Finite Group Databasebeta

← #13155695

Subgroups — #13155695

Order 1728.

578 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 130 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
1C3×Dic3×Dic6C212yesmaximal
2C2×C3×#3718C211yesmaximal, characteristic
3#11722079C211yesmaximal, characteristic
4#11722037C212yesmaximal
5#11722019C211yesmaximal, characteristic
6#10591576—32nomaximal
7#10590381—31nomaximal
8C2×C32×Dic6C2212yes—
9C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
10C3×C4⋊(C3⋊Dic3)C2212yes—
11C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
12C32×C4×Dic3C2211yescharacteristic
13C3×Dic3⋊Dic3—22no—
14C3×Dic32—22no—
15C3×#3718C414yescomplemented
16C3×Q8×Dic3—32no—
17C2×C3×#216—32no—
18C3×C4×Dic6—32no—
19Dic3×Dic6D312yes—
20C2×#3718D311yescharacteristic
21#63662—32no—
22#63630—31no—
23#63574—31no—
24#63573—32no—
25#63565—32no—
26#63565—32no—
27#63565—62no—
28#63554—32no—
29#63554—61no—
30C2×C33×C4C2311yescharacteristic
31C2×C3×C3⋊Dic3—22no—
32C2×C32×Dic3—22no—
33C2×C32×Dic3D412yes—
34C32×Dic6—22no—
35C32×Dic6C2×C414yes—
36C3⋊(C3⋊(C3⋊C8))—21no—
37C3×C3⋊(C3⋊C8)C2×C412yes—
38#5184—91no—
39#5030—92no—
40C2×C32×Q8—32no—
41C2×C3×Dic6—22no—
42C2×C3×Dic6C2×D312yes—
43C2×C3×Dic6C2×D312yes—
44C32×C42—31no—
45C4⋊(C3⋊Dic3)C2×D312yes—
46C2×C3⋊(C3⋊C8)—32no—
47C2×C3⋊(C3⋊C8)—31no—
48C2×C3⋊(C3⋊C8)—61no—
49C2×C3⋊(C3⋊C8)—62no—
50C2×C3⋊(C3⋊C8)—121no—
51C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
52C3×#216—38no—
53C3×C4⋊Dic3—32no—
54C3×C4⋊Dic3—32no—
55C3×C4⋊Dic3—62no—
56C3×C4⋊Dic3—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—
64Dic3⋊Dic3—22no—
65Dic32—22no—
66#3718Dic314yescomplemented
67C22×C33C2×D411yescharacteristic
68C33×C4C22×C411yescharacteristic
69C33×C4#4111yescharacteristic
70C3×C3⋊Dic3—42no—
71C32×Dic3—24no—
72C32×Dic3—22no—
73C2×C3×Q16—91no—
74C3×C4×Q8—92no—
75Q8×Dic3—32no—
76C2×#216—32no—
77C4×Dic6—32no—
78#835—92no—
79#835—92no—
80#835—182no—
81#817—91no—
82#815—91no—
83#810—92no—
84#808—92no—
85#803—92no—
86#803—181no—
87C32×Q8—34no—
88C32×Q8—62no—
89C2×C32×C4—21no—
90C2×C32×C4—22no—
91C2×C32×C4—32no—
92C2×C32×C4—41no—
93C2×C32×C4—62no—
94C2×C32×C4C22×D312yes—
95C2×C32×C4C22×D311yescharacteristic
96C2×C3⋊Dic3—22no—
97C2×C3×Dic3—22no—
98C2×C3×Dic3—22no—
99C2×C3×Dic3—22no—
100C2×C3×Dic3—42no—
101C2×C3×Dic3—62no—
102C2×C3×Dic3—62no—
103C2×C3×Dic3—122no—
104C2×C3×Dic3#6712yes—
105C2×C3×Dic3#6712yes—
106C3×Dic6—22no—
107C3×Dic6—22no—
108C3×Dic6—24no—
109C3×Dic6—42no—
110C3×Dic6C4×D314yes—
111C3×Dic6C2×Dic314yes—
112C3⋊(C3⋊C8)—61no—
113C3⋊(C3⋊C8)—62no—
114C3⋊(C3⋊C8)—62no—
115C3⋊(C3⋊C8)—122no—
116C3⋊(C3⋊C8)—122no—
117C3⋊(C3⋊C8)C2×Dic312yes—
118C3×C3⋊C8—34no—
119C3×C3⋊C8—62no—
120#441—271noSylow
121C2×C33C4×D411yescharacteristic
122C2×C33#13712yes—
123C2×C3×Q8—32no—
124C2×C3×Q8—32no—
125C2×C3×Q8—62no—
126C2×Dic6D3212yes—
127C3×Q16—94no—
128C2×C3×C8—91no—
129C3×C4⋊C4—92no—
130C3×C4⋊C4—182no—
131C3×C42—32no—
132C3×C42—61no—
133C3×C42—182no—
134#216—38no—
135C4⋊Dic3—32no—
136C4⋊Dic3—32no—
137C4⋊Dic3—62no—
138C4⋊Dic3—62no—
139Dic3⋊C4—62no—
140C4×Dic3—62no—
141C4×Dic3—62no—
142C4×Dic3D3211yescharacteristic
143C2×C3⋊C8—32no—
144C2×C3⋊C8—61no—
145C2×C3⋊C8—92no—
146C2×C3⋊C8—91no—
147C2×C3⋊C8—181no—
148C2×C3⋊C8—182no—
149C2×C3⋊C8—361no—
150C22×C32—21no—
151C22×C32—22no—
152C22×C32—41no—
153C22×C32D3×D411yescharacteristic
