Finite Group Databasebeta

← #13166243

Subgroups — #13166243

Order 1728.

456 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 88 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
1#11724254C211yesmaximal, characteristic
2#11724246C211yesmaximal, characteristic
3D3×C8×C3⋊D3C211yesmaximal, characteristic
4C8×C3⋊#157C211yesmaximal, characteristic, complemented
5C3×C8×#157C211yesmaximal, characteristic, complemented
6#11722705C211yesmaximal, characteristic, complemented
7#11722704C211yesmaximal, characteristic, complemented
8#10593765—31nomaximal
9C4×D3×C3⋊D3C2211yescharacteristic
10C4×C3⋊#157C2211yescharacteristic
11#88988C2211yescharacteristic
12C3×C4×#157C2211yescharacteristic
13C3×#3787C2211yescharacteristic
14#88926C412yescomplemented
15#88925C412yescomplemented
16Dic3⋊#157C412yes—
17(C2×D3)⋊#157C412yes—
18C8×C3⋊(C3⋊D3)C411yescharacteristic, complemented
19C3×C8×C3⋊D3C2211yescharacteristic
20C32×D3×C8C411yescharacteristic, complemented
21#88785C411yescharacteristic, complemented
22C3⋊D3×C3⋊C8C2211yescharacteristic
23D3×C3⋊(C3⋊C8)C411yescharacteristic, complemented
24#64286—31no—
25#64283—31no—
26C2×C8×C3⋊D3—31no—
27D32×C8—62no—
28#63767—31no—
29#63767—31no—
30C8×#157—31no—
31C8×#157D311yescharacteristic, complemented
32C2×D3×C3⋊D3C2×C411yescharacteristic
33C2×C3⋊#157C2×C412yes—
34C2×C3×#157C2×C412yes—
35C4×C3⋊(C3⋊D3)C2×C411yescharacteristic
36C3×C4×C3⋊D3C2311yescharacteristic
37C32×C4×D3C2×C411yescharacteristic
38#6289C2×C411yescharacteristic
39Dic3×C3⋊D3C2×C411yescharacteristic
40D3×C3⋊Dic3C2×C411yescharacteristic
41C3⋊#622C2×C412yes—
42C3×#622C2×C412yes—
43C33×C8C2×C411yescharacteristic
44C3⋊(C3⋊(C3⋊C8))C2×C411yescharacteristic
45C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
46C32×C3⋊C8C2×C411yescharacteristic
47#4689—91nomaximal
48C2×C4×C3⋊D3—31no—
49C4×D32—62no—
50C22⋊#157—32no—
51#3792—32no—
52C4⋊#157—32no—
53C4×#157—31no—
54C4×#157C2×D311yescharacteristic
55#3788—32no—
56#3787—31no—
57#3787C2×D311yescharacteristic
58C2×C32×C8—31no—
59C2×C3⋊(C3⋊C8)—31no—
60C8×C3⋊D3—31no—
61C8×C3⋊D3—31no—
62C8×C3⋊D3—31no—
63C8×C3⋊D3—62no—
64C8×C3⋊D3—122no—
65C8×C3⋊D3C2×D311yescharacteristic
66C3×D3×C8—22no—
67C3×D3×C8—62no—
68#3710—62no—
69D3×C3⋊C8—62no—
70D3×C3⋊C8—62no—
71C2×C3⋊(C3⋊D3)C4211yescharacteristic
72C2×C3×C3⋊D3C22×C411yescharacteristic
73C2×C32×D3C4211yescharacteristic
74D3×C3⋊D3—22no—
75C3⋊#157—22no—
76C3×#157—22no—
77C33×C4C22×C411yescharacteristic
