Finite Group Databasebeta

← #13138902

Subgroups — #13138902

Order 1728.

453 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 453 conjugacy classes of subgroups — too many to draw. Only its 77 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×#88527C211yesmaximal, characteristic
2C2×#3746⋊C3C211yesmaximal, characteristic, complemented
3#11721094C212yesmaximal
4#63644⋊C3C211yesmaximal, characteristic, complemented
5#11720313C212yesmaximal, complemented
6#10590854—31nomaximal
7#10590853—31nomaximal
8C2×C4×#272C2211yescharacteristic
9C2×#634⋊C3C2212yes—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11C2×#6180C2211yescharacteristic
12#88527—22no—
13C4⋊((C3⋊Dic3)⋊C3)C2212yes—
14#3746⋊C3—22no—
15#88451—22no—
16#88448—22no—
17C2×#3746D311yescharacteristic, complemented
18C2×C3×Dic12—31no—
19C2×C3×ΓL(1,𝔽25)—31no—
20C2×C3×ΓL(1,𝔽25)—31no—
21C2×C3×ΓL(1,𝔽25)—61no—
22#63857—32no—
23#63856—32no—
24#63644D311yescharacteristic, complemented
25C3×#817—31no—
26#63567—32no—
27#63566—32no—
28C22×#272D411yescharacteristic
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30C2×#1084D411yescharacteristic
31C4×#272—21no—
32C4×#272—21no—
33C2×(C3⋊Dic3)⋊C3—22no—
34C2×(C3⋊Dic3)⋊C3—22no—
35#634⋊C3—22no—
36#634⋊C3—22no—
37C8×H(3,𝔽3)D412yes—
38#6180—21no—
39#5189—91no—
40#5137—91no—
41#4867—91no—
42C2×#634C2×D312yes—
43C2×C3×C4×D3—31no—
44C2×C3×C4×D3—31no—
45C2×C3×C4×D3—61no—
46C2×C3×Dic6—32no—
47C2×C32×C8—21no—
48C2×C32×C8C2×D311yescharacteristic
49C2×C32×C8C2×D311yescharacteristic
50C4⋊(C3⋊Dic3)C2×D312yes—
51#3746—22no—
52C3×C4⋊Dic3—32no—
53C2×C3×C3⋊C8—31no—
54C2×C3×C3⋊C8—31no—
55C2×C3×C3⋊C8—61no—
56C3×Dic12—62no—
57C3×ΓL(1,𝔽25)—64no—
58C3×ΓL(1,𝔽25)—62no—
59C3×ΓL(1,𝔽25)—62no—
60#3724—62no—
61Dic3⋊Dic3—62no—
62(C2×D3)⋊Dic3—62no—
63(C2×D3)⋊Dic3—62no—
64C22×H(3,𝔽3)C2×D411yescharacteristic
65C2×#272—21no—
66C2×#272—22no—
67C4×H(3,𝔽3)C2×D411yescharacteristic
68C4×H(3,𝔽3)#4111yescharacteristic
69#1084—21no—
70(C3⋊Dic3)⋊C3—42no—
71(C3⋊Dic3)⋊C3—42no—
72C2×C3×Q16—91no—
73C2×C3×M16—91no—
74#946—92no—
75#924—92no—
76C2×Dic12—31no—
77C2×Dic12—91no—
78C2×ΓL(1,𝔽25)—31no—
79C2×ΓL(1,𝔽25)—31no—
80C2×ΓL(1,𝔽25)—61no—
81#897—92no—
82C3×#106—91no—
83#835—92no—
84#817—31no—
85#817—91no—
86#816—92no—
87#810—92no—
88C22×C3×D3—31no—
89C22×C3×D3—31no—
90C22×C3×D3—61no—
91C2×C32×C4—21no—
92C2×C32×C4C22×D311yescharacteristic
93C2×C32×C4C22×D311yescharacteristic
94C2×C3⋊Dic3—22no—
95C2×C3⋊Dic3—22no—
96#634—22no—
97#634—22no—
98C2×C3×Dic3—31no—
99C2×C3×Dic3—31no—
100C2×C3×Dic3—61no—
101C2×C3×Dic3—62no—
102C2×C3×Dic3—62no—
103C3×C4×D3—62no—
104C3×C4×D3—62no—
105C3×C4×D3—61no—
106C3×C4×D3—61no—
107C3×C4×D3—61no—
108C3×C4×D3—61no—
109C3×Dic6—62no—
110C3×Dic6—62no—
111C32×C8—22no—
112C32×C8D1212yes—
113C32×C8#6712yes—
114C3×C3⋊C8—62no—
115C3×C3⋊C8—61no—
116C3×C3⋊C8—61no—
117#470—271noSylow
118C2×H(3,𝔽3)#12111yescharacteristic
119C2×H(3,𝔽3)#13712yes—
120#272—41no—
121C2×C3×Q8—92no—
122C22×C3×C4—91no—
123C2×C4×D3—31no—
124C2×C4×D3—31no—
125C2×C4×D3—61no—
126C2×Dic6—32no—
127C2×Dic6—92no—
128C3×Q16—182no—
129C3×M16—182no—
130C2×C3×C8—31no—
131C2×C3×C8—31no—
132C2×C3×C8—61no—
133C2×C3×C8—91no—
134C2×C3×C8D3211yescharacteristic
135C3×C4⋊C4—92no—
136#217—182no—
137(C2×D3)⋊C4—182no—
138(C2×D3)⋊C4—182no—
139C4⋊Dic3—32no—
140C4⋊Dic3—92no—
141C4⋊Dic3—182no—
142Dic3⋊C4—182no—
143Dic3⋊C4—182no—
144C2×C3⋊C8—31no—
145C2×C3⋊C8—31no—
146C2×C3⋊C8—61no—
147Dic12—62no—
148Dic12—182no—
149ΓL(1,𝔽25)—62no—
150ΓL(1,𝔽25)—62no—
