Finite Group Databasebeta

← #13139515

Subgroups — #13139515

Order 1728.

518 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 109 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×C3×Dic18C211yesmaximal, characteristic
2#11721813C211yesmaximal, characteristic
3C2×#88418C212yesmaximal, complemented
4C9×#830C211yesmaximal, characteristic
5#11720276C212yesmaximal, complemented
6#10591186—31nomaximal
7#10588405—31nomaximal
8C22×C3×C4×C9C2211yescharacteristic
9C2×#6226C2211yescharacteristic
10C22×C3×Dic9—21no—
11C2×C3×Dic18—21no—
12C2×C3×Dic18—24no—
13C2×C3×Dic18C2212yes—
14#88551—21no—
15C4⋊(C3⋊Dic9)C2211yescharacteristic
16#88534—21no—
17C2×C9×C3⋊C8C2212yes—
18#88418—24no—
19C22×C3×Dic6—31no—
20#64133—31no—
21C2×#3719—32no—
22C22×Dic18D311yescharacteristic
23C3×#830D311yescharacteristic
24#63566—32no—
25#63486—31no—
26#63486—61no—
27C2×Dic36—32no—
28C9×C22⋊C8—31no—
29#63376—32no—
30C23×C3×C9D411yescharacteristic
31C2×C3×C4×C9—22no—
32C2×C3×C4×C9D411yescharacteristic
33C2×C3×C4×C9C2311yescharacteristic
34C2×C3⋊Dic9—21no—
35C2×C3⋊Dic9—21no—
36#6226—22no—
37C2×C3×Dic9—24no—
38C2×C3×Dic9—22no—
39C3×Dic18—24no—
40C3×Dic18—22no—
41C3×Dic18D414yes—
42C9×C3⋊C8—22no—
43#5228—91no—
44#4739—91no—
45C22×C32×C4C2×D311yescharacteristic
46C2×#634—31no—
47C22×C3×Dic3—61no—
48C2×C3×Dic6—32no—
49C2×C3×Dic6—64no—
50C2×C3×Dic6—61no—
51#3757—61no—
52C4⋊(C3⋊Dic3)—31no—
53#3750—61no—
54C2×C3×C3⋊C8C2×D312yes—
55#3719—64no—
56C22×C4×C9—21no—
57C22×C4×C9C2×D311yescharacteristic
58C22×Dic9—21no—
59C2×Dic18—21no—
60C2×Dic18—24no—
61C2×Dic18—31no—
62C2×Dic18—61no—
63C2×Dic18C2×D312yes—
64C2×C8×C9—32no—
65#3676—62no—
66#3676—61no—
67C4⋊Dic9—31no—
68C4⋊Dic9—61no—
69Dic9⋊C4—62no—
70Dic9⋊C4—61no—
71Dic36—64no—
72C22×C3×C9—21no—
73C22×C3×C9—21no—
74C22×C3×C9Q1612yescomplemented
75C22×C3×C9C2×D411yescharacteristic
76C3×C4×C9—21no—
77C3×C4×C9C2×D412yes—
78C3⋊Dic9—41no—
79C3⋊Dic9—41no—
80C3×Dic9—24no—
81C22×C3×Q8—91no—
82C22×Dic6—31no—
83C2×#216—92no—
84#924—91no—
85#924—91no—
86#924—181no—
87C2×Dic12—92no—
88C3×C22⋊C8—31no—
89#835—92no—
90#830D911yescharacteristic
91#816—92no—
92C23×C32D1211yescharacteristic
93C2×C32×C4—22no—
94C2×C32×C4D1211yescharacteristic
95C2×C32×C4C22×D311yescharacteristic
96C2×C3⋊Dic3—61no—
97C2×C3⋊Dic3—61no—
98#634—62no—
99C2×C3×Dic3—64no—
100C2×C3×Dic3—62no—
101C3×Dic6—34no—
102C3×Dic6—64no—
103C3×Dic6—62no—
104C23×C9—21no—
105C23×C9#6711yescharacteristic
106C3×C3⋊C8—22no—
107C2×C4×C9—22no—
108C2×C4×C9—21no—
109C2×C4×C9—21no—
110C2×C4×C9—42no—
111C2×C4×C9#6711yescharacteristic
112C2×C4×C9C22×D311yescharacteristic
113C2×Dic9—24no—
114C2×Dic9—22no—
115C2×Dic9—62no—
116C2×Dic9—62no—
117C2×Dic9—61no—
118C2×Dic9—61no—
119Dic18—24no—
120Dic18—22no—
121Dic18—62no—
122Dic18—122no—
123Dic18#6714yes—
124C8×C9—62no—
125#451—271noSylow
126C2×C3×C9—22no—
127C2×C3×C9#12011yescharacteristic
128C2×C3×C9C2×Q1611yescharacteristic
129C2×C3×C9#13711yescharacteristic
130C2×C3×Q8—92no—
131C2×C3×Q8—184no—
132C2×C3×Q8—181no—
133C22×C3×C4—21no—
134C22×C3×C4—181no—
135C22×C3×C4C2×D911yescharacteristic
136C22×C3×C4D3211yescharacteristic
137C22×Dic3—61no—
138C2×Dic6—32no—
139C2×Dic6—64no—
140C2×Dic6—61no—
141C2×Dic6—91no—
142C2×Dic6—91no—
143C2×Dic6—181no—
144C2×C3×C8—32no—
145#217—182no—
146#217—181no—
147#217—181no—
148#216—184no—
149C4⋊Dic3—91no—
150C4⋊Dic3—91no—
151C4⋊Dic3—181no—
152Dic3⋊C4—182no—
153Dic3⋊C4—181no—
154Dic3⋊C4—181no—
155C2×C3⋊C8C2×D912yes—
156Dic12—184no—
157C22×C32—21no—
158C22×C32—21no—
159C22×C32Dic1212yes—
160C22×C32C2×D1211yescharacteristic
161C32×C4—21no—
162C32×C4D3×D412yes—
163C3⋊Dic3—121no—
164C3⋊Dic3—121no—
165C3×Dic3—64no—
166C22×C9—21no—
167C22×C9—21no—
