Finite Group Databasebeta

← D3×H(3,𝔽3)×3−1+2 #72243511

Subgroups — D3×H(3,𝔽3)×3−1+2 #72243511

Order 4374.

5869 conjugacy classes of subgroups; only the 777 normal subgroups are shown.

normal subgroups only

This group has 5869 conjugacy classes of subgroups — too many to draw. Only its 777 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×H(3,𝔽3)×3−1+2C211yesmaximal, characteristic, complemented, Sylow
2C32×D3×3−1+2C314yesmaximal, complemented
3C32×D3×H(3,𝔽3)C311yesmaximal, characteristic
4D3×C9⋊H(3,𝔽3)C3124yesmaximal, complemented
5D3×#6891C318yesmaximal, complemented
6D3×C9×H(3,𝔽3)C313yesmaximal, complemented
7C2×H(3,𝔽3)×3−1+2—31nomaximal
8C33×3−1+2C2×C314yescomplemented
9C33×H(3,𝔽3)C2×C311yescharacteristic
10H(3,𝔽3)×3−1+2—224no—
11H(3,𝔽3)×3−1+2—28no—
12H(3,𝔽3)×3−1+2—23no—
13H(3,𝔽3)×3−1+2—24no—
14H(3,𝔽3)×3−1+2—21no—
15H(3,𝔽3)×3−1+2D311yescharacteristic, direct product factor
16C3×C9⋊H(3,𝔽3)C2×C3124yescomplemented
17C3×#6891C2×C318yescomplemented
18C3×C9×H(3,𝔽3)C2×C313yescomplemented
19C34×D3C3214yes—
20C2×C32×3−1+2—34no—
21C2×C32×H(3,𝔽3)—31no—
22C3×D3×3−1+2—336no—
23C3×D3×3−1+2C3218yes—
24C3×D3×3−1+2C3211yescharacteristic
25C3×D3×3−1+2C32124yescomplemented
26C3×D3×H(3,𝔽3)—39no—
27C3×D3×H(3,𝔽3)C3218yes—
28C3×D3×H(3,𝔽3)C3211yescharacteristic
29C32×D3×C9C32112yescomplemented
30C2×C9⋊H(3,𝔽3)—324no—
31C2×#6891—38no—
32C2×C9×H(3,𝔽3)—33no—
33D3×C32⋊C9C32124yescomplemented
34D3×C32⋊C9C32148yescomplemented
35C35C2×C3214yes—
36C32×3−1+2—24no—
37C32×3−1+2—28no—
38C32×3−1+2—24no—
39C32×3−1+2—224no—
40C32×3−1+2—212no—
41C32×3−1+2—336no—
42C32×3−1+2—636no—
43C32×3−1+2C3×D314yescomplemented
44C32×3−1+2C2×C3211yescharacteristic
45C32×3−1+2C2×C3218yes—
46C32×3−1+2C2×C32124yescomplemented
47C32×H(3,𝔽3)—24no—
48C32×H(3,𝔽3)—28no—
49C32×H(3,𝔽3)—21no—
50C32×H(3,𝔽3)—39no—
51C32×H(3,𝔽3)—69no—
52C32×H(3,𝔽3)C3×D311yescharacteristic
53C32×H(3,𝔽3)C2×C3211yescharacteristic
54C32×H(3,𝔽3)C2×C3218yes—
55C33×C9C2×C32112yescomplemented
56C9⋊H(3,𝔽3)—272no—
57C9⋊H(3,𝔽3)—224no—
58C9⋊H(3,𝔽3)—224no—
59C9⋊H(3,𝔽3)—248no—
60C9⋊H(3,𝔽3)—2144no—
61C9⋊H(3,𝔽3)C3×D3124yescomplemented
62#6891—28no—
63#6891—224no—
64#6891—224no—
65#6891—248no—
66#6891C3×D318yescomplemented
67C9×H(3,𝔽3)—23no—
68C9×H(3,𝔽3)—212no—
69C9×H(3,𝔽3)—224no—
70C9×H(3,𝔽3)C3×D313yescomplemented
71C3×C32⋊C9C2×C32124yescomplemented
72C3×C32⋊C9C2×C32148yescomplemented
73C2×C34—34no—
74C33×D3—312no—
75C33×D3—312no—
76C33×D3—98no—
77C33×D3C3318yes—
78C33×D3C3311yescharacteristic
79C33×D3C3314yes—
80C2×C3×3−1+2—31no—
81C2×C3×3−1+2—38no—
82C2×C3×3−1+2—324no—
83C2×C3×3−1+2—936no—
