Finite Group Databasebeta

← #13137522

Subgroups — #13137522

Order 1728.

350 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C9×#888C211yesmaximal, characteristic, complemented
2#11720952C211yesmaximal, characteristic
3#11720528C211yesmaximal, characteristic
4C3×#63368C211yesmaximal, characteristic, complemented
5#11720268C211yesmaximal, characteristic, complemented
6#11720267C211yesmaximal, characteristic
7#11720258C211yesmaximal, characteristic, complemented
8#10590535—31nomaximal
9#10588774—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×C9×Dic6—21no—
12C2×C3×Dic18C2211yescharacteristic
13C3×C9×C4⋊C4C2211yescharacteristic
14C2×C3⋊(C9⋊C8)C2211yescharacteristic
15C9×C4⋊Dic3C2211yescharacteristic
16C9×Dic3⋊C4—21no—
17C4×C9×Dic3C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19#88445—21no—
20#88445—21no—
21C3×#888D311yescharacteristic
22#63864—31no—
23#63636—31no—
24C3×#810—31no—
25#63565—31no—
26#63564—31no—
27#63554—31no—
28C9×C4⋊Q8—31no—
29#63437—31no—
30#63369—31no—
31#63369—31no—
32#63369—61no—
33#63368—31no—
34#63368D311yescharacteristic, complemented
35#63362—31no—
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×C9×Dic3D411yescharacteristic
43C2×C9×Dic3D411yescharacteristic
44C2×C3×Dic9—21no—
45C9×Dic6—42no—
46C3×Dic18—21no—
47C3×Dic18—21no—
48C3⋊(C9⋊C8)—21no—
49C3×C9⋊C8—21no—
50#5089—91no—
51#4794—91no—
52C2×C4⋊(C3⋊D3)—31no—
53C2×C3×Dic6—21no—
54C2×C3×Dic6—31no—
55C32×C4⋊C4C2×D311yescharacteristic
56C2×C3⋊(C3⋊C8)—31no—
57C3×C4⋊Dic3C2×D311yescharacteristic
58C3×Dic3⋊C4—21no—
59C3×C4×Dic3C2×D311yescharacteristic
60C2×C3×C3⋊C8—31no—
61#3722—61no—
62#3722—61no—
63C2×Q8×C9—61no—
64C2×D36—31no—
65C2×D36—61no—
66C2×Dic18C2×D311yescharacteristic
67C9×C4⋊C4—21no—
68C9×C4⋊C4—31no—
69C9×C4⋊C4—61no—
70C9×C4⋊C4C2×D311yescharacteristic
71C42×C9—31no—
72(C2×D9)⋊C4—61no—
73(C2×D9)⋊C4—61no—
74C2×C9⋊C8—31no—
75C2×C9⋊C8—61no—
76C2×C9⋊C8C2×D311yescharacteristic
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—21no—
85C9×Dic3—21no—
86C9×Dic3—41no—
87C3×Dic9—41no—
88C3×C4⋊Q8—31no—
89#947—91no—
90#888D911yescharacteristic, complemented
91#875—91no—
92C3×#103—91no—
93#835—91no—
94#821—91no—
95#815—91no—
96#810—31no—
97#810—91no—
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—61no—
110C2×C3×Dic3#6711yescharacteristic
111C2×C3×Dic3#6711yescharacteristic
112C3×Dic6—42no—
113C3×Dic6—61no—
114C3×Dic6—61no—
115C22×D9—61no—
116C22×D9—121no—
117C3⋊(C3⋊C8)—61no—
118C3×C3⋊C8—61no—
119Q8×C9—122no—
120C2×C4×C9—21no—
121C2×C4×C9—21no—
122C2×C4×C9—31no—
123C2×C4×C9—31no—
124C2×C4×C9—41no—
125C2×C4×C9—61no—
126C2×C4×C9C22×D311yescharacteristic
127C2×Dic9—21no—
128D36—62no—
129D36—61no—
130D36—61no—
131D36—121no—
132Dic18—21no—
133Dic18—21no—
134C9⋊C8—21no—
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—61no—
143C2×C3×Q8—91no—
144C2×D12—91no—
145C2×D12—91no—
146C2×D12—181no—
147C2×Dic6—21no—
148C2×Dic6—31no—
149C2×C3×C8—91no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—31no—
152C3×C4⋊C4—61no—
153C3×C4⋊C4C2×D911yescharacteristic
154C3×C4⋊C4D3211yescharacteristic
155C3×C42—31no—
156(C2×D3)⋊C4—181no—
157(C2×D3)⋊C4—181no—
158(C2×D3)⋊C4—181no—
159(C2×D3)⋊C4—181no—
160C4⋊Dic3C2×D911yescharacteristic
161Dic3⋊C4—21no—
162C4×Dic3C2×D911yescharacteristic
163C2×C3⋊C8—31no—
164C2×C3⋊C8—91no—
165C2×C3⋊C8—91no—
166C2×C3⋊C8—181no—
167C22×C32C2×#6711yescharacteristic
168C2×C3⋊D3—121no—
169C2×C3⋊D3—121no—
170C2×C3⋊D3—121no—
171C32×C4—41no—
