Finite Group Databasebeta

← C32×D3×(C3⋊D3)⋊C3 #72243749

Subgroups — C32×D3×(C3⋊D3)⋊C3 #72243749

Order 2916.

3328 conjugacy classes of subgroups; only the 298 normal subgroups are shown.

normal subgroups only

This group has 3328 conjugacy classes of subgroups — too many to draw. Only its 298 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
1C32×C3⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
2C32×D3×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3C33×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
4C32×D3×C3⋊D3C311yesmaximal, characteristic, complemented
5C33×D32—31nomaximal
6C2×C32×(C3⋊D3)⋊C3—31nomaximal
7C3×D3×(C3⋊D3)⋊C3C3112yesmaximal, direct product factor
8C33×H(3,𝔽3)C2211yescharacteristic, complemented, Sylow
9C32×C3⋊(C3⋊D3)C2×C311yescharacteristic, complemented
10C33×C3⋊D3—31no—
11C33×C3⋊D3C2×C311yescharacteristic, complemented
12C34×D3—21no—
13C34×D3—31no—
14C34×D3D311yescharacteristic, complemented
15C34×D3C2×C311yescharacteristic, complemented
16C2×C32×H(3,𝔽3)—31no—
17C3×C3⋊((C3⋊D3)⋊C3)C2×C3112yescomplemented
18C3×D3×H(3,𝔽3)—212no—
19C3×D3×H(3,𝔽3)C2×C3112yescomplemented
20C32×(C3⋊D3)⋊C3—21no—
21C32×(C3⋊D3)⋊C3—212no—
22C32×(C3⋊D3)⋊C3—31no—
23C32×(C3⋊D3)⋊C3—61no—
24C32×(C3⋊D3)⋊C3D311yescharacteristic, direct product factor
25C32×(C3⋊D3)⋊C3C2×C3112yescomplemented
26C2×C32×C3⋊D3—31no—
27C2×C33×D3—91no—
28C2×C33×D3—91no—
29C3×D3×C3⋊D3C3214yescomplemented
30C32×D32—31no—
31C32×D32—312no—
32C32×D32—91no—
33C2×C3×(C3⋊D3)⋊C3—312no—
34D3×(C3⋊D3)⋊C3C3219yesdirect product factor
35C35—21no—
36C35C2×D311yescharacteristic, complemented
37C35C22×C311yescharacteristic, complemented
38C32×H(3,𝔽3)—21no—
39C32×H(3,𝔽3)—21no—
40C32×H(3,𝔽3)—212no—
41C32×H(3,𝔽3)—212no—
42C32×H(3,𝔽3)—412no—
43C32×H(3,𝔽3)—41no—
44C32×H(3,𝔽3)C2×D311yescharacteristic, complemented
45C32×H(3,𝔽3)C22×C3112yescomplemented
46C2×C34—31no—
47C2×C34—31no—
48C2×C34—61no—
49C2×C34—91no—
50C3×C3⋊(C3⋊D3)C2×C3214yescomplemented
51C32×C3⋊D3—24no—
52C32×C3⋊D3—31no—
53C32×C3⋊D3—312no—
54C32×C3⋊D3—31no—
55C32×C3⋊D3—61no—
56C32×C3⋊D3—91no—
57C32×C3⋊D3—181no—
58C32×C3⋊D3C3×D311yescharacteristic, complemented
59C32×C3⋊D3C2×C3214yescomplemented
60C33×D3—24no—
61C33×D3—212no—
62C33×D3—31no—
63C33×D3—31no—
64C33×D3—31no—
65C33×D3—312no—
66C33×D3—31no—
67C33×D3—64no—
68C33×D3—64no—
69C33×D3—68no—
70C33×D3—61no—
71C33×D3—612no—
