Finite Group Databasebeta

← #13137796

Subgroups — #13137796

Order 1728.

348 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 85 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×C4⋊C4×D9C211yesmaximal, characteristic
2#11720944C211yesmaximal, characteristic
3#11720525C211yesmaximal, characteristic, complemented
4C9×#810C211yesmaximal, characteristic, complemented
5#11720297C211yesmaximal, characteristic, complemented
6(C2×D9)⋊(C3⋊C8)C211yesmaximal, characteristic
7#11720279C211yesmaximal, characteristic, complemented
8#10590517—31nomaximal
9#10588491—31nomaximal
10C2×C3×C4×D9—21no—
11C2×C3×C4×D9C2211yescharacteristic
12C2×C9×Dic6C2211yescharacteristic
13C3×C9×C4⋊C4C2211yescharacteristic
14C4⋊(C3⋊Dic9)C2211yescharacteristic
15C2×C3⋊(C9⋊C8)C2211yescharacteristic
16C3×C4⋊Dic9C2211yescharacteristic
17C3×Dic9⋊C4—21no—
18C2×C9×C3⋊C8C2211yescharacteristic
19(C2×D9)⋊Dic3—21no—
20#88442—21no—
21C3×D3×C4⋊C4—31no—
22#63856—31no—
23#63634—31no—
24C3×#810D311yescharacteristic
25#63575—31no—
26#63565—31no—
27(C2×D3)⋊(C3⋊C8)—31no—
28#63508—31no—
29C4⋊C4×D9D311yescharacteristic
30C9×#103—31no—
31#63395—31no—
32(C2×D9)⋊C8—31no—
33#63378—31no—
34#63366—31no—
35#63366—61no—
36C22×C3×D9D411yescharacteristic
37C2×C3×C4×C9—21no—
38C2×C3×C4×C9C2311yescharacteristic
39C2×C3⋊Dic9—21no—
40C2×C9×Dic3—21no—
41C2×C3×Dic9—21no—
42C2×C3×Dic9D411yescharacteristic
43C3×C4×D9—21no—
44C3×C4×D9—21no—
45C3×C4×D9—42no—
46C9×Dic6—21no—
47C9×Dic6—21no—
48C3⋊(C9⋊C8)—21no—
49C9×C3⋊C8—21no—
50#4970—91no—
51#4811—91no—
52C2×C3×C4×D3—31no—
53C2×C3×C4×D3—61no—
54C2×C3×Dic6C2×D311yescharacteristic
55C32×C4⋊C4C2×D311yescharacteristic
56C4⋊(C3⋊Dic3)—31no—
57C2×C3⋊(C3⋊C8)—31no—
58C3×C4⋊Dic3—31no—
59C3×Dic3⋊C4—61no—
60C2×C3×C3⋊C8C2×D311yescharacteristic
61#3724—61no—
62(C2×D3)⋊Dic3—61no—
63C2×Q8×C9—31no—
64C2×C4×D9—21no—
65C2×C4×D9C2×D311yescharacteristic
66C2×C8×C9—31no—
67C9×C4⋊C4—21no—
68C9×C4⋊C4C2×D311yescharacteristic
69(C2×D9)⋊C4—61no—
70C4⋊Dic9—31no—
71C4⋊Dic9—61no—
72C4⋊Dic9C2×D311yescharacteristic
73Dic9⋊C4—21no—
74Dic9⋊C4—61no—
75C2×C9⋊C8—31no—
76C2×C9⋊C8—61no—
77C22×C3×C9C2×D411yescharacteristic
78C2×C3×D9—21no—
79C2×C3×D9—21no—
80C2×C3×D9Q1612yes—
81C3×C4×C9—41no—
82C3×C4×C9#4111yescharacteristic
83C3×C4×C9#4111yescharacteristic
84C3⋊Dic9—41no—
85C9×Dic3—41no—
86C3×Dic9—21no—
87C3×Dic9—41no—
88C2×C3×C4⋊C4—91no—
89#946—91no—
90#924—91no—
91D3×C4⋊C4—31no—
92C3×#103—31no—
93#835—91no—
94#830—91no—
95(C2×D3)⋊C8—91no—
96#818—91no—
97#810—91no—
98#810D911yescharacteristic, complemented
99#807—91no—
100#807—91no—
101#807—91no—
102#807—181no—
103C22×C3×D3—31no—
