Finite Group Databasebeta

← C2×#11721113 #13150632

Subgroups — C2×#11721113 #13150632

Order 1728.

1263 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 1263 conjugacy classes of subgroups — too many to draw. Only its 219 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×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
2C22×#634⋊C3C211yesmaximal, characteristic, complemented
3C2×C4×#1084C211yesmaximal, characteristic, complemented
4C2×#88450C214yesmaximal, complemented
5#11721113C218yesmaximal, direct product factor
6C2×#63864—31nomaximal
7C2×#63860—31nomaximal
8C23×(C3⋊D3)⋊C3C2212yes—
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×#1084C2212yes—
11C22×(C3⋊Dic3)⋊C3C2212yes—
12C2×C4⋊((C3⋊D3)⋊C3)—24no—
13C2×C4⋊((C3⋊D3)⋊C3)C2214yes—
14C2×#634⋊C3—24no—
15C2×#634⋊C3C2214yescomplemented
16C4×#1084C2214yescomplemented
17#88450C22116yescomplemented
18C22×C4⋊(C3⋊D3)D311yescharacteristic
19C22×#634D311yescharacteristic, complemented
20C22×C3×D12—31no—
21C22×C3×Dic6—31no—
22C2×C3×C4×Dic3—31no—
23C2×C3×C4×Dic3—31no—
24C2×C3×C4×Dic3—61no—
25C2×#3722—34no—
26C2×(C2×D3)⋊Dic3—34no—
27#63864—38no—
28#63860—38no—
29C23×H(3,𝔽3)C2311yescharacteristic
30C22×(C3⋊D3)⋊C3—24no—
31C22×(C3⋊D3)⋊C3—28no—
32C22×(C3⋊D3)⋊C3C2314yes—
33C2×C4×H(3,𝔽3)D414yes—
34C2×C4×H(3,𝔽3)C2312yes—
35C2×#1084—24no—
36C2×#1084C2314yes—
37C2×(C3⋊Dic3)⋊C3—24no—
38C2×(C3⋊Dic3)⋊C3C2314yes—
39C4⋊((C3⋊D3)⋊C3)—28no—
40#634⋊C3—28no—
41C2×#947—91no—
42C2×#936—91no—
43C2×#875—91no—
44C23×C3⋊D3C2×D312yes—
45C23×C3×D3—32no—
46C22×C32×C4—21no—
47C22×C32×C4C2×D311yescharacteristic
48C22×C32×C4C2×D311yescharacteristic
49C22×C3⋊Dic3C2×D312yes—
50C2×C4⋊(C3⋊D3)—24no—
51C2×C4⋊(C3⋊D3)C2×D314yes—
52C2×#634—24no—
53C2×#634C2×D314yescomplemented
54C22×C3×Dic3—32no—
55C22×C3×Dic3—32no—
56C22×C3×Dic3—32no—
57C22×C3×Dic3—62no—
58C2×C3×D12—34no—
59C2×C3×D12—64no—
60C2×C3×Dic6—34no—
61C2×C3×Dic6—64no—
62C3×C4×Dic3—34no—
63C3×C4×Dic3—34no—
64C3×C4×Dic3—64no—
65#3722—316no—
66(C2×D3)⋊Dic3—316no—
67C22×H(3,𝔽3)C2×D412yes—
68C22×H(3,𝔽3)#4114yes—
69C22×H(3,𝔽3)C2411yescharacteristic
70C2×(C3⋊D3)⋊C3—24no—
71C2×(C3⋊D3)⋊C3—28no—
72C2×(C3⋊D3)⋊C3—48no—
73C4×H(3,𝔽3)C2×D414yes—
74#1084—24no—
75(C3⋊Dic3)⋊C3—24no—
76C22×C3×Q8—91no—
77C22×C3×D4—91no—
78C22×D12—31no—
79C22×D12—91no—
80C22×Dic6—31no—
81C22×Dic6—91no—
82C2×C3×C42—91no—
83C2×#217—94no—
84#947—98no—
85#936—98no—
86C2×(C2×D3)⋊C4—94no—
87C2×(C2×D3)⋊C4—94no—
88C2×C4×Dic3—31no—
89C2×C4×Dic3—31no—
90C2×C4×Dic3—61no—
91#875—98no—
92C23×C32—21no—
93C23×C32C22×D311yescharacteristic
94C23×C32C22×D311yescharacteristic
95C22×C3⋊D3—28no—
96C22×C3⋊D3—24no—
97C22×C3⋊D3C22×D314yes—
98C22×C3×D3—34no—
