Finite Group Databasebeta

← #13138858

Subgroups — #13138858

Order 1728.

421 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 421 conjugacy classes of subgroups — too many to draw. Only its 79 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×#3743⋊C3C211yesmaximal, characteristic
2C2×#3746⋊C3C211yesmaximal, characteristic, complemented
3#11721106C212yesmaximal
4#11720550C211yesmaximal, characteristic, complemented
5#11720309C212yesmaximal, complemented
6#10590858—31nomaximal
7#10590853—31nomaximal
8C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
9C2×#634⋊C3C2212yes—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11C4⋊#1084C2212yes—
12C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
13#3743⋊C3—22no—
14#3746⋊C3—22no—
15#88450—22no—
16#88449—22no—
17C2×#3746D311yescharacteristic, complemented
18C2×#3743D311yescharacteristic
19C2×C3×Dic12—31no—
20C2×C3×ΓL(1,𝔽25)—31no—
21#63871—32no—
22#63857—32no—
23C3×#817—31no—
24C3×#817—31no—
25C3×#817—61no—
26#63567—32no—
27#63565—32no—
28C22×(C3⋊D3)⋊C3D411yescharacteristic
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30C2×#1084—22no—
31C2×(C3⋊Dic3)⋊C3—22no—
32C2×(C3⋊Dic3)⋊C3D411yescharacteristic
33C4×(C3⋊D3)⋊C3—21no—
34C4×(C3⋊D3)⋊C3—21no—
35#634⋊C3—22no—
36#634⋊C3—22no—
37C8×H(3,𝔽3)D412yes—
38(C3⋊(C3⋊C8))⋊C3—21no—
39#5189—91no—
40#5137—91no—
41#4867—91no—
42C2×C4×C3⋊D3C2×D311yescharacteristic
43C2×#634C2×D312yes—
44C2×C3×C4×D3—31no—
45C2×C3×Dic6—32no—
46C2×C32×C8—21no—
47C2×C32×C8C2×D311yescharacteristic
48C2×C32×C8C2×D311yescharacteristic
49C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
50#3746—22no—
51#3743—22no—
52C3×C4⋊Dic3—32no—
53C3×C4⋊Dic3—32no—
54C3×C4⋊Dic3—62no—
55C2×C3×C3⋊C8—31no—
56C3×Dic12—62no—
57C3×ΓL(1,𝔽25)—62no—
58Dic3⋊Dic3—62no—
59Dic3⋊Dic3—62no—
60#3722—62no—
61(C2×D3)⋊Dic3—62no—
62C22×H(3,𝔽3)C2×D411yescharacteristic
63C2×(C3⋊D3)⋊C3—21no—
64C2×(C3⋊D3)⋊C3—22no—
65C4×H(3,𝔽3)C2×D411yescharacteristic
66C4×H(3,𝔽3)#4111yescharacteristic
67#1084—42no—
68(C3⋊Dic3)⋊C3—21no—
69(C3⋊Dic3)⋊C3—42no—
70C2×C3×Q16—91no—
71C2×C3×M16—91no—
72#946—92no—
73#924—92no—
74C2×Dic12—31no—
75C2×Dic12—91no—
76C2×ΓL(1,𝔽25)—31no—
77C2×ΓL(1,𝔽25)—91no—
78#897—92no—
79C3×#106—91no—
80#835—92no—
81#817—31no—
82#817—31no—
83#817—61no—
84#816—92no—
85#810—92no—
86C22×C3⋊D3D1211yescharacteristic
87C22×C3×D3—31no—
88C2×C32×C4—21no—
89C2×C32×C4C22×D311yescharacteristic
90C2×C32×C4C22×D311yescharacteristic
91C2×C3⋊Dic3—22no—
92C2×C3⋊Dic3D1211yescharacteristic
93C4×C3⋊D3—21no—
94C4×C3⋊D3—21no—
95#634—22no—
96#634—22no—
97C2×C3×Dic3—31no—
98C2×C3×Dic3—62no—
99C2×C3×Dic3—62no—
100C2×C3×Dic3—62no—
101C2×C3×Dic3—122no—
102C3×C4×D3—61no—
103C3×C4×D3—61no—
104C3×Dic6—62no—
105C3×Dic6—62no—
106C32×C8—22no—
107C32×C8#6712yes—
108C32×C8#6712yes—
109C3⋊(C3⋊C8)—21no—
110C3×C3⋊C8—61no—
111#470—271noSylow
112C2×H(3,𝔽3)#12111yescharacteristic
113C2×H(3,𝔽3)#13712yes—
114(C3⋊D3)⋊C3—41no—
115C2×C3×Q8—92no—
116C22×C3×C4—91no—
117C2×C4×D3—31no—
118C2×C4×D3—91no—
119C2×Dic6—32no—
120C2×Dic6—92no—
121C3×Q16—182no—
122C3×M16—182no—
123C2×C3×C8—31no—
124C2×C3×C8—31no—
125C2×C3×C8—61no—
126C2×C3×C8—91no—
127C2×C3×C8D3211yescharacteristic
128C3×C4⋊C4—92no—
129#217—182no—
130(C2×D3)⋊C4—182no—
131(C2×D3)⋊C4—182no—
132C4⋊Dic3—32no—
133C4⋊Dic3—32no—
134C4⋊Dic3—62no—
135C4⋊Dic3—182no—
136Dic3⋊C4—182no—
137Dic3⋊C4—182no—
138C2×C3⋊C8—31no—
139C2×C3⋊C8—91no—
140Dic12—62no—
141Dic12—182no—
142ΓL(1,𝔽25)—62no—
