Finite Group Databasebeta

← #11713975

Subgroups — #11713975

Order 768.

589 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 589 conjugacy classes of subgroups — too many to draw. Only its 59 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
1C4×Q8×A4C211yesmaximal, characteristic, complemented
2C2×#5417C211yesmaximal, characteristic
3C2×#5416C211yesmaximal, characteristic, complemented
4#85290C211yesmaximal, characteristic
5#73053C211yesmaximal, characteristic
6#73052C211yesmaximal, characteristic, complemented
7C4⋊(A4⋊C8)C211yesmaximal, characteristic, complemented
8#32909—31nomaximal, Sylow
9C2×Q8×A4C2211yescharacteristic
10C2×C4⋊S4C2211yescharacteristic
11C2×A4⋊Q8C2211yescharacteristic
12A4×C4⋊C4—21no—
13A4×C4⋊C4C2211yescharacteristic
14C42×A4—21no—
15C42×A4C2211yescharacteristic
16#5417—22no—
17#5416—22no—
18(C2×S4)⋊C4—21no—
19(C2×S4)⋊C4—21no—
20C2×A4⋊C8C2211yescharacteristic
21C2×A4⋊C8C2211yescharacteristic
22#5028—41nomaximal
23C22×C4×Q8D311yescharacteristic, complemented
24C2×#464—31no—
25C2×#464—31no—
26C2×#451—31no—
27C2×#448—31no—
28#2850—31no—
29#1880—31no—
30#1880—31no—
31#1878—31no—
32#1878—31no—
33#1874—31no—
34#1872—31no—
35#1805—31no—
36#1805—31no—
37#1797—31no—
38C22×S4—21no—
39Q8×A4—21no—
40Q8×A4D412yes—
41C2×C4×A4—21no—
42C2×C4×A4—21no—
43C2×C4×A4D411yescharacteristic
44C2×C4×A4D411yescharacteristic
45C2×C4×A4C2311yescharacteristic
46C2×SL(2,ℤ/4)—21no—
47C4⋊S4—21no—
48C4⋊S4—21no—
49A4⋊Q8—21no—
50A4⋊Q8—21no—
51C3×C4×Q8—41no—
52C2×#216—41no—
53C2×#215—41no—
54#875—41no—
55A4⋊C8—21no—
56A4⋊C8—21no—
57#810—41no—
58#809—41no—
59C4⋊(C3⋊C8)—41no—
60C23×Q8C2×D311yescharacteristic
61C22×Q16—31no—
62C22×Q16—31no—
63C22×SD16—31no—
64C22×SD16—31no—
65C2×#124—31no—
66C2×C22⋊Q8—31no—
67C2×#121—31no—
68C2×C4×Q8—31no—
69C2×C4×Q8—31no—
70C2×C4×Q8—31no—
71C2×C4×Q8—31no—
72C2×C4×Q8—61no—
73C2×C4×Q8—61no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4C2×D311yescharacteristic
76C22×C42—21no—
77C22×C42C2×D311yescharacteristic
78#464—62no—
79#464—62no—
80#464—61no—
81#464—61no—
82#464—61no—
83#464—61no—
84#451—62no—
85#451—62no—
86#448—62no—
87#448—62no—
88C2×C4⋊C8—31no—
89C2×C4⋊C8—31no—
90C2×#103—31no—
91C2×#103—31no—
92C2×#103—31no—
93C2×#103—31no—
94C2×#103—31no—
95C2×#103—31no—
96C2×#102—31no—
97C2×#102—31no—
98C2×C22⋊C8—31no—
99C2×C22⋊C8—31no—
100#391—31no—
101#390—31no—
102#388—61no—
103#388—61no—
104#386—61no—
105#386—61no—
106#382—31no—
