Finite Group Databasebeta

← #13137808

Subgroups — #13137808

Order 1728.

338 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 81 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×#63452C211yesmaximal, characteristic, complemented
2#11720953C211yesmaximal, characteristic
3#11720528C211yesmaximal, characteristic
4C9×#810C211yesmaximal, characteristic, complemented
5#11720297C211yesmaximal, characteristic, complemented
6#11720284C211yesmaximal, characteristic
7C3⋊C8×Dic9C211yesmaximal, characteristic, complemented
8#10590537—31nomaximal
9#10588480—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×C9×Dic6C2211yescharacteristic
12C3×C9×C4⋊C4C2211yescharacteristic
13C2×C3⋊(C9⋊C8)C2211yescharacteristic
14C3×C4⋊Dic9—21no—
15C3×C4⋊Dic9C2211yescharacteristic
16C3×Dic9⋊C4—21no—
17C3×C4×Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19#88445—21no—
20#88445—21no—
21C3×#890—31no—
22#63864—31no—
23#63636—31no—
24C3×#810D311yescharacteristic
25#63565—31no—
26#63564—31no—
27Dic3×C3⋊C8—31no—
28#63509—31no—
29#63452D311yescharacteristic, complemented
30C9×#103—31no—
31#63395—31no—
32#63380—31no—
33C8×Dic9—31no—
34#63369—31no—
35#63369—61no—
36C22×C3⋊D9—21no—
37C2×C3×C4×C9—21no—
38C2×C3×C4×C9C2311yescharacteristic
39C4⋊(C3⋊D9)—21no—
40C4⋊(C3⋊D9)—21no—
41C2×C9×Dic3—21no—
42C2×C3×Dic9—21no—
43C2×C3×Dic9D411yescharacteristic
44C2×C3×Dic9D411yescharacteristic
45C9×Dic6—21no—
46C9×Dic6—21no—
47C3⋊(C9⋊C8)—21no—
48C9×C3⋊C8—21no—
49#5068—91no—
50#4800—91no—
51C2×C4⋊(C3⋊D3)—31no—
52C2×C3×Dic6C2×D311yescharacteristic
53C32×C4⋊C4C2×D311yescharacteristic
54C2×C3⋊(C3⋊C8)—31no—
55C3×C4⋊Dic3—31no—
56C3×C4⋊Dic3—61no—
57C3×Dic3⋊C4—61no—
58C3×C4×Dic3—31no—
59C2×C3×C3⋊C8C2×D311yescharacteristic
60#3722—61no—
61#3722—61no—
62C2×Q8×C9—31no—
63C2×D36—31no—
64C2×D36—61no—
65C2×C8×C9—31no—
66C9×C4⋊C4—21no—
67C9×C4⋊C4C2×D311yescharacteristic
68(C2×D9)⋊C4—61no—
69(C2×D9)⋊C4—61no—
70C4⋊Dic9—21no—
71C4⋊Dic9C2×D311yescharacteristic
72Dic9⋊C4—21no—
73C4×Dic9C2×D311yescharacteristic
74C2×C9⋊C8—31no—
75C2×C9⋊C8—61no—
76C22×C3×C9C2×D411yescharacteristic
77C2×C3⋊D9—41no—
78C2×C3⋊D9—41no—
79C2×C3⋊D9—41no—
80C3×C4×C9—41no—
81C3×C4×C9#4111yescharacteristic
82C3×C4×C9#4111yescharacteristic
83C9×Dic3—41no—
84C3×Dic9—21no—
85C3×Dic9—21no—
86C3×Dic9—41no—
87C3×#125—91no—
88#947—91no—
89#890—31no—
90#875—91no—
91C3×#103—31no—
92#835—91no—
93#821—91no—
94C8×Dic3—91no—
95#810—91no—
96#810D911yescharacteristic, complemented
97#809—91no—
98#809—91no—
99#809—91no—
100#809—181no—
101C4×C3⋊C8—91no—
102C22×C3⋊D3—61no—
103C2×C32×C4—21no—
