Finite Group Databasebeta

← C2×#11721105 #13150634

Subgroups — C2×#11721105 #13150634

Order 1728.

1263 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C2×C4×#1084C211yesmaximal, characteristic, complemented
3C2×C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
4C2×#88450C212yesmaximal
5C2×#88448C212yesmaximal, complemented
6#11721105C218yesmaximal, direct product factor
7C2×#63874—31nomaximal
8C2×#63855—31nomaximal
9C23×(C3⋊D3)⋊C3C2211yescharacteristic
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C22×#1084C2212yes—
12C22×(C3⋊Dic3)⋊C3C2212yes—
13C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
14C2×C4×(C3⋊D3)⋊C3C418yescomplemented
15C2×C4×(C3⋊D3)⋊C3C2214yes—
16C4×#1084C2214yescomplemented
17C4×(C3⋊Dic3)⋊C3C2214yescomplemented
18#88450C2218yes—
19#88448C2218yescomplemented
20C22×C4×C3⋊D3D311yescharacteristic
21C22×C3×C4×D3—31no—
22C2×C4×C3⋊Dic3D311yescharacteristic, complemented
23C2×C3×C4×Dic3—31no—
24C2×C3×C4×Dic3—31no—
25C2×C3×C4×Dic3—31no—
26C2×C3×C4×Dic3—61no—
27C2×#3724—32no—
28C2×Dic3⋊Dic3—32no—
29C2×#3722—32no—
30C2×(C2×D3)⋊Dic3—32no—
31#63874—38no—
32#63855—38no—
33C23×H(3,𝔽3)C2311yescharacteristic
34C22×(C3⋊D3)⋊C3—24no—
35C22×(C3⋊D3)⋊C3C2×C414yes—
36C22×(C3⋊D3)⋊C3C2312yes—
37C2×C4×H(3,𝔽3)C2×C414yes—
38C2×C4×H(3,𝔽3)C2312yes—
39C2×#1084—24no—
40C2×#1084C2314yes—
41C2×(C3⋊Dic3)⋊C3—24no—
42C2×(C3⋊Dic3)⋊C3C2×C414yes—
43C2×(C3⋊Dic3)⋊C3C2314yes—
44C2×(C3⋊Dic3)⋊C3C2312yes—
45C4×(C3⋊D3)⋊C3C2×C4116yescomplemented
46C2×#926—91no—
47C2×#872—91no—
48C2×#872—91no—
49C23×C3⋊D3C2×D311yescharacteristic
50C23×C3×D3—31no—
51C22×C32×C4—21no—
52C22×C32×C4C2×D311yescharacteristic
53C22×C32×C4C2×D311yescharacteristic
54C22×C3⋊Dic3C2×D312yes—
55C22×C3⋊Dic3C2×D311yescharacteristic
56C2×C4×C3⋊D3Dic318yescomplemented
57C2×C4×C3⋊D3C2×D314yes—
58C22×C3×Dic3—31no—
59C22×C3×Dic3—32no—
60C22×C3×Dic3—32no—
61C22×C3×Dic3—32no—
62C22×C3×Dic3—62no—
63C2×C3×C4×D3—38no—
64C2×C3×C4×D3—34no—
65C4×C3⋊Dic3C2×D314yescomplemented
66C3×C4×Dic3—34no—
67C3×C4×Dic3—34no—
68C3×C4×Dic3—34no—
69C3×C4×Dic3—64no—
70#3724—38no—
71Dic3⋊Dic3—38no—
72#3722—38no—
73(C2×D3)⋊Dic3—38no—
74C22×H(3,𝔽3)C22×C412yes—
75C22×H(3,𝔽3)#4114yes—
76C22×H(3,𝔽3)C2411yescharacteristic
77C2×(C3⋊D3)⋊C3—28no—
78C2×(C3⋊D3)⋊C3—24no—
79C2×(C3⋊D3)⋊C3C22×C414yes—
80C4×H(3,𝔽3)C22×C414yes—
81#1084—24no—
82(C3⋊Dic3)⋊C3—24no—
83(C3⋊Dic3)⋊C3C22×C414yes—
84C23×C3×C4—91no—
85C22×C4×D3—31no—
86C22×C4×D3—91no—
87C2×C3×C42—91no—
88C2×C3×C42—91no—
89C2×#217—92no—
90C2×(C2×D3)⋊C4—92no—
91C2×(C2×D3)⋊C4—92no—
92#926—98no—
93C2×C4⋊Dic3—92no—
94C2×Dic3⋊C4—92no—
95C2×Dic3⋊C4—92no—
96C2×C4×Dic3—31no—
97C2×C4×Dic3—31no—
98C2×C4×Dic3—31no—
99C2×C4×Dic3—61no—
100C2×C4×Dic3—91no—
101#872—98no—
102#872—98no—
103C23×C32—21no—
104C23×C32C22×D311yescharacteristic
105C23×C32C22×D311yescharacteristic
106C22×C3⋊D3—24no—
107C22×C3⋊D3C2×Dic314yes—
108C22×C3⋊D3C22×D312yes—
109C22×C3×D3—34no—
110C22×C3×D3—32no—
111C22×C3×D3—64no—
112C2×C32×C4—24no—
