Finite Group Databasebeta

← Q16×SL(2,ℤ/4) #11712939

Subgroups — Q16×SL(2,ℤ/4) #11712939

Order 768.

585 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 585 conjugacy classes of subgroups — too many to draw. Only its 81 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×A4×Q16C211yesmaximal, characteristic
2Q8×SL(2,ℤ/4)C212yesmaximal
3#73147C212yesmaximal
4#73111C211yesmaximal, characteristic
5C8×SL(2,ℤ/4)C211yesmaximal, characteristic
6C22⋊C4×Q16—31nomaximal, Sylow
7C2×Q8×A4C2212yes—
8A4×Q16C414yescomplemented
9C2×C8×A4C2211yescharacteristic
10C4⋊SL(2,ℤ/4)—22no—
11C4⋊SL(2,ℤ/4)C2212yes—
12C4×SL(2,ℤ/4)—22no—
13C4×SL(2,ℤ/4)C2211yescharacteristic
14C2×A4⋊C8C2211yescharacteristic
15Dic3×Q16—41nomaximal
16C23×Q16D311yescharacteristic
17C2×C4×Q16—31no—
18C2×C4×Q16—31no—
19Q8×C22⋊C4—32no—
20#1969—31no—
21#1969—31no—
22#1878—32no—
23#1878—32no—
24#1808—31no—
25#1787—32no—
26C8×C22⋊C4—31no—
27Q8×A4—22no—
28Q8×A4C2×C414yes—
29C2×C4×A4—22no—
30C2×C4×A4C2311yescharacteristic
31C2×SL(2,ℤ/4)—22no—
32C2×SL(2,ℤ/4)D411yescharacteristic
33C2×SL(2,ℤ/4)D411yescharacteristic
34C2×C3×Q16—41no—
35Q8×Dic3—42no—
36C8×A4C2×C412yes—
37A4⋊C8—21no—
38#835—42no—
39#818—41no—
40C8×Dic3—41no—
41C23×Q8C2×D312yes—
42C22×Q16—31no—
43C22×Q16—32no—
44C22×Q16—31no—
45C22×Q16—34no—
46C22×Q16Dic314yescomplemented
47C23×C8C2×D311yescharacteristic
48C2×C4×Q8—32no—
49C2×C4×Q8—32no—
50C4×Q16—64no—
51C4×Q16—62no—
52C4×Q16—61no—
53C4×Q16—61no—
54C2×#107—31no—
55C2×#107—31no—
56C2×#103—32no—
57C2×#103—32no—
58C2×#103—32no—
59C2×#103—32no—
60C2×C22⋊C8—31no—
61C2×C4×C8—31no—
62C2×C4×C8—31no—
63#391—32no—
64#391—32no—
65#391—62no—
66#391—62no—
67#380—32no—
68#380—62no—
69C4×C22⋊C4—31no—
70C4×C22⋊C4—62no—
71#340—32no—
72#336—31no—
73#336—31no—
74C22×A4C2×D411yescharacteristic
75C2×C3×Q8—42no—
76C4×A4—24no—
77C4×A4C22×C411yescharacteristic
78C4×A4#4111yescharacteristic
79SL(2,ℤ/4)—21no—
80SL(2,ℤ/4)—42no—
81SL(2,ℤ/4)Q1612yesdirect product factor
82C3×Q16—44no—
83C2×C3×C8—41no—
84C4⋊Dic3—42no—
85C4⋊Dic3—82no—
86C4×Dic3—41no—
87C4×Dic3—82no—
88C2×C3⋊C8—41no—
89C22×Q8—22no—
90C22×Q8—32no—
91C22×Q8—32no—
92C22×Q8—34no—
93C22×Q8—62no—
94C22×Q8—62no—
95C22×Q8C2×Dic314yes—
96C23×C4—22no—
97C23×C4C22×D311yescharacteristic
98C2×Q16—32no—
99C2×Q16—34no—
100C2×Q16—34no—
101C2×Q16—34no—
102C2×Q16—38no—
103C2×Q16—31no—
104C2×Q16—62no—
105C2×Q16—61no—
106C2×Q16—64no—
107C2×Q16S411yescharacteristic
108C22×C8—31no—
109C22×C8—32no—
110C22×C8—31no—
111C22×C8—31no—
112C22×C8—31no—
113C22×C8C2×Dic312yes—
114C4×Q8—64no—
115C4×Q8—62no—
116C4×Q8—62no—
117C4×Q8—122no—
118C4×Q8—122no—
119C2×C4⋊C4—32no—
120C2×C4⋊C4—32no—
121C2×C4⋊C4—62no—
122C2×C4⋊C4—62no—
123C2×C22⋊C4—31no—
124C2×C22⋊C4—31no—
125C2×C22⋊C4—62no—
126C2×C42—31no—
127C2×C42—31no—
128C2×C42—62no—
129C2×C42—62no—
130#107—62no—
131#107—61no—
132#107—61no—
133#103—64no—
134#103—62no—
135#103—62no—
136#103—122no—
137#103—122no—
138C22⋊C8—61no—
139C22⋊C8—61no—
140C4×C8—62no—
141C4×C8—61no—
142C4×C8—61no—
143#95—31no—
144#95—31no—
145#95—62no—
146#95—62no—
147#95—62no—
148#95—62no—
149#95—62no—
150C2×A4C4×D411yescharacteristic
151C2×A4C2×Q1611yescharacteristic
152C2×A4#13511yescharacteristic
153C3×Q8—44no—
154C3×Q8—82no—
155C2×C3×C4—41no—
156C2×C3×C4—82no—
157C2×Dic3—41no—
158C2×Dic3—41no—
159C2×Dic3—82no—
160C3×C8—42no—
161C3⋊C8—81no—
162C24D3×D411yescharacteristic
163C2×Q8—34no—