72C33×D3—61no—
73C33×D3—91no—
74C33×D3—91no—
75C33×D3—91no—
76C33×D3—181no—
77C33×D3C3×D311yescharacteristic
78C33×D3C3×D3112yescomplemented
79C33×D3C2×C3214yescomplemented
80C2×C3×H(3,𝔽3)—312no—
81C2×C3×H(3,𝔽3)—612no—
82C3⋊((C3⋊D3)⋊C3)C2×C3219yescomplemented
83D3×H(3,𝔽3)—236no—
84D3×H(3,𝔽3)C2×C3219yescomplemented
85C3×(C3⋊D3)⋊C3—212no—
86C3×(C3⋊D3)⋊C3—236no—
87C3×(C3⋊D3)⋊C3—312no—
88C3×(C3⋊D3)⋊C3—612no—
89C3×(C3⋊D3)⋊C3C3×D3112yesdirect product factor
90C3×(C3⋊D3)⋊C3C2×C3219yescomplemented
91C22×C33—271no—
92C2×C3×C3⋊D3—34no—
93C2×C32×D3—912no—
94C2×C32×D3—91no—
95C2×C32×D3—912no—
96C2×C32×D3—91no—
97C2×C32×D3—271no—
98D3×C3⋊D3C3311yescharacteristic, complemented
99C3×D32—34no—
100C3×D32—39no—
101C3×D32—94no—
102C2×(C3⋊D3)⋊C3—39no—
103C34—21no—
104C34—21no—
105C34—212no—
106C34—24no—
107C34—212no—
108C34—21no—
109C34—24no—
110C34—31no—
111C34—31no—
112C34—44no—
113C34—412no—
114C34—41no—
115C34—64no—
116C34—61no—
117C34—64no—
118C34—61no—
119C34—68no—
120C34—61no—
121C34—121no—
122C34—124no—
123C34—124no—
124C34—128no—
125C34D3211yescharacteristic, complemented
126C34C2×C3×D311yescharacteristic
127C34C2×C3×D311yescharacteristic, complemented
128C34C2×C3×D3112yescomplemented
129C34C22×C3214yescomplemented
130C3×H(3,𝔽3)—212no—
131C3×H(3,𝔽3)—212no—
132C3×H(3,𝔽3)—212no—
133C3×H(3,𝔽3)—236no—
134C3×H(3,𝔽3)—236no—
135C3×H(3,𝔽3)—436no—
136C3×H(3,𝔽3)—4108no—
137C3×H(3,𝔽3)—412no—
138C3×H(3,𝔽3)—412no—
139C3×H(3,𝔽3)—424no—
140C3×H(3,𝔽3)—412no—
141C3×H(3,𝔽3)C2×C3×D3112yescomplemented
142C3×H(3,𝔽3)C22×C3219yescomplemented
143C2×C33—31no—
144C2×C33—312no—
145C2×C33—34no—
146C2×C33—612no—
147C2×C33—64no—
148C2×C33—91no—
149C2×C33—91no—
150C2×C33—91no—
151C2×C33—912no—
152C2×C33—91no—
153C2×C33—181no—
154C2×C33—184no—
155C2×C33—184no—
156C2×C33—188no—
157C2×C33—1812no—
158C2×C33—181no—
159C2×C33—271no—
160C3⋊(C3⋊D3)C2×C3311yescharacteristic, complemented
161C3×C3⋊D3—24no—
162C3×C3⋊D3—34no—
163C3×C3⋊D3—39no—
164C3×C3⋊D3—34no—
165C3×C3⋊D3—64no—
166C3×C3⋊D3—94no—
167C3×C3⋊D3—184no—
168C3×C3⋊D3C32×D314yescomplemented
169C3×C3⋊D3C2×C3311yescharacteristic, complemented
170C32×D3—24no—
171C32×D3—24no—
172C32×D3—29no—
173C32×D3—31no—
174C32×D3—312no—
175C32×D3—312no—
176C32×D3—34no—
177C32×D3—39no—
178C32×D3—34no—
179C32×D3—612no—
180C32×D3—64no—
181C32×D3—68no—
182C32×D3—624no—
183C32×D3—612no—
184C32×D3—636no—