154C22×C32C2×#6712yes—
155C32×C4—21no—
156C32×C4—21no—
157C32×C4—22no—
158C32×C4—22no—
159C32×C4—41no—
160C32×C4—41no—
161C32×C4—62no—
162C32×C4—64no—
163C32×C4C2×C4×D312yes—
164C32×C4#23512yes—
165C32×C4#23711yescharacteristic
166C32×C4C22×Dic311yescharacteristic
167C3⋊Dic3—42no—
168C3×Dic3—24no—
169C3×Dic3—24no—
170C3×Dic3—22no—
171C3×Dic3—22no—
172C3×Dic3—44no—
173C3×Dic3—42no—
174C3×Dic3—122no—
175C3×Dic3—122no—
176C3×Dic3—242no—
177C2×Q16—91no—
178C4×Q8—92no—
179#106—271no—
180#103—272no—
181#97—271no—
182C33#44111yescharacteristic, complemented, Sylow
183C3×Q8—34no—
184C3×Q8—34no—
185C3×Q8—64no—
186C3×Q8—62no—
187C3×Q8—62no—
188C3×Q8—122no—
189C2×C3×C4—21no—
190C2×C3×C4—22no—
191C2×C3×C4—32no—
192C2×C3×C4—32no—
193C2×C3×C4—41no—
194C2×C3×C4—62no—
195C2×C3×C4—62no—
196C2×C3×C4—62no—
197C2×C3×C4—122no—
198C2×C3×C4—182no—
199C2×C3×C4C2×D3211yescharacteristic
200C2×C3×C4C2×D3212yes—
201C2×Dic3—22no—
202C2×Dic3—62no—
203C2×Dic3—62no—
204C2×Dic3—122no—
205C2×Dic3#62512yes—
206Dic6—22no—
207Dic6D3×Dic314yes—
208C3×C8—92no—
209C3⋊C8—34no—
210C3⋊C8—62no—
211C3⋊C8—182no—
212C3⋊C8—181no—
213C3⋊C8—182no—
214C3⋊C8—362no—
215C3⋊C8—362no—
216C2×C32—22no—
217C2×C32—21no—
218C2×C32—22no—
219C2×C32—22no—
220C2×C32—22no—
221C2×C32—41no—
222C2×C32—42no—
223C2×C32#91812yes—
224C2×C32C4×#6712yes—
225C2×C32D4×Dic311yescharacteristic
226C2×C32#94212yes—
227C2×C32#95112yes—
228C2×Q8—32no—
229Q16—94no—
230C2×C8—91no—
231C2×C8—271no—
232C4⋊C4—92no—
233C4⋊C4—182no—
234C42—31no—
235C42—182no—
236C22×C3—22no—
237C22×C3—21no—
238C22×C3—41no—
239C22×C3C2×#62511yescharacteristic
240C22×C3D3×#6712yes—
241C3×C4—21no—
242C3×C4—22no—
243C3×C4—21no—
244C3×C4—22no—
245C3×C4—41no—
246C3×C4—41no—
247C3×C4—64no—
248C3×C4—62no—
249C3×C4—62no—
250C3×C4—64no—
251C3×C4—122no—
252C3×C4—124no—
253C3×C4—362no—
254C3×C4#379811yescharacteristic
255C3×C4C4×D3211yescharacteristic
256C3×C4C2×D3×Dic312yes—
257C3×C4#380412yes—
258Dic3—22no—
259Dic3—24no—
260Dic3—122no—
261Dic3—122no—
262Dic3—242no—
263C32—22no—
264C32—21no—
265C32—41no—
266C32#503012yescomplemented
267C32#518411yescharacteristic, complemented
268Q8—34no—
269Q8—62no—
270C2×C4—32no—
271C2×C4—62no—
272C2×C4—182no—
273C2×C4D3311yescharacteristic
274C8—92no—
275C8—541no—
276C2×C3—22no—
277C2×C3—22no—
278C2×C3—22no—
279C2×C3—22no—
280C2×C3—21no—
281C2×C3—42no—
282C2×C3—41no—
283C2×C3#6382712yes—
284C2×C3C4×#62511yescharacteristic
285C2×C3#6394312yes—
286C2×C3#6394812yes—
287C2×C3Dic3×#6712yes—
288C22D3×#62511yescharacteristic
289C4—62no—
290C4—64no—
291C4—362no—
292C4D32×Dic311yescharacteristic
293C4#8895911yescharacteristic
294C3—21no—
295C3—22no—
296C3—41no—
297C3#1059038111yescharacteristic, complemented
298C3#1059157612yescomplemented
299C2#1172288312yes—
300C2Dic3×#62511yescharacteristic
subgroupG/H
centerC22D3×#625
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socle, prime radicalC22×C33C2×D4
hypocenterC33#441
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
15. C3×#3718C4semidirect
66. #3718Dic3semidirect
182. C33#441semidirect
266. C32#5030semidirect
267. C32#5184semidirect
297. C3#10590381semidirect
298. C3#10591576semidirect

Maximal subgroups

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

subgroupindex
C3×Dic3×Dic62
C2×C3×#37182
#117220792
#117220372
#117220192
#105915763
#105903813