Finite Group Databasebeta

← D3×3−1+22 #72243249

Subgroups — D3×3−1+22 #72243249

Order 4374.

3790 conjugacy classes of subgroups; only the 777 normal subgroups are shown.

normal subgroups only

This group has 3790 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
1C3×3−1+22C211yesmaximal, characteristic, complemented, Sylow
2C32×D3×3−1+2C312yesmaximal
3D3×#6900C319yesmaximal, complemented
4D3×#6896C319yesmaximal, complemented
5D3×#6894C3112yesmaximal, complemented
6D3×#6891C312yesmaximal
7D3×C9×3−1+2C316yesmaximal, complemented
8C2×3−1+22—31nomaximal
9C33×3−1+2C2×C312yes—
103−1+22—29no—
113−1+22—29no—
123−1+22—212no—
133−1+22—22no—
143−1+22—26no—
153−1+22—22no—
163−1+22D311yescharacteristic, direct product factor
17C3×#6900C2×C319yescomplemented
18C3×#6896C2×C319yescomplemented
19C3×#6894C2×C3112yescomplemented
20C3×#6891C2×C312yes—
21C3×C9×3−1+2C2×C316yescomplemented
22C34×D3C3211yescharacteristic
23C2×C32×3−1+2—32no—
24C3×D3×3−1+2—318no—
25C3×D3×3−1+2C3214yes—
26C3×D3×3−1+2C3216yes—
27C3×D3×3−1+2C3216yes—
28C3×D3×3−1+2C3216yes—
29C3×D3×3−1+2C3212yes—
30C32×D3×C9C3216yes—
31C2×#6900—39no—
32C2×#6896—39no—
33C2×#6894—312no—
34C2×#6891—32no—
35C2×C9×3−1+2—36no—
36D3×C9⋊C9C32136yescomplemented
37D3×C9⋊C9C32136yescomplemented
38D3×C32⋊C9C32112yes—
39D3×C32⋊C9C3216yes—
40D3×C92C3219yescomplemented
41C35C2×C3211yescharacteristic
42C32×3−1+2—22no—
43C32×3−1+2—24no—
44C32×3−1+2—22no—
45C32×3−1+2—26no—
46C32×3−1+2—26no—
47C32×3−1+2—26no—
48C32×3−1+2—318no—
49C32×3−1+2—618no—
50C32×3−1+2C3×D312yes—
51C32×3−1+2C2×C3212yes—
52C32×3−1+2C2×C3214yes—
53C32×3−1+2C2×C3216yes—
54C32×3−1+2C2×C3216yes—
55C32×3−1+2C2×C3216yes—
56C33×C9C2×C3216yes—
57#6900—218no—
58#6900—218no—
59#6900—236no—
60#6900—236no—
61#6900—29no—
62#6900C3×D319yescomplemented
63#6896—218no—
64#6896—236no—
65#6896—236no—
66#6896—218no—
67#6896—29no—
68#6896C3×D319yescomplemented
69#6894—212no—
70#6894—212no—
71#6894—272no—
72#6894—236no—
73#6894—224no—
74#6894C3×D3112yescomplemented
75#6891—22no—
76#6891—26no—
77#6891—212no—
78#6891—26no—
79#6891C3×D312yes—
80C9×3−1+2—26no—
81C9×3−1+2—26no—
82C9×3−1+2—236no—
83C9×3−1+2—212no—
84C9×3−1+2—218no—
85C9×3−1+2C3×D316yescomplemented
86C3×C9⋊C9C2×C32136yescomplemented
87C3×C9⋊C9C2×C32136yescomplemented
88C3×C32⋊C9C2×C32112yes—
89C3×C32⋊C9C2×C3216yes—
90C3×C92C2×C3219yescomplemented
91C2×C34—31no—
92C33×D3—36no—
93C33×D3—92no—
94C33×D3C3311yescharacteristic
95C33×D3C3311yescharacteristic
96C33×D3C3312yes—
97C2×C3×3−1+2—32no—
98C2×C3×3−1+2—34no—
99C2×C3×3−1+2—36no—
100C2×C3×3−1+2—36no—
101C2×C3×3−1+2—36no—
102C2×C3×3−1+2—918no—
103C2×C32×C9—36no—
104D3×3−1+2—312no—
105D3×3−1+2—318no—
106D3×3−1+2—318no—
107D3×3−1+2—96no—
108D3×3−1+23−1+2118yesdirect product factor
109C3×D3×C9—318no—
110C3×D3×C9C33118yes—
111C3×D3×C9C33112yes—
112C3×D3×C9C3316yes—
113C2×C9⋊C9—336no—
114C2×C9⋊C9—336no—
115C2×C32⋊C9—312no—
116C2×C32⋊C9—36no—
117C2×C92—39no—
118C34—21no—
119C34—21no—
120C34—22no—
121C34—36no—
122C34—66no—
123C34—92no—
124C34—182no—
125C34C32×D311yescharacteristic
126C34C2×C3312yes—
127C34C2×C3311yescharacteristic
128C34C2×C3311yescharacteristic
129C3×3−1+2—22no—
130C3×3−1+2—218no—
131C3×3−1+2—24no—
132C3×3−1+2—26no—
133C3×3−1+2—26no—
134C3×3−1+2—26no—
135C3×3−1+2—26no—
136C3×3−1+2—212no—
137C3×3−1+2—212no—
138C3×3−1+2—26no—
139C3×3−1+2—212no—
140C3×3−1+2—26no—
141C3×3−1+2—218no—
142C3×3−1+2—318no—
143C3×3−1+2—312no—
144C3×3−1+2—318no—
145C3×3−1+2—318no—
146C3×3−1+2—618no—
147C3×3−1+2—618no—
148C3×3−1+2—654no—
149C3×3−1+2—636no—
150C3×3−1+2—654no—
151C3×3−1+2—654no—
152C3×3−1+2—618no—
153C3×3−1+2—612no—
154C3×3−1+2—618no—
155C3×3−1+2—96no—
156C3×3−1+2—186no—