84C2×C3×H(3,𝔽3)—31no—
85C2×C3×H(3,𝔽3)—38no—
86C2×C3×H(3,𝔽3)—99no—
87C2×C32×C9—312no—
88D3×3−1+2—324no—
89D3×3−1+2—372no—
90D3×3−1+2H(3,𝔽3)19yesdirect product factor
91D3×H(3,𝔽3)—324no—
92D3×H(3,𝔽3)3−1+219yesdirect product factor
93C3×D3×C9—336no—
94C3×D3×C9C33124yes—
95C3×D3×C9C3313yes—
96C2×C32⋊C9—324no—
97C2×C32⋊C9—348no—
98C34—28no—
99C34—24no—
100C34—24no—
101C34—312no—
102C34—312no—
103C34—612no—
104C34—612no—
105C34—98no—
106C34—188no—
107C34C32×D314yes—
108C34C2×C3311yescharacteristic
109C34C2×C3314yes—
110C34C2×C3318yes—
111C3×3−1+2—21no—
112C3×3−1+2—29no—
113C3×3−1+2—28no—
114C3×3−1+2—224no—
115C3×3−1+2—23no—
116C3×3−1+2—224no—
117C3×3−1+2—248no—
118C3×3−1+2—224no—
119C3×3−1+2—336no—
120C3×3−1+2—324no—
121C3×3−1+2—372no—
122C3×3−1+2—672no—
123C3×3−1+2—6108no—
124C3×3−1+2—672no—
125C3×3−1+2—6216no—
126C3×3−1+2—636no—
127C3×3−1+2—624no—
128C3×3−1+2—636no—
129C3×3−1+2C2×H(3,𝔽3)19yescomplemented
130C3×3−1+2C32×D318yes—
131C3×3−1+2C32×D311yescharacteristic
132C3×3−1+2C32×D3124yescomplemented
133C3×H(3,𝔽3)—24no—
134C3×H(3,𝔽3)—29no—
135C3×H(3,𝔽3)—224no—
136C3×H(3,𝔽3)—28no—
137C3×H(3,𝔽3)—39no—
138C3×H(3,𝔽3)—324no—
139C3×H(3,𝔽3)—672no—
140C3×H(3,𝔽3)—636no—
141C3×H(3,𝔽3)—624no—
142C3×H(3,𝔽3)—69no—
143C3×H(3,𝔽3)C2×3−1+219yescomplemented
144C3×H(3,𝔽3)C32×D318yes—
145C3×H(3,𝔽3)C32×D311yescharacteristic
146C32×C9—212no—
147C32×C9—224no—
148C32×C9—212no—
149C32×C9—336no—
150C32×C9—636no—
151C32×C9C32×D3112yescomplemented
152C32×C9C2×C3313yes—
153C32×C9C2×C33124yes—
154C32⋊C9—224no—
155C32⋊C9—248no—
156C32⋊C9—272no—
157C32⋊C9—2144no—
158C32⋊C9C32×D3124yescomplemented
159C32⋊C9C32×D3148yescomplemented
160C2×C33—34no—
161C2×C33—31no—
162C2×C33—38no—
163C2×C33—912no—
164C2×C33—912no—
165C2×C33—278no—
166C32×D3—34no—
167C32×D3—31no—
168C32×D3—32no—
169C32×D3—38no—
170C32×D3—38no—
171C32×D3—38no—
172C32×D3—316no—
173C32×D3—936no—
174C32×D3C3×H(3,𝔽3)13yes—
175C32×D3C3×3−1+2112yescomplemented
176C32×D3C3411yescharacteristic
177C2×3−1+2—39no—
178C2×3−1+2—972no—
179C2×3−1+2—924no—
180C2×H(3,𝔽3)—39no—
181C2×H(3,𝔽3)—924no—
182C2×C3×C9—33no—
183C2×C3×C9—324no—
184C2×C3×C9—936no—
185D3×C9—324no—
186D3×C9C3×H(3,𝔽3)19yescomplemented
187C33—23no—
188C33—212no—
189C33—24no—
190C33—21no—
191C33—28no—
192C33—34no—
193C33—31no—
194C33—32no—
195C33—38no—
196C33—312no—
197C33—312no—
198C33—38no—
199C33—38no—
200C33—316no—
201C33—64no—
202C33—61no—
203C33—62no—
204C33—68no—
205C33—612no—
206C33—624no—
207C33—612no—
208C33—624no—
209C33—68no—
210C33—68no—
211C33—616no—
212C33—612no—
213C33—612no—
214C33—98no—
215C33—936no—