107#340—31no—
108#340—31no—
109#340—31no—
110#340—31no—
111#336—31no—
112#336—31no—
113C22×A4C2×D411yescharacteristic
114C2×S4—41no—
115C2×S4—41no—
116C2×S4—41no—
117C2×C3×Q8—41no—
118C2×D12—41no—
119C2×Dic6—41no—
120C4×A4—22no—
121C4×A4—21no—
122C4×A4—21no—
123C4×A4—41no—
124C4×A4C2×D411yescharacteristic
125C4×A4#4111yescharacteristic
126SL(2,ℤ/4)—41no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—81no—
129C3×C42—41no—
130C3×C42—81no—
131#216—82no—
132#215—82no—
133(C2×D3)⋊C4—81no—
134(C2×D3)⋊C4—81no—
135C2×C3⋊C8—41no—
136C2×C3⋊C8—41no—
137C22×Q8—21no—
138C22×Q8—31no—
139C22×Q8—31no—
140C22×Q8—32no—
141C22×Q8—31no—
142C22×Q8—61no—
143C22×Q8—61no—
144C22×Q8D1212yes—
145C22×D4—31no—
146C22×D4—61no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4#6711yescharacteristic
150C23×C4#6711yescharacteristic
151C23×C4C22×D311yescharacteristic
152C2×Q16—62no—
153C2×Q16—62no—
154C2×Q16—62no—
155C2×Q16—62no—
156C2×Q16—62no—
157C2×Q16—61no—
158C2×Q16—61no—
159C2×SD16—62no—
160C2×SD16—62no—
161C2×SD16—62no—
162C2×SD16—62no—
163C2×SD16—62no—
164C2×SD16—61no—
165C2×SD16—61no—
166C22×C8—31no—
167C22×C8—31no—
168C22×C8—31no—
169C22×C8—31no—
170#124—61no—
171#124—61no—
172#124—61no—
173#124—61no—
174C22⋊Q8—61no—
175C22⋊Q8—61no—
176C22⋊Q8—61no—
177C22⋊Q8—61no—
178#121—61no—
179#121—61no—
180#121—61no—
181#121—61no—
182C4×Q8—32no—
183C4×Q8—32no—
184C4×Q8—31no—
185C4×Q8—31no—
186C4×Q8—31no—
187C4×Q8—61no—
188C4×Q8—61no—
189C4×Q8—61no—
190C4×Q8—62no—
191C4×Q8—62no—
192C4×Q8S411yescharacteristic, complemented
193C2×C4⋊C4—31no—
194C2×C4⋊C4—31no—
195C2×C4⋊C4—31no—
196C2×C4⋊C4—31no—
197C2×C4⋊C4—61no—
198C2×C4⋊C4—61no—
199C2×C4⋊C4—61no—
200C2×C4⋊C4—61no—
201C2×C4⋊C4—61no—
202C2×C4⋊C4—61no—
203C2×C22⋊C4—61no—
204C2×C22⋊C4—61no—
205C2×C22⋊C4—61no—
206C2×C22⋊C4—61no—
207C2×C22⋊C4—61no—
208C2×C22⋊C4—61no—
209C2×C42—31no—
210C2×C42—31no—
211C2×C42—31no—
212C2×C42—31no—
213C2×C42—61no—
214C2×C42—61no—
215C2×C42—61no—
216C2×C42—61no—
217C4⋊C8—61no—
218C4⋊C8—61no—
219C4⋊C8—61no—
220C4⋊C8—61no—
221#103—62no—
222#103—61no—
223#103—62no—
224#103—62no—
225#103—62no—
226#103—62no—
227#103—61no—
228#102—62no—
229#102—61no—
230#102—61no—
231C22⋊C8—61no—
232C22⋊C8—61no—
233C22⋊C8—61no—
234C22⋊C8—61no—
235#95—61no—
236#95—61no—
237#95—61no—
238#95—61no—
239C22×D3—81no—
240C2×A4#12111yescharacteristic
241C2×A4#13511yescharacteristic
242C2×A4#13711yescharacteristic
243S4—81no—
244C3×Q8—42no—
