Finite Group Databasebeta

← #11923725

Subgroups — #11923725

Order 1152.

1818 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1818 conjugacy classes of subgroups — too many to draw. Only its 187 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, complemented
2#10596582C214yesmaximal, complemented
3Q8×C3⋊S4C214yesmaximal, complemented
4#10596575C216yesmaximal, complemented
5#87483—33nomaximal
6#87251—31nomaximal
7#64364—41nomaximal
8C3×Q8×A4C2214yes—
9C2×C3×C4×A4C2213yes—
10#64269C2214yes—
11C4⋊(C3⋊S4)C2216yes—
12C4×C3⋊S4C22112yes—
13#64259C2216yes—
14C23×C3×Q8D311yescharacteristic, complemented
15C2×#1007—31no—
16C2×Q8×A4D313yescomplemented
17#5919—312no—
18Q8×S4—312no—
19#5912—318no—
20#5817—34no—
21#5817—34no—
22#5816—36no—
23Q8×#67—34no—
24Q8×#67—34no—
25#5812—31no—
26#5791—36no—
27C22×C3×A4C2311yescharacteristic
28C2×C3⋊S4C2314yes—
29C2×C32×Q8—41no—
30#3832—44no—
31Q8×C3⋊D3—44no—
32#3828—46no—
33C3×C4×A4C2316yes—
34C3⋊SL(2,ℤ/4)C2314yes—
35#3486—91noSylow
36C22×C3×Q8—31no—
37C22×C3×Q8—34no—
38C22×C3×Q8—33no—
39C22×C3×Q8C2×D314yes—
40C23×C3×C4C2×D313yes—
41#1007—64no—
42#1007—63no—
43#1007—61no—
44#1007—123no—
45C2×#239—34no—
46C2×D3×Q8—34no—
47C2×#235—36no—
48Q8×A4C2×D3112yes—
49C2×C4×A4C2×D319yes—
50GL(2,ℤ/4)—312no—
51C4⋊S4—318no—
52C4×S4—336no—
53A4⋊Q8—318no—
54#953—34no—
55#947—312no—
56#946—312no—
57#946—312no—
58Q8×Dic3—34no—
59Q8×Dic3—34no—
60#944—312no—
61#930—36no—
62#930—36no—
63#929—312no—
64C4×#67—312no—
65C4×#67—312no—
66#926—36no—
67#924—36no—
68#924—36no—
69C2×C3×A4—21no—
70C2×C3×A4C2411yescharacteristic
71C3⋊S4—24no—
72C32×Q8—44no—
73C2×C32×C4—43no—
74#638—44no—
75C4⋊(C3⋊D3)—46no—
76C4×C3⋊D3—412no—
77#634—46no—
78C2×2−1+4—91no—
79C23×Q8C3⋊D311yescharacteristic, complemented
80D4×Q8—94no—
81D4×Q8—94no—
82#548—94no—
83#548—94no—
84#536—96no—
85#525—96no—
86#519—91no—
87C24×C3C22×D311yescharacteristic
88C22×A4C22×D313yes—
89C2×S4—312no—
90C2×C3×Q8—34no—
91C2×C3×Q8—34no—
92C2×C3×Q8—312no—
93C2×C3×Q8—33no—
94C2×C3×Q8—43no—
95C2×C3×Q8—61no—
96C2×C3×Q8—66no—
97C2×C3×Q8S411yescharacteristic, complemented
98C22×C3×C4—33no—
99C22×C3×C4—36no—
100C22×C3×C4—33no—
101C22×C3×C4C22×D316yes—
102C2×#67—34no—
103C2×#67—34no—
104C2×#67—34no—
105#239—612no—
106#239—64no—
107#239—1212no—
108D3×Q8—612no—
109D3×Q8—64no—
110D3×Q8—1212no—
111#235—66no—
112#235—66no—
113#235—66no—
114#235—66no—
115#235—1218no—
116C2×D12—36no—
117C2×C4×D3—312no—
118C2×Dic6—36no—
119C4×A4C22×D3118yes—
120SL(2,ℤ/4)—312no—
121#217—34no—
122#217—34no—
123#217—34no—
124(C2×D3)⋊C4—312no—
125(C2×D3)⋊C4—312no—
126C4⋊Dic3—36no—
127C4⋊Dic3—36no—
128Dic3⋊C4—312no—
129Dic3⋊C4—312no—
130C4×Dic3—312no—
131C22×C32—41no—
132C2×C3⋊D3—44no—
133C3×A42−1+411yescharacteristic, complemented
134C32×C4—46no—
135C3⋊Dic3—44no—
1362−1+4—184no—
1372−1+4—183no—
1382−1+4—181no—
139C2×#41—96no—
140C2×#41—94no—
141C22×Q8—31no—
142C22×Q8—33no—
143C22×Q8—34no—
144C22×Q8—94no—
145C22×Q8C2×C3⋊D314yes—
146C23×C4C2×C3⋊D313yes—
147C4⋊Q8—912no—
148#124—912no—
149#123—912no—
150C22⋊Q8—96no—
151C22⋊Q8—912no—
152C22⋊Q8—912no—
153C22⋊Q8—96no—
154#121—96no—
155#121—96no—
156#120—94no—
157C4×Q8—94no—
158C4×Q8—94no—
159C4×D4—912no—
160C4×D4—912no—
161#117—96no—
162C23×C3—21no—
163C23×C3—31no—
164C23×C3—31no—
165C23×C3—61no—
166C23×C3C23×D311yescharacteristic
167C22×D3—34no—
168C2×A4—23no—
169C2×A4C23×D313yes—
