Finite Group Databasebeta

← #11713001

Subgroups — #11713001

Order 768.

534 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 69 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×#5400C211yesmaximal, characteristic
2C2×C8×S4C211yesmaximal, characteristic
3#85287C211yesmaximal, characteristic
4A4×C4⋊C8C211yesmaximal, characteristic, complemented
5SL(2,ℤ/4)⋊C8C211yesmaximal, characteristic, complemented
6SL(2,ℤ/4)⋊C8C211yesmaximal, characteristic, complemented
7C4⋊(A4⋊C8)C211yesmaximal, characteristic, complemented
8#22647—31nomaximal, Sylow
9C2×C4×S4C2211yescharacteristic
10C2×C8×A4C2211yescharacteristic
11C2×C8×A4C2211yescharacteristic
12C42×A4C2211yescharacteristic
13(C2×S4)⋊C4C411yescharacteristic
14(C2×S4)⋊C4C411yescharacteristic
15SL(2,ℤ/4)⋊C4C411yescharacteristic
16SL(2,ℤ/4)⋊C4C411yescharacteristic
17C4×SL(2,ℤ/4)C2211yescharacteristic
18C2×A4⋊C8C2211yescharacteristic
19C2×A4⋊C8C2211yescharacteristic
20#5400—22no—
21C8×S4—22no—
22#4839—41nomaximal
23C2×#435—31no—
24C2×C8×D4—31no—
25C2×#424—31no—
26C22×C4⋊C8D311yescharacteristic, complemented
27#2290—31no—
28#1940—31no—
29#1939—31no—
30#1851—31no—
31#1850—31no—
32#1850—31no—
33#1850—31no—
34#1843—31no—
35#1798—31no—
36#1797—31no—
37#1768—31no—
38C22×S4C2×C411yescharacteristic
39C2×C4×A4C2×C411yescharacteristic
40C2×C4×A4C2×C411yescharacteristic
41C2×C4×A4C2311yescharacteristic
42C2×SL(2,ℤ/4)C2×C411yescharacteristic
43C2×SL(2,ℤ/4)C2×C411yescharacteristic
44C2×SL(2,ℤ/4)C2×C411yescharacteristic
45C4×S4D412yes—
46C4×S4Q812yes—
47C2×ΓL(1,𝔽25)—41no—
48C2×D3×C8—41no—
49#872—41no—
50C8×A4—21no—
51C8×A4—21no—
52A4⋊C8—21no—
53A4⋊C8—21no—
54C3×C4⋊C8—41no—
55Dic3⋊C8—41no—
56Dic3⋊C8—41no—
57C4⋊(C3⋊C8)—41no—
58C22×M16—31no—
59C23×C8—31no—
60C23×C8C2×D311yescharacteristic
61C23×C8C2×D311yescharacteristic
62C2×C4×D4—31no—
63C2×#117—31no—
64C22×C42C2×D311yescharacteristic
65#435—62no—
66#435—62no—
67C8×D4—62no—
68C8×D4—62no—
69#424—61no—
70#424—61no—
71#424—61no—
72#424—61no—
73C2×C4⋊C8—31no—
74C2×C4⋊C8—31no—
75C2×C4⋊C8—31no—
76C2×C4⋊C8—31no—
77C2×C4⋊C8—31no—
78C2×C4⋊C8—31no—
79C2×C4⋊C8—31no—
80C2×C4⋊C8—31no—
81C2×C4⋊C8—31no—
82C2×C4⋊C8—31no—
83C2×C4⋊C8—31no—
84C2×C4⋊C8—31no—
85C2×C22⋊C8—31no—
86C2×C22⋊C8—31no—
87C2×C22⋊C8—31no—
88C2×C22⋊C8—31no—
89C2×#97—31no—
90C2×C4×C8—31no—
91#388—31no—
92#388—31no—
93#387—31no—
94#387—31no—
95#386—31no—
96#386—31no—
97#385—31no—
98#385—31no—
99C4×C4⋊C4—31no—
100C4×C22⋊C4—31no—
101C4×C22⋊C4—31no—
102#376—31no—
103#336—31no—
104#336—31no—
105#336—31no—
106#336—31no—
107#336—31no—
108#336—31no—
109C22×A4C22×C411yescharacteristic
110C2×S4—21no—
111C2×S4—21no—
112C2×S4C4⋊C412yes—
113C2×C4×D3—41no—
114C4×A4—21no—
115C4×A4—21no—
116C4×A4C2×D411yescharacteristic
117C4×A4C2×Q811yescharacteristic
118SL(2,ℤ/4)—21no—
119SL(2,ℤ/4)—21no—
120SL(2,ℤ/4)C4⋊C412yes—
121C2×C3×C8—41no—
122C2×C3×C8—41no—
123C3×C42—41no—
124(C2×D3)⋊C4—41no—
125(C2×D3)⋊C4—41no—
126Dic3⋊C4—41no—
127Dic3⋊C4—41no—
128C4×Dic3—41no—
129C2×C3⋊C8—41no—
130C2×C3⋊C8—41no—
131ΓL(1,𝔽25)—82no—
132D3×C8—82no—
133C22×D4—31no—
134C23×C4—31no—