104C2×C32×C4—21no—
105C2×C32×C4C22×D311yescharacteristic
106C2×C3⋊Dic3—61no—
107C2×C3×Dic3—21no—
108C2×C3×Dic3—31no—
109C2×C3×Dic3—61no—
110C3×C4×D3—61no—
111C3×C4×D3—61no—
112C3×C4×D3—122no—
113C3×Dic6—21no—
114C3×Dic6—21no—
115C22×D9#6711yescharacteristic
116C3⋊(C3⋊C8)—61no—
117C3×C3⋊C8—21no—
118Q8×C9—61no—
119Q8×C9—61no—
120C2×C4×C9—21no—
121C2×C4×C9—21no—
122C2×C4×C9—41no—
123C2×C4×C9—61no—
124C2×C4×C9C22×D311yescharacteristic
125C2×Dic9—21no—
126C2×Dic9—61no—
127C2×Dic9—121no—
128C2×Dic9#6711yescharacteristic
129C4×D9—21no—
130C4×D9—21no—
131C4×D9—42no—
132C8×C9—61no—
133C9⋊C8—62no—
134C9⋊C8—61no—
135#484—271noSylow
136C2×C3×C9#12311yescharacteristic
137C2×C3×C9C2×Q1611yescharacteristic
138C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
139C3×D9—22no—
140C2×C3×Q8—31no—
141C22×C3×C4—91no—
142C22×C3×C4—181no—
143C2×C4×D3—31no—
144C2×C4×D3—61no—
145C2×Dic6C2×D911yescharacteristic
146C2×C3×C8—31no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—91no—
149C3×C4⋊C4—181no—
150C3×C4⋊C4C2×D911yescharacteristic
151C3×C4⋊C4D3211yescharacteristic
152#217—181no—
153(C2×D3)⋊C4—181no—
154C4⋊Dic3—31no—
155C4⋊Dic3—91no—
156C4⋊Dic3—91no—
157C4⋊Dic3—181no—
158Dic3⋊C4—61no—
159Dic3⋊C4—181no—
160Dic3⋊C4—181no—
161C2×C3⋊C8—91no—
162C2×C3⋊C8—91no—
163C2×C3⋊C8—181no—
164C2×C3⋊C8C2×D911yescharacteristic
165C22×C32D3×D411yescharacteristic
166C2×C3×D3—32no—
167C2×C3×D3—61no—
168C2×C3×D3—61no—
169C32×C4—41no—
170C32×C4#23511yescharacteristic
171C32×C4#23711yescharacteristic
172C3⋊Dic3—121no—
173C3×Dic3—41no—
174C3×Dic3—61no—
175C3×Dic3—121no—
176C22×C9—21no—
177C22×C9C2×#6711yescharacteristic
178C2×D9—21no—
179C2×D9—21no—
180C2×D9#21612yes—
181C4×C9—21no—
182C4×C9—21no—
183C4×C9—41no—
184C4×C9—81no—
185C4×C9—121no—
186C4×C9#23511yescharacteristic
187C4×C9#23511yescharacteristic
188Dic9—21no—
189Dic9—41no—
190Dic9—121no—
191Dic9—241no—
192C22⋊Q8—271no—
193C2×C4⋊C4—91no—
194#107—271no—
195#107—271no—
196#103—31no—
197#103—271no—
198C22⋊C8—271no—
199C3×C9#48411yescharacteristic, complemented, Sylow
200C23×C3—91no—
201C22×D3—31no—
202C3×Q8—61no—
203C3×Q8—61no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—41no—
208C2×C3×C4—61no—
209C2×C3×C4—91no—
210C2×C3×C4—181no—
211C2×C3×C4—181no—
212C2×C3×C4—181no—
213C2×C3×C4—362no—
214C2×C3×C4C22×D911yescharacteristic
215C2×C3×C4C2×D3211yescharacteristic
216C2×Dic3—21no—
217C2×Dic3—31no—
218C2×Dic3—61no—
219C2×Dic3—181no—
220C2×Dic3—181no—
221C2×Dic3—361no—
222C4×D3—61no—
223C4×D3—61no—
224C4×D3—122no—
225Dic6—21no—
226Dic6—21no—
227C3×C8—61no—
228C3⋊C8—21no—
229C3⋊C8—182no—
