Finite Group Databasebeta

← #14553759

Subgroups — #14553759

Order 2187.

680 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10624905C311yesmaximal, characteristic, complemented
2#10624903C311yesmaximal, characteristic, complemented
3#10624897C311yesmaximal, characteristic, complemented
4#10624883C311yesmaximal, characteristic, complemented
5#10624669C313yesmaximal, complemented
6#10624669C313yesmaximal, complemented
7#10624669C313yesmaximal, complemented
8#10624654C313yesmaximal, complemented
9#10624654C313yesmaximal, complemented
10#10624638C313yesmaximal, complemented
11#10624632C311yesmaximal, characteristic, complemented
12#10624628C311yesmaximal, characteristic, complemented
13#10624622C313yesmaximal, complemented
14#10624617C311yesmaximal, characteristic, complemented
15#10624614C311yesmaximal, characteristic, complemented
16#10624602C311yesmaximal, characteristic, complemented
17#10624601C313yesmaximal, complemented
18#10624601C313yesmaximal, complemented
19#10624600C311yesmaximal, characteristic, complemented
20#10624598C311yesmaximal, characteristic, complemented
21#10624596C311yesmaximal, characteristic, complemented
22C3×#6856C311yesmaximal, characteristic, complemented
23C32×H(3,𝔽3)C3211yescharacteristic
24C32×H(3,𝔽3)C3211yescharacteristic
25#6899—33no—
26#6897—33no—
27#6895C3213yescomplemented
28#6895C3213yescomplemented
29#6894—31no—
30#6894—31no—
31#6893—33no—
32C9⋊H(3,𝔽3)—33no—
33C9⋊H(3,𝔽3)—33no—
34C9⋊H(3,𝔽3)—33no—
35C9⋊H(3,𝔽3)—33no—
36C9⋊H(3,𝔽3)—33no—
37C9⋊H(3,𝔽3)—31no—
38C9⋊H(3,𝔽3)—31no—
39C9⋊H(3,𝔽3)C3213yescomplemented
40C9⋊H(3,𝔽3)C3213yescomplemented
41C9⋊H(3,𝔽3)C3213yescomplemented
42C9⋊H(3,𝔽3)C3213yescomplemented
43C9⋊H(3,𝔽3)C3213yescomplemented
44C9⋊H(3,𝔽3)C3213yescomplemented
45C9⋊H(3,𝔽3)C3213yescomplemented
46#6891—31no—
47#6891—31no—
48#6891—31no—
49#6891—31no—
50#6891—31no—
51#6891C3211yescharacteristic
52#6891C3211yescharacteristic
53#6891C3211yescharacteristic
54#6891C3211yescharacteristic
55#6891C3211yescharacteristic
56C32⋊H(3,𝔽3)C3211yescharacteristic
57C9×3−1+2—31no—
58C9×H(3,𝔽3)—31no—
59C9×H(3,𝔽3)C3213yescomplemented
60C9×H(3,𝔽3)C3213yescomplemented
61#6887C3213yescomplemented
62C3×C32⋊C9—31no—
63C3×C32⋊C9C3211yescharacteristic
64C3×C32⋊C9C3211yescharacteristic
65C3×C32⋊C9C3211yescharacteristic
66C3×C32⋊C9C3211yescharacteristic
67C3×C32⋊C9C3211yescharacteristic
68#6861C3213yescomplemented
69#6861C3213yescomplemented
70#6861C3213yescomplemented
71#6861C3213yescomplemented
72#6861C3213yescomplemented
73#6861C3213yescomplemented
74#6861C3213yescomplemented
75#6861C3213yescomplemented
76#6861C3213yescomplemented
77#6859C3213yescomplemented
78#6859C3213yescomplemented
79#6859C3213yescomplemented
80#6859C3213yescomplemented
81#6859C3213yescomplemented
82#6859C3213yescomplemented
83#6859C3213yescomplemented
84#6859C3213yescomplemented
85#6859C3213yescomplemented
86#6856C3211yescharacteristic
87#6856C3211yescharacteristic
88#6856C3211yescharacteristic
89#6856C3211yescharacteristic
90#6856C3211yescharacteristic
91#6856C3211yescharacteristic
92#6856C3211yescharacteristic
93#6856C3211yescharacteristic
94#6856C3211yescharacteristic
95#6856C3213yescomplemented
96#6856C3213yescomplemented
