Finite Group Databasebeta

← C2×#11720794 #13147432

Subgroups — C2×#11720794 #13147432

Order 1728.

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

normal subgroups only

This group has 751 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×#88631C212yesmaximal, complemented
3#11720794C214yesmaximal, direct product factor
4C24×H(3,𝔽3)C411yescharacteristic
5C23×#272C411yescharacteristic
6C2×#6235—22no—
7C2×#6235—22no—
8C2×#6235C412yes—
9C2×#6235C2212yes—
10C22×#1084C2211yescharacteristic
11#88631—22no—
12#88631C2214yescomplemented
13C2×#6188—22no—
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×C412yes—
20C22×#272—21no—
21C22×#272—21no—
22C22×#272—42no—
23C22×#272C2×C412yes—
24#6235—24no—
25#6235—24no—
26C2×#1084C2×C411yescharacteristic
27C2×#1084D412yes—
28C2×#1084D412yes—
29C2×#1084C2311yescharacteristic
30#6188—24no—
31C2×C3×#100—91nomaximal
32C24×C32—22no—
33C23×C3×D3—62no—
34C2×C3×#67—64no—
35C2×C3×#67—64no—
36C2×C3×#67—68no—
37C2×C3×#67—68no—
38C22×C3×Dic3—62no—
39C22×H(3,𝔽3)—22no—
40C22×H(3,𝔽3)—22no—
41C22×H(3,𝔽3)—42no—
42C22×H(3,𝔽3)—42no—
43C22×H(3,𝔽3)—42no—
44C22×H(3,𝔽3)C22⋊C411yescharacteristic
45C22×H(3,𝔽3)C22⋊C411yescharacteristic
46C22×H(3,𝔽3)C22×C411yescharacteristic
47C2×#272—22no—
48C2×#272—42no—
49C2×#272—42no—
50C2×#272—42no—
51#1084C22⋊C412yes—
52#1084C2×D412yes—
53C22×C3×D4—91no—
54C22×#67—62no—
55C2×C3×M16—92no—
56C3×#100—94no—
57C23×C32—24no—
58C23×C32—24no—
59C23×C32—24no—
60C23×C32—22no—
61C23×C32—48no—
62C22×C3×D3—64no—
63C22×C3×D3—64no—
64C22×C3×D3—64no—
65C22×C3×D3—128no—
66C3×#67—616no—
67C3×#67—616no—
68C2×C3×Dic3—62no—
69C2×C3×Dic3—64no—
70C2×C3×Dic3—64no—
71C2×C3×Dic3—62no—
72C2×#100—91noSylow
73C2×H(3,𝔽3)—21no—
74C2×H(3,𝔽3)—21no—
75C2×H(3,𝔽3)—42no—
76C2×H(3,𝔽3)#10012yescomplemented
77C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
78#272—42no—
79C24×C3—62no—
80C24×C3—91no—
81C24×C3#15711yescharacteristic
82C23×D3—62no—
83C2×C3×D4—92no—
84C2×C3×D4—92no—
85C2×C3×D4—182no—
86C2×C3×D4—182no—
87C22×C3×C4—91no—
88C2×#67—64no—
89C2×#67—64no—
90C2×#67—68no—
91C2×#67—68no—
92C22×Dic3—62no—
93C3×M16—94no—
94C3×M16—182no—
95C2×C3×C8—182no—
96C22×C32—28no—
97C22×C32—28no—
98C22×C32—22no—
99C22×C32—22no—
100C22×C32—22no—
101C22×C32—48no—
102C22×C32—48no—
103C22×C32—48no—
104C2×C3×D3—68no—
105C2×C3×D3—128no—
106C2×C3×D3—128no—
107C2×C3×D3—128no—
108C3×Dic3—64no—
109C3×Dic3—64no—
110C22×D4—91no—
111C2×M16—92no—
112#100—94no—
113H(3,𝔽3)C2×#10011yescharacteristic, complemented, Sylow
114C23×C3—21no—
115C23×C3—21no—
116C23×C3—42no—
117C23×C3—64no—
