Finite Group Databasebeta

← #13155659

Subgroups — #13155659

Order 1728.

590 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 126 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×#88812C211yesmaximal, characteristic
2C3×#63844C212yesmaximal
3#11722039C212yesmaximal
4Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic
5C3×#63576C211yesmaximal, characteristic
6#10590481—32nomaximal
7C4×#3718—31nomaximal
8C2×C3×#634C2212yes—
9C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
10C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
11C32×C4⋊Dic3C2212yes—
12C32×C4×Dic3C2211yescharacteristic
13#88812C414yescomplemented
14C3×#3724—22no—
15C3×Dic32—22no—
16C3×#887—32no—
17C3×C4×Dic6—32no—
18#63844D312yes—
19C2×#3719—32no—
20C2×#3718—31no—
21#63637—32no—
22C4×C3⋊(C3⋊C8)—31no—
23C3×#807—32no—
24#63576D311yescharacteristic
25#63565—32no—
26#63565—32no—
27#63565—62no—
28Dic3×C3⋊C8—32no—
29Dic3×C3⋊C8—61no—
30C2×C33×C4C2311yescharacteristic
31C2×C3×C3⋊Dic3—22no—
32C3×#634—22no—
33C3×#634C2×C414yes—
34C2×C32×Dic3—22no—
35C2×C32×Dic3D411yescharacteristic
36C2×C32×Dic3D411yescharacteristic
37C3⋊(C3⋊(C3⋊C8))—21no—
38C3×C3⋊(C3⋊C8)C2×C412yes—
39C4×#216—92no—
40#4873—91no—
41C2×#634C2×D312yes—
42C2×C3×Dic6—32no—
43C2×C3×Dic6—32no—
44C2×C3×Dic6—62no—
45C32×C4⋊C4—32no—
46C32×C42—31no—
47C2×C3⋊(C3⋊C8)—32no—
48C2×C3⋊(C3⋊C8)—31no—
49C2×C3⋊(C3⋊C8)—61no—
50C2×C3⋊(C3⋊C8)—62no—
51C2×C3⋊(C3⋊C8)—121no—
52C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
53C3×C4⋊Dic3—22no—
54C3×C4⋊Dic3C2×D312yes—
55C3×C4⋊Dic3C2×D312yes—
56C3×Dic3⋊C4—62no—
57C3×Dic3⋊C4—62no—
58C3×C4×Dic3—21no—
59C3×C4×Dic3—62no—
60C3×C4×Dic3—62no—
61C3×C4×Dic3C2×D312yes—
62C2×C3×C3⋊C8—32no—
63C2×C3×C3⋊C8—61no—
64#3724—22no—
65Dic32—22no—
66#3719—38no—
67#3718—34no—
68C22×C33C2×D411yescharacteristic
69C33×C4C22×C411yescharacteristic
70C33×C4#4111yescharacteristic
71C3×C3⋊Dic3—24no—
72C32×Dic3—21no—
73C32×Dic3—42no—
74C32×Dic3Q1612yescomplemented
75C3×C4×Q8—92no—
76C2×#216—92no—
77C2×Dic12—91no—
78#887—32no—
79C4×Dic6—32no—
80C3×#107—91no—
81#835—92no—
82C8×Dic3—91no—
83#810—92no—
84#810—92no—
85#810—182no—
86#807—32no—
87C4×C3⋊C8—92no—
88C4×C3⋊C8—181no—
89C2×C32×C4—21no—
90C2×C32×C4—22no—
91C2×C32×C4—31no—
92C2×C32×C4—31no—
93C2×C32×C4—41no—
94C2×C32×C4—62no—
95C2×C32×C4C22×D312yes—
96C2×C32×C4C22×D311yescharacteristic
97C2×C3⋊Dic3—22no—
98#634—22no—
99#634C4×D314yes—
100C2×C3×Dic3—22no—
101C2×C3×Dic3—22no—
102C2×C3×Dic3—21no—