97#6856C3213yescomplemented
98#6856C3213yescomplemented
99#6856C3213yescomplemented
100#6856C3213yescomplemented
101C34—31no—
102C34—31no—
103C34C3311yescharacteristic
104C3×3−1+2—31no—
105C3×3−1+2—31no—
106C3×3−1+2—31no—
107C3×3−1+2—31no—
108C3×3−1+2—31no—
109C3×3−1+2—33no—
110C3×3−1+2—33no—
111C3×3−1+2—33no—
112C3×3−1+2—33no—
113C3×3−1+2—33no—
114C3×3−1+2—33no—
115C3×3−1+2—33no—
116C3×H(3,𝔽3)—31no—
117C3×H(3,𝔽3)—31no—
118C3×H(3,𝔽3)—31no—
119C3×H(3,𝔽3)—31no—
120C3×H(3,𝔽3)—31no—
121C3×H(3,𝔽3)—31no—
122C3×H(3,𝔽3)—31no—
123C3×H(3,𝔽3)—93no—
124C3×H(3,𝔽3)—93no—
125C3×H(3,𝔽3)C3311yescharacteristic
126C3×H(3,𝔽3)C3311yescharacteristic
127C3×H(3,𝔽3)C3311yescharacteristic
128C3×H(3,𝔽3)C3311yescharacteristic
129C3×H(3,𝔽3)C3311yescharacteristic
130C3×H(3,𝔽3)C3311yescharacteristic
131C3×H(3,𝔽3)C3313yes—
132C32×C9—31no—
133C32×C9—31no—
134C32×C9—31no—
135C32×C9—31no—
136C32×C9—33no—
137C32×C9—33no—
138C32×C9C3313yes—
139C9⋊C9—33no—
140C9⋊C9—33no—
141C9⋊C9—33no—
142C9⋊C9—33no—
143C9⋊C9—33no—
144C9⋊C9—33no—
145C9⋊C9—33no—
146C9⋊C9—33no—
147C32⋊C9—31no—
148C32⋊C9—31no—
149C32⋊C9—31no—
150C32⋊C9—31no—
151C32⋊C9—31no—
152C32⋊C9—31no—
153C32⋊C9—31no—
154C32⋊C9—31no—
155C32⋊C9—31no—
156C32⋊C9—31no—
157C32⋊C9—31no—
158C32⋊C9—31no—
159C32⋊C9—31no—
160C32⋊C9—31no—
161C32⋊C9—31no—
162C32⋊C9—31no—
163C32⋊C9—31no—
164C32⋊C9—31no—
165C32⋊C9—33no—
166C32⋊C9—33no—
167C32⋊C9—33no—
168C32⋊C9—33no—
169C32⋊C9—33no—
170C32⋊C9—33no—
171C32⋊C9—33no—
172C32⋊C9—33no—
173C32⋊C9—33no—
174C32⋊C9—33no—
175C32⋊C9—33no—
176C32⋊C9—33no—
177C32⋊C9—33no—
178C32⋊C9—33no—
179C32⋊C9—33no—
180C32⋊C9—33no—
181C32⋊C9—33no—
182C32⋊C9—33no—
183C32⋊C9H(3,𝔽3)11yescharacteristic
184C32⋊C9H(3,𝔽3)11yescharacteristic
185C32⋊C9H(3,𝔽3)11yescharacteristic
186C32⋊C9H(3,𝔽3)13yes—
187C32⋊C9H(3,𝔽3)13yes—
188C32⋊C9C3311yescharacteristic
189C32⋊C9C3311yescharacteristic
190C32⋊C9C3311yescharacteristic
191C32⋊C9C3311yescharacteristic
192C32⋊C9C3311yescharacteristic
193C32⋊C9C3311yescharacteristic
194C32⋊C9C3313yes—
195C32⋊C9C3313yes—
196C32⋊C9C3313yes—
197C32⋊C9C3313yes—
198C32⋊C9C3313yes—
199C32⋊C9C3313yes—
200C32⋊C9C3313yes—
201C92—33no—
202C33—31no—
203C33—33no—
204C33—33no—
205C33—31no—
206C33—31no—
207C33—31no—
208C33—31no—
209C33—31no—
210C33—91no—
211C33—91no—
212C33—91no—
213C33—93no—
214C33—93no—
215C33—93no—
216C33—271no—
217C33—271no—
218C33—271no—
219C33—271no—
220C33—271no—
221C33C3×H(3,𝔽3)11yescharacteristic
222C33C3×H(3,𝔽3)11yescharacteristic
223C33C3×H(3,𝔽3)11yescharacteristic
224C33C3411yescharacteristic
2253−1+2—93no—
2263−1+2—273no—
2273−1+2—273no—
2283−1+2—273no—
2293−1+2—273no—
2303−1+2—273no—
2313−1+2—273no—
2323−1+2—273no—
2333−1+2—271no—
2343−1+2—271no—
2353−1+2—271no—
2363−1+2—271no—
237H(3,𝔽3)—33no—
238H(3,𝔽3)—93no—
239H(3,𝔽3)—93no—
240H(3,𝔽3)—91no—
241H(3,𝔽3)—91no—
242H(3,𝔽3)—91no—
243H(3,𝔽3)—91no—
244H(3,𝔽3)—91no—
245H(3,𝔽3)—93no—
246H(3,𝔽3)—93no—
247H(3,𝔽3)—93no—
248H(3,𝔽3)—93no—
249H(3,𝔽3)—271no—
250H(3,𝔽3)—271no—
251C3×C9—31no—
252C3×C9—33no—
253C3×C9—33no—
254C3×C9—31no—
255C3×C9—33no—
256C3×C9—33no—
257C3×C9—33no—
258C3×C9—33no—
259C3×C9—31no—
260C3×C9—31no—
261C3×C9—33no—
262C3×C9—31no—
263C3×C9—31no—
264C3×C9—93no—
265C3×C9—273no—
266C3×C9—273no—
