Finite Group Databasebeta

← C2×#11721090 #13150616

Subgroups — C2×#11721090 #13150616

Order 1728.

1171 conjugacy classes of subgroups; only the 251 normal subgroups are shown.

normal subgroups only

This group has 1171 conjugacy classes of subgroups — too many to draw. Only its 251 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
1C22×#6229C211yesmaximal, characteristic
2C22×#634⋊C3C211yesmaximal, characteristic
3C2×C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
4C2×#88448C214yesmaximal
5#11721090C218yesmaximal, direct product factor
6C2×#63873—31nomaximal
7C2×#63866—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084C2212yes—
10C2×#6229—24no—
11C2×#6229C2214yes—
12C22×(C3⋊Dic3)⋊C3C2212yes—
13C22×(C3⋊Dic3)⋊C3C2212yes—
14C2×#634⋊C3—24no—
15C2×#634⋊C3C2214yes—
16C4×(C3⋊Dic3)⋊C3C2214yes—
17#88448C22116yes—
18C22×#634D311yescharacteristic
19C22×C3×Dic6—31no—
20C22×C3×Dic6—31no—
21C22×C3×Dic6—31no—
22C22×C3×Dic6—61no—
23C2×C4×C3⋊Dic3D311yescharacteristic
24C2×C3×C4×Dic3—31no—
25C2×#3724—34no—
26C2×Dic3⋊Dic3—34no—
27#63873—38no—
28#63866—38no—
29C23×H(3,𝔽3)C2311yescharacteristic
30C2×C4×H(3,𝔽3)D414yes—
31C2×C4×H(3,𝔽3)C2312yes—
32C2×#1084—24no—
33C2×#1084C2314yes—
34#6229—28no—
35C2×(C3⋊Dic3)⋊C3—24no—
36C2×(C3⋊Dic3)⋊C3Q818yescomplemented
37C2×(C3⋊Dic3)⋊C3C2314yes—
38C2×(C3⋊Dic3)⋊C3C2314yes—
39#634⋊C3—28no—
40C2×#944—91no—
41C2×#944—91no—
42C2×#869—91no—
43C22×C32×C4—21no—
44C22×C32×C4C2×D311yescharacteristic
45C22×C32×C4C2×D311yescharacteristic
46C22×C3⋊Dic3C2×D312yes—
47C22×C3⋊Dic3C2×D312yes—
48C2×#634—24no—
49C2×#634C2×D314yes—
50C22×C3×Dic3—32no—
51C22×C3×Dic3—32no—
52C22×C3×Dic3—32no—
53C22×C3×Dic3—32no—
54C22×C3×Dic3—62no—
55C2×C3×Dic6—34no—
56C2×C3×Dic6—34no—
57C2×C3×Dic6—34no—
58C2×C3×Dic6—64no—
59C2×C3×Dic6—68no—
60C2×C3×Dic6—64no—
61C2×C3×Dic6—64no—
62C2×C3×Dic6—64no—
63C4×C3⋊Dic3C2×D314yes—
64C3×C4×Dic3—34no—
65#3724—316no—
66Dic3⋊Dic3—316no—
67C22×H(3,𝔽3)C2×D412yes—
68C22×H(3,𝔽3)C2×Q814yes—
69C22×H(3,𝔽3)C2411yescharacteristic
70C4×H(3,𝔽3)C2×D414yes—
71#1084—24no—
72(C3⋊Dic3)⋊C3—24no—
73(C3⋊Dic3)⋊C3C2×Q818yescomplemented
74C22×C3×Q8—91no—
75C22×C3×Q8—91no—
76C22×Dic6—31no—
77C22×Dic6—31no—
78C22×Dic6—31no—
79C22×Dic6—61no—
80C22×Dic6—91no—
81C2×C3×C42—91no—
82#944—98no—
83#944—98no—
84C2×C4⋊Dic3—94no—
85C2×Dic3⋊C4—94no—
86C2×Dic3⋊C4—94no—
87C2×C4×Dic3—31no—
88C2×C4×Dic3—91no—
89#869—98no—
90C23×C32—21no—
91C23×C32C22×D311yescharacteristic
92C23×C32C22×D311yescharacteristic
93C2×C32×C4—24no—
94C2×C32×C4—22no—
