Finite Group Databasebeta

← #13147021

Subgroups — #13147021

Order 1728.

358 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 45 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
1C4×#6258C211yesmaximal, characteristic
2C8×C22⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
3#11720736C211yesmaximal, characteristic, complemented
4C3×#5400—33nomaximal
5C3×#5111—31nomaximal
6C2×#6258C411yescharacteristic
7C4×C22⋊H(3,𝔽3)C2211yescharacteristic
8#88613C411yescharacteristic
9#88527—41nomaximal
10C3×C4×S4—33no—
11C22×C32×C8D311yescharacteristic, complemented
12C3×C4×#67—31no—
13C2×C3×ΓL(1,𝔽25)—31no—
14C3×C8×A4D313yescomplemented
15C3×A4⋊C8—33no—
16C3×#830—31no—
17C3×(C2×D3)⋊C8—31no—
18C3×#815—31no—
19C3×Dic3⋊C8—31no—
20C2×C22⋊H(3,𝔽3)C2×C411yescharacteristic
21#6258—21no—
22C4×#272—41no—
23C8×H(3,𝔽3)—41no—
24#6180—41no—
25#5400—33no—
26C3×#435—91no—
27#5111—31no—
28C2×C3×S4—33no—
29C22×C32×C4C2×D311yescharacteristic
30C2×C3×#67—31no—
31C2×C3×C4×D3—31no—
32C3×C4×A4C2×D313yes—
33C3×SL(2,ℤ/4)—33no—
34C2×C32×C8—31no—
35C2×C32×C8—31no—
36C3×#217—31no—
37C3×(C2×D3)⋊C4—31no—
38C3×Dic3⋊C4—31no—
39C3×C4×Dic3—31no—
40C2×C3×C3⋊C8—31no—
41C2×C3×C3⋊C8—31no—
42C3×ΓL(1,𝔽25)—61no—
43C3×ΓL(1,𝔽25)—61no—
44C3×ΓL(1,𝔽25)—123no—
45C2×#272—41no—
46C22⋊H(3,𝔽3)M1611yescharacteristic, complemented
47C4×H(3,𝔽3)—41no—
48#1084—41no—
49C4×S4—33no—
50C2×C3×M16—91no—
51C22×C3×C8—31no—
52C22×C3×C8C3⋊D311yescharacteristic
53C3×C4×D4—91no—
54C4×#67—31no—
55C2×ΓL(1,𝔽25)—31no—
56C8×A4—33no—
57A4⋊C8—33no—
58C3×C4⋊C8—91no—
59C3×C22⋊C8—91no—
60C3×C22⋊C8—91no—
61C3×#97—91no—
62#830—31no—
63(C2×D3)⋊C8—31no—
64#815—31no—
65Dic3⋊C8—31no—
66C23×C32C4×D311yescharacteristic
67C22×C3×D3—31no—
68C2×C3×A4C4×D313yes—
69C3×S4—63no—
70C2×C32×C4—31no—
71C2×C32×C4—31no—
72C3×#67—61no—
73C3×#67—61no—
74C2×C3×Dic3—31no—
75C2×C3×Dic3—31no—
76C2×C3×Dic3—31no—
77C3×C4×D3—61no—
78C3×C4×D3—61no—
79C3×C4×D3—123no—
80C32×C8—31no—
81C32×C8—43no—
82C32×C8S411yescharacteristic, complemented
83C3×C3⋊C8—61no—
84C3×C3⋊C8—61no—
85C3×C3⋊C8—123no—
86#435—91noSylow
87C2×H(3,𝔽3)—41no—
88#272—81no—
89C2×S4—33no—
90C2×C3×D4—91no—
91C22×C3×C4—31no—
92C22×C3×C4—91no—
93C22×C3×C4C2×C3⋊D311yescharacteristic
94C2×#67—31no—
95C2×C4×D3—31no—
96C4×A4—33no—
97SL(2,ℤ/4)—33no—
98C3×M16—181no—
99C3×M16—181no—
100C2×C3×C8—31no—
101C2×C3×C8—31no—
102C2×C3×C8—31no—
103C2×C3×C8—31no—
104C2×C3×C8—61no—
105C2×C3×C8—61no—
106C2×C3×C8—91no—
107C2×C3×C8—91no—
108C3×C4⋊C4—91no—
109C3×C22⋊C4—91no—
110C3×C22⋊C4—91no—
111C3×C42—91no—
112#217—31no—
113(C2×D3)⋊C4—31no—
114Dic3⋊C4—31no—
115C4×Dic3—31no—
116C2×C3⋊C8—31no—
117C2×C3⋊C8—31no—
118ΓL(1,𝔽25)—61no—
119ΓL(1,𝔽25)—61no—
120ΓL(1,𝔽25)—123no—
121C22×C32—31no—
122C22×C32—31no—
123C22×C32ΓL(1,𝔽25)11yescharacteristic, complemented
124C2×C3×D3—61no—
125C2×C3×D3—61no—
126C2×C3×D3—61no—
127C2×C3×D3—123no—
128C3×A4ΓL(1,𝔽25)13yescomplemented
129C32×C4—31no—
130C32×C4—43no—
131C32×C4C2×S411yescharacteristic
132C3×Dic3—61no—
133C3×Dic3—61no—
134C3×Dic3—61no—
135C3×Dic3—123no—
136C2×M16—91no—
137C22×C8#27211yescharacteristic, complemented
138C4×D4—91no—
139C4⋊C8—91no—
140C22⋊C8—91no—
141C22⋊C8—91no—
142#97—91no—
143H(3,𝔽3)—41noSylow
