Finite Group Databasebeta

← #11711752

Subgroups — #11711752

Order 768.

1045 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 1045 conjugacy classes of subgroups — too many to draw. Only its 91 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×A4×C4⋊C4C211yesmaximal, characteristic
3C2×SL(2,ℤ/4)⋊C4C212yesmaximal
4A4⋊#95C212yesmaximal
5A4⋊#95C211yesmaximal, characteristic
6#14720—31nomaximal, Sylow
7C22×C4×A4C2211yescharacteristic
8C22×C4×A4C2211yescharacteristic
9C22×C4×A4C2211yescharacteristic
10C22×SL(2,ℤ/4)C2212yes—
11C22×SL(2,ℤ/4)C2212yes—
12C2×A4⋊Q8—22no—
13C2×A4⋊Q8—24no—
14A4×C4⋊C4—22no—
15SL(2,ℤ/4)⋊C4—24no—
16#4979—41nomaximal
17C22×C22⋊Q8—31no—
18C23×C4⋊C4D311yescharacteristic
19C2×#395—31no—
20C2×#395—31no—
21C2×#385—32no—
22#1444—32no—
23#1444—32no—
24#1441—31no—
25#1441—31no—
26C22⋊#95—32no—
27C22⋊#95—31no—
28C23×A4C2311yescharacteristic
29C22×Dic6—41no—
30C2×C4×A4—22no—
31C2×C4×A4—21no—
32C2×C4×A4—21no—
33C2×C4×A4—22no—
34C2×C4×A4—21no—
35C2×C4×A4D412yes—
36C2×C4×A4Q812yescomplemented
37C2×SL(2,ℤ/4)—22no—
38C2×SL(2,ℤ/4)—22no—
39C2×SL(2,ℤ/4)—22no—
40C2×SL(2,ℤ/4)—22no—
41C2×SL(2,ℤ/4)D414yes—
42C2×SL(2,ℤ/4)Q814yescomplemented
43A4⋊Q8—28no—
44C2×C3×C4⋊C4—41no—
45C2×Dic3⋊C4—42no—
46C3⋊#95—41no—
47C3⋊#95—42no—
48C23×Q8—31no—
49C24×C4C2×D311yescharacteristic
50C24×C4C2×D311yescharacteristic
51C24×C4C2×D311yescharacteristic
52C2×C22⋊Q8—64no—
53C2×C22⋊Q8—64no—
54C2×C22⋊Q8—62no—
55C2×C22⋊Q8—62no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—31no—
58C22×C4⋊C4—31no—
59C22×C4⋊C4—31no—
60C22×C4⋊C4—31no—
61C22×C4⋊C4—31no—
62C22×C4⋊C4—32no—
63C22×C4⋊C4—32no—
64C22×C4⋊C4—32no—
65C22×C4⋊C4—62no—
66C22×C22⋊C4—32no—
67C22×C22⋊C4—32no—
68#395—64no—
69#395—61no—
70#395—61no—
71#395—61no—
72#395—61no—
73#385—64no—
74#385—64no—
75C2×#95—31no—
76C2×#95—31no—
77C2×#95—32no—
78C2×#95—32no—
79C2×#95—32no—
80C2×#95—32no—
81C2×#95—32no—
82C2×#95—32no—
83C2×#95—32no—
84C22×A4C2×D412yes—
85C22×A4C2×D411yescharacteristic
86C22×A4C2×Q811yescharacteristic
87C22×A4C2×Q812yes—
88C22×A4#4111yescharacteristic
89C22×C3×C4—41no—
90C22×C3×C4—41no—
91C22×C3×C4—41no—
92C22×Dic3—42no—
93C22×Dic3—42no—
94C2×Dic6—84no—
95C2×Dic6—82no—
96C4×A4—22no—
97C4×A4—22no—
98C4×A4—41no—
99SL(2,ℤ/4)—24no—
100SL(2,ℤ/4)—24no—
101C3×C4⋊C4—82no—
102Dic3⋊C4—84no—
103C25C22×D311yescharacteristic
104C22×Q8—62no—
105C22×Q8—64no—
106C22×Q8—61no—
107C22×Q8—62no—
108C22×Q8—61no—
109C22×Q8—122no—
110C23×C4—21no—
111C23×C4—22no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—31no—
116C23×C4—31no—
117C23×C4—31no—
118C23×C4—31no—
119C23×C4—31no—
120C23×C4—32no—
121C23×C4—32no—
122C23×C4—32no—
123C23×C4—32no—
124C23×C4—32no—
125C23×C4—32no—
126C23×C4—31no—
127C23×C4—61no—
128C23×C4—61no—
129C23×C4—61no—
130C23×C4—62no—
131C23×C4—62no—
132C23×C4Dic612yescomplemented
133C23×C4#6712yes—
134C22⋊Q8—68no—
135C22⋊Q8—68no—
136C22⋊Q8—616no—
137C2×C4⋊C4—34no—
138C2×C4⋊C4—31no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4—31no—
141C2×C4⋊C4—62no—
142C2×C4⋊C4—62no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4—62no—
145C2×C4⋊C4—62no—
146C2×C4⋊C4—62no—