78C3⋊(C3⋊Dic3)C4211yescharacteristic
79C3×C3⋊Dic3C22×C411yescharacteristic
80C32×Dic3C4211yescharacteristic
81C2×ΓL(1,𝔽25)—91no—
82C2×D3×C8—91no—
83C2×D3×C8—182no—
84#872—91no—
85C3×C4×C8—91no—
86#815—91no—
87C8×Dic3—91no—
88#803—91no—
89C22×C3⋊D3—31no—
90C2×D32—62no—
91C2×#157—32no—
92C2×#157C4×D312yes—
93C2×C32×C4—31no—
94C2×C3⋊Dic3—31no—
95C4×C3⋊D3—31no—
96C4×C3⋊D3—31no—
97C4×C3⋊D3—31no—
98C4×C3⋊D3—62no—
99C4×C3⋊D3—122no—
100C4×C3⋊D3C22×D311yescharacteristic
101C3×C4×D3—22no—
102C3×C4×D3—62no—
103#624—62no—
104D3×Dic3—62no—
105D3×Dic3—62no—
106#622—32no—
107#622C4×D312yes—
108C32×C8—22no—
109C32×C8—31no—
110C32×C8—42no—
111C32×C8C4×D311yescharacteristic
112C3⋊(C3⋊C8)—31no—
113C3⋊(C3⋊C8)—62no—
114C3⋊(C3⋊C8)—122no—
115C3⋊(C3⋊C8)C4×D311yescharacteristic
116C3×C3⋊C8—22no—
117C3×C3⋊C8—62no—
118#405—271noSylow
119C2×C33C2×C4211yescharacteristic
120C3⋊(C3⋊D3)—21no—
121C3×C3⋊D3#13112yes—
122C32×D3—21no—
123C2×C4×D3—91no—
124C2×C4×D3—182no—
125C2×C3×C8—62no—
126C2×C3×C8—91no—
127C2×C3×C8—91no—
128C3×C42—91no—
129(C2×D3)⋊C4—92no—
130Dic3⋊C4—92no—
131C4×Dic3—91no—
132C2×C3⋊C8—91no—
133C2×C3⋊C8—91no—
134C2×C3⋊C8—182no—
135ΓL(1,𝔽25)—92no—
136ΓL(1,𝔽25)—92no—
137D3×C8—62no—
138D3×C8—91no—
139D3×C8—91no—
140D3×C8—91no—
141D3×C8—182no—
142D3×C8—182no—
143D3×C8—182no—
144D3×C8—362no—
145D3×C8#15711yescharacteristic, complemented
146C22×C32—31no—
147C2×C3⋊D3—31no—
148C2×C3⋊D3—62no—
149C2×C3⋊D3—62no—
150C2×C3⋊D3—122no—
151C2×C3⋊D3C2×C4×D311yescharacteristic
152C2×C3×D3—22no—
153C2×C3×D3—62no—
154D32—68no—
155#157—22no—
156#157—62no—
157C32×C4—22no—
158C32×C4—31no—
159C32×C4—42no—
160C32×C4C2×C4×D311yescharacteristic
161C3⋊Dic3—31no—
162C3⋊Dic3—62no—
163C3⋊Dic3—122no—
164C3⋊Dic3C2×C4×D311yescharacteristic
165C3×Dic3—22no—
166C3×Dic3—62no—
167C2×M16—271no—
168C22×C8—271no—
169#117—271no—
170#97—271no—
171#97—271no—
172C4×C8—91no—
173C4×C8—271no—
174C33#40511yescharacteristic, complemented, Sylow
175C22×D3—91no—
176C22×D3—182no—
177C2×C3×C4—62no—
178C2×C3×C4—91no—
179C2×C3×C4—92no—
180C2×Dic3—91no—
181C2×Dic3—92no—
182C2×Dic3—182no—
183C4×D3—62no—
184C4×D3—91no—
185C4×D3—91no—
186C4×D3—91no—
187C4×D3—182no—
188C4×D3—182no—
189C4×D3—182no—
190C4×D3—362no—
191C4×D3C2×#15711yescharacteristic
192C3×C8—22no—
193C3×C8—42no—
194C3×C8—62no—
195C3×C8—92no—
196C3×C8—91no—
197C3×C8C2×#15711yescharacteristic
198C3⋊C8—62no—
199C3⋊C8—92no—
200C3⋊C8—91no—
201C3⋊C8—182no—
202C3⋊C8—362no—
203C3⋊C8C2×#15711yescharacteristic
204C2×C32—22no—
