Finite Group Databasebeta

← #13166572

Subgroups — #13166572

Order 1728.

1016 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 1016 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C22×Dic3×C3⋊D3C211yesmaximal, characteristic
2C2×#88994C211yesmaximal, characteristic
3C2×C3×C22⋊#157C211yesmaximal, characteristic
4#11722722C214yesmaximal, complemented
5#10593811—31nomaximal
6C23×C3×C3⋊D3C2211yescharacteristic
7C22×C3⋊#157C2212yes—
8C22×C3×#157C2212yes—
9C22×C3⋊(C3⋊Dic3)C411yescharacteristic, complemented
10C22×C32×Dic3C411yescharacteristic, complemented
11C2×Dic3×C3⋊D3—22no—
12C2×Dic3×C3⋊D3—22no—
13C2×Dic3×C3⋊D3C412yescomplemented
14C2×Dic3×C3⋊D3C2212yes—
15#88994—22no—
16C3×C22⋊#157—22no—
17C22×C4×C3⋊D3—31no—
18C22×D3×Dic3—62no—
19C2×C22⋊#157—31no—
20C2×C22⋊#157D311yescharacteristic
21#63781—34no—
22C23×C33C2×C411yescharacteristic
23C22×C3×C3⋊D3—24no—
24C22×C3×C3⋊D3C2×C411yescharacteristic
25C22×C3×C3⋊D3D412yes—
26C22×C3×C3⋊D3D412yes—
27C22×C3×C3⋊D3C2311yescharacteristic
28C2×C3⋊#157—22no—
29C2×C3⋊#157—24no—
30C2×C3×#157—22no—
31C2×C3×#157—24no—
32C2×C3⋊(C3⋊Dic3)—22no—
33C2×C3⋊(C3⋊Dic3)C2×C412yes—
34C2×C32×Dic3—22no—
35C2×C32×Dic3C2×C412yes—
36Dic3×C3⋊D3—24no—
37Dic3×C3⋊D3—22no—
38Dic3×C3⋊D3—22no—
39#4943—91nomaximal
40C23×C3⋊D3C2×D311yescharacteristic
41C23×C3×D3—62no—
42C22×#157—32no—
43C22×#157C2×D312yes—
44C22×C32×C4—31no—
45C22×C3⋊Dic3—31no—
46C22×C3⋊Dic3—62no—
47C22×C3⋊Dic3—122no—
48C2×C4×C3⋊D3—32no—
49C2×C4×C3⋊D3—32no—
50C2×C4×C3⋊D3—62no—
51C2×C4×C3⋊D3—62no—
52C22×C3×Dic3—22no—
53C2×D3×Dic3—64no—
54C2×D3×Dic3—64no—
55C2×D3×Dic3—68no—
56C2×D3×Dic3—68no—
57C22⋊#157—22no—
58C22⋊#157—62no—
59C22×C33—22no—
60C22×C33C22⋊C412yes—
61C22×C33C22×C411yescharacteristic
62C2×C3×C3⋊D3—22no—
63C2×C3×C3⋊D3—24no—
64C2×C3×C3⋊D3—24no—
65C2×C3×C3⋊D3C22⋊C412yes—
66C2×C3×C3⋊D3C2×D411yescharacteristic
67C2×C3×C3⋊D3C2×D411yescharacteristic
68C2×C3×C3⋊D3#4114yes—
69C3⋊#157—42no—
70C3×#157—42no—
71C3⋊(C3⋊Dic3)—22no—
72C32×Dic3—22no—
73C23×Dic3—91no—
74C22×C4×D3—182no—
75C2×C3×C22⋊C4—91no—
76C2×#217—91no—
77C3⋊#95—94no—
78C23×C32—22no—
79C23×C32—42no—
80C23×C32C4×D311yescharacteristic
81C22×C3⋊D3—24no—
82C22×C3⋊D3C4×D311yescharacteristic
83C22×C3⋊D3D1212yes—
84C22×C3⋊D3#6712yes—
85C22×C3⋊D3C22×D311yescharacteristic
86C22×C3×D3—62no—
87C22×C3×D3—64no—
88C22×C3×D3—64no—
89C22×C3×D3—62no—
90C22×C3×D3—616no—
91C2×#157—24no—
92C2×#157—22no—
