Finite Group Databasebeta

← #13139928

Subgroups — #13139928

Order 1728.

475 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 475 conjugacy classes of subgroups — too many to draw. Only its 123 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
1#11721826C211yesmaximal, characteristic
2C2×#3748⋊C3C211yesmaximal, characteristic
3C22×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic
4#11720304C214yesmaximal, complemented
5#10591117—31nomaximal
6#10591116—31nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8#88552C412yes—
9C4⋊#1084C412yes—
10C4×#1084C2212yes—
11#3748⋊C3—22no—
12C2×(C3⋊(C3⋊C8))⋊C3—22no—
13C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
14C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
15C2×#3748D311yescharacteristic
16C22×C3⋊(C3⋊C8)D311yescharacteristic
17C3×#926—31no—
18C3×#926—31no—
19C3×#926—61no—
20C2×C3×#208—31no—
21C22×C3×C3⋊C8—31no—
22Dic3⋊(C3⋊C8)—34no—
23Dic3⋊(C3⋊C8)—34no—
24C23×H(3,𝔽3)C2×C411yescharacteristic
25C2×C4×H(3,𝔽3)C2×C411yescharacteristic
26C2×C4×H(3,𝔽3)D412yes—
27C2×C4×H(3,𝔽3)Q812yes—
28C2×C4×H(3,𝔽3)C2311yescharacteristic
29C2×#1084C2×C414yes—
30(C3⋊(C3⋊C8))⋊C3—22no—
31(C3⋊(C3⋊C8))⋊C3—22no—
32#5119—91no—
33#5102—91no—
34#4999—91no—
35C22×C32×C4—21no—
36C22×C32×C4C2×D311yescharacteristic
37C22×C32×C4C2×D311yescharacteristic
38#3748—22no—
39C2×C3⋊(C3⋊C8)—22no—
40C2×C3⋊(C3⋊C8)C2×D312yes—
41C2×C3⋊(C3⋊C8)C2×D312yes—
42C3×#217—32no—
43C3×#217—32no—
44C3×#217—62no—
45C3×C4⋊Dic3—32no—
46C3×C4⋊Dic3—32no—
47C3×C4⋊Dic3—62no—
48C3×C4×Dic3—32no—
49C3×C4×Dic3—32no—
50C3×C4×Dic3—62no—
51C3×#208—62no—
52C2×C3×C3⋊C8—32no—
53C2×C3×C3⋊C8—32no—
54C2×C3×C3⋊C8—62no—
55C22×H(3,𝔽3)—22no—
56C22×H(3,𝔽3)C4⋊C412yes—
57C22×H(3,𝔽3)C22×C411yescharacteristic
58C4×H(3,𝔽3)C4⋊C412yes—
59C4×H(3,𝔽3)C2×D411yescharacteristic
60C4×H(3,𝔽3)C2×Q811yescharacteristic
61#1084—24no—
62C2×C3×M16—91no—
63C22×C3×C8—91no—
64C3×#117—91no—
65#926—31no—
66#926—31no—
67#926—61no—
68C2×#208—31no—
69C2×#208—91no—
70C22×C3⋊C8—31no—
71C22×C3⋊C8—91no—
72Dic3⋊C8—94no—
73Dic3⋊C8—94no—
74C4⋊(C3⋊C8)—94no—
75C23×C32—21no—
76C23×C32C4×D311yescharacteristic
77C23×C32C4×D311yescharacteristic
78C2×C32×C4—21no—
79C2×C32×C4—22no—
80C2×C32×C4—22no—
81C2×C32×C4—21no—
82C2×C32×C4Dic612yes—
83C2×C32×C4Dic612yes—
84C2×C32×C4C4×D311yescharacteristic
85C2×C32×C4C4×D311yescharacteristic
86C2×C32×C4#6712yes—
87C2×C32×C4#6712yes—
88C2×C32×C4C22×D311yescharacteristic
89C2×C32×C4C22×D311yescharacteristic
90C2×C3×Dic3—34no—
91C2×C3×Dic3—34no—
92C2×C3×Dic3—64no—
93C3⋊(C3⋊C8)—22no—
94C3⋊(C3⋊C8)—22no—
95C3×C3⋊C8—62no—
96C3×C3⋊C8—62no—
97#424—271noSylow
98C2×H(3,𝔽3)—22no—
99C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
100C2×H(3,𝔽3)#13112yes—
101C22×C3×C4—31no—
102C22×C3×C4—31no—
103C22×C3×C4—61no—
104C22×C3×C4D3211yescharacteristic
105C3×M16—182no—
106C2×C3×C8—92no—
107C2×C3×C8—92no—
108C2×C3×C8—182no—
109C3×C4⋊C4—92no—
110C3×C22⋊C4—92no—
111C3×C42—92no—
112#217—32no—
113#217—32no—
114#217—62no—
115C4⋊Dic3—32no—
116C4⋊Dic3—32no—
117C4⋊Dic3—62no—
118C4×Dic3—32no—
119C4×Dic3—32no—
120C4×Dic3—62no—
121#208—62no—
122#208—182no—
123C2×C3⋊C8—32no—
124C2×C3⋊C8—32no—
125C2×C3⋊C8—62no—
126C2×C3⋊C8—92no—
127C2×C3⋊C8—92no—
128C2×C3⋊C8—182no—
129C22×C32—22no—
130C22×C32—22no—
131C22×C32—22no—
132C22×C32—21no—
