Finite Group Databasebeta

← C2×C32×#907 #13179121

Subgroups — C2×C32×#907 #13179121

Order 1728.

1868 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#235C211yesmaximal, characteristic
2C2×C33×M16C211yesmaximal, characteristic, complemented
3C22×C32×C3⋊C8C211yesmaximal, characteristic, complemented
4C32×#907C218yesmaximal, direct product factor
5C2×C32×ΓL(1,𝔽25)C212yesmaximal, complemented
6C2×C32×D3×C8C212yesmaximal, complemented
7C2×C32×#131—31nomaximal
8C2×C3×#907C314yesmaximal, direct product factor
9C22×C33×C4C2211yescharacteristic
10C2×C32×#67C412yes—
11C32×#235C414yes—
12C32×#235C2214yes—
13C2×C32×D12C411yescharacteristic
14C2×C32×C4×D3C2212yes—
15C2×C32×Dic6C411yescharacteristic
16C33×M16C2214yescomplemented
17C2×C33×C8C2212yes—
18C2×C32×C3⋊C8C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C2×C32×C3⋊C8C2214yescomplemented
21C32×ΓL(1,𝔽25)C2218yescomplemented
22C32×D3×C8C2218yescomplemented
23C2×C32×#41—31no—
24C2×C3×#235C2×C314yes—
25C32×#131—38no—
26C2×C32×M16—24no—
27C2×C32×M16—32no—
28C2×C32×M16D311yescharacteristic, complemented
29C2×C32×M16C2×C314yescomplemented
30C22×C32×C8—31no—
31C22×C32×C8—32no—
32C22×C3×C3⋊C8C2×C314yescomplemented
33C3×#907C2×C3132yesdirect product factor
34C2×C3×ΓL(1,𝔽25)C2×C318yescomplemented
35C2×C3×D3×C8C2×C318yescomplemented
36C23×C33C2×C411yescharacteristic
37C22×C32×D3C2×C412yes—
38C2×C33×C4C2×C411yescharacteristic
39C2×C33×C4C2×C412yes—
40C2×C33×C4C2312yes—
41C2×C33×C4C2311yescharacteristic
42C32×#67C2×C418yes—
43C2×C32×Dic3C2×C412yes—
44C32×D12C2×C414yes—
45C32×C4×D3C2×C414yes—
46C32×C4×D3C2314yes—
47C32×Dic6C2×C414yes—
48C33×C8C2314yes—
49C32×C3⋊C8C2312yes—
50C32×C3⋊C8C2312yes—
51C2×C3×#131—34no—
52C2×#907C3211yescharacteristic, direct product factor
53C32×#41—34no—
54C32×#41—34no—
55C2×C32×Q8—31no—
56C2×C32×D4—32no—
57C2×C32×D4—31no—
58C22×C32×C4—24no—
59C22×C32×C4—32no—
60C22×C32×C4C2×D311yescharacteristic
61C22×C32×C4C22×C314yes—
62C2×C3×#67C3×C418yes—
63C3×#235C3×C4116yes—
64C3×#235C22×C3116yes—
65C2×C3×D12C3×C414yes—
66C2×C3×C4×D3C22×C318yes—
67C2×C3×Dic6C3×C414yes—
68C32×M16—216no—
69C32×M16—38no—
70C32×M16C2×D314yescomplemented
71C32×M16C22×C3116yescomplemented
72C2×C32×C8—28no—
73C2×C32×C8—34no—
74C2×C32×C8—31no—
75C2×C32×C8—31no—
76C2×C32×C8—38no—
77C2×C32×C8C2×D312yes—
78C2×C32×C8C22×C318yes—
79C2×C3×C3⋊C8C22×C314yes—
80C2×C3×C3⋊C8C22×C314yes—
81C2×C3×C3⋊C8C22×C3116yescomplemented
82C3×ΓL(1,𝔽25)C22×C3132yescomplemented
83C3×D3×C8C22×C3132yescomplemented
84C22×C33—22no—
85C22×C33C22×C412yes—
86C22×C33C22×C411yescharacteristic
87C2×C32×D3—24no—
88C2×C32×D3C22×C414yes—
89C33×C4C22×C412yes—
90C33×C4C22×C411yescharacteristic
91C33×C4C2411yescharacteristic
92C32×Dic3C22×C414yes—
93C2×C3×#41—34no—
94C2×#235C2×C3211yescharacteristic
95C3×#131—332no—
96C2×C3×M16—24no—
97C2×C3×M16—38no—
98C2×C3×M16C3×D314yescomplemented
99C2×C3×M16C2×C3211yescharacteristic, complemented
100C22×C3×C8—34no—
101C22×C3×C8—38no—