216C33—188no—
217C33—188no—
218C33—1816no—
219C33—1836no—
220C33—1836no—
221C33—1872no—
222C33C2×C3×H(3,𝔽3)13yes—
223C33C2×C3×3−1+2112yescomplemented
224C33C33×D318yes—
225C33C33×D311yescharacteristic
226C33C33×D314yes—
227C33C2×C3411yescharacteristic
2283−1+2—29no—
2293−1+2—227no—
2303−1+2—324no—
2313−1+2—372no—
2323−1+2—6144no—
2333−1+2—672no—
2343−1+2—672no—
2353−1+2—624no—
2363−1+2—672no—
2373−1+2D3×H(3,𝔽3)19yesdirect product factor
238H(3,𝔽3)—236no—
239H(3,𝔽3)—324no—
240H(3,𝔽3)—672no—
241H(3,𝔽3)—624no—
242H(3,𝔽3)D3×3−1+219yesdirect product factor
243C3×C9—23no—
244C3×C9—29no—
245C3×C9—224no—
246C3×C9—336no—
247C3×C9—324no—
248C3×C9—672no—
249C3×C9—636no—
250C3×C9—624no—
251C3×C9—636no—
252C3×C9C2×C3×H(3,𝔽3)19yescomplemented
253C3×C9C33×D3124yes—
254C3×C9C33×D313yes—
255C2×C32—31no—
256C2×C32—33no—
257C2×C32—312no—
258C2×C32—94no—
259C2×C32—91no—
260C2×C32—98no—
261C2×C32—92no—
262C2×C32—98no—
263C2×C32—98no—
264C2×C32—916no—
265C2×C32—2736no—
266C3×D3—33no—
267C3×D3—312no—
268C3×D3—98no—
269C3×D3C32×H(3,𝔽3)11yescharacteristic
270C3×D3C32×3−1+211yescharacteristic
271C3×D33−1+412yes—
272C2×C9—39no—
273C2×C9—924no—
274C32—21no—
275C32—21no—
276C32—22no—
277C32—23no—
278C32—212no—
279C32—34no—
280C32—31no—
281C32—32no—
282C32—38no—
283C32—33no—
284C32—312no—
285C32—38no—
286C32—38no—
287C32—316no—
288C32—63no—
289C32—612no—
290C32—612no—
291C32—624no—
292C32—63no—
293C32—66no—
294C32—64no—
295C32—61no—
296C32—68no—
297C32—62no—
298C32—68no—
299C32—68no—
300C32—616no—
301C32—936no—
302C32—98no—
303C32—188no—
304C32—188no—
305C32—1816no—
306C32—1836no—
307C32—1836no—
308C32—1872no—
309C32C3×D3×H(3,𝔽3)13yes—
310C32C3×D3×3−1+2112yescomplemented
311C32C2×C32×H(3,𝔽3)11yescharacteristic
312C32C2×C32×3−1+211yescharacteristic
313C32C2×3−1+412yes—
314C32C34×D311yescharacteristic
315C9—29no—
316C9—324no—
317C9—624no—
318C9C3×D3×H(3,𝔽3)19yescomplemented
319C2×C3—31no—
320C2×C3—31no—
321C2×C3—32no—
322C2×C3—912no—
323C2×C3—93no—
324C2×C3—278no—
325D3H(3,𝔽3)×3−1+211yescharacteristic, direct product factor
326C3—21no—
327C3—21no—
328C3—22no—
329C3—33no—
330C3—312no—
331C3—612no—
332C3—63no—
333C3—98no—
334C3—188no—
335C3C2×H(3,𝔽3)×3−1+211yescharacteristic, complemented
336C3C32×D3×H(3,𝔽3)11yescharacteristic
337C3C32×D3×3−1+211yescharacteristic
338C3D3×3−1+412yes—
339C2—31noSylow
subgroupG/H
center, FrattiniC32C34×D3
commutator, socle, cosocleC33C2×C34
FittingC3×H(3,𝔽3)×3−1+2C2
prime radicalC32×D3×H(3,𝔽3)C3
hypercenterH(3,𝔽3)×3−1+2D3
hypocenterC3C2×H(3,𝔽3)×3−1+2
LCSG ▷ C33 ▷ C3
UCSC1 ◁ C32 ◁ H(3,𝔽3)×3−1+2

