Finite Group Databasebeta

← #11713034

Subgroups — #11713034

Order 768.

550 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 77 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
1C42×S4C211yesmaximal, characteristic
2A4×C4⋊C8C211yesmaximal, characteristic, complemented
3(C2×S4)⋊C8C212yesmaximal
4C8×SL(2,ℤ/4)C212yesmaximal, complemented
5C4⋊(A4⋊C8)C211yesmaximal, characteristic, complemented
6#22748—31nomaximal, Sylow
7C2×C4×S4C411yescharacteristic
8C2×C4×S4C411yescharacteristic
9C2×C4×S4C2211yescharacteristic
10C2×C8×A4C2212yes—
11C42×A4C2211yescharacteristic
12C4×SL(2,ℤ/4)C411yescharacteristic
13C4×SL(2,ℤ/4)C411yescharacteristic
14C4×SL(2,ℤ/4)C2211yescharacteristic
15C2×A4⋊C8C2212yes—
16#4833—41nomaximal
17C2×#431—31no—
18C22×C4⋊C8D311yescharacteristic, complemented
19C42×D4—31no—
20#1923—32no—
21#1916—32no—
22#1869—32no—
23#1831—31no—
24#1797—31no—
25#1797—31no—
26C4×C4⋊C8—31no—
27C8×C22⋊C4—32no—
28C22×S4C2×C411yescharacteristic
29C2×C4×A4C2×C411yescharacteristic
30C2×C4×A4C2×C411yescharacteristic
31C2×C4×A4C2311yescharacteristic
32C2×SL(2,ℤ/4)C2×C411yescharacteristic
33C2×SL(2,ℤ/4)C2×C411yescharacteristic
34C2×SL(2,ℤ/4)C2×C411yescharacteristic
35C4×S4—21no—
36C4×S4—21no—
37C4×S4—22no—
38C4×S4C814yescomplemented
39C42×D3—41no—
40C8×A4—22no—
41A4⋊C8—22no—
42C3×C4⋊C8—41no—
43(C2×D3)⋊C8—42no—
44C8×Dic3—42no—
45C4⋊(C3⋊C8)—41no—
46C23×C8C2×D312yes—
47C2×C4×D4—31no—
48C2×C4×D4—31no—
49C2×C4×D4—31no—
50C22×C42—31no—
51C22×C42C2×D311yescharacteristic
52#431—61no—
53#431—62no—
54#431—61no—
55C2×C4⋊C8—31no—
56C2×C4⋊C8—31no—
57C2×C4⋊C8—32no—
58C2×C4⋊C8—31no—
59C2×C4⋊C8—31no—
60C2×C22⋊C8—32no—
61C2×C22⋊C8—32no—
62C2×C22⋊C8—32no—
63C2×C4×C8—32no—
64C2×C4×C8—32no—
65C4×C4⋊C4—31no—
66C4×C4⋊C4—31no—
67C4×C4⋊C4—31no—
68C4×C22⋊C4—31no—
69C4×C22⋊C4—31no—
70C4×C22⋊C4—31no—
71C4×C22⋊C4—31no—
72C4×C22⋊C4—31no—
73C4×C22⋊C4—31no—
74C43—31no—
75#336—32no—
76#336—32no—
77#336—32no—
78#336—32no—
79#336—32no—
80C22×A4C22×C411yescharacteristic
81C2×S4—21no—
82C2×S4—21no—
83C2×S4C2×C812yes—
84C2×C4×D3—41no—
85C2×C4×D3—41no—
86C2×C4×D3—41no—
87C4×A4—21no—
88C4×A4C2×C812yes—
89C4×A4#4111yescharacteristic
90C4×A4#4111yescharacteristic
91SL(2,ℤ/4)—21no—
92SL(2,ℤ/4)C2×C812yes—
93SL(2,ℤ/4)M1612yes—
94C2×C3×C8—42no—
95C3×C42—41no—
96C4×Dic3—41no—
97C4×Dic3—41no—
98C4×Dic3—41no—
99C2×C3⋊C8—42no—
100C22×D4—31no—
101C23×C4—31no—
102C23×C4—31no—
103C23×C4—31no—
104C23×C4C4×D311yescharacteristic
105C23×C4C4×D311yescharacteristic
106C23×C4C22×D311yescharacteristic
107C22×C8—22no—
108C22×C8—32no—
109C22×C8—32no—
110C22×C8—32no—
111C22×C8—32no—
112C22×C8—62no—
113C4×D4—34no—
114C4×D4—34no—
115C4×D4—62no—
116C4×D4—62no—
