Finite Group Databasebeta

← #13147436

Subgroups — #13147436

Order 1728.

670 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6226C211yesmaximal, characteristic
2C9×#869C211yesmaximal, characteristic
3C3×C4×Dic18C211yesmaximal, characteristic
4#11720937C212yesmaximal
5#11720924C212yesmaximal
6#11720917C214yesmaximal
7#11720910C212yesmaximal
8Dic6×Dic9C212yesmaximal
9#10593861—31nomaximal
10#10589823—31nomaximal
11C2×#6226C2211yescharacteristic
12C2×C9×Dic6C2212yes—
13C2×C3×Dic18C2211yescharacteristic
14C3×C42×C9C2211yescharacteristic
15C4⋊(C3⋊Dic9)C2211yescharacteristic
16#88534C2212yes—
17C4×C3⋊Dic9C2212yes—
18C9×C4⋊Dic3C2212yes—
19C9×C4⋊Dic3C2212yes—
20C3×C4⋊Dic9C2211yescharacteristic
21C3×Dic9⋊C4C2212yes—
22C3×C4×Dic9C2212yes—
23Dic3⋊Dic9C2214yes—
24Dic9⋊Dic3C2214yes—
25Dic9⋊Dic3C2214yes—
26Dic3×Dic9C2214yes—
27C4×#634—31no—
28C3×#869D311yescharacteristic
29C3×C4×Dic6—31no—
30#63872—32no—
31#63866—32no—
32#63846—34no—
33#63843—32no—
34Dic3×Dic6—32no—
35C9×C4⋊Q8—31no—
36Q8×Dic9—32no—
37#63506—32no—
38#63452—32no—
39#63451—34no—
40#63449—32no—
41C4×Dic18—31no—
42C4×Dic18—61no—
43C4×Dic18D311yescharacteristic
44C2×C3×C4×C9C2311yescharacteristic
45C2×C3×C4×C9C2311yescharacteristic
46C2×C3×C4×C9C2311yescharacteristic
47C2×C3⋊Dic9C2312yes—
48C2×C3⋊Dic9C2312yes—
49#6226—22no—
50C2×C9×Dic3C2314yes—
51C2×C3×Dic9C2312yes—
52C2×C3×Dic9C2312yes—
53C9×Dic6—24no—
54C3×Dic18Q814yescomplemented
55#5721—91no—
56#5570—91no—
57C2×#634—31no—
58C2×C3×Dic6—31no—
59C2×C3×Dic6C2×D312yes—
60C32×C42C2×D311yescharacteristic
61C4⋊(C3⋊Dic3)—31no—
62#3750—32no—
63C4×C3⋊Dic3—32no—
64C3×C4⋊Dic3—31no—
65C3×C4⋊Dic3C2×D312yes—
66C3×C4⋊Dic3C2×D312yes—
67C3×Dic3⋊C4—32no—
68C3×C4×Dic3—32no—
69Dic3⋊Dic3—34no—
70Dic3⋊Dic3—34no—
71Dic3⋊Dic3—34no—
72Dic32—34no—
73C2×Q8×C9—32no—
74C2×Dic18—31no—
75C2×Dic18—61no—
76C2×Dic18C2×D311yescharacteristic
77C9×C4⋊C4—32no—
78C9×C4⋊C4—32no—
79C42×C9—21no—
80C42×C9C2×D311yescharacteristic
81C4⋊Dic9—31no—
82C4⋊Dic9—34no—
83C4⋊Dic9—61no—
84C4⋊Dic9C2×D311yescharacteristic
85Dic9⋊C4—32no—
86Dic9⋊C4—34no—
87Dic9⋊C4—34no—
88Dic9⋊C4—62no—
89Dic9⋊C4C2×D312yes—
90C4×Dic9—32no—
91C4×Dic9—34no—
92C4×Dic9—62no—
93C4×Dic9C2×D312yes—
94C22×C3×C9C2411yescharacteristic
95C3×C4×C9—21no—
96C3×C4×C9C2×Q812yes—
97C3×C4×C9#4112yes—
98C3⋊Dic9—22no—
99C3⋊Dic9—22no—
100C9×Dic3—24no—
101C3×Dic9—22no—
102C3×Dic9C2×Q814yes—
103C3×C4⋊Q8—31no—
104C3×C4×Q8—91no—
105Q8×Dic3—92no—
106Q8×Dic3—92no—
107#944—92no—
108#890—92no—
109#890—94no—
110#890—92no—
111#889—94no—
112#887—92no—
113#869—92no—
114#869D911yescharacteristic