102C22×C3⋊C8C2×C3211yescharacteristic, complemented
103#907C2×C3218yesdirect product factor
104C2×ΓL(1,𝔽25)C2×C3212yescomplemented
105C2×D3×C8C2×C3212yescomplemented
106C23×C32—24no—
107C23×C32—32no—
108C23×C32C4×D311yescharacteristic
109C23×C32C2×C3×C414yes—
110C22×C3×D3C2×C3×C418yes—
111C32×Q8—34no—
112C32×D4—38no—
113C32×D4—34no—
114C2×C32×C4—24no—
115C2×C32×C4—28no—
116C2×C32×C4—24no—
117C2×C32×C4—28no—
118C2×C32×C4—32no—
119C2×C32×C4—34no—
120C2×C32×C4—34no—
121C2×C32×C4C4×D311yescharacteristic
122C2×C32×C4C4×D312yes—
123C2×C32×C4C2×C3×C414yes—
124C2×C32×C4C2×C3×C418yes—
125C2×C32×C4C22×D312yes—
126C2×C32×C4C22×D311yescharacteristic
127C2×C32×C4C23×C314yes—
128C2×C32×C4C23×C318yes—
129C3×#67C2×C3×C4132yes—
130C2×C3×Dic3C2×C3×C418yes—
131C3×D12C2×C3×C4116yes—
132C3×C4×D3C2×C3×C4116yes—
133C3×C4×D3C23×C3116yes—
134C3×Dic6C2×C3×C4116yes—
135C32×C8—216no—
136C32×C8—32no—
137C32×C8—32no—
138C32×C8C22×D314yes—
139C32×C8C23×C3116yes—
140C3×C3⋊C8C23×C318yes—
141C3×C3⋊C8C23×C318yes—
142C2×#131—31noSylow
143C2×C33—22no—
144C2×C33#13112yescomplemented
145C2×C33C23×C411yescharacteristic
146C32×D3—24no—
147C3×#41—316no—
148C3×#41—316no—
149C2×C3×Q8—34no—
150C2×C3×D4—38no—
151C2×C3×D4—34no—
152C22×C3×C4—24no—
153C22×C3×C4—38no—
154C22×C3×C4C2×C3×D314yes—
155C22×C3×C4C22×C3211yescharacteristic
156C2×#67C32×C412yes—
157#235C32×C414yes—
158#235C22×C3214yes—
159C2×D12C32×C411yescharacteristic
160C2×C4×D3C22×C3212yes—
161C2×Dic6C32×C411yescharacteristic
162C3×M16—216no—
163C3×M16—332no—
164C3×M16C2×C3×D3116yescomplemented
165C3×M16C22×C3214yescomplemented
166C2×C3×C8—28no—
167C2×C3×C8—34no—
168C2×C3×C8—34no—
169C2×C3×C8—316no—
170C2×C3×C8—332no—
171C2×C3×C8C2×C3×D318yes—
172C2×C3×C8C22×C3212yes—
173C2×C3⋊C8C22×C3211yescharacteristic
174C2×C3⋊C8C22×C3211yescharacteristic
175C2×C3⋊C8C22×C3214yescomplemented
176ΓL(1,𝔽25)C22×C3218yescomplemented
177D3×C8C22×C3218yescomplemented
178C22×C32—22no—
179C22×C32—24no—
180C22×C32—28no—
181C22×C32—28no—
182C22×C32—34no—
183C22×C32—48no—
184C22×C32—64no—
185C22×C32C2×C4×D312yes—
186C22×C32C2×C4×D311yescharacteristic
187C22×C32C22×C3×C418yes—
188C22×C32C22×C3×C414yes—
189C2×C3×D3—216no—
190C2×C3×D3C22×C3×C4116yes—
191C32×C4—24no—
192C32×C4—24no—
193C32×C4—28no—
194C32×C4—34no—
195C32×C4C2×C4×D311yescharacteristic
196C32×C4C2×C4×D312yes—
197C32×C4C22×C3×C418yes—
198C32×C4C22×C3×C414yes—
199C32×C4C23×D311yescharacteristic
200C32×C4C24×C314yes—
201C3×Dic3C22×C3×C4116yes—
202C2×#41—31no—
203#131—38no—
204C2×M16—32no—
205C2×M16C32×D311yescharacteristic, complemented
206C22×C8—31no—
207C22×C8—32no—
208C33C2×#13111yescharacteristic, complemented, Sylow
209C23×C3—24no—
210C23×C3—38no—
211C23×C3C3×C4×D314yes—
212C23×C3C2×C32×C411yescharacteristic
213C22×D3C2×C32×C412yes—
214C3×Q8—316no—
215C3×D4—332no—
216C3×D4—316no—
217C2×C3×C4—24no—
218C2×C3×C4—28no—
219C2×C3×C4—24no—
220C2×C3×C4—28no—
221C2×C3×C4—38no—
222C2×C3×C4—316no—
223C2×C3×C4—316no—
