Finite Group Databasebeta

← #14552214

Subgroups — #14552214

Order 2187.

1046 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#6895C311yesmaximal, characteristic, complemented
2C3×#6894C311yesmaximal, characteristic, complemented
3C3×#6893C311yesmaximal, characteristic
4C3×C9⋊H(3,𝔽3)C311yesmaximal, characteristic
5C3×C9×3−1+2C311yesmaximal, characteristic, complemented
6#10624566C313yesmaximal, complemented
7#10624566C313yesmaximal, complemented
8#10624566C313yesmaximal, complemented
9#10624566C313yesmaximal, complemented
10#10624565C313yesmaximal, complemented
11#10624565C313yesmaximal, complemented
12#10624560C313yesmaximal, complemented
13#10624558C311yesmaximal, characteristic, complemented
14#10624558C311yesmaximal, characteristic, complemented
15#10624558C311yesmaximal, characteristic, complemented
16#10624558C311yesmaximal, characteristic, complemented
17#10624557C313yesmaximal, complemented
18#10624557C313yesmaximal, complemented
19#10624555C311yesmaximal, characteristic
20#10624555C311yesmaximal, characteristic
21#10624549C311yesmaximal, characteristic, complemented
22C3×#6855C311yesmaximal, characteristic, complemented
23C32×3−1+2C3211yescharacteristic
24C32×3−1+2C3211yescharacteristic
25C32×3−1+2C3213yes—
26C32×H(3,𝔽3)C3211yescharacteristic
27C33×C9C3211yescharacteristic
28C33×C9C3211yescharacteristic
29C33×C9C3211yescharacteristic
30C33×C9C3211yescharacteristic
31#6895—39no—
32#6894—39no—
33#6893—39no—
34C9⋊H(3,𝔽3)—39no—
35C9×3−1+2—39no—
36C3×C9⋊C9C3213yes—
37C3×C9⋊C9C3213yes—
38C3×C9⋊C9C3213yes—
39C3×C9⋊C9C3213yes—
40C3×C9⋊C9C3213yes—
41C3×C9⋊C9C3213yes—
42C3×C9⋊C9C3213yes—
43C3×C9⋊C9C3213yes—
44C3×C32⋊C9C3211yescharacteristic
45C3×C32⋊C9C3211yescharacteristic
46C3×C32⋊C9C3211yescharacteristic
47C3×C32⋊C9C3213yes—
48C3×C32⋊C9C3213yes—
49C3×C32⋊C9C3211yescharacteristic
50C3×C32⋊C9C3211yescharacteristic
51C3×C32⋊C9C3213yes—
52C3×C32⋊C9C3213yes—
53C3×C32⋊C9C3213yes—
54C3×C32⋊C9C3213yes—
55C3×C32⋊C9C3213yes—
56C3×C32⋊C9C3213yes—
57C3×C32⋊C9C3213yes—
58C3×C32⋊C9C3213yes—
59C3×C32⋊C9C3213yes—
60C3×C32⋊C9C3211yescharacteristic
61C3×C92C3213yes—
62#6855C3211yescharacteristic
63#6855C3211yescharacteristic
64#6855C3211yescharacteristic
65#6855C3211yescharacteristic
66#6855C3211yescharacteristic
67#6855C3211yescharacteristic
68#6855C3211yescharacteristic
69#6855C3211yescharacteristic
70#6855C3211yescharacteristic
71#6855C3213yes—
72#6855C3213yes—
73#6855C3213yes—
74#6855C3213yes—
75#6855C3213yes—
76#6855C3213yes—
77#6855C3213yes—
78#6855C3213yes—
79#6855C3213yes—
80#6855C3213yes—
81#6855C3213yes—
82#6855C3213yes—
83#6855C3213yes—
84#6855C3213yes—
85#6855C3213yes—
86C34C3313yes—
87C34C3311yescharacteristic
88C3×3−1+2—33no—
89C3×3−1+2—33no—
90C3×3−1+2—33no—
91C3×3−1+2—33no—
92C3×3−1+2—33no—
93C3×3−1+2—33no—
