Finite Group Databasebeta

← S4×C4⋊Q8 #11714409

Subgroups — S4×C4⋊Q8 #11714409

Order 768.

1579 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 1579 conjugacy classes of subgroups — too many to draw. Only its 157 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×Q8×S4C212yesmaximal
2A4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#85502C212yesmaximal, complemented
4C4⋊C4×S4C214yesmaximal
5#85303C214yesmaximal, complemented
6C42×S4C211yesmaximal, characteristic
7#85283C211yesmaximal, characteristic, complemented
8D4×C4⋊Q8—31nomaximal, Sylow
9C2×Q8×A4C2212yes—
10Q8×S4—28no—
11C2×C4×S4C2214yes—
12C2×C4×S4C2211yescharacteristic
13C2×C4×S4C2212yes—
14C2×A4⋊Q8C2212yes—
15C2×A4⋊Q8C2214yes—
16D3×C4⋊Q8—41nomaximal
17A4×C4⋊C4C2214yes—
18C42×A4C2211yescharacteristic
19C4⋊SL(2,ℤ/4)C2214yes—
20SL(2,ℤ/4)⋊C4C2218yes—
21C4×SL(2,ℤ/4)C2212yes—
22C4×SL(2,ℤ/4)C2211yescharacteristic
23C2×D4×Q8—32no—
24C22×C4⋊Q8—31no—
25C22×C4⋊Q8D311yescharacteristic, complemented
26#2868—31no—
27#2857—31no—
28#2857—31no—
29#2551—34no—
30#2546—32no—
31#2440—34no—
32#2440—34no—
33#2434—32no—
34#2434—32no—
35D4×C4⋊C4—34no—
36C4×C4⋊Q8—31no—
37C42×D4—31no—
38C22×S4C2311yescharacteristic
39C2×D3×Q8—42no—
40Q8×A4—24no—
41C2×C4×A4C2314yes—
42C2×C4×A4C2311yescharacteristic
43C2×C4×A4C2312yes—
44C2×SL(2,ℤ/4)C2314yes—
45C2×SL(2,ℤ/4)C2311yescharacteristic
46C2×SL(2,ℤ/4)C2312yes—
47C4×S4—28no—
48C4×S4D414yes—
49C4×S4Q818yescomplemented
50A4⋊Q8—24no—
51A4⋊Q8—28no—
52C3×C4⋊Q8—41no—
53#944—42no—
54D3×C4⋊C4—44no—
55#888—44no—
56C42×D3—41no—
57#869—41no—
58C23×Q8—32no—
59C23×Q8C2×D312yes—
60D4×Q8—68no—
61D4×Q8—68no—
62C2×C4⋊Q8—31no—
63C2×C4⋊Q8—31no—
64C2×C4⋊Q8—32no—
65C2×C4⋊Q8—34no—
66C2×C4⋊Q8—31no—
67C2×C4⋊Q8—32no—
68C2×C4⋊Q8—32no—
69C2×C4⋊Q8—34no—
70C2×C4⋊Q8—34no—
71C2×C4⋊Q8—32no—
72C2×C4⋊Q8—62no—
73C2×C4⋊Q8—64no—
74C2×C4⋊Q8—61no—
75C2×C4⋊Q8—61no—
76C2×C22⋊Q8—32no—
77C2×C22⋊Q8—34no—
78C2×C22⋊Q8—32no—
79C2×C22⋊Q8—34no—
80C2×C4×Q8—32no—
81C2×C4×D4—32no—
82C2×C4×D4—31no—
83C2×C4×D4—34no—
84C22×C4⋊C4—34no—
85C22×C4⋊C4C2×D314yes—
86C22×C42—31no—
87C22×C42C2×D311yescharacteristic
88#398—38no—
89#395—38no—
90#395—38no—
91#391—32no—
92#391—32no—
93#391—32no—
94#389—34no—
95#389—34no—
