Finite Group Databasebeta

← C22×#63564 #11857733

Subgroups — C22×#63564 #11857733

Order 1152.

2825 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊(C3⋊D3)C211yesmaximal, characteristic
2C22×C3×C4⋊Dic3C211yesmaximal, characteristic, complemented
3C23×C3×C3⋊C8C211yesmaximal, characteristic, complemented
4C2×#63564C2112yesmaximal, direct product factor
5C22×#821—31nomaximal
6C22×#809—31nomaximal
7C24×C3⋊D3—21no—
8C23×C32×C4C2211yescharacteristic
9C22×C4⋊(C3⋊D3)—24no—
10C22×C4⋊(C3⋊D3)—23no—
11C22×C4⋊(C3⋊D3)C418yescomplemented
12C22×C4⋊(C3⋊D3)C2216yes—
13C23×C3×Dic3—21no—
14C2×C3×C4⋊Dic3—23no—
15C2×C3×C4⋊Dic3C2216yescomplemented
16C22×C3×C3⋊C8—24no—
17C22×C3×C3⋊C8C2216yescomplemented
18#63564C22116yesdirect product factor
19C23×D12—31no—
20C23×D12—31no—
21C23×D12—61no—
22C23×C3×C8—31no—
23C22×C3×C4⋊C4—31no—
24C22×C4⋊Dic3D311yescharacteristic, complemented
25C23×C3⋊C8D311yescharacteristic, complemented
26C2×#821—312no—
27C2×#809—312no—
28C24×C32D411yescharacteristic
29C23×C3⋊D3—28no—
30C23×C3⋊D3—26no—
31C23×C3⋊D3—44no—
32C23×C3⋊D3—44no—
33C22×C32×C4C2×C414yes—
34C22×C32×C4D413yes—
35C22×C32×C4D414yes—
36C22×C32×C4C2313yes—
37C2×C4⋊(C3⋊D3)—224no—
38C2×C4⋊(C3⋊D3)—26no—
39C2×C4⋊(C3⋊D3)—212no—
40C2×C4⋊(C3⋊D3)C2×C4124yescomplemented
41C2×C4⋊(C3⋊D3)C2314yes—
42C22×C3×Dic3—26no—
43C22×C3×Dic3—44no—
44C3×C4⋊Dic3—26no—
45C3×C4⋊Dic3C2314yescomplemented
46C2×C3×C3⋊C8—212no—
47C2×C3×C3⋊C8C2314yescomplemented
48C22×#102—91noSylow
49C24×D3—62no—
50C24×D3—61no—
51C24×D3—61no—
52C23×C3×C4—21no—
53C23×C3×C4—61no—
54C23×C3×C4C2×D311yescharacteristic
55C23×C3×C4C2×D311yescharacteristic
56C23×Dic3—21no—
57C22×D12—38no—
58C22×D12—38no—
59C22×D12—36no—
60C22×D12—36no—
61C22×D12—68no—
62C22×D12—66no—
63C22×D12—66no—
64C22×D12—68no—
65C22×D12—63no—
66C22×D12—64no—
67C22×D12—63no—
68C22×D12—64no—
69C22×C3×C8—36no—
70C22×C3×C8—64no—
71C2×C3×C4⋊C4—36no—
72C2×C3×C4⋊C4—63no—
73C2×C4⋊Dic3—23no—
74C2×C4⋊Dic3C2×D316yescomplemented
75C22×C3⋊C8—24no—
76C22×C3⋊C8C2×D316yescomplemented
77#821—316no—
78#809—316no—
79C23×C32C22⋊C414yes—
80C23×C32D814yes—
81C23×C32SD1614yes—
82C23×C32C2×D413yes—
83C22×C3⋊D3—224no—
84C22×C3⋊D3—24no—
85C22×C3⋊D3—432no—
86C22×C3⋊D3—412no—
87C22×C3⋊D3—412no—
88C2×C32×C4C22⋊C4112yes—
89C2×C32×C4C22×C416yes—
90C2×C32×C4C2×D413yes—
91C2×C32×C4C2×D416yes—
92C2×C32×C4C2411yescharacteristic
93C4⋊(C3⋊D3)—224no—
94C4⋊(C3⋊D3)—24no—
95C4⋊(C3⋊D3)C22×C418yescomplemented
96C2×C3×Dic3—24no—
97C2×C3×Dic3—412no—
98C3×C3⋊C8—24no—
99C23×D4—91no—
100C23×C8—31no—
101C22×C4⋊C4—31no—
102C2×#102—912no—
103C24×C3—21no—
104C24×C3D1211yescharacteristic
105C24×C3#6711yescharacteristic
106C23×D3—616no—
107C23×D3—612no—
108C23×D3—68no—
109C23×D3—68no—
110C23×D3—66no—
111C23×D3—66no—
112C23×D3—128no—
113C23×D3—128no—
114C23×D3—124no—
115C23×D3—124no—
116C23×D3—124no—
117C23×D3—124no—
118C22×C3×C4—24no—
119C22×C3×C4—23no—
120C22×C3×C4—24no—
121C22×C3×C4—23no—
122C22×C3×C4—66no—
123C22×C3×C4—124no—
124C22×C3×C4C4×D314yes—
125C22×C3×C4C4×D314yes—
126C22×C3×C4D1213yes—
127C22×C3×C4D1214yes—
128C22×C3×C4#6713yes—
129C22×C3×C4#6714yes—
