Finite Group Databasebeta

← C2×#11720791 #13147414

Subgroups — C2×#11720791 #13147414

Order 1728.

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

normal subgroups only

This group has 583 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
1C2×#88748C211yesmaximal, characteristic
2C2×C4×#1088C211yesmaximal, characteristic, complemented
3C2×#88627C211yesmaximal, characteristic, complemented
4#11720791C214yesmaximal, direct product factor
5C22×#1088—21no—
6C2×Q8×H(3,𝔽3)C411yescharacteristic, complemented
7#88748—21no—
8#88748—21no—
9#88748C412yescomplemented
10#88748C2212yes—
11C2×C4⋊#272—21no—
12C2×C4⋊#272C411yescharacteristic, complemented
13C2×C4×#272—21no—
14C2×C4×#272C2211yescharacteristic
15C2×#6188—21no—
16C4×#1088—21no—
17C4×#1088C2212yescomplemented
18#88627—21no—
19#88627C2212yescomplemented
20C2×C3×#239—62no—
21C22×#272—21no—
22C22×#272D411yescharacteristic
23C2×#1088—24no—
24C2×#1088—22no—
25Q8×H(3,𝔽3)—21no—
26Q8×H(3,𝔽3)C2×C412yescomplemented
27C2×C4×H(3,𝔽3)—21no—
28C2×C4×H(3,𝔽3)C2×C411yescharacteristic
29C2×#1084D411yescharacteristic
30C4⋊#272—22no—
31C4⋊#272—22no—
32C4⋊#272—21no—
33C4⋊#272C2×C412yescomplemented
34C4×#272—22no—
35C4×#272—22no—
36C4×#272C2×C411yescharacteristic
37C4×#272D411yescharacteristic
38C4×#272D411yescharacteristic
39C4×#272C2311yescharacteristic
40#6188—22no—
41C2×C3×#104—91nomaximal
42C2×C32×Q8—22no—
43C3×#239—64no—
44C3×#239—64no—
45C3×#239—64no—
46C3×#239—64no—
47C2×C3×D12—62no—
48C2×C3×D12—64no—
49C2×C3×C4×D3—62no—
50C2×C3×C4×D3—64no—
51C22×H(3,𝔽3)C22⋊C411yescharacteristic
52C2×#272—22no—
53C2×#272—22no—
54C2×#272—42no—
55C2×#272C22⋊C411yescharacteristic
56C2×#272C2×D411yescharacteristic
57#1088—24no—
58C4×H(3,𝔽3)—22no—
59C4×H(3,𝔽3)C22⋊C411yescharacteristic
60C4×H(3,𝔽3)C22×C411yescharacteristic
61#1084C22⋊C411yescharacteristic
62#1084C2×D411yescharacteristic
63C2×C3×#41—91no—
64C2×#239—62no—
65C2×C3×M16—91no—
66C2×C3×C42—91no—
67C3×#104—94no—
68C22×C3×D3—62no—
69C22×C3×D3—64no—
70C32×Q8—24no—
71C32×Q8—24no—
72C2×C32×C4—24no—
73C2×C32×C4—22no—
74C2×C3×Dic3—62no—
75C3×D12—64no—
76C3×D12—68no—
77C3×D12—68no—
78C3×D12—64no—
79C3×C4×D3—62no—
80C3×C4×D3—62no—
81C3×C4×D3—62no—
82C3×C4×D3—62no—
83C3×C4×D3—68no—
84C3×C4×D3—68no—
85C2×#104—91noSylow
86C2×H(3,𝔽3)#10412yescomplemented
87C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
88#272—21no—
89#272—21no—
90#272—42no—
91C3×#41—92no—
92C3×#41—92no—
93C3×#41—181no—
94C3×#41—181no—
95C2×C3×Q8—62no—
96C2×C3×Q8#15711yescharacteristic
97C2×C3×D4—91no—
98C2×C3×D4—181no—
99C22×C3×C4—91no—
100C22×C3×C4—181no—
101C22×C3×C4—181no—
102#239—64no—
103#239—64no—
104#239—64no—
105#239—64no—
106C2×D12—62no—
107C2×D12—64no—
108C2×C4×D3—62no—
109C2×C4×D3—64no—
110C3×M16—92no—
111C3×M16—181no—
112C2×C3×C8—181no—
113C3×C42—92no—
114C3×C42—181no—
115C22×C32—22no—
116C2×C3×D3—62no—
117C2×C3×D3—62no—
118C2×C3×D3—68no—
119C2×C3×D3—124no—
120C2×C3×D3—128no—
121C32×C4—28no—
122C32×C4—22no—
123C32×C4—22no—
124C3×Dic3—62no—
125C3×Dic3—62no—
126C2×#41—91no—
127C2×M16—91no—
128C2×C42—91no—
129#104—94no—
130H(3,𝔽3)C2×#10411yescharacteristic, complemented, Sylow