224C2×C3×C4C3×C4×D314yes—
225C2×C3×C4C3×C4×D318yes—
226C2×C3×C4C2×C32×C411yescharacteristic
227C2×C3×C4C2×C32×C412yes—
228C2×C3×C4C22×C3×D318yes—
229C2×C3×C4C22×C3×D314yes—
230C2×C3×C4C23×C3211yescharacteristic
231C2×C3×C4C23×C3212yes—
232#67C2×C32×C418yes—
233C2×Dic3C2×C32×C412yes—
234D12C2×C32×C414yes—
235C4×D3C2×C32×C414yes—
236C4×D3C23×C3214yes—
237Dic6C2×C32×C414yes—
238C3×C8—216no—
239C3×C8—38no—
240C3×C8—38no—
241C3×C8C22×C3×D3116yes—
242C3×C8C23×C3214yes—
243C3⋊C8C23×C3212yes—
244C3⋊C8C23×C3212yes—
245C2×C32—22no—
246C2×C32—28no—
247C2×C32—24no—
248C2×C32—28no—
249C2×C32—48no—
250C2×C32—64no—
251C2×C32#90712yesdirect product factor
252C2×C32C3×#13118yescomplemented
253C2×C32C22×C4×D311yescharacteristic
254C2×C32C23×C3×C414yes—
255C3×D3—216no—
256#41—34no—
257#41—34no—
258C2×Q8—31no—
259C2×D4—32no—
260C2×D4—31no—
261C22×C4—32no—
262C22×C4C2×C32×D311yescharacteristic
263M16—38no—
264M16C2×C32×D314yescomplemented
265C2×C8—31no—
266C2×C8—31no—
267C2×C8—34no—
268C2×C8—38no—
269C2×C8C2×C32×D312yes—
270C22×C3—28no—
271C22×C3—22no—
272C22×C3—24no—
273C22×C3—28no—
274C22×C3—316no—
275C22×C3—48no—
276C22×C3—616no—
277C22×C3C2×C3×C4×D318yes—
278C22×C3C2×C3×C4×D314yes—
279C22×C3C22×C32×C412yes—
280C22×C3C22×C32×C411yescharacteristic
281C2×D3—24no—
282C2×D3C22×C32×C414yes—
283C3×C4—24no—
284C3×C4—24no—
285C3×C4—28no—
286C3×C4—316no—
287C3×C4C2×C3×C4×D314yes—
288C3×C4C2×C3×C4×D318yes—
289C3×C4C22×C32×C412yes—
290C3×C4C22×C32×C411yescharacteristic
291C3×C4C23×C3×D314yes—
292C3×C4C24×C3211yescharacteristic
293Dic3C22×C32×C414yes—
294C32—24no—
295C32C2×#90711yescharacteristic, direct product factor
296C32C2×C3×#13114yescomplemented
297C23—32no—
298C23C32×C4×D311yescharacteristic
299Q8—34no—
300D4—38no—
301D4—34no—
302C2×C4—32no—
303C2×C4—34no—
304C2×C4—34no—
305C2×C4C32×C4×D311yescharacteristic
306C2×C4C32×C4×D312yes—
307C2×C4C22×C32×D311yescharacteristic
308C2×C4C22×C32×D312yes—
309C8—32no—
310C8—32no—
311C8C22×C32×D314yes—
312C2×C3—28no—
313C2×C3—22no—
314C2×C3—28no—
315C2×C3—24no—
316C2×C3—48no—
317C2×C3—616no—
318C2×C3C3×#90718yesdirect product factor
319C2×C3C32×#13112yescomplemented
320C2×C3C22×C3×C4×D314yes—
321C2×C3C23×C32×C411yescharacteristic
322D3—24no—
323C22—22no—
324C22—34no—
325C22—64no—
326C22C2×C32×C4×D312yes—
327C22C2×C32×C4×D311yescharacteristic
328C4—34no—
329C4C2×C32×C4×D311yescharacteristic
330C4C2×C32×C4×D312yes—
331C4C23×C32×D311yescharacteristic
332C3—24no—
333C3C2×C3×#90714yesdirect product factor
334C3C2×C32×#13111yescharacteristic, complemented
335C2—22no—
336C2—64no—
337C2C32×#90712yesdirect product factor
338C2C22×C32×C4×D311yescharacteristic
subgroupG/H
centerC2×C32×C4C22×D3
commutatorC2×C3C23×C32×C4
FrattiniC4C23×C32×D3
FittingC2×C33×M16C2
socleC22×C33C22×C4
prime radicalC2×C32×#235C2
cosocleC3×C4C24×C32
hypercenterC2×C32×M16D3
hypocenterC3C2×C32×#131
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C32×C4 ◁ C2×C32×M16