147C2×C4⋊C4—64no—
148C2×C4⋊C4—64no—
149C2×C4⋊C4—64no—
150C2×C4⋊C4—64no—
151C2×C4⋊C4—62no—
152C2×C4⋊C4—62no—
153C2×C4⋊C4—62no—
154C2×C4⋊C4—64no—
155C2×C4⋊C4—64no—
156C2×C4⋊C4—64no—
157C2×C4⋊C4—62no—
158C2×C4⋊C4—62no—
159C2×C4⋊C4—122no—
160C2×C4⋊C4—122no—
161C2×C4⋊C4S411yescharacteristic
162C2×C22⋊C4—34no—
163C2×C22⋊C4—34no—
164C2×C22⋊C4—34no—
165C2×C22⋊C4—34no—
166C2×C22⋊C4—62no—
167C2×C22⋊C4—62no—
168C2×C22⋊C4—62no—
169C2×C22⋊C4—62no—
170C2×C22⋊C4—62no—
171C2×C22⋊C4—62no—
172C2×C22⋊C4—62no—
173C2×C22⋊C4—62no—
174#95—64no—
175#95—64no—
176#95—64no—
177#95—64no—
178#95—64no—
179#95—62no—
180#95—61no—
181#95—62no—
182#95—61no—
183#95—62no—
184#95—122no—
185C23×C3—41no—
186C2×A4#12011yescharacteristic
187C2×A4C22⋊Q812yes—
188C2×A4C22⋊Q811yescharacteristic
189C2×A4C4⋊Q812yes—
190C2×A4C4⋊Q811yescharacteristic
191C2×C3×C4—42no—
192C2×C3×C4—42no—
193C2×C3×C4—81no—
194C2×C3×C4—82no—
195C2×C3×C4—81no—
196C2×C3×C4—82no—
197C2×C3×C4—81no—
198C2×Dic3—44no—
199C2×Dic3—44no—
200C2×Dic3—82no—
201C2×Dic3—82no—
202C2×Dic3—82no—
203C2×Dic3—82no—
204Dic6—88no—
205C24—31no—
206C24—32no—
207C24—32no—
208C24—31no—
209C24—31no—
210C24—31no—
211C24C2×Dic611yescharacteristic
212C24#23511yescharacteristic
213C24D3×D412yes—
214C24D3×Q812yes—
215C24C2×#6711yescharacteristic
216C2×Q8—68no—
217C2×Q8—124no—
218C2×Q8—122no—
219C2×Q8—128no—
220C2×Q8—124no—
221C2×Q8—124no—
222C2×Q8—122no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—32no—
226C22×C4—32no—
227C22×C4—32no—
228C22×C4—31no—
229C22×C4—31no—
230C22×C4—32no—
231C22×C4—32no—
232C22×C4—32no—
233C22×C4—32no—
234C22×C4—32no—
235C22×C4—34no—
236C22×C4—34no—
237C22×C4—34no—
238C22×C4—34no—
239C22×C4—32no—
240C22×C4—31no—
241C22×C4—41no—
242C22×C4—61no—
243C22×C4—61no—
244C22×C4—62no—
245C22×C4—61no—
246C22×C4—61no—
247C22×C4—62no—
248C22×C4—62no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4—62no—
252C22×C4—61no—
253C22×C4—62no—
254C22×C4—61no—
255C22×C4—64no—
256C22×C4—61no—
257C22×C4—62no—
258C22×C4—62no—
259C22×C4—61no—
260C22×C4—62no—
261C22×C4—62no—
262C22×C4—62no—
263C22×C4—62no—
264C22×C4—62no—
265C22×C4—64no—
266C22×C4—62no—
267C22×C4—62no—
268C22×C4—62no—
269C22×C4—62no—
270C22×C4—62no—
271C22×C4—62no—
272C22×C4—62no—
273C22×C4—62no—
274C22×C4—62no—
275C22×C4—62no—
276C22×C4—62no—
277C22×C4—62no—
278C22×C4—62no—
279C22×C4—62no—
280C22×C4—62no—
281C22×C4—62no—
282C22×C4—62no—
283C22×C4—62no—
284C22×C4—62no—
285C22×C4—62no—
286C22×C4—62no—
287C22×C4—62no—
288C22×C4—62no—
289C22×C4—62no—
290C22×C4—61no—
291C22×C4—122no—
292C22×C4—122no—
293C22×C4—122no—
294C22×C4—121no—
295C22×C4C2×S411yescharacteristic
296C22×C4C2×S411yescharacteristic
297C22×C4C2×S411yescharacteristic
298C4⋊C4—22no—
299C4⋊C4—62no—
300C4⋊C4—62no—
301C4⋊C4—64no—
302C4⋊C4—68no—
303C4⋊C4—616no—
304C4⋊C4—62no—
305C4⋊C4—124no—
306C4⋊C4—128no—
307C4⋊C4—124no—
308C22⋊C4—64no—
309C22⋊C4—64no—
310C22⋊C4—64no—
311C22⋊C4—64no—
312C22⋊C4—64no—
313C22⋊C4—64no—
314C22⋊C4—64no—
315C22⋊C4—64no—
316C22×C3—41no—
317C22×C3—41no—
318C22×C3—42no—
319C22×C3—42no—
320C22×C3—41no—
321A4#39511yescharacteristic, complemented
