Finite Group Databasebeta

← C4×#88941 #13167487

Subgroups — C4×#88941 #13167487

Order 1728.

564 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C2×#88941C211yesmaximal, characteristic
2C4×#6295C212yesmaximal, complemented
3C3×C4×#622C211yesmaximal, characteristic, complemented
4#11722797C211yesmaximal, characteristic, complemented
5#11722793C212yesmaximal, complemented
6C4×#3774—31nomaximal
7C2×C3×C4×C3⋊D3—22no—
8C2×#6295C2212yes—
9C2×C3×#622C2211yescharacteristic
10C2×C3×#622C2211yescharacteristic
11#88941C414yesdirect product factor
12C3×C4×C3⋊Dic3C2211yescharacteristic
13#88815C2212yes—
14#88814—22no—
15C2×#3774—31no—
16C4×#626—62no—
17C4×#625—32no—
18C4×#622D311yescharacteristic, complemented
19#63743—31no—
20#63739—32no—
21C22×C3×C3⋊D3—22no—
22C2×C33×C4D411yescharacteristic
23C2×C3×C3⋊Dic3—22no—
24C2×C3×C3⋊Dic3D411yescharacteristic
25C2×C3×C3⋊Dic3C2311yescharacteristic
26C3×C4×C3⋊D3—44no—
27#6295—22no—
28#6295C2×C414yescomplemented
29C3×#622—21no—
30C3×#622C2×C412yescomplemented
31C4×D24—91nomaximal
32C2×C4×C3⋊D3—22no—
33C2×C3×C4×D3—62no—
34C2×C3×C4×D3—62no—
35C2×C3×C4×D3—122no—
36C2×#626—62no—
37C2×#625—32no—
38C2×#622C2×D311yescharacteristic
39C2×#622C2×D311yescharacteristic
40#3774—34no—
41C4×C3⋊Dic3C2×D311yescharacteristic
42C3×C4×Dic3—62no—
43#3724—32no—
44Dic3⋊Dic3—62no—
45#3722—62no—
46(C2×D3)⋊Dic3—62no—
47(C2×D3)⋊Dic3—62no—
48C22×C33C2×D411yescharacteristic
49C2×C3×C3⋊D3—24no—
50C2×C3×C3⋊D3—42no—
51C2×C3×C3⋊D3—42no—
52C33×C4D812yescomplemented
53C3×C3⋊Dic3—21no—
54C3×C3⋊Dic3—42no—
55C3×C3⋊Dic3C22×C411yescharacteristic
56C3×C3⋊Dic3#4111yescharacteristic
57C4×#67—182no—
58C2×D24—91no—
59C4×D12—92no—
60C3×C4×C8—91no—
61#821—92no—
62#818—91no—
63C22×C3⋊D3—22no—
64C22×C3×D3—62no—
65C22×C3×D3—62no—
66C22×C3×D3—122no—
67C2×C32×C4—22no—
68C2×C32×C4—42no—
69C2×C32×C4D1211yescharacteristic
70C2×C3⋊Dic3—22no—
71C2×C3⋊Dic3D1211yescharacteristic
72C2×C3⋊Dic3C22×D311yescharacteristic
73C4×C3⋊D3—44no—
74C2×C3×Dic3—62no—
75C2×C3×Dic3—62no—
76C2×C3×Dic3—62no—
77C2×C3×Dic3—62no—
78C2×C3×Dic3—122no—
79C3×C4×D3—124no—
80C3×C4×D3—128no—
81C3×C4×D3—124no—
82#626—64no—
83#626—64no—
84#625—34no—
85#625—62no—
86#622—21no—
87#622C4×D312yescomplemented
88C4×D8—271noSylow
89C2×C33C4×D411yescharacteristic
90C2×C33C2×D811yescharacteristic
91C2×C33#13511yescharacteristic
92C3×C3⋊D3—44no—
93C22×C3×C4—182no—
94C2×#67—182no—
95C2×D12—92no—
96C2×C4×D3—62no—
97C2×C4×D3—62no—
98C2×C4×D3—122no—
99C2×C3×C8—91no—
100C2×C3×C8—91no—
101C3×C42—91no—
102#217—182no—
103(C2×D3)⋊C4—182no—
104(C2×D3)⋊C4—182no—
105C4⋊Dic3—92no—
106Dic3⋊C4—182no—
107C4×Dic3—62no—
108D24—94no—
109C22×C32—22no—
110C22×C32—42no—
111C22×C32C2×D1211yescharacteristic
112C2×C3⋊D3—24no—
113C2×C3⋊D3—42no—
114C2×C3⋊D3—42no—
115C2×C3×D3—64no—
116C2×C3×D3—64no—
117C2×C3×D3—124no—
118C2×C3×D3—122no—
119C2×C3×D3—122no—
120C2×C3×D3—124no—
121C2×C3×D3—124no—
122C2×C3×D3—122no—
123C2×C3×D3—122no—
124C32×C4—24no—
125C32×C4—44no—
126C32×C4D2412yescomplemented
127C3⋊Dic3—21no—
128C3⋊Dic3—42no—
129C3⋊Dic3C2×C4×D311yescharacteristic
130C3⋊Dic3#23511yescharacteristic
131C3×Dic3—62no—
132C3×Dic3—62no—
133C3×Dic3—122no—
134C3×Dic3—122no—