94C3×3−1+2—33no—
95C3×3−1+2—39no—
96C3×3−1+2—39no—
97C3×3−1+2—93no—
98C3×3−1+2—93no—
99C3×3−1+23−1+219yescomplemented
100C3×H(3,𝔽3)—33no—
101C3×H(3,𝔽3)—33no—
102C3×H(3,𝔽3)—33no—
103C3×H(3,𝔽3)—33no—
104C32×C9—33no—
105C32×C9—33no—
106C32×C9—33no—
107C32×C9—33no—
108C32×C9—33no—
109C32×C9—33no—
110C32×C9—33no—
111C32×C9—33no—
112C32×C9—33no—
113C32×C9—33no—
114C32×C9—91no—
115C32×C9—91no—
116C32×C9—91no—
117C32×C9—91no—
118C32×C9—91no—
119C32×C9—91no—
120C32×C9C3311yescharacteristic
121C32×C9C3311yescharacteristic
122C32×C9C3311yescharacteristic
123C32×C9C3311yescharacteristic
124C32×C9C3311yescharacteristic
125C32×C9C3311yescharacteristic
126C32×C9C3311yescharacteristic
127C32×C9C3311yescharacteristic
128C32×C9C3311yescharacteristic
129C32×C9C3313yes—
130C32×C9C3313yes—
131C32×C9C3313yes—
132C32×C9C3313yes—
133C32×C9C3313yes—
134C32×C9C3313yes—
135C32×C9C3313yes—
136C32×C9C3313yes—
137C32×C9C3311yescharacteristic
138C32×C9C3311yescharacteristic
139C32×C9C3311yescharacteristic
140C9⋊C9—39no—
141C9⋊C9—39no—
142C9⋊C9—39no—
143C9⋊C9—39no—
144C9⋊C9—39no—
145C9⋊C9—39no—
146C9⋊C9—39no—
147C9⋊C9—39no—
148C32⋊C9—33no—
149C32⋊C9—33no—
150C32⋊C9—33no—
151C32⋊C9—33no—
152C32⋊C9—33no—
153C32⋊C9—39no—
154C32⋊C9—39no—
155C32⋊C9—39no—
156C32⋊C9—39no—
157C32⋊C9—39no—
158C32⋊C9—39no—
159C32⋊C9—39no—
160C32⋊C9—93no—
161C32⋊C9—93no—
162C32⋊C9—93no—
163C32⋊C9—93no—
164C32⋊C9H(3,𝔽3)19yes—
165C92—39no—
166C33—31no—
167C33—31no—
168C33—31no—
169C33—31no—
170C33—31no—
171C33—31no—
172C33—31no—
173C33—31no—
174C33—31no—
175C33—33no—
176C33—33no—
177C33—33no—
178C33—33no—
179C33—33no—
180C33—33no—
181C33—33no—
182C33—33no—
183C33—33no—
184C33—33no—
185C33—33no—
186C33—33no—
187C33—33no—
188C33C3×H(3,𝔽3)13yes—
189C33C3×3−1+213yes—
190C33#73713yes—
191C33#73713yes—
192C33C3411yescharacteristic
1933−1+2—327no—
1943−1+2—927no—
1953−1+2—99no—
196H(3,𝔽3)—99no—
197C3×C9—31no—
198C3×C9—33no—
199C3×C9—31no—
200C3×C9—31no—
201C3×C9—33no—
202C3×C9—33no—
203C3×C9—33no—
204C3×C9—31no—
205C3×C9—31no—
206C3×C9—31no—
207C3×C9—31no—
208C3×C9—31no—
209C3×C9—31no—
210C3×C9—33no—
211C3×C9—33no—
212C3×C9—33no—
213C3×C9—31no—
214C3×C9—31no—
215C3×C9—33no—
216C3×C9—33no—
217C3×C9—33no—
218C3×C9—33no—
219C3×C9—31no—
220C3×C9—31no—
221C3×C9—31no—
222C3×C9—31no—
223C3×C9—31no—
224C3×C9—31no—
225C3×C9—31no—
226C3×C9—33no—
227C3×C9—31no—
228C3×C9—31no—
229C3×C9—31no—
230C3×C9—33no—
231C3×C9—33no—
232C3×C9—33no—
233C3×C9—33no—
234C3×C9—33no—
235C3×C9—31no—
236C3×C9—31no—
237C3×C9—31no—
238C3×C9—31no—
239C3×C9—31no—
240C3×C9—31no—
241C3×C9—31no—
242C3×C9—31no—
243C3×C9—31no—
244C3×C9—31no—
245C3×C9—31no—
246C3×C9—31no—
247C3×C9—33no—
248C3×C9—33no—
249C3×C9—33no—
250C3×C9—31no—
251C3×C9—31no—
252C3×C9—31no—
253C3×C9—31no—
254C3×C9—33no—
255C3×C9—33no—
256C3×C9—33no—
