Finite Group Databasebeta

← C4×C22⋊#1088 #13147396

Subgroups — C4×C22⋊#1088 #13147396

Order 1728.

859 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 859 conjugacy classes of subgroups — too many to draw. Only its 99 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
2C2×C22⋊#1088C212yesmaximal
3C2×C4×#1088C212yesmaximal, complemented
4#11720787C212yesmaximal, complemented
5C23×#272C2211yescharacteristic
6C22×#1088C2214yes—
7C22×C4×H(3,𝔽3)C411yescharacteristic, complemented
8C22×#1084C411yescharacteristic, complemented
9C2×C4×#272—24no—
10C2×C4×#272C411yescharacteristic, complemented
11C2×C4×#272C411yescharacteristic, complemented
12C2×C4×#272C2211yescharacteristic
13C2×C4×#272C2211yescharacteristic
14C22⋊#1088C418yesdirect product factor
15C4×#1088—24no—
16C22×C3×C4×D3—62no—
17C23×H(3,𝔽3)C2×C411yescharacteristic
18C22×#272—22no—
19C22×#272—22no—
20C22×#272C2×C411yescharacteristic
21C22×#272C2×C414yes—
22C22×#272C2311yescharacteristic
23C2×#1088—24no—
24C2×#1088—24no—
25C2×#1088C2×C418yescomplemented
26C2×C4×H(3,𝔽3)—22no—
27C2×C4×H(3,𝔽3)C2×C411yescharacteristic
28C2×C4×H(3,𝔽3)C2×C411yescharacteristic
29C2×#1084—22no—
30C2×#1084C2×C411yescharacteristic
31C2×#1084C2×C411yescharacteristic
32C4×#272—22no—
33C4×#272—22no—
34C4×#272—22no—
35C4×#272D414yes—
36C3×C4×C22⋊C4—91nomaximal
37C23×C3×D3—62no—
38C22×C32×C4—22no—
39C22×C3×Dic3—62no—
40C2×C3×C4×D3—62no—
41C2×C3×C4×D3—62no—
42C2×C3×C4×D3—62no—
43C2×C3×C4×D3—62no—
44C2×C3×C4×D3—616no—
45C22×H(3,𝔽3)—21no—
46C22×H(3,𝔽3)—21no—
47C22×H(3,𝔽3)C4212yescomplemented
48C22×H(3,𝔽3)C22×C411yescharacteristic
49C2×#272—21no—
50C2×#272—21no—
51C2×#272—28no—
52C2×#272C4212yescomplemented
53C2×#272C22×C412yes—
54C2×#272C2×D412yes—
55C2×#272#4112yes—
56#1088—28no—
57C4×H(3,𝔽3)—21no—
58C4×H(3,𝔽3)C22⋊C412yescomplemented
59#1084—21no—
60#1084C22⋊C412yescomplemented
61C23×C3×C4—91no—
62C22×C4×D3—62no—
63C2×C3×C22⋊C4—92no—
64C2×C3×C42—92no—
65C3×#95—92no—
66C23×C32—22no—
67C22×C3×D3—68no—
68C22×C3×D3—62no—
69C22×C3×D3—62no—
70C22×C3×D3—68no—
71C22×C3×D3—68no—
72C2×C32×C4—28no—
73C2×C32×C4—22no—
74C2×C32×C4—22no—
75C2×C3×Dic3—62no—
76C2×C3×Dic3—62no—
77C2×C3×Dic3—68no—
78C3×C4×D3—68no—
79C3×C4×D3—68no—
80C3×C4×D3—68no—
81C3×C4×D3—68no—
82C4×C22⋊C4—91noSylow
83C2×H(3,𝔽3)—22no—
84C2×H(3,𝔽3)C2×C4211yescharacteristic
85C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
86C2×H(3,𝔽3)#11711yescharacteristic
87#272—22no—
88#272C4×D414yes—
89C24×C3—91no—
90C23×D3—62no—
91C22×C3×C4—62no—
92C22×C3×C4—91no—
93C22×C3×C4—94no—
94C22×C3×C4—91no—
95C22×C3×C4—91no—
96C22×C3×C4—91no—
97C22×C3×C4—91no—
98C22×C3×C4—184no—
99C22×C3×C4#15711yescharacteristic
100C22×Dic3—62no—
101C2×C4×D3—62no—
102C2×C4×D3—62no—
103C2×C4×D3—62no—
104C2×C4×D3—62no—
105C2×C4×D3—616no—
106C3×C22⋊C4—98no—
107C3×C42—184no—
108C22×C32—24no—
109C22×C32—24no—
110C22×C32—22no—
111C22×C32—24no—
112C2×C3×D3—64no—
113C2×C3×D3—64no—
114C2×C3×D3—64no—
115C2×C3×D3—64no—
116C2×C3×D3—64no—
117C2×C3×D3—632no—
118C2×C3×D3—64no—
119C32×C4—24no—
120C32×C4—24no—
121C3×Dic3—64no—
122C3×Dic3—64no—
123C23×C4—91no—
124C2×C22⋊C4—92no—
125C2×C42—92no—
126#95—92no—
127H(3,𝔽3)C4×C22⋊C411yescharacteristic, complemented, Sylow
128C23×C3—62no—
129C23×C3—91no—
130C23×C3—91no—
131C23×C3—94no—
132C23×C3—182no—
133C23×C3—182no—