93C2×#157—62no—
94C2×#157—64no—
95C2×C32×C4—32no—
96C2×C32×C4—62no—
97C2×C3⋊Dic3—32no—
98C2×C3⋊Dic3—64no—
99C2×C3⋊Dic3—62no—
100C2×C3⋊Dic3—68no—
101C2×C3⋊Dic3—128no—
102C2×C3⋊Dic3—124no—
103C4×C3⋊D3—64no—
104C4×C3⋊D3—62no—
105C4×C3⋊D3—62no—
106C2×C3×Dic3—28no—
107C2×C3×Dic3—24no—
108D3×Dic3—616no—
109D3×Dic3—68no—
110D3×Dic3—68no—
111#381—271noSylow
112C2×C33—22no—
113C2×C33C2×C22⋊C411yescharacteristic
114C2×C33#11712yes—
115C3×C3⋊D3—22no—
116C3×C3⋊D3#12312yes—
117C3×C3⋊D3#12312yes—
118C24×C3—91no—
119C23×D3—62no—
120C22×C3×C4—62no—
121C22×C3×C4—92no—
122C22×Dic3—91no—
123C22×Dic3—92no—
124C22×Dic3—92no—
125C22×Dic3—92no—
126C22×Dic3—182no—
127C22×Dic3—182no—
128C22×Dic3—182no—
129C22×Dic3—362no—
130C22×Dic3#15711yescharacteristic, complemented
131C2×C4×D3—184no—
132C2×C4×D3—184no—
133C2×C4×D3—188no—
134C2×C4×D3—188no—
135C3×C22⋊C4—182no—
136#217—182no—
137C22×C32—22no—
138C22×C32—28no—
139C22×C32—22no—
140C22×C32—24no—
141C22×C32—48no—
142C22×C32—44no—
143C22×C32—42no—
144C22×C32(C2×D3)⋊C412yes—
145C22×C32C2×C4×D311yescharacteristic
146C2×C3⋊D3—22no—
147C2×C3⋊D3—24no—
148C2×C3⋊D3—24no—
149C2×C3⋊D3(C2×D3)⋊C412yes—
150C2×C3⋊D3C2×D1211yescharacteristic
151C2×C3⋊D3#23714yes—
152C2×C3⋊D3C2×#6711yescharacteristic
153C2×C3×D3—64no—
154C2×C3×D3—62no—
155C2×C3×D3—62no—
156C2×C3×D3—68no—
157C2×C3×D3—616no—
158C2×C3×D3—616no—
159C2×C3×D3—68no—
160#157—42no—
161#157—122no—
162C32×C4—62no—
163C3⋊Dic3—62no—
164C3⋊Dic3—68no—
165C3⋊Dic3—128no—
166C3×Dic3—28no—
167C23×C4—271no—
168C2×C22⋊C4—91no—
169C2×C22⋊C4—271no—
170#95—274no—
171C33#38111yescharacteristic, complemented, Sylow
172C23×C3—22no—
173C23×C3—42no—
174C23×C3—91no—
175C23×C3—91no—
176C23×C3—92no—
177C23×C3—92no—
178C23×C3—184no—
179C23×C3C2×#15711yescharacteristic
180C22×D3—62no—
181C22×D3—64no—
182C22×D3—64no—
183C22×D3—62no—
184C22×D3—616no—
185C2×C3×C4—64no—
186C2×C3×C4—68no—
187C2×C3×C4—182no—
188C2×C3×C4—184no—
189C2×Dic3—22no—
190C2×Dic3—92no—
191C2×Dic3—184no—
192C2×Dic3—182no—
193C2×Dic3—182no—
194C2×Dic3—184no—
195C2×Dic3—188no—
196C2×Dic3—182no—
197C2×Dic3—182no—
198C2×Dic3—184no—
199C2×Dic3—364no—
200C2×Dic3—368no—
201C2×Dic3C2×#15712yes—
202C4×D3—1816no—
203C4×D3—188no—
204C4×D3—188no—
205C2×C32—22no—
206C2×C32—28no—
207C2×C32—24no—
208C2×C32—22no—
209C2×C32—48no—
210C2×C32—42no—
211C2×C32—44no—
212C2×C32#87712yes—
213C2×C32C2×(C2×D3)⋊C411yescharacteristic
