Finite Group Databasebeta

← #13137806

Subgroups — #13137806

Order 1728.

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

normal subgroups only

This group has 350 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×#63450C211yesmaximal, characteristic, complemented
2#11720953C211yesmaximal, characteristic
3#11720528C211yesmaximal, characteristic
4C9×#810C211yesmaximal, characteristic, complemented
5#11720297C211yesmaximal, characteristic, complemented
6#11720284C211yesmaximal, characteristic
7#11720274C211yesmaximal, characteristic, complemented
8#10590535—31nomaximal
9#10588474—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×C9×Dic6C2211yescharacteristic
12C2×C3×Dic18—21no—
13C3×C9×C4⋊C4C2211yescharacteristic
14C2×C3⋊(C9⋊C8)C2211yescharacteristic
15C3×C4⋊Dic9C2211yescharacteristic
16C3×Dic9⋊C4—21no—
17C3×C4×Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19#88445—21no—
20#88445—21no—
21C3×#888—31no—
22#63864—31no—
23#63636—31no—
24C3×#810D311yescharacteristic
25#63565—31no—
26#63564—31no—
27#63554—31no—
28#63509—31no—
29#63450D311yescharacteristic, complemented
30C9×#103—31no—
31#63395—31no—
32#63380—31no—
33#63375—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×Dic18—42no—
48C3⋊(C9⋊C8)—21no—
49C9×C3⋊C8—21no—
50#5089—91no—
51#4794—91no—
52C2×C4⋊(C3⋊D3)—31no—
53C2×C3×Dic6—61no—
54C2×C3×Dic6C2×D311yescharacteristic
55C32×C4⋊C4C2×D311yescharacteristic
56C2×C3⋊(C3⋊C8)—31no—
57C3×C4⋊Dic3—31no—
58C3×Dic3⋊C4—61no—
59C3×C4×Dic3—31no—
60C2×C3×C3⋊C8C2×D311yescharacteristic
61#3722—61no—
62#3722—61no—
63C2×Q8×C9—31no—
64C2×D36—31no—
65C2×D36—61no—
66C2×Dic18—21no—
67C2×C8×C9—31no—
68C9×C4⋊C4—21no—
69C9×C4⋊C4C2×D311yescharacteristic
70(C2×D9)⋊C4—61no—
71(C2×D9)⋊C4—61no—
72C4⋊Dic9C2×D311yescharacteristic
73Dic9⋊C4—21no—
74C4×Dic9C2×D311yescharacteristic
75C2×C9⋊C8—31no—
76C2×C9⋊C8—61no—
77C22×C3×C9C2×D411yescharacteristic
78C2×C3⋊D9—41no—
79C2×C3⋊D9—41no—
80C2×C3⋊D9—41no—
81C3×C4×C9—41no—
82C3×C4×C9#4111yescharacteristic
83C3×C4×C9#4111yescharacteristic
84C9×Dic3—41no—
85C3×Dic9—21no—
86C3×Dic9—21no—
87C3×Dic9—41no—
88C3×C4⋊Q8—91no—
89#947—91no—
90#888—31no—
91#875—91no—
92C3×#103—31no—
93#835—91no—
94#821—91no—
95#815—91no—
96#810—91no—
97#810D911yescharacteristic, complemented
98#809—91no—
99#809—91no—
100#809—91no—
101#809—181no—
102#803—91no—
103C22×C3⋊D3—61no—
104C2×C32×C4—21no—
105C2×C32×C4C22×D311yescharacteristic
106C4⋊(C3⋊D3)—61no—
107C4⋊(C3⋊D3)—61no—
108C2×C3×Dic3—21no—
109C2×C3×Dic3—31no—
110C2×C3×Dic3—31no—
111C2×C3×Dic3—61no—
112C3×Dic6—21no—
113C3×Dic6—21no—
114C3×Dic6—122no—
115C22×D9—61no—
116C22×D9—121no—
117C3⋊(C3⋊C8)—61no—
118C3×C3⋊C8—21no—
119Q8×C9—61no—
120Q8×C9—61no—
121C2×C4×C9—21no—
122C2×C4×C9—21no—
123C2×C4×C9—41no—
124C2×C4×C9—61no—
125C2×C4×C9C22×D311yescharacteristic
126C2×Dic9—21no—
127C2×Dic9#6711yescharacteristic
128C2×Dic9#6711yescharacteristic
129D36—62no—
130D36—61no—
131D36—61no—
132D36—121no—
133Dic18—42no—
134C8×C9—61no—
135C9⋊C8—61no—
136C9⋊C8—121no—
137#489—271noSylow
138C2×C3×C9#12411yescharacteristic
139C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
140C2×C3×C9#13711yescharacteristic
141C3⋊D9—81no—
142C2×C3×Q8—31no—
143C2×C3×Q8—181no—
144C2×D12—91no—
145C2×D12—91no—
146C2×D12—181no—
147C2×Dic6—61no—
148C2×Dic6C2×D911yescharacteristic
149C2×C3×C8—31no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—91no—
152C3×C4⋊C4—181no—
153C3×C4⋊C4C2×D911yescharacteristic
154C3×C4⋊C4D3211yescharacteristic
155C3×C42—91no—
156(C2×D3)⋊C4—181no—
157(C2×D3)⋊C4—181no—
158(C2×D3)⋊C4—181no—
159(C2×D3)⋊C4—181no—
160C4⋊Dic3—31no—
161Dic3⋊C4—61no—
162C4×Dic3—31no—
163C2×C3⋊C8—91no—
164C2×C3⋊C8—91no—
165C2×C3⋊C8—181no—
166C2×C3⋊C8C2×D911yescharacteristic
167C22×C32D3×D411yescharacteristic
168C2×C3⋊D3—121no—
169C2×C3⋊D3—121no—
170C2×C3⋊D3—121no—
171C32×C4—41no—
