Finite Group Databasebeta

← S4×#108 #11713009

Subgroups — S4×#108 #11713009

Order 768.

483 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 483 conjugacy classes of subgroups — too many to draw. Only its 65 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
1M16×S4C212yesmaximal
2C2×C8×S4C211yesmaximal, characteristic
3A4×#108C211yesmaximal, characteristic, complemented
4#73114C212yesmaximal, complemented
5#73105C211yesmaximal, characteristic, complemented
6D4×#108—31nomaximal, Sylow
7C2×C4×S4C2211yescharacteristic
8A4×M16C2212yes—
9C2×C8×A4C2211yescharacteristic
10#5409C2212yes—
11C2×A4⋊C8C2211yescharacteristic
12#5400—22no—
13C8×S4—22no—
14C8×S4C412yes—
15C8×S4C412yes—
16D3×#108—41nomaximal
17D4×M16—32no—
18C2×C8×D4—31no—
19C22×#108—31no—
20C22×#108D311yescharacteristic, complemented
21#1950—32no—
22#1855—32no—
23#1855—32no—
24#1848—31no—
25#1810—31no—
26#1810—31no—
27C4×#108—31no—
28C22×S4Q811yescharacteristic
29C2×C4×A4C2311yescharacteristic
30C2×SL(2,ℤ/4)Q811yescharacteristic
31C4×S4C2×C411yescharacteristic
32C4×S4C2×C411yescharacteristic
33C4×S4D411yescharacteristic
34C4×S4D411yescharacteristic
35D3×M16—42no—
36C2×D3×C8—41no—
37C8×A4—22no—
38C8×A4C2×C412yes—
39A4⋊C8—22no—
40A4⋊C8C2×C412yes—
41C3×#108—41no—
42#822—42no—
43#819—41no—
44C22×M16—32no—
45C22×M16C2×D312yes—
46C23×C8—31no—
47C23×C8C2×D311yescharacteristic
48C2×C4×D4—31no—
49#436—62no—
50#435—62no—
51#435—62no—
52C8×D4—32no—
53C8×D4—32no—
54C8×D4—32no—
55C8×D4—32no—
56C8×D4—62no—
57C2×#108—31no—
58C2×#108—31no—
59C2×#108—32no—
60C2×#108—31no—
61C2×#108—31no—
62C2×#108—32no—
63C2×#108—32no—
64C2×#108—62no—
65C2×#108—61no—
66C2×#108—61no—
67#423—32no—
68C2×C4⋊C8—31no—
69#407—32no—
70#407—32no—
71C2×C22⋊C8—31no—
72C2×C22⋊C8—31no—
73C4×M16—32no—
74C2×C4×C8—31no—
75#341—32no—
76#341—32no—
77#341—32no—
78C22×A4C2×Q811yescharacteristic
79C2×S4—22no—
80C2×S4C4⋊C411yescharacteristic
81C2×S4C4⋊C411yescharacteristic
82C2×C4×D3—41no—
83C4×A4C22×C411yescharacteristic
84C4×A4C2×D411yescharacteristic
85SL(2,ℤ/4)C4⋊C411yescharacteristic
86SL(2,ℤ/4)C4⋊C411yescharacteristic
87C3×M16—42no—
88C2×C3×C8—41no—
89#208—42no—
90C2×C3⋊C8—41no—
91ΓL(1,𝔽25)—82no—
92D3×C8—42no—
93D3×C8—42no—
94D3×C8—82no—
95C22×D4—31no—
96C23×C4—31no—
97C23×C4C22×D311yescharacteristic
98C2×M16—32no—
99C2×M16—32no—
100C2×M16—32no—
101C2×M16—32no—
102C2×M16—62no—
103C2×M16—62no—
104C2×M16—62no—
105C2×M16—62no—
106C2×M16—122no—
107C22×C8—22no—
108C22×C8—31no—
109C22×C8—31no—
110C22×C8—32no—
111C22×C8—31no—
112C22×C8—32no—
113C22×C8—32no—
114C22×C8—31no—
115C22×C8—62no—
116C22×C8—61no—
117C22×C8—61no—
118C22×C8—62no—
119C22×C8C4×D312yes—
120C4×D4—31no—
121C4×D4—31no—
122C4×D4—31no—
123C4×D4—31no—
124C4×D4—31no—
125C4×D4—31no—
126C4×D4—31no—
127C4×D4—31no—
128C2×C4⋊C4—31no—
129C2×C22⋊C4—31no—
130C2×C22⋊C4—31no—
131C2×C42—31no—
132#108—32no—
133#108—31no—
134#108—61no—
135#108—62no—
136#108—61no—
137#108—62no—
138#108—62no—
139#108—61no—
140#108—122no—
141#108S411yescharacteristic, direct product factor
142C4⋊C8—32no—
143C4⋊C8—32no—
144C4⋊C8—62no—
145C4⋊C8—62no—
146C22⋊C8—32no—
147C22⋊C8—32no—
148C22⋊C8—32no—
149C22⋊C8—32no—
150C22⋊C8—62no—
151C22⋊C8—62no—
152C22⋊C8—62no—
153C22⋊C8—62no—
154#97—62no—
155C4×C8—32no—
156C4×C8—32no—
157C4×C8—62no—
158C22×D3—41no—
159C2×A4—21no—
160C2×A4C2×C4⋊C411yescharacteristic
161S4—21no—
162S4#10812yesdirect product factor
163C2×C3×C4—41no—
164C2×Dic3—41no—
165C4×D3—41no—
166C4×D3—41no—
167C4×D3—41no—
168C4×D3—41no—