Complements

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

NH ≅ G/Ntype
2. C2×C33×M16C2semidirect
3. C22×C32×C3⋊C8C2semidirect
4. C32×#907C2direct (N × H)
5. C2×C32×ΓL(1,𝔽25)C2semidirect
6. C2×C32×D3×C8C2semidirect
8. C2×C3×#907C3direct (N × H)
16. C33×M16C22semidirect
20. C2×C32×C3⋊C8C22semidirect
21. C32×ΓL(1,𝔽25)C22semidirect
22. C32×D3×C8C22semidirect
28. C2×C32×M16D3semidirect
29. C2×C32×M16C2×C3semidirect
32. C22×C3×C3⋊C8C2×C3semidirect
33. C3×#907C2×C3direct (N × H)
34. C2×C3×ΓL(1,𝔽25)C2×C3semidirect
35. C2×C3×D3×C8C2×C3semidirect
52. C2×#907C32direct (N × H)
70. C32×M16C2×D3semidirect
71. C32×M16C22×C3semidirect
81. C2×C3×C3⋊C8C22×C3semidirect
82. C3×ΓL(1,𝔽25)C22×C3semidirect
83. C3×D3×C8C22×C3semidirect
98. C2×C3×M16C3×D3semidirect
99. C2×C3×M16C2×C32semidirect
102. C22×C3⋊C8C2×C32semidirect
103. #907C2×C32direct (N × H)
104. C2×ΓL(1,𝔽25)C2×C32semidirect
105. C2×D3×C8C2×C32semidirect
144. C2×C33#131semidirect
164. C3×M16C2×C3×D3semidirect
165. C3×M16C22×C32semidirect
175. C2×C3⋊C8C22×C32semidirect
176. ΓL(1,𝔽25)C22×C32semidirect
177. D3×C8C22×C32semidirect
205. C2×M16C32×D3semidirect
208. C33C2×#131semidirect
251. C2×C32#907direct (N × H)
252. C2×C32C3×#131semidirect
264. M16C2×C32×D3semidirect
295. C32C2×#907direct (N × H)
296. C32C2×C3×#131semidirect
318. C2×C3C3×#907direct (N × H)
319. C2×C3C32×#131semidirect
333. C3C2×C3×#907direct (N × H)
334. C3C2×C32×#131semidirect
337. C2C32×#907direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#2352
C2×C33×M162
C22×C32×C3⋊C82
C32×#9072
C2×C32×ΓL(1,𝔽25)2
C2×C32×D3×C82
C2×C32×#1313
C2×C3×#9073