Finite Group Databasebeta

← C3×Q8×SL(2,ℤ/4) #11922788

Subgroups — C3×Q8×SL(2,ℤ/4) #11922788

Order 1152.

740 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 140 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×C3×Q8×A4C211yesmaximal, characteristic
2C3×C4⋊SL(2,ℤ/4)C213yesmaximal
3C3×C4×SL(2,ℤ/4)C213yesmaximal
4Q8×SL(2,ℤ/4)C311yesmaximal, characteristic, direct product factor
5C3×Q8×C22⋊C4—31nomaximal
6C3×Q8×A4C414yescomplemented
7C2×C3×C4×A4C2213yes—
8C2×C3×SL(2,ℤ/4)C2213yes—
9C2×C3×SL(2,ℤ/4)C2211yescharacteristic
10C3×Q8×Dic3—41nomaximal
11C23×C3×Q8D311yescharacteristic
12C2×Q8×A4—21no—
13C2×Q8×A4C2×C311yescharacteristic
14C2×C3×C4×Q8—31no—
15C2×C3×C4×Q8—31no—
16C4⋊SL(2,ℤ/4)C2×C313yes—
17C4×SL(2,ℤ/4)C2×C313yes—
18C3×#391—33no—
19C3×#391—33no—
20C3×#380—33no—
21C3×C4×C22⋊C4—33no—
22C22×C3×A4C2311yescharacteristic
23C2×C32×Q8—41no—
24C3×C4×A4C2×C416yes—
25C3×SL(2,ℤ/4)—23no—
26C3×SL(2,ℤ/4)Q812yesdirect product factor
27C3×C4⋊Dic3—43no—
28C3×C4×Dic3—43no—
29Q8×C22⋊C4—31noSylow
30C22×C3×Q8—31no—
31C22×C3×Q8—34no—
32C22×C3×Q8—33no—
33C22×C3×Q8Dic314yescomplemented
34C23×C3×C4C2×D313yes—
35Q8×A4—24no—
36Q8×A4C3×C414yescomplemented
37C2×C4×A4—23no—
38C2×C4×A4C22×C313yes—
39C2×SL(2,ℤ/4)C22×C313yes—
40C2×SL(2,ℤ/4)C22×C311yescharacteristic
41C3×C4×Q8—64no—
42C3×C4×Q8—63no—
43C3×C4×Q8—61no—
44C2×C3×C4⋊C4—33no—
45C2×C3×C4⋊C4—33no—
46C2×C3×C22⋊C4—31no—
47C2×C3×C22⋊C4—33no—
48C2×C3×C42—33no—
49C2×C3×C42—33no—
50Q8×Dic3—41no—
51C3×#95—33no—
52C3×#95—33no—
53C3×#95—36no—
54C2×C3×A4C22×C411yescharacteristic
55C2×C3×A4C2×Q811yescharacteristic
56C2×C3×A4#4111yescharacteristic
57C32×Q8—44no—
58C2×C32×C4—43no—
59C2×C3×Dic3—41no—
60C2×C3×Dic3—43no—
61C23×Q8C3×D311yescharacteristic
62C2×C4×Q8—31no—
63C2×C4×Q8—31no—
64#391—33no—
65#391—33no—
66#380—33no—
67C4×C22⋊C4—33no—
68C24×C3C22×D311yescharacteristic
69C22×A4—21no—
70C22×A4C23×C311yescharacteristic
71C2×C3×Q8—34no—
72C2×C3×Q8—34no—
73C2×C3×Q8—312no—
74C2×C3×Q8—33no—
75C2×C3×Q8—41no—
76C2×C3×Q8—61no—
77C2×C3×Q8—66no—
78C2×C3×Q8—81no—
79C2×C3×Q8S411yescharacteristic
80C22×C3×C4—33no—
81C22×C3×C4—33no—
82C22×C3×C4—36no—
83C22×C3×C4—31no—
84C22×C3×C4—31no—
85C22×C3×C4—33no—
86C22×C3×C4—33no—
87C22×C3×C4C2×Dic316yes—
88C4×A4—26no—
89C4×A4C2×C3×C416yes—
90SL(2,ℤ/4)—23no—
91SL(2,ℤ/4)C3×Q812yesdirect product factor
