Finite Group Databasebeta

← C3×#10592440 #13162779

Subgroups — C3×#10592440 #13162779

Order 1728.

1396 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 1396 conjugacy classes of subgroups — too many to draw. Only its 174 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×#64361C211yesmaximal, characteristic, complemented
2C3×#64146C211yesmaximal, characteristic, complemented
3C3×#64091C212yesmaximal, complemented
4C33×#99C211yesmaximal, characteristic, complemented
5C3×#63633C212yesmaximal, complemented
6#10592440C311yesmaximal, characteristic, direct product factor
7C3×#4741—34nomaximal
8C23×C3×C3⋊D3C2211yescharacteristic, complemented
9C2×C33×D4C2211yescharacteristic
10C2×C3×#638—21no—
11C2×C3×#638—21no—
12C2×C3×#638C2212yes—
13C3×#3830—22no—
14C3×D4×C3⋊D3—22no—
15C3×#3828—21no—
16C2×C3×C4⋊(C3⋊D3)—22no—
17C33×C22⋊C4C2212yescomplemented
18C3×#3757C2211yescharacteristic, complemented
19C3×#3752—21no—
20C3×#3752—22no—
21#64361C2×C311yescharacteristic, complemented
22C3×#1004—34no—
23#64146C2×C311yescharacteristic, complemented
24#64091C2×C312yescomplemented
25C3×#937—34no—
26C3×#882—38no—
27C32×#99—24no—
28C32×#99D314yescomplemented
29C32×#99C2×C311yescharacteristic, complemented
30#63633C2×C312yescomplemented
31C3×#806—38no—
32C23×C33D411yescharacteristic, complemented
33C23×C33C2311yescharacteristic
34C22×C3×C3⋊D3—22no—
35C22×C3×C3⋊D3—21no—
36C22×C3×C3⋊D3—21no—
37C22×C3×C3⋊D3—42no—
38C22×C3×C3⋊D3D412yescomplemented
39C33×D4—22no—
40C2×C33×C4—22no—
41C2×C33×C4D411yescharacteristic, complemented
42C3×#638—22no—
43C3×#638—22no—
44C3×#638—22no—
45C3×#638—42no—
46C2×C3×C3⋊Dic3—21no—
47C2×C3×C3⋊Dic3—21no—
48C2×C3×C3⋊Dic3D412yescomplemented
49C3×C4⋊(C3⋊D3)—21no—
50C3×C4⋊(C3⋊D3)—44no—
51C3×C4×C3⋊D3—22no—
52C3×#634—21no—
53C3×B(4,𝔽2)—91no—
54#4741—34no—
55C23×C3⋊D3C22×C311yescharacteristic, complemented
56C23×C3×D3—34no—
57C2×C32×D4—24no—
58C2×C32×D4C2×D314yes—
59C2×C32×D4C22×C311yescharacteristic
60C2×#638—21no—
61C2×#638—21no—
62C2×#638C22×C312yes—
63#3830—22no—
64D4×C3⋊D3—22no—
65#3828—21no—
66C2×C4⋊(C3⋊D3)—22no—
67C2×C3×#67—38no—
68C2×C3×#67—64no—
69C2×C3×#67—64no—
70C3×#237—68no—
71C3×D3×D4—68no—
72C3×#235—64no—
73C2×C3×D12—68no—
74C32×C22⋊C4—28no—
75C32×C22⋊C4C2×D318yescomplemented
76C32×C22⋊C4C22×C312yescomplemented
77#3757C22×C311yescharacteristic, complemented
78#3752—21no—
79#3752—22no—
80C3×#217—34no—
81C3×(C2×D3)⋊C4—68no—
82C3×(C2×D3)⋊C4—64no—
83C22×C33—21no—
84C22×C33—41no—
85C22×C33—41no—
86C22×C33C2×D411yescharacteristic
87C22×C33C2×D412yes—
88C2×C3×C3⋊D3—24no—
89C2×C3×C3⋊D3—22no—
90C2×C3×C3⋊D3—22no—
91C2×C3×C3⋊D3—41no—
92C2×C3×C3⋊D3—42no—
93C2×C3×C3⋊D3—41no—
94C2×C3×C3⋊D3—42no—
95C2×C3×C3⋊D3—81no—
96C33×C4—21no—
97C33×C4—42no—
98C3×C3⋊Dic3—22no—
99C3×C3⋊Dic3—41no—
100C3×H(5,𝔽2)—91no—
101#1004—34no—
102C3×#120—91no—
103C3×#120—92no—
104#937—34no—
105#882—38no—
106C3×#99—24no—
107C3×#99—92no—
108C3×#99C3×D314yescomplemented
109C3×#99C3⋊D311yescharacteristic, complemented
110#806—38no—
111C23×C32—24no—
112C23×C32—24no—
113C23×C32D1214yescomplemented
114C23×C32C3×D411yescharacteristic, complemented
