Finite Group Databasebeta

← #11714469

Subgroups — #11714469

Order 768.

1797 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 1797 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×#5912C212yesmaximal, complemented
2C2×C42×A4C211yesmaximal, characteristic, complemented
3#85486C214yesmaximal, complemented
4#85290C212yesmaximal, complemented
5#85289C211yesmaximal, characteristic, complemented
6C4×C4⋊S4C214yesmaximal, complemented
7#85283C211yesmaximal, characteristic, complemented
8#36624—31nomaximal, Sylow
9C22×C4×A4C2212yes—
10C22×C4×A4C2211yescharacteristic
11C2×GL(2,ℤ/4)C2214yes—
12#5912—28no—
13C2×C4⋊S4C2214yes—
14C2×C4⋊S4C2212yes—
15C2×C4×S4C2214yes—
16C2×A4⋊Q8C2212yes—
17#5477—41nomaximal
18C42×A4C2211yescharacteristic
19C42×A4C2212yes—
20C42×A4C2211yescharacteristic
21(C2×S4)⋊C4C2218yes—
22C4⋊SL(2,ℤ/4)C2214yes—
23C2×#532—31no—
24C2×#525—32no—
25C23×C42D311yescharacteristic, complemented
26#2857—31no—
27#2857—31no—
28#2856—31no—
29#2856—31no—
30#2854—34no—
31#2850—32no—
32#2850—32no—
33#2847—34no—
34C4×#121—34no—
35C4×#121—34no—
36#2297—32no—
37#2285—31no—
38C23×A4C2311yescharacteristic
39C22×S4C2314yes—
40C2×#235—42no—
41C2×C4×A4—24no—
42C2×C4×A4D414yescomplemented
43C2×C4×A4C2312yes—
44C2×C4×A4C2311yescharacteristic
45C2×C4×A4C2311yescharacteristic
46C2×C4×A4C2312yes—
47GL(2,ℤ/4)—28no—
48C2×SL(2,ℤ/4)C2314yes—
49C4⋊S4—24no—
50C4⋊S4—28no—
51C4×S4—28no—
52A4⋊Q8—24no—
53C2×C3×C42—41no—
54#930—44no—
55#875—42no—
56#874—41no—
57C4×D12—44no—
58#869—41no—
59C22×#41—32no—
60C24×C4C2×D312yes—
61C24×C4C2×D311yescharacteristic
62#532—61no—
63#532—62no—
64#532—61no—
65#532—61no—
66#532—62no—
67#532—61no—
68C2×C4⋊Q8—31no—
69C2×#127—31no—
70C2×#124—32no—
71#525—68no—
72#525—68no—
73C2×#123—34no—
74C2×C22⋊Q8—32no—
75C2×C22⋊Q8—32no—
76C2×#121—34no—
77C2×#121—34no—
78C2×#121—32no—
79C2×#121—34no—
80C2×#121—32no—
81C2×#120—34no—
82C2×C4×D4—34no—
83C2×C4×D4—34no—
84C2×C4×D4—34no—
85C2×#117—32no—
86C22×C42—31no—
87C22×C42—31no—
88C22×C42—32no—
89C22×C42—31no—
90C22×C42—31no—
91C22×C42—32no—
92C22×C42C2×D311yescharacteristic
93C22×C42C2×D311yescharacteristic
94C22×C42C2×D312yes—
95#391—32no—
96#391—32no—
97#390—32no—
98#390—32no—
99#390—34no—
100#390—34no—
101#389—34no—
102#389—34no—
103#388—38no—
104#388—38no—
105#386—38no—
106#386—38no—
107#386—38no—
108#384—31no—
109#384—31no—
110#382—32no—
111#380—34no—
112#380—34no—
113#380—34no—
114C4×C4⋊C4—34no—
115C4×C22⋊C4—38no—
116C22×A4—22no—
