Finite Group Databasebeta

← S4×C4⋊C8 #11713000

Subgroups — S4×C4⋊C8 #11713000

Order 768.

644 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 89 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×C8×S4C212yesmaximal
2C42×S4C211yesmaximal, characteristic
3A4×C4⋊C8C211yesmaximal, characteristic, complemented
4SL(2,ℤ/4)⋊C8C212yesmaximal, complemented
5C4⋊(A4⋊C8)C211yesmaximal, characteristic, complemented
6D4×C4⋊C8—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—
16C8×S4—24no—
17D3×C4⋊C8—41nomaximal
18C2×C8×D4—32no—
19C22×C4⋊C8—31no—
20C22×C4⋊C8D311yescharacteristic, complemented
21C42×D4—31no—
22#1939—32no—
23#1850—32no—
24#1850—32no—
25#1842—31no—
26#1797—31no—
27#1797—31no—
28C4×C4⋊C8—31no—
29C22×S4C2×C411yescharacteristic
30C2×C4×A4C2×C411yescharacteristic
31C2×C4×A4C2×C411yescharacteristic
32C2×C4×A4C2311yescharacteristic
33C2×SL(2,ℤ/4)C2×C411yescharacteristic
34C2×SL(2,ℤ/4)C2×C411yescharacteristic
35C2×SL(2,ℤ/4)C2×C411yescharacteristic
36C4×S4—22no—
37C4×S4C814yescomplemented
38C4×S4D412yes—
39C4×S4Q812yes—
40C2×D3×C8—42no—
41C42×D3—41no—
42C8×A4—22no—
43A4⋊C8—22no—
44C3×C4⋊C8—41no—
45Dic3⋊C8—42no—
46C4⋊(C3⋊C8)—41no—
47C23×C8—32no—
48C23×C8C2×D312yes—
49C2×C4×D4—31no—
50C2×C4×D4—31no—
51C2×C4×D4—31no—
52C22×C42—31no—
53C22×C42C2×D311yescharacteristic
54C8×D4—64no—
55C8×D4—64no—
56C2×C4⋊C8—31no—
57C2×C4⋊C8—31no—
58C2×C4⋊C8—32no—
59C2×C4⋊C8—31no—
60C2×C4⋊C8—31no—
61C2×C4⋊C8—32no—
62C2×C4⋊C8—32no—
63C2×C4⋊C8—32no—
64C2×C4⋊C8—62no—
65C2×C4⋊C8—61no—
66C2×C4⋊C8—61no—
67C2×C22⋊C8—32no—
68C2×C22⋊C8—32no—
69C2×C4×C8—32no—
70C4×C4⋊C4—31no—
71C4×C4⋊C4—31no—
72C4×C4⋊C4—31no—
73C4×C22⋊C4—31no—
74C4×C22⋊C4—31no—
75C4×C22⋊C4—31no—
76C4×C22⋊C4—31no—
77C4×C22⋊C4—31no—
78C4×C22⋊C4—31no—
79C43—31no—
80#336—32no—
81#336—32no—
82#336—32no—
83C22×A4C22×C411yescharacteristic
84C2×S4C4⋊C412yes—
85C2×S4C2×C812yes—
86C2×S4M1612yes—
87C2×C4×D3—41no—
88C2×C4×D3—41no—
89C2×C4×D3—41no—
90C4×A4—21no—
91C4×A4C2×C812yescomplemented
92C4×A4C2×D411yescharacteristic
93C4×A4C2×Q811yescharacteristic
94SL(2,ℤ/4)—21no—
95SL(2,ℤ/4)C4⋊C412yes—
96SL(2,ℤ/4)C2×C812yescomplemented
97C2×C3×C8—42no—
98C3×C42—41no—
99C4×Dic3—41no—
100C4×Dic3—41no—
101C4×Dic3—41no—
102C2×C3⋊C8—42no—
103D3×C8—84no—
104C22×D4—31no—
105C23×C4—31no—
106C23×C4—31no—
107C23×C4—31no—
108C23×C4C4×D311yescharacteristic
109C23×C4C4×D311yescharacteristic
110C23×C4C22×D311yescharacteristic
111C22×C8—22no—
112C22×C8—32no—
113C22×C8—32no—
114C22×C8—32no—
115C22×C8—32no—
116C22×C8—62no—
117C22×C8—62no—
118C22×C8—64no—
119C22×C8—62no—
120C22×C8—122no—
121C4×D4—34no—
122C4×D4—32no—
123C4×D4—32no—
124C4×D4—34no—
125C4×D4—32no—
126C4×D4—32no—
127C4×D4—62no—
128C4×D4—62no—