170S4—612no—
171C3×Q8—312no—
172C3×Q8—412no—
173C3×Q8—64no—
174C3×Q8C2×S414yes—
175C2×C3×C4—33no—
176C2×C3×C4—36no—
177C2×C3×C4—36no—
178C2×C3×C4—36no—
179C2×C3×C4—49no—
180C2×C3×C4—63no—
181C2×C3×C4C2×S413yes—
182#67—64no—
183#67—64no—
184#67—68no—
185#67—64no—
186#67—64no—
187#67—1212no—
188C2×Dic3—34no—
189C2×Dic3—34no—
190C2×Dic3—34no—
191D12—66no—
192D12—66no—
193D12—1218no—
194C4×D3—612no—
195C4×D3—612no—
196C4×D3—1236no—
197Dic6—66no—
198Dic6—66no—
199Dic6—1218no—
200C2×C32—41no—
201C2×C32—81no—
202C3⋊D3—84no—
203C24C22×C3⋊D311yescharacteristic
204#41—186no—
205#41—186no—
206#41—186no—
207#41—1812no—
208#41—184no—
209#41—186no—
210C2×Q8—34no—
211C2×Q8—34no—
212C2×Q8—312no—
213C2×Q8—33no—
214C2×Q8—61no—
215C2×Q8—66no—
216C2×Q8—96no—
217C2×Q8—1812no—
218C2×Q8—184no—
219C2×Q8C3⋊S411yescharacteristic, complemented
220C2×D4—96no—
221C2×D4—94no—
222C2×D4—94no—
223C2×D4—94no—
224C22×C4—33no—
225C22×C4—36no—
226C22×C4—33no—
227C22×C4—912no—
228C22×C4C22×C3⋊D316yes—
229C4⋊C4—96no—
230C4⋊C4—96no—
231C4⋊C4—912no—
232C4⋊C4—912no—
233C22⋊C4—912no—
234C22⋊C4—912no—
235C22⋊C4—94no—
236C22⋊C4—94no—
237C22⋊C4—94no—
238C42—912no—
239C22×C3—31no—
240C22×C3—31no—
241C22×C3—32no—
242C22×C3—31no—
243C22×C3—43no—
244C22×C3—61no—
245C22×C3—61no—
246C22×C3—61no—
247C22×C3#100711yescharacteristic, complemented
248C22×C3C22×S411yescharacteristic
249C2×D3—64no—
250C2×D3—64no—
251C2×D3—64no—
252C2×D3—1212no—
253A4#100713yescomplemented
254C3×C4—36no—
255C3×C4—418no—
256C3×C4C22×S416yes—
257Dic3—64no—
258Dic3—64no—
259Dic3—64no—
260Dic3—1212no—
261C32—41noSylow
262C23—21no—
263C23—31no—
264C23—31no—
265C23—61no—
266C23—94no—
267C23C23×C3⋊D311yescharacteristic
268Q8—312no—
269Q8—64no—
270Q8—186no—
271Q8—186no—
272Q8C2×C3⋊S414yes—
273D4—186no—
274D4—184no—
275D4—184no—
276D4—188no—
277D4—184no—
278D4—184no—
279D4—186no—
280C2×C4—33no—
281C2×C4—36no—
282C2×C4—36no—
283C2×C4—36no—
284C2×C4—63no—
285C2×C4—94no—
286C2×C4—94no—
287C2×C4—94no—
288C2×C4—1812no—
289C2×C4—1812no—
290C2×C4C2×C3⋊S413yes—
291C2×C3—21no—
292C2×C3—31no—
293C2×C3—31no—
294C2×C3—43no—
295C2×C3—61no—
296C2×C3—83no—
297C2×C3C23×S411yescharacteristic
298D3—124no—
299D3—2412no—
300C22—31no—
301C22—31no—
302C22—32no—
303C22—31no—
304C22—61no—
305C22—61no—
306C22—61no—
307C22—184no—
308C22—184no—
309C22—184no—
310C22#6436411yescharacteristic, complemented
311C22C22×C3⋊S411yescharacteristic
312C4—36no—
313C4—184no—
314C4—184no—
315C4—184no—
316C4C22×C3⋊S416yes—
317C3—43no—
318C3#8748311yescharacteristic, complemented
319C2—21no—
320C2—31no—
321C2—31no—
322C2—61no—
323C2—364no—
324C2C23×C3⋊S411yescharacteristic
subgroupG/H
center, FrattiniC2C23×C3⋊S4
commutator, cosocleC2×C3×A4C24
FittingC23×C3×Q8D3
socleC23×C3C23×D3
hypercenterC2×Q8C3⋊S4
hypocenterC3×A42−1+4
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C2 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×Q8×A4C2semidirect
2. #10596582C2semidirect
3. Q8×C3⋊S4C2semidirect
4. #10596575C2semidirect
14. C23×C3×Q8D3semidirect
16. C2×Q8×A4D3semidirect
79. C23×Q8C3⋊D3semidirect
97. C2×C3×Q8S4semidirect
133. C3×A42−1+4semidirect
219. C2×Q8C3⋊S4semidirect
247. C22×C3#1007semidirect
253. A4#1007semidirect
310. C22#64364semidirect
318. C3#87483semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×A42
#105965822
Q8×C3⋊S42
#105965752
#874833
#872513
#643644