Finite Group Databasebeta

← C2×#11720792 #13147429

Subgroups — C2×#11720792 #13147429

Order 1728.

787 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 787 conjugacy classes of subgroups — too many to draw. Only its 63 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×#6235C211yesmaximal, characteristic
2C2×C22⋊#1088C212yesmaximal, complemented
3#11720792C214yesmaximal, direct product factor
4C24×H(3,𝔽3)C411yescharacteristic, complemented
5C23×#272C2211yescharacteristic
6C22×#1088—22no—
7C2×#6235—22no—
8C2×#6235—22no—
9C2×#6235C412yescomplemented
10C2×#6235C2212yes—
11C22×#1084C411yescharacteristic, complemented
12C22⋊#1088—22no—
13C22⋊#1088C2214yescomplemented
14C22×C3×#67—62no—
15C23×H(3,𝔽3)—21no—
16C23×H(3,𝔽3)—21no—
17C23×H(3,𝔽3)—42no—
18C23×H(3,𝔽3)C2×C411yescharacteristic
19C23×H(3,𝔽3)C2×C412yescomplemented
20C22×#272—42no—
21C22×#272C2×C411yescharacteristic
22C22×#272D412yes—
23C22×#272D412yes—
24C22×#272C2311yescharacteristic
25C2×#1088—24no—
26C2×#1088—44no—
27#6235—24no—
28#6235—24no—
29C2×#1084—21no—
30C2×#1084—21no—
31C2×#1084C2×C412yescomplemented
32C2×C3×#99—91nomaximal
33C24×C32—22no—
34C23×C3×D3—62no—
35C2×C3×#67—64no—
36C2×C3×#67—64no—
37C2×C3×#67—68no—
38C2×C3×#67—68no—
39C22×C3×Dic3—62no—
40C22×H(3,𝔽3)—22no—
41C22×H(3,𝔽3)—22no—
42C22×H(3,𝔽3)—42no—
43C22×H(3,𝔽3)—42no—
44C22×H(3,𝔽3)—42no—
45C22×H(3,𝔽3)C22⋊C411yescharacteristic
46C22×H(3,𝔽3)C22⋊C411yescharacteristic
47C22×H(3,𝔽3)C22×C411yescharacteristic
48C2×#272—24no—
49C2×#272—41no—
50C2×#272—42no—
51C2×#272—41no—
52C2×#272C22⋊C412yes—
53C2×#272C2×D412yes—
54#1088—44no—
55#1084—22no—
56C22×C3×D4—91no—
57C22×#67—62no—
58C2×C3×C22⋊C4—92no—
59C3×#99—94no—
60C23×C32—24no—
61C23×C32—24no—
62C23×C32—24no—
63C23×C32—22no—
64C23×C32—48no—
65C22×C3×D3—62no—
66C22×C3×D3—64no—
67C22×C3×D3—64no—
68C22×C3×D3—62no—
69C22×C3×D3—128no—
70C3×#67—616no—
71C3×#67—616no—
72C2×C3×Dic3—64no—
73C2×C3×Dic3—64no—
74C2×C3×Dic3—64no—
75C2×#99—91noSylow
76C2×H(3,𝔽3)—21no—
77C2×H(3,𝔽3)—21no—
78C2×H(3,𝔽3)—42no—
79C2×H(3,𝔽3)#9912yescomplemented
80C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
81#272—22no—
82#272—22no—
83C24×C3—62no—
84C24×C3—91no—
85C24×C3#15711yescharacteristic
86C23×D3—62no—
87C2×C3×D4—92no—
88C2×C3×D4—92no—
89C2×C3×D4—182no—
90C2×C3×D4—182no—
91C22×C3×C4—91no—
92C22×C3×C4—182no—
93C2×#67—64no—
94C2×#67—64no—
95C2×#67—68no—
96C2×#67—68no—
97C22×Dic3—62no—
98C3×C22⋊C4—94no—
99C3×C22⋊C4—182no—
100C22×C32—28no—
101C22×C32—28no—
102C22×C32—22no—
103C22×C32—22no—
104C22×C32—22no—
105C22×C32—48no—
106C22×C32—48no—
107C22×C32—48no—
108C2×C3×D3—64no—
109C2×C3×D3—64no—
110C2×C3×D3—128no—
111C2×C3×D3—124no—
112C2×C3×D3—128no—
113C2×C3×D3—124no—
114C3×Dic3—68no—
115C22×D4—91no—
116C2×C22⋊C4—92no—
117#99—94no—
118H(3,𝔽3)C2×#9911yescharacteristic, complemented, Sylow
119C23×C3—21no—
120C23×C3—21no—
121C23×C3—42no—
122C23×C3—64no—
123C23×C3—64no—
124C23×C3—62no—
125C23×C3—64no—
126C23×C3—92no—
127C23×C3—91no—
128C23×C3—91no—
129C23×C3—92no—
130C23×C3—128no—