130C22×C3×C4C22×D313yes—
131C22×C3×C4C22×D313yes—
132C22×Dic3—26no—
133C22×Dic3—44no—
134C2×D12—324no—
135C2×D12—324no—
136C2×D12—34no—
137C2×D12—34no—
138C2×D12—624no—
139C2×D12—64no—
140C2×D12—648no—
141C2×D12—624no—
142C2×D12—612no—
143C2×D12—624no—
144C2×D12—624no—
145C2×D12—66no—
146C2×D12—612no—
147C2×D12—66no—
148C2×D12—612no—
149C2×C3×C8—34no—
150C2×C3×C8—612no—
151C3×C4⋊C4—34no—
152C3×C4⋊C4—66no—
153C4⋊Dic3—26no—
154C4⋊Dic3C22×D314yescomplemented
155C2×C3⋊C8—212no—
156C2×C3⋊C8C22×D314yescomplemented
157C22×C32#102116yescomplemented
158C22×C32C2×C22⋊C416yes—
159C22×C32C2×D816yes—
160C22×C32C2×SD1616yes—
161C22×C32C22×D411yescharacteristic
162C2×C3⋊D3—28no—
163C2×C3⋊D3—44no—
164C2×C3⋊D3—44no—
165C2×C3⋊D3—448no—
166C32×C4C2×C22⋊C416yes—
167C32×C4C23×C411yescharacteristic
168C32×C4C22×D411yescharacteristic
169C3×Dic3—44no—
170C25—181no—
171C22×D4—98no—
172C22×D4—96no—
173C22×D4—183no—
174C22×D4—184no—
175C23×C4—61no—
176C23×C4D3211yescharacteristic
177C22×C8—36no—
178C22×C8—64no—
179C2×C4⋊C4—36no—
180C2×C4⋊C4—63no—
181#102—916no—
182C23×C3—24no—
183C23×C3—24no—
184C23×C3—24no—
185C23×C3—23no—
186C23×C3#20414yes—
187C23×C3D2414yes—
188C23×C3(C2×D3)⋊C414yes—
189C23×C3(C2×D3)⋊C414yes—
190C23×C3#21314yes—
191C23×C3#21514yes—
192C23×C3C2×D1213yes—
193C23×C3C2×#6713yes—
194C22×D3—68no—
195C22×D3—648no—
196C22×D3—624no—
197C22×D3—624no—
198C22×D3—64no—
199C22×D3—64no—
200C22×D3—1264no—
201C22×D3—1224no—
202C22×D3—1224no—
203C22×D3—1232no—
204C22×D3—1232no—
205C22×D3—1212no—
206C22×D3—1212no—
207C22×D3—1212no—
208C22×D3—1212no—
209C2×C3×C4—21no—
210C2×C3×C4—212no—
211C2×C3×C4—26no—
212C2×C3×C4—26no—
213C2×C3×C4—23no—
214C2×C3×C4—64no—
215C2×C3×C4—1212no—
216C2×C3×C4(C2×D3)⋊C4112yes—
217C2×C3×C4(C2×D3)⋊C4112yes—
218C2×C3×C4C2×C4×D316yes—
219C2×C3×C4C2×C4×D316yes—
220C2×C3×C4C2×D1213yes—
221C2×C3×C4C2×D1216yes—
222C2×C3×C4C2×#6713yes—
223C2×C3×C4C2×#6716yes—
224C2×C3×C4C23×D311yescharacteristic
225C2×C3×C4C23×D311yescharacteristic
226C2×Dic3—24no—
227C2×Dic3—412no—
228D12—38no—
229D12—38no—
230D12—68no—
231D12—68no—
232D12—648no—
233D12—624no—
234D12—624no—
235D12—64no—
236D12—64no—
237C3×C8—64no—
238C3⋊C8—24no—
239C2×C32C2×#102112yescomplemented
240C2×C32C22×C22⋊C411yescharacteristic
241C2×C32C22×D811yescharacteristic
242C2×C32C22×SD1611yescharacteristic
243C3⋊D3—48no—
244C24—188no—
245C24—186no—
246C24—364no—
247C24—364no—
248C24#62611yescharacteristic
249C2×D4—94no—
250C2×D4—924no—
251C2×D4—1824no—
252C2×D4—1812no—
253C2×D4—186no—
254C22×C4—66no—
255C22×C4—124no—
256C22×C4#62414yes—
257C22×C4#62613yes—
258C22×C4#62614yes—
259C22×C4C2×D3213yes—
260C2×C8—34no—
261C2×C8—612no—
262C4⋊C4—34no—
263C4⋊C4—66no—
264C22×C3—21no—
265C22×C3—216no—
266C22×C3—26no—
267C22×C3—26no—
268C22×C3—26no—
269C22×C3#809116yescomplemented
270C22×C3#821116yescomplemented
271C22×C3C2×#20416yes—
272C22×C3C2×D2416yes—
273C22×C3C2×(C2×D3)⋊C416yes—
274C22×C3C2×(C2×D3)⋊C416yes—
275C22×C3C2×#21316yes—
276C22×C3C2×#21516yes—
277C22×C3C22×D1211yescharacteristic
278C22×C3C22×#6711yescharacteristic
279C2×D3—616no—
280C2×D3—68no—