169C3×C8—42no—
170C3×C8—82no—
171C3⋊C8—42no—
172C3⋊C8—82no—
173C24—31no—
174C24D3×Q811yescharacteristic
175C2×D4—31no—
176C2×D4—31no—
177C2×D4—31no—
178C2×D4—31no—
179C2×D4—62no—
180C2×D4—62no—
181C22×C4—31no—
182C22×C4—31no—
183C22×C4—31no—
184C22×C4—31no—
185C22×C4—31no—
186C22×C4—31no—
187C22×C4—31no—
188C22×C4—31no—
189C22×C4—31no—
190C22×C4—31no—
191C22×C4—31no—
192C22×C4—61no—
193C22×C4—61no—
194C22×C4—61no—
195C22×C4—61no—
196C22×C4C2×C4×D311yescharacteristic
197C22×C4D3×D411yescharacteristic
198M16—32no—
199M16—62no—
200M16—62no—
201M16—62no—
202M16—62no—
203M16—62no—
204M16—62no—
205M16—122no—
206M16—122no—
207M16C2×S412yes—
208C2×C8—31no—
209C2×C8—32no—
210C2×C8—32no—
211C2×C8—32no—
212C2×C8—32no—
213C2×C8—32no—
214C2×C8—62no—
215C2×C8—61no—
216C2×C8—62no—
217C2×C8—62no—
218C2×C8—61no—
219C2×C8—61no—
220C2×C8—61no—
221C2×C8—61no—
222C2×C8—62no—
223C2×C8—62no—
224C2×C8—62no—
225C2×C8—62no—
226C2×C8—62no—
227C2×C8—122no—
228C2×C8—122no—
229C2×C8C2×S411yescharacteristic
230C4⋊C4—31no—
231C4⋊C4—31no—
232C4⋊C4—31no—
233C4⋊C4—31no—
234C22⋊C4—31no—
235C22⋊C4—31no—
236C22⋊C4—31no—
237C22⋊C4—31no—
238C22⋊C4—31no—
239C22⋊C4—31no—
240C22⋊C4—31no—
241C22⋊C4—31no—
242C42—31no—
243C42—31no—
244C42—31no—
245C42—31no—
246C22×C3—41no—
247C2×D3—41no—
248C2×D3—41no—
249C2×D3—82no—
250A4C2×#10811yescharacteristic, complemented
251C3×C4—41no—
252C3×C4—41no—
253Dic3—41no—
254Dic3—41no—
255C23—21no—
256C23—31no—
257C23—31no—
258C23—31no—
259C23—31no—
260C23—61no—
261C23—61no—
262C23—62no—
263C23—61no—
264C23—121no—
265C23D3×C4⋊C411yescharacteristic
266D4—32no—
267D4—32no—
268D4—62no—
269D4—61no—
270D4—62no—
271D4—61no—
272C2×C4—31no—
273C2×C4—31no—
274C2×C4—31no—
275C2×C4—31no—
276C2×C4—31no—
277C2×C4—31no—
278C2×C4—31no—
279C2×C4—31no—
280C2×C4—31no—
281C2×C4—31no—
282C2×C4—31no—
283C2×C4—31no—
284C2×C4—31no—
285C2×C4—61no—
286C2×C4—61no—
287C2×C4—61no—
288C2×C4—61no—
289C2×C4—61no—
290C2×C4—62no—
291C2×C4—61no—
292C2×C4—61no—
293C2×C4—61no—
294C2×C4—61no—
295C2×C4—61no—
296C2×C4—61no—
297C2×C4—121no—
298C2×C4C22×S411yescharacteristic
299C8—22no—
300C8—32no—
301C8—62no—
302C8—62no—
303C8—62no—
304C8—122no—
305C8—122no—
306C8C4×S412yes—
307C2×C3—41no—
308C2×C3—81no—
309D3—42no—
310D3—81no—
311C22—31no—
312C22—31no—
313C22—32no—
314C22—31no—
315C22—61no—
316C22—61no—
317C22—61no—
318C22—61no—
319C22—61no—
320C22—61no—
321C22—61no—
322C22—61no—
323C22—61no—
324C22—122no—
325C22—122no—
326C22D3×#10811yescharacteristic, complemented
327C22Q8×S411yescharacteristic
328C4—32no—
329C4—31no—
330C4—31no—
331C4—61no—
332C4—61no—
333C4—61no—
334C4—61no—
335C4—61no—
336C4C2×C4×S411yescharacteristic
337C4D4×S411yescharacteristic
338C3—41noSylow
339C2—21no—
340C2—31no—
341C2—31no—
342C2—61no—
343C2—62no—
344C2—121no—
345C2C4⋊C4×S411yescharacteristic
subgroupG/H
centerC4D4×S4
commutatorC4×A4C22×C4
FrattiniC2×C4C22×S4
FittingC22×#108D3
socleC23D3×C4⋊C4
prime radicalC22×S4Q8
cosocleC2×C4×A4C23
hypercenter#108S4
hypocenterA4C2×#108
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C4 ◁ C2×C4 ◁ #108

Complements

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

NH ≅ G/Ntype
3. A4×#108C2semidirect
4. #73114C2semidirect
5. #73105C2semidirect
20. C22×#108D3semidirect
141. #108S4direct (N × H)
162. S4#108direct (N × H)
250. A4C2×#108semidirect
326. C22D3×#108semidirect

Maximal subgroups

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

subgroupindex
M16×S42
C2×C8×S42
A4×#1082
#731142
#731052
D4×#1083
D3×#1084