Finite Group Databasebeta

← #14550596

Subgroups — #14550596

Order 2187.

1247 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 1247 conjugacy classes of subgroups — too many to draw. Only its 98 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×#6911C311yesmaximal, characteristic, complemented
2C3×#6911C312yesmaximal, complemented
3C3×#6910C312yesmaximal, complemented
4C3×#6909C312yesmaximal, complemented
5C3×#6909C311yesmaximal, characteristic, complemented
6C32×#732C311yesmaximal, characteristic, complemented
7C32×#731C311yesmaximal, characteristic, complemented
8#10624881C311yesmaximal, characteristic, complemented
9C3×#6893C312yesmaximal, complemented
10C32×3−1+2C3212yes—
11C32×3−1+2C3211yescharacteristic
12C32×3−1+2C3211yescharacteristic
13C32×3−1+2C3212yes—
14C32×H(3,𝔽3)C3212yes—
15C32×H(3,𝔽3)C3211yescharacteristic
16C32×H(3,𝔽3)C3211yescharacteristic
17C32×H(3,𝔽3)C3212yes—
18C33×C9C3211yescharacteristic
19#6911—33no—
20#6911—36no—
21#6911—36no—
22#6911—36no—
23#6911—36no—
24#6910—36no—
25#6910—36no—
26#6910—36no—
27#6909—36no—
28#6909—36no—
29#6909—33no—
30#6909—36no—
31#6909—36no—
32C3×#732—33no—
33C3×#732—36no—
34C3×#732—33no—
35C3×#732—33no—
36C3×#732—36no—
37C3×#731—33no—
38C3×#731—33no—
39C3×#731—33no—
40C3×#731—33no—
41C3×#731—36no—
42C3×AΣL(1,𝔽27)—36no—
43C3×AΣL(1,𝔽27)—33no—
44#6893—36no—
45#6893—36no—
46#6893—33no—
47#6893—36no—
48#6893—36no—
49#6891—33no—
50C3×C32⋊C9—36no—
51C34—32no—
52C34—31no—
53C34—31no—
54C34—32no—
55C34C3311yescharacteristic
56C3×3−1+2—32no—
57C3×3−1+2—32no—
58C3×3−1+2—32no—
59C3×3−1+2—32no—
60C3×3−1+2—32no—
61C3×3−1+2—32no—
62C3×3−1+2—32no—
63C3×3−1+2—32no—
64C3×3−1+2—32no—
65C3×3−1+2—32no—
66C3×3−1+2—32no—
67C3×3−1+2—32no—
68C3×3−1+2—32no—
69C3×3−1+2—32no—
70C3×3−1+2—32no—
71C3×3−1+2—32no—
72C3×3−1+2—32no—
73C3×3−1+2—32no—
74C3×3−1+2—31no—
75C3×3−1+2—31no—
76C3×3−1+2—31no—
77C3×3−1+2—31no—
78C3×3−1+2—31no—
79C3×3−1+2—32no—
80C3×3−1+2—32no—
81C3×3−1+2—31no—
82C3×3−1+2—32no—
83C3×3−1+2—32no—
84C3×3−1+2—32no—
85C3×3−1+2—32no—
86C3×3−1+2—32no—
87C3×3−1+2—32no—
88C3×3−1+2—32no—
89C3×3−1+2—32no—
90C3×3−1+2—32no—
91C3×3−1+2—32no—
92C3×3−1+2—32no—
93C3×3−1+2H(3,𝔽3)13yes—
94C3×3−1+2H(3,𝔽3)16yes—
95C3×3−1+2H(3,𝔽3)13yes—
96C3×H(3,𝔽3)—32no—
97C3×H(3,𝔽3)—31no—
