Finite Group Databasebeta

← C2×D4×#1088 #13164673

Subgroups — C2×D4×#1088 #13164673

Order 1728.

1111 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 1111 conjugacy classes of subgroups — too many to draw. Only its 175 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
1C23×#1088C212yesmaximal, complemented
2C2×D4×#272C211yesmaximal, characteristic
3C2×C22⋊#1088C212yesmaximal, complemented
4D4×#1088C218yesmaximal, direct product factor
5C2×C4⋊#1088C211yesmaximal, characteristic, complemented
6C2×C4×#1088C211yesmaximal, characteristic, complemented
7C23×#272C2212yes—
8C22×#1088—28no—
9C22×#1088C2212yes—
10C22×#1088C2211yescharacteristic
11C22×#1088C2211yescharacteristic
12C22×#1088C2218yescomplemented
13C2×D4×H(3,𝔽3)C411yescharacteristic, complemented
14C2×#6235C412yescomplemented
15D4×#272C414yescomplemented
16D4×#272C2214yes—
17C2×C4⋊#272C411yescharacteristic, complemented
18C2×C4×#272C2211yescharacteristic
19C22⋊#1088C2218yescomplemented
20C4⋊#1088C2214yescomplemented
21C4×#1088C2214yescomplemented
22C2×C3×D3×D4—62no—
23C23×H(3,𝔽3)C2×C412yescomplemented
24C22×#272—24no—
25C22×#272—24no—
26C22×#272C2×C414yescomplemented
27C22×#272C2×C412yescomplemented
28C22×#272C2314yes—
29C22×#272C2311yescharacteristic
30C2×#1088—24no—
31C2×#1088—22no—
32C2×#1088—216no—
33C2×#1088D414yesdirect product factor
34C2×#1088C2312yes—
35C2×#1088C2314yes—
36C2×#1088C2312yes—
37D4×H(3,𝔽3)C2×C414yescomplemented
38C2×C4×H(3,𝔽3)C2×C411yescharacteristic, complemented
39#6235C2×C418yescomplemented
40C2×#1084C2×C411yescharacteristic, complemented
41C4⋊#272C2×C414yescomplemented
42C4×#272C2×C412yescomplemented
43C4×#272C2312yes—
44C2×C3×C4×D4—91nomaximal
45C23×C3×D3—64no—
46C2×C32×D4—22no—
47C2×C3×#67—64no—
48C3×D3×D4—68no—
49C3×D3×D4—68no—
50C2×C3×D12—62no—
51C2×C3×C4×D3—62no—
52C22×H(3,𝔽3)—24no—
53C22×H(3,𝔽3)C22×C411yescharacteristic
54C22×H(3,𝔽3)C22×C414yescomplemented
55C2×#272—24no—
56C2×#272—28no—
57C2×#272—24no—
58C2×#272—24no—
59C2×#272C22×C411yescharacteristic
60C2×#272C22×C414yescomplemented
61C2×#272C2×D412yes—
62C2×#272#4112yes—
63C2×#272C2411yescharacteristic
64#1088—24no—
65#1088—22no—
66#1088C2×D414yesdirect product factor
67C4×H(3,𝔽3)C22×C412yescomplemented
68#1084C22×C412yescomplemented
69C22×C3×D4—91no—
70C23×C3×C4—92no—
71C2×D3×D4—62no—
72C3×C4×D4—98no—
73C2×C3×C4⋊C4—91no—
74C2×C3×C22⋊C4—92no—
75C2×C3×C42—91no—
76C23×C32—24no—
77C22×C3×D3—62no—
78C22×C3×D3—64no—
79C22×C3×D3—68no—
80C22×C3×D3—68no—
81C22×C3×D3—128no—
82C22×C3×D3—128no—
83C32×D4—28no—
84C2×C32×C4—22no—
85C3×#67—616no—
86C2×C3×Dic3—62no—
87C3×D12—68no—