99C22×C3×D3—68no—
100C22×C3×D3—64no—
101C2×C32×C4—24no—
102C2×C32×C4—22no—
103C2×C32×C4#6714yes—
104C2×C32×C4#6714yes—
105C2×C32×C4C22×D312yes—
106C2×C32×C4C22×D312yes—
107C2×C3⋊Dic3—24no—
108C2×C3⋊Dic3C22×D314yes—
109C4⋊(C3⋊D3)—28no—
110#634—28no—
111C2×C3×Dic3—34no—
112C2×C3×Dic3—34no—
113C2×C3×Dic3—34no—
114C2×C3×Dic3—64no—
115C2×C3×Dic3—68no—
116C2×C3×Dic3—64no—
117C2×C3×Dic3—64no—
118C2×C3×Dic3—64no—
119C3×D12—68no—
120C3×Dic6—68no—
121C2×#124—271noSylow
122C2×H(3,𝔽3)#12414yescomplemented
123C2×H(3,𝔽3)C22×D411yescharacteristic
124C2×H(3,𝔽3)C2×#4112yes—
125(C3⋊D3)⋊C3—44no—
126C24×C3—92no—
127C23×D3—32no—
128C23×D3—92no—
129C2×C3×Q8—94no—
130C2×C3×Q8—184no—
131C2×C3×D4—94no—
132C2×C3×D4—184no—
133C22×C3×C4—31no—
134C22×C3×C4—31no—
135C22×C3×C4—61no—
136C22×C3×C4—92no—
137C22×C3×C4—92no—
138C22×C3×C4D3211yescharacteristic
139C22×Dic3—32no—
140C22×Dic3—32no—
141C22×Dic3—32no—
142C22×Dic3—62no—
143C22×Dic3—92no—
144C2×D12—34no—
145C2×D12—64no—
146C2×D12—94no—
147C2×D12—184no—
148C2×Dic6—34no—
149C2×Dic6—64no—
150C2×Dic6—94no—
151C2×Dic6—184no—
152C3×C42—94no—
153#217—916no—
154(C2×D3)⋊C4—916no—
155(C2×D3)⋊C4—916no—
156C4×Dic3—34no—
157C4×Dic3—34no—
158C4×Dic3—64no—
159C22×C32—21no—
160C22×C32—24no—
161C22×C32—22no—
162C22×C32#23714yes—
163C22×C32#23914yes—
164C22×C32C2×#6712yes—
165C22×C32C2×#6712yes—
166C22×C32C23×D311yescharacteristic
167C22×C32C23×D311yescharacteristic
168C2×C3⋊D3—28no—
169C2×C3⋊D3—24no—
170C2×C3⋊D3—48no—
171C2×C3×D3—68no—
172C2×C3×D3—64no—
173C2×C3×D3—128no—
174C32×C4—24no—
175C32×C4C2×#6714yes—
176C32×C4C2×#6714yes—
177C3⋊Dic3—24no—
178C3×Dic3—68no—
179C3×Dic3—64no—
180C3×Dic3—64no—
181C3×Dic3—64no—
182C22×Q8—91no—
183C22×D4—91no—
184#124—278no—
185C2×C22⋊C4—274no—
186C2×C42—91no—
187H(3,𝔽3)C2×#12411yescharacteristic, complemented, Sylow
188C23×C3—31no—
189C23×C3—31no—
190C23×C3—61no—
191C23×C3—94no—
192C23×C3—184no—
193C23×C3—188no—
194C23×C3C2×D3211yescharacteristic
195C22×D3—34no—
196C22×D3—68no—
197C22×D3—64no—
198C22×D3—94no—
199C22×D3—184no—
200C22×D3—188no—
201C3×Q8—188no—
202C3×D4—188no—
203C2×C3×C4—34no—
204C2×C3×C4—34no—
205C2×C3×C4—32no—
206C2×C3×C4—32no—
207C2×C3×C4—64no—
208C2×C3×C4—62no—
209C2×C3×C4—94no—
210C2×C3×C4—94no—
211C2×C3×C4—184no—
212C2×C3×C4—184no—
213C2×C3×C4#62514yes—
214C2×C3×C4C2×D3212yes—
215C2×Dic3—34no—
216C2×Dic3—34no—
217C2×Dic3—34no—
218C2×Dic3—64no—
219C2×Dic3—64no—
220C2×Dic3—64no—
221C2×Dic3—68no—
222C2×Dic3—64no—
223C2×Dic3—94no—
224C2×Dic3—184no—
225D12—68no—
226D12—188no—
227Dic6—68no—
228Dic6—188no—
229C2×C32—22no—
230C2×C32—21no—
231C2×C32—24no—
232C2×C32#93614yescomplemented
233C2×C32#94714yescomplemented