117C4×D4—61no—
118C4×D4—61no—
119C4×D4—61no—
120C4×D4—61no—
121C2×C4⋊C4—31no—
122C2×C4⋊C4—31no—
123C2×C4⋊C4—31no—
124C2×C22⋊C4—31no—
125C2×C22⋊C4—31no—
126C2×C22⋊C4—31no—
127C2×C22⋊C4—31no—
128C2×C22⋊C4—31no—
129C2×C22⋊C4—31no—
130C2×C42—31no—
131C2×C42—31no—
132C2×C42—31no—
133C2×C42—31no—
134C2×C42—31no—
135C2×C42—31no—
136C2×C42—31no—
137C2×C42—31no—
138C2×C42—31no—
139C2×C42—31no—
140C2×C42—31no—
141C2×C42—31no—
142C2×C42—31no—
143C2×C42—61no—
144C2×C42—61no—
145C2×C42—61no—
146C2×C42—61no—
147C4⋊C8—32no—
148C4⋊C8—31no—
149C4⋊C8—61no—
150C4⋊C8—61no—
151C4⋊C8—61no—
152C4⋊C8—61no—
153C4⋊C8—61no—
154C4⋊C8S411yescharacteristic, complemented
155C22⋊C8—62no—
156C22⋊C8—62no—
157C22⋊C8—62no—
158C22⋊C8—62no—
159C22⋊C8—122no—
160C4×C8—62no—
161C4×C8—62no—
162C4×C8—122no—
163#95—31no—
164#95—31no—
165#95—31no—
166#95—31no—
167#95—31no—
168#95—31no—
169C22×D3—41no—
170C2×A4#11711yescharacteristic
171C2×A4C22×C811yescharacteristic
172C2×A4C2×M1611yescharacteristic
173S4—22no—
174C2×C3×C4—41no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×Dic3—41no—
178C2×Dic3—41no—
179C2×Dic3—41no—
180C4×D3—44no—
181C4×D3—82no—
182C4×D3—81no—
183C4×D3—81no—
184C3×C8—82no—
185C3⋊C8—82no—
186C24—31no—
187C24C2×C4×D311yescharacteristic
188C2×D4—32no—
189C2×D4—32no—
190C2×D4—61no—
191C2×D4—61no—
192C2×D4—61no—
193C2×D4—61no—
194C22×C4—21no—
195C22×C4—31no—
196C22×C4—31no—
197C22×C4—31no—
198C22×C4—31no—
199C22×C4—31no—
200C22×C4—32no—
201C22×C4—31no—
202C22×C4—31no—
203C22×C4—31no—
204C22×C4—31no—
205C22×C4—31no—
206C22×C4—31no—
207C22×C4—34no—
208C22×C4—31no—
209C22×C4—31no—
210C22×C4—31no—
211C22×C4—31no—
212C22×C4—61no—
213C22×C4—62no—
214C22×C4—61no—
215C22×C4—61no—
216C22×C4—61no—
217C22×C4—61no—
218C22×C4—61no—
219C22×C4—62no—
220C22×C4—61no—
221C22×C4—61no—
222C22×C4—61no—
223C22×C4—61no—
224C22×C4—61no—
225C22×C4—121no—
226C22×C4D3×C812yes—
227C22×C4#23711yescharacteristic
228C22×C4#23911yescharacteristic
229C2×C8—32no—
230C2×C8—62no—
231C2×C8—62no—
232C2×C8—62no—
233C2×C8—62no—
234C2×C8—62no—
235C2×C8—62no—
236C2×C8—62no—
237C2×C8—62no—
238C2×C8—62no—
239C2×C8—62no—
240C2×C8C2×S412yes—
241C4⋊C4—34no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—62no—
245C22⋊C4—34no—
246C22⋊C4—32no—
247C22⋊C4—32no—
248C22⋊C4—32no—
249C22⋊C4—32no—
250C22⋊C4—61no—
251C22⋊C4—61no—
252C22⋊C4—62no—
253C22⋊C4—61no—
254C22⋊C4—61no—
255C42—34no—
256C42—31no—
257C42—31no—
258C42—31no—
259C42—61no—
260C42—61no—
261C42—62no—
262C42—61no—
263C42—61no—
264C42—61no—
265C42—61no—
266C42—61no—
267C42—61no—
268C42—62no—
269C42—61no—
270C42—61no—
271C42—61no—
272C42—121no—
273C42C2×S411yescharacteristic
274C22×C3—41no—
275C2×D3—42no—
