Finite Group Databasebeta

← C2×#11722392 #13164496

Subgroups — C2×#11722392 #13164496

Order 1728.

692 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 70 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×#41⋊H(3,𝔽3)C211yesmaximal, characteristic
2#11722392C214yesmaximal, direct product factor
3C2×#88610C212yesmaximal, complemented
4C2×C3×#984—33nomaximal
5C2×C3×#951—31nomaximal
6C2×#6229—41nomaximal
7#41⋊H(3,𝔽3)C2212yes—
8C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
9#88610C2214yescomplemented
10C2×C32×#41D311yescharacteristic
11C22×C3×Dic6—31no—
12C2×C3×#41⋊C3D313yes—
13C3×#984—312no—
14C2×C3×#226—36no—
15C3×#951—34no—
16C3×#951—34no—
17C2×C3×#216—32no—
18C2×C3×#214—32no—
19C2×C3×#208—31no—
20C2×C4×H(3,𝔽3)—41no—
21C2×#1084—42no—
22#6229—44no—
23Q8⋊H(3,𝔽3)C2311yescharacteristic
24C2×#984—33no—
25C2×C3×#137—91no—
26C2×#951—31no—
27C32×#41—32no—
28C32×#41C2×D312yes—
29C2×C32×Q8C2×D311yescharacteristic
30C2×C32×D4—31no—
31C22×C32×C4—31no—
32C22×C3×Dic3—61no—
33C2×C3×Dic6—32no—
34C2×C3×Dic6—34no—
35C2×C3×Dic6—62no—
36C2×C3×Dic6—61no—
37C2×C3×Dic6—123no—
38C3×#41⋊C3C2×D316yes—
39C2×C3×SL(2,𝔽3)C2×D313yes—
40C3×#226—312no—
41C3×#216—34no—
42C3×#216—34no—
43C3×#214—38no—
44C3×#208—34no—
45C2×C3×C3⋊C8—32no—
46C22×H(3,𝔽3)—41no—
47C4×H(3,𝔽3)—42no—
48#1084—44no—
49C2×C3×#41—31no—
50C2×C3×#41C3⋊D311yescharacteristic
51C22×C3×Q8—91no—
52C22×Dic6—31no—
53C2×#41⋊C3—33no—
54#984—312no—
55C2×#226—36no—
56C3×#137—94no—
57C3×#137—94no—
58C2×C3×Q16—92no—
59C2×C3×SD16—92no—
60C2×C3×M16—91no—
61#951—34no—
62#951—34no—
63C2×#216—32no—
64C2×#214—32no—
65C2×#208—31no—
66C23×C32—31no—
67C32×Q8—31no—
68C32×Q8C22×D311yescharacteristic
69C32×D4—32no—
70C32×D4—61no—
71C2×C32×C4—31no—
72C2×C32×C4—32no—
73C2×C32×C4—32no—
74C2×C32×C4—43no—
75C2×C32×C4S411yescharacteristic
76C2×C3×Dic3—64no—
77C2×C3×Dic3—62no—
78C2×C3×Dic3—126no—
79C3×Dic6—34no—
80C3×Dic6—62no—
81C3×Dic6—64no—
82C3×Dic6—1212no—
83C3×SL(2,𝔽3)C22×D313yes—
84C3×C3⋊C8—34no—
85C2×#137—91noSylow
86C2×H(3,𝔽3)—42no—
87C2×H(3,𝔽3)—41no—
88C3×#41—32no—
89C3×#41—32no—
90C3×#41—32no—
91C3×#41—62no—
92C3×#41C2×C3⋊D312yes—
93C2×C3×Q8—31no—
94C2×C3×Q8—92no—
95C2×C3×Q8—94no—
96C2×C3×Q8—182no—
97C2×C3×Q8—181no—
98C2×C3×Q8C2×C3⋊D311yescharacteristic
99C2×C3×D4—31no—
100C2×C3×D4—31no—
101C2×C3×D4—61no—
102C22×C3×C4—31no—
103C22×C3×C4—31no—
104C22×C3×C4—61no—
105C22×C3×C4—181no—
106C22×Dic3—61no—
107C2×Dic6—32no—
108C2×Dic6—34no—
109C2×Dic6—62no—
110C2×Dic6—61no—
111C2×Dic6—123no—
112#41⋊C3—36no—
113C2×SL(2,𝔽3)—33no—
114#226—312no—
115C3×Q16—94no—
116C3×Q16—94no—
117C3×SD16—98no—
118C3×M16—94no—
119C2×C3×C8—92no—
120#216—34no—
121#216—34no—
122#214—38no—
123#208—34no—
124C2×C3⋊C8—32no—
125C22×C32—32no—
126C22×C32—43no—
127C22×C32—62no—
128C22×C32C2×S411yescharacteristic
129C32×C4—31no—
130C32×C4—31no—
131C32×C4—46no—
132C32×C4C2×S412yes—
133C3×Dic3—64no—