157C3×3−1+2C2×3−1+2118yescomplemented
158C3×3−1+2C32×D314yes—
159C3×3−1+2C32×D316yes—
160C3×3−1+2C32×D316yes—
161C3×3−1+2C32×D316yes—
162C3×3−1+2C32×D312yes—
163C32×C9—26no—
164C32×C9—212no—
165C32×C9—26no—
166C32×C9—318no—
167C32×C9—618no—
168C32×C9C32×D316yes—
169C32×C9C2×C3316yes—
170C32×C9C2×C33112yes—
171C32×C9C2×C33118yes—
172C9⋊C9—236no—
173C9⋊C9—272no—
174C9⋊C9—236no—
175C9⋊C9—272no—
176C9⋊C9—236no—
177C9⋊C9—236no—
178C9⋊C9C32×D3136yescomplemented
179C9⋊C9C32×D3136yescomplemented
180C32⋊C9—212no—
181C32⋊C9—26no—
182C32⋊C9—236no—
183C32⋊C9—218no—
184C32⋊C9C32×D3112yes—
185C32⋊C9C32×D316yes—
186C92—218no—
187C92—218no—
188C92C32×D319yescomplemented
189C2×C33—32no—
190C2×C33—31no—
191C2×C33—31no—
192C2×C33—96no—
193C2×C33—272no—
194C32×D3—32no—
195C32×D3—34no—
196C32×D3—32no—
197C32×D3—32no—
198C32×D3—32no—
199C32×D3—32no—
200C32×D3—99no—
201C32×D3C3×3−1+216yes—
202C32×D3C3411yescharacteristic
203C2×3−1+2—318no—
204C2×3−1+2—918no—
205C2×3−1+2—918no—
206C2×3−1+2—912no—
207C2×3−1+2—276no—
208C2×C3×C9—36no—
209C2×C3×C9—318no—
210C2×C3×C9—312no—
211C2×C3×C9—918no—
212D3×C9—312no—
213D3×C9—318no—
214D3×C9C3×3−1+2118yescomplemented
215C33—26no—
216C33—22no—
217C33—21no—
218C33—21no—
219C33—32no—
220C33—34no—
221C33—36no—
222C33—32no—
223C33—32no—
224C33—32no—
225C33—32no—
226C33—62no—
227C33—64no—
228C33—66no—
229C33—66no—
230C33—66no—
231C33—62no—
232C33—62no—
233C33—62no—
234C33—62no—
235C33—66no—
236C33—92no—
237C33—99no—
238C33—184no—
239C33—182no—
240C33—182no—
241C33—1818no—
242C33—1818no—
243C33C2×C3×3−1+216yes—
244C33C33×D311yescharacteristic
245C33C33×D311yescharacteristic
246C33C33×D312yes—
247C33C2×C3411yescharacteristic
2483−1+2—218no—
2493−1+2—254no—
2503−1+2—312no—
2513−1+2—318no—
2523−1+2—318no—
2533−1+2—636no—
2543−1+2—636no—
2553−1+2—618no—
2563−1+2—618no—
2573−1+2—636no—
2583−1+2—612no—
2593−1+2—618no—
2603−1+2—618no—
2613−1+2—96no—
2623−1+2—1818no—
2633−1+2—186no—
2643−1+2D3×3−1+2118yesdirect product factor
265C3×C9—26no—
266C3×C9—218no—
267C3×C9—218no—
268C3×C9—212no—
269C3×C9—318no—
270C3×C9—312no—
271C3×C9—318no—
272C3×C9—636no—
273C3×C9—618no—
274C3×C9—612no—
275C3×C9—618no—
276C3×C9—618no—
277C3×C9C2×C3×3−1+2118yescomplemented
278C3×C9C33×D3118yes—
279C3×C9C33×D3112yes—
280C3×C9C33×D316yes—
281C2×C32—31no—
282C2×C32—36no—
283C2×C32—92no—
284C2×C32—94no—
285C2×C32—92no—
286C2×C32—92no—
287C2×C32—92no—
288C2×C32—92no—
289C2×C32—279no—
290C3×D3—36no—
291C3×D3—91no—
292C3×D3—91no—
293C3×D3C32×3−1+212yes—
294C3×D33−1+412yes—
295C2×C9—318no—
296C2×C9—912no—
297C2×C9—918no—
298C32—22no—
299C32—22no—
300C32—26no—
301C32—32no—
302C32—34no—
303C32—36no—
304C32—32no—
305C32—32no—
306C32—32no—
307C32—32no—
308C32—66no—
309C32—66no—
310C32—612no—
311C32—62no—
312C32—64no—
313C32—62no—
314C32—62no—
315C32—62no—
316C32—62no—
317C32—99no—
318C32—91no—
319C32—91no—
320C32—182no—
321C32—182no—
322C32—182no—
323C32—182no—
324C32—1818no—
325C32—189no—
326C32—189no—
327C32C3×D3×3−1+216yes—
328C32C2×C32×3−1+212yes—
329C32C2×3−1+412yes—
330C32C34×D311yescharacteristic
331C9—218no—
332C9—312no—
333C9—318no—
334C9—612no—
335C9—618no—
336C9C3×D3×3−1+2118yescomplemented
337C2×C3—32no—
338C2×C3—32no—
339C2×C3—96no—
340C2×C3—271no—
341C2×C3—271no—
342D33−1+2211yescharacteristic, direct product factor
343C3—22no—
344C3—22no—
345C3—36no—
346C3—66no—
347C3—91no—
348C3—91no—
349C3—181no—
350C3—181no—
351C3C2×3−1+2211yescharacteristic, complemented
352C3C32×D3×3−1+212yes—
353C3D3×3−1+412yes—
354C2—31noSylow
subgroupG/H
center, FrattiniC32C34×D3
commutator, socle, cosocleC33C2×C34
FittingC3×3−1+22C2
prime radicalC34×D3C32
hypercenter3−1+22D3
hypocenterC3C2×3−1+22
LCSG ▷ C33 ▷ C3
UCSC1 ◁ C32 ◁ 3−1+22

