Finite Group Databasebeta

← #13154844

Subgroups — #13154844

Order 1728.

728 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 48 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×#88804C211yesmaximal, characteristic
2#11723112C211yesmaximal, characteristic, complemented
3C3×C22⋊#622C211yesmaximal, characteristic, complemented
4#10590204—31nomaximal
5C23×C3⋊(C3⋊D3)C411yescharacteristic
6C22×C3×C3⋊Dic3C2211yescharacteristic
7C22×C32×Dic3C411yescharacteristic
8C2×C3⋊#622—21no—
9C2×C3×#622—21no—
10#88804—21no—
11#88804—21no—
12C2×#3752—31no—
13C2×#3722—62no—
14C22⋊#622—31no—
15C22⋊#622D311yescharacteristic, complemented
16C23×C33C2×C411yescharacteristic
17C22×C3⋊(C3⋊D3)—21no—
18C22×C3⋊(C3⋊D3)—21no—
19C22×C3⋊(C3⋊D3)—41no—
20C22×C3⋊(C3⋊D3)—41no—
21C22×C3⋊(C3⋊D3)C812yescomplemented
22C2×C3×C3⋊Dic3—41no—
23C2×C3×C3⋊Dic3D411yescharacteristic
24C2×C3×C3⋊Dic3D411yescharacteristic
25C2×C32×Dic3—21no—
26C2×C32×Dic3—41no—
27C3⋊#622—41no—
28C3×#622—41no—
29#4468—91nomaximal
30C23×C3⋊D3—31no—
31C23×C3⋊D3—62no—
32C23×C3⋊D3—122no—
33C22×C32×C4—31no—
34C22×C3⋊Dic3C2×D311yescharacteristic
35C22×C3×Dic3—22no—
36C22×C3×Dic3—62no—
37C2×#622—21no—
38C2×#622—61no—
39#3752—61no—
40#3752—61no—
41#3722—64no—
42#3722—64no—
43C22×C33—21no—
44C22×C33—21no—
45C22×C33C22⋊C411yescharacteristic
46C22×C33C2×C811yescharacteristic
47C22×C33M1611yescharacteristic
48C2×C3⋊(C3⋊D3)—22no—
49C2×C3⋊(C3⋊D3)—42no—
50C2×C3⋊(C3⋊D3)—41no—
51C2×C3⋊(C3⋊D3)—41no—
52C2×C3⋊(C3⋊D3)—42no—
53C3×C3⋊Dic3—21no—
54C3×C3⋊Dic3—21no—
55C32×Dic3—41no—
56C2×(C2×D3)⋊C4—91no—
57C2×(C2×D3)⋊C4—182no—
58C3×C22⋊C8—91no—
59#830—91no—
60C23×C32—22no—
61C23×C32—42no—
62C23×C32C4×D311yescharacteristic
63C22×C3⋊D3—32no—
64C22×C3⋊D3—64no—
65C22×C3⋊D3—61no—
66C22×C3⋊D3—61no—
67C22×C3⋊D3—64no—
68C22×C3⋊D3—64no—
69C22×C3⋊D3—124no—
70C22×C3⋊D3—124no—
71C22×C3⋊D3—124no—
72C22×C3⋊D3—124no—
73C22×C3⋊D3—124no—
74C22×C3⋊D3—128no—
75C22×C3⋊D3—128no—
76C22×C3⋊D3—121no—
77C22×C3⋊D3—121no—
78C2×C32×C4—61no—
79C2×C32×C4—121no—
80C2×C3⋊Dic3—41no—
81C2×C3⋊Dic3D1211yescharacteristic
82C2×C3⋊Dic3#6711yescharacteristic
83C2×C3×Dic3—24no—
84C2×C3×Dic3—44no—
85C2×C3×Dic3—62no—
86C2×C3×Dic3—62no—
87C2×C3×Dic3—124no—
88#622—41no—
89#622—121no—
90#323—271noSylow
91C2×C33—21no—
92C2×C33—21no—
93C2×C33C22⋊C811yescharacteristic
94C2×C33#9911yescharacteristic
95C2×C33#10011yescharacteristic
96C3⋊(C3⋊D3)—42no—
97C23×D3—91no—
98C23×D3—182no—
99C23×D3—362no—
100C22×C3×C4—62no—
101C22×C3×C4—91no—
102C22×Dic3—62no—
103C22×Dic3#15711yescharacteristic
104C2×C3×C8—181no—
105(C2×D3)⋊C4—181no—
106(C2×D3)⋊C4—181no—
107(C2×D3)⋊C4—184no—
108(C2×D3)⋊C4—184no—
109C2×C3⋊C8—181no—
110C22×C32—21no—
111C22×C32—21no—
112C22×C32—24no—
113C22×C32—24no—
114C22×C32—22no—
115C22×C32—22no—
116C22×C32—22no—
117C22×C32—44no—
118C22×C32—44no—
119C22×C32—42no—
120C22×C32—42no—
121C22×C32—42no—
122C22×C32D3×C811yescharacteristic
123C22×C32ΓL(1,𝔽25)11yescharacteristic
124C22×C32(C2×D3)⋊C411yescharacteristic
125C2×C3⋊D3—62no—
126C2×C3⋊D3—68no—
127C2×C3⋊D3—128no—
128C2×C3⋊D3—122no—
129C2×C3⋊D3—128no—
130C2×C3⋊D3—124no—
131C2×C3⋊D3—124no—
132C2×C3⋊D3—128no—
133C2×C3⋊D3—1216no—
134C2×C3⋊D3—1216no—
135C2×C3⋊D3—128no—
136C2×C3⋊D3—128no—
137C2×C3⋊D3—121no—
138C2×C3⋊D3—121no—
139C2×C3⋊D3—122no—
140C32×C4—121no—
141C3⋊Dic3—21no—
142C3⋊Dic3—21no—
143C3×Dic3—44no—
144C3×Dic3—122no—
145C3×Dic3—122no—
146C2×C22⋊C4—271no—
