Finite Group Databasebeta

← C2×#11722416 #13164587

Subgroups — C2×#11722416 #13164587

Order 1728.

638 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 87 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×C9×#41⋊C3C211yesmaximal, characteristic, complemented
2#11722416C214yesmaximal, direct product factor
3C2×D9×SL(2,𝔽3)C212yesmaximal, complemented
4C2×#1010⋊C3—31nomaximal
5C2×#63716C311yesmaximal, characteristic, complemented
6C2×C3×D36—41nomaximal
7C9×#41⋊C3C2212yescomplemented
8C2×C9×SL(2,𝔽3)C2211yescharacteristic
9D9×SL(2,𝔽3)C2214yescomplemented
10C2×C3×#41⋊C3D311yescharacteristic
11#1010⋊C3—34no—
12C2×D3×SL(2,𝔽3)—32no—
13C2×C9×#41C2×C311yescharacteristic, complemented
14#63716—34no—
15#63716C2×C314yescomplemented
16C2×Q8×D9C2×C312yescomplemented
17C2×#3699—32no—
18C2×#3697—31no—
19C22×Dic18—31no—
20C2×#41⋊C9—21no—
21C22×C3×D9—42no—
22C2×C3×C4×C9—41no—
23C3×D36—44no—
24C9×SL(2,𝔽3)C2311yescharacteristic
25C2×#1010—31no—
26C2×2−1+4⋊C3—91no—
27C2×C3×D12—121no—
28C3×#41⋊C3C2×D312yes—
29C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
30D3×SL(2,𝔽3)—34no—
31C9×#41—32no—
32C9×#41C22×C312yescomplemented
33C2×Q8×C9C22×C311yescharacteristic
34C2×D4×C9—31no—
35C22×C4×C9—31no—
36C2×#611—32no—
37C22×Dic9—32no—
38Q8×D9—34no—
39Q8×D9C22×C314yescomplemented
40#3699—38no—
41#3699—38no—
42#3697—34no—
43#3697—34no—
44C2×D36A411yescharacteristic
45C2×C4×D9—32no—
46C2×Dic18—34no—
47C2×Dic18—31no—
48C2×Dic18—32no—
49C2×Dic18—34no—
50#41⋊C9—22no—
51C2×Q8⋊C9—21no—
52C22×C3×C9—41no—
53C2×C3×D9—44no—
54C2×C3×D9—84no—
55C3×C4×C9—42no—
56C2×C3×#41C3×D311yescharacteristic
57#1010—34no—
58#1010—94no—
59C2×D3×Q8—32no—
60C2×#237—92no—
61C2×#235—91no—
62C22×Dic6—91no—
632−1+4⋊C3—94no—
64C2×#41⋊C3—21no—
65C2×#41⋊C3D911yescharacteristic, complemented
66C22×SL(2,𝔽3)—92no—
67C22×C3×D3—122no—
68C2×C32×C4—41no—
69C3×D12—124no—
70C3×SL(2,𝔽3)C22×D311yescharacteristic
71C23×C9—31no—
72C22×D9C2×A412yes—
73Q8×C9—31no—
74Q8×C9C23×C311yescharacteristic
75D4×C9—32no—
76D4×C9—32no—
77C2×C4×C9—31no—
78C2×C4×C9—32no—
79C2×C4×C9—32no—
80C2×C4×C9—81no—
81C2×C4×C9C2×A411yescharacteristic
82#611—38no—
83C2×Dic9—38no—
84C2×Dic9—32no—
85D36C2×A414yes—
86C4×D9—34no—
87C4×D9—34no—
88Dic18—34no—
89Dic18—34no—
90Dic18—34no—
91Q8⋊C9—21no—
92C2×2−1+4—91noSylow
93C2×C3×C9—42no—
94C2×C3×C9—41no—
95C3×D9—84no—
96C3×#41—32no—
97C3×#41C2×C3×D312yes—
98C2×C3×Q8C2×C3×D311yescharacteristic
99C2×C3×D4—31no—
100C2×C3×D4—361no—
101C22×C3×C4—31no—
102C2×#67—92no—
103C22×Dic3—92no—
104D3×Q8—34no—
105D3×Q8—94no—
106#237—98no—
107#237—98no—
108#235—94no—
109#235—94no—
110C2×D12—31no—
111C2×C4×D3—92no—
112C2×Dic6—94no—
113C2×Dic6—91no—
114C2×Dic6—92no—
115C2×Dic6—94no—
116#41⋊C3—22no—
117#41⋊C3C2×D912yescomplemented
118C2×SL(2,𝔽3)—21no—
119C2×SL(2,𝔽3)—94no—
120C2×SL(2,𝔽3)C2×D911yescharacteristic
121C22×C32—41no—
122C2×C3×D3—124no—
123C2×C3×D3—244no—
124C32×C4—42no—
125C22×C9—32no—
126C22×C9—62no—
127C22×C9—81no—
128C22×C9C22×A411yescharacteristic
129C2×D9—24no—
130C2×D9C22×A414yes—
131C4×C9—31no—
132C4×C9—31no—
133C4×C9—82no—
134C4×C9C22×A412yes—
135Dic9—34no—