164C2×Q8—34no—
165C2×Q8—34no—
166C2×Q8—32no—
167C2×Q8—62no—
168C2×Q8—62no—
169C2×Q8—62no—
170C2×Q8—64no—
171C2×Q8—62no—
172C2×Q8—64no—
173C2×Q8—62no—
174C2×Q8—62no—
175C2×Q8—62no—
176C2×Q8—124no—
177C2×Q8C2×S412yes—
178C22×C4—24no—
179C22×C4—31no—
180C22×C4—31no—
181C22×C4—31no—
182C22×C4—31no—
183C22×C4—31no—
184C22×C4—31no—
185C22×C4—31no—
186C22×C4—31no—
187C22×C4—62no—
188C22×C4—62no—
189C22×C4—64no—
190C22×C4—62no—
191C22×C4—62no—
192C22×C4#23711yescharacteristic
193C22×C4C22×Dic311yescharacteristic
194Q16—38no—
195Q16—34no—
196Q16—64no—
197Q16SL(2,ℤ/4)14yesdirect product factor
198C2×C8—31no—
199C2×C8—32no—
200C2×C8—32no—
201C2×C8—32no—
202C2×C8—61no—
203C2×C8—61no—
204C2×C8—61no—
205C2×C8—61no—
206C2×C8—61no—
207C2×C8—61no—
208C2×C8—61no—
209C2×C8C2×S411yescharacteristic
210C4⋊C4—62no—
211C4⋊C4—62no—
212C4⋊C4—62no—
213C4⋊C4—62no—
214C4⋊C4—122no—
215C4⋊C4—124no—
216C4⋊C4—122no—
217C22⋊C4—32no—
218C22⋊C4—32no—
219C22⋊C4—61no—
220C22⋊C4—61no—
221C22⋊C4—122no—
222C22⋊C4—122no—
223C42—61no—
224C42—61no—
225C42—61no—
226C42—61no—
227C42—122no—
228C42—124no—
229C42—122no—
230C22×C3—41no—
231A4C4×Q1611yescharacteristic, complemented
232C3×C4—41no—
233C3×C4—41no—
234C3×C4—84no—
235Dic3—42no—
236Dic3—81no—
237Dic3—162no—
238C23—31no—
239C23—31no—
240C23—31no—
241C23—31no—
242C23D3×Q1611yescharacteristic
243C23#91911yescharacteristic
244C23D4×Dic311yescharacteristic
245Q8—22no—
246Q8—34no—
247Q8—64no—
248Q8—64no—
249Q8—64no—
250Q8—62no—
251Q8—122no—
252Q8C2×SL(2,ℤ/4)14yes—
253C2×C4—22no—
254C2×C4—31no—
255C2×C4—31no—
256C2×C4—31no—
257C2×C4—31no—
258C2×C4—31no—
259C2×C4—31no—
260C2×C4—31no—
261C2×C4—32no—
262C2×C4—32no—
263C2×C4—64no—
264C2×C4—64no—
265C2×C4—61no—
266C2×C4—61no—
267C2×C4—61no—
268C2×C4—61no—
269C2×C4—61no—
270C2×C4—62no—
271C2×C4—64no—
272C2×C4—61no—
273C2×C4—61no—
274C2×C4—61no—
275C2×C4—61no—
276C2×C4—61no—
277C2×C4—61no—
278C2×C4—122no—
279C2×C4—122no—
280C2×C4—122no—
281C2×C4—122no—
282C2×C4—122no—
283C2×C4—122no—
284C2×C4—122no—
285C2×C4C22×S411yescharacteristic
286C8—32no—
287C8—121no—
288C8—121no—
289C8C2×SL(2,ℤ/4)12yes—
290C2×C3—41no—
291C2×C3—41no—
292C2×C3—41no—
293C22—31no—
294C22—31no—
295C22—31no—
296C22—31no—
297C22—31no—
298C22—31no—
299C22—31no—
300C22—31no—
301C22—31no—
302C22—61no—
303C22Dic3×Q1611yescharacteristic, complemented
304C22D4×S411yescharacteristic
305C4—24no—
306C4—31no—
307C4—31no—
308C4—62no—
309C4—62no—
310C4—64no—
311C4—121no—
312C4—121no—
313C4—242no—
314C4—242no—
315C4#591411yescharacteristic
316C4C22×SL(2,ℤ/4)11yescharacteristic
317C3—41noSylow
318C2—31no—
319C2—31no—
320C2—31no—
321C2—31no—
322C2Q16×S411yescharacteristic
323C2#8546511yescharacteristic
324C2D4×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC22D4×S4
commutatorC4×A4C22×C4
FrattiniC2×C4C22×S4
FittingC23×Q16D3
socleC24D3×D4
prime radicalC22×A4C2×D4
cosocleC2×C4×A4C23
hypercenterC2×Q16S4
hypocenterA4C4×Q16
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q16

Complements

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

NH ≅ G/Ntype
8. A4×Q16C4semidirect
46. C22×Q16Dic3semidirect
81. SL(2,ℤ/4)Q16direct (N × H)
197. Q16SL(2,ℤ/4)direct (N × H)
231. A4C4×Q16semidirect
303. C22Dic3×Q16semidirect

Maximal subgroups

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

subgroupindex
C2×A4×Q162
Q8×SL(2,ℤ/4)2
#731472
#731112
C8×SL(2,ℤ/4)2
C22⋊C4×Q163
Dic3×Q164