172C32×C4#23911yescharacteristic
173C32×C4#23911yescharacteristic
174C3×Dic3—21no—
175C3×Dic3—21no—
176C3×Dic3—41no—
177C3×Dic3—121no—
178C22×C9—21no—
179C22×C9D3×D411yescharacteristic
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—61no—
190C4×C9—61no—
191C4×C9—81no—
192C4×C9—121no—
193C4×C9#23511yescharacteristic
194C4×C9#23711yescharacteristic
195Dic9—41no—
196C4⋊Q8—31no—
197#124—271no—
198#103—91no—
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—122no—
208C3×Q8—181no—
209C3×Q8—181no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—31no—
214C2×C3×C4—31no—
215C2×C3×C4—41no—
216C2×C3×C4—61no—
217C2×C3×C4—181no—
218C2×C3×C4C22×D911yescharacteristic
219C2×C3×C4C2×D3211yescharacteristic
220C2×Dic3—21no—
221C2×Dic3—61no—
222C2×Dic3#61111yescharacteristic
223C2×Dic3#61111yescharacteristic
224D12—182no—
225D12—181no—
226D12—181no—
227D12—181no—
228D12—181no—
229D12—361no—
230Dic6—42no—
231Dic6—61no—
232Dic6—61no—
233C3×C8—181no—
234C3⋊C8—61no—
235C3⋊C8—181no—
236C3⋊C8—181no—
237C3⋊C8—361no—
238C2×C32#93211yescharacteristic
239C2×C32#94211yescharacteristic
240C2×C32#94711yescharacteristic
241C3⋊D3—241no—
242C2×C9—21no—
243C2×C9—21no—
244C2×C9—21no—
245C2×C9#88511yescharacteristic
246C2×C9#91411yescharacteristic
247C2×C9#91811yescharacteristic
248D9—242no—
249D9—241no—
250C2×Q8—61no—
251C2×Q8—91no—
252C2×D4—271no—
253C2×C8—91no—
254C2×C8—271no—
255C4⋊C4—31no—
256C4⋊C4—61no—
257C4⋊C4D3×D911yescharacteristic
258C22⋊C4—541no—
259C22⋊C4—541no—
260C42—31no—
261C22×C3—21no—
262C22×C3C2×#61111yescharacteristic
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—61no—
278C3×C4—61no—
279C3×C4—81no—
280C3×C4—121no—
281C3×C4—361no—
282C3×C4#370111yescharacteristic
283C3×C4#370111yescharacteristic
284C3×C4#379611yescharacteristic
285C3×C4#379911yescharacteristic
286Dic3—21no—
287Dic3—21no—
288Dic3—41no—
289Dic3—121no—
290C32#508911yescharacteristic
291C9—21no—
292C9#479411yescharacteristic, complemented
293C23—541no—
294Q8—122no—
295Q8—181no—
296Q8—181no—
297D4—541no—
298D4—541no—
299C2×C4—21no—
300C2×C4—31no—
301C2×C4—31no—
302C2×C4—61no—
303C2×C4—181no—
304C2×C4C2×D3×D911yescharacteristic
305C8—181no—
306C8—541no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—21no—
310C2×C3#6349511yescharacteristic
311C2×C3#6350511yescharacteristic
312C2×C3#6350911yescharacteristic
313C2×C3#6390711yescharacteristic
314C2×C3#6393411yescharacteristic
315C2×C3#6394811yescharacteristic
316D3—722no—
317D3—721no—
318D3—721no—
319C22—1081no—
320C22—1081no—
321C22—1081no—
322C22D3×#61111yescharacteristic
323C4—41no—
324C4—61no—
325C4—61no—
326C4—121no—
327C4—361no—
328C4#8863411yescharacteristic
329C4#8863511yescharacteristic
330C3—21no—
331C3#1058877411yescharacteristic, complemented
332C3#1059053511yescharacteristic
333C2—2161no—
334C2#1172100311yescharacteristic
335C2#1172117011yescharacteristic
336C2#1172120811yescharacteristic
subgroupG/H
centerC22D3×#611
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C2×#67
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. C9×#888C2semidirect
4. C3×#63368C2semidirect
5. #11720268C2semidirect
7. #11720258C2semidirect
34. #63368D3semidirect
90. #888D9semidirect
203. C3×C9#489semidirect
292. C9#4794semidirect
331. C3#10588774semidirect

Maximal subgroups

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

subgroupindex
C9×#8882
#117209522
#117205282
C3×#633682
#117202682
#117202672
#117202582
#105905353
#105887743