Complements

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

NH ≅ G/Ntype
1. C3×H(3,𝔽3)×3−1+2C2semidirect
2. C32×D3×3−1+2C3semidirect
4. D3×C9⋊H(3,𝔽3)C3semidirect
5. D3×#6891C3semidirect
6. D3×C9×H(3,𝔽3)C3semidirect
8. C33×3−1+2C2×C3semidirect
15. H(3,𝔽3)×3−1+2D3direct (N × H)
16. C3×C9⋊H(3,𝔽3)C2×C3semidirect
17. C3×#6891C2×C3semidirect
18. C3×C9×H(3,𝔽3)C2×C3semidirect
25. C3×D3×3−1+2C32semidirect
29. C32×D3×C9C32semidirect
33. D3×C32⋊C9C32semidirect
34. D3×C32⋊C9C32semidirect
43. C32×3−1+2C3×D3semidirect
46. C32×3−1+2C2×C32semidirect
55. C33×C9C2×C32semidirect
61. C9⋊H(3,𝔽3)C3×D3semidirect
66. #6891C3×D3semidirect
70. C9×H(3,𝔽3)C3×D3semidirect
71. C3×C32⋊C9C2×C32semidirect
72. C3×C32⋊C9C2×C32semidirect
90. D3×3−1+2H(3,𝔽3)direct (N × H)
92. D3×H(3,𝔽3)3−1+2direct (N × H)
129. C3×3−1+2C2×H(3,𝔽3)semidirect
132. C3×3−1+2C32×D3semidirect
143. C3×H(3,𝔽3)C2×3−1+2semidirect
151. C32×C9C32×D3semidirect
158. C32⋊C9C32×D3semidirect
159. C32⋊C9C32×D3semidirect
175. C32×D3C3×3−1+2semidirect
186. D3×C9C3×H(3,𝔽3)semidirect
223. C33C2×C3×3−1+2semidirect
237. 3−1+2D3×H(3,𝔽3)direct (N × H)
242. H(3,𝔽3)D3×3−1+2direct (N × H)
252. C3×C9C2×C3×H(3,𝔽3)semidirect
310. C32C3×D3×3−1+2semidirect
318. C9C3×D3×H(3,𝔽3)semidirect
325. D3H(3,𝔽3)×3−1+2direct (N × H)
335. C3C2×H(3,𝔽3)×3−1+2semidirect

Maximal subgroups

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

subgroupindex
C3×H(3,𝔽3)×3−1+22
C32×D3×3−1+23
C32×D3×H(3,𝔽3)3
D3×C9⋊H(3,𝔽3)3
D3×#68913
D3×C9×H(3,𝔽3)3
C2×H(3,𝔽3)×3−1+23