135C23×C4C4×D311yescharacteristic
136C23×C4C4×D311yescharacteristic
137C23×C4C22×D311yescharacteristic
138C2×M16—61no—
139C2×M16—62no—
140C2×M16—61no—
141C2×M16—121no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—31no—
145C22×C8—31no—
146C22×C8—31no—
147C22×C8—31no—
148C22×C8—31no—
149C22×C8—31no—
150C22×C8—31no—
151C22×C8—31no—
152C22×C8—61no—
153C22×C8—61no—
154C22×C8—61no—
155C22×C8—62no—
156C22×C8—61no—
157C22×C8—121no—
158C4×D4—32no—
159C4×D4—32no—
160C4×D4—32no—
161C4×D4—32no—
162#117—61no—
163#117—61no—
164#117—61no—
165#117—61no—
166C2×C4⋊C4—31no—
167C2×C4⋊C4—31no—
168C2×C4⋊C4—31no—
169C2×C4⋊C4—31no—
170C2×C4⋊C4—31no—
171C2×C22⋊C4—31no—
172C2×C22⋊C4—31no—
173C2×C22⋊C4—31no—
174C2×C22⋊C4—31no—
175C2×C22⋊C4—31no—
176C2×C22⋊C4—31no—
177C2×C22⋊C4—31no—
178C2×C22⋊C4—31no—
179C2×C42—31no—
180C2×C42—31no—
181C2×C42—31no—
182C2×C42—31no—
183C2×C42—31no—
184C2×C42—31no—
185C2×C42—31no—
186C4⋊C8—31no—
187C4⋊C8—31no—
188C4⋊C8—31no—
189C4⋊C8—61no—
190C4⋊C8—61no—
191C4⋊C8—61no—
192C4⋊C8—61no—
193C4⋊C8—62no—
194C4⋊C8—62no—
195C4⋊C8—61no—
196C4⋊C8—61no—
197C4⋊C8—61no—
198C4⋊C8—61no—
199C4⋊C8—61no—
200C4⋊C8—121no—
201C4⋊C8—121no—
202C4⋊C8S411yescharacteristic, complemented
203C22⋊C8—62no—
204C22⋊C8—61no—
205C22⋊C8—62no—
206C22⋊C8—61no—
207C22⋊C8—61no—
208C22⋊C8—61no—
209#97—62no—
210C4×C8—62no—
211#95—31no—
212#95—31no—
213#95—31no—
214#95—31no—
215#95—31no—
216#95—31no—
217#95—31no—
218#95—31no—
219#95—31no—
220#95—31no—
221C22×D3—41no—
222C2×A4C2×C4⋊C411yescharacteristic
223C2×A4#13111yescharacteristic
224C2×A4#13111yescharacteristic
225S4—22no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231C2×Dic3—41no—
232C4×D3—42no—
233C4×D3—42no—
234C3×C8—81no—
235C3×C8—81no—
236C3⋊C8—81no—
237C3⋊C8—81no—
238C24—31no—
239C24C2×C4×D311yescharacteristic
240C2×D4—32no—
241C2×D4—32no—
242C2×D4—61no—
243C2×D4—61no—
244C2×D4—61no—
245C2×D4—61no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—31no—
249C22×C4—31no—
250C22×C4—31no—
251C22×C4—31no—
252C22×C4—31no—
253C22×C4—31no—
254C22×C4—31no—
255C22×C4—31no—
256C22×C4—32no—
257C22×C4—32no—
258C22×C4—31no—
259C22×C4—31no—
260C22×C4—31no—
261C22×C4—31no—
262C22×C4—31no—
263C22×C4—31no—
264C22×C4—31no—
265C22×C4—61no—
266C22×C4—61no—
267C22×C4—61no—
268C22×C4—61no—
269C22×C4—61no—
270C22×C4—61no—
271C22×C4D3×D411yescharacteristic
272C22×C4D3×Q811yescharacteristic
273M16—122no—
274M16—122no—
275C2×C8—31no—
276C2×C8—31no—
277C2×C8—61no—
278C2×C8—61no—
279C2×C8—61no—
280C2×C8—61no—
281C2×C8—61no—
282C2×C8—61no—
283C2×C8—61no—
284C2×C8—61no—
285C2×C8—61no—
286C2×C8—61no—
287C2×C8—61no—
288C2×C8—61no—
289C2×C8—61no—
290C2×C8—61no—
291C2×C8—61no—
292C2×C8—61no—
293C2×C8—61no—
294C2×C8—61no—
295C2×C8—61no—
296C2×C8—61no—
297C2×C8—122no—
298C2×C8—122no—
299C2×C8C2×S411yescharacteristic
300C2×C8C2×S411yescharacteristic
301C4⋊C4—32no—
302C4⋊C4—32no—
303C4⋊C4—62no—
304C4⋊C4—61no—
305C4⋊C4—61no—
306C4⋊C4—61no—
307C4⋊C4—61no—
308C4⋊C4—121no—
309C22⋊C4—32no—
310C22⋊C4—32no—
311C22⋊C4—32no—
312C22⋊C4—32no—
313C22⋊C4—62no—
314C22⋊C4—61no—
315C22⋊C4—61no—