185C32×D3—64no—
186C32×D3—64no—
187C32×D3—612no—
188C32×D3—91no—
189C32×D3—912no—
190C32×D3—91no—
191C32×D3—91no—
192C32×D3—912no—
193C32×D3—94no—
194C32×D3—184no—
195C32×D3—181no—
196C32×D3—1812no—
197C32×D3—271no—
198C32×D3—541no—
199C32×D3(C3⋊D3)⋊C311yescharacteristic, direct product factor
200C32×D3C32×D314yes—
201C32×D3C32×D319yescomplemented
202C32×D3C2×C3311yescharacteristic, complemented
203C2×H(3,𝔽3)—39no—
204C2×H(3,𝔽3)—636no—
205(C3⋊D3)⋊C3—29no—
206(C3⋊D3)⋊C3—39no—
207(C3⋊D3)⋊C3—69no—
208(C3⋊D3)⋊C3C32×D319yesdirect product factor
209C22×C32—271no—
210C22×C32—2712no—
211C2×C3⋊D3—31no—
212C2×C3×D3—99no—
213C2×C3×D3—94no—
214C2×C3×D3—99no—
215C2×C3×D3—94no—
216C2×C3×D3—274no—
217D32—31no—
218D32—91no—
219C33—21no—
220C33—24no—
221C33—24no—
222C33—24no—
223C33—212no—
224C33—212no—
225C33—236no—
226C33—24no—
227C33—24no—
228C33—29no—
229C33—24no—
230C33—31no—
231C33—31no—
232C33—312no—
233C33—34no—
234C33—44no—
235C33—44no—
236C33—436no—
237C33—412no—
238C33—412no—
239C33—44no—
240C33—64no—
241C33—612no—
242C33—64no—
243C33—612no—
244C33—612no—
245C33—64no—
246C33—64no—
247C33—64no—
248C33—68no—
249C33—68no—
250C33—624no—
251C33—61no—
252C33—61no—
253C33—61no—
254C33—64no—
255C33—612no—
256C33—124no—
257C33—124no—
258C33—124no—
259C33—124no—
260C33—1212no—
261C33—1212no—
262C33—128no—
263C33—1212no—
264C33—124no—
265C33—1212no—
266C33—1236no—
267C33—124no—
268C33—1212no—
269C33—128no—
270C33—128no—
271C33—1224no—
272C33—1224no—
273C33—1272no—
274C33—1224no—
275C33—121no—
276C33C2×(C3⋊D3)⋊C311yescharacteristic, complemented
277C33C3×D3211yescharacteristic
278C33C3×D32112yescomplemented
279C33C2×C32×D314yes—
280C33C2×C32×D314yescomplemented
281C33C2×C32×D319yescomplemented
282C33C22×C3311yescharacteristic, complemented
283H(3,𝔽3)—236no—
284H(3,𝔽3)—29no—
285H(3,𝔽3)—29no—
286H(3,𝔽3)—49no—
287H(3,𝔽3)—436no—
288H(3,𝔽3)—472no—
289H(3,𝔽3)—436no—
290H(3,𝔽3)C2×C32×D319yescomplemented
291C2×C32—31no—
292C2×C32—34no—
293C2×C32—39no—
294C2×C32—31no—
295C2×C32—64no—
296C2×C32—69no—
297C2×C32—64no—
298C2×C32—912no—
299C2×C32—94no—
300C2×C32—912no—
301C2×C32—91no—
302C2×C32—99no—
303C2×C32—94no—
304C2×C32—184no—
305C2×C32—1812no—
306C2×C32—1812no—
307C2×C32—184no—
308C2×C32—188no—
309C2×C32—1824no—
310C2×C32—184no—
311C2×C32—1836no—
312C2×C32—1812no—
313C2×C32—2712no—
314C2×C32—271no—
315C3⋊D3—31no—
316C3⋊D3—31no—
