Finite Group Databasebeta

← C2×#11723397 #13176803

Subgroups — C2×#11723397 #13176803

Order 1728.

2220 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×Dic6C211yesmaximal, characteristic
2C2×C4×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic
3C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic
4C2×Dic3⋊(C3⋊Dic3)C212yesmaximal
5C2×Dic3×C3⋊Dic3C212yesmaximal
6#11723397C218yesmaximal, direct product factor
7C2×Q8×C3⋊Dic3—31nomaximal
8C2×#63843—34nomaximal
9C22×C33×C4C2211yescharacteristic
10C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
11C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
12C22×C3×C3⋊Dic3C2212yes—
13C22×C32×Dic3C2212yes—
14C2×C32×Dic6C418yescomplemented
15C2×C32×Dic6C2214yes—
16C4×C3⋊(C3⋊Dic3)C2214yes—
17C3×C4⋊(C3⋊Dic3)C2214yes—
18Dic3⋊(C3⋊Dic3)C2218yes—
19Dic3×C3⋊Dic3C2218yes—
20C22×C32×Q8—31no—
21C22×C3×Dic6D314yes—
22Q8×C3⋊Dic3—38no—
23C2×C4⋊(C3⋊Dic3)—32no—
24C2×C4⋊(C3⋊Dic3)D311yescharacteristic
25C2×C4×C3⋊Dic3—31no—
26C2×C4×C3⋊Dic3—32no—
27C2×C4×C3⋊Dic3—34no—
28C2×C4×C3⋊Dic3—64no—
29C2×C3×C4⋊Dic3—34no—
30C2×Dic3⋊Dic3—38no—
31C2×Dic32—38no—
32#63843—332no—
33C23×C33C2311yescharacteristic
34C2×C33×C4C2×C414yes—
35C2×C33×C4C2312yes—
36C2×C3⋊(C3⋊Dic3)—22no—
37C2×C3⋊(C3⋊Dic3)Q814yescomplemented
38C2×C3⋊(C3⋊Dic3)C2312yes—
39C2×C3⋊(C3⋊Dic3)C2312yes—
40C2×C3×C3⋊Dic3—24no—
41C2×C3×C3⋊Dic3C2314yes—
42C2×C32×Dic3C2×C418yes—
43C2×C32×Dic3C2314yes—
44C32×Dic6C2×C4116yescomplemented
45C2×Q8×Dic3—94no—
46C2×#887—91no—
47C2×C32×Q8—38no—
48C2×C32×Q8—34no—
49C22×C32×C4—24no—
50C22×C32×C4—32no—
51C22×C32×C4C2×D311yescharacteristic
52C22×C32×C4C2×D314yes—
53C22×C3⋊Dic3—31no—
54C22×C3⋊Dic3—34no—
55C22×C3⋊Dic3—31no—
56C22×C3⋊Dic3—34no—
57C22×C3⋊Dic3—64no—
58C22×C3⋊Dic3—64no—
59C22×C3⋊Dic3C2×D312yes—
60C22×C3×Dic3—38no—
61C22×C3×Dic3C2×D318yes—
62C2×C3×Dic6Dic3132yescomplemented
63C2×C3×Dic6C2×D3116yes—
64C4⋊(C3⋊Dic3)—38no—
65C4⋊(C3⋊Dic3)C2×D314yes—
66C4×C3⋊Dic3—34no—
67C4×C3⋊Dic3—316no—
68C4×C3⋊Dic3—38no—
69C4×C3⋊Dic3—616no—
70C3×C4⋊Dic3—316no—
71Dic3⋊Dic3—332no—
72Dic32—332no—
73C22×C33C22×C412yes—
74C22×C33C2×Q812yes—
75C22×C33#4112yes—
76C22×C33C2411yescharacteristic
77C33×C4C22×C414yes—
78C3⋊(C3⋊Dic3)—22no—
79C3⋊(C3⋊Dic3)C2×Q814yescomplemented
80C3×C3⋊Dic3—24no—
81C32×Dic3C22×C418yes—
82C22×C3×Q8—34no—
83C22×Dic6C3⋊D311yescharacteristic
84C2×C3×C4⋊C4—91no—
85Q8×Dic3—932no—
86C2×C4⋊Dic3—34no—
87C2×C4⋊Dic3—98no—
88C2×Dic3⋊C4—92no—
89C2×C4×Dic3—98no—
90C2×C4×Dic3—94no—
91C2×C4×Dic3—91no—
92C2×C4×Dic3—92no—
93C2×C4×Dic3—184no—
94#887—98no—
95C23×C32—24no—
96C23×C32C22×D314yes—
97C23×C32C22×D311yescharacteristic
98C32×Q8—316no—
99C2×C32×C4—216no—
100C2×C32×C4—28no—
101C2×C32×C4—38no—
102C2×C32×C4—34no—
103C2×C32×C4C4×D314yes—
104C2×C32×C4C2×Dic3116yes—
105C2×C32×C4C22×D318yes—