133C22×C32—42no—
134C22×C32Dic3⋊C412yes—
135C22×C32Dic3⋊C412yes—
136C22×C32C2×C4×D311yescharacteristic
137C22×C32C2×C4×D311yescharacteristic
138C32×C4—22no—
139C32×C4—21no—
140C32×C4—21no—
141C32×C4Dic3⋊C412yes—
142C32×C4Dic3⋊C412yes—
143C32×C4C2×Dic611yescharacteristic
144C32×C4C2×Dic611yescharacteristic
145C32×C4C2×#6711yescharacteristic
146C32×C4C2×#6711yescharacteristic
147C3×Dic3—64no—
148C3×Dic3—64no—
149C3×Dic3—124no—
150C2×M16—91no—
151C22×C8—91no—
152#117—91no—
153C4⋊C8—274no—
154H(3,𝔽3)#42411yescharacteristic, complemented, Sylow
155C23×C3—31no—
156C23×C3—31no—
157C23×C3—61no—
158C23×C3#62411yescharacteristic
159C2×C3×C4—32no—
160C2×C3×C4—32no—
161C2×C3×C4—32no—
162C2×C3×C4—32no—
163C2×C3×C4—31no—
164C2×C3×C4—31no—
165C2×C3×C4—31no—
166C2×C3×C4—31no—
167C2×C3×C4—62no—
168C2×C3×C4—62no—
169C2×C3×C4—61no—
170C2×C3×C4—61no—
171C2×C3×C4—94no—
172C2×C3×C4#62411yescharacteristic
173C2×C3×C4#62512yes—
174C2×C3×C4#62712yes—
175C2×C3×C4C2×D3211yescharacteristic
176C2×Dic3—34no—
177C2×Dic3—34no—
178C2×Dic3—64no—
179C3×C8—182no—
180C3×C8—182no—
181C3⋊C8—62no—
182C3⋊C8—62no—
183C3⋊C8—182no—
184C3⋊C8—182no—
185C2×C32—22no—
186C2×C32—22no—
187C2×C32—22no—
188C2×C32—21no—
189C2×C32—42no—
190C2×C32#90112yes—
191C2×C32#90712yes—
192C2×C32C2×Dic3⋊C411yescharacteristic
193C2×C32C2×Dic3⋊C411yescharacteristic
194C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
195M16—182no—
196C2×C8—92no—
197C2×C8—92no—
198C2×C8—182no—
199C4⋊C4—92no—
200C22⋊C4—92no—
201C42—92no—
202C22×C3—22no—
203C22×C3—31no—
204C22×C3—31no—
205C22×C3—32no—
206C22×C3—32no—
207C22×C3—62no—
208C22×C3—62no—
209C22×C3—61no—
210C22×C3—62no—
211C22×C3—122no—
212C22×C3#372412yes—
213C22×C3C2×#62411yescharacteristic
214C3×C4—31no—
215C3×C4—31no—
216C3×C4—31no—
217C3×C4—31no—
218C3×C4—32no—
219C3×C4—32no—
220C3×C4—61no—
221C3×C4—61no—
222C3×C4—62no—
223C3×C4—184no—
224C3×C4#372412yes—
225C3×C4C2×#62511yescharacteristic
226C3×C4C2×#62711yescharacteristic
227Dic3—64no—
228Dic3—64no—
229Dic3—124no—
230C32—21no—
231C32#510211yescharacteristic, complemented
232C32#511911yescharacteristic, complemented
233C23#619911yescharacteristic
234C2×C4—94no—
235C2×C4#619612yes—
236C2×C4#619812yes—
237C2×C4#619911yescharacteristic
238C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
239C8—182no—
240C8—182no—
241C2×C3—22no—
242C2×C3—31no—
243C2×C3—31no—
244C2×C3—32no—
245C2×C3—32no—
246C2×C3—62no—
247C2×C3—62no—
248C2×C3—61no—
249C2×C3—62no—
250C2×C3—122no—
251C2×C3#6381712yes—
252C2×C3C2×#372411yescharacteristic
253C22—22no—
254C22#8844912yes—
255C22C2×#619911yescharacteristic
256C4—184no—
257C4#8844912yes—
258C4C2×#619611yescharacteristic
259C4C2×#619811yescharacteristic
260C3—31no—
261C3—31no—
262C3—61no—
263C3#1059111911yescharacteristic
264C2—22no—
265C2#1172106412yes—
266C2C2×#8844911yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#624
prime radicalC23×H(3,𝔽3)C2×C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#424
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11720304C2semidirect
154. H(3,𝔽3)#424semidirect
231. C32#5102semidirect
232. C32#5119semidirect

Maximal subgroups

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

subgroupindex
#117218262
C2×#3748⋊C32
C22×(C3⋊(C3⋊C8))⋊C32
#117203042
#105911173
#105911163