Finite Group Databasebeta

← #10592579⋊C3 #13142221

Subgroups — #10592579⋊C3 #13142221

Order 1728.

435 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 435 conjugacy classes of subgroups — too many to draw. Only its 124 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×C8×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2#64101⋊C3C212yesmaximal
3H(3,𝔽3)×#107C211yesmaximal, characteristic, complemented
4#63645⋊C3C211yesmaximal, characteristic, complemented
5#63635⋊C3C212yesmaximal, complemented
6#10592579C311yesmaximal, characteristic, complemented
7C3×#4880—31nomaximal
8C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
9C2×C8×H(3,𝔽3)C2211yescharacteristic
10C4⋊C4×H(3,𝔽3)C2212yes—
11#3752⋊C3—22no—
12C4⋊((C3⋊Dic3)⋊C3)C2212yes—
13C4×(C3⋊Dic3)⋊C3—22no—
14C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
15C8×(C3⋊D3)⋊C3C414yescomplemented
16C2×C8×C3⋊D3C2×C311yescharacteristic
17#64101C2×C312yes—
18C2×C3×D3×C8—31no—
19C3×#892—32no—
20C32×#107—21no—
21C32×#107D311yescharacteristic, complemented
22C32×#107C2×C311yescharacteristic, complemented
23#63645C2×C311yescharacteristic, complemented
24#63635C2×C312yescomplemented
25C3×#818—31no—
26C3×#808—32no—
27C22×(C3⋊D3)⋊C3D411yescharacteristic
28C2×C4×H(3,𝔽3)—22no—
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30C2×(C3⋊Dic3)⋊C3—22no—
31C2×(C3⋊Dic3)⋊C3D411yescharacteristic
32C4×(C3⋊D3)⋊C3C2×C412yes—
33C4×(C3⋊D3)⋊C3Q812yes—
34C8×H(3,𝔽3)C2×C412yes—
35(C3⋊(C3⋊C8))⋊C3C2×C412yes—
36C3×#427—91no—
37#4880—31no—
38#4880—91no—
39C2×C4×C3⋊D3C22×C311yescharacteristic
40C2×C3×C4×D3—31no—
41C2×C32×C8—21no—
42C2×C32×C8C2×D311yescharacteristic
43C2×C32×C8C22×C311yescharacteristic
44C32×C4⋊C4—22no—
45C32×C4⋊C4C2×D312yes—
46C32×C4⋊C4C22×C312yes—
47#3752—22no—
48C4⋊(C3⋊Dic3)C22×C312yes—
49C4×C3⋊Dic3—22no—
50C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
51C8×C3⋊D3C3×C414yescomplemented
52C3×(C2×D3)⋊C4—62no—
53C3×C4⋊Dic3—32no—
54C3×C4×Dic3—62no—
55C2×C3×C3⋊C8—31no—
56C3×D3×C8—34no—
57C22×H(3,𝔽3)C2×D411yescharacteristic
58C2×(C3⋊D3)⋊C3—21no—
59C2×(C3⋊D3)⋊C3—21no—
60C2×(C3⋊D3)⋊C3C4⋊C412yes—
61C4×H(3,𝔽3)—42no—
62C4×H(3,𝔽3)C22×C411yescharacteristic
63C4×H(3,𝔽3)C2×Q811yescharacteristic
64(C3⋊Dic3)⋊C3—42no—
65(C3⋊Dic3)⋊C3C4⋊C412yes—
66C22×C3×C8—91no—
67C3×#117—92no—
68C2×D3×C8—31no—
69C2×D3×C8—91no—
70#892—32no—
71#892—92no—
72C3×#107—31no—
73C3×#107—31no—
74C3×#107—61no—
75C3×#107—91no—
76C3×#107C3×D311yescharacteristic
77C3×#106—92no—
78#818—31no—
79#818—91no—
80#808—32no—
81#808—92no—
82C22×C3⋊D3C3×D411yescharacteristic
