4—34no—
234C22×C4—34no—
235C22×C4—34no—
236C22×C4—34no—
237C22×C4—32no—
238C22×C4—32no—
239C22×C4—38no—
240C22×C4—32no—
241C22×C4—32no—
242C22×C4—32no—
243C22×C4—32no—
244C22×C4—38no—
245C22×C4—38no—
246C22×C4—64no—
247C22×C4—64no—
248C22×C4—62no—
249C22×C4—62no—
250C22×C4—61no—
251C22×C4—62no—
252C22×C4—62no—
253C22×C4—62no—
254C22×C4—64no—
255C22×C4—64no—
256C22×C4—64no—
257C22×C4—64no—
258C22×C4—64no—
259C22×C4C2×Dic612yes—
260C22×C4C2×Dic612yes—
261C22×C4#23514yes—
262C22×C4C2×S412yes—
263C22×C4C2×S411yescharacteristic
264C4⋊C4—68no—
265C4⋊C4—64no—
266C4⋊C4—68no—
267C4⋊C4—64no—
268C4⋊C4—68no—
269C4⋊C4—68no—
270C4⋊C4—616no—
271C4⋊C4—616no—
272C4⋊C4—64no—
273C4⋊C4—68no—
274C4⋊C4—64no—
275C22⋊C4—68no—
276C22⋊C4—64no—
277C22⋊C4—68no—
278C22⋊C4—616no—
279C22⋊C4—68no—
280C22⋊C4—64no—
281C42—32no—
282C42—32no—
283C42—32no—
284C42—31no—
285C42—32no—
286C42—31no—
287C42—32no—
288C42—62no—
289C42—61no—
290C42—61no—
291C42—68no—
292C42—64no—
293C42—64no—
294C42C2×S411yescharacteristic
295C42C2×S411yescharacteristic
296C42C2×S412yes—
297C22×C3—42no—
298C22×C3—41no—
299C22×C3—82no—
300A4#53311yescharacteristic, complemented
301C3×C4—42no—
302C3×C4—42no—
303C3×C4—44no—
304C3×C4—82no—
305Dic3—88no—
306C23—22no—
307C23—31no—
308C23—32no—
309C23—32no—
310C23—31no—
311C23—34no—
312C23—32no—
313C23—32no—
314C23—31no—
315C23—62no—
316C23—62no—
317C23—62no—
318C23—61no—
319C23—62no—
320C23—64no—
321C23—64no—
322C23C22×Dic611yescharacteristic
323C23C2×#23512yes—
324C23C22×S411yescharacteristic
325Q8—128no—
326Q8—128no—
327C2×C4—24no—
328C2×C4—34no—
329C2×C4—31no—
330C2×C4—31no—
331C2×C4—32no—
332C2×C4—32no—
333C2×C4—32no—
334C2×C4—32no—
335C2×C4—32no—
336C2×C4—34no—
337C2×C4—34no—
338C2×C4—34no—
339C2×C4—34no—
340C2×C4—34no—
341C2×C4—32no—
342C2×C4—32no—
343C2×C4—34no—
344C2×C4—32no—
345C2×C4—64no—
346C2×C4—62no—
347C2×C4—62no—
348C2×C4—61no—
349C2×C4—61no—
350C2×C4—64no—
351C2×C4—62no—
352C2×C4—64no—
353C2×C4—68no—
354C2×C4—64no—
355C2×C4—62no—
356C2×C4—68no—
357C2×C4—68no—
358C2×C4—68no—
359C2×C4—68no—
360C2×C4—68no—
361C2×C4—68no—
362C2×C4A4⋊Q814yescomplemented
363C2×C4C22×S412yes—
364C2×C4C22×S411yescharacteristic
365C2×C4C22×S412yes—
366C2×C4C22×S411yescharacteristic
367C2×C3—41no—
368C2×C3—42no—
369C2×C3—82no—
370C22—22no—
371C22—31no—
372C22—32no—
373C22—31no—
374C22—31no—
375C22—32no—
376C22—32no—
377C22—34no—
378C22—32no—
379C22—62no—
380C22—64no—
381C22—62no—
382C22—61no—
383C22—62no—
384C22—64no—
385C22—62no—
386C22#547011yescharacteristic, complemented
387C22C2×A4⋊Q812yes—
388C22C23×S411yescharacteristic
389C4—22no—
390C4—32no—
391C4—32no—
392C4—34no—
393C4—62no—
394C4—128no—
395C4—128no—
396C4C2×A4⋊Q812yes—
397C4C2×A4⋊Q812yes—
398C4#591214yes—
399C3—41noSylow
400C2—22no—
401C2—32no—
402C2—31no—
403C2—31no—
404C2—62no—
405C2C22×A4⋊Q811yescharacteristic
406C2C2×#591212yes—
subgroupG/H
centerC2×C4C22×S4
commutator, cosocleC22×A4C24
FrattiniC22C23×S4
FittingC23×C42D3
socleC24C23×D3
prime radicalC23×A4C23
hypercenterC2×C42S4
hypocenterA4#533
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C2×C4 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
4. #85284C2semidirect
5. #85283C2semidirect
6. C4×A4⋊Q8C2semidirect
35. C2×C4×A4Q8semidirect
129. C23×C4Dic6semidirect
300. A4#533semidirect
362. C2×C4A4⋊Q8semidirect
386. C22#5470semidirect

Maximal subgroups

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

subgroupindex
C2×C42×A42
#854822
#854802
#852842
#852832
C4×A4⋊Q82
#366253
#54704