117C22×A4C2×D412yes—
118C22×A4C2411yescharacteristic
119C2×S4—24no—
120C2×S4—28no—
121C22×C3×C4—41no—
122C22×C3×C4—42no—
123C2×#67—44no—
124#235—88no—
125C2×D12—44no—
126C2×D12—42no—
127C2×C4×D3—44no—
128C2×Dic6—42no—
129C4×A4—22no—
130C4×A4C2×D412yes—
131C4×A4C2×D412yes—
132C4×A4#4114yescomplemented
133SL(2,ℤ/4)—24no—
134C3×C42—41no—
135C3×C42—42no—
136C3×C42—41no—
137(C2×D3)⋊C4—48no—
138C4⋊Dic3—44no—
139C25C22×D311yescharacteristic
140C2×#41—68no—
141C2×#41—62no—
142C2×#41—62no—
143C2×#41—62no—
144C2×#41—68no—
145C2×#41—62no—
146C22×Q8—32no—
147C22×D4—34no—
148C22×D4—34no—
149C22×D4—34no—
150C22×D4—32no—
151C22×D4—34no—
152C23×C4—24no—
153C23×C4—31no—
154C23×C4—31no—
155C23×C4—31no—
156C23×C4—32no—
157C23×C4—31no—
158C23×C4—34no—
159C23×C4—32no—
160C23×C4—32no—
161C23×C4—32no—
162C23×C4—34no—
163C23×C4—64no—
164C23×C4D1214yescomplemented
165C23×C4C22×D312yes—
166C23×C4C22×D311yescharacteristic
167C23×C4C22×D311yescharacteristic
168C23×C4C22×D312yes—
169C4⋊Q8—61no—
170C4⋊Q8—62no—
171C4⋊Q8—61no—
172#127—61no—
173#127—62no—
174#127—61no—
175#124—62no—
176#124—64no—
177#124—62no—
178#123—616no—
179C22⋊Q8—68no—
180C22⋊Q8—64no—
181C22⋊Q8—64no—
182#121—68no—
183#121—616no—
184#121—68no—
185#121—64no—
186#121—64no—
187#121—64no—
188#121—64no—
189#121—64no—
190#121—68no—
191#121—64no—
192#120—68no—
193#120—68no—
194C4×D4—64no—
195C4×D4—64no—
196C4×D4—64no—
197C4×D4—616no—
198C4×D4—68no—
199C4×D4—68no—
200C4×D4—64no—
201#117—68no—
202C2×C4⋊C4—32no—
203C2×C4⋊C4—32no—
204C2×C4⋊C4—34no—
205C2×C4⋊C4—34no—
206C2×C4⋊C4—34no—
207C2×C4⋊C4—34no—
208C2×C4⋊C4—34no—
209C2×C22⋊C4—34no—
210C2×C22⋊C4—34no—
211C2×C22⋊C4—34no—
212C2×C22⋊C4—34no—
213C2×C22⋊C4—38no—
214C2×C22⋊C4—38no—
215C2×C22⋊C4—38no—
216C2×C22⋊C4—34no—
217C2×C42—32no—
218C2×C42—31no—
219C2×C42—32no—
220C2×C42—31no—
221C2×C42—31no—
222C2×C42—31no—
223C2×C42—31no—
224C2×C42—32no—
225C2×C42—32no—
226C2×C42—32no—
227C2×C42—32no—
228C2×C42—32no—
229C2×C42—32no—
230C2×C42—31no—
231C2×C42—31no—
232C2×C42—34no—
233C2×C42—62no—
234C2×C42—62no—
235C2×C42—61no—
236C2×C42—61no—
237C2×C42—61no—
238C2×C42S411yescharacteristic, complemented
239#95—38no—
240#95—38no—
241#95—38no—
242C23×C3—41no—
243C22×D3—44no—
244C2×A4—22no—
245C2×A4C22×D411yescharacteristic
246C2×A4C2×#4112yes—
247S4—44no—
248C2×C3×C4—44no—
249C2×C3×C4—42no—
250C2×C3×C4—41no—
251C2×C3×C4—42no—
252C2×C3×C4—41no—
253C2×C3×C4—84no—
254#67—88no—
255C2×Dic3—44no—
256D12—88no—
257D12—84no—
258C4×D3—88no—