104C2×C32×C4C22×D311yescharacteristic
105C4⋊(C3⋊D3)—61no—
106C4⋊(C3⋊D3)—61no—
107C2×C3×Dic3—21no—
108C2×C3×Dic3—31no—
109C2×C3×Dic3—31no—
110C2×C3×Dic3—61no—
111C3×Dic6—21no—
112C3×Dic6—21no—
113C22×D9—61no—
114C22×D9—121no—
115C3⋊(C3⋊C8)—61no—
116C3×C3⋊C8—21no—
117Q8×C9—61no—
118Q8×C9—61no—
119C2×C4×C9—21no—
120C2×C4×C9—21no—
121C2×C4×C9—41no—
122C2×C4×C9—61no—
123C2×C4×C9C22×D311yescharacteristic
124C2×Dic9—21no—
125C2×Dic9#6711yescharacteristic
126C2×Dic9#6711yescharacteristic
127D36—62no—
128D36—61no—
129D36—61no—
130D36—121no—
131C8×C9—61no—
132C9⋊C8—61no—
133C9⋊C8—121no—
134#488—271noSylow
135C2×C3×C9#12411yescharacteristic
136C2×C3×C9#13511yescharacteristic
137C2×C3×C9#13511yescharacteristic
138C3⋊D9—81no—
139C2×C3×Q8—31no—
140C2×D12—91no—
141C2×D12—91no—
142C2×D12—181no—
143C2×Dic6C2×D911yescharacteristic
144C2×C3×C8—31no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—91no—
147C3×C4⋊C4—181no—
148C3×C4⋊C4—181no—
149C3×C4⋊C4C2×D911yescharacteristic
150C3×C4⋊C4D3211yescharacteristic
151C3×C42—91no—
152(C2×D3)⋊C4—181no—
153(C2×D3)⋊C4—181no—
154(C2×D3)⋊C4—181no—
155(C2×D3)⋊C4—181no—
156C4⋊Dic3—31no—
157C4⋊Dic3—61no—
158Dic3⋊C4—61no—
159C4×Dic3—31no—
160C2×C3⋊C8—91no—
161C2×C3⋊C8—91no—
162C2×C3⋊C8—181no—
163C2×C3⋊C8C2×D911yescharacteristic
164C22×C32D3×D411yescharacteristic
165C2×C3⋊D3—121no—
166C2×C3⋊D3—121no—
167C2×C3⋊D3—121no—
168C32×C4—41no—
169C32×C4#23511yescharacteristic
170C32×C4#23711yescharacteristic
171C3×Dic3—41no—
172C3×Dic3—61no—
173C3×Dic3—61no—
174C3×Dic3—121no—
175C22×C9—21no—
176C22×C9C2×#6711yescharacteristic
177C2×D9—122no—
178C2×D9—121no—
179C2×D9—121no—
180C2×D9—121no—
181C2×D9—241no—
182C2×D9—241no—
183C4×C9—21no—
184C4×C9—21no—
185C4×C9—41no—
186C4×C9—81no—
187C4×C9—121no—
188C4×C9#23911yescharacteristic
189C4×C9#23911yescharacteristic
190Dic9—21no—
191Dic9—21no—
192Dic9—41no—
193#125—91no—
194#124—271no—
195#103—31no—
196#103—271no—
197#102—271no—
198#102—271no—
199C4×C8—271no—
200C3×C9#48811yescharacteristic, complemented, Sylow
201C22×D3—181no—
202C22×D3—181no—
203C22×D3—361no—
204C3×Q8—61no—
205C3×Q8—61no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—41no—
210C2×C3×C4—61no—
211C2×C3×C4—91no—
212C2×C3×C4—91no—
213C2×C3×C4—181no—
214C2×C3×C4C22×D911yescharacteristic
215C2×C3×C4C2×D3211yescharacteristic
216C2×Dic3—21no—
217C2×Dic3—31no—
218C2×Dic3—31no—
219C2×Dic3—61no—
220D12—182no—
221D12—181no—
222D12—181no—
223D12—181no—
224D12—181no—
225D12—361no—
226Dic6—21no—
227Dic6—21no—
228C3×C8—61no—
229C3⋊C8—21no—
230C3⋊C8—181no—
231C3⋊C8—181no—
232C3⋊C8—361no—
233C2×C32#88511yescharacteristic