113C2×C32×C4—22no—
114C2×C32×C4C4×D314yes—
115C2×C32×C4C2×Dic314yes—
116C2×C32×C4C22×D312yes—
117C2×C32×C4C22×D312yes—
118C2×C3⋊Dic3—24no—
119C2×C3⋊Dic3C2×Dic314yes—
120C2×C3⋊Dic3C22×D314yes—
121C2×C3⋊Dic3C22×D312yes—
122C4×C3⋊D3C2×Dic3116yescomplemented
123C2×C3×Dic3—34no—
124C2×C3×Dic3—32no—
125C2×C3×Dic3—34no—
126C2×C3×Dic3—34no—
127C2×C3×Dic3—34no—
128C2×C3×Dic3—64no—
129C2×C3×Dic3—68no—
130C2×C3×Dic3—64no—
131C2×C3×Dic3—64no—
132C2×C3×Dic3—64no—
133C3×C4×D3—316no—
134C2×#117—271noSylow
135C2×H(3,𝔽3)#11714yescomplemented
136C2×H(3,𝔽3)C23×C411yescharacteristic
137C2×H(3,𝔽3)C2×#4112yes—
138(C3⋊D3)⋊C3—24no—
139C24×C3—91no—
140C23×D3—31no—
141C23×D3—91no—
142C22×C3×C4—31no—
143C22×C3×C4—31no—
144C22×C3×C4—61no—
145C22×C3×C4—94no—
146C22×C3×C4—98no—
147C22×C3×C4—91no—
148C22×C3×C4—92no—
149C22×C3×C4—92no—
150C22×C3×C4D3211yescharacteristic
151C22×Dic3—31no—
152C22×Dic3—32no—
153C22×Dic3—32no—
154C22×Dic3—32no—
155C22×Dic3—62no—
156C22×Dic3—91no—
157C22×Dic3—92no—
158C2×C4×D3—38no—
159C2×C4×D3—34no—
160C2×C4×D3—94no—
161C2×C4×D3—98no—
162C3×C42—94no—
163C3×C42—94no—
164#217—98no—
165(C2×D3)⋊C4—98no—
166(C2×D3)⋊C4—98no—
167C4⋊Dic3—98no—
168Dic3⋊C4—98no—
169Dic3⋊C4—98no—
170C4×Dic3—34no—
171C4×Dic3—34no—
172C4×Dic3—34no—
173C4×Dic3—64no—
174C4×Dic3—94no—
175C22×C32—21no—
176C22×C32—24no—
177C22×C32—22no—
178C22×C32C2×C4×D312yes—
179C22×C32#23514yes—
180C22×C32#23514yes—
181C22×C32C22×Dic312yes—
182C22×C32C23×D311yescharacteristic
183C22×C32C23×D311yescharacteristic
184C2×C3⋊D3—28no—
185C2×C3⋊D3—24no—
186C2×C3⋊D3C22×Dic314yes—
187C2×C3×D3—34no—
188C2×C3×D3—68no—
189C2×C3×D3—64no—
190C32×C4—24no—
191C32×C4C2×C4×D314yes—
192C32×C4C22×Dic314yes—
193C3⋊Dic3—24no—
194C3⋊Dic3C22×Dic314yes—
195C3×Dic3—34no—
196C3×Dic3—68no—
197C3×Dic3—64no—
198C3×Dic3—64no—
199C3×Dic3—64no—
200C23×C4—91no—
201#117—278no—
202C2×C4⋊C4—272no—
203C2×C22⋊C4—272no—
204C2×C42—91no—
205C2×C42—91no—
206H(3,𝔽3)C2×#11711yescharacteristic, complemented, Sylow
207C23×C3—31no—
208C23×C3—31no—
209C23×C3—61no—
210C23×C3—92no—
211C23×C3—94no—
212C23×C3—184no—
213C23×C3C2×D3211yescharacteristic
214C22×D3—32no—
215C22×D3—34no—
216C22×D3—64no—
217C22×D3—94no—
218C22×D3—92no—
219C22×D3—184no—
220C2×C3×C4—34no—
221C2×C3×C4—34no—
222C2×C3×C4—32no—
223C2×C3×C4—32no—
224C2×C3×C4—64no—
225C2×C3×C4—62no—
226C2×C3×C4—916no—
227C2×C3×C4—92no—
228C2×C3×C4—94no—
229C2×C3×C4—94no—
230C2×C3×C4—94no—
231C2×C3×C4—184no—
232C2×C3×C4—184no—
233C2×C3×C4D3×Dic314yes—
234C2×C3×C4C2×D3212yes—
235C2×Dic3—32no—
236C2×Dic3—34no—
237C2×Dic3—34no—
238C2×Dic3—34no—
239C2×Dic3—34no—
240C2×Dic3—64no—
241C2×Dic3—64no—
242C2×Dic3—64no—
243C2×Dic3—68no—
244C2×Dic3—64no—
245C2×Dic3—94no—
246C2×Dic3—92no—
247C2×Dic3—94no—
248C2×Dic3—184no—
249C4×D3—316no—
250C4×D3—916no—
251C2×C32—22no—
252C2×C32—21no—
253C2×C32—24no—
254C2×C32#87214yescomplemented
255C2×C32#92614yescomplemented
256C2×C32C22×C4×D311yescharacteristic
257C2×C32C2×#23512yes—
258C2×C32C2×#23512yes—