172C32×C4#23511yescharacteristic
173C32×C4#23711yescharacteristic
174C3×Dic3—41no—
175C3×Dic3—61no—
176C3×Dic3—61no—
177C3×Dic3—121no—
178C22×C9—21no—
179C22×C9C2×#6711yescharacteristic
180C2×D9—122no—
181C2×D9—121no—
182C2×D9—121no—
183C2×D9—121no—
184C2×D9—241no—
185C2×D9—241no—
186C4×C9—21no—
187C4×C9—21no—
188C4×C9—41no—
189C4×C9—81no—
190C4×C9—121no—
191C4×C9#23911yescharacteristic
192C4×C9#23911yescharacteristic
193Dic9—21no—
194Dic9—21no—
195Dic9—41no—
196C4⋊Q8—91no—
197#124—271no—
198#103—31no—
199#103—271no—
200#102—271no—
201#102—271no—
202#97—271no—
203C3×C9#48911yescharacteristic, complemented, Sylow
204C22×D3—181no—
205C22×D3—181no—
206C22×D3—361no—
207C3×Q8—61no—
208C3×Q8—61no—
209C3×Q8—362no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—41no—
214C2×C3×C4—61no—
215C2×C3×C4—91no—
216C2×C3×C4—91no—
217C2×C3×C4—181no—
218C2×C3×C4C22×D911yescharacteristic
219C2×C3×C4C2×D3211yescharacteristic
220C2×Dic3—21no—
221C2×Dic3—31no—
222C2×Dic3—31no—
223C2×Dic3—61no—
224D12—182no—
225D12—181no—
226D12—181no—
227D12—181no—
228D12—181no—
229D12—361no—
230Dic6—21no—
231Dic6—21no—
232Dic6—122no—
233C3×C8—61no—
234C3⋊C8—21no—
235C3⋊C8—181no—
236C3⋊C8—181no—
237C3⋊C8—361no—
238C2×C32#88511yescharacteristic
239C2×C32#91411yescharacteristic
240C2×C32#91811yescharacteristic
241C3⋊D3—241no—
242C2×C9—21no—
243C2×C9—21no—
244C2×C9—21no—
245C2×C9#93211yescharacteristic
246C2×C9#94211yescharacteristic
247C2×C9#94711yescharacteristic
248D9—242no—
249D9—241no—
250C2×Q8—31no—
251C2×Q8—181no—
252C2×D4—271no—
253C2×C8—31no—
254C2×C8—271no—
255C4⋊C4—91no—
256C4⋊C4—181no—
257C4⋊C4D3×D911yescharacteristic
258C22⋊C4—541no—
259C22⋊C4—541no—
260C42—91no—
261C22×C3—21no—
262C22×C3D4×D911yescharacteristic
263C22×C3D3×#6711yescharacteristic
264C2×D3—362no—
265C2×D3—361no—
266C2×D3—361no—
267C2×D3—361no—
268C2×D3—361no—
269C2×D3—361no—
270C2×D3—361no—
271C2×D3—721no—
272C2×D3—721no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—41no—
276C3×C4—41no—
277C3×C4—81no—
278C3×C4—121no—
279C3×C4—181no—
280C3×C4—181no—
281C3×C4—361no—
282C3×C4#369711yescharacteristic
283C3×C4#369911yescharacteristic
284C3×C4#379611yescharacteristic
285C3×C4#379911yescharacteristic
286Dic3—41no—
287Dic3—61no—
288Dic3—61no—
289Dic3—121no—
290C32#479411yescharacteristic
291C9—21no—
292C9#508911yescharacteristic, complemented
293C23—541no—
294Q8—61no—
295Q8—61no—
296Q8—362no—
297D4—541no—
298D4—541no—
299C2×C4—21no—
300C2×C4—61no—
301C2×C4—91no—
302C2×C4—91no—
303C2×C4—181no—
304C2×C4C2×D3×D911yescharacteristic
305C8—61no—
306C8—541no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—21no—
310C2×C3#6344711yescharacteristic
311C2×C3#6347611yescharacteristic
312C2×C3#6348011yescharacteristic
313C2×C3#6390711yescharacteristic
314C2×C3#6393411yescharacteristic
315C2×C3#6394811yescharacteristic
316D3—722no—
317D3—721no—
318D3—721no—
319C22—1081no—
320C22—1081no—
321C22—1081no—
322C22D9×#6711yescharacteristic
323C4—41no—
324C4—121no—
325C4—181no—
326C4—181no—
327C4—361no—
328C4#8863911yescharacteristic
329C4#8864211yescharacteristic
330C3—21no—
331C3#1058847411yescharacteristic, complemented
332C3#1059053511yescharacteristic
333C2—2161no—
334C2#1172103511yescharacteristic
335C2#1172118411yescharacteristic
336C2#1172122211yescharacteristic
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#489
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×#63450C2semidirect
4. C9×#810C2semidirect
5. #11720297C2semidirect
7. #11720274C2semidirect
29. #63450D3semidirect
97. #810D9semidirect
203. C3×C9#489semidirect
292. C9#5089semidirect
331. C3#10588474semidirect

Maximal subgroups

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

subgroupindex
C3×#634502
#117209532
#117205282
C9×#8102
#117202972
#117202842
#117202742
#105905353
#105884743