214C3⋊D3—22no—
215C3⋊D3#88612yes—
216C3⋊D3#93512yes—
217C3×D3—64no—
218C3×D3—64no—
219C3×D3—68no—
220C24—91no—
221C22×C4—31no—
222C22×C4—92no—
223C22×C4—272no—
224C22×C4—272no—
225C22×C4—271no—
226C22×C4—272no—
227C22×C4—542no—
228C22×C4—542no—
229C22⋊C4—182no—
230C22⋊C4—542no—
231C22×C3—28no—
232C22×C3—22no—
233C22×C3—22no—
234C22×C3—24no—
235C22×C3—48no—
236C22×C3—42no—
237C22×C3—44no—
238C22×C3—94no—
239C22×C3—91no—
240C22×C3—91no—
241C22×C3—92no—
242C22×C3—184no—
243C22×C3—184no—
244C22×C3—182no—
245C22×C3C22⋊#15712yes—
246C22×C3C22×#15711yescharacteristic
247C2×D3—64no—
248C2×D3—62no—
249C2×D3—62no—
250C2×D3—68no—
251C2×D3—616no—
252C2×D3—616no—
253C2×D3—68no—
254C3×C4—68no—
255C3×C4—362no—
256Dic3—22no—
257Dic3—188no—
258Dic3—182no—
259Dic3—368no—
260Dic3—362no—
261C32—22no—
262C32—42no—
263C32#494311yescharacteristic, complemented
264C23—91no—
265C23—91no—
266C23—92no—
267C23—92no—
268C23—184no—
269C23D3×#15711yescharacteristic
270C2×C4—32no—
271C2×C4—62no—
272C2×C4—182no—
273C2×C4—184no—
274C2×C4—272no—
275C2×C4—542no—
276C2×C4—542no—
277C2×C4—544no—
278C2×C4—542no—
279C2×C4—542no—
280C2×C4—544no—
281C2×C3—28no—
282C2×C3—22no—
283C2×C3—24no—
284C2×C3—22no—
285C2×C3—48no—
286C2×C3—44no—
287C2×C3—42no—
288C2×C3—92no—
289C2×C3—92no—
290C2×C3—182no—
291C2×C3#6428612yes—
292C2×C3C2×C22⋊#15711yescharacteristic
293D3—64no—
294D3—64no—
295D3—68no—
296C22—22no—
297C22—94no—
298C22—91no—
299C22—91no—
300C22—92no—
301C22—184no—
302C22—184no—
303C22—182no—
304C22(C2×D3)⋊#15712yes—
305C22C2×D3×#15711yescharacteristic
306C4—62no—
307C4—362no—
308C4—542no—
309C4—1082no—
310C3—22no—
311C3—42no—
312C3#1059381111yescharacteristic, complemented
313C2—22no—
314C2—92no—
315C2—92no—
316C2—182no—
317C2#1172426112yes—
318C2C2×(C2×D3)⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutator, socleC22×C33C22×C4
FittingC23×C33C2×C4
prime radicalC23×C3×C3⋊D3C22
cosocleC22×C3×C3⋊D3C23
hypercenterC23D3×#157
hypocenterC33#381
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
4. #11722722C2semidirect
9. C22×C3⋊(C3⋊Dic3)C4semidirect
10. C22×C32×Dic3C4semidirect
13. C2×Dic3×C3⋊D3C4semidirect
130. C22×Dic3#157semidirect
171. C33#381semidirect
263. C32#4943semidirect
312. C3#10593811semidirect

Maximal subgroups

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

subgroupindex
C22×Dic3×C3⋊D32
C2×#889942
C2×C3×C22⋊#1572
#117227222
#105938113
#49439