115C23×C32C22×D314yes—
116C23×C32C23×C311yescharacteristic
117C22×C3⋊D3—22no—
118C22×C3⋊D3—21no—
119C22×C3⋊D3—21no—
120C22×C3⋊D3—42no—
121C22×C3⋊D3C3×D412yescomplemented
122C22×C3×D3—38no—
123C22×C3×D3—64no—
124C22×C3×D3—68no—
125C22×C3×D3—64no—
126C22×C3×D3—128no—
127C32×D4—22no—
128C32×D4—28no—
129C32×D4—48no—
130C2×C32×C4—22no—
131C2×C32×C4—24no—
132C2×C32×C4—28no—
133C2×C32×C4—48no—
134C2×C32×C4D1214yescomplemented
135C2×C32×C4C3×D411yescharacteristic, complemented
136#638—22no—
137#638—22no—
138#638—22no—
139#638—42no—
140C2×C3⋊Dic3—21no—
141C2×C3⋊Dic3—21no—
142C2×C3⋊Dic3C3×D412yescomplemented
143C4⋊(C3⋊D3)—21no—
144C4⋊(C3⋊D3)—44no—
145C4×C3⋊D3—22no—
146#634—21no—
147C3×#67—68no—
148C3×#67—68no—
149C3×#67—68no—
150C3×#67—128no—
151C2×C3×Dic3—38no—
152C2×C3×Dic3—64no—
153C2×C3×Dic3—64no—
154C3×D12—64no—
155C3×D12—1216no—
156C3×C4×D3—68no—
157C3×Dic6—64no—
158B(4,𝔽2)—91noSylow
159C2×C33—21no—
160C2×C33—22no—
161C2×C33—41no—
162C2×C33#12011yescharacteristic
163C3×C3⋊D3—42no—
164C3×C3⋊D3—42no—
165C24×C3—91no—
166C23×D3—34no—
167C3×#41—182no—
168C3×#41—181no—
169C2×C3×D4—24no—
170C2×C3×D4—92no—
171C2×C3×D4—182no—
172C2×C3×D4—182no—
173C2×C3×D4—181no—
174C2×C3×D4—181no—
175C2×C3×D4C2×C3×D314yes—
176C2×C3×D4C2×C3⋊D311yescharacteristic
177C2×#67—38no—
178C2×#67—64no—
179C2×#67—64no—
180#237—68no—
181D3×D4—68no—
182#235—64no—
183C2×D12—68no—
184C3×C22⋊C4—28no—
185C3×C22⋊C4—91no—
186C3×C22⋊C4—182no—
187C3×C22⋊C4—181no—
188C3×C22⋊C4C2×C3×D318yescomplemented
189C3×C22⋊C4C2×C3⋊D312yescomplemented
190#217—34no—
191(C2×D3)⋊C4—68no—
192(C2×D3)⋊C4—64no—
193C22×C32—21no—
194C22×C32—24no—
195C22×C32—28no—
196C22×C32—24no—
197C22×C32—44no—
198C22×C32—41no—
199C22×C32—41no—
200C22×C32—44no—
201C22×C32—44no—
202C22×C32—84no—
203C22×C32—84no—
204C22×C32C2×D1214yes—
205C22×C32D3×D418yes—
206C22×C32C2×C3×D411yescharacteristic
207C22×C32C2×C3×D412yes—
208C2×C3⋊D3—24no—
209C2×C3⋊D3—22no—
210C2×C3⋊D3—22no—
211C2×C3⋊D3—41no—
212C2×C3⋊D3—42no—
213C2×C3⋊D3—41no—
214C2×C3⋊D3—42no—
215C2×C3⋊D3—81no—
216C2×C3×D3—616no—
217C2×C3×D3—68no—
218C2×C3×D3—68no—
219C2×C3×D3—124no—
220C2×C3×D3—128no—
221C2×C3×D3—124no—
222C2×C3×D3—128no—
223C2×C3×D3—244no—
224C32×C4—21no—
225C32×C4—24no—
226C32×C4—44no—
227C32×C4—42no—
228C32×C4—48no—
229C32×C4—88no—
230C3⋊Dic3—22no—
231C3⋊Dic3—41no—
232C3×Dic3—68no—
233C3×Dic3—124no—
234H(5,𝔽2)—91no—
235#120—91no—
236#120—92no—
237#99—92no—
238#99C3×C3⋊D311yescharacteristic, complemented
239C33B(4,𝔽2)11yescharacteristic, complemented, Sylow
240C23×C3—24no—
241C23×C3—24no—
242C23×C3—92no—
243C23×C3—181no—
244C23×C3—181no—
245C23×C3—182no—
246C23×C3—362no—
247C23×C3C3×D1214yescomplemented
248C23×C3C4⋊(C3⋊D3)11yescharacteristic, complemented
249C23×C3C22×C3×D314yes—
250C23×C3C22×C3⋊D311yescharacteristic
251C22×D3—38no—
252C22×D3—64no—
253C22×D3—68no—
254C22×D3—64no—
255C22×D3—128no—
256C3×Q8—181no—
257C3×D4—28no—
258C3×D4—22no—
259C3×D4—48no—
260C3×D4—181no—
261C3×D4—182no—
262C3×D4—182no—
263C3×D4—182no—
264C3×D4—364no—
265C3×D4—362no—
266C2×C3×C4—28no—
267C2×C3×C4—22no—
268C2×C3×C4—24no—