96#389—34no—
97#389—38no—
98#385—38no—
99#385—38no—
100#384—31no—
101#384—34no—
102#384—34no—
103#380—34no—
104#380—34no—
105C4×C4⋊C4—34no—
106C4×C4⋊C4—31no—
107C4×C4⋊C4—32no—
108C4×C22⋊C4—31no—
109C4×C22⋊C4—32no—
110C4×C22⋊C4—31no—
111C4×C22⋊C4—32no—
112C43—31no—
113C22×A4C2411yescharacteristic
114C2×S4C2×D412yes—
115C2×S4C2×Q814yes—
116C2×C3×Q8—42no—
117D3×Q8—88no—
118C2×C4×D3—42no—
119C2×C4×D3—41no—
120C2×C4×D3—44no—
121C2×Dic6—42no—
122C2×Dic6—44no—
123C4×A4—24no—
124C4×A4C2×D412yes—
125C4×A4C2×Q814yescomplemented
126SL(2,ℤ/4)—24no—
127SL(2,ℤ/4)C2×D412yes—
128SL(2,ℤ/4)C2×Q814yescomplemented
129C3×C4⋊C4—44no—
130C3×C42—41no—
131C4⋊Dic3—44no—
132Dic3⋊C4—48no—
133C4×Dic3—42no—
134C4×Dic3—41no—
135C22×Q8—24no—
136C22×Q8—32no—
137C22×Q8—32no—
138C22×Q8—34no—
139C22×Q8—32no—
140C22×Q8—34no—
141C22×Q8—64no—
142C22×Q8—68no—
143C22×Q8—64no—
144C22×Q8—62no—
145C22×Q8—62no—
146C22×Q8—124no—
147C22×D4—31no—
148C23×C4—32no—
149C23×C4—31no—
150C23×C4—34no—
151C23×C4C22×D314yes—
152C23×C4C22×D311yescharacteristic
153C23×C4C22×D312yes—
154C4⋊Q8—32no—
155C4⋊Q8—34no—
156C4⋊Q8—31no—
157C4⋊Q8—62no—
158C4⋊Q8—61no—
159C4⋊Q8—61no—
160C4⋊Q8—62no—
161C4⋊Q8—68no—
162C4⋊Q8—64no—
163C4⋊Q8—68no—
164C4⋊Q8—64no—
165C4⋊Q8—616no—
166C4⋊Q8—64no—
167C4⋊Q8—62no—
168C4⋊Q8—62no—
169C4⋊Q8—64no—
170C4⋊Q8—61no—
171C4⋊Q8—124no—
172C4⋊Q8S411yescharacteristic, direct product factor
173C22⋊Q8—68no—
174C22⋊Q8—68no—
175C22⋊Q8—616no—
176C22⋊Q8—64no—
177C22⋊Q8—68no—
178C22⋊Q8—64no—
179C4×Q8—68no—
180C4×D4—38no—
181C4×D4—34no—
182C4×D4—34no—
183C4×D4—38no—
184C4×D4—68no—
185C4×D4—68no—
186C2×C4⋊C4—31no—
187C2×C4⋊C4—32no—
188C2×C4⋊C4—34no—
189C2×C4⋊C4—34no—
190C2×C4⋊C4—38no—
191C2×C4⋊C4—34no—
192C2×C4⋊C4—38no—
193C2×C4⋊C4—34no—
194C2×C4⋊C4—32no—
195C2×C4⋊C4—34no—
196C2×C4⋊C4—34no—
197C2×C4⋊C4—34no—
198C2×C4⋊C4—34no—
199C2×C4⋊C4—34no—
200C2×C4⋊C4—38no—
201C2×C4⋊C4—68no—
202C2×C4⋊C4—64no—
203C2×C4⋊C4—64no—
204C2×C22⋊C4—31no—
205C2×C22⋊C4—32no—
206C2×C22⋊C4—31no—
207C2×C22⋊C4—32no—
208C2×C22⋊C4—34no—
209C2×C22⋊C4—34no—
210C2×C42—31no—
211C2×C42—32no—
212C2×C42—31no—
213C2×C42—32no—
214C2×C42—31no—
215C2×C42—34no—
216C2×C42—31no—
217C2×C42—32no—
218C2×C42—31no—
219C2×C42—32no—
220C2×C42—61no—
