Finite Group Databasebeta

← #13140003

Subgroups — #13140003

Order 1728.

559 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 559 conjugacy classes of subgroups — too many to draw. Only its 131 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×C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
2#11720574C211yesmaximal, characteristic
3#63658⋊C3C211yesmaximal, characteristic
4#11720334C212yesmaximal
5#11720334C212yesmaximal
6#10591229—31nomaximal
7#10591223—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084C2212yes—
10C22×(C3⋊Dic3)⋊C3C2212yes—
11C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
12C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
13C4×(C3⋊Dic3)⋊C3C414yescomplemented
14C2×C4×C3⋊Dic3D311yescharacteristic
15C2×C3×C4×Dic3—31no—
16#63658D311yescharacteristic
17C3×C3⋊#95—31no—
18C3×C3⋊#95—31no—
19C3×C3⋊#95—31no—
20C3×C3⋊#95—61no—
21#63579—32no—
22#63579—32no—
23#63579—32no—
24#63579—32no—
25C23×H(3,𝔽3)C2311yescharacteristic
26C2×C4×H(3,𝔽3)—22no—
27C2×C4×H(3,𝔽3)C2×C412yes—
28C2×#1084—22no—
29C2×#1084—22no—
30C2×#1084—22no—
31C2×(C3⋊Dic3)⋊C3—22no—
32C2×(C3⋊Dic3)⋊C3—21no—
33C2×(C3⋊Dic3)⋊C3—21no—
34C2×(C3⋊Dic3)⋊C3—22no—
35C2×(C3⋊Dic3)⋊C3—22no—
36C2×(C3⋊Dic3)⋊C3—22no—
37C2×(C3⋊Dic3)⋊C3C2×C412yes—
38C2×(C3⋊Dic3)⋊C3C2×C412yes—
39#4937—91no—
40#4651—91no—
41#4651—91no—
42C22×C32×C4—21no—
43C22×C32×C4C2×D311yescharacteristic
44C22×C32×C4C2×D311yescharacteristic
45C22×C3⋊Dic3C2×D312yes—
46C22×C3⋊Dic3C2×D311yescharacteristic
47C22×C3⋊Dic3C2×D311yescharacteristic
48C22×C3×Dic3—31no—
49C22×C3×Dic3—31no—
50C22×C3×Dic3—32no—
51C22×C3×Dic3—32no—
52C22×C3×Dic3—32no—
53C22×C3×Dic3—62no—
54C4×C3⋊Dic3Dic314yescomplemented
55C3×C4×Dic3—34no—
56C22×H(3,𝔽3)C22×C411yescharacteristic
57C22×H(3,𝔽3)#4112yes—
58C22×H(3,𝔽3)#4111yescharacteristic
59C22×H(3,𝔽3)#4112yes—
60C22×H(3,𝔽3)#4111yescharacteristic
61C4×H(3,𝔽3)—22no—
62#1084—42no—
63(C3⋊Dic3)⋊C3—22no—
64(C3⋊Dic3)⋊C3—22no—
65(C3⋊Dic3)⋊C3—42no—
66C2×C3×C42—91no—
67C2×C4×Dic3—31no—
68C2×C4×Dic3—91no—
69C3×#95—91no—
70C3×#95—91no—
71C3⋊#95—31no—
72C3⋊#95—31no—
73C3⋊#95—31no—
74C3⋊#95—61no—
75C3⋊#95—91no—
76C3⋊#95—92no—
77C3⋊#95—92no—
78C3⋊#95—92no—
79C3⋊#95—92no—
80C3⋊#95—92no—
81C3⋊#95—92no—
82C23×C32—21no—
83C23×C32C22×D311yescharacteristic
84C23×C32C22×D311yescharacteristic
85C2×C32×C4—22no—
86C2×C32×C4—22no—
87C2×C32×C4—22no—
88C2×C32×C4—42no—
89C2×C32×C4C4×D312yes—
90C2×C32×C4C2×Dic312yes—
91C2×C3⋊Dic3—21no—
92C2×C3⋊Dic3—22no—
93C2×C3⋊Dic3—21no—
94C2×C3⋊Dic3—22no—
95C2×C3⋊Dic3—22no—
96C2×C3⋊Dic3—22no—
97C2×C3⋊Dic3C2×Dic312yes—
98C2×C3⋊Dic3C2×Dic312yes—
99C2×C3×Dic3—32no—
100C2×C3×Dic3—32no—
101C2×C3×Dic3—62no—