276C2×D3—81no—
277C2×D3—81no—
278A4#43111yescharacteristic, complemented
279C3×C4—42no—
280C3×C4—41no—
281C3×C4—41no—
282C3×C4—81no—
283Dic3—42no—
284Dic3—42no—
285Dic3—81no—
286C23—31no—
287C23—31no—
288C23—31no—
289C23—31no—
290C23—32no—
291C23—61no—
292C23—61no—
293C23—61no—
294C23—61no—
295C23—61no—
296C23—61no—
297C23#89211yescharacteristic
298C23C2×D3×C811yescharacteristic
299C23D3×M1611yescharacteristic
300D4—62no—
301D4—62no—
302D4—62no—
303D4—62no—
304C2×C4—31no—
305C2×C4—31no—
306C2×C4—32no—
307C2×C4—32no—
308C2×C4—31no—
309C2×C4—31no—
310C2×C4—31no—
311C2×C4—31no—
312C2×C4—32no—
313C2×C4—31no—
314C2×C4—31no—
315C2×C4—32no—
316C2×C4—32no—
317C2×C4—32no—
318C2×C4—32no—
319C2×C4—32no—
320C2×C4—31no—
321C2×C4—61no—
322C2×C4—61no—
323C2×C4—61no—
324C2×C4—61no—
325C2×C4—64no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—61no—
331C2×C4—61no—
332C2×C4—61no—
333C2×C4—61no—
334C2×C4—61no—
335C2×C4—61no—
336C2×C4—64no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—122no—
346C2×C4—121no—
347C2×C4—121no—
348C2×C4—122no—
349C2×C4—121no—
350C2×C4—121no—
351C2×C4C4×S411yescharacteristic
352C2×C4C4×S411yescharacteristic
353C2×C4C22×S411yescharacteristic
354C8—22no—
355C8—62no—
356C8—122no—
357C8—122no—
358C2×C3—41no—
359C2×C3—41no—
360C2×C3—41no—
361D3—82no—
362C22—31no—
363C22—31no—
364C22—31no—
365C22—31no—
366C22—31no—
367C22—31no—
368C22—31no—
369C22—31no—
370C22—31no—
371C22—61no—
372C22—62no—
373C22—62no—
374C22—62no—
375C22—121no—
376C22—121no—
377C22—121no—
378C22—121no—
379C22#483311yescharacteristic, complemented
380C22C2×C4×S411yescharacteristic
381C4—21no—
382C4—31no—
383C4—31no—
384C4—32no—
385C4—62no—
386C4—62no—
387C4—62no—
388C4—62no—
389C4—61no—
390C4—62no—
391C4—121no—
392C4—121no—
393C4C8×S412yes—
394C4#591411yescharacteristic
395C4#591911yescharacteristic
396C3—41noSylow
397C2—31no—
398C2—31no—
399C2—31no—
400C2—31no—
401C2—122no—
402C2#8530711yescharacteristic
403C2C2×C8×S411yescharacteristic
404C2M16×S411yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×S4
commutatorC2×A4C22×C8
FittingC22×C4⋊C8D3
socleC24C2×C4×D3
prime radicalC22×S4C2×C4
cosocleC2×C4×A4C23
hypercenterC4⋊C8S4
hypocenterA4#431
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C2×C4 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
2. A4×C4⋊C8C2semidirect
4. C8×SL(2,ℤ/4)C2semidirect
5. C4⋊(A4⋊C8)C2semidirect
18. C22×C4⋊C8D3semidirect
38. C4×S4C8semidirect
154. C4⋊C8S4semidirect
278. A4#431semidirect
379. C22#4833semidirect

Maximal subgroups

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

subgroupindex
C42×S42
A4×C4⋊C82
(C2×S4)⋊C82
C8×SL(2,ℤ/4)2
C4⋊(A4⋊C8)2
#227483
#48334