135C3×Dic3—124no—
136C3×Dic3—122no—
137C2×D8—271no—
138C4×D4—272no—
139#107—271no—
140#102—272no—
141C4×C8—91no—
142C33C4×D811yescharacteristic, complemented, Sylow
143C23×C3—182no—
144C22×D3—62no—
145C22×D3—62no—
146C22×D3—122no—
147C2×C3×C4—22no—
148C2×C3×C4—42no—
149C2×C3×C4—91no—
150C2×C3×C4—91no—
151C2×C3×C4—182no—
152C2×C3×C4—364no—
153C2×C3×C4GO+(4,𝔽2)11yescharacteristic
154#67—184no—
155#67—184no—
156C2×Dic3—62no—
157C2×Dic3—62no—
158C2×Dic3—62no—
159C2×Dic3—62no—
160C2×Dic3—122no—
161D12—94no—
162D12—182no—
163C4×D3—124no—
164C4×D3—128no—
165C4×D3—124no—
166C3×C8—92no—
167C3×C8—181no—
168C2×C32—22no—
169C2×C32—22no—
170C2×C32—22no—
171C2×C32—42no—
172C2×C32—42no—
173C2×C32—42no—
174C2×C32C4×D1211yescharacteristic
175C2×C32C2×D2411yescharacteristic
176C2×C32#90411yescharacteristic
177C3⋊D3—44no—
178C3×D3—124no—
179C3×D3—128no—
180C3×D3—124no—
181C2×D4—272no—
182C22×C4—182no—
183D8—274no—
184C2×C8—91no—
185C2×C8—91no—
186C4⋊C4—272no—
187C22⋊C4—542no—
188C42—91no—
189C22×C3—22no—
190C22×C3—42no—
191C22×C3—184no—
192C22×C3—362no—
193C22×C3—362no—
194C22×C3C2×GO+(4,𝔽2)11yescharacteristic
195C2×D3—64no—
196C2×D3—64no—
197C2×D3—124no—
198C2×D3—122no—
199C2×D3—122no—
200C2×D3—124no—
201C2×D3—124no—
202C2×D3—122no—
203C2×D3—122no—
204C3×C4—24no—
205C3×C4—44no—
206C3×C4—91no—
207C3×C4—91no—
208C3×C4—181no—
209C3×C4—362no—
210C3×C4#377412yescomplemented
211Dic3—62no—
212Dic3—62no—
213Dic3—122no—
214Dic3—122no—
215Dic3—124no—
216Dic3—122no—
217C32—22no—
218C32—42no—
219C32C4×D2411yescharacteristic, complemented
220C23—182no—
221D4—274no—
222D4—542no—
223C2×C4—91no—
224C2×C4—91no—
225C2×C4—182no—
226C2×C4—364no—
227C2×C4#157⋊D311yescharacteristic
228C8—92no—
229C8—181no—
230C2×C3—22no—
231C2×C3—22no—
232C2×C3—22no—
233C2×C3—42no—
234C2×C3—42no—
235C2×C3—42no—
236C2×C3—364no—
237C2×C3#6422311yescharacteristic
238C2×C3C4×GO+(4,𝔽2)11yescharacteristic
239C2×C3C2×#377411yescharacteristic
240D3—124no—
241D3—128no—
242D3—124no—
243C22—184no—
244C22—362no—
245C22—362no—
246C22C2×#157⋊D311yescharacteristic
247C4—91no—
248C4—91no—
249C4—181no—
250C4—362no—
251C4#8894112yesdirect product factor
252C3—22no—
253C3—42no—
254C3C4×#377411yescharacteristic, complemented
255C2—364no—
256C2#1172433711yescharacteristic
257C2C4×#157⋊D311yescharacteristic
258C2C2×#8894111yescharacteristic
subgroupG/H
center, hypercenterC2×C4#157⋊D3
commutatorC3×C3⋊Dic3C22×C4
FrattiniC22C2×#157⋊D3
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×#88941C2
cosocleC2×C3×C3⋊Dic3C23
hypocenterC33C4×D8
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C4×#6295C2semidirect
3. C3×C4×#622C2semidirect
4. #11722797C2semidirect
5. #11722793C2semidirect
11. #88941C4direct (N × H)
18. C4×#622D3semidirect
28. #6295C2×C4semidirect
30. C3×#622C2×C4semidirect
52. C33×C4D8semidirect
87. #622C4×D3semidirect
126. C32×C4D24semidirect
142. C33C4×D8semidirect
210. C3×C4#3774semidirect
219. C32C4×D24semidirect
251. C4#88941direct (N × H)
254. C3C4×#3774semidirect

Maximal subgroups

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

subgroupindex
C2×#889412
C4×#62952
C3×C4×#6222
#117227972
#117227932
C4×#37743
C4×D249