317C3⋊D3—61no—
318C3⋊D3—91no—
319C3⋊D3—181no—
320C3⋊D3C33×D311yescharacteristic, complemented
321C3×D3—24no—
322C3×D3—34no—
323C3×D3—39no—
324C3×D3—31no—
325C3×D3—39no—
326C3×D3—31no—
327C3×D3—69no—
328C3×D3—64no—
329C3×D3—64no—
330C3×D3—69no—
331C3×D3—91no—
332C3×D3—94no—
333C3×D3—99no—
334C3×D3—94no—
335C3×D3—94no—
336C3×D3—99no—
337C3×D3—184no—
338C3×D3—184no—
339C3×D3—189no—
340C3×D3—274no—
341C3×D3—544no—
342C3×D3C3×(C3⋊D3)⋊C314yesdirect product factor
343C3×D3C33×D311yescharacteristic
344C22×C3—274no—
345C22×C3—279no—
346C2×D3—91no—
347C2×D3—91no—
348C2×D3—271no—
349C32—24no—
350C32—24no—
351C32—24no—
352C32—24no—
353C32—24no—
354C32—29no—
355C32—21no—
356C32—24no—
357C32—29no—
358C32—312no—
359C32—31no—
360C32—39no—
361C32—34no—
362C32—44no—
363C32—412no—
364C32—44no—
365C32—49no—
366C32—44no—
367C32—612no—
368C32—69no—
369C32—61no—
370C32—64no—
371C32—636no—
372C32—64no—
373C32—68no—
374C32—624no—
375C32—612no—
376C32—64no—
377C32—612no—
378C32—64no—
379C32—69no—
380C32—64no—
381C32—1212no—
382C32—124no—
383C32—124no—
384C32—124no—
385C32—129no—
386C32—1212no—
387C32—1236no—
388C32—128no—
389C32—1224no—
390C32—1236no—
391C32—124no—
392C32—128no—
393C32—1224no—
394C32—1272no—
395C32—124no—
396C32—1212no—
397C32D3×(C3⋊D3)⋊C311yescharacteristic, direct product factor
398C32C2×C3×(C3⋊D3)⋊C314yescomplemented
399C32C32×D3214yes—
400C32C32×D3219yescomplemented
401C32C2×C33×D311yescharacteristic
402C32C2×C33×D311yescharacteristic, complemented
403C2×C3—31no—
404C2×C3—34no—
405C2×C3—64no—
406C2×C3—91no—
407C2×C3—99no—
408C2×C3—99no—
409C2×C3—91no—
410C2×C3—94no—
411C2×C3—184no—
412C2×C3—189no—
413C2×C3—184no—
414C2×C3—189no—
415C2×C3—279no—
416C2×C3—274no—
417D3—31no—
418D3—91no—
419D3—91no—
420D3—91no—
421D3—181no—
422D3—271no—
423D3—541no—
424D3C32×(C3⋊D3)⋊C311yescharacteristic, direct product factor
425C22—271noSylow
426C3—21no—
427C3—24no—
428C3—24no—
429C3—31no—
430C3—39no—
431C3—44no—
432C3—61no—
433C3—69no—
434C3—64no—
435C3—69no—
436C3—124no—
437C3—129no—
438C3C3×D3×(C3⋊D3)⋊C314yesdirect product factor
439C3C2×C32×(C3⋊D3)⋊C311yescharacteristic, complemented
440C3C33×D3211yescharacteristic
441C2—31no—
442C2—91no—
443C2—271no—
subgroupG/H
center, hypercenterC32D3×(C3⋊D3)⋊C3
commutator, cosocle, hypocenterC33C22×C33
FrattiniC3C33×D32
FittingC33×H(3,𝔽3)C22
socleC34C2×C3×D3
LCSG ▷ C33
UCSC1 ◁ C32