129C2×C4⋊C4—31no—
130C2×C4⋊C4—31no—
131C2×C4⋊C4—31no—
132C2×C22⋊C4—31no—
133C2×C22⋊C4—31no—
134C2×C22⋊C4—31no—
135C2×C22⋊C4—31no—
136C2×C22⋊C4—31no—
137C2×C22⋊C4—31no—
138C2×C42—31no—
139C2×C42—31no—
140C2×C42—31no—
141C2×C42—31no—
142C2×C42—31no—
143C2×C42—31no—
144C2×C42—31no—
145C2×C42—31no—
146C2×C42—31no—
147C2×C42—31no—
148C2×C42—31no—
149C2×C42—31no—
150C2×C42—31no—
151C2×C42—61no—
152C2×C42—61no—
153C2×C42—61no—
154C2×C42—61no—
155C4⋊C8—32no—
156C4⋊C8—31no—
157C4⋊C8—61no—
158C4⋊C8—61no—
159C4⋊C8—61no—
160C4⋊C8—61no—
161C4⋊C8—64no—
162C4⋊C8—62no—
163C4⋊C8—62no—
164C4⋊C8—61no—
165C4⋊C8—122no—
166C4⋊C8S411yescharacteristic, direct product factor
167C22⋊C8—64no—
168C22⋊C8—62no—
169C22⋊C8—62no—
170C4×C8—64no—
171#95—31no—
172#95—31no—
173#95—31no—
174#95—31no—
175#95—31no—
176#95—31no—
177C22×D3—41no—
178C2×A4C2×C4⋊C411yescharacteristic
179C2×A4C22×C811yescharacteristic
180C2×A4C2×M1611yescharacteristic
181S4C4⋊C814yesdirect product factor
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×Dic3—41no—
186C2×Dic3—41no—
187C2×Dic3—41no—
188C4×D3—42no—
189C4×D3—44no—
190C4×D3—42no—
191C4×D3—82no—
192C3×C8—82no—
193C3⋊C8—82no—
194C24—31no—
195C24C2×C4×D311yescharacteristic
196C2×D4—32no—
197C2×D4—32no—
198C2×D4—32no—
199C2×D4—32no—
200C2×D4—32no—
201C2×D4—32no—
202C22×C4—21no—
203C22×C4—31no—
204C22×C4—31no—
205C22×C4—31no—
206C22×C4—31no—
207C22×C4—31no—
208C22×C4—31no—
209C22×C4—32no—
210C22×C4—31no—
211C22×C4—31no—
212C22×C4—31no—
213C22×C4—31no—
214C22×C4—31no—
215C22×C4—32no—
216C22×C4—34no—
217C22×C4—32no—
218C22×C4—31no—
219C22×C4—31no—
220C22×C4—31no—
221C22×C4—31no—
222C22×C4—61no—
223C22×C4—61no—
224C22×C4—61no—
225C22×C4—62no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4—61no—
229C22×C4—61no—
230C22×C4—61no—
231C22×C4—62no—
232C22×C4—61no—
233C22×C4—121no—
234C22×C4D3×C812yescomplemented
235C22×C4D3×D411yescharacteristic
236C22×C4D3×Q811yescharacteristic
237C2×C8—32no—
238C2×C8—62no—
239C2×C8—62no—
240C2×C8—62no—
241C2×C8—62no—
242C2×C8—62no—
243C2×C8—62no—
244C2×C8—62no—
245C2×C8—62no—
246C2×C8—62no—
247C2×C8—62no—
248C2×C8—124no—
249C2×C8—124no—
250C2×C8C2×S412yes—
251C4⋊C4—34no—
252C4⋊C4—32no—
253C4⋊C4—32no—
254C4⋊C4—62no—
255C22⋊C4—34no—
256C22⋊C4—32no—
257C22⋊C4—32no—
258C22⋊C4—32no—
259C22⋊C4—32no—
260C22⋊C4—32no—
261C22⋊C4—32no—
262C22⋊C4—62no—
263C22⋊C4—61no—
264C22⋊C4—61no—
265C42—32no—
266C42—34no—
267C42—32no—
268C42—31no—
269C42—31no—
270C42—31no—
271C42—61no—
272C42—61no—
273C42—62no—
274C42—61no—
275C42—61no—
276C42—61no—
277C42—61no—
278C42—62no—
279C42—61no—
280C42—61no—
281C42—61no—
282C42—121no—
283C42C2×S411yescharacteristic
284C22×C3—41no—
285C2×D3—42no—
286C2×D3—42no—
287C2×D3—42no—