134C23×C3C2×#15711yescharacteristic
135C22×D3—62no—
136C22×D3—62no—
137C22×D3—68no—
138C22×D3—68no—
139C22×D3—68no—
140C2×C3×C4—22no—
141C2×C3×C4—68no—
142C2×C3×C4—62no—
143C2×C3×C4—62no—
144C2×C3×C4—91no—
145C2×C3×C4—94no—
146C2×C3×C4—98no—
147C2×C3×C4—91no—
148C2×C3×C4—182no—
149C2×C3×C4—182no—
150C2×C3×C4—182no—
151C2×C3×C4—184no—
152C2×C3×C4—184no—
153C2×C3×C4—182no—
154C2×C3×C4C2×#15711yescharacteristic
155C2×C3×C4C2×#15711yescharacteristic
156C2×Dic3—62no—
157C2×Dic3—62no—
158C2×Dic3—68no—
159C4×D3—68no—
160C4×D3—68no—
161C4×D3—68no—
162C4×D3—68no—
163C2×C32—28no—
164C2×C32—22no—
165C2×C32—22no—
166C2×C32—22no—
167C3×D3—68no—
168C3×D3—68no—
169C24—91no—
170C22×C4—91no—
171C22×C4—94no—
172C22×C4—91no—
173C22×C4—91no—
174C22×C4—91no—
175C22×C4—91no—
176C22×C4—184no—
177C22×C4#108811yescharacteristic, complemented
178C22⋊C4—98no—
179C42—184no—
180C22×C3—21no—
181C22×C3—21no—
182C22×C3—64no—
183C22×C3—64no—
184C22×C3—64no—
185C22×C3—62no—
186C22×C3—92no—
187C22×C3—92no—
188C22×C3—92no—
189C22×C3—92no—
190C22×C3—188no—
191C22×C3—181no—
192C22×C3—181no—
193C22×C3C4×#15712yes—
194C22×C3C22×#15711yescharacteristic
195C2×D3—64no—
196C2×D3—64no—
197C2×D3—64no—
198C2×D3—64no—
199C2×D3—64no—
200C2×D3—632no—
201C2×D3—64no—
202C3×C4—21no—
203C3×C4—64no—
204C3×C4—64no—
205C3×C4—92no—
206C3×C4—188no—
207C3×C4—181no—
208C3×C4C22⋊#15712yes—
209Dic3—64no—
210Dic3—64no—
211C32—22no—
212C23—91no—
213C23—94no—
214C23—91no—
215C23—182no—
216C23—182no—
217C23C2×#108811yescharacteristic
218C2×C4—22no—
219C2×C4—91no—
220C2×C4—94no—
221C2×C4—98no—
222C2×C4—91no—
223C2×C4—182no—
224C2×C4—182no—
225C2×C4—182no—
226C2×C4—184no—
227C2×C4—184no—
228C2×C4—182no—
229C2×C4C2×#108811yescharacteristic
230C2×C4C2×#108811yescharacteristic
231C2×C3—22no—
232C2×C3—68no—
233C2×C3—62no—
234C2×C3—62no—
235C2×C3—62no—
236C2×C3—94no—
237C2×C3—182no—
238C2×C3C2×C4×#15711yescharacteristic
239C2×C3#6428611yescharacteristic
240C2×C3C2×C22⋊#15711yescharacteristic
241D3—68no—
242D3—68no—
243C22—21no—
244C22—21no—
245C22—92no—
246C22—92no—
247C22—92no—
248C22—92no—
249C22—188no—
250C22—181no—
251C22—181no—
252C22C4×#108812yescomplemented
253C22C22×#108811yescharacteristic
254C4—21no—
255C4—92no—
256C4—188no—
257C4—181no—
258C4C22⋊#108812yesdirect product factor
259C3—62no—
260C3C4×C22⋊#15711yescharacteristic
261C2—22no—
262C2—94no—
263C2—182no—
264C2C2×C4×#108811yescharacteristic
265C2#1172244811yescharacteristic
266C2C2×C22⋊#108811yescharacteristic
subgroupG/H
centerC2×C3×C4C2×#157
commutatorC2×H(3,𝔽3)C2×C42
Frattini, socleC22×C3C22×#157
FittingC22×C4×H(3,𝔽3)C4
prime radicalC23×#272C22
cosocleC22×#272C23
hypercenterC22×C3×C4#157
hypocenterH(3,𝔽3)C4×C22⋊C4
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
3. C2×C4×#1088C2semidirect
4. #11720787C2semidirect
7. C22×C4×H(3,𝔽3)C4semidirect
8. C22×#1084C4semidirect
10. C2×C4×#272C4semidirect
11. C2×C4×#272C4semidirect
14. C22⋊#1088C4direct (N × H)
25. C2×#1088C2×C4semidirect
47. C22×H(3,𝔽3)C42semidirect
52. C2×#272C42semidirect
58. C4×H(3,𝔽3)C22⋊C4semidirect
60. #1084C22⋊C4semidirect
127. H(3,𝔽3)C4×C22⋊C4semidirect
177. C22×C4#1088semidirect
252. C22C4×#1088semidirect
258. C4C22⋊#1088direct (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
C2×C22⋊#10882
C2×C4×#10882
#117207872
C3×C4×C22⋊C49