267C3×C9—271no—
268C3×C9—271no—
269C3×C9—271no—
270C3×C9—271no—
271C3×C9C3×H(3,𝔽3)13yes—
272C3×C9C3×H(3,𝔽3)13yes—
273C3×C9C3×H(3,𝔽3)13yes—
274C32—31no—
275C32—31no—
276C32—31no—
277C32—31no—
278C32—31no—
279C32—31no—
280C32—31no—
281C32—31no—
282C32—31no—
283C32—31no—
284C32—33no—
285C32—31no—
286C32—31no—
287C32—31no—
288C32—31no—
289C32—31no—
290C32—91no—
291C32—91no—
292C32—91no—
293C32—91no—
294C32—91no—
295C32—91no—
296C32—93no—
297C32—91no—
298C32—91no—
299C32—93no—
300C32—91no—
301C32—91no—
302C32—91no—
303C32—91no—
304C32—91no—
305C32—91no—
306C32—91no—
307C32—93no—
308C32—91no—
309C32—91no—
310C32—91no—
311C32—93no—
312C32—91no—
313C32—91no—
314C32—91no—
315C32—91no—
316C32—91no—
317C32—273no—
318C32—279no—
319C32—279no—
320C32#691313yes—
321C32C32×H(3,𝔽3)11yescharacteristic
322C9—33no—
323C9—33no—
324C9—33no—
325C9—93no—
326C9—91no—
327C9—93no—
328C9—93no—
329C9—93no—
330C9—93no—
331C9—93no—
332C9—91no—
333C9—91no—
334C9—93no—
335C9—91no—
336C9—91no—
337C9—91no—
338C3—33no—
339C3—91no—
340C3—91no—
341C3—91no—
342C3—93no—
343C3—273no—
344C3—271no—
345C3—271no—
346C3—271no—
347C3—271no—
348C3—271no—
349C3C3×#691311yescharacteristic
350C3#1062493211yescharacteristic
351C3#1062493411yescharacteristic
352C3#1062493611yescharacteristic
subgroupG/H
center, socleC32C32×H(3,𝔽3)
commutator, Frattini, cosocleC33C34
LCSG ▷ C33 ▷ C32 ▷ C1
UCSC1 ◁ C32 ◁ C3×C32⋊C9 ◁ G
LECSG ▷ C33 ▷ C32 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #10624905C3semidirect
2. #10624903C3semidirect
3. #10624897C3semidirect
4. #10624883C3semidirect
5. #10624669C3semidirect
6. #10624669C3semidirect
7. #10624669C3semidirect
8. #10624654C3semidirect
9. #10624654C3semidirect
10. #10624638C3semidirect
11. #10624632C3semidirect
12. #10624628C3semidirect
13. #10624622C3semidirect
14. #10624617C3semidirect
15. #10624614C3semidirect
16. #10624602C3semidirect
17. #10624601C3semidirect
18. #10624601C3semidirect
19. #10624600C3semidirect
20. #10624598C3semidirect
21. #10624596C3semidirect
22. C3×#6856C3semidirect
27. #6895C32semidirect
28. #6895C32semidirect
39. C9⋊H(3,𝔽3)C32semidirect
40. C9⋊H(3,𝔽3)C32semidirect
41. C9⋊H(3,𝔽3)C32semidirect
42. C9⋊H(3,𝔽3)C32semidirect
43. C9⋊H(3,𝔽3)C32semidirect
44. C9⋊H(3,𝔽3)C32semidirect
45. C9⋊H(3,𝔽3)C32semidirect
59. C9×H(3,𝔽3)C32semidirect
60. C9×H(3,𝔽3)C32semidirect
61. #6887C32semidirect
68. #6861C32semidirect
69. #6861C32semidirect
70. #6861C32semidirect
71. #6861C32semidirect
72. #6861C32semidirect
73. #6861C32semidirect
74. #6861C32semidirect
75. #6861C32semidirect
76. #6861C32semidirect
77. #6859C32semidirect
78. #6859C32semidirect
79. #6859C32semidirect
80. #6859C32semidirect
81. #6859C32semidirect
82. #6859C32semidirect
83. #6859C32semidirect
84. #6859C32semidirect
85. #6859C32semidirect
95. #6856C32semidirect
96. #6856C32semidirect
97. #6856C32semidirect
98. #6856C32semidirect
99. #6856C32semidirect
100. #6856C32semidirect

Maximal subgroups

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

subgroupindex
#106249053
#106249033
#106248973
#106248833
#106246693
#106246543
#106246383
#106246323
#106246283
#106246223
#106246173
#106246143
#106246023
#106246013
#106246003
#106245983
#106245963
C3×#68563