144C23×C3—31no—
145C23×C3—91no—
146C23×C3C4×C3⋊D311yescharacteristic
147C22×D3—31no—
148C2×A4—33no—
149S4—63no—
150C3×D4—181no—
151C3×D4—181no—
152C2×C3×C4—31no—
153C2×C3×C4—31no—
154C2×C3×C4—31no—
155C2×C3×C4—31no—
156C2×C3×C4—61no—
157C2×C3×C4—61no—
158C2×C3×C4—91no—
159C2×C3×C4—91no—
160C2×C3×C4—91no—
161C2×C3×C4—181no—
162C2×C3×C4—181no—
163#67—61no—
164#67—61no—
165C2×Dic3—31no—
166C2×Dic3—31no—
167C2×Dic3—31no—
168C4×D3—61no—
169C4×D3—61no—
170C4×D3—123no—
171C3×C8—31no—
172C3×C8—31no—
173C3×C8—31no—
174C3×C8—61no—
175C3×C8—123no—
176C3×C8—181no—
177C3×C8—181no—
178C3×C8C3⋊S411yescharacteristic
179C3⋊C8—61no—
180C3⋊C8—61no—
181C3⋊C8—123no—
182C2×C32—31no—
183C2×C32—31no—
184C2×C32—43no—
185C2×C32C4×S411yescharacteristic
186C3×D3—121no—
187C3×D3—243no—
188C2×D4—91no—
189C22×C4—91no—
190C22×C4C2×#27211yescharacteristic
191M16—181no—
192M16—181no—
193C2×C8—31no—
194C2×C8—31no—
195C2×C8—91no—
196C2×C8—91no—
197C4⋊C4—91no—
198C22⋊C4—91no—
199C22⋊C4—91no—
200C42—91no—
201C22×C3—31no—
202C22×C3—31no—
203C22×C3—31no—
204C22×C3—31no—
205C22×C3—31no—
206C22×C3—61no—
207C22×C3—61no—
208C22×C3—181no—
209C22×C3—181no—
210C22×C3—181no—
211C22×C3#374311yescharacteristic
212C2×D3—61no—
213C2×D3—61no—
214C2×D3—61no—
215C2×D3—123no—
216A4—33no—
217C3×C4—31no—
218C3×C4—31no—
219C3×C4—31no—
220C3×C4—61no—
221C3×C4—123no—
222C3×C4—181no—
223C3×C4—181no—
224C3×C4—181no—
225C3×C4C2×C3⋊S411yescharacteristic
226Dic3—61no—
227Dic3—61no—
228Dic3—61no—
229Dic3—123no—
230C32—43no—
231C32#540011yescharacteristic, complemented
232C23—91no—
233C23C4×#27211yescharacteristic
234D4—181no—
235D4—181no—
236C2×C4—31no—
237C2×C4—31no—
238C2×C4—91no—
239C2×C4—91no—
240C2×C4—91no—
241C2×C4—181no—
242C2×C4—181no—
243C8—31no—
244C8—181no—
245C8—181no—
246C8#625811yescharacteristic, complemented
247C2×C3—31no—
248C2×C3—31no—
249C2×C3—31no—
250C2×C3—31no—
251C2×C3—31no—
252C2×C3—61no—
253C2×C3—61no—
254C2×C3—123no—
255C2×C3—361no—
256C2×C3C4×C3⋊S411yescharacteristic
257D3—121no—
258D3—243no—
259C22—31no—
260C22—31no—
261C22—181no—
262C22—181no—
263C22—181no—
264C22#8852711yescharacteristic, complemented
265C4—31no—
266C4—181no—
267C4—181no—
268C4—181no—
269C4C2×#625811yescharacteristic
270C3—31no—
271C3—123no—
272C3#1059363411yescharacteristic
273C2—31no—
274C2—31no—
275C2—361no—
276C2C4×#625811yescharacteristic
subgroupG/H
center, FrattiniC3×C4C2×C3⋊S4
commutatorC2×C22⋊H(3,𝔽3)C2×C4
FittingC22×C32×C8D3
socleC23×C3C4×C3⋊D3
prime radicalC2×#6258C4
cosocleC4×C22⋊H(3,𝔽3)C22
hypercenterC3×C8C3⋊S4
hypocenterC22⋊H(3,𝔽3)M16
LCSG ▷ C2×C22⋊H(3,𝔽3) ▷ C22⋊H(3,𝔽3)
UCSC1 ◁ C3×C4 ◁ C3×C8

Complements

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

NH ≅ G/Ntype
2. C8×C22⋊H(3,𝔽3)C2semidirect
3. #11720736C2semidirect
11. C22×C32×C8D3semidirect
14. C3×C8×A4D3semidirect
46. C22⋊H(3,𝔽3)M16semidirect
82. C32×C8S4semidirect
123. C22×C32ΓL(1,𝔽25)semidirect
128. C3×A4ΓL(1,𝔽25)semidirect
137. C22×C8#272semidirect
231. C32#5400semidirect
246. C8#6258semidirect
264. C22#88527semidirect

Maximal subgroups

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

subgroupindex
C4×#62582
C8×C22⋊H(3,𝔽3)2
#117207362
C3×#54003
C3×#51113
#885274