269C2×C3×C4—48no—
270C2×C3×C4—92no—
271C2×C3×C4—182no—
272C2×C3×C4—181no—
273C2×C3×C4—181no—
274C2×C3×C4C3×D1214yescomplemented
275C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic, complemented
276#67—68no—
277#67—68no—
278#67—68no—
279#67—128no—
280C2×Dic3—38no—
281C2×Dic3—64no—
282C2×Dic3—64no—
283D12—64no—
284D12—1216no—
285C4×D3—68no—
286Dic6—64no—
287C2×C32—21no—
288C2×C32—22no—
289C2×C32—24no—
290C2×C32—24no—
291C2×C32—28no—
292C2×C32—48no—
293C2×C32—44no—
294C2×C32—41no—
295C2×C32—44no—
296C2×C32—84no—
297C2×C32#88214yes—
298C2×C32C3×#12011yescharacteristic
299C3⋊D3—42no—
300C3⋊D3—42no—
301C3×D3—128no—
302C3×D3—128no—
303C24—91no—
304#41—182no—
305#41—181no—
306C2×D4—92no—
307C2×D4—182no—
308C2×D4—182no—
309C2×D4—181no—
310C2×D4—181no—
311C2×D4C2×C3×C3⋊D311yescharacteristic
312C22⋊C4—91no—
313C22⋊C4—182no—
314C22⋊C4—181no—
315C22⋊C4C2×C3×C3⋊D312yescomplemented
316C22×C3—24no—
317C22×C3—21no—
318C22×C3—24no—
319C22×C3—28no—
320C22×C3—44no—
321C22×C3—44no—
322C22×C3—41no—
323C22×C3—44no—
324C22×C3—41no—
325C22×C3—84no—
326C22×C3—84no—
327C22×C3—184no—
328C22×C3—182no—
329C22×C3—182no—
330C22×C3—362no—
331C22×C3—362no—
332C22×C3—361no—
333C22×C3—361no—
334C22×C3—721no—
335C22×C3C2×C3×D1214yes—
336C22×C3C3×D3×D418yes—
337C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
338C22×C3D4×C3⋊D312yes—
339C2×D3—616no—
340C2×D3—68no—
341C2×D3—68no—
342C2×D3—124no—
343C2×D3—128no—
344C2×D3—124no—
345C2×D3—128no—
346C2×D3—244no—
347C3×C4—24no—
348C3×C4—21no—
349C3×C4—44no—
350C3×C4—48no—
351C3×C4—42no—
352C3×C4—88no—
353C3×C4—182no—
354C3×C4—361no—
355Dic3—68no—
356Dic3—124no—
357C32—24no—
358C32#474114yescomplemented
359C32C3×B(4,𝔽2)11yescharacteristic, complemented
360C23—92no—
361C23—181no—
362C23—181no—
363C23—182no—
364C23—362no—
365C23C3×C4⋊(C3⋊D3)11yescharacteristic, complemented
366C23C22×C3×C3⋊D311yescharacteristic
367Q8—181no—
368D4—22no—
369D4—181no—
370D4—182no—
371D4—182no—
372D4—182no—
373D4—364no—
374D4—362no—
375C2×C4—22no—
376C2×C4—92no—
377C2×C4—182no—
378C2×C4—181no—
379C2×C4—181no—
380C2×C4C3×C4⋊(C3⋊D3)11yescharacteristic, complemented
381C2×C3—24no—
382C2×C3—28no—
383C2×C3—22no—
384C2×C3—21no—
385C2×C3—24no—
386C2×C3—48no—
387C2×C3—44no—
388C2×C3—44no—
389C2×C3—41no—
390C2×C3—84no—
391C2×C3—362no—
392C2×C3—362no—
393C2×C3C3×#88214yes—
394C2×C3#6409111yescharacteristic
395D3—128no—
396D3—128no—
397C22—21no—
398C22—41no—
399C22—41no—
400C22—184no—
401C22—182no—
402C22—182no—
403C22—362no—
404C22—362no—
405C22—361no—
406C22—361no—
407C22—721no—
408C22C2×C3×C4⋊(C3⋊D3)11yescharacteristic
409C22C3×D4×C3⋊D312yes—
410C4—21no—
411C4—42no—
412C4—182no—
413C4—361no—
414C3—24no—
415C3C3×#474114yescomplemented
416C3#1059244011yescharacteristic, direct product factor
417C2—22no—
418C2—21no—
419C2—41no—
420C2—362no—
421C2—362no—
422C2C3×#6409111yescharacteristic
subgroupG/H
centerC2×C3#64091
commutator, cosocleC23×C32C23×C3
FrattiniC23C22×C3×C3⋊D3
FittingC33×#99C2
socleC2×C33#120
hypercenterC3×#99C3⋊D3
hypocenterC32C3×B(4,𝔽2)
LCSG ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C3×#99