234C2×C32C2×#23712yes—
235C2×C32C2×#23912yes—
236C2×C32C22×#6711yescharacteristic
237C2×C32C22×#6711yescharacteristic
238C3⋊D3—44no—
239C3×D3—124no—
240C24—92no—
241C2×Q8—94no—
242C2×Q8—184no—
243C2×D4—94no—
244C2×D4—184no—
245C22×C4—92no—
246C22×C4—92no—
247C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
248C22⋊C4—2716no—
249C42—94no—
250C22×C3—31no—
251C22×C3—31no—
252C22×C3—34no—
253C22×C3—34no—
254C22×C3—32no—
255C22×C3—32no—
256C22×C3—61no—
257C22×C3—64no—
258C22×C3—62no—
259C22×C3—184no—
260C22×C3—188no—
261C22×C3—368no—
262C22×C3#379614yes—
263C22×C3C2×#62512yes—
264C22×C3C22×D3211yescharacteristic
265C2×D3—68no—
266C2×D3—64no—
267C2×D3—128no—
268C2×D3—184no—
269C2×D3—188no—
270C2×D3—368no—
271C3×C4—34no—
272C3×C4—34no—
273C3×C4—64no—
274C3×C4—184no—
275C3×C4—184no—
276C3×C4C2×#62514yes—
277Dic3—64no—
278Dic3—64no—
279Dic3—68no—
280Dic3—64no—
281Dic3—184no—
282C32—21no—
283C32C2×#93611yescharacteristic, complemented
284C32C2×#94711yescharacteristic, complemented
285C23—94no—
286C23—188no—
287C23—184no—
288C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
289Q8—188no—
290D4—188no—
291C2×C4—94no—
292C2×C4—94no—
293C2×C4—184no—
294C2×C4—184no—
295C2×C4#619814yes—
296C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
297C2×C3—31no—
298C2×C3—31no—
299C2×C3—34no—
300C2×C3—34no—
301C2×C3—32no—
302C2×C3—32no—
303C2×C3—61no—
304C2×C3—64no—
305C2×C3—62no—
306C2×C3—364no—
307C2×C3#6386314yes—
308C2×C3C2×#379612yes—
309C2×C3C22×#62511yescharacteristic
310D3—124no—
311D3—364no—
312C22—184no—
313C22—188no—
314C22—368no—
315C22#8864914yes—
316C22C2×#619812yes—
317C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
318C4—184no—
319C4—184no—
320C4C2×#619814yes—
321C3—31no—
322C3—31no—
323C3—61no—
324C3C2×#6386311yescharacteristic
325C2—364no—
326C2#1172111314yesdirect product factor
327C2C2×#8864912yes—
328C2C22×#619811yescharacteristic
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 radicalC22×C4⋊((C3⋊D3)⋊C3)C2
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#124
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
2. C22×#634⋊C3C2semidirect
3. C2×C4×#1084C2semidirect
4. C2×#88450C2semidirect
5. #11721113C2direct (N × H)
15. C2×#634⋊C3C22semidirect
16. C4×#1084C22semidirect
17. #88450C22semidirect
19. C22×#634D3semidirect
53. C2×#634C2×D3semidirect
122. C2×H(3,𝔽3)#124semidirect
187. H(3,𝔽3)C2×#124semidirect
232. C2×C32#936semidirect
233. C2×C32#947semidirect
283. C32C2×#936semidirect
284. C32C2×#947semidirect
326. C2#11721113direct (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)⋊C3)2
C22×#634⋊C32
C2×C4×#10842
C2×#884502
#117211132
C2×#638643
C2×#638603