92C3×C4⋊C4—63no—
93C3×C4⋊C4—66no—
94C3×C4⋊C4—63no—
95C3×C22⋊C4—32no—
96C3×C22⋊C4—32no—
97C3×C22⋊C4—63no—
98C3×C22⋊C4—63no—
99C3×C42—63no—
100C3×C42—66no—
101C3×C42—63no—
102C4⋊Dic3—43no—
103C4×Dic3—43no—
104C22×C32—41no—
105C3×A4C4×Q811yescharacteristic, complemented
106C32×C4—46no—
107C3×Dic3—42no—
108C3×Dic3—83no—
109C22×Q8—31no—
110C22×Q8—33no—
111C22×Q8—34no—
112C22×Q8C3×Dic314yescomplemented
113C23×C4C2×C3×D313yes—
114C4×Q8—64no—
115C4×Q8—63no—
116C4×Q8—61no—
117C2×C4⋊C4—33no—
118C2×C4⋊C4—33no—
119C2×C22⋊C4—31no—
120C2×C22⋊C4—33no—
121C2×C42—33no—
122C2×C42—33no—
123#95—33no—
124#95—33no—
125#95—36no—
126C23×C3—31no—
127C23×C3—31no—
128C23×C3—31no—
129C23×C3—31no—
130C23×C3D3×Q811yescharacteristic
131C23×C3#23911yescharacteristic
132C23×C3C22×Dic311yescharacteristic
133C2×A4—21no—
134C2×A4—21no—
135C2×A4—21no—
136C2×A4C22×C3×C411yescharacteristic
137C2×A4C2×C3×Q811yescharacteristic
138C2×A4C3×#4111yescharacteristic
139C3×Q8—312no—
140C3×Q8—44no—
141C3×Q8—64no—
142C3×Q8—84no—
143C3×Q8SL(2,ℤ/4)14yesdirect product factor
144C2×C3×C4—33no—
145C2×C3×C4—36no—
146C2×C3×C4—36no—
147C2×C3×C4—36no—
148C2×C3×C4—32no—
149C2×C3×C4—32no—
150C2×C3×C4—43no—
151C2×C3×C4—63no—
152C2×C3×C4—61no—
153C2×C3×C4—61no—
154C2×C3×C4—61no—
155C2×C3×C4—61no—
156C2×C3×C4—63no—
157C2×C3×C4—63no—
158C2×C3×C4—63no—
159C2×C3×C4—63no—
160C2×C3×C4—63no—
161C2×C3×C4—63no—
162C2×C3×C4—83no—
163C2×C3×C4C2×S413yes—
164C2×Dic3—41no—
165C2×Dic3—43no—
166C2×C32—41no—
167C2×C32—41no—
168C2×C32—41no—
169C24C22×C3×D311yescharacteristic
170C2×Q8—34no—
171C2×Q8—34no—
172C2×Q8—312no—
173C2×Q8—33no—
174C2×Q8—61no—
175C2×Q8—66no—
176C2×Q8C3×S411yescharacteristic
177C22×C4—33no—
178C22×C4—33no—
179C22×C4—36no—
180C22×C4—31no—
181C22×C4—31no—
182C22×C4—33no—
183C22×C4—33no—
184C22×C4C2×C3×Dic316yes—
185C4⋊C4—63no—
186C4⋊C4—66no—
187C4⋊C4—63no—
188C22⋊C4—32no—
189C22⋊C4—32no—
190C22⋊C4—63no—
191C22⋊C4—63no—
192C42—63no—
193C42—66no—
194C42—63no—
195C22×C3—31no—
196C22×C3—31no—
197C22×C3—31no—
198C22×C3—31no—
199C22×C3—31no—
200C22×C3—31no—
201C22×C3—31no—
202C22×C3—31no—
203C22×C3—31no—
204C22×C3—41no—
205C22×C3—61no—
206C22×C3—81no—
207C22×C3Q8×Dic311yescharacteristic, complemented
208C22×C3C22×S411yescharacteristic
209A4—21no—
210A4C3×C4×Q811yescharacteristic, complemented
211C3×C4—36no—
212C3×C4—46no—
213C3×C4—62no—