205C2×C32—42no—
206C2×C32—61no—
207C2×C32C42×D311yescharacteristic
208C3⋊D3—61no—
209C3⋊D3—64no—
210C3⋊D3—124no—
211C3⋊D3#90112yes—
212C3×D3—24no—
213C3×D3—64no—
214C22×C4—271no—
215M16—272no—
216M16—272no—
217C2×C8—31no—
218C2×C8—91no—
219C2×C8—91no—
220C2×C8—271no—
221C2×C8—271no—
222C2×C8—271no—
223C2×C8—271no—
224C2×C8—271no—
225C4⋊C4—272no—
226C22⋊C4—272no—
227C42—91no—
228C42—271no—
229C22×C3—62no—
230C22×C3—91no—
231C2×D3—62no—
232C2×D3—91no—
233C2×D3—182no—
234C2×D3—182no—
235C2×D3—188no—
236C2×D3—362no—
237C2×D3C4×#15711yescharacteristic
238C3×C4—22no—
239C3×C4—42no—
240C3×C4—62no—
241C3×C4—91no—
242C3×C4—182no—
243C3×C4C22×#15711yescharacteristic
244Dic3—62no—
245Dic3—91no—
246Dic3—182no—
247Dic3—182no—
248Dic3—362no—
249Dic3C4×#15711yescharacteristic
250C32—22no—
251C32—42no—
252C32#468911yescharacteristic, complemented
253C23—271no—
254C2×C4—31no—
255C2×C4—91no—
256C2×C4—92no—
257C2×C4—271no—
258C2×C4—271no—
259C2×C4—271no—
260C2×C4—271no—
261C2×C4—272no—
262C8—31no—
263C8—92no—
264C8—91no—
265C8—272no—
266C8—271no—
267C8D3×#15711yescharacteristic
268C2×C3—22no—
269C2×C3—42no—
270C2×C3—64no—
271C2×C3—92no—
272C2×C3C2×C4×#15711yescharacteristic
273D3—21no—
274D3—64no—
275D3—184no—
276D3—181no—
277D3—364no—
278C22—31no—
279C22—91no—
280C22—271no—
281C22—542no—
282C4—31no—
283C4—91no—
284C4—182no—
285C4—271no—
286C4—542no—
287C4C2×D3×#15711yescharacteristic
288C3—22no—
289C3—42no—
290C3#1059376511yescharacteristic, complemented
291C2—61no—
292C2—92no—
293C2—541no—
294C2C4×D3×#15711yescharacteristic
subgroupG/H
center, hypercenterC8D3×#157
commutator, socleC2×C33C2×C42
FrattiniC4C2×D3×#157
FittingC33×C8C2×C4
prime radicalC2×D3×C3⋊D3C2×C4
cosocleC3×C4×C3⋊D3C23
hypocenterC33#405
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C8

Complements

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

NH ≅ G/Ntype
4. C8×C3⋊#157C2semidirect
5. C3×C8×#157C2semidirect
6. #11722705C2semidirect
7. #11722704C2semidirect
14. #88926C4semidirect
15. #88925C4semidirect
18. C8×C3⋊(C3⋊D3)C4semidirect
20. C32×D3×C8C4semidirect
21. #88785C4semidirect
23. D3×C3⋊(C3⋊C8)C4semidirect
31. C8×#157D3semidirect
145. D3×C8#157semidirect
174. C33#405semidirect
252. C32#4689semidirect
290. C3#10593765semidirect

Maximal subgroups

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

subgroupindex
#117242542
#117242462
D3×C8×C3⋊D32
C8×C3⋊#1572
C3×C8×#1572
#117227052
#117227042
#105937653
#46899