95C2×C32×C4D1214yes—
96C2×C32×C4#6714yes—
97C2×C32×C4C22×D312yes—
98C2×C32×C4C22×D312yes—
99C2×C3⋊Dic3—24no—
100C2×C3⋊Dic3Dic618yescomplemented
101C2×C3⋊Dic3C22×D314yes—
102C2×C3⋊Dic3C22×D314yes—
103#634—28no—
104C2×C3×Dic3—38no—
105C2×C3×Dic3—34no—
106C2×C3×Dic3—34no—
107C2×C3×Dic3—34no—
108C2×C3×Dic3—34no—
109C2×C3×Dic3—64no—
110C2×C3×Dic3—68no—
111C2×C3×Dic3—64no—
112C2×C3×Dic3—64no—
113C2×C3×Dic3—64no—
114C3×Dic6—616no—
115C3×Dic6—68no—
116C3×Dic6—68no—
117C3×Dic6—68no—
118C2×C4⋊Q8—271noSylow
119C2×H(3,𝔽3)C4⋊Q814yescomplemented
120C2×H(3,𝔽3)C22×D411yescharacteristic
121C2×H(3,𝔽3)C22×Q812yes—
122C2×C3×Q8—94no—
123C2×C3×Q8—94no—
124C2×C3×Q8—184no—
125C2×C3×Q8—184no—
126C22×C3×C4—31no—
127C22×C3×C4—31no—
128C22×C3×C4—61no—
129C22×C3×C4—92no—
130C22×C3×C4—92no—
131C22×C3×C4—92no—
132C22×C3×C4D3211yescharacteristic
133C22×Dic3—32no—
134C22×Dic3—32no—
135C22×Dic3—32no—
136C22×Dic3—32no—
137C22×Dic3—62no—
138C22×Dic3—92no—
139C22×Dic3—92no—
140C2×Dic6—34no—
141C2×Dic6—34no—
142C2×Dic6—34no—
143C2×Dic6—64no—
144C2×Dic6—64no—
145C2×Dic6—64no—
146C2×Dic6—68no—
147C2×Dic6—64no—
148C2×Dic6—94no—
149C2×Dic6—184no—
150C3×C42—94no—
151C4⋊Dic3—916no—
152Dic3⋊C4—916no—
153Dic3⋊C4—916no—
154C4×Dic3—34no—
155C4×Dic3—94no—
156C22×C32—21no—
157C22×C32—24no—
158C22×C32—22no—
159C22×C32C2×Dic614yes—
160C22×C32C2×D1212yes—
161C22×C32D3×Q814yes—
162C22×C32C2×#6712yes—
163C22×C32C23×D311yescharacteristic
164C22×C32C23×D311yescharacteristic
165C32×C4—24no—
166C32×C4C2×D1214yes—
167C32×C4C2×#6714yes—
168C3⋊Dic3—24no—
169C3⋊Dic3C2×Dic618yescomplemented
170C3×Dic3—38no—
171C3×Dic3—68no—
172C3×Dic3—64no—
173C3×Dic3—64no—
174C3×Dic3—64no—
175C22×Q8—91no—
176C22×Q8—91no—
177C4⋊Q8—278no—
178C2×C4⋊C4—274no—
179C2×C42—91no—
180H(3,𝔽3)C2×C4⋊Q811yescharacteristic, complemented, Sylow
181C23×C3—31no—
182C23×C3—31no—
183C23×C3—61no—
184C23×C3C2×D3211yescharacteristic
185C3×Q8—188no—
186C3×Q8—188no—
187C2×C3×C4—34no—
188C2×C3×C4—34no—
189C2×C3×C4—32no—
190C2×C3×C4—32no—
191C2×C3×C4—64no—
192C2×C3×C4—62no—
193C2×C3×C4—94no—
194C2×C3×C4—98no—
195C2×C3×C4—94no—
196C2×C3×C4—94no—
197C2×C3×C4—184no—
198C2×C3×C4—184no—
199C2×C3×C4#62614yes—
200C2×C3×C4C2×D3212yes—
201C2×Dic3—34no—
202C2×Dic3—38no—
203C2×Dic3—34no—
204C2×Dic3—34no—
205C2×Dic3—34no—
206C2×Dic3—64no—
207C2×Dic3—64no—
208C2×Dic3—64no—
209C2×Dic3—68no—
210C2×Dic3—64no—
211C2×Dic3—98no—
212C2×Dic3—94no—
213C2×Dic3—94no—
214C2×Dic3—184no—
215Dic6—68no—
216Dic6—68no—
217Dic6—616no—
218Dic6—68no—
219Dic6—188no—
220C2×C32—22no—