134C3×Dic3—1212no—
135C2×#41#27211yescharacteristic
136C22×Q8—91no—
137#137—94no—
138#137—94no—
139C2×Q16—92no—
140C2×SD16—92no—
141C2×M16—91no—
142H(3,𝔽3)—41noSylow
143C23×C3—31no—
144C23×C3—31no—
145C23×C3—61no—
146C3×Q8—31no—
147C3×Q8—31no—
148C3×Q8—31no—
149C3×Q8—61no—
150C3×Q8—94no—
151C3×Q8—182no—
152C3×Q8—184no—
153C3×Q8C22×C3⋊D311yescharacteristic
154C3×D4—32no—
155C3×D4—32no—
156C3×D4—62no—
157C3×D4—61no—
158C3×D4—61no—
159C3×D4—62no—
160C2×C3×C4—31no—
161C2×C3×C4—32no—
162C2×C3×C4—32no—
163C2×C3×C4—31no—
164C2×C3×C4—32no—
165C2×C3×C4—32no—
166C2×C3×C4—31no—
167C2×C3×C4—62no—
168C2×C3×C4—62no—
169C2×C3×C4—61no—
170C2×C3×C4—123no—
171C2×C3×C4—184no—
172C2×C3×C4—182no—
173C2×C3×C4C3⋊S411yescharacteristic
174C2×Dic3—64no—
175C2×Dic3—62no—
176C2×Dic3—126no—
177Dic6—34no—
178Dic6—62no—
179Dic6—64no—
180Dic6—1212no—
181SL(2,𝔽3)—33no—
182C3×C8—94no—
183C3⋊C8—34no—
184C2×C32—46no—
185C2×C32—43no—
186C2×C32—62no—
187C2×C32#98412yescomplemented
188C2×C32C22×S411yescharacteristic
189#41—32no—
190#41C2×#27212yes—
191C2×Q8—92no—
192C2×Q8—94no—
193C2×Q8—182no—
194C2×Q8—181no—
195C2×Q8C2×#27211yescharacteristic
196C2×D4—31no—
197C22×C4—31no—
198C22×C4—181no—
199Q16—94no—
200Q16—94no—
201SD16—98no—
202M16—94no—
203C2×C8—92no—
204C22×C3—32no—
205C22×C3—32no—
206C22×C3—31no—
207C22×C3—62no—
208C22×C3—62no—
209C22×C3—62no—
210C22×C3—122no—
211C22×C3—123no—
212C22×C3C2×C3⋊S411yescharacteristic
213C3×C4—31no—
214C3×C4—31no—
215C3×C4—31no—
216C3×C4—31no—
217C3×C4—32no—
218C3×C4—61no—
219C3×C4—61no—
220C3×C4—126no—
221C3×C4—184no—
222C3×C4C2×C3⋊S412yes—
223Dic3—64no—
224Dic3—1212no—
225C32—43no—
226C32C2×#98411yescharacteristic, complemented
227C23—31no—
228Q8—31no—
229Q8—94no—
230Q8—182no—
231Q8—184no—
232Q8C22×#27211yescharacteristic
233D4—32no—
234D4—61no—
235C2×C4—31no—
236C2×C4—32no—
237C2×C4—32no—
238C2×C4—184no—
239C2×C4—182no—
240C2×C4#625811yescharacteristic
241C8—94no—
242C2×C3—31no—
243C2×C3—32no—
244C2×C3—62no—
245C2×C3—62no—
246C2×C3—122no—
247C2×C3—123no—
248C2×C3—126no—
249C2×C3#6426512yes—
250C2×C3C22×C3⋊S411yescharacteristic
251C22—32no—
252C22—62no—
253C22C2×#625811yescharacteristic
254C4—31no—
255C4—31no—
256C4—184no—
257C4C2×#625812yes—
258C3—31no—
259C3—123no—
260C3C2×#6426511yescharacteristic
261C2—62no—
262C2#1172239212yesdirect product factor
263C2C22×#625811yescharacteristic
subgroupG/H
center, socleC22×C3C2×C3⋊S4
commutator, cosocle, hypocenterQ8⋊H(3,𝔽3)C23
FrattiniC2×C3C22×C3⋊S4
FittingC2×C32×#41D3
prime radicalC2×#41⋊H(3,𝔽3)C2
hypercenterC2×C3×C4C3⋊S4
LCSG ▷ Q8⋊H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. #11722392C2direct (N × H)
3. C2×#88610C2semidirect
9. #88610C22semidirect
187. C2×C32#984semidirect
226. C32C2×#984semidirect
262. C2#11722392direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#41⋊H(3,𝔽3)2
#117223922
C2×#886102
C2×C3×#9843
C2×C3×#9513
C2×#62294