Finite Group Databasebeta

← C2×#11721763 #13153856

Subgroups — C2×#11721763 #13153856

Order 1728.

2343 conjugacy classes of subgroups; only the 325 normal subgroups are shown.

normal subgroups only

This group has 2343 conjugacy classes of subgroups — too many to draw. Only its 325 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⋊#272C211yesmaximal, characteristic, complemented
2C22×C4⋊#272C212yesmaximal, complemented
3C22×C4×#272C211yesmaximal, characteristic, complemented
4C2×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
5C2×#88542C212yesmaximal, complemented
6#11721763C218yesmaximal, direct product factor
7C2×C3×#895—34nomaximal
8C23×#272C2212yes—
9C23×#272C2211yescharacteristic
10C22×C4×H(3,𝔽3)C2212yes—
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C22×#1084C2211yescharacteristic
13C2×C4⋊#272—28no—
14C2×C4⋊#272—24no—
15C2×C4⋊#272C2218yescomplemented
16C2×C4⋊#272C2214yescomplemented
17C2×C4×#272—24no—
18C2×C4×#272C2214yescomplemented
19C4⋊C4×H(3,𝔽3)C2214yescomplemented
20#88542C2218yescomplemented
21C22×C3×D12—38no—
22C22×C3×D12—34no—
23C22×C3×C4×D3—34no—
24C2×C32×C4⋊C4D314yescomplemented
25C2×C3×(C2×D3)⋊C4—38no—
26C3×#895—332no—
27C23×H(3,𝔽3)C2311yescharacteristic
28C22×#272—22no—
29C22×#272—22no—
30C22×#272—24no—
31C22×#272—24no—
32C22×#272—24no—
33C22×#272D414yescomplemented
34C22×#272C2314yes—
35C22×#272C2312yes—
36C2×C4×H(3,𝔽3)—24no—
37C2×C4×H(3,𝔽3)D414yescomplemented
38C2×C4×H(3,𝔽3)C2314yes—
39C2×C4×H(3,𝔽3)C2312yes—
40C2×#1084—22no—
41C2×#1084C2312yes—
42C4⋊#272—216no—
43C4⋊#272—28no—
44C4×#272—28no—
45C2×C3×#121—91no—
46C2×#895—34no—
47C23×C3×D3—34no—
48C23×C3×D3—38no—
49C22×C32×C4C2×D318yes—
50C22×C32×C4C2×D314yes—
51C22×C3×Dic3—34no—
52C2×C3×D12—332no—
53C2×C3×D12—316no—
54C2×C3×D12—632no—
55C2×C3×D12—616no—
56C2×C3×C4×D3—316no—
57C2×C3×C4×D3—616no—
58C32×C4⋊C4C2×D3116yescomplemented
59C3×(C2×D3)⋊C4—332no—
60C22×H(3,𝔽3)C2×D412yes—
61C22×H(3,𝔽3)C2×D412yes—
62C22×H(3,𝔽3)#4112yes—
63C22×H(3,𝔽3)C2411yescharacteristic
64C2×#272—28no—
65C2×#272—22no—
66C2×#272—22no—
67C2×#272—24no—
68C2×#272—24no—
69C2×#272—24no—
70C2×#272—48no—
71C2×#272C2×D414yescomplemented
72C4×H(3,𝔽3)—24no—
73C4×H(3,𝔽3)C2×D414yescomplemented
74#1084—22no—
75C22×C3×D4—92no—
76C22×C3×D4—91no—
77C23×C3×C4—91no—
78C22×D12—34no—
79C22×D12—38no—
80C22×C4×D3—34no—
81C3×#121—98no—
82C2×C3×C4⋊C4—34no—
83C2×C3×C4⋊C4C3⋊D311yescharacteristic
84C2×C3×C22⋊C4—92no—
85C2×(C2×D3)⋊C4—38no—
86#895—332no—
87C23×C32C22×D314yes—
88C22×C3×D3—316no—
89C22×C3×D3—38no—
90C22×C3×D3—316no—
91C22×C3×D3—68no—
92C22×C3×D3—68no—
93C22×C3×D3—616no—
94C22×C3×D3—616no—
95C22×C3×D3—616no—