83C22×C3×D3—31no—
84C2×C32×C4—22no—
85C2×C32×C4—22no—
86C2×C32×C4—21no—
87C2×C32×C4—42no—
88C2×C32×C4C22×D311yescharacteristic
89C2×C32×C4C23×C311yescharacteristic
90C2×C3⋊Dic3—22no—
91C2×C3⋊Dic3C3×D411yescharacteristic
92C4×C3⋊D3C2×C3×C412yes—
93C4×C3⋊D3C3×Q812yes—
94C2×C3×Dic3—31no—
95C2×C3×Dic3—62no—
96C3×C4×D3—32no—
97C3×C4×D3—32no—
98C32×C8—22no—
99C32×C8C4×D312yes—
100C32×C8C2×C3×C412yes—
101C3⋊(C3⋊C8)C2×C3×C412yes—
102C3×C3⋊C8—32no—
103#427—91noSylow
104C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
105C2×H(3,𝔽3)#13511yescharacteristic
106C2×H(3,𝔽3)#13511yescharacteristic
107(C3⋊D3)⋊C3—22no—
108C22×C3×C4—91no—
109C2×C4×D3—31no—
110C2×C4×D3—91no—
111C2×C3×C8—31no—
112C2×C3×C8—31no—
113C2×C3×C8—61no—
114C2×C3×C8—94no—
115C2×C3×C8—91no—
116C2×C3×C8C2×C3×D311yescharacteristic
117C3×C4⋊C4—32no—
118C3×C4⋊C4—32no—
119C3×C4⋊C4—62no—
120C3×C4⋊C4—92no—
121C3×C4⋊C4C2×C3×D312yes—
122C3×C22⋊C4—182no—
123C3×C42—182no—
124(C2×D3)⋊C4—62no—
125(C2×D3)⋊C4—182no—
126C4⋊Dic3—32no—
127C4⋊Dic3—92no—
128C4×Dic3—62no—
129C4×Dic3—182no—
130C2×C3⋊C8—31no—
131C2×C3⋊C8—91no—
132D3×C8—34no—
133D3×C8—94no—
134C22×C32—21no—
135C22×C32D3×D411yescharacteristic
136C22×C32C2×C3×D411yescharacteristic
137C2×C3⋊D3—21no—
138C2×C3⋊D3—21no—
139C2×C3⋊D3C3×C4⋊C412yes—
140C2×C3×D3—32no—
141C2×C3×D3—61no—
142C2×C3×D3—61no—
143C32×C4—21no—
144C32×C4—21no—
145C32×C4—42no—
146C32×C4—42no—
147C32×C4—82no—
148C32×C4C2×C4×D311yescharacteristic
149C32×C4D3×Q811yescharacteristic
150C32×C4C22×C3×C411yescharacteristic
151C32×C4C2×C3×Q811yescharacteristic
152C3⋊Dic3—42no—
153C3⋊Dic3C3×C4⋊C412yes—
154C3×Dic3—32no—
155C3×Dic3—122no—
156C22×C8—91no—
157#117—92no—
158#107—91no—
159#107(C3⋊D3)⋊C311yescharacteristic, complemented
160#106—92no—
161H(3,𝔽3)#42711yescharacteristic, complemented, Sylow
162C23×C3—91no—
163C22×D3—31no—
164C22×D3—91no—
165C2×C3×C4—22no—
166C2×C3×C4—31no—
167C2×C3×C4—31no—
168C2×C3×C4—62no—
169C2×C3×C4—62no—
170C2×C3×C4—61no—
171C2×C3×C4—92no—
172C2×C3×C4—92no—
173C2×C3×C4—91no—
174C2×C3×C4—122no—
175C2×C3×C4—182no—
176C2×C3×C4C22×C3×D311yescharacteristic
177C2×Dic3—31no—
178C2×Dic3—62no—
179C2×Dic3—91no—
180C2×Dic3—182no—
181C4×D3—32no—
182C4×D3—32no—
183C4×D3—92no—
184C4×D3—92no—
185C3×C8—32no—
186C3×C8—32no—
187C3×C8—62no—
188C3×C8—92no—
189C3×C8C3×C4×D312yes—
190C3⋊C8—32no—
191C3⋊C8—92no—
192C2×C32—21no—
193C2×C32—21no—
194C2×C32—21no—
195C2×C32D3×C4⋊C411yescharacteristic
196C2×C32#91211yescharacteristic
197C2×C32#91911yescharacteristic
198C2×C32C2×C3×C4⋊C411yescharacteristic
199C2×C32C3×#13511yescharacteristic