103C2×C3×Dic3—21no—
104C2×C3×Dic3—42no—
105C2×C3×Dic3—62no—
106C2×C3×Dic3—62no—
107C2×C3×Dic3—122no—
108C2×C3×Dic3#6712yes—
109C2×C3×Dic3#6712yes—
110C3×Dic6—34no—
111C3×Dic6—34no—
112C3×Dic6—64no—
113C3×Dic6—62no—
114C3×Dic6—62no—
115C3×Dic6—122no—
116C3⋊(C3⋊C8)—61no—
117C3⋊(C3⋊C8)—62no—
118C3⋊(C3⋊C8)—62no—
119C3⋊(C3⋊C8)—122no—
120C3⋊(C3⋊C8)—122no—
121C3⋊(C3⋊C8)C4×D312yes—
122C3×C3⋊C8—34no—
123C3×C3⋊C8—62no—
124C4×Q16—271noSylow
125C2×C33C4×D411yescharacteristic
126C2×C33C2×Q1611yescharacteristic
127C2×C33#13511yescharacteristic
128C2×C3×Q8—92no—
129C2×Dic6—32no—
130C2×Dic6—32no—
131C2×Dic6—62no—
132C2×C3×C8—91no—
133C3×C4⋊C4—32no—
134C3×C4⋊C4—32no—
135C3×C4⋊C4—62no—
136C3×C4⋊C4—182no—
137C3×C42—32no—
138C3×C42—61no—
139C3×C42—182no—
140#216—98no—
141C4⋊Dic3D3212yes—
142Dic3⋊C4—62no—
143Dic3⋊C4—62no—
144C4×Dic3—62no—
145C4×Dic3—62no—
146C4×Dic3D3211yescharacteristic
147C2×C3⋊C8—32no—
148C2×C3⋊C8—61no—
149C2×C3⋊C8—92no—
150C2×C3⋊C8—91no—
151C2×C3⋊C8—181no—
152C2×C3⋊C8—182no—
153C2×C3⋊C8—361no—
154Dic12—94no—
155C22×C32—21no—
156C22×C32—22no—
157C22×C32—41no—
158C22×C32D3×D411yescharacteristic
159C22×C32C2×#6712yes—
160C32×C4—21no—
161C32×C4—22no—
162C32×C4—21no—
163C32×C4—22no—
164C32×C4—32no—
165C32×C4—41no—
166C32×C4—41no—
167C32×C4—61no—
168C32×C4—122no—
169C32×C4C2×C4×D311yescharacteristic
170C32×C4C2×C4×D312yes—
171C32×C4#23512yes—
172C32×C4#23911yescharacteristic
173C3⋊Dic3—24no—
174C3×Dic3—22no—
175C3×Dic3—22no—
176C3×Dic3—22no—
177C3×Dic3—42no—
178C3×Dic3—42no—
179C3×Dic3—64no—
180C3×Dic3—64no—
181C3×Dic3—82no—
182C3×Dic3—124no—
183C3×Dic3#21614yescomplemented
184C2×Q16—271no—
185C4×Q8—92no—
186#107—91no—
187#103—272no—
188C4×C8—271no—
189C33C4×Q1611yescharacteristic, complemented, Sylow
190C3×Q8—94no—
191C3×Q8—182no—
192C2×C3×C4—21no—
193C2×C3×C4—22no—
194C2×C3×C4—32no—
195C2×C3×C4—32no—
196C2×C3×C4—41no—
197C2×C3×C4—61no—
198C2×C3×C4—61no—
199C2×C3×C4—62no—
200C2×C3×C4—62no—
201C2×C3×C4—122no—
202C2×C3×C4—182no—
203C2×C3×C4C2×D3211yescharacteristic
204C2×C3×C4C2×D3212yes—
205C2×Dic3—22no—
206C2×Dic3—62no—
207C2×Dic3—62no—
208C2×Dic3—122no—
209C2×Dic3#62511yescharacteristic
210C2×Dic3#62511yescharacteristic
211Dic6—34no—
212Dic6—34no—
213Dic6—64no—
214Dic6—62no—
215Dic6—62no—
216Dic6—122no—
217C3×C8—92no—
218C3⋊C8—34no—
219C3⋊C8—62no—
220C3⋊C8—182no—
221C3⋊C8—181no—
222C3⋊C8—182no—
223C3⋊C8—362no—
224C3⋊C8—362no—
225C2×C32—21no—
226C2×C32—21no—
227C2×C32—21no—