214C3×C4—62no—
215C3×C4—86no—
216C3×C4—123no—
217C3×C4—123no—
218C3×C4C2×SL(2,ℤ/4)16yes—
219Dic3—42no—
220Dic3—83no—
221C32—41noSylow
222C23—31no—
223C23—31no—
224C23—31no—
225C23—31no—
226C23C3×D3×Q811yescharacteristic
227C23C3×#23911yescharacteristic
228C23C22×C3×Dic311yescharacteristic
229Q8—312no—
230Q8—64no—
231Q8C3×SL(2,ℤ/4)14yesdirect product factor
232C2×C4—33no—
233C2×C4—36no—
234C2×C4—36no—
235C2×C4—36no—
236C2×C4—32no—
237C2×C4—32no—
238C2×C4—63no—
239C2×C4—61no—
240C2×C4—61no—
241C2×C4—61no—
242C2×C4—61no—
243C2×C4—63no—
244C2×C4—63no—
245C2×C4—63no—
246C2×C4—63no—
247C2×C4—63no—
248C2×C4—63no—
249C2×C4C2×C3×S413yes—
250C2×C3—31no—
251C2×C3—31no—
252C2×C3—31no—
253C2×C3—31no—
254C2×C3—41no—
255C2×C3—41no—
256C2×C3—41no—
257C2×C3—81no—
258C2×C3—81no—
259C2×C3—81no—
260C2×C3Q8×S411yescharacteristic
261C2×C3#591911yescharacteristic
262C2×C3C22×SL(2,ℤ/4)11yescharacteristic
263C22—31no—
264C22—31no—
265C22—31no—
266C22—31no—
267C22—31no—
268C22—31no—
269C22—31no—
270C22—31no—
271C22—31no—
272C22—61no—
273C22C3×Q8×Dic311yescharacteristic, complemented
274C22C22×C3×S411yescharacteristic
275C4—36no—
276C4—62no—
277C4—62no—
278C4—123no—
279C4—123no—
280C4C2×C3×SL(2,ℤ/4)16yes—
281C3—41no—
282C3—81no—
283C3Q8×SL(2,ℤ/4)11yescharacteristic, direct product factor
284C2—31no—
285C2—31no—
286C2—31no—
287C2—31no—
288C2C3×Q8×S411yescharacteristic
289C2C3×#591911yescharacteristic
290C2C22×C3×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC22×C3C22×S4
commutatorC2×A4C22×C3×C4
FrattiniC22C22×C3×S4
FittingC23×C3×Q8D3
socleC24×C3C22×D3
prime radicalC22×C3×A4C23
cosocleC22×A4C23×C3
hypercenterC2×C3×Q8S4
hypocenterA4C3×C4×Q8
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C22×C3 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
4. Q8×SL(2,ℤ/4)C3direct (N × H)
6. C3×Q8×A4C4semidirect
26. C3×SL(2,ℤ/4)Q8direct (N × H)
33. C22×C3×Q8Dic3semidirect
36. Q8×A4C3×C4semidirect
91. SL(2,ℤ/4)C3×Q8direct (N × H)
105. C3×A4C4×Q8semidirect
112. C22×Q8C3×Dic3semidirect
143. C3×Q8SL(2,ℤ/4)direct (N × H)
207. C22×C3Q8×Dic3semidirect
210. A4C3×C4×Q8semidirect
231. Q8C3×SL(2,ℤ/4)direct (N × H)
273. C22C3×Q8×Dic3semidirect
283. C3Q8×SL(2,ℤ/4)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Q8×A42
C3×C4⋊SL(2,ℤ/4)2
C3×C4×SL(2,ℤ/4)2
Q8×SL(2,ℤ/4)3
C3×Q8×C22⋊C43
C3×Q8×Dic34