96C2×C32×C4—216no—
97C2×C32×C4D12116yescomplemented
98C2×C32×C4C22×D3116yes—
99C2×C32×C4C22×D318yes—
100C2×C3×Dic3—38no—
101C2×C3×Dic3—68no—
102C3×D12—664no—
103C3×D12—632no—
104C3×C4×D3—632no—
105C2×#121—91noSylow
106C2×H(3,𝔽3)#12114yescomplemented
107C2×H(3,𝔽3)C22×D411yescharacteristic
108C2×H(3,𝔽3)C22×D411yescharacteristic
109C2×H(3,𝔽3)C2×#4111yescharacteristic
110#272—24no—
111#272—44no—
112C24×C3—91no—
113C24×C3—92no—
114C23×D3—34no—
115C23×D3—38no—
116C2×C3×D4—98no—
117C2×C3×D4—94no—
118C2×C3×D4—188no—
119C2×C3×D4—184no—
120C22×C3×C4—34no—
121C22×C3×C4—38no—
122C22×C3×C4—91no—
123C22×C3×C4—94no—
124C22×C3×C4—184no—
125C22×C3×C4C2×C3⋊D312yes—
126C22×C3×C4C2×C3⋊D311yescharacteristic
127C22×Dic3—34no—
128C2×D12—316no—
129C2×D12—332no—
130C2×D12—616no—
131C2×D12—632no—
132C2×C4×D3—316no—
133C2×C4×D3—616no—
134C3×C4⋊C4—316no—
135C3×C4⋊C4C2×C3⋊D314yes—
136C3×C22⋊C4—98no—
137(C2×D3)⋊C4—332no—
138C22×C32C2×D1218yes—
139C22×C32D3×D418yes—
140C22×C32#23918yes—
141C22×C32C23×D314yes—
142C2×C3×D3—316no—
143C2×C3×D3—632no—
144C2×C3×D3—68no—
145C2×C3×D3—68no—
146C2×C3×D3—616no—
147C2×C3×D3—616no—
148C2×C3×D3—616no—
149C2×C3×D3—1232no—
150C32×C4—216no—
151C32×C4C2×D12116yescomplemented
152C3×Dic3—68no—
153C22×D4—92no—
154C22×D4—91no—
155C23×C4—91no—
156#121—98no—
157C2×C4⋊C4#27211yescharacteristic, complemented
158C2×C22⋊C4—92no—
159H(3,𝔽3)C2×#12111yescharacteristic, complemented, Sylow
160C23×C3—34no—
161C23×C3—92no—
162C23×C3—94no—
163C23×C3—94no—
164C23×C3—182no—
165C23×C3—182no—
166C23×C3—184no—
167C23×C3—184no—
168C23×C3—184no—
169C23×C3C22×C3⋊D311yescharacteristic
170C22×D3—38no—
171C22×D3—316no—
172C22×D3—316no—
173C22×D3—68no—
174C22×D3—68no—
175C22×D3—616no—
176C22×D3—616no—
177C22×D3—616no—
178C3×D4—1816no—
179C3×D4—188no—
180C2×C3×C4—24no—
181C2×C3×C4—316no—
182C2×C3×C4—38no—
183C2×C3×C4—316no—
184C2×C3×C4—616no—
185C2×C3×C4—92no—
186C2×C3×C4—182no—
187C2×C3×C4—188no—
188C2×C3×C4C4⋊(C3⋊D3)14yes—
189C2×C3×C4C22×C3⋊D314yes—
190C2×C3×C4C22×C3⋊D312yes—
191C2×Dic3—38no—
192C2×Dic3—68no—
193D12—632no—
194D12—664no—
195C4×D3—632no—
196C2×C32#895116yescomplemented
197C2×C32C22×D1214yes—
198C2×C32C2×D3×D414yes—
199C2×C32C2×#23914yes—
200C3×D3—616no—
201C3×D3—1216no—
202C24—91no—
203C24—92no—
204C2×D4—98no—
205C2×D4—94no—
206C2×D4—188no—
207C2×D4—184no—
208C22×C4—91no—
209C22×C4—94no—
210C22×C4—184no—
211C22×C4C2×#27212yes—
212C22×C4C2×#27211yescharacteristic
213C4⋊C4C2×#27214yescomplemented
214C22⋊C4—98no—
215C22×C3—34no—
216C22×C3—38no—
217C22×C3—38no—
218C22×C3—38no—
219C22×C3—94no—
220C22×C3—182no—
221C22×C3—188no—
222C22×C3—182no—
223C22×C3—184no—
224C22×C3—184no—
