Finite Group Databasebeta

← #10588474⋊C3 #13141937

Subgroups — #10588474⋊C3 #13141937

Order 1728.

321 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 321 conjugacy classes of subgroups — too many to draw. Only its 86 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
1#63509⋊C3C211yesmaximal, characteristic
2#63450⋊C3C211yesmaximal, characteristic, complemented
33−1+2×#103C211yesmaximal, characteristic, complemented
4#63395⋊C3C211yesmaximal, characteristic, complemented
5#63380⋊C3C211yesmaximal, characteristic
6#63375⋊C3C211yesmaximal, characteristic, complemented
7#63369⋊C3C211yesmaximal, characteristic
8C3×#4794—31nomaximal
9#10588474C311yesmaximal, characteristic, complemented
10C2×Q8×3−1+2C2211yescharacteristic
11C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
12C2×3−1+2⋊Q8—21no—
13C2×C8×3−1+2C2211yescharacteristic
14C4⋊C4×3−1+2C2211yescharacteristic
15(C2×AGL(1,ℤ/9))⋊C4—21no—
16(C2×AGL(1,ℤ/9))⋊C4—21no—
17C4⋊(3−1+2⋊C4)C2211yescharacteristic
18(3−1+2⋊C4)⋊C4—21no—
19C4×3−1+2⋊C4C2211yescharacteristic
20C2×3−1+2⋊C8C2211yescharacteristic
21C3×#947—31no—
22C3×#888—31no—
23C32×#103D311yescharacteristic
24C3×#835—31no—
25C3×#821—31no—
26C3×#815—31no—
27C3×#809—31no—
28#63509C2×C311yescharacteristic
29#63450C2×C311yescharacteristic, complemented
30C9×#103—21no—
31C9×#103C2×C311yescharacteristic, complemented
32#63395C2×C311yescharacteristic, complemented
33#63380C2×C311yescharacteristic
34#63375C2×C311yescharacteristic, complemented
35#63369C2×C311yescharacteristic
36C22×AGL(1,ℤ/9)—21no—
37Q8×3−1+2—21no—
38Q8×3−1+2—21no—
39C2×C4×3−1+2—21no—
40C2×C4×3−1+2—21no—
41C2×C4×3−1+2C2311yescharacteristic
42C2×3−1+2⋊C4—21no—
43C2×3−1+2⋊C4D411yescharacteristic
44C2×3−1+2⋊C4D411yescharacteristic
45C4⋊AGL(1,ℤ/9)—21no—
46C4⋊AGL(1,ℤ/9)—21no—
473−1+2⋊Q8—42no—
48C8×3−1+2—21no—
493−1+2⋊C8—21no—
50C3×#489—91no—
51#4794—31no—
52C2×C32×Q8C2×D311yescharacteristic
53C2×C3×D12—31no—
54C2×C3×Dic6—61no—
55C2×C32×C8C2×D311yescharacteristic
56C32×C4⋊C4C2×D311yescharacteristic
57C3×(C2×D3)⋊C4—61no—
58C3×(C2×D3)⋊C4—61no—
59C3×C4⋊Dic3—31no—
60C3×Dic3⋊C4—61no—
61C3×C4×Dic3—31no—
62C2×C3×C3⋊C8—31no—
63C2×Q8×C9—21no—
64C2×Q8×C9C22×C311yescharacteristic
65C2×D36C22×C311yescharacteristic
66C2×Dic18—21no—
67C2×C8×C9—21no—
68C2×C8×C9C22×C311yescharacteristic
69C9×C4⋊C4—21no—
70C9×C4⋊C4C22×C311yescharacteristic
71(C2×D9)⋊C4—21no—
72(C2×D9)⋊C4—21no—
73C4⋊Dic9C22×C311yescharacteristic
74Dic9⋊C4—21no—
75C4×Dic9C22×C311yescharacteristic
76C2×C9⋊C8C22×C311yescharacteristic
77C22×3−1+2C2×D411yescharacteristic
78C2×AGL(1,ℤ/9)—41no—
79C2×AGL(1,ℤ/9)—41no—
80C2×AGL(1,ℤ/9)—41no—
81C4×3−1+2—41no—
82C4×3−1+2—41no—
83C4×3−1+2#4111yescharacteristic
84C4×3−1+2#4111yescharacteristic