322C3×C4—82no—
323C3×C4—82no—
324C3×C4—161no—
325Dic3—84no—
326Dic3—84no—
327C23—31no—
328C23—31no—
329C23—31no—
330C23—31no—
331C23—32no—
332C23—32no—
333C23—32no—
334C23—32no—
335C23—31no—
336C23—31no—
337C23—32no—
338C23—31no—
339C23—32no—
340C23—32no—
341C23—32no—
342C23—31no—
343C23—31no—
344C23—31no—
345C23—32no—
346C23—32no—
347C23—32no—
348C23—32no—
349C23—31no—
350C23—61no—
351C23—61no—
352C23—62no—
353C23—62no—
354C23—61no—
355C23#88812yes—
356C23#89612yes—
357C23#92411yescharacteristic
358C23#93711yescharacteristic
359C23#94411yescharacteristic
360C23C22×S411yescharacteristic
361Q8—128no—
362Q8—128no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—34no—
369C2×C4—32no—
370C2×C4—32no—
371C2×C4—62no—
372C2×C4—62no—
373C2×C4—62no—
374C2×C4—62no—
375C2×C4—62no—
376C2×C4—64no—
377C2×C4—62no—
378C2×C4—62no—
379C2×C4—62no—
380C2×C4—61no—
381C2×C4—62no—
382C2×C4—61no—
383C2×C4—62no—
384C2×C4—62no—
385C2×C4—62no—
386C2×C4—64no—
387C2×C4—64no—
388C2×C4—64no—
389C2×C4—64no—
390C2×C4—64no—
391C2×C4—64no—
392C2×C4—64no—
393C2×C4—64no—
394C2×C4—64no—
395C2×C4—64no—
396C2×C4—64no—
397C2×C4—62no—
398C2×C4—64no—
399C2×C4—64no—
400C2×C4—61no—
401C2×C4—121no—
402C2×C4—122no—
403C2×C4—121no—
404C2×C4—122no—
405C2×C4—122no—
406C2×C4—122no—
407C2×C4—122no—
408C2×C4—122no—
409C2×C4—122no—
410C2×C4—122no—
411C2×C4—122no—
412C2×C4—122no—
413C2×C4—122no—
414C2×C4—122no—
415C2×C4—122no—
416C2×C4—122no—
417C2×C4—122no—
418C2×C4—122no—
419C2×C4—122no—
420C2×C4—122no—
421C2×C4—121no—
422C2×C4—121no—
423C2×C4A4⋊Q812yescomplemented
424C2×C4GL(2,ℤ/4)12yes—
425C2×C3—42no—
426C2×C3—42no—
427C2×C3—41no—
428C2×C3—41no—
429C2×C3—41no—
430C22—32no—
431C22—32no—
432C22—31no—
433C22—31no—
434C22—31no—
435C22—32no—
436C22—32no—
437C22—32no—
438C22—32no—
439C22—32no—
440C22—32no—
441C22—32no—
442C22—32no—
443C22—31no—
444C22—31no—
445C22—31no—
446C22—31no—
447C22—31no—
448C22—32no—
449C22—31no—
450C22—32no—
451C22—31no—
452C22—31no—
453C22—62no—
454C22—61no—
455C22—61no—
456C22—62no—
457C22—61no—
458C22#497911yescharacteristic, complemented
459C22C2×A4⋊Q811yescharacteristic
460C22#591211yescharacteristic
461C22D4×S412yes—
462C22Q8×S412yes—
463C22C2×GL(2,ℤ/4)11yescharacteristic
464C4—22no—
465C4—22no—
466C4—41no—
467C4—62no—
468C4—62no—
469C4—124no—
470C4—124no—
471C4—124no—
472C4—121no—
473C4—124no—
474C3—41noSylow
475C2—31no—
476C2—32no—
477C2—31no—
478C2—32no—
479C2—31no—
480C2—31no—
481C2#8530312yes—
482C2#8531112yes—
483C2#8548011yescharacteristic
484C2#8549311yescharacteristic
485C2#8550211yescharacteristic
subgroupG/H
center, FrattiniC23C22×S4
commutator, prime radical, cosocleC23×A4C23
FittingC23×C4⋊C4D3
socleC25C22×D3
hypercenterC2×C4⋊C4S4
hypocenterA4#395
LCSG ▷ C23×A4 ▷ A4
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
36. C2×C4×A4Q8semidirect
42. C2×SL(2,ℤ/4)Q8semidirect
132. C23×C4Dic6semidirect
321. A4#395semidirect
423. C2×C4A4⋊Q8semidirect
458. C22#4979semidirect

Maximal subgroups

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

subgroupindex
C22×A4⋊Q82
C2×A4×C4⋊C42
C2×SL(2,ℤ/4)⋊C42
A4⋊#952
#147203
#49794