225C22×C3—184no—
226C22×C3—368no—
227C22×C3C2×C4⋊(C3⋊D3)12yes—
228C22×C3D4×C3⋊D312yes—
229C22×C3#383212yes—
230C22×C3C23×C3⋊D311yescharacteristic
231C2×D3—316no—
232C2×D3—68no—
233C2×D3—632no—
234C2×D3—68no—
235C2×D3—616no—
236C2×D3—616no—
237C2×D3—616no—
238C2×D3—1232no—
239C3×C4—24no—
240C3×C4—316no—
241C3×C4—616no—
242C3×C4—182no—
243C3×C4C2×C4⋊(C3⋊D3)14yes—
244Dic3—68no—
245C32C2×#89514yescomplemented
246C23—92no—
247C23—94no—
248C23—94no—
249C23—182no—
250C23—182no—
251C23—184no—
252C23—184no—
253C23—184no—
254C23C22×#27211yescharacteristic
255D4—1816no—
256D4—188no—
257C2×C4—24no—
258C2×C4—92no—
259C2×C4—182no—
260C2×C4—188no—
261C2×C4C4⋊#27214yescomplemented
262C2×C4C22×#27214yes—
263C2×C4C22×#27212yes—
264C2×C3—316no—
265C2×C3—34no—
266C2×C3—34no—
267C2×C3—34no—
268C2×C3—184no—
269C2×C3—364no—
270C2×C3#6410414yes—
271C2×C3C22×C4⋊(C3⋊D3)11yescharacteristic
272C2×C3C2×D4×C3⋊D311yescharacteristic
273C2×C3C2×#383211yescharacteristic
274D3—616no—
275D3—1216no—
276C22—94no—
277C22—188no—
278C22—182no—
279C22—182no—
280C22—184no—
281C22—184no—
282C22—184no—
283C22—368no—
284C22C2×C4⋊#27212yes—
285C22D4×#27212yes—
286C22#8874812yes—
287C22C23×#27211yescharacteristic
288C4—24no—
289C4—182no—
290C4C2×C4⋊#27214yescomplemented
291C3—34no—
292C3C2×#6410411yescharacteristic
293C2—184no—
294C2—364no—
295C2#1172176314yesdirect product factor
296C2C22×C4⋊#27211yescharacteristic
297C2C2×D4×#27211yescharacteristic
298C2C2×#8874811yescharacteristic
subgroupG/H
center, socleC23×C3C22×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FrattiniC22×C3C23×C3⋊D3
FittingC2×C4⋊C4×H(3,𝔽3)C2
hypercenterC2×C3×C4⋊C4C3⋊D3
hypocenterH(3,𝔽3)C2×#121
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊#272C2semidirect
2. C22×C4⋊#272C2semidirect
3. C22×C4×#272C2semidirect
4. C2×C4⋊C4×H(3,𝔽3)C2semidirect
5. C2×#88542C2semidirect
6. #11721763C2direct (N × H)
15. C2×C4⋊#272C22semidirect
16. C2×C4⋊#272C22semidirect
18. C2×C4×#272C22semidirect
19. C4⋊C4×H(3,𝔽3)C22semidirect
20. #88542C22semidirect
24. C2×C32×C4⋊C4D3semidirect
33. C22×#272D4semidirect
37. C2×C4×H(3,𝔽3)D4semidirect
58. C32×C4⋊C4C2×D3semidirect
71. C2×#272C2×D4semidirect
73. C4×H(3,𝔽3)C2×D4semidirect
97. C2×C32×C4D12semidirect
106. C2×H(3,𝔽3)#121semidirect
151. C32×C4C2×D12semidirect
157. C2×C4⋊C4#272semidirect
159. H(3,𝔽3)C2×#121semidirect
196. C2×C32#895semidirect
213. C4⋊C4C2×#272semidirect
245. C32C2×#895semidirect
261. C2×C4C4⋊#272semidirect
290. C4C2×C4⋊#272semidirect
295. C2#11721763direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊#2722
C22×C4×#2722
C2×C4⋊C4×H(3,𝔽3)2
C2×#885422
#117217632
C2×C3×#8953