102C2×C3×Dic3—61no—
103C2×C3×Dic3—61no—
104C2×C3×Dic3—64no—
105C2×C3×Dic3—62no—
106C2×C3×Dic3—62no—
107C2×C3×Dic3—62no—
108C2×C3×Dic3—62no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—62no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—62no—
114C2×C3×Dic3—122no—
115C2×C3×Dic3—122no—
116#383—271noSylow
117C2×H(3,𝔽3)#11711yescharacteristic
118C2×H(3,𝔽3)#11711yescharacteristic
119C2×H(3,𝔽3)#11711yescharacteristic
120C2×H(3,𝔽3)#12612yes—
121C2×H(3,𝔽3)#12612yes—
122C22×C3×C4—31no—
123C22×C3×C4—31no—
124C22×C3×C4—61no—
125C22×C3×C4—91no—
126C22×C3×C4—91no—
127C22×C3×C4—92no—
128C22×C3×C4—92no—
129C22×C3×C4D3211yescharacteristic
130C22×Dic3—31no—
131C22×Dic3—31no—
132C22×Dic3—32no—
133C22×Dic3—32no—
134C22×Dic3—32no—
135C22×Dic3—62no—
136C22×Dic3—91no—
137C22×Dic3—91no—
138C22×Dic3—92no—
139C3×C42—94no—
140C4×Dic3—34no—
141C4×Dic3—94no—
142C22×C32—21no—
143C22×C32—22no—
144C22×C32—21no—
145C22×C32—21no—
146C22×C32—22no—
147C22×C32C2×C4×D311yescharacteristic
148C22×C32#23511yescharacteristic
149C22×C32#23512yes—
150C22×C32#23512yes—
151C22×C32#23511yescharacteristic
152C22×C32#23512yes—
153C22×C32#23711yescharacteristic
154C22×C32#23712yes—
155C22×C32#23911yescharacteristic
156C22×C32C22×Dic311yescharacteristic
157C32×C4—22no—
158C32×C4—22no—
159C32×C4—42no—
160C3⋊Dic3—22no—
161C3⋊Dic3—22no—
162C3⋊Dic3—42no—
163C3×Dic3—62no—
164C3×Dic3—62no—
165C3×Dic3—124no—
166C3×Dic3—122no—
167C3×Dic3—122no—
168C3×Dic3—122no—
169C2×C42—91no—
170#95—91no—
171#95—91no—
172#95—272no—
173#95—272no—
174H(3,𝔽3)#38311yescharacteristic, complemented, Sylow
175C23×C3—31no—
176C23×C3—31no—
177C23×C3—61no—
178C23×C3C2×D3211yescharacteristic
179C2×C3×C4—22no—
180C2×C3×C4—32no—
181C2×C3×C4—32no—
182C2×C3×C4—62no—
183C2×C3×C4—62no—
184C2×C3×C4—62no—
185C2×C3×C4—92no—
186C2×C3×C4—92no—
187C2×C3×C4—122no—
188C2×C3×C4—181no—
189C2×C3×C4—182no—
190C2×C3×C4—181no—
191C2×C3×C4—182no—
192C2×C3×C4—182no—
193C2×C3×C4—182no—
194C2×C3×C4—182no—
195C2×C3×C4—182no—
196C2×C3×C4—182no—
197C2×C3×C4D3×Dic312yes—
198C2×Dic3—32no—
199C2×Dic3—32no—
200C2×Dic3—61no—
201C2×Dic3—62no—
202C2×Dic3—61no—
203C2×Dic3—62no—
204C2×Dic3—62no—
205C2×Dic3—62no—
206C2×Dic3—62no—
207C2×Dic3—62no—
208C2×Dic3—62no—
209C2×Dic3—64no—
210C2×Dic3—62no—
211C2×Dic3—62no—
212C2×Dic3—62no—
213C2×Dic3—92no—
214C2×Dic3—92no—
215C2×Dic3—122no—
216C2×Dic3—122no—
217C2×Dic3—181no—
218C2×Dic3—182no—
219C2×Dic3—181no—
220C2×Dic3—182no—
221C2×Dic3—182no—
222C2×Dic3—182no—
223C2×C32—21no—
224C2×C32—21no—
225C2×C32—22no—
226C2×C32—22no—
227C2×C32—21no—
228C2×C32#87211yescharacteristic
229C2×C32#87612yes—
230C2×C32#87612yes—
231C2×C32#87711yescharacteristic