106C2×C32×C4C22×D312yes—
107C2×C3⋊Dic3—24no—
108C2×C3⋊Dic3—34no—
109C2×C3⋊Dic3—32no—
110C2×C3⋊Dic3—32no—
111C2×C3⋊Dic3—316no—
112C2×C3⋊Dic3—38no—
113C2×C3⋊Dic3—38no—
114C2×C3⋊Dic3—68no—
115C2×C3⋊Dic3—68no—
116C2×C3⋊Dic3—616no—
117C2×C3⋊Dic3—616no—
118C2×C3⋊Dic3—62no—
119C2×C3⋊Dic3—68no—
120C2×C3⋊Dic3C22×D314yes—
121C2×C3×Dic3—316no—
122C2×C3×Dic3—616no—
123C2×C3×Dic3C2×Dic3132yes—
124C2×C3×Dic3C22×D3116yes—
125C3×Dic6C2×Dic3164yescomplemented
126C2×C4×Q8—271noSylow
127C2×C33C4×Q814yescomplemented
128C2×C33C23×C411yescharacteristic
129C2×C33C22×Q811yescharacteristic
130C2×C33C2×#4111yescharacteristic
131C2×C3×Q8—332no—
132C2×C3×Q8—316no—
133C22×C3×C4—24no—
134C22×C3×C4—38no—
135C22×C3×C4—92no—
136C22×C3×C4D3214yes—
137C22×C3×C4C2×C3⋊D311yescharacteristic
138C22×Dic3—38no—
139C22×Dic3—94no—
140C22×Dic3—91no—
141C22×Dic3—94no—
142C22×Dic3—91no—
143C22×Dic3—184no—
144C22×Dic3—184no—
145C22×Dic3C2×C3⋊D312yes—
146C2×Dic6C3⋊Dic318yescomplemented
147C2×Dic6C2×C3⋊D314yes—
148C3×C4⋊C4—94no—
149C4⋊Dic3—316no—
150C4⋊Dic3—932no—
151Dic3⋊C4—98no—
152C4×Dic3—932no—
153C4×Dic3—916no—
154C4×Dic3—94no—
155C4×Dic3—98no—
156C4×Dic3—1816no—
157C22×C32—24no—
158C22×C32—28no—
159C22×C32—28no—
160C22×C32—28no—
161C22×C32C2×C4×D312yes—
162C22×C32#23712yes—
163C22×C32D3×Q818yes—
164C22×C32D3×Q812yes—
165C22×C32#23918yes—
166C22×C32C22×Dic318yes—
167C22×C32C23×D314yes—
168C22×C32C23×D311yescharacteristic
169C32×C4—216no—
170C32×C4—38no—
171C32×C4C2×C4×D314yes—
172C32×C4C22×Dic3116yes—
173C3⋊Dic3—24no—
174C3⋊Dic3—34no—
175C3⋊Dic3—316no—
176C3⋊Dic3—616no—
177C3⋊Dic3—616no—
178C3⋊Dic3—62no—
179C3⋊Dic3—68no—
180C3×Dic3—616no—
181C3×Dic3C22×Dic3132yes—
182C22×Q8—31no—
183C4×Q8—278no—
184C2×C4⋊C4—91no—
185C2×C4⋊C4—272no—
186C2×C42—272no—
187C2×C42—271no—
188C33C2×C4×Q811yescharacteristic, complemented, Sylow
189C23×C3—24no—
190C23×C3C2×D3214yes—
191C23×C3C22×C3⋊D311yescharacteristic
192C3×Q8—364no—
193C2×C3×C4—216no—
194C2×C3×C4—28no—
195C2×C3×C4—316no—
196C2×C3×C4—332no—
197C2×C3×C4—94no—
198C2×C3×C4—184no—
199C2×C3×C4D3×Dic3116yes—
200C2×C3×C4C2×C3⋊Dic314yes—
201C2×C3×C4C2×D3218yes—
202C2×C3×C4C22×C3⋊D312yes—
203C2×Dic3—316no—
204C2×Dic3—616no—
205C2×Dic3—98no—
206C2×Dic3—92no—
207C2×Dic3—916no—
208C2×Dic3—94no—
209C2×Dic3—98no—
210C2×Dic3—92no—
211C2×Dic3—188no—
212C2×Dic3—188no—
213C2×Dic3—1816no—
214C2×Dic3—1816no—
215C2×Dic3—188no—
216C2×Dic3—182no—
217C2×Dic3C2×C3⋊Dic318yes—
218C2×Dic3C22×C3⋊D314yes—
219Dic6C2×C3⋊Dic3116yescomplemented
220C2×C32—24no—
221C2×C32—24no—
222C2×C32—24no—
223C2×C32—216no—
224C2×C32#88714yescomplemented
225C2×C32Q8×Dic3116yescomplemented
226C2×C32C22×C4×D311yescharacteristic
227C2×C32C2×#23711yescharacteristic
228C2×C32C2×D3×Q814yes—
229C2×C32C2×D3×Q811yescharacteristic
230C2×C32C2×#23914yes—