168C22×C9—22no—
169C22×C9—21no—
170C22×C9—41no—
171C22×C9—41no—
172C22×C9#21612yescomplemented
173C22×C9C2×#6711yescharacteristic
174C4×C9—21no—
175C4×C9—22no—
176C4×C9—41no—
177C4×C9C2×#6712yes—
178Dic9—24no—
179Dic9—122no—
180Dic9—122no—
181Dic9—121no—
182Dic9—121no—
183C22×Q8—91no—
184C2×Q16—272no—
185C22⋊Q8—271no—
186#103—272no—
187C22⋊C8—31no—
188C3×C9#45111yescharacteristic, complemented, Sylow
189C23×C3—21no—
190C23×C3D3611yescharacteristic
191C23×C3#62611yescharacteristic
192C3×Q8—94no—
193C3×Q8—184no—
194C3×Q8—182no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—42no—
200C2×C3×C4—184no—
201C2×C3×C4—182no—
202C2×C3×C4D3611yescharacteristic
203C2×C3×C4C22×D911yescharacteristic
204C2×C3×C4#62611yescharacteristic
205C2×C3×C4C2×D3211yescharacteristic
206C2×Dic3—64no—
207C2×Dic3—62no—
208C2×Dic3—182no—
209C2×Dic3—182no—
210C2×Dic3—181no—
211C2×Dic3—181no—
212C2×Dic3—181no—
213C2×Dic3—181no—
214Dic6—34no—
215Dic6—64no—
216Dic6—62no—
217Dic6—182no—
218Dic6—182no—
219Dic6—362no—
220C3×C8—62no—
221C3⋊C8—22no—
222C2×C32—22no—
223C2×C32#88211yescharacteristic
224C2×C32C2×Dic1211yescharacteristic
225C2×C32#90911yescharacteristic
226C2×C9—22no—
227C2×C9—21no—
228C2×C9—21no—
229C2×C9—21no—
230C2×C9—42no—
231C2×C9#94211yescharacteristic
232C2×C9C2×#21611yescharacteristic
233C2×C9#95311yescharacteristic
234C2×Q8—92no—
235C2×Q8—184no—
236C2×Q8—181no—
237C2×Q8—271no—
238C22×C4—181no—
239C22×C4D3×D911yescharacteristic
240Q16—544no—
241C2×C8—32no—
242C4⋊C4—271no—
243C4⋊C4—541no—
244C22⋊C4—541no—
245C22×C3—21no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—22no—
251C22×C3—41no—
252C22×C3—41no—
253C22×C3Dic3612yescomplemented
254C22×C3C2×D3611yescharacteristic
255C22×C3#371912yes—
256C22×C3C2×#62611yescharacteristic
257C3×C4—21no—
258C3×C4—21no—
259C3×C4—22no—
260C3×C4—41no—
261C3×C4—184no—
262C3×C4D4×D912yes—
263C3×C4D3×#6712yes—
264Dic3—64no—
265Dic3—362no—
266Dic3—362no—
267Dic3—361no—
268Dic3—361no—
269Dic3—361no—
270Dic3—361no—
271C32#473911yescharacteristic
272C9—21no—
273C9#522811yescharacteristic, complemented
274C23#619211yescharacteristic
275Q8—94no—
276Q8—184no—
277Q8—182no—
278Q8—542no—
279C2×C4—22no—
280C2×C4—184no—
281C2×C4—182no—
282C2×C4—541no—
283C2×C4—541no—
284C2×C4#619211yescharacteristic
285C2×C4C2×D3×D911yescharacteristic
286C8—62no—
287C2×C3—22no—
288C2×C3—22no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3—21no—
292C2×C3—42no—
293C2×C3#6344411yescharacteristic
294C2×C3C2×Dic3611yescharacteristic
295C2×C3#6347111yescharacteristic
296C2×C3#6383311yescharacteristic
297C2×C3C2×#371911yescharacteristic
298C2×C3#6397711yescharacteristic
299C22—21no—
300C22—21no—
301C22#8841812yescomplemented
302C22C2×#619211yescharacteristic
303C4—21no—
304C4—184no—
305C4—1081no—
306C4—1081no—
307C4D9×#6712yes—
308C3—21no—
309C3#1058840511yescharacteristic, complemented
310C3#1059118611yescharacteristic
311C2—22no—
312C2#1172087111yescharacteristic
313C2C2×#8841811yescharacteristic
314C2#1172127611yescharacteristic
subgroupG/H
centerC22C2×#6192
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC22×C3×C4×C9C22
socleC22×C32C2×D12
prime radicalC23×C32D12
hypercenterC22×C4D3×D9
hypocenterC3×C9#451
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. C2×#88418C2semidirect
5. #11720276C2semidirect
74. C22×C3×C9Q16semidirect
172. C22×C9#216semidirect
188. C3×C9#451semidirect
253. C22×C3Dic36semidirect
273. C9#5228semidirect
301. C22#88418semidirect
309. C3#10588405semidirect

Maximal subgroups

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

subgroupindex
C22×C3×Dic182
#117218132
C2×#884182
C9×#8302
#117202762
#105911863
#105884053