98C3×H(3,𝔽3)—31no—
99C3×H(3,𝔽3)—32no—
100C3×H(3,𝔽3)—31no—
101C3×H(3,𝔽3)—32no—
102C3×H(3,𝔽3)—32no—
103C3×H(3,𝔽3)—31no—
104C3×H(3,𝔽3)—32no—
105C3×H(3,𝔽3)—32no—
106C3×H(3,𝔽3)—32no—
107C3×H(3,𝔽3)—32no—
108C3×H(3,𝔽3)—32no—
109C3×H(3,𝔽3)—32no—
110C3×H(3,𝔽3)—32no—
111C3×H(3,𝔽3)—32no—
112C3×H(3,𝔽3)—32no—
113C3×H(3,𝔽3)—31no—
114C3×H(3,𝔽3)—32no—
115C3×H(3,𝔽3)—32no—
116C3×H(3,𝔽3)—32no—
117C3×H(3,𝔽3)—31no—
118C3×H(3,𝔽3)—32no—
119C3×H(3,𝔽3)—32no—
120C3×H(3,𝔽3)—32no—
121C3×H(3,𝔽3)—32no—
122C3×H(3,𝔽3)—32no—
123C3×H(3,𝔽3)—32no—
124C3×H(3,𝔽3)—32no—
125C3×H(3,𝔽3)—32no—
126C3×H(3,𝔽3)—32no—
127C3×H(3,𝔽3)—32no—
128C3×H(3,𝔽3)—32no—
129C3×H(3,𝔽3)—32no—
130C3×H(3,𝔽3)—32no—
131C3×H(3,𝔽3)—32no—
132C3×H(3,𝔽3)H(3,𝔽3)13yes—
133C3×H(3,𝔽3)H(3,𝔽3)16yes—
134C3×H(3,𝔽3)H(3,𝔽3)13yes—
135C3×H(3,𝔽3)H(3,𝔽3)16yes—
136C32×C9—32no—
137C32×C9—32no—
138C32×C9—32no—
139C32×C9—31no—
140C32×C9—31no—
141C32×C9—31no—
142C32×C9—32no—
143C32×C9—31no—
144C32×C9—32no—
145C32×C9—31no—
146C32×C9—31no—
147C32×C9H(3,𝔽3)13yes—
148C32×C9H(3,𝔽3)16yes—
149#732—327no—
150#732—327no—
151#732—354no—
152#731—327no—
153#731—354no—
154AΣL(1,𝔽27)—354no—
155AΣL(1,𝔽27)—327no—
156C32⋊C9—354no—
157C32⋊C9—327no—
158C33—36no—
159C33—33no—
160C33—33no—
161C33—36no—
162C33—93no—
163C33—918no—
164C33—92no—
165C33—92no—
166C33—91no—
167C33—91no—
168C33—91no—
169C33—92no—
170C33—92no—
171C33—92no—
172C33—92no—
173C33—92no—
174C33—91no—
175C33—273no—
176C33—276no—
177C33—273no—
178C33C3×H(3,𝔽3)11yescharacteristic
179C33C3×H(3,𝔽3)12yes—
180C33C3×H(3,𝔽3)11yescharacteristic
181C33C3×H(3,𝔽3)12yes—
182C33C3×H(3,𝔽3)11yescharacteristic
183C33C3×H(3,𝔽3)12yes—
184C33C3×H(3,𝔽3)12yes—
185C33C3×H(3,𝔽3)11yescharacteristic
186C33C3×H(3,𝔽3)11yescharacteristic
1873−1+2—36no—
1883−1+2—36no—
1893−1+2—36no—
1903−1+2—36no—
1913−1+2—36no—
1923−1+2—33no—
1933−1+2—33no—
1943−1+2—92no—
1953−1+2—92no—
1963−1+2—92no—
1973−1+2—92no—
1983−1+2—92no—
1993−1+2—92no—
2003−1+2—92no—
2013−1+2—92no—
2023−1+2—92no—
2033−1+2—92no—
2043−1+2—92no—
2053−1+2—92no—
2063−1+2—92no—
2073−1+2—92no—
2083−1+2—92no—
2093−1+2—92no—
2103−1+2—92no—
2113−1+2—92no—
2123−1+2—92no—
2133−1+2—92no—
2143−1+2—92no—
215H(3,𝔽3)—33no—
216H(3,𝔽3)—36no—
217H(3,𝔽3)—36no—
218H(3,𝔽3)—33no—
219H(3,𝔽3)—33no—
220H(3,𝔽3)—36no—
221H(3,𝔽3)—36no—