316C22⋊C4—61no—
317C22⋊C4—61no—
318C22⋊C4—61no—
319C22⋊C4—61no—
320C22⋊C4—61no—
321C22⋊C4—61no—
322C22⋊C4—121no—
323C42—32no—
324C42—32no—
325C42—31no—
326C42—31no—
327C42—31no—
328C42—61no—
329C42—61no—
330C42—61no—
331C42—61no—
332C42—61no—
333C42C2×S411yescharacteristic
334C22×C3—41no—
335C2×D3—42no—
336C2×D3—81no—
337C2×D3—81no—
338A4#42411yescharacteristic, complemented
339C3×C4—41no—
340C3×C4—41no—
341C3×C4—81no—
342C3×C4—81no—
343Dic3—42no—
344Dic3—81no—
345Dic3—81no—
346C23—31no—
347C23—31no—
348C23—31no—
349C23—31no—
350C23—32no—
351C23—61no—
352C23—61no—
353C23—61no—
354C23—61no—
355C23—61no—
356C23—61no—
357C23D3×C4⋊C411yescharacteristic
358C23#90111yescharacteristic
359C23#90711yescharacteristic
360D4—62no—
361D4—62no—
362D4—62no—
363D4—62no—
364C2×C4—31no—
365C2×C4—31no—
366C2×C4—31no—
367C2×C4—31no—
368C2×C4—31no—
369C2×C4—31no—
370C2×C4—31no—
371C2×C4—32no—
372C2×C4—32no—
373C2×C4—32no—
374C2×C4—32no—
375C2×C4—31no—
376C2×C4—31no—
377C2×C4—61no—
378C2×C4—61no—
379C2×C4—61no—
380C2×C4—61no—
381C2×C4—61no—
382C2×C4—62no—
383C2×C4—62no—
384C2×C4—61no—
385C2×C4—61no—
386C2×C4—61no—
387C2×C4—61no—
388C2×C4—61no—
389C2×C4—61no—
390C2×C4—62no—
391C2×C4—62no—
392C2×C4—61no—
393C2×C4—61no—
394C2×C4—61no—
395C2×C4—61no—
396C2×C4—61no—
397C2×C4—61no—
398C2×C4—61no—
399C2×C4—61no—
400C2×C4—61no—
401C2×C4—61no—
402C2×C4—61no—
403C2×C4—61no—
404C2×C4—61no—
405C2×C4—61no—
406C2×C4—61no—
407C2×C4C4×S411yescharacteristic
408C2×C4C4×S411yescharacteristic
409C2×C4C22×S411yescharacteristic
410C8—21no—
411C8—21no—
412C8—61no—
413C8—61no—
414C8—121no—
415C8—121no—
416C8—121no—
417C8—121no—
418C2×C3—41no—
419C2×C3—41no—
420C2×C3—41no—
421D3—82no—
422C22—31no—
423C22—31no—
424C22—31no—
425C22—31no—
426C22—31no—
427C22—31no—
428C22—31no—
429C22—31no—
430C22—31no—
431C22—61no—
432C22—62no—
433C22—62no—
434C22—62no—
435C22—121no—
436C22—121no—
437C22—121no—
438C22—121no—
439C22#483911yescharacteristic, complemented
440C22C2×C4×S411yescharacteristic
441C4—21no—
442C4—21no—
443C4—31no—
444C4—31no—
445C4—62no—
446C4—62no—
447C4—62no—
448C4—61no—
449C4—61no—
450C4—121no—
451C4—121no—
452C4—121no—
453C4—121no—
454C4D4×S411yescharacteristic
455C4Q8×S411yescharacteristic
456C3—41noSylow
457C2—31no—
458C2—31no—
459C2—31no—
460C2—31no—
461C2—122no—
462C2C4⋊C4×S411yescharacteristic
463C2#8543711yescharacteristic
464C2#8544211yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×S4
commutatorC22×A4C22×C4
FittingC22×C4⋊C8D3
socleC24C2×C4×D3
prime radicalC22×S4C2×C4
cosocleC2×C4×A4C23
hypercenterC4⋊C8S4
hypocenterA4#424
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C2×C4 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
4. A4×C4⋊C8C2semidirect
5. SL(2,ℤ/4)⋊C8C2semidirect
6. SL(2,ℤ/4)⋊C8C2semidirect
7. C4⋊(A4⋊C8)C2semidirect
26. C22×C4⋊C8D3semidirect
202. C4⋊C8S4semidirect
338. A4#424semidirect
439. C22#4839semidirect

Maximal subgroups

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

subgroupindex
C2×#54002
C2×C8×S42
#852872
A4×C4⋊C82
SL(2,ℤ/4)⋊C82
C4⋊(A4⋊C8)2
#226473
#48394