147C22⋊C8—91no—
148C22⋊C8—271no—
149C33#32311yescharacteristic, complemented, Sylow
150C23×C3—22no—
151C23×C3—42no—
152C23×C3C2×#15711yescharacteristic
153C22×D3—92no—
154C22×D3—184no—
155C22×D3—184no—
156C22×D3—184no—
157C22×D3—181no—
158C22×D3—181no—
159C22×D3—364no—
160C22×D3—364no—
161C22×D3—364no—
162C22×D3—364no—
163C22×D3—364no—
164C22×D3—368no—
165C22×D3—368no—
166C22×D3—361no—
167C22×D3—361no—
168C2×C3×C4—64no—
169C2×C3×C4—91no—
170C2×C3×C4—91no—
171C2×C3×C4—124no—
172C2×C3×C4—361no—
173C2×Dic3—21no—
174C2×Dic3—41no—
175C2×Dic3—62no—
176C2×Dic3—62no—
177C2×Dic3—124no—
178C3×C8—361no—
179C3⋊C8—361no—
180C2×C32—21no—
181C2×C32—21no—
182C2×C32—24no—
183C2×C32—24no—
184C2×C32—22no—
185C2×C32—22no—
186C2×C32—22no—
187C2×C32—44no—
188C2×C32—44no—
189C2×C32—42no—
190C2×C32—42no—
191C2×C32—42no—
192C2×C32#80611yescharacteristic
193C2×C32(C2×D3)⋊C811yescharacteristic
194C2×C32#82311yescharacteristic
195C3⋊D3—128no—
196C3⋊D3—1216no—
197C3⋊D3—122no—
198C24—271no—
199C22×C4—31no—
200C22×C4—91no—
201C2×C8—181no—
202C2×C8—541no—
203C22⋊C4—541no—
204C22⋊C4—541no—
205C22×C3—24no—
206C22×C3—24no—
207C22×C3—21no—
208C22×C3—21no—
209C22×C3—22no—
210C22×C3—22no—
211C22×C3—22no—
212C22×C3—44no—
213C22×C3—44no—
214C22×C3—42no—
215C22×C3—42no—
216C22×C3—42no—
217C22×C3#378711yescharacteristic
218C22×C3#378811yescharacteristic
219C22×C3C22⋊#15711yescharacteristic
220C2×D3—188no—
221C2×D3—182no—
222C2×D3—368no—
223C2×D3—368no—
224C2×D3—368no—
225C2×D3—362no—
226C2×D3—364no—
227C2×D3—364no—
228C2×D3—3616no—
229C2×D3—3616no—
230C2×D3—368no—
231C2×D3—368no—
232C2×D3—361no—
233C2×D3—361no—
234C2×D3—362no—
235C3×C4—124no—
236C3×C4—181no—
237C3×C4—181no—
238Dic3—41no—
239Dic3—122no—
240Dic3—122no—
241C32—22no—
242C32—42no—
243C32#446811yescharacteristic, complemented
244C23—272no—
245C23—541no—
246C23—541no—
247C23—1081no—
248C23—1081no—
249C23D3×#15711yescharacteristic
250C2×C4—61no—
251C2×C4—91no—
252C2×C4—91no—
253C2×C4—121no—
254C2×C4—361no—
255C8—361no—
256C8—1081no—
257C2×C3—24no—
258C2×C3—24no—
259C2×C3—21no—
260C2×C3—21no—
261C2×C3—22no—
262C2×C3—22no—
263C2×C3—22no—
264C2×C3—44no—
265C2×C3—44no—
266C2×C3—42no—
267C2×C3—42no—
268C2×C3—42no—
269C2×C3#6377411yescharacteristic
270C2×C3#6377811yescharacteristic
271C2×C3#6378811yescharacteristic
272D3—368no—
273D3—3616no—
274D3—362no—
275C22—21no—
276C22—21no—
277C22—542no—
278C22—1082no—
279C22—1082no—
280C22—1081no—
281C22—1081no—
282C22(C2×D3)⋊#15711yescharacteristic
283C22#8892411yescharacteristic
284C22#8892611yescharacteristic
285C4—121no—
286C4—181no—
287C4—181no—
288C3—22no—
289C3—42no—
290C3#1059020411yescharacteristic, complemented
291C2—21no—
292C2—21no—
293C2—1082no—
294C2#1172272611yescharacteristic
295C2#1172272811yescharacteristic
296C2#1172272911yescharacteristic
subgroupG/H
centerC22(C2×D3)⋊#157
commutatorC22×C33C2×C8
Frattini, hypercenterC23D3×#157
FittingC23×C33C2×C4
socleC22×C33C22⋊C4
prime radicalC23×C3⋊(C3⋊D3)C4
cosocleC22×C3×C3⋊Dic3C22
hypocenterC33#323
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. #11723112C2semidirect
3. C3×C22⋊#622C2semidirect
15. C22⋊#622D3semidirect
21. C22×C3⋊(C3⋊D3)C8semidirect
149. C33#323semidirect
243. C32#4468semidirect
290. C3#10590204semidirect

Maximal subgroups

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

subgroupindex
C2×#888042
#117231122
C3×C22⋊#6222
#105902043
#44689