88C3×C4×D3—64no—
89C3×C4×D3—64no—
90C2×C4×D4—91noSylow
91C2×H(3,𝔽3)—24no—
92C2×H(3,𝔽3)C4×D412yescomplemented
93C2×H(3,𝔽3)C23×C411yescharacteristic
94#272—24no—
95#272C4×D412yescomplemented
96#272C22×D411yescharacteristic
97#272C2×#4111yescharacteristic
98C24×C3—92no—
99C23×D3—64no—
100C2×C3×D4—62no—
101C2×C3×D4—92no—
102C2×C3×D4—94no—
103C2×C3×D4—94no—
104C2×C3×D4—91no—
105C2×C3×D4#15711yescharacteristic
106C22×C3×C4—91no—
107C22×C3×C4—91no—
108C22×C3×C4—91no—
109C22×C3×C4—92no—
110C22×C3×C4—98no—
111C22×C3×C4—188no—
112C2×#67—64no—
113D3×D4—68no—
114D3×D4—68no—
115C2×D12—62no—
116C2×C4×D3—62no—
117C3×C4⋊C4—94no—
118C3×C22⋊C4—98no—
119C3×C42—94no—
120C22×C32—28no—
121C22×C32—22no—
122C22×C32—48no—
123C2×C3×D3—62no—
124C2×C3×D3—64no—
125C2×C3×D3—64no—
126C2×C3×D3—62no—
127C2×C3×D3—68no—
128C2×C3×D3—128no—
129C2×C3×D3—1216no—
130C2×C3×D3—128no—
131C2×C3×D3—128no—
132C32×C4—24no—
133C3×Dic3—64no—
134C22×D4—91no—
135C23×C4—92no—
136C4×D4—98no—
137C2×C4⋊C4—91no—
138C2×C22⋊C4—92no—
139C2×C42—91no—
140H(3,𝔽3)C2×C4×D411yescharacteristic, complemented, Sylow
141C23×C3—64no—
142C23×C3—91no—
143C23×C3—94no—
144C23×C3—94no—
145C23×C3—92no—
146C23×C3—184no—
147C23×C3—184no—
148C23×C3C2×#15712yes—
149C22×D3—62no—
150C22×D3—64no—
151C22×D3—68no—
152C22×D3—68no—
153C22×D3—128no—
154C22×D3—128no—
155C3×D4—68no—
156C3×D4—98no—
157C3×D4—94no—
158C3×D4C2×#15714yes—
159C2×C3×C4—62no—
160C2×C3×C4—94no—
161C2×C3×C4—92no—
162C2×C3×C4—91no—
163C2×C3×C4—92no—
164C2×C3×C4—92no—
165C2×C3×C4—92no—
166C2×C3×C4—94no—
167C2×C3×C4—1816no—
168C2×C3×C4—182no—
169C2×C3×C4—184no—
170C2×C3×C4C2×#15711yescharacteristic
171#67—616no—
172C2×Dic3—62no—
173D12—68no—
174C4×D3—64no—
175C4×D3—64no—
176C2×C32—24no—
177C2×C32—22no—
178C2×C32—48no—
179C3×D3—64no—
180C3×D3—62no—
181C3×D3—62no—
182C3×D3—128no—
183C24—92no—
184C2×D4—92no—
185C2×D4—94no—
186C2×D4—94no—
187C2×D4—91no—
188C2×D4#108811yescharacteristic, direct product factor
189C22×C4—91no—
190C22×C4—91no—
191C22×C4—91no—
192C22×C4—92no—
193C22×C4—98no—
194C22×C4—188no—
195C4⋊C4—94no—
196C22⋊C4—98no—
197C42—94no—
198C22×C3—24no—
199C22×C3—62no—
200C22×C3—68no—
201C22×C3—92no—
202C22×C3—92no—
203C22×C3—91no—
204C22×C3—91no—
205C22×C3—94no—
206C22×C3—128no—
207C22×C3—184no—
208C22×C3—184no—
209C22×C3—188no—
210C22×C3—184no—
211C22×C3C22×#15714yes—
212C22×C3C22×#15711yescharacteristic
213C2×D3—62no—
214C2×D3—62no—
215C2×D3—64no—
216C2×D3—64no—
217C2×D3—68no—
218C2×D3—128no—
219C2×D3—128no—
220C2×D3—1216no—
221C2×D3—128no—
222C3×C4—64no—
223C3×C4—94no—
224C3×C4—92no—
225C3×C4—182no—
226C3×C4—184no—