228C2×C32—22no—
229C2×C32—22no—
230C2×C32—22no—
231C2×C32—41no—
232C2×C32—41no—
233C2×C32—41no—
234C2×C32#89311yescharacteristic
235C2×C32#91211yescharacteristic
236C2×C32D3×Q1611yescharacteristic
237C2×C32C4×#6712yes—
238C2×C32C2×#21612yes—
239C2×C32#95012yes—
240C2×Q8—92no—
241Q16—274no—
242C2×C8—91no—
243C2×C8—271no—
244C4⋊C4—32no—
245C4⋊C4—182no—
246C42—31no—
247C42—182no—
248C22×C3—22no—
249C22×C3—21no—
250C22×C3—41no—
251C22×C3C2×#62511yescharacteristic
252C22×C3D3×#6712yes—
253C3×C4—21no—
254C3×C4—22no—
255C3×C4—21no—
256C3×C4—22no—
257C3×C4—34no—
258C3×C4—41no—
259C3×C4—41no—
260C3×C4—62no—
261C3×C4—62no—
262C3×C4—62no—
263C3×C4—122no—
264C3×C4—122no—
265C3×C4—184no—
266C3×C4—242no—
267C3×C4#379811yescharacteristic
268C3×C4#379912yes—
269C3×C4C4×D3211yescharacteristic
270C3×C4C4×D3212yes—
271Dic3—21no—
272Dic3—42no—
273Dic3—64no—
274Dic3—64no—
275Dic3—124no—
276Dic3#371812yescomplemented
277C32—22no—
278C32—21no—
279C32—41no—
280C32#487311yescharacteristic, complemented
281C32C4×#21612yescomplemented
282Q8—94no—
283Q8—182no—
284C2×C4—31no—
285C2×C4—31no—
286C2×C4—62no—
287C2×C4—182no—
288C2×C4D3311yescharacteristic
289C8—92no—
290C8—541no—
291C2×C3—21no—
292C2×C3—22no—
293C2×C3—21no—
294C2×C3—22no—
295C2×C3—22no—
296C2×C3—21no—
297C2×C3—41no—
298C2×C3—41no—
299C2×C3—41no—
300C2×C3#6382211yescharacteristic
301C2×C3C2×#371811yescharacteristic
302C2×C3C4×#62511yescharacteristic
303C2×C3#6390412yes—
304C2×C3#6393312yes—
305C2×C3D3×#21612yes—
306C22D3×#62511yescharacteristic
307C4—32no—
308C4—61no—
309C4—122no—
310C4—184no—
311C4#8894911yescharacteristic
312C4#8896511yescharacteristic
313C3—21no—
314C3—22no—
315C3—41no—
316C3C4×#371811yescharacteristic, complemented
317C3#1059048112yescomplemented
318C2D3×#371811yescharacteristic
319C2#1172285011yescharacteristic
320C2#1172293111yescharacteristic
subgroupG/H
centerC22D3×#625
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socle, prime radicalC22×C33C2×D4
hypocenterC33C4×Q16
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
13. #88812C4semidirect
74. C32×Dic3Q16semidirect
183. C3×Dic3#216semidirect
189. C33C4×Q16semidirect
276. Dic3#3718semidirect
280. C32#4873semidirect
281. C32C4×#216semidirect
316. C3C4×#3718semidirect
317. C3#10590481semidirect

Maximal subgroups

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

subgroupindex
C2×#888122
C3×#638442
#117220392
Dic3×C3⋊(C3⋊C8)2
C3×#635762
#105904813
C4×#37183