115C4×Dic6—31no—
116C4×Dic6—92no—
117C4×Dic6—91no—
118C4×Dic6—91no—
119C4×Dic6—181no—
120C2×C32×C4C22×D311yescharacteristic
121C2×C32×C4C22×D311yescharacteristic
122C2×C32×C4C22×D311yescharacteristic
123C2×C3⋊Dic3—32no—
124C2×C3⋊Dic3—32no—
125#634—62no—
126C2×C3×Dic3—32no—
127C2×C3×Dic3—32no—
128C2×C3×Dic3C22×D314yes—
129C3×Dic6—24no—
130C3×Dic6—34no—
131Q8×C9—64no—
132C2×C4×C9—21no—
133C2×C4×C9—21no—
134C2×C4×C9—21no—
135C2×C4×C9—34no—
136C2×C4×C9C22×D311yescharacteristic
137C2×C4×C9C22×D311yescharacteristic
138C2×C4×C9C22×D311yescharacteristic
139C2×Dic9—32no—
140C2×Dic9—32no—
141C2×Dic9—62no—
142C2×Dic9—62no—
143C2×Dic9C22×D312yes—
144C2×Dic9C22×D312yes—
145Dic18—64no—
146Dic18—62no—
147Dic18Dic614yescomplemented
148#557—271noSylow
149C2×C3×C9C22×Q811yescharacteristic
150C2×C3×C9C2×#4111yescharacteristic
151C2×C3×C92−1+411yescharacteristic
152C2×C3×Q8—32no—
153C2×C3×Q8—91no—
154C2×Dic6—31no—
155C2×Dic6—91no—
156C2×Dic6—91no—
157C2×Dic6—181no—
158C2×Dic6C2×D912yes—
159C3×C4⋊C4—32no—
160C3×C4⋊C4—32no—
161C3×C4⋊C4—92no—
162C3×C4⋊C4—91no—
163C3×C42—21no—
164C3×C42—92no—
165C3×C42C2×D911yescharacteristic
166C3×C42D3211yescharacteristic
167C4⋊Dic3—31no—
168C4⋊Dic3—94no—
169C4⋊Dic3—91no—
170C4⋊Dic3—91no—
171C4⋊Dic3—94no—
172C4⋊Dic3—94no—
173C4⋊Dic3—181no—
174C4⋊Dic3C2×D912yes—
175C4⋊Dic3C2×D912yes—
176Dic3⋊C4—32no—
177Dic3⋊C4—94no—
178Dic3⋊C4—94no—
179Dic3⋊C4—92no—
180Dic3⋊C4—92no—
181Dic3⋊C4—94no—
182Dic3⋊C4—182no—
183C4×Dic3—32no—
184C4×Dic3—94no—
185C4×Dic3—92no—
186C4×Dic3—92no—
187C4×Dic3—94no—
188C4×Dic3—182no—
189C22×C32C23×D311yescharacteristic
190C32×C4—21no—
191C32×C4#23712yes—
192C32×C4D3×Q812yes—
193C3⋊Dic3—62no—
194C3⋊Dic3—62no—
195C3×Dic3—24no—
196C3×Dic3—34no—
197C3×Dic3—62no—
198C22×C9—21no—
199C22×C9C23×D311yescharacteristic
200C4×C9—21no—
201C4×C9—22no—
202C4×C9—22no—
203C4×C9—41no—
204C4×C9—64no—
205C4×C9C2×Dic612yes—
206C4×C9#23912yes—
207Dic9—22no—
208Dic9—64no—
209Dic9—62no—
210Dic9—62no—
211Dic9—122no—
212Dic9C2×Dic614yes—
213C4⋊Q8—31no—
214C4⋊Q8—272no—
215#125—274no—
216#125—272no—
217C4×Q8—91no—
218C4×Q8—272no—
219C4×Q8—272no—
220C4×Q8—271no—
221C3×C9#55711yescharacteristic, complemented, Sylow
222C3×Q8—64no—
223C3×Q8—94no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—34no—
228C2×C3×C4—92no—
229C2×C3×C4—92no—
230C2×C3×C4C22×D911yescharacteristic
231C2×C3×C4C22×D911yescharacteristic
232C2×C3×C4C22×D911yescharacteristic
233C2×C3×C4C2×D3211yescharacteristic
234C2×C3×C4C2×D3211yescharacteristic
235C2×C3×C4C2×D3211yescharacteristic
236C2×Dic3—32no—
237C2×Dic3—32no—