231C2×C32C23×Dic314yes—
232C2×Q8—38no—
233C2×Q8—34no—
234C22×C4—32no—
235C22×C4—92no—
236C22×C4—271no—
237C22×C4—271no—
238C22×C4D3×C3⋊D311yescharacteristic
239C4⋊C4—94no—
240C4⋊C4—278no—
241C42—278no—
242C42—274no—
243C22×C3—24no—
244C22×C3—28no—
245C22×C3—28no—
246C22×C3—28no—
247C22×C3#379618yes—
248C22×C3#379718yes—
249C22×C3C2×D3×Dic318yes—
250C22×C3Q8×C3⋊D312yes—
251C22×C3#383212yes—
252C22×C3C22×C3⋊Dic312yes—
253C22×C3C22×D3214yes—
254C22×C3C23×C3⋊D311yescharacteristic
255C3×C4—216no—
256C3×C4—332no—
257C3×C4—184no—
258C3×C4C2×D3×Dic3116yes—
259C3×C4C22×C3⋊Dic314yes—
260Dic3—616no—
261Dic3—916no—
262Dic3—94no—
263Dic3—1816no—
264Dic3—1816no—
265Dic3—188no—
266Dic3—182no—
267Dic3C22×C3⋊Dic318yes—
268C32—24no—
269C32C2×#88711yescharacteristic, complemented
270C32C2×Q8×Dic314yescomplemented
271C23C2×D3×C3⋊D311yescharacteristic
272Q8—316no—
273C2×C4—34no—
274C2×C4—38no—
275C2×C4—94no—
276C2×C4—184no—
277C2×C4—272no—
278C2×C4—274no—
279C2×C4—272no—
280C2×C4—542no—
281C2×C4D3×C3⋊Dic314yes—
282C2×C4C2×D3×C3⋊D312yes—
283C2×C3—24no—
284C2×C3—24no—
285C2×C3—24no—
286C2×C3—216no—
287C2×C3#63843116yescomplemented
288C2×C3Q8×C3⋊Dic314yescomplemented
289C2×C3C2×#379614yes—
290C2×C3C2×#379714yes—
291C2×C3C22×D3×Dic314yes—
292C2×C3C2×Q8×C3⋊D311yescharacteristic
293C2×C3C2×#383211yescharacteristic
294C2×C3C23×C3⋊Dic311yescharacteristic
295C22#8901712yes—
296C22#8901912yes—
297C22C2×D3×C3⋊Dic312yes—
298C22C22×D3×C3⋊D311yescharacteristic
299C4—38no—
300C4—184no—
301C4—274no—
302C4—542no—
303C4C2×D3×C3⋊Dic314yes—
304C3—24no—
305C3C2×#6384314yescomplemented
306C3C2×Q8×C3⋊Dic311yescharacteristic, complemented
307C2#1172339714yesdirect product factor
308C2C2×#8901711yescharacteristic
309C2C2×#8901911yescharacteristic
310C2C22×D3×C3⋊Dic311yescharacteristic
subgroupG/H
centerC23C2×D3×C3⋊D3
commutatorC2×C33C23×C4
FrattiniC22C22×D3×C3⋊D3
FittingC22×C33×C4C22
socle, prime radicalC23×C33C23
cosocleC22×C33C24
hypercenterC22×C4D3×C3⋊D3
hypocenterC33C2×C4×Q8
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
6. #11723397C2direct (N × H)
14. C2×C32×Dic6C4semidirect
37. C2×C3⋊(C3⋊Dic3)Q8semidirect
44. C32×Dic6C2×C4semidirect
62. C2×C3×Dic6Dic3semidirect
79. C3⋊(C3⋊Dic3)C2×Q8semidirect
125. C3×Dic6C2×Dic3semidirect
127. C2×C33C4×Q8semidirect
146. C2×Dic6C3⋊Dic3semidirect
188. C33C2×C4×Q8semidirect
219. Dic6C2×C3⋊Dic3semidirect
224. C2×C32#887semidirect
225. C2×C32Q8×Dic3semidirect
269. C32C2×#887semidirect
270. C32C2×Q8×Dic3semidirect
287. C2×C3#63843semidirect
288. C2×C3Q8×C3⋊Dic3semidirect
305. C3C2×#63843semidirect
306. C3C2×Q8×C3⋊Dic3semidirect
307. C2#11723397direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×Dic62
C2×C4×C3⋊(C3⋊Dic3)2
C2×C3×C4⋊(C3⋊Dic3)2
C2×Dic3⋊(C3⋊Dic3)2
C2×Dic3×C3⋊Dic32
#117233972
C2×Q8×C3⋊Dic33
C2×#638433