259C2×C32C23×Dic311yescharacteristic
260C3⋊D3—24no—
261C3×D3—64no—
262C24—91no—
263C22×C4—91no—
264C22×C4—98no—
265C22×C4—94no—
266C22×C4—92no—
267C22×C4—92no—
268C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
269C4⋊C4—278no—
270C22⋊C4—278no—
271C42—94no—
272C42—94no—
273C22×C3—31no—
274C22×C3—31no—
275C22×C3—34no—
276C22×C3—34no—
277C22×C3—32no—
278C22×C3—32no—
279C22×C3—61no—
280C22×C3—64no—
281C22×C3—62no—
282C22×C3—94no—
283C22×C3—188no—
284C22×C3—184no—
285C22×C3#379814yes—
286C22×C3C2×D3×Dic312yes—
287C22×C3C22×D3211yescharacteristic
288C2×D3—34no—
289C2×D3—64no—
290C2×D3—68no—
291C2×D3—94no—
292C2×D3—188no—
293C2×D3—184no—
294C3×C4—34no—
295C3×C4—34no—
296C3×C4—64no—
297C3×C4—94no—
298C3×C4—184no—
299C3×C4—184no—
300C3×C4C2×D3×Dic314yes—
301Dic3—34no—
302Dic3—64no—
303Dic3—64no—
304Dic3—68no—
305Dic3—64no—
306Dic3—94no—
307Dic3—184no—
308C32—21no—
309C32C2×#87211yescharacteristic, complemented
310C32C2×#92611yescharacteristic, complemented
311C23—92no—
312C23—94no—
313C23—184no—
314C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
315C2×C4—92no—
316C2×C4—94no—
317C2×C4—916no—
318C2×C4—94no—
319C2×C4—94no—
320C2×C4—184no—
321C2×C4—184no—
322C2×C4((C3⋊D3)⋊C3)⋊C414yes—
323C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
324C2×C3—31no—
325C2×C3—31no—
326C2×C3—34no—
327C2×C3—34no—
328C2×C3—32no—
329C2×C3—32no—
330C2×C3—61no—
331C2×C3—64no—
332C2×C3—62no—
333C2×C3—184no—
334C2×C3#6385514yes—
335C2×C3C2×#379812yes—
336C2×C3C22×D3×Dic311yescharacteristic
337D3—64no—
338D3—184no—
339C22—94no—
340C22—188no—
341C22—184no—
342C22#8865014yes—
343C22C2×((C3⋊D3)⋊C3)⋊C412yes—
344C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
345C4—94no—
346C4—184no—
347C4—184no—
348C4C2×((C3⋊D3)⋊C3)⋊C414yes—
349C3—31no—
350C3—31no—
351C3—61no—
352C3C2×#6385511yescharacteristic
353C2—184no—
354C2#1172110514yesdirect product factor
355C2C2×#8865012yes—
356C2C22×((C3⋊D3)⋊C3)⋊C411yescharacteristic
subgroupG/H
center, hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
FrattiniC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC23×C3C2×D32
prime radicalC23×(C3⋊D3)⋊C3C22
cosocleC22×H(3,𝔽3)C24
hypocenterH(3,𝔽3)C2×#117
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×#1084C2semidirect
3. C2×C4×(C3⋊Dic3)⋊C3C2semidirect
5. C2×#88448C2semidirect
6. #11721105C2direct (N × H)
14. C2×C4×(C3⋊D3)⋊C3C4semidirect
16. C4×#1084C22semidirect
17. C4×(C3⋊Dic3)⋊C3C22semidirect
19. #88448C22semidirect
22. C2×C4×C3⋊Dic3D3semidirect
45. C4×(C3⋊D3)⋊C3C2×C4semidirect
56. C2×C4×C3⋊D3Dic3semidirect
65. C4×C3⋊Dic3C2×D3semidirect
122. C4×C3⋊D3C2×Dic3semidirect
135. C2×H(3,𝔽3)#117semidirect
206. H(3,𝔽3)C2×#117semidirect
254. C2×C32#872semidirect
255. C2×C32#926semidirect
309. C32C2×#872semidirect
310. C32C2×#926semidirect
354. C2#11721105direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×(C3⋊D3)⋊C32
C2×C4×#10842
C2×C4×(C3⋊Dic3)⋊C32
C2×#884502
C2×#884482
#117211052
C2×#638743
C2×#638553