222H(3,𝔽3)—36no—
223H(3,𝔽3)—36no—
224H(3,𝔽3)—36no—
225H(3,𝔽3)—33no—
226H(3,𝔽3)—92no—
227H(3,𝔽3)—92no—
228H(3,𝔽3)—92no—
229H(3,𝔽3)—92no—
230H(3,𝔽3)—92no—
231H(3,𝔽3)—92no—
232H(3,𝔽3)—92no—
233H(3,𝔽3)—92no—
234H(3,𝔽3)—92no—
235H(3,𝔽3)—92no—
236H(3,𝔽3)—92no—
237H(3,𝔽3)—92no—
238H(3,𝔽3)—92no—
239H(3,𝔽3)—92no—
240H(3,𝔽3)—92no—
241H(3,𝔽3)—92no—
242H(3,𝔽3)—92no—
243H(3,𝔽3)—92no—
244C3×C9—36no—
245C3×C9—36no—
246C3×C9—36no—
247C3×C9—33no—
248C3×C9—36no—
249C3×C9—33no—
250C3×C9—33no—
251C3×C9—36no—
252C3×C9—33no—
253C3×C9—36no—
254C3×C9—92no—
255C3×C9—91no—
256C3×C9—92no—
257C3×C9—91no—
258C3×C9—92no—
259C3×C9—92no—
260C3×C9—91no—
261C3×C9—91no—
262C3×C9—91no—
263C3×C9—92no—
264C3×C9—91no—
265C3×C9—91no—
266C3×C9—91no—
267C3×C9—92no—
268C3×C9—91no—
269C32—36no—
270C32—33no—
271C32—33no—
272C32—91no—
273C32—92no—
274C32—92no—
275C32—91no—
276C32—92no—
277C32—91no—
278C32—918no—
279C32—96no—
280C32—93no—
281C32—96no—
282C32—93no—
283C32—99no—
284C32—276no—
285C32—273no—
286C32—276no—
287C32—276no—
288C32—276no—
289C32—273no—
290C32—276no—
291C32—276no—
292C32—273no—
293C32—273no—
294C32—276no—
295C32—276no—
296C32—273no—
297C32C32⋊H(3,𝔽3)12yes—
298C32C32⋊H(3,𝔽3)11yescharacteristic
299C32C32⋊H(3,𝔽3)11yescharacteristic
300C32C32⋊H(3,𝔽3)12yes—
301C32C32⋊H(3,𝔽3)12yes—
302C32C32⋊H(3,𝔽3)11yescharacteristic
303C32C32⋊H(3,𝔽3)11yescharacteristic
304C32C32⋊H(3,𝔽3)12yes—
305C32C32⋊H(3,𝔽3)11yescharacteristic
306C9—93no—
307C9—96no—
308C9—96no—
309C9—93no—
310C9—93no—
311C3—32no—
312C3—32no—
313C3—31no—
314C3—32no—
315C3—32no—
316C3—31no—
317C3—31no—
318C3—31no—
319C3—276no—
320C3—273no—
321C3—276no—
322C3—273no—
323C3#1062456211yescharacteristic
324C3#1062475313yes—
subgroupG/H
center, socleC32C32⋊H(3,𝔽3)
commutator, Frattini, cosocleC34C33
LCSG ▷ C34 ▷ C3 ▷ C1
UCSC1 ◁ C32 ◁ C34 ◁ G
LECSG ▷ C34 ▷ C3 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C3×#6911C3semidirect
2. C3×#6911C3semidirect
3. C3×#6910C3semidirect
4. C3×#6909C3semidirect
5. C3×#6909C3semidirect
6. C32×#732C3semidirect
7. C32×#731C3semidirect
8. #10624881C3semidirect
9. C3×#6893C3semidirect

Maximal subgroups

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

subgroupindex
C3×#69113
C3×#69103
C3×#69093
C32×#7323
C32×#7313
#106248813
C3×#68933