853−1+2⋊C4—21no—
863−1+2⋊C4—21no—
873−1+2⋊C4—41no—
88C3×C4⋊Q8—91no—
89C3×#124—91no—
90#947—31no—
91#888—31no—
92C3×#103—31no—
93C3×#103—91no—
94C3×#103C3×D311yescharacteristic
95C3×#102—91no—
96C3×#102—91no—
97C3×#97—91no—
98#835—31no—
99#821—31no—
100#815—31no—
101#809—31no—
102C22×C3×D3—61no—
103C32×Q8—21no—
104C32×Q8—21no—
105C2×C32×C4—21no—
106C2×C32×C4—21no—
107C2×C32×C4C22×D311yescharacteristic
108C2×C3×Dic3—31no—
109C2×C3×Dic3—31no—
110C2×C3×Dic3—61no—
111C3×D12—61no—
112C3×D12—61no—
113C3×Dic6—122no—
114C22×D9—21no—
115C32×C8—21no—
116C3×C3⋊C8—61no—
117Q8×C9—22no—
118Q8×C9—21no—
119Q8×C9—21no—
120Q8×C9—41no—
121C2×C4×C9—21no—
122C2×C4×C9—21no—
123C2×C4×C9—21no—
124C2×C4×C9—41no—
125C2×C4×C9—41no—
126C2×C4×C9C23×C311yescharacteristic
127C2×Dic9—21no—
128C2×Dic9C3×D411yescharacteristic
129C2×Dic9C3×D411yescharacteristic
130D36—21no—
131D36—21no—
132Dic18—42no—
133C8×C9—21no—
134C8×C9—41no—
135C9⋊C8—21no—
136#489—91noSylow
137C2×3−1+2#12411yescharacteristic
138C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
139C2×3−1+2#13711yescharacteristic
140AGL(1,ℤ/9)—81no—
141C2×C3×Q8—31no—
142C2×C3×Q8—181no—
143C2×C3×Q8C2×C3×D311yescharacteristic
144C2×C3×D4—91no—
145C2×D12—31no—
146C2×Dic6—61no—
147C2×C3×C8—31no—
148C2×C3×C8—91no—
149C2×C3×C8C2×C3×D311yescharacteristic
150C3×C4⋊C4—31no—
151C3×C4⋊C4—91no—
152C3×C4⋊C4—181no—
153C3×C4⋊C4C2×C3×D311yescharacteristic
154C3×C22⋊C4—181no—
155C3×C22⋊C4—181no—
156C3×C42—91no—
157(C2×D3)⋊C4—61no—
158(C2×D3)⋊C4—61no—
159C4⋊Dic3—31no—
160Dic3⋊C4—61no—
161C4×Dic3—31no—
162C2×C3⋊C8—31no—
163C22×C32D3×D411yescharacteristic
164C2×C3×D3—121no—
165C2×C3×D3—121no—
166C2×C3×D3—121no—
167C32×C4—41no—
168C32×C4—41no—
169C32×C4#23511yescharacteristic
170C32×C4#23711yescharacteristic
171C3×Dic3—61no—
172C3×Dic3—61no—
173C3×Dic3—121no—
174C22×C9—21no—
175C22×C9C2×C3×D411yescharacteristic
176C2×D9—41no—
177C2×D9—41no—
178C2×D9—41no—
179C4×C9—21no—
180C4×C9—21no—
181C4×C9—42no—
182C4×C9—41no—
183C4×C9—41no—
184C4×C9—81no—
185C4×C9C3×#4111yescharacteristic
186C4×C9C3×#4111yescharacteristic
187Dic9—21no—
188Dic9—21no—
189Dic9—41no—
190C4⋊Q8—91no—
191#124—91no—
192#103—91no—
193#103AGL(1,ℤ/9)11yescharacteristic, complemented
194#102—91no—
195#102—91no—
196#97—91no—
1973−1+2#48911yescharacteristic, complemented, Sylow
198C23×C3—181no—
199C22×D3—61no—
200C3×Q8—21no—
201C3×Q8—21no—
202C3×Q8—61no—
203C3×Q8—61no—
204C3×Q8—362no—
205C3×D4—181no—
206C3×D4—181no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—31no—
210C2×C3×C4—61no—
211C2×C3×C4—61no—
212C2×C3×C4—91no—
213C2×C3×C4—91no—
214C2×C3×C4—181no—
215C2×C3×C4C22×C3×D311yescharacteristic
216C2×Dic3—31no—
217C2×Dic3—31no—
218C2×Dic3—61no—
219D12—61no—
220D12—61no—
221Dic6—122no—