131C23×C3—91no—
132C23×C3—181no—
133C22×D3—62no—
134C22×D3—64no—
135C3×Q8—21no—
136C3×Q8—64no—
137C3×Q8—64no—
138C3×Q8C2×#15712yes—
139C3×D4—92no—
140C3×D4—182no—
141C3×D4—182no—
142C3×D4—181no—
143C2×C3×C4—21no—
144C2×C3×C4—64no—
145C2×C3×C4—62no—
146C2×C3×C4—91no—
147C2×C3×C4—91no—
148C2×C3×C4—91no—
149C2×C3×C4—91no—
150C2×C3×C4—91no—
151C2×C3×C4—182no—
152C2×C3×C4—182no—
153C2×C3×C4—182no—
154C2×C3×C4—184no—
155C2×C3×C4C2×#15711yescharacteristic
156C2×Dic3—62no—
157D12—64no—
158D12—68no—
159D12—68no—
160D12—64no—
161C4×D3—62no—
162C4×D3—62no—
163C4×D3—62no—
164C4×D3—62no—
165C4×D3—68no—
166C4×D3—68no—
167C3×C8—182no—
168C2×C32—22no—
169C2×C32—24no—
170C3×D3—122no—
171C3×D3—122no—
172C3×D3—128no—
173#41—92no—
174#41—92no—
175#41—181no—
176#41—181no—
177C2×Q8#108811yescharacteristic, complemented
178C2×D4—91no—
179C2×D4—181no—
180C22×C4—91no—
181C22×C4—181no—
182C22×C4—181no—
183M16—92no—
184M16—181no—
185C2×C8—181no—
186C42—92no—
187C42—181no—
188C22×C3—62no—
189C22×C3—91no—
190C22×C3—91no—
191C22×C3—182no—
192C22×C3—182no—
193C22×C3—362no—
194C22×C3C22⋊#15711yescharacteristic
195C2×D3—62no—
196C2×D3—62no—
197C2×D3—68no—
198C2×D3—124no—
199C2×D3—128no—
200C3×C4—22no—
201C3×C4—68no—
202C3×C4—62no—
203C3×C4—62no—
204C3×C4—91no—
205C3×C4—91no—
206C3×C4—184no—
207C3×C4C22⋊#15711yescharacteristic
208C3×C4C22×#15711yescharacteristic
209Dic3—62no—
210Dic3—62no—
211C32—22no—
212C23—91no—
213C23—181no—
214Q8—21no—
215Q8C2×#108812yescomplemented
216D4—92no—
217D4—182no—
218D4—182no—
219D4—181no—
220C2×C4—21no—
221C2×C4—91no—
222C2×C4—91no—
223C2×C4—91no—
224C2×C4—91no—
225C2×C4—91no—
226C2×C4—182no—
227C2×C4—182no—
228C2×C4—182no—
229C2×C4—184no—
230C2×C4C2×#108811yescharacteristic
231C8—182no—
232C2×C3—64no—
233C2×C3—62no—
234C2×C3—181no—
235C2×C3—181no—
236C2×C3—362no—
237C2×C3#6378512yes—
238C2×C3C2×C22⋊#15711yescharacteristic
239D3—122no—
240D3—122no—
241D3—128no—
242C22—91no—
243C22—91no—
244C22—182no—
245C22—182no—
246C22—362no—
247C22C22⋊#108811yescharacteristic
248C4—22no—
249C4—91no—
250C4—91no—
251C4—184no—
252C4C22⋊#108811yescharacteristic
253C4C22×#108811yescharacteristic
254C3—62no—
255C3C2×#6378511yescharacteristic
256C2—181no—
257C2—181no—
258C2—362no—
259C2#1172079112yesdirect product factor
260C2C2×C22⋊#108811yescharacteristic
subgroupG/H
center, socleC22×C3C22⋊#157
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC3×C4C22×#157
FittingC2×Q8×H(3,𝔽3)C4
prime radicalC2×#88748C2
cosocleC4×#272C23
hypercenterC2×C3×Q8#157
hypocenterH(3,𝔽3)C2×#104
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C4×#1088C2semidirect
3. C2×#88627C2semidirect
4. #11720791C2direct (N × H)
6. C2×Q8×H(3,𝔽3)C4semidirect
9. #88748C4semidirect
12. C2×C4⋊#272C4semidirect
17. C4×#1088C22semidirect
19. #88627C22semidirect
26. Q8×H(3,𝔽3)C2×C4semidirect
33. C4⋊#272C2×C4semidirect
86. C2×H(3,𝔽3)#104semidirect
130. H(3,𝔽3)C2×#104semidirect
177. C2×Q8#1088semidirect
215. Q8C2×#1088semidirect
259. C2#11720791direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#887482
C2×C4×#10882
C2×#886272
#117207912
C2×C3×#1049