259Dic6—84no—
260C24—22no—
261C24—31no—
262C24—31no—
263C24—32no—
264C24—34no—
265C24—62no—
266C24C2×D1212yes—
267C24C23×D311yescharacteristic
268#41—128no—
269#41—128no—
270#41—128no—
271#41—128no—
272C2×Q8—64no—
273C2×Q8—62no—
274C2×Q8—64no—
275C2×Q8—62no—
276C2×D4—64no—
277C2×D4—64no—
278C2×D4—68no—
279C2×D4—68no—
280C2×D4—62no—
281C2×D4—64no—
282C2×D4—68no—
283C2×D4—64no—
284C2×D4—68no—
285C2×D4—64no—
286C2×D4—68no—
287C2×D4—616no—
288C2×D4—68no—
289C2×D4—64no—
290C2×D4—68no—
291C2×D4—64no—
292C2×D4—64no—
293C2×D4—62no—
294C22×C4—22no—
295C22×C4—31no—
296C22×C4—34no—
297C22×C4—34no—
298C22×C4—32no—
299C22×C4—31no—
300C22×C4—31no—
301C22×C4—31no—
302C22×C4—31no—
303C22×C4—32no—
304C22×C4—32no—
305C22×C4—31no—
306C22×C4—31no—
307C22×C4—34no—
308C22×C4—32no—
309C22×C4—34no—
310C22×C4—32no—
311C22×C4—34no—
312C22×C4—34no—
313C22×C4—38no—
314C22×C4—32no—
315C22×C4—32no—
316C22×C4—32no—
317C22×C4—32no—
318C22×C4—34no—
319C22×C4—34no—
320C22×C4—34no—
321C22×C4—64no—
322C22×C4—64no—
323C22×C4—62no—
324C22×C4—62no—
325C22×C4—62no—
326C22×C4—61no—
327C22×C4—62no—
328C22×C4—62no—
329C22×C4—64no—
330C22×C4—64no—
331C22×C4—64no—
332C22×C4—64no—
333C22×C4—64no—
334C22×C4—68no—
335C22×C4—68no—
336C22×C4—64no—
337C22×C4—64no—
338C22×C4C2×D1212yes—
339C22×C4C2×D1212yes—
340C22×C4#23514yescomplemented
341C22×C4C2×S412yes—
342C22×C4C2×S411yescharacteristic
343C4⋊C4—64no—
344C4⋊C4—64no—
345C4⋊C4—64no—
346C4⋊C4—68no—
347C4⋊C4—68no—
348C4⋊C4—68no—
349C4⋊C4—68no—
350C4⋊C4—64no—
351C22⋊C4—68no—
352C22⋊C4—68no—
353C22⋊C4—68no—
354C22⋊C4—616no—
355C22⋊C4—616no—
356C22⋊C4—68no—
357C22⋊C4—68no—
358C22⋊C4—64no—
359C22⋊C4—64no—
360C22⋊C4—68no—
361C42—32no—
362C42—32no—
363C42—32no—
364C42—31no—
365C42—31no—
366C42—32no—
367C42—32no—
368C42—61no—
369C42—62no—
370C42—61no—
371C42—68no—
372C42C2×S411yescharacteristic
373C42C2×S411yescharacteristic
374C42C2×S412yes—
375C22×C3—42no—
376C22×C3—41no—
377C22×C3—82no—
378C2×D3—84no—
379C2×D3—88no—
380A4#53211yescharacteristic, complemented
381C3×C4—42no—
382C3×C4—42no—
383C3×C4—44no—
384C3×C4—82no—
385Dic3—84no—
386C23—22no—
387C23—31no—
388C23—32no—
389C23—31no—
390C23—32no—
391C23—34no—
392C23—32no—
393C23—32no—
394C23—31no—
395C23—62no—
396C23—62no—
397C23—62no—
398C23—61no—
399C23—62no—
400C23—64no—
401C23—64no—
402C23—64no—
403C23—68no—
404C23—68no—
405C23—64no—
406C23—64no—
407C23C22×D1211yescharacteristic
408C23C2×#23512yes—
409C23C22×S411yescharacteristic
410Q8—124no—