230C3⋊C8—181no—
231C3⋊C8—181no—
232C2×C32#88311yescharacteristic
233C2×C32#91411yescharacteristic
234C2×C32D3×Q1611yescharacteristic
235C3×D3—62no—
236C2×C9—21no—
237C2×C9—21no—
238C2×C9—21no—
239C2×C9#92911yescharacteristic
240C2×C9#93211yescharacteristic
241C2×C9C2×#21611yescharacteristic
242D9—22no—
243C2×Q8—31no—
244C22×C4—91no—
245C22×C4—181no—
246C2×C8—31no—
247C2×C8—271no—
248C4⋊C4—91no—
249C4⋊C4—181no—
250C4⋊C4—271no—
251C4⋊C4—541no—
252C4⋊C4D3×D911yescharacteristic
253C22⋊C4—541no—
254C22×C3—21no—
255C22×C3—92no—
256C22×C3—181no—
257C22×C3—181no—
258C22×C3D4×D911yescharacteristic
259C22×C3D3×#6711yescharacteristic
260C2×D3—32no—
261C2×D3—61no—
262C2×D3—61no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—41no—
266C3×C4—41no—
267C3×C4—81no—
268C3×C4—121no—
269C3×C4—181no—
270C3×C4—361no—
271C3×C4#369711yescharacteristic
272C3×C4#369911yescharacteristic
273C3×C4#379511yescharacteristic
274C3×C4#379811yescharacteristic
275Dic3—41no—
276Dic3—61no—
277Dic3—121no—
278Dic3—361no—
279Dic3—361no—
280Dic3—721no—
281C32#481111yescharacteristic
282C9—21no—
283C9#497011yescharacteristic, complemented
284C23—91no—
285Q8—61no—
286Q8—61no—
287C2×C4—21no—
288C2×C4—61no—
289C2×C4—91no—
290C2×C4—181no—
291C2×C4—181no—
292C2×C4—181no—
293C2×C4—362no—
294C2×C4—541no—
295C2×C4C2×D3×D911yescharacteristic
296C8—61no—
297C8—541no—
298C2×C3—21no—
299C2×C3—21no—
300C2×C3—21no—
301C2×C3—182no—
302C2×C3#6344511yescharacteristic
303C2×C3#6347611yescharacteristic
304C2×C3Q16×D911yescharacteristic
305C2×C3#6390511yescharacteristic
306C2×C3#6393411yescharacteristic
307C2×C3D3×#21611yescharacteristic
308D3—62no—
309C22—92no—
310C22—181no—
311C22—181no—
312C22D9×#6711yescharacteristic
313C4—41no—
314C4—121no—
315C4—181no—
316C4—361no—
317C4—1081no—
318C4#8863811yescharacteristic
319C4#8864011yescharacteristic
320C3—21no—
321C3#1058849111yescharacteristic, complemented
322C3#1059051711yescharacteristic
323C2—182no—
324C2#1172103111yescharacteristic
325C2#1172118411yescharacteristic
326C2D9×#21611yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC22×C3×D9D4
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#484
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #11720525C2semidirect
4. C9×#810C2semidirect
5. #11720297C2semidirect
7. #11720279C2semidirect
98. #810D9semidirect
199. C3×C9#484semidirect
283. C9#4970semidirect
321. C3#10588491semidirect

Maximal subgroups

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

subgroupindex
C3×C4⋊C4×D92
#117209442
#117205252
C9×#8102
#117202972
(C2×D9)⋊(C3⋊C8)2
#117202792
#105905173
#105884913