245C3×Q8—81no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—81no—
250C2×C3×C4—81no—
251C2×Dic3—81no—
252D12—81no—
253D12—81no—
254Dic6—81no—
255Dic6—81no—
256C3⋊C8—81no—
257C3⋊C8—81no—
258C24—61no—
259C24C2×#6711yescharacteristic
260C2×Q8—32no—
261C2×Q8—32no—
262C2×Q8—32no—
263C2×Q8—32no—
264C2×Q8—32no—
265C2×Q8—31no—
266C2×Q8—61no—
267C2×Q8—61no—
268C2×Q8—61no—
269C2×Q8—62no—
270C2×Q8—61no—
271C2×Q8—61no—
272C2×Q8—62no—
273C2×Q8—61no—
274C2×Q8—61no—
275C2×Q8—61no—
276C2×Q8—61no—
277C2×Q8—61no—
278C2×Q8—64no—
279C2×Q8—61no—
280C2×Q8C2×S411yescharacteristic
281C2×D4—61no—
282C2×D4—61no—
283C2×D4—61no—
284C2×D4—61no—
285C2×D4—61no—
286C2×D4—61no—
287C2×D4—121no—
288C2×D4—121no—
289C2×D4—121no—
290C2×D4—121no—
291C2×D4—121no—
292C2×D4—121no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—31no—
297C22×C4—31no—
298C22×C4—31no—
299C22×C4—31no—
300C22×C4—31no—
301C22×C4—31no—
302C22×C4—31no—
303C22×C4—31no—
304C22×C4—41no—
305C22×C4—61no—
306C22×C4—61no—
307C22×C4—61no—
308C22×C4—61no—
309C22×C4—62no—
310C22×C4—61no—
311C22×C4—61no—
312C22×C4—61no—
313C22×C4—61no—
314C22×C4—61no—
315C22×C4—121no—
316C22×C4C2×D1211yescharacteristic
317C22×C4#23511yescharacteristic
318Q16—124no—
319Q16—122no—
320Q16—122no—
321SD16—124no—
322SD16—122no—
323SD16—122no—
324C2×C8—61no—
325C2×C8—61no—
326C2×C8—61no—
327C2×C8—61no—
328C2×C8—61no—
329C2×C8—61no—
330C2×C8—61no—
331C2×C8—61no—
332C2×C8—61no—
333C2×C8—61no—
334C2×C8—61no—
335C2×C8—61no—
336C4⋊C4—21no—
337C4⋊C4—32no—
338C4⋊C4—31no—
339C4⋊C4—61no—
340C4⋊C4—61no—
341C4⋊C4—61no—
342C4⋊C4—62no—
343C4⋊C4—61no—
344C4⋊C4—61no—
345C4⋊C4—61no—
346C4⋊C4—61no—
347C4⋊C4—61no—
348C4⋊C4—121no—
349C4⋊C4—121no—
350C4⋊C4C2×S411yescharacteristic
351C22⋊C4—122no—
352C22⋊C4—121no—
353C22⋊C4—121no—
354C22⋊C4—121no—
355C22⋊C4—121no—
356C22⋊C4—122no—
357C22⋊C4—121no—
358C22⋊C4—121no—
359C22⋊C4—121no—
360C22⋊C4—121no—
361C42—21no—
362C42—31no—
363C42—31no—
364C42—31no—
365C42—61no—
366C42—61no—
367C42—61no—
368C42—62no—
369C42—61no—
370C42C2×S411yescharacteristic
371C22×C3—41no—
372C2×D3—161no—
373C2×D3—161no—
374C2×D3—161no—
375A4#46411yescharacteristic, complemented
376C3×C4—41no—
377C3×C4—41no—
378C3×C4—81no—
379C3×C4—81no—
380C3×C4—82no—
381C3×C4—161no—
382Dic3—161no—
383C23—31no—
384C23—31no—
385C23—31no—
386C23—31no—
387C23—121no—
388C23—121no—
389C23—121no—
390C23—121no—
391C23—121no—
392C23—121no—
393C23—121no—
394C23#93011yescharacteristic