411Q8—124no—
412D4—128no—
413D4—124no—
414D4—124no—
415D4—128no—
416D4—1216no—
417D4—128no—
418D4—128no—
419D4—124no—
420D4—124no—
421D4—128no—
422C2×C4—24no—
423C2×C4—34no—
424C2×C4—32no—
425C2×C4—31no—
426C2×C4—31no—
427C2×C4—32no—
428C2×C4—32no—
429C2×C4—32no—
430C2×C4—32no—
431C2×C4—34no—
432C2×C4—34no—
433C2×C4—34no—
434C2×C4—34no—
435C2×C4—32no—
436C2×C4—32no—
437C2×C4—34no—
438C2×C4—34no—
439C2×C4—32no—
440C2×C4—64no—
441C2×C4—62no—
442C2×C4—62no—
443C2×C4—61no—
444C2×C4—64no—
445C2×C4—61no—
446C2×C4—62no—
447C2×C4—64no—
448C2×C4—68no—
449C2×C4—64no—
450C2×C4—62no—
451C2×C4—64no—
452C2×C4—64no—
453C2×C4—64no—
454C2×C4—68no—
455C2×C4—68no—
456C2×C4—64no—
457C2×C4—64no—
458C2×C4—128no—
459C2×C4—128no—
460C2×C4C4⋊S414yescomplemented
461C2×C4C22×S412yes—
462C2×C4C22×S411yescharacteristic
463C2×C4C22×S412yes—
464C2×C4C22×S411yescharacteristic
465C2×C3—41no—
466C2×C3—42no—
467C2×C3—82no—
468D3—164no—
469C22—22no—
470C22—31no—
471C22—32no—
472C22—31no—
473C22—31no—
474C22—32no—
475C22—32no—
476C22—34no—
477C22—32no—
478C22—62no—
479C22—62no—
480C22—64no—
481C22—61no—
482C22—62no—
483C22—64no—
484C22—62no—
485C22—124no—
486C22—124no—
487C22—128no—
488C22—124no—
489C22—128no—
490C22#547711yescharacteristic, complemented
491C22C2×C4⋊S412yes—
492C22C23×S411yescharacteristic
493C4—22no—
494C4—32no—
495C4—32no—
496C4—34no—
497C4—62no—
498C4—124no—
499C4—124no—
500C4—124no—
501C4C2×C4⋊S412yes—
502C4C2×C4⋊S412yes—
503C4#591214yescomplemented
504C3—41noSylow
505C2—22no—
506C2—31no—
507C2—31no—
508C2—32no—
509C2—62no—
510C2—244no—
511C2C22×C4⋊S411yescharacteristic
512C2C2×#591212yes—
subgroupG/H
centerC2×C4C22×S4
commutator, cosocleC22×A4C24
FrattiniC22C23×S4
FittingC23×C42D3
socleC24C23×D3
hypercenterC2×C42S4
hypocenterA4#532
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C2×C4 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
1. C2×#5912C2semidirect
2. C2×C42×A4C2semidirect
3. #85486C2semidirect
4. #85290C2semidirect
5. #85289C2semidirect
6. C4×C4⋊S4C2semidirect
7. #85283C2semidirect
25. C23×C42D3semidirect
42. C2×C4×A4D4semidirect
132. C4×A4#41semidirect
164. C23×C4D12semidirect
238. C2×C42S4semidirect
340. C22×C4#235semidirect
380. A4#532semidirect
460. C2×C4C4⋊S4semidirect
490. C22#5477semidirect
503. C4#5912semidirect

Maximal subgroups

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

subgroupindex
C2×#59122
C2×C42×A42
#854862
#852902
#852892
C4×C4⋊S42
#852832
#366243
#54774