281C2×D3—68no—
282C2×D3—1296no—
283C2×D3—128no—
284C2×D3—128no—
285C2×D3—1248no—
286C2×D3—124no—
287C2×D3—124no—
288C2×D3—124no—
289C2×D3—124no—
290C2×D3—1248no—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—26no—
294C3×C4—124no—
295C3×C4C2×(C2×D3)⋊C416yes—
296C3×C4C2×(C2×D3)⋊C416yes—
297C3×C4C22×C4×D311yescharacteristic
298C3×C4C22×C4×D311yescharacteristic
299C3×C4C22×D1211yescharacteristic
300C3×C4C22×#6711yescharacteristic
301Dic3—44no—
302C32C22×#10211yescharacteristic, complemented, Sylow
303C23—184no—
304C23—1824no—
305C23—3632no—
306C23—3612no—
307C23—3612no—
308C23#371414yes—
309C23#371714yes—
310C23#372214yes—
311C23C2×#62613yes—
312D4—98no—
313D4—184no—
314D4—1824no—
315C2×C4—64no—
316C2×C4—1212no—
317C2×C4#3722112yes—
318C2×C4C2×#62416yes—
319C2×C4C2×#62616yes—
320C2×C4C2×#62613yes—
321C2×C4C22×D3211yescharacteristic
322C8—64no—
323C2×C3—212no—
324C2×C3—21no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3C2×#809112yescomplemented
328C2×C3C2×#821112yescomplemented
329C2×C3C22×#20411yescharacteristic
330C2×C3C22×D2411yescharacteristic
331C2×C3C22×(C2×D3)⋊C411yescharacteristic
332C2×C3C22×(C2×D3)⋊C411yescharacteristic
333C2×C3C22×#21311yescharacteristic
334C2×C3C22×#21511yescharacteristic
335D3—1216no—
336D3—128no—
337D3—128no—
338C22—188no—
339C22—3648no—
340C22—364no—
341C22—364no—
342C22#63564116yesdirect product factor
343C22C2×#371416yes—
344C22C2×#371716yes—
345C22C2×#372216yes—
346C22C22×#62611yescharacteristic
347C4—124no—
348C4C2×#372216yes—
349C4C22×#62411yescharacteristic
350C4C22×#62611yescharacteristic
351C3—21no—
352C3C22×#80911yescharacteristic, complemented
353C3C22×#82111yescharacteristic, complemented
354C2—368no—
355C2C2×#63564112yesdirect product factor
356C2C22×#371411yescharacteristic
357C2C22×#371711yescharacteristic
358C2C22×#372211yescharacteristic
subgroupG/H
centerC24#626
commutatorC32×C4C23×C4
FrattiniC2×C4C22×D32
FittingC23×C32×C4C22
socleC24×C32D4
prime radicalC23×C4⋊(C3⋊D3)C2
cosocleC2×C32×C4C24
hypercenterC23×C4D32
hypocenterC32C22×#102
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C24 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. C22×C3×C4⋊Dic3C2semidirect
3. C23×C3×C3⋊C8C2semidirect
4. C2×#63564C2direct (N × H)
11. C22×C4⋊(C3⋊D3)C4semidirect
15. C2×C3×C4⋊Dic3C22semidirect
17. C22×C3×C3⋊C8C22semidirect
18. #63564C22direct (N × H)
24. C22×C4⋊Dic3D3semidirect
25. C23×C3⋊C8D3semidirect
40. C2×C4⋊(C3⋊D3)C2×C4semidirect
45. C3×C4⋊Dic3C23semidirect
47. C2×C3×C3⋊C8C23semidirect
74. C2×C4⋊Dic3C2×D3semidirect
76. C22×C3⋊C8C2×D3semidirect
95. C4⋊(C3⋊D3)C22×C4semidirect
154. C4⋊Dic3C22×D3semidirect
156. C2×C3⋊C8C22×D3semidirect
157. C22×C32#102semidirect
239. C2×C32C2×#102semidirect
269. C22×C3#809semidirect
270. C22×C3#821semidirect
302. C32C22×#102semidirect
327. C2×C3C2×#809semidirect
328. C2×C3C2×#821semidirect
342. C22#63564direct (N × H)
352. C3C22×#809semidirect
353. C3C22×#821semidirect
355. C2C2×#63564direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C4⋊(C3⋊D3)2
C22×C3×C4⋊Dic32
C23×C3×C3⋊C82
C2×#635642
C22×#8213
C22×#8093