221C2×C42—62no—
222C2×C42—61no—
223#95—32no—
224#95—32no—
225#95—32no—
226#95—38no—
227#95—38no—
228#95—38no—
229C22×D3—41no—
230C2×A4C22×D411yescharacteristic
231C2×A4C22×Q812yes—
232S4C4⋊Q814yesdirect product factor
233C3×Q8—84no—
234C2×C3×C4—42no—
235C2×C3×C4—41no—
236C2×C3×C4—44no—
237C2×Dic3—42no—
238C2×Dic3—41no—
239C2×Dic3—44no—
240C4×D3—44no—
241C4×D3—48no—
242C4×D3—88no—
243Dic6—88no—
244Dic6—84no—
245C24—31no—
246C24C23×D311yescharacteristic
247C2×Q8—34no—
248C2×Q8—32no—
249C2×Q8—62no—
250C2×Q8—64no—
251C2×Q8—64no—
252C2×Q8—64no—
253C2×Q8—68no—
254C2×Q8—64no—
255C2×Q8—64no—
256C2×Q8—68no—
257C2×Q8—62no—
258C2×Q8—64no—
259C2×Q8—64no—
260C2×Q8—64no—
261C2×Q8—62no—
262C2×Q8—124no—
263C2×Q8—124no—
264C2×Q8—128no—
265C2×Q8—1216no—
266C2×Q8—128no—
267C2×Q8C2×S412yes—
268C2×D4—32no—
269C2×D4—34no—
270C2×D4—32no—
271C2×D4—34no—
272C22×C4—24no—
273C22×C4—31no—
274C22×C4—32no—
275C22×C4—31no—
276C22×C4—32no—
277C22×C4—31no—
278C22×C4—32no—
279C22×C4—34no—
280C22×C4—32no—
281C22×C4—32no—
282C22×C4—31no—
283C22×C4—31no—
284C22×C4—34no—
285C22×C4—38no—
286C22×C4—34no—
287C22×C4—34no—
288C22×C4—34no—
289C22×C4—34no—
290C22×C4—62no—
291C22×C4—61no—
292C22×C4—64no—
293C22×C4—62no—
294C22×C4—64no—
295C22×C4—62no—
296C22×C4—61no—
297C22×C4—64no—
298C22×C4—68no—
299C22×C4—64no—
300C22×C4—124no—
301C22×C4D3×D412yes—
302C22×C4D3×Q814yescomplemented
303C4⋊C4—38no—
304C4⋊C4—34no—
305C4⋊C4—38no—
306C4⋊C4—34no—
307C4⋊C4—64no—
308C4⋊C4—68no—
309C4⋊C4—64no—
310C4⋊C4—68no—
311C4⋊C4—68no—
312C4⋊C4—64no—
313C4⋊C4—68no—
314C4⋊C4—68no—
315C4⋊C4—68no—
316C4⋊C4—68no—
317C4⋊C4—68no—
318C4⋊C4—64no—
319C4⋊C4—128no—
320C4⋊C4C2×S414yes—
321C22⋊C4—38no—
322C22⋊C4—34no—
323C22⋊C4—34no—
324C22⋊C4—32no—
325C22⋊C4—32no—
326C22⋊C4—34no—
327C22⋊C4—68no—
328C22⋊C4—64no—
329C22⋊C4—64no—
330C42—38no—
331C42—34no—
332C42—31no—
333C42—32no—
334C42—61no—
335C42—64no—
336C42—62no—
337C42—61no—
338C42—68no—
339C42—62no—
340C42—61no—
341C42—62no—
342C42C2×S411yescharacteristic
343C22×C3—41no—
344C2×D3—42no—
345C2×D3—44no—
346A4C2×C4⋊Q811yescharacteristic, complemented
347C3×C4—42no—
348C3×C4—44no—
349C3×C4—84no—
350Dic3—42no—
351Dic3—44no—
352Dic3—84no—
353C23—31no—
354C23—31no—
355C23—32no—
