Finite Group Databasebeta

← #13139012

Subgroups — #13139012

Order 1728.

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

normal subgroups only

This group has 445 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×#88530C211yesmaximal, characteristic, complemented
2C2×#3743⋊C3C211yesmaximal, characteristic
3#11721107C212yesmaximal
4#63644⋊C3C211yesmaximal, characteristic, complemented
5#11720326C212yesmaximal, complemented
6#10590858—31nomaximal
7#10590854—31nomaximal
8C2×#6229C2212yes—
9C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11#88530—22no—
12C4⋊((C3⋊Dic3)⋊C3)C2212yes—
13C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
14#3743⋊C3—22no—
15#88450—22no—
16#88448—22no—
17C2×#3743D311yescharacteristic
18C2×C3×Dic12—31no—
19C2×C3×Dic12—31no—
20C2×C3×Dic12—61no—
21C2×C3×ΓL(1,𝔽25)—31no—
22#63871—32no—
23#63856—32no—
24#63644D311yescharacteristic, complemented
25C3×#817—31no—
26#63566—32no—
27#63565—32no—
28C22×(C3⋊D3)⋊C3D411yescharacteristic
29C2×C4×H(3,𝔽3)C2311yescharacteristic
30C2×#1084—22no—
31#6229—22no—
32#6229—22no—
33C2×(C3⋊Dic3)⋊C3—22no—
34C2×(C3⋊Dic3)⋊C3D411yescharacteristic
35C4×(C3⋊D3)⋊C3—21no—
36C4×(C3⋊D3)⋊C3—21no—
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×C3×C4×D3—31no—
44C2×C3×Dic6—32no—
45C2×C3×Dic6—32no—
46C2×C3×Dic6—62no—
47C2×C32×C8—21no—
48C2×C32×C8C2×D311yescharacteristic
49C2×C32×C8C2×D311yescharacteristic
50C4⋊(C3⋊Dic3)C2×D312yes—
51C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
52#3743—22no—
53C3×C4⋊Dic3—32no—
54C2×C3×C3⋊C8—31no—
55C3×Dic12—64no—
56C3×Dic12—62no—
57C3×Dic12—62no—
58C3×ΓL(1,𝔽25)—62no—
59#3724—62no—
60Dic3⋊Dic3—62no—
61#3722—62no—
62(C2×D3)⋊Dic3—62no—
63C22×H(3,𝔽3)C2×D411yescharacteristic
64C2×(C3⋊D3)⋊C3—21no—
65C2×(C3⋊D3)⋊C3—22no—
66C4×H(3,𝔽3)C2×D411yescharacteristic
67C4×H(3,𝔽3)#4111yescharacteristic
68#1084—42no—
69(C3⋊Dic3)⋊C3—21no—
70(C3⋊Dic3)⋊C3—42no—
71C2×C3×Q16—91no—
72C2×C3×M16—91no—
73#946—92no—
74#924—92no—
75C2×Dic12—31no—
76C2×Dic12—31no—
77C2×Dic12—61no—
78C2×ΓL(1,𝔽25)—31no—
79C2×ΓL(1,𝔽25)—91no—
80#897—92no—
81C3×#106—91no—
82#835—92no—
83#817—31no—
84#817—91no—
85#816—92no—
86#810—92no—
87C22×C3⋊D3D1211yescharacteristic
88C22×C3×D3—31no—
89C2×C32×C4—21no—
90C2×C32×C4C22×D311yescharacteristic
91C2×C32×C4C22×D311yescharacteristic
92C2×C3⋊Dic3—22no—
93C2×C3⋊Dic3D1211yescharacteristic
94C4×C3⋊D3—21no—
95C4×C3⋊D3—21no—
96C2×C3×Dic3—31no—
97C2×C3×Dic3—62no—
98C2×C3×Dic3—62no—
99C2×C3×Dic3—62no—
100C2×C3×Dic3—122no—
101C3×C4×D3—61no—
102C3×C4×D3—61no—
103C3×Dic6—64no—
104C3×Dic6—62no—
105C3×Dic6—62no—
106C3×Dic6—62no—
107C3×Dic6—62no—
108C3×Dic6—122no—
109C32×C8—22no—
110C32×C8D1212yes—
111C32×C8#6712yes—
112C3⋊(C3⋊C8)—21no—
113C3×C3⋊C8—61no—
114#470—271noSylow
115C2×H(3,𝔽3)#12111yescharacteristic
116C2×H(3,𝔽3)#13712yes—
117(C3⋊D3)⋊C3—41no—
118C2×C3×Q8—92no—
119C22×C3×C4—91no—
120C2×C4×D3—31no—
121C2×C4×D3—91no—
122C2×Dic6—32no—
123C2×Dic6—32no—
124C2×Dic6—62no—
125C3×Q16—182no—
126C3×M16—182no—
127C2×C3×C8—31no—
128C2×C3×C8—31no—
129C2×C3×C8—61no—
130C2×C3×C8—91no—
131C2×C3×C8D3211yescharacteristic
132C3×C4⋊C4—92no—
133#217—182no—
134(C2×D3)⋊C4—182no—
135(C2×D3)⋊C4—182no—
136C4⋊Dic3—32no—
137C4⋊Dic3—92no—
138C4⋊Dic3—182no—
139Dic3⋊C4—182no—
140Dic3⋊C4—182no—
141C2×C3⋊C8—31no—
142C2×C3⋊C8—91no—
143Dic12—62no—
144Dic12—62no—
145Dic12—64no—