1362−1+4—94no—
1372−1+4—274no—
138C2×#41—271no—
139C2×#41—272no—
140C2×#41C3×D911yescharacteristic, complemented
141C22×Q8—92no—
142C22×Q8—271no—
143C3×C9—41noSylow
144C23×C3—31no—
145C23×C3—362no—
146C22×D3—32no—
147C3×Q8—31no—
148C3×Q8C22×C3×D311yescharacteristic
149C3×D4—32no—
150C3×D4—32no—
151C3×D4—364no—
152C2×C3×C4—31no—
153C2×C3×C4—32no—
154C2×C3×C4—32no—
155C2×C3×C4—41no—
156C2×C3×C4—81no—
157C2×C3×C4D3×A411yescharacteristic
158#67—98no—
159C2×Dic3—98no—
160C2×Dic3—92no—
161D12—34no—
162C4×D3—94no—
163C4×D3—94no—
164Dic6—94no—
165Dic6—94no—
166Dic6—94no—
167SL(2,𝔽3)—21no—
168SL(2,𝔽3)C22×D911yescharacteristic
169C2×C32—42no—
170C2×C32—41no—
171C3×D3—244no—
172C2×C9—62no—
173C2×C9—82no—
174C2×C9—81no—
175C2×C92−1+4⋊C312yescomplemented
176C2×C9C23×A411yescharacteristic
177D9—24no—
178#41—32no—
179#41—274no—
180#41—274no—
181#41—278no—
182#41—278no—
183#41C2×C3×D912yescomplemented
184C2×Q8—94no—
185C2×Q8—274no—
186C2×Q8—274no—
187C2×Q8—272no—
188C2×Q8—271no—
189C2×Q8—274no—
190C2×Q8C2×C3×D911yescharacteristic
191C2×D4—31no—
192C2×D4—91no—
193C2×D4—272no—
194C22×C4—31no—
195C22×C4—272no—
196C22×C4—272no—
197C22×C3—32no—
198C22×C3—41no—
199C22×C3—62no—
200C22×C3—81no—
201C22×C3—364no—
202C22×C3—724no—
203C22×C3C2×D3×A411yescharacteristic
204C2×D3—34no—
205C2×D3—64no—
206C3×C4—31no—
207C3×C4—31no—
208C3×C4—42no—
209C3×C4—82no—
210C3×C4C2×D3×A412yes—
211Dic3—94no—
212C32—41no—
213C9—81no—
214C9C2×2−1+4⋊C311yescharacteristic, complemented
215C23—31no—
216C23—92no—
217Q8—31no—
218Q8—274no—
219Q8—274no—
220Q8—274no—
221Q8C22×C3×D911yescharacteristic
222D4—32no—
223D4—32no—
224D4—94no—
225D4—278no—
226C2×C4—31no—
227C2×C4—32no—
228C2×C4—32no—
229C2×C4—274no—
230C2×C4—274no—
231C2×C4—278no—
232C2×C4—272no—
233C2×C4A4×D911yescharacteristic
234C2×C3—41no—
235C2×C3—42no—
236C2×C3—62no—
237C2×C3—81no—
238C2×C3—82no—
239C2×C3—724no—
240C2×C3#1010⋊C312yes—
241C2×C3C22×D3×A411yescharacteristic
242D3—64no—
243C22—32no—
244C22—62no—
245C22—94no—
246C22—184no—
247C22C2×A4×D911yescharacteristic
248C4—31no—
249C4—31no—
250C4—274no—
251C4C2×A4×D912yes—
252C3—41no—
253C3—81no—
254C3C2×#1010⋊C311yescharacteristic
255C2—62no—
256C2—184no—
257C2#1172241612yesdirect product factor
258C2C22×A4×D911yescharacteristic
subgroupG/H
centerC22C2×A4×D9
commutator, cosocle, hypocenterQ8×C9C23×C3
FrattiniC2×C3C22×D3×A4
FittingC2×C9×#41C2×C3
socleC22×C3C2×D3×A4
hypercenterC2×C4A4×D9
LCSG ▷ Q8×C9
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C9×#41⋊C3C2semidirect
2. #11722416C2direct (N × H)
3. C2×D9×SL(2,𝔽3)C2semidirect
5. C2×#63716C3semidirect
7. C9×#41⋊C3C22semidirect
9. D9×SL(2,𝔽3)C22semidirect
13. C2×C9×#41C2×C3semidirect
15. #63716C2×C3semidirect
16. C2×Q8×D9C2×C3semidirect
32. C9×#41C22×C3semidirect
39. Q8×D9C22×C3semidirect
65. C2×#41⋊C3D9semidirect
117. #41⋊C3C2×D9semidirect
140. C2×#41C3×D9semidirect
175. C2×C92−1+4⋊C3semidirect
183. #41C2×C3×D9semidirect
214. C9C2×2−1+4⋊C3semidirect
257. C2#11722416direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C9×#41⋊C32
#117224162
C2×D9×SL(2,𝔽3)2
C2×#1010⋊C33
C2×#637163
C2×C3×D364