356C23—32no—
357C23—34no—
358C23—61no—
359C23—62no—
360C23—61no—
361C23C2×D3×D411yescharacteristic
362C23C2×D3×Q812yes—
363Q8—24no—
364Q8—68no—
365Q8—64no—
366Q8—128no—
367Q8—124no—
368Q8—124no—
369Q8—128no—
370Q8—124no—
371D4—34no—
372D4—38no—
373D4—34no—
374C2×C4—32no—
375C2×C4—34no—
376C2×C4—32no—
377C2×C4—31no—
378C2×C4—32no—
379C2×C4—32no—
380C2×C4—34no—
381C2×C4—34no—
382C2×C4—32no—
383C2×C4—34no—
384C2×C4—32no—
385C2×C4—34no—
386C2×C4—32no—
387C2×C4—34no—
388C2×C4—34no—
389C2×C4—64no—
390C2×C4—64no—
391C2×C4—62no—
392C2×C4—61no—
393C2×C4—68no—
394C2×C4—64no—
395C2×C4—62no—
396C2×C4—62no—
397C2×C4—62no—
398C2×C4—62no—
399C2×C4—61no—
400C2×C4—62no—
401C2×C4—61no—
402C2×C4—64no—
403C2×C4—68no—
404C2×C4—64no—
405C2×C4—64no—
406C2×C4—64no—
407C2×C4—64no—
408C2×C4—68no—
409C2×C4—64no—
410C2×C4—64no—
411C2×C4—128no—
412C2×C4—128no—
413C2×C4C22×S414yes—
414C2×C4C22×S411yescharacteristic
415C2×C4C22×S412yes—
416C2×C3—41no—
417C2×C3—42no—
418D3—44no—
419C22—31no—
420C22—32no—
421C22—31no—
422C22—31no—
423C22—32no—
424C22—32no—
425C22—34no—
426C22—61no—
427C22—62no—
428C22—64no—
429C22—62no—
430C22—64no—
431C22D3×C4⋊Q811yescharacteristic, complemented
432C22C23×S411yescharacteristic
433C4—24no—
434C4—34no—
435C4—32no—
436C4—34no—
437C4—62no—
438C4—64no—
439C4—62no—
440C4—64no—
441C4—64no—
442C4—124no—
443C4—124no—
444C4D4×S412yes—
445C4Q8×S414yescomplemented
446C3—41noSylow
447C2—31no—
448C2—31no—
449C2—32no—
450C2—64no—
451C2C2×D4×S411yescharacteristic
452C2C2×Q8×S412yes—
subgroupG/H
center, FrattiniC22C23×S4
commutator, cosocleC22×A4C24
FittingC22×C4⋊Q8D3
socleC24C23×D3
prime radicalC22×S4C23
hypercenterC4⋊Q8S4
hypocenterA4C2×C4⋊Q8
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. A4×C4⋊Q8C2semidirect
3. #85502C2semidirect
5. #85303C2semidirect
7. #85283C2semidirect
25. C22×C4⋊Q8D3semidirect
49. C4×S4Q8semidirect
125. C4×A4C2×Q8semidirect
128. SL(2,ℤ/4)C2×Q8semidirect
172. C4⋊Q8S4direct (N × H)
232. S4C4⋊Q8direct (N × H)
302. C22×C4D3×Q8semidirect
346. A4C2×C4⋊Q8semidirect
431. C22D3×C4⋊Q8semidirect
445. C4Q8×S4semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×S42
A4×C4⋊Q82
#855022
C4⋊C4×S42
#853032
C42×S42
#852832
D4×C4⋊Q83
D3×C4⋊Q84