288A4C2×C4⋊C811yescharacteristic, complemented
289C3×C4—42no—
290C3×C4—41no—
291C3×C4—41no—
292C3×C4—81no—
293Dic3—42no—
294Dic3—42no—
295Dic3—81no—
296C23—31no—
297C23—31no—
298C23—31no—
299C23—31no—
300C23—32no—
301C23—32no—
302C23—32no—
303C23—61no—
304C23—61no—
305C23—61no—
306C23—61no—
307C23D3×C4⋊C411yescharacteristic
308C23C2×D3×C811yescharacteristic
309C23D3×M1611yescharacteristic
310D4—34no—
311D4—34no—
312D4—34no—
313D4—34no—
314C2×C4—32no—
315C2×C4—32no—
316C2×C4—32no—
317C2×C4—31no—
318C2×C4—31no—
319C2×C4—32no—
320C2×C4—32no—
321C2×C4—31no—
322C2×C4—31no—
323C2×C4—31no—
324C2×C4—31no—
325C2×C4—32no—
326C2×C4—31no—
327C2×C4—31no—
328C2×C4—32no—
329C2×C4—32no—
330C2×C4—32no—
331C2×C4—32no—
332C2×C4—31no—
333C2×C4—61no—
334C2×C4—61no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—62no—
338C2×C4—62no—
339C2×C4—64no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—64no—
349C2×C4—62no—
350C2×C4—62no—
351C2×C4—61no—
352C2×C4—61no—
353C2×C4—61no—
354C2×C4—61no—
355C2×C4—61no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—122no—
360C2×C4—122no—
361C2×C4C4×S411yescharacteristic
362C2×C4C4×S411yescharacteristic
363C2×C4C22×S411yescharacteristic
364C8—22no—
365C8—62no—
366C8—122no—
367C8—122no—
368C2×C3—41no—
369C2×C3—41no—
370C2×C3—41no—
371D3—44no—
372C22—31no—
373C22—31no—
374C22—31no—
375C22—31no—
376C22—31no—
377C22—31no—
378C22—31no—
379C22—31no—
380C22—31no—
381C22—34no—
382C22—61no—
383C22—62no—
384C22—62no—
385C22—62no—
386C22—62no—
387C22—62no—
388C22—62no—
389C22D3×C4⋊C811yescharacteristic, complemented
390C22C2×C4×S411yescharacteristic
391C4—21no—
392C4—34no—
393C4—31no—
394C4—31no—
395C4—32no—
396C4—62no—
397C4—62no—
398C4—62no—
399C4—62no—
400C4—61no—
401C4—121no—
402C4—121no—
403C4C8×S412yescomplemented
404C4D4×S411yescharacteristic
405C4Q8×S411yescharacteristic
406C3—41noSylow
407C2—31no—
408C2—31no—
409C2—31no—
410C2—31no—
411C2—64no—
412C2C4⋊C4×S411yescharacteristic
413C2C2×C8×S411yescharacteristic
414C2M16×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
hypocenterA4C2×C4⋊C8
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C2×C4 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
3. A4×C4⋊C8C2semidirect
4. SL(2,ℤ/4)⋊C8C2semidirect
5. C4⋊(A4⋊C8)C2semidirect
20. C22×C4⋊C8D3semidirect
37. C4×S4C8semidirect
91. C4×A4C2×C8semidirect
96. SL(2,ℤ/4)C2×C8semidirect
166. C4⋊C8S4direct (N × H)
181. S4C4⋊C8direct (N × H)
234. C22×C4D3×C8semidirect
288. A4C2×C4⋊C8semidirect
389. C22D3×C4⋊C8semidirect
403. C4C8×S4semidirect

Maximal subgroups

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

subgroupindex
C2×C8×S42
C42×S42
A4×C4⋊C82
SL(2,ℤ/4)⋊C82
C4⋊(A4⋊C8)2
D4×C4⋊C83
D3×C4⋊C84