146ΓL(1,𝔽25)—62no—
147ΓL(1,𝔽25)—182no—
148C22×C32—21no—
149C22×C32C2×D1211yescharacteristic
150C22×C32D3×D411yescharacteristic
151C2×C3⋊D3—22no—
152C2×C3⋊D3—21no—
153C2×C3×D3—61no—
154C2×C3×D3—62no—
155C32×C4—21no—
156C32×C4—21no—
157C32×C4C2×D1211yescharacteristic
158C32×C4#23511yescharacteristic
159C32×C4#23911yescharacteristic
160C32×C4C2×#6711yescharacteristic
161C3⋊Dic3—21no—
162C3⋊Dic3—42no—
163C3×Dic3—61no—
164C3×Dic3—124no—
165C3×Dic3—122no—
166C3×Dic3—122no—
167C3×Dic3—122no—
168C2×Q16—91no—
169C2×M16—91no—
170C22⋊Q8—272no—
171#106—91no—
172#103—272no—
173H(3,𝔽3)#47011yescharacteristic, complemented, Sylow
174C23×C3—91no—
175C22×D3—31no—
176C22×D3—91no—
177C3×Q8—182no—
178C3×Q8—182no—
179C2×C3×C4—31no—
180C2×C3×C4—31no—
181C2×C3×C4—61no—
182C2×C3×C4—91no—
183C2×C3×C4—181no—
184C2×C3×C4—181no—
185C2×C3×C4—182no—
186C2×C3×C4—182no—
187C2×C3×C4C2×D3211yescharacteristic
188C2×Dic3—31no—
189C2×Dic3—62no—
190C2×Dic3—62no—
191C2×Dic3—62no—
192C2×Dic3—91no—
193C2×Dic3—122no—
194C2×Dic3—182no—
195C4×D3—61no—
196C4×D3—61no—
197C4×D3—181no—
198C4×D3—181no—
199Dic6—62no—
200Dic6—62no—
201Dic6—62no—
202Dic6—62no—
203Dic6—64no—
204Dic6—122no—
205C3×C8—32no—
206C3×C8—32no—
207C3×C8—62no—
208C3×C8—181no—
209C3×C8#62612yes—
210C3⋊C8—61no—
211C3⋊C8—181no—
212C2×C32—22no—
213C2×C32—21no—
214C2×C32#89511yescharacteristic
215C2×C32#90912yes—
216C2×C32#91512yes—
217C2×C32#93011yescharacteristic
218C3⋊D3—41no—
219C3×D3—121no—
220C2×Q8—92no—
221C22×C4—91no—
222Q16—182no—
223M16—182no—
224C2×C8—91no—
225C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
226C4⋊C4—92no—
227C4⋊C4—542no—
228C22⋊C4—542no—
229C22×C3—31no—
230C22×C3—31no—
231C22×C3—61no—
232C22×C3—181no—
233C22×C3—182no—
234C22×C3D3×D1211yescharacteristic
235C2×D3—62no—
236C2×D3—61no—
237C2×D3—181no—
238C2×D3—182no—
239C3×C4—31no—
240C3×C4—31no—
241C3×C4—31no—
242C3×C4—31no—
243C3×C4—61no—
244C3×C4—61no—
245C3×C4—181no—
246C3×C4—362no—
247C3×C4—362no—
248C3×C4#379911yescharacteristic
249C3×C4C2×#62611yescharacteristic
250Dic3—61no—
251Dic3—122no—
252Dic3—122no—
253Dic3—124no—
254Dic3—122no—
255Dic3—181no—
256Dic3—362no—
257C32—21no—
258C32#486711yescharacteristic, complemented
259C32#513711yescharacteristic, complemented
260C23—91no—
261Q8—182no—
262Q8—182no—
263C2×C4—91no—
264C2×C4—181no—
265C2×C4—181no—
266C2×C4—182no—
267C2×C4—182no—
268C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
269C8—181no—
270C8#620012yes—
271C2×C3—32no—
272C2×C3—32no—
273C2×C3—31no—
274C2×C3—31no—
275C2×C3—62no—
276C2×C3—61no—
277C2×C3—361no—
278C2×C3#6380012yes—
279C2×C3#6390911yescharacteristic
280D3—121no—
281D3—361no—
282C22—182no—
283C22—181no—
284C22#8865411yescharacteristic
285C4—181no—
286C4—362no—
287C4—362no—
288C4#8865211yescharacteristic
289C4C2×#620011yescharacteristic
290C3—31no—
291C3—31no—
292C3—61no—
293C3#1059085311yescharacteristic
294C2—361no—
295C2#1172085212yes—
296C2#1172115611yescharacteristic
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
1. C2×#88530C2semidirect
4. #63644⋊C3C2semidirect
5. #11720326C2semidirect
24. #63644D3semidirect
173. H(3,𝔽3)#470semidirect
258. C32#4867semidirect
259. C32#5137semidirect

Maximal subgroups

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

subgroupindex
C2×#885302
C2×#3743⋊C32
#117211072
#63644⋊C32
#117203262
#105908583
#105908543