131C23×C3—362no—
132C23×C3C2×#15712yes—
133C23×C3C2×#15711yescharacteristic
134C22×D3—64no—
135C22×D3—64no—
136C22×D3—62no—
137C22×D3—62no—
138C22×D3—128no—
139C3×D4—184no—
140C3×D4—184no—
141C2×C3×C4—92no—
142C2×C3×C4—181no—
143C2×C3×C4—181no—
144C2×C3×C4—184no—
145C2×C3×C4—364no—
146#67—616no—
147#67—616no—
148C2×Dic3—64no—
149C2×Dic3—64no—
150C2×Dic3—64no—
151C2×C32—24no—
152C2×C32—24no—
153C2×C32—22no—
154C2×C32—24no—
155C2×C32—48no—
156C3×D3—124no—
157C3×D3—124no—
158C24—91no—
159C24#108811yescharacteristic, complemented
160C2×D4—92no—
161C2×D4—92no—
162C2×D4—182no—
163C2×D4—182no—
164C22×C4—91no—
165C22×C4—182no—
166C22⋊C4—94no—
167C22⋊C4—182no—
168C22×C3—22no—
169C22×C3—22no—
170C22×C3—42no—
171C22×C3—42no—
172C22×C3—42no—
173C22×C3—68no—
174C22×C3—68no—
175C22×C3—62no—
176C22×C3—62no—
177C22×C3—62no—
178C22×C3—92no—
179C22×C3—92no—
180C22×C3—128no—
181C22×C3—128no—
182C22×C3—128no—
183C22×C3—184no—
184C22×C3—361no—
185C22×C3—361no—
186C22×C3—362no—
187C22×C3C22⋊#15711yescharacteristic
188C22×C3C22⋊#15711yescharacteristic
189C22×C3C22×#15711yescharacteristic
190C2×D3—64no—
191C2×D3—64no—
192C2×D3—128no—
193C2×D3—128no—
194C2×D3—124no—
195C2×D3—124no—
196C3×C4—182no—
197C3×C4—364no—
198Dic3—68no—
199C32—22no—
200C23—21no—
201C23—21no—
202C23—42no—
203C23—92no—
204C23—92no—
205C23—91no—
206C23—91no—
207C23—362no—
208C23C2×#108811yescharacteristic
209C23C2×#108812yescomplemented
210D4—184no—
211D4—184no—
212C2×C4—92no—
213C2×C4—181no—
214C2×C4—181no—
215C2×C4—184no—
216C2×C4—364no—
217C2×C3—21no—
218C2×C3—21no—
219C2×C3—42no—
220C2×C3—64no—
221C2×C3—64no—
222C2×C3—64no—
223C2×C3—62no—
224C2×C3—128no—
225C2×C3—182no—
226C2×C3—182no—
227C2×C3#6378612yes—
228C2×C3C2×C22⋊#15711yescharacteristic
229D3—124no—
230D3—124no—
231C22—22no—
232C22—22no—
233C22—42no—
234C22—42no—
235C22—42no—
236C22—92no—
237C22—92no—
238C22—184no—
239C22—362no—
240C22—361no—
241C22—361no—
242C22C22⋊#108811yescharacteristic
243C22C22⋊#108811yescharacteristic
244C22C22×#108811yescharacteristic
245C4—182no—
246C4—364no—
247C3—62no—
248C3C2×#6378611yescharacteristic
249C2—21no—
250C2—21no—
251C2—42no—
252C2—182no—
253C2—182no—
254C2#1172079212yesdirect product factor
255C2C2×C22⋊#108811yescharacteristic
subgroupG/H
center, socleC22×C3C22⋊#157
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC22×C3C22×#157
FittingC24×H(3,𝔽3)C4
prime radicalC22×#6235C2
cosocleC22×#272C23
hypercenterC24×C3#157
hypocenterH(3,𝔽3)C2×#99
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C23×C3 ◁ C24×C3

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊#1088C2semidirect
3. #11720792C2direct (N × H)
4. C24×H(3,𝔽3)C4semidirect
9. C2×#6235C4semidirect
11. C22×#1084C4semidirect
13. C22⋊#1088C22semidirect
19. C23×H(3,𝔽3)C2×C4semidirect
31. C2×#1084C2×C4semidirect
79. C2×H(3,𝔽3)#99semidirect
118. H(3,𝔽3)C2×#99semidirect
159. C24#1088semidirect
209. C23C2×#1088semidirect
254. C2#11720792direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62352
C2×C22⋊#10882
#117207922
C2×C3×#999