Complements

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

NH ≅ G/Ntype
1. C32×C3⋊((C3⋊D3)⋊C3)C2semidirect
2. C32×D3×H(3,𝔽3)C2semidirect
3. C33×(C3⋊D3)⋊C3C2semidirect
4. C32×D3×C3⋊D3C3semidirect
7. C3×D3×(C3⋊D3)⋊C3C3direct (N × H)
8. C33×H(3,𝔽3)C22semidirect
9. C32×C3⋊(C3⋊D3)C2×C3semidirect
11. C33×C3⋊D3C2×C3semidirect
14. C34×D3D3semidirect
15. C34×D3C2×C3semidirect
17. C3×C3⋊((C3⋊D3)⋊C3)C2×C3semidirect
19. C3×D3×H(3,𝔽3)C2×C3semidirect
24. C32×(C3⋊D3)⋊C3D3direct (N × H)
25. C32×(C3⋊D3)⋊C3C2×C3semidirect
29. C3×D3×C3⋊D3C32semidirect
34. D3×(C3⋊D3)⋊C3C32direct (N × H)
36. C35C2×D3semidirect
37. C35C22×C3semidirect
44. C32×H(3,𝔽3)C2×D3semidirect
45. C32×H(3,𝔽3)C22×C3semidirect
50. C3×C3⋊(C3⋊D3)C2×C32semidirect
58. C32×C3⋊D3C3×D3semidirect
59. C32×C3⋊D3C2×C32semidirect
78. C33×D3C3×D3semidirect
79. C33×D3C2×C32semidirect
82. C3⋊((C3⋊D3)⋊C3)C2×C32semidirect
84. D3×H(3,𝔽3)C2×C32semidirect
89. C3×(C3⋊D3)⋊C3C3×D3direct (N × H)
90. C3×(C3⋊D3)⋊C3C2×C32semidirect
98. D3×C3⋊D3C33semidirect
125. C34D32semidirect
127. C34C2×C3×D3semidirect
128. C34C2×C3×D3semidirect
129. C34C22×C32semidirect
141. C3×H(3,𝔽3)C2×C3×D3semidirect
142. C3×H(3,𝔽3)C22×C32semidirect
160. C3⋊(C3⋊D3)C2×C33semidirect
168. C3×C3⋊D3C32×D3semidirect
169. C3×C3⋊D3C2×C33semidirect
199. C32×D3(C3⋊D3)⋊C3direct (N × H)
201. C32×D3C32×D3semidirect
202. C32×D3C2×C33semidirect
208. (C3⋊D3)⋊C3C32×D3direct (N × H)
276. C33C2×(C3⋊D3)⋊C3semidirect
278. C33C3×D32semidirect
280. C33C2×C32×D3semidirect
281. C33C2×C32×D3semidirect
282. C33C22×C33semidirect
290. H(3,𝔽3)C2×C32×D3semidirect
320. C3⋊D3C33×D3semidirect
342. C3×D3C3×(C3⋊D3)⋊C3direct (N × H)
397. C32D3×(C3⋊D3)⋊C3direct (N × H)
398. C32C2×C3×(C3⋊D3)⋊C3semidirect
400. C32C32×D32semidirect
402. C32C2×C33×D3semidirect
424. D3C32×(C3⋊D3)⋊C3direct (N × H)
438. C3C3×D3×(C3⋊D3)⋊C3direct (N × H)
439. C3C2×C32×(C3⋊D3)⋊C3semidirect

Maximal subgroups

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

subgroupindex
C32×C3⋊((C3⋊D3)⋊C3)2
C32×D3×H(3,𝔽3)2
C33×(C3⋊D3)⋊C32
C32×D3×C3⋊D33
C33×D323
C2×C32×(C3⋊D3)⋊C33
C3×D3×(C3⋊D3)⋊C33