200C2×C32C3×#13511yescharacteristic
201C3⋊D3—22no—
202C3×D3—62no—
203C22×C4—91no—
204C2×C8—91no—
205C2×C8—94no—
206C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
207C4⋊C4—92no—
208C4⋊C4C2×(C3⋊D3)⋊C312yes—
209C22⋊C4—182no—
210C42—182no—
211C22×C3—31no—
212C22×C3—31no—
213C22×C3—61no—
214C22×C3—92no—
215C22×C3—181no—
216C22×C3—181no—
217C22×C3C3×D3×D411yescharacteristic
218C2×D3—32no—
219C2×D3—61no—
220C2×D3—61no—
221C2×D3—92no—
222C2×D3—181no—
223C2×D3—181no—
224C3×C4—31no—
225C3×C4—31no—
226C3×C4—31no—
227C3×C4—31no—
228C3×C4—42no—
229C3×C4—61no—
230C3×C4—61no—
231C3×C4—92no—
232C3×C4—122no—
233C3×C4—122no—
234C3×C4—242no—
235C3×C4—362no—
236C3×C4C2×C3×C4×D311yescharacteristic
237C3×C4C3×D3×Q811yescharacteristic
238Dic3—32no—
239Dic3—92no—
240Dic3—122no—
241Dic3—362no—
242C32—21no—
243C32#488011yescharacteristic, complemented
244C32C3×#42711yescharacteristic, complemented
245C23—91no—
246C2×C4—22no—
247C2×C4—91no—
248C2×C4—92no—
249C2×C4—92no—
250C2×C4—182no—
251C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
252C8—92no—
253C8C4×(C3⋊D3)⋊C312yes—
254C2×C3—31no—
255C2×C3—31no—
256C2×C3—31no—
257C2×C3—31no—
258C2×C3—31no—
259C2×C3—31no—
260C2×C3—61no—
261C2×C3—61no—
262C2×C3—61no—
263C2×C3—182no—
264C2×C3C3×D3×C4⋊C411yescharacteristic
265C2×C3C3×#91211yescharacteristic
266C2×C3C3×#91911yescharacteristic
267D3—62no—
268D3—182no—
269C22—92no—
270C22—181no—
271C22—181no—
272C22D4×(C3⋊D3)⋊C311yescharacteristic
273C4—42no—
274C4—92no—
275C4—362no—
276C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
277C4Q8×(C3⋊D3)⋊C311yescharacteristic
278C3—31no—
279C3—31no—
280C3—61no—
281C3C3×#488011yescharacteristic
282C2—182no—
283C2C4⋊C4×(C3⋊D3)⋊C311yescharacteristic
284C2#64121⋊C311yescharacteristic
285C2#64128⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#107C2
socleC22×C3C3×D3×D4
prime radicalC22×(C3⋊D3)⋊C3D4
cosocleC2×C32×C4C23×C3
hypercenter#107(C3⋊D3)⋊C3
hypocenterC32C3×#427
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
3. H(3,𝔽3)×#107C2semidirect
4. #63645⋊C3C2semidirect
5. #63635⋊C3C2semidirect
6. #10592579C3semidirect
15. C8×(C3⋊D3)⋊C3C4semidirect
21. C32×#107D3semidirect
22. C32×#107C2×C3semidirect
23. #63645C2×C3semidirect
24. #63635C2×C3semidirect
51. C8×C3⋊D3C3×C4semidirect
159. #107(C3⋊D3)⋊C3semidirect
161. H(3,𝔽3)#427semidirect
243. C32#4880semidirect
244. C32C3×#427semidirect

Maximal subgroups

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

subgroupindex
C2×C8×(C3⋊D3)⋊C32
#64101⋊C32
H(3,𝔽3)×#1072
#63645⋊C32
#63635⋊C32
#105925793
C3×#48803