234C2×C32#91611yescharacteristic
235C2×C32#91911yescharacteristic
236C3⋊D3—241no—
237C2×C9—21no—
238C2×C9—21no—
239C2×C9—21no—
240C2×C9#94711yescharacteristic
241C2×C9#95011yescharacteristic
242C2×C9#95011yescharacteristic
243D9—242no—
244D9—241no—
245C2×Q8—31no—
246C2×D4—271no—
247C2×C8—31no—
248C2×C8—271no—
249C4⋊C4—91no—
250C4⋊C4—181no—
251C4⋊C4—181no—
252C4⋊C4D3×D911yescharacteristic
253C22⋊C4—541no—
254C22⋊C4—541no—
255C42—91no—
256C22×C3—21no—
257C22×C3D4×D911yescharacteristic
258C22×C3D3×#6711yescharacteristic
259C2×D3—362no—
260C2×D3—361no—
261C2×D3—361no—
262C2×D3—361no—
263C2×D3—361no—
264C2×D3—361no—
265C2×D3—361no—
266C2×D3—721no—
267C2×D3—721no—
268C3×C4—21no—
269C3×C4—21no—
270C3×C4—41no—
271C3×C4—41no—
272C3×C4—81no—
273C3×C4—121no—
274C3×C4—181no—
275C3×C4—181no—
276C3×C4—361no—
277C3×C4#369711yescharacteristic
278C3×C4#369911yescharacteristic
279C3×C4#379611yescharacteristic
280C3×C4#379911yescharacteristic
281Dic3—41no—
282Dic3—61no—
283Dic3—61no—
284Dic3—121no—
285C32#480011yescharacteristic
286C9—21no—
287C9#506811yescharacteristic, complemented
288C23—541no—
289Q8—61no—
290Q8—61no—
291D4—541no—
292D4—541no—
293C2×C4—21no—
294C2×C4—61no—
295C2×C4—91no—
296C2×C4—91no—
297C2×C4—181no—
298C2×C4C2×D3×D911yescharacteristic
299C8—61no—
300C8—541no—
301C2×C3—21no—
302C2×C3—21no—
303C2×C3—21no—
304C2×C3#6344711yescharacteristic
305C2×C3#6347811yescharacteristic
306C2×C3#6348111yescharacteristic
307C2×C3#6390711yescharacteristic
308C2×C3#6393511yescharacteristic
309C2×C3#6394911yescharacteristic
310D3—722no—
311D3—721no—
312D3—721no—
313C22—1081no—
314C22—1081no—
315C22—1081no—
316C22D9×#6711yescharacteristic
317C4—41no—
318C4—121no—
319C4—181no—
320C4—181no—
321C4—361no—
322C4#8863911yescharacteristic
323C4#8864211yescharacteristic
324C3—21no—
325C3#1058848011yescharacteristic, complemented
326C3#1059053711yescharacteristic
327C2—2161no—
328C2#1172103511yescharacteristic
329C2#1172118511yescharacteristic
330C2#1172122311yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC2×C4⋊(C3⋊D9)C22
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#488
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
1. C3×#63452C2semidirect
4. C9×#810C2semidirect
5. #11720297C2semidirect
7. C3⋊C8×Dic9C2semidirect
29. #63452D3semidirect
96. #810D9semidirect
200. C3×C9#488semidirect
287. C9#5068semidirect
325. C3#10588480semidirect

Maximal subgroups

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

subgroupindex
C3×#634522
#117209532
#117205282
C9×#8102
#117202972
#117202842
C3⋊C8×Dic92
#105905373
#105884803