395C23#94211yescharacteristic
396C23#95011yescharacteristic
397Q8—21no—
398Q8—34no—
399Q8—32no—
400Q8—62no—
401Q8—62no—
402Q8—62no—
403Q8—61no—
404Q8—121no—
405Q8—121no—
406Q8—121no—
407Q8—121no—
408Q8C4⋊S412yes—
409D4—121no—
410D4—121no—
411D4—121no—
412D4—121no—
413D4—241no—
414D4—241no—
415D4—241no—
416D4—241no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—31no—
420C2×C4—31no—
421C2×C4—31no—
422C2×C4—31no—
423C2×C4—31no—
424C2×C4—31no—
425C2×C4—31no—
426C2×C4—32no—
427C2×C4—31no—
428C2×C4—31no—
429C2×C4—61no—
430C2×C4—61no—
431C2×C4—62no—
432C2×C4—61no—
433C2×C4—61no—
434C2×C4—62no—
435C2×C4—61no—
436C2×C4—62no—
437C2×C4—61no—
438C2×C4—61no—
439C2×C4—61no—
440C2×C4—61no—
441C2×C4—62no—
442C2×C4—61no—
443C2×C4—121no—
444C2×C4—121no—
445C2×C4—121no—
446C2×C4—121no—
447C2×C4—121no—
448C2×C4—121no—
449C2×C4—121no—
450C2×C4—121no—
451C2×C4—121no—
452C2×C4—121no—
453C2×C4—121no—
454C2×C4—121no—
455C2×C4—121no—
456C2×C4GL(2,ℤ/4)11yescharacteristic
457C2×C4GL(2,ℤ/4)11yescharacteristic
458C2×C4C22×S411yescharacteristic
459C8—121no—
460C8—121no—
461C8—121no—
462C8—121no—
463C2×C3—41no—
464C2×C3—41no—
465C2×C3—41no—
466D3—321no—
467C22—31no—
468C22—31no—
469C22—31no—
470C22—31no—
471C22—31no—
472C22—31no—
473C22—31no—
474C22—31no—
475C22—31no—
476C22—61no—
477C22—241no—
478C22—241no—
479C22—241no—
480C22—241no—
481C22—241no—
482C22—241no—
483C22—241no—
484C22#502811yescharacteristic, complemented
485C22C2×GL(2,ℤ/4)11yescharacteristic
486C4—21no—
487C4—22no—
488C4—21no—
489C4—31no—
490C4—31no—
491C4—41no—
492C4—61no—
493C4—62no—
494C4—61no—
495C4—121no—
496C4—241no—
497C4—241no—
498C4—241no—
499C4C2×C4⋊S411yescharacteristic
500C4#591211yescharacteristic
501C3—41noSylow
502C2—31no—
503C2—31no—
504C2—31no—
505C2—31no—
506C2—481no—
507C2#8548611yescharacteristic
508C2#8549911yescharacteristic
509C2#8551411yescharacteristic
subgroupG/H
centerC22C2×GL(2,ℤ/4)
commutator, cosocleC2×C4×A4C23
FrattiniC2×C4C22×S4
FittingC22×C4×Q8D3
socleC24C2×#67
prime radicalC2×C4⋊S4C22
hypercenterC4×Q8S4
hypocenterA4#464
LCSG ▷ C2×C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. C4×Q8×A4C2semidirect
3. C2×#5416C2semidirect
6. #73052C2semidirect
7. C4⋊(A4⋊C8)C2semidirect
23. C22×C4×Q8D3semidirect
192. C4×Q8S4semidirect
375. A4#464semidirect
484. C22#5028semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×A42
C2×#54172
C2×#54162
#852902
#730532
#730522
C4⋊(A4⋊C8)2
#329093
#50284