238C2×Dic3—92no—
239C2×Dic3—92no—
240C2×Dic3—92no—
241C2×Dic3—92no—
242C2×Dic3—182no—
243C2×Dic3—182no—
244C2×Dic3C22×D914yes—
245Dic6—24no—
246Dic6—34no—
247Dic6—184no—
248Dic6—182no—
249Dic6—182no—
250C2×C32C2×#23711yescharacteristic
251C2×C32C2×D3×Q811yescharacteristic
252C2×C32#100711yescharacteristic
253C2×C9—21no—
254C2×C9—21no—
255C2×C9—21no—
256C2×C9C22×Dic611yescharacteristic
257C2×C9C2×#23911yescharacteristic
258C2×C9#101011yescharacteristic
259C2×Q8—32no—
260C2×Q8—91no—
261C2×Q8—271no—
262C4⋊C4—32no—
263C4⋊C4—32no—
264C4⋊C4—92no—
265C4⋊C4—91no—
266C4⋊C4—274no—
267C4⋊C4—274no—
268C4⋊C4—274no—
269C4⋊C4—271no—
270C4⋊C4—272no—
271C42—92no—
272C42—274no—
273C42—272no—
274C42D3×D911yescharacteristic
275C22×C3—21no—
276C22×C3C23×D911yescharacteristic
277C22×C3C22×D3211yescharacteristic
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—22no—
281C3×C4—22no—
282C3×C4—41no—
283C3×C4—64no—
284C3×C4—94no—
285C3×C4—182no—
286C3×C4#369912yes—
287C3×C4Q8×D912yes—
288C3×C4D3×Dic612yes—
289C3×C4#379612yes—
290Dic3—24no—
291Dic3—34no—
292Dic3—62no—
293Dic3—184no—
294Dic3—182no—
295Dic3—182no—
296Dic3—182no—
297Dic3—182no—
298Dic3—362no—
299C32#572111yescharacteristic
300C9—21no—
301C9#557011yescharacteristic, complemented
302Q8—64no—
303Q8—94no—
304Q8—542no—
305C2×C4—34no—
306C2×C4—92no—
307C2×C4—92no—
308C2×C4—272no—
309C2×C4—272no—
310C2×C4C2×D3×D911yescharacteristic
311C2×C4C2×D3×D911yescharacteristic
312C2×C4C2×D3×D911yescharacteristic
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3C2×#369911yescharacteristic
317C2×C3C2×Q8×D911yescharacteristic
318C2×C3#6371311yescharacteristic
319C2×C3C2×D3×Dic611yescharacteristic
320C2×C3C2×#379611yescharacteristic
321C2×C3#6429711yescharacteristic
322C22C22×D3×D911yescharacteristic
323C4—21no—
324C4—64no—
325C4—94no—
326C4—182no—
327C4—542no—
328C4—542no—
329C4D9×Dic612yes—
330C4#8863912yes—
331C3—21no—
332C3#1058982311yescharacteristic, complemented
333C3#1059386111yescharacteristic
334C2C2×D9×Dic611yescharacteristic
335C2#1172245811yescharacteristic
336C2C2×#8863911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socle, prime radicalC22×C32C23×D3
hypercenterC42D3×D9
hypocenterC3×C9#557
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
54. C3×Dic18Q8semidirect
147. Dic18Dic6semidirect
221. C3×C9#557semidirect
301. C9#5570semidirect
332. C3#10589823semidirect

Maximal subgroups

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

subgroupindex
C4×#62262
C9×#8692
C3×C4×Dic182
#117209372
#117209242
#117209172
#117209102
Dic6×Dic92
#105938613
#105898233