118C23×C3—64no—
119C23×C3—62no—
120C23×C3—64no—
121C23×C3—92no—
122C23×C3—128no—
123C23×C3—181no—
124C23×C3—181no—
125C23×C3—362no—
126C23×C3C2×#15712yes—
127C23×C3C2×#15711yescharacteristic
128C22×D3—64no—
129C22×D3—64no—
130C22×D3—64no—
131C22×D3—128no—
132C3×D4—184no—
133C3×D4—184no—
134C2×C3×C4—91no—
135C2×C3×C4—91no—
136C2×C3×C4—92no—
137C2×C3×C4—92no—
138#67—616no—
139#67—616no—
140C2×Dic3—64no—
141C2×Dic3—64no—
142C2×Dic3—62no—
143C2×Dic3—62no—
144C3×C8—184no—
145C2×C32—24no—
146C2×C32—24no—
147C2×C32—22no—
148C2×C32—24no—
149C2×C32—48no—
150C3×D3—128no—
151C24—91no—
152C24#108811yescharacteristic
153C2×D4—92no—
154C2×D4—92no—
155C2×D4—182no—
156C2×D4—182no—
157C22×C4—91no—
158M16—94no—
159M16—182no—
160C2×C8—182no—
161C22×C3—22no—
162C22×C3—22no—
163C22×C3—42no—
164C22×C3—42no—
165C22×C3—42no—
166C22×C3—68no—
167C22×C3—68no—
168C22×C3—62no—
169C22×C3—62no—
170C22×C3—62no—
171C22×C3—128no—
172C22×C3—128no—
173C22×C3—128no—
174C22×C3—182no—
175C22×C3—362no—
176C22×C3—362no—
177C22×C3—362no—
178C22×C3C22⋊#15711yescharacteristic
179C22×C3C22⋊#15711yescharacteristic
180C22×C3C22×#15711yescharacteristic
181C2×D3—68no—
182C2×D3—128no—
183C2×D3—128no—
184C2×D3—128no—
185C3×C4—92no—
186C3×C4—92no—
187Dic3—64no—
188Dic3—64no—
189C32—22no—
190C23—21no—
191C23—21no—
192C23—42no—
193C23—92no—
194C23—181no—
195C23—181no—
196C23—362no—
197C23C2×#108811yescharacteristic
198C23C2×#108812yes—
199D4—184no—
200D4—184no—
201C2×C4—92no—
202C2×C4—92no—
203C2×C4—91no—
204C2×C4—91no—
205C8—184no—
206C2×C3—21no—
207C2×C3—21no—
208C2×C3—42no—
209C2×C3—64no—
210C2×C3—64no—
211C2×C3—64no—
212C2×C3—62no—
213C2×C3—128no—
214C2×C3—362no—
215C2×C3#6378812yes—
216C2×C3C2×C22⋊#15711yescharacteristic
217D3—128no—
218C22—22no—
219C22—22no—
220C22—42no—
221C22—42no—
222C22—42no—
223C22—182no—
224C22—362no—
225C22—362no—
226C22—362no—
227C22C22⋊#108811yescharacteristic
228C22C22⋊#108811yescharacteristic
229C22C22×#108811yescharacteristic
230C4—92no—
231C4—92no—
232C3—62no—
233C3C2×#6378811yescharacteristic
234C2—21no—
235C2—21no—
236C2—42no—
237C2—362no—
238C2#1172079412yesdirect product factor
239C2C2×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
cosocleC2×#1084C23
hypercenterC24×C3#157
hypocenterH(3,𝔽3)C2×#100
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×#88631C2semidirect
3. #11720794C2direct (N × H)
12. #88631C22semidirect
76. C2×H(3,𝔽3)#100semidirect
113. H(3,𝔽3)C2×#100semidirect
238. C2#11720794direct (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×#886312
#117207942
C2×C3×#1009