227C3×C4C22×#15712yes—
228Dic3—64no—
229C32—22no—
230C23—91no—
231C23—92no—
232C23—94no—
233C23—94no—
234C23—184no—
235C23—184no—
236C23C2×#108812yescomplemented
237D4—98no—
238D4—94no—
239D4C2×#108814yesdirect product factor
240C2×C4—92no—
241C2×C4—94no—
242C2×C4—91no—
243C2×C4—92no—
244C2×C4—92no—
245C2×C4—92no—
246C2×C4—94no—
247C2×C4—1816no—
248C2×C4—184no—
249C2×C4—182no—
250C2×C4C2×#108811yescharacteristic, complemented
251C2×C3—24no—
252C2×C3—62no—
253C2×C3—64no—
254C2×C3—91no—
255C2×C3—91no—
256C2×C3—92no—
257C2×C3—128no—
258C2×C3—184no—
259C2×C3D4×#15712yes—
260C2×C3C23×#15711yescharacteristic
261D3—62no—
262D3—62no—
263D3—64no—
264D3—128no—
265C22—24no—
266C22—91no—
267C22—91no—
268C22—92no—
269C22—92no—
270C22—94no—
271C22—184no—
272C22—184no—
273C22—188no—
274C22—184no—
275C22C22×#108811yescharacteristic
276C22C22×#108814yescomplemented
277C4—94no—
278C4—92no—
279C4—184no—
280C4—182no—
281C4C22×#108812yescomplemented
282C3—62no—
283C3C2×D4×#15711yescharacteristic
284C2—24no—
285C2—91no—
286C2—91no—
287C2—92no—
288C2—184no—
289C2D4×#108812yesdirect product factor
290C2C23×#108811yescharacteristic
subgroupG/H
center, socleC22×C3C22×#157
commutatorC2×H(3,𝔽3)C23×C4
FrattiniC2×C3C23×#157
FittingC2×D4×H(3,𝔽3)C4
prime radicalC2×D4×#272C2
cosocleC2×#272C24
hypercenterC2×C3×D4#157
hypocenterH(3,𝔽3)C2×C4×D4
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C23×#1088C2semidirect
3. C2×C22⋊#1088C2semidirect
4. D4×#1088C2direct (N × H)
5. C2×C4⋊#1088C2semidirect
6. C2×C4×#1088C2semidirect
12. C22×#1088C22semidirect
13. C2×D4×H(3,𝔽3)C4semidirect
14. C2×#6235C4semidirect
15. D4×#272C4semidirect
17. C2×C4⋊#272C4semidirect
19. C22⋊#1088C22semidirect
20. C4⋊#1088C22semidirect
21. C4×#1088C22semidirect
23. C23×H(3,𝔽3)C2×C4semidirect
26. C22×#272C2×C4semidirect
27. C22×#272C2×C4semidirect
33. C2×#1088D4direct (N × H)
37. D4×H(3,𝔽3)C2×C4semidirect
38. C2×C4×H(3,𝔽3)C2×C4semidirect
39. #6235C2×C4semidirect
40. C2×#1084C2×C4semidirect
41. C4⋊#272C2×C4semidirect
42. C4×#272C2×C4semidirect
54. C22×H(3,𝔽3)C22×C4semidirect
60. C2×#272C22×C4semidirect
66. #1088C2×D4direct (N × H)
67. C4×H(3,𝔽3)C22×C4semidirect
68. #1084C22×C4semidirect
92. C2×H(3,𝔽3)C4×D4semidirect
95. #272C4×D4semidirect
140. H(3,𝔽3)C2×C4×D4semidirect
188. C2×D4#1088direct (N × H)
236. C23C2×#1088semidirect
239. D4C2×#1088direct (N × H)
250. C2×C4C2×#1088semidirect
276. C22C22×#1088semidirect
281. C4C22×#1088semidirect
289. C2D4×#1088direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#10882
C2×D4×#2722
C2×C22⋊#10882
D4×#10882
C2×C4⋊#10882
C2×C4×#10882
C2×C3×C4×D49