151ΓL(1,𝔽25)—64no—
152C22×C32—21no—
153C22×C32D3×D411yescharacteristic
154C22×C32D3×D411yescharacteristic
155C2×C3×D3—62no—
156C2×C3×D3—61no—
157C2×C3×D3—62no—
158C2×C3×D3—62no—
159C2×C3×D3—61no—
160C2×C3×D3—122no—
161C32×C4—21no—
162C32×C4—21no—
163C32×C4C2×D1211yescharacteristic
164C32×C4#23711yescharacteristic
165C32×C4#23911yescharacteristic
166C32×C4C2×#6711yescharacteristic
167C3⋊Dic3—42no—
168C3⋊Dic3—42no—
169C3×Dic3—62no—
170C3×Dic3—61no—
171C3×Dic3—61no—
172C3×Dic3—122no—
173C3×Dic3—122no—
174C2×Q16—91no—
175C2×M16—91no—
176C22⋊Q8—272no—
177#106—91no—
178#103—272no—
179H(3,𝔽3)#47011yescharacteristic, complemented, Sylow
180C23×C3—91no—
181C22×D3—31no—
182C22×D3—31no—
183C22×D3—61no—
184C3×Q8—182no—
185C3×Q8—182no—
186C2×C3×C4—31no—
187C2×C3×C4—31no—
188C2×C3×C4—61no—
189C2×C3×C4—91no—
190C2×C3×C4—181no—
191C2×C3×C4—181no—
192C2×C3×C4—182no—
193C2×C3×C4—182no—
194C2×C3×C4C2×D3211yescharacteristic
195C2×Dic3—31no—
196C2×Dic3—31no—
197C2×Dic3—61no—
198C2×Dic3—62no—
199C2×Dic3—62no—
200C2×Dic3—182no—
201C2×Dic3—182no—
202C4×D3—61no—
203C4×D3—61no—
204C4×D3—61no—
205C4×D3—61no—
206C4×D3—62no—
207C4×D3—62no—
208Dic6—62no—
209Dic6—62no—
210Dic6—182no—
211Dic6—182no—
212C3×C8—32no—
213C3×C8—32no—
214C3×C8—62no—
215C3×C8—181no—
216C3×C8#62612yes—
217C3⋊C8—61no—
218C3⋊C8—61no—
219C3⋊C8—62no—
220C2×C32—22no—
221C2×C32—21no—
222C2×C32#89511yescharacteristic
223C2×C32#91512yes—
224C2×C32#91812yes—
225C2×C32#93811yescharacteristic
226C3×D3—122no—
227C3×D3—121no—
228C3×D3—121no—
229C2×Q8—92no—
230C22×C4—91no—
231Q16—182no—
232M16—182no—
233C2×C8—91no—
234C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
235C4⋊C4—92no—
236C4⋊C4—542no—
237C22⋊C4—542no—
238C22×C3—31no—
239C22×C3—31no—
240C22×C3—61no—
241C22×C3—182no—
242C22×C3—181no—
243C22×C3#380211yescharacteristic
244C2×D3—61no—
245C2×D3—61no—
246C2×D3—62no—
247C2×D3—62no—
248C2×D3—62no—
249C2×D3—122no—
250C3×C4—31no—
251C3×C4—31no—
252C3×C4—31no—
253C3×C4—31no—
254C3×C4—61no—
255C3×C4—61no—
256C3×C4—181no—
257C3×C4—362no—
258C3×C4—362no—
259C3×C4#379611yescharacteristic
260C3×C4C2×#62611yescharacteristic
261Dic3—61no—
262Dic3—61no—
263Dic3—62no—
264Dic3—122no—
265Dic3—122no—
266Dic3—362no—
267Dic3—362no—
268C32—21no—
269C32#486711yescharacteristic, complemented
270C32#518911yescharacteristic, complemented
271C23—91no—
272Q8—182no—
273Q8—182no—
274C2×C4—91no—
275C2×C4—181no—
276C2×C4—181no—
277C2×C4—182no—
278C2×C4—182no—
279C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
280C8—181no—
281C8#620012yes—
282C2×C3—32no—
283C2×C3—32no—
284C2×C3—31no—
285C2×C3—31no—
286C2×C3—62no—
287C2×C3—61no—
288C2×C3—361no—
289C2×C3#6381112yes—
290C2×C3#6391611yescharacteristic
291D3—121no—
292D3—121no—
293D3—122no—
294C22—182no—
295C22—181no—
296C22#8865511yescharacteristic
297C4—181no—
298C4—362no—
299C4—362no—
300C4#8864911yescharacteristic
301C4C2×#620011yescharacteristic
302C3—31no—
303C3—31no—
304C3—61no—
305C3#1059085811yescharacteristic
306C2—361no—
307C2#1172084512yes—
308C2#1172115911yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC22×#272D4
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. #63644⋊C3C2semidirect
5. #11720313C2semidirect
17. C2×#3746D3semidirect
24. #63644D3semidirect
179. H(3,𝔽3)#470semidirect
269. C32#4867semidirect
270. C32#5189semidirect

Maximal subgroups

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

subgroupindex
C2×#885272
C2×#3746⋊C32
#117210942
#63644⋊C32
#117203132
#105908543
#105908533