222C3×C8—21no—
223C3×C8—61no—
224C3×C8—181no—
225C3⋊C8—61no—
226C2×C32#88511yescharacteristic
227C2×C32#91411yescharacteristic
228C2×C32#91811yescharacteristic
229C3×D3—241no—
230C2×C9—21no—
231C2×C9—21no—
232C2×C9—21no—
233C2×C9C3×#12411yescharacteristic
234C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
235C2×C9C3×#13711yescharacteristic
236D9—81no—
237C2×Q8—181no—
238C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
239C2×D4—91no—
240C2×C8—91no—
241C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
242C4⋊C4—91no—
243C4⋊C4—181no—
244C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
245C22⋊C4—181no—
246C22⋊C4—181no—
247C42—91no—
248C22×C3—31no—
249C22×C3—361no—
250C22×C3—361no—
251C22×C3—361no—
252C22×C3C3×D3×D411yescharacteristic
253C2×D3—121no—
254C2×D3—121no—
255C2×D3—121no—
256C3×C4—31no—
257C3×C4—31no—
258C3×C4—41no—
259C3×C4—41no—
260C3×C4—121no—
261C3×C4—121no—
262C3×C4—181no—
263C3×C4—181no—
264C3×C4—361no—
265C3×C4C3×#23511yescharacteristic
266C3×C4C3×#23711yescharacteristic
267Dic3—61no—
268Dic3—61no—
269Dic3—121no—
270C32#479411yescharacteristic
271C9—21no—
272C9C3×#48911yescharacteristic, complemented
273C23—181no—
274Q8—21no—
275Q8—21no—
276Q8—362no—
277D4—181no—
278D4—181no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—91no—
282C2×C4—91no—
283C2×C4—181no—
284C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
285C8—21no—
286C8—181no—
287C2×C3—31no—
288C2×C3—31no—
289C2×C3—31no—
290C2×C3—721no—
291C2×C3C3×#88511yescharacteristic
292C2×C3C3×#91411yescharacteristic
293C2×C3C3×#91811yescharacteristic
294D3—241no—
295C22—361no—
296C22—361no—
297C22—361no—
298C22D4×AGL(1,ℤ/9)11yescharacteristic
299C4—41no—
300C4—41no—
301C4—181no—
302C4—181no—
303C4—361no—
304C4#3697⋊C311yescharacteristic
305C4#3699⋊C311yescharacteristic
306C3—31no—
307C3C3×#479411yescharacteristic
308C2—721no—
309C2#63447⋊C311yescharacteristic
310C2#63476⋊C311yescharacteristic
311C2#63480⋊C311yescharacteristic
subgroupG/H
centerC22D4×AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#103C2
socleC22×C3C3×D3×D4
prime radicalC2×C4⋊AGL(1,ℤ/9)C22
hypercenter#103AGL(1,ℤ/9)
hypocenterC9C3×#489
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
2. #63450⋊C3C2semidirect
3. 3−1+2×#103C2semidirect
4. #63395⋊C3C2semidirect
6. #63375⋊C3C2semidirect
9. #10588474C3semidirect
29. #63450C2×C3semidirect
31. C9×#103C2×C3semidirect
32. #63395C2×C3semidirect
34. #63375C2×C3semidirect
193. #103AGL(1,ℤ/9)semidirect
197. 3−1+2#489semidirect
272. C9C3×#489semidirect

Maximal subgroups

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

subgroupindex
#63509⋊C32
#63450⋊C32
3−1+2×#1032
#63395⋊C32
#63380⋊C32
#63375⋊C32
#63369⋊C32
C3×#47943
#105884743