143ΓL(1,𝔽25)—182no—
144C22×C32—21no—
145C22×C32C2×D1211yescharacteristic
146C22×C32D3×D411yescharacteristic
147C2×C3⋊D3—22no—
148C2×C3⋊D3—21no—
149C2×C3×D3—62no—
150C2×C3×D3—61no—
151C32×C4—21no—
152C32×C4—21no—
153C32×C4#23511yescharacteristic
154C32×C4#23711yescharacteristic
155C32×C4C2×#6711yescharacteristic
156C32×C4C2×#6711yescharacteristic
157C3⋊Dic3—21no—
158C3⋊Dic3—42no—
159C3×Dic3—61no—
160C3×Dic3—122no—
161C3×Dic3—122no—
162C3×Dic3—122no—
163C3×Dic3—242no—
164C2×Q16—91no—
165C2×M16—91no—
166C22⋊Q8—272no—
167#106—91no—
168#103—272no—
169H(3,𝔽3)#47011yescharacteristic, complemented, Sylow
170C23×C3—91no—
171C22×D3—31no—
172C22×D3—91no—
173C3×Q8—182no—
174C3×Q8—182no—
175C2×C3×C4—31no—
176C2×C3×C4—31no—
177C2×C3×C4—61no—
178C2×C3×C4—91no—
179C2×C3×C4—181no—
180C2×C3×C4—181no—
181C2×C3×C4—182no—
182C2×C3×C4—182no—
183C2×C3×C4C2×D3211yescharacteristic
184C2×Dic3—31no—
185C2×Dic3—62no—
186C2×Dic3—62no—
187C2×Dic3—62no—
188C2×Dic3—91no—
189C2×Dic3—122no—
190C2×Dic3—182no—
191C4×D3—61no—
192C4×D3—61no—
193C4×D3—181no—
194C4×D3—181no—
195Dic6—62no—
196Dic6—62no—
197Dic6—182no—
198Dic6—182no—
199C3×C8—32no—
200C3×C8—32no—
201C3×C8—62no—
202C3×C8—181no—
203C3×C8#62512yes—
204C3⋊C8—61no—
205C3⋊C8—181no—
206C2×C32—22no—
207C2×C32—21no—
208C2×C32#90912yes—
209C2×C32#91812yes—
210C2×C32#93011yescharacteristic
211C2×C32#93811yescharacteristic
212C3⋊D3—41no—
213C3×D3—121no—
214C2×Q8—92no—
215C22×C4—91no—
216Q16—182no—
217M16—182no—
218C2×C8—91no—
219C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
220C4⋊C4—92no—
221C4⋊C4—542no—
222C22⋊C4—542no—
223C22×C3—31no—
224C22×C3—31no—
225C22×C3—61no—
226C22×C3—181no—
227C22×C3—182no—
228C22×C3D3×D1211yescharacteristic
229C2×D3—62no—
230C2×D3—61no—
231C2×D3—181no—
232C2×D3—182no—
233C3×C4—31no—
234C3×C4—31no—
235C3×C4—31no—
236C3×C4—31no—
237C3×C4—61no—
238C3×C4—61no—
239C3×C4—181no—
240C3×C4—362no—
241C3×C4—362no—
242C3×C4#379511yescharacteristic
243C3×C4C2×#62511yescharacteristic
244Dic3—61no—
245Dic3—122no—
246Dic3—122no—
247Dic3—122no—
248Dic3—181no—
249Dic3—242no—
250Dic3—362no—
251C32—21no—
252C32#513711yescharacteristic, complemented
253C32#518911yescharacteristic, complemented
254C23—91no—
255Q8—182no—
256Q8—182no—
257C2×C4—91no—
258C2×C4—181no—
259C2×C4—181no—
260C2×C4—182no—
261C2×C4—182no—
262C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
263C8—181no—
264C8#619812yes—
265C2×C3—32no—
266C2×C3—32no—
267C2×C3—31no—
268C2×C3—31no—
269C2×C3—62no—
270C2×C3—61no—
271C2×C3—361no—
272C2×C3#6380112yes—
273C2×C3#6390811yescharacteristic
274D3—121no—
275D3—361no—
276C22—182no—
277C22—181no—
278C22#8865411yescharacteristic
279C4—181no—
280C4—362no—
281C4—362no—
282C4#8864811yescharacteristic
283C4C2×#619811yescharacteristic
284C3—31no—
285C3—31no—
286C3—61no—
287C3#1059085411yescharacteristic
288C2—361no—
289C2#1172085312yes—
290C2#1172114111yescharacteristic
subgroupG/H
centerC22#88654
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radicalC22×(C3⋊D3)⋊C3D4
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#470
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×#3746⋊C3C2semidirect
4. #11720550C2semidirect
5. #11720309C2semidirect
17. C2×#3746D3semidirect
169. H(3,𝔽3)#470semidirect
252. C32#5137semidirect
253. C32#5189semidirect

Maximal subgroups

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

subgroupindex
C2×#3743⋊C32
C2×#3746⋊C32
#117211062
#117205502
#117203092
#105908583
#105908533