221C2×C32—21no—
222C2×C32—24no—
223C2×C32#86914yescomplemented
224C2×C32#94414yescomplemented
225C2×C32C22×Dic612yes—
226C2×C32C22×D1211yescharacteristic
227C2×C32C2×D3×Q812yes—
228C2×C32C22×#6711yescharacteristic
229C2×Q8—94no—
230C2×Q8—94no—
231C2×Q8—184no—
232C2×Q8—184no—
233C22×C4—92no—
234C22×C4—92no—
235C22×C4—92no—
236C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
237C4⋊C4—2716no—
238C42—94no—
239C22×C3—31no—
240C22×C3—31no—
241C22×C3—34no—
242C22×C3—34no—
243C22×C3—32no—
244C22×C3—32no—
245C22×C3—61no—
246C22×C3—64no—
247C22×C3—62no—
248C22×C3D3×Dic614yes—
249C22×C3C2×#62612yes—
250C22×C3C22×D3211yescharacteristic
251C3×C4—34no—
252C3×C4—34no—
253C3×C4—64no—
254C3×C4—98no—
255C3×C4—184no—
256C3×C4—184no—
257C3×C4C2×#62614yes—
258Dic3—38no—
259Dic3—64no—
260Dic3—64no—
261Dic3—68no—
262Dic3—64no—
263Dic3—98no—
264Dic3—184no—
265C32—21no—
266C32C2×#86911yescharacteristic, complemented
267C32C2×#94411yescharacteristic, complemented
268C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
269Q8—188no—
270Q8—188no—
271C2×C4—94no—
272C2×C4—98no—
273C2×C4—94no—
274C2×C4—94no—
275C2×C4—184no—
276C2×C4—184no—
277C2×C4#620014yes—
278C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
279C2×C3—31no—
280C2×C3—31no—
281C2×C3—34no—
282C2×C3—34no—
283C2×C3—32no—
284C2×C3—32no—
285C2×C3—61no—
286C2×C3—64no—
287C2×C3—62no—
288C2×C3#6386614yes—
289C2×C3C2×D3×Dic612yes—
290C2×C3C22×#62611yescharacteristic
291C22((C3⋊D3)⋊C3)⋊Q814yes—
292C22C2×#620012yes—
293C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
294C4—98no—
295C4—184no—
296C4—184no—
297C4C2×#620014yes—
298C3—31no—
299C3—31no—
300C3—61no—
301C3C2×#6386611yescharacteristic
302C2#1172109014yesdirect product factor
303C2C2×((C3⋊D3)⋊C3)⋊Q812yes—
304C2C22×#620011yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
FrattiniC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC23×C3C2×D32
prime radicalC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×C4⋊Q8
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
5. #11721090C2direct (N × H)
36. C2×(C3⋊Dic3)⋊C3Q8semidirect
73. (C3⋊Dic3)⋊C3C2×Q8semidirect
100. C2×C3⋊Dic3Dic6semidirect
119. C2×H(3,𝔽3)C4⋊Q8semidirect
169. C3⋊Dic3C2×Dic6semidirect
180. H(3,𝔽3)C2×C4⋊Q8semidirect
223. C2×C32#869semidirect
224. C2×C32#944semidirect
266. C32C2×#869semidirect
267. C32C2×#944semidirect
302. C2#11721090direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62292
C22×#634⋊C32
C2×C4×(C3⋊Dic3)⋊C32
C2×#884482
#117210902
C2×#638733
C2×#638663