232C2×C32#87912yes—
233C2×C32#89211yescharacteristic
234C2×C32#89812yes—
235C2×C32#92611yescharacteristic
236C2×C32#92611yescharacteristic
237C2×C32#92611yescharacteristic
238C22×C4—91no—
239C22×C4—91no—
240C22×C4—92no—
241C22×C4—92no—
242C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
243C42—94no—
244C22×C3—31no—
245C22×C3—31no—
246C22×C3—32no—
247C22×C3—32no—
248C22×C3—31no—
249C22×C3—31no—
250C22×C3—31no—
251C22×C3—31no—
252C22×C3—32no—
253C22×C3—32no—
254C22×C3—61no—
255C22×C3—62no—
256C22×C3—61no—
257C22×C3—61no—
258C22×C3—62no—
259C22×C3#379511yescharacteristic
260C22×C3#379812yes—
261C22×C3#379911yescharacteristic
262C22×C3C2×D3×Dic311yescharacteristic
263C22×C3#380412yes—
264C3×C4—22no—
265C3×C4—62no—
266C3×C4—62no—
267C3×C4—122no—
268C3×C4—182no—
269C3×C4—182no—
270C3×C4—362no—
271C3×C4—362no—
272Dic3—62no—
273Dic3—62no—
274Dic3—122no—
275Dic3—122no—
276Dic3—124no—
277Dic3—122no—
278Dic3—182no—
279Dic3—182no—
280Dic3—362no—
281C32—21no—
282C32#465111yescharacteristic, complemented
283C32#493711yescharacteristic, complemented
284C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
285C2×C4—22no—
286C2×C4—92no—
287C2×C4—92no—
288C2×C4—181no—
289C2×C4—182no—
290C2×C4—181no—
291C2×C4—182no—
292C2×C4—182no—
293C2×C4—182no—
294C2×C4—182no—
295C2×C4—182no—
296C2×C4—182no—
297C2×C4((C3⋊D3)⋊C3)⋊C412yes—
298C2×C3—31no—
299C2×C3—31no—
300C2×C3—31no—
301C2×C3—31no—
302C2×C3—32no—
303C2×C3—32no—
304C2×C3—31no—
305C2×C3—31no—
306C2×C3—32no—
307C2×C3—32no—
308C2×C3—61no—
309C2×C3—61no—
310C2×C3—62no—
311C2×C3—61no—
312C2×C3—62no—
313C2×C3#6384111yescharacteristic
314C2×C3#6385511yescharacteristic
315C2×C3#6385912yes—
316C2×C3#6386812yes—
317C2×C3#6395511yescharacteristic
318C22#8864811yescharacteristic
319C22#8865012yes—
320C22#8865211yescharacteristic
321C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
322C22#8867112yes—
323C4—22no—
324C4—182no—
325C4—182no—
326C4—362no—
327C4—362no—
328C3—31no—
329C3—31no—
330C3—61no—
331C3#1059122311yescharacteristic
332C2#1172108111yescharacteristic
333C2#1172109212yes—
334C2#1172110511yescharacteristic
335C2#1172110912yes—
336C2#1172129611yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radical, cosocleC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#383
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
13. C4×(C3⋊Dic3)⋊C3C4semidirect
54. C4×C3⋊Dic3Dic3semidirect
174. H(3,𝔽3)#383semidirect
282. C32#4651semidirect
283. C32#4937semidirect

Maximal subgroups

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

subgroupindex
C2×C4×(C3⋊Dic3)⋊C32
#117205742
#63658⋊C32
#117203342
#105912293
#105912233