Complements

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

NH ≅ G/Ntype
1. C3×3−1+22C2semidirect
3. D3×#6900C3semidirect
4. D3×#6896C3semidirect
5. D3×#6894C3semidirect
7. D3×C9×3−1+2C3semidirect
16. 3−1+22D3direct (N × H)
17. C3×#6900C2×C3semidirect
18. C3×#6896C2×C3semidirect
19. C3×#6894C2×C3semidirect
21. C3×C9×3−1+2C2×C3semidirect
36. D3×C9⋊C9C32semidirect
37. D3×C9⋊C9C32semidirect
40. D3×C92C32semidirect
62. #6900C3×D3semidirect
68. #6896C3×D3semidirect
74. #6894C3×D3semidirect
85. C9×3−1+2C3×D3semidirect
86. C3×C9⋊C9C2×C32semidirect
87. C3×C9⋊C9C2×C32semidirect
90. C3×C92C2×C32semidirect
108. D3×3−1+23−1+2direct (N × H)
157. C3×3−1+2C2×3−1+2semidirect
178. C9⋊C9C32×D3semidirect
179. C9⋊C9C32×D3semidirect
188. C92C32×D3semidirect
214. D3×C9C3×3−1+2semidirect
264. 3−1+2D3×3−1+2direct (N × H)
277. C3×C9C2×C3×3−1+2semidirect
336. C9C3×D3×3−1+2semidirect
342. D33−1+22direct (N × H)
351. C3C2×3−1+22semidirect

Maximal subgroups

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

subgroupindex
C3×3−1+222
C32×D3×3−1+23
D3×#69003
D3×#68963
D3×#68943
D3×#68913
D3×C9×3−1+23
C2×3−1+223