257C3×C9—33no—
258C3×C9—33no—
259C3×C9—33no—
260C3×C9—33no—
261C3×C9—33no—
262C3×C9—31no—
263C3×C9—31no—
264C3×C9—31no—
265C3×C9—31no—
266C3×C9—31no—
267C3×C9—31no—
268C3×C9—31no—
269C3×C9—31no—
270C3×C9—33no—
271C3×C9—33no—
272C3×C9—99no—
273C3×C9—99no—
274C3×C9—99no—
275C3×C9—99no—
276C3×C9C3×H(3,𝔽3)19yes—
277C3×C9C3×3−1+219yes—
278C3×C9#73713yes—
279C3×C9#73713yes—
280C3×C9#73713yes—
281C32—33no—
282C32—33no—
283C32—33no—
284C32—33no—
285C32—33no—
286C32—33no—
287C32—33no—
288C32—33no—
289C32—33no—
290C32—33no—
291C32—33no—
292C32—33no—
293C32—93no—
294C32—93no—
295C32—93no—
296C32—93no—
297C32—93no—
298C32—93no—
299C32—93no—
300C32—93no—
301C32—93no—
302C32—93no—
303C32—93no—
304C32—93no—
305C32—93no—
306C32C9⋊H(3,𝔽3)19yes—
307C32C32×H(3,𝔽3)11yescharacteristic
308C32C32×3−1+211yescharacteristic
309C32C3×#73711yescharacteristic
310C32C3×#73711yescharacteristic
311C32C3×#73711yescharacteristic
312C32C3×#73711yescharacteristic
313C32C3×#73711yescharacteristic
314C323−1+411yescharacteristic
315C323−1+411yescharacteristic
316C323−1+411yescharacteristic
317C323−1+411yescharacteristic
318C323−1+411yescharacteristic
319C323−1+411yescharacteristic
320C9—39no—
321C9—33no—
322C9—39no—
323C9—33no—
324C9—33no—
325C9—91no—
326C9—91no—
327C9—91no—
328C9—93no—
329C9—91no—
330C9—91no—
331C9—93no—
332C9—93no—
333C9—93no—
334C9—93no—
335C9—93no—
336C9—91no—
337C9—91no—
338C9—91no—
339C9—91no—
340C3—39no—
341C3—273no—
342C3C3×C9⋊H(3,𝔽3)11yescharacteristic
343C3#1062487711yescharacteristic
344C3#1062489111yescharacteristic
345C3#1062489111yescharacteristic
346C3#1062489411yescharacteristic
347C3#1062489611yescharacteristic
348C3#1062489611yescharacteristic
349C3#1062490111yescharacteristic
350C3#1062490111yescharacteristic
351C3#1062490111yescharacteristic
352C3#1062490411yescharacteristic
353C3#1062490711yescharacteristic
354C3#1062490711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC33C34
prime radicalC32×H(3,𝔽3)C32
LCSG ▷ C33 ▷ C1
UCSC1 ◁ C33 ◁ G
LECSG ▷ C33 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C3×#6895C3semidirect
2. C3×#6894C3semidirect
5. C3×C9×3−1+2C3semidirect
6. #10624566C3semidirect
7. #10624566C3semidirect
8. #10624566C3semidirect
9. #10624566C3semidirect
10. #10624565C3semidirect
11. #10624565C3semidirect
12. #10624560C3semidirect
13. #10624558C3semidirect
14. #10624558C3semidirect
15. #10624558C3semidirect
16. #10624558C3semidirect
17. #10624557C3semidirect
18. #10624557C3semidirect
21. #10624549C3semidirect
22. C3×#6855C3semidirect
99. C3×3−1+23−1+2semidirect

Maximal subgroups

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

subgroupindex
C3×#68953
C3×#68943
C3×#68933
C3×C9⋊H(3,𝔽3)3
C3×C9×3−1+23
#106245663
#106245653
#106245603
#106245583
#106245573
#106245553
#106245493
C3×#68553