Complements

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

NH ≅ G/Ntype
1. C3×#64361C2semidirect
2. C3×#64146C2semidirect
3. C3×#64091C2semidirect
4. C33×#99C2semidirect
5. C3×#63633C2semidirect
6. #10592440C3direct (N × H)
8. C23×C3×C3⋊D3C22semidirect
17. C33×C22⋊C4C22semidirect
18. C3×#3757C22semidirect
21. #64361C2×C3semidirect
23. #64146C2×C3semidirect
24. #64091C2×C3semidirect
28. C32×#99D3semidirect
29. C32×#99C2×C3semidirect
30. #63633C2×C3semidirect
32. C23×C33D4semidirect
38. C22×C3×C3⋊D3D4semidirect
41. C2×C33×C4D4semidirect
48. C2×C3×C3⋊Dic3D4semidirect
55. C23×C3⋊D3C22×C3semidirect
75. C32×C22⋊C4C2×D3semidirect
76. C32×C22⋊C4C22×C3semidirect
77. #3757C22×C3semidirect
108. C3×#99C3×D3semidirect
109. C3×#99C3⋊D3semidirect
113. C23×C32D12semidirect
114. C23×C32C3×D4semidirect
121. C22×C3⋊D3C3×D4semidirect
134. C2×C32×C4D12semidirect
135. C2×C32×C4C3×D4semidirect
142. C2×C3⋊Dic3C3×D4semidirect
188. C3×C22⋊C4C2×C3×D3semidirect
189. C3×C22⋊C4C2×C3⋊D3semidirect
238. #99C3×C3⋊D3semidirect
239. C33B(4,𝔽2)semidirect
247. C23×C3C3×D12semidirect
248. C23×C3C4⋊(C3⋊D3)semidirect
274. C2×C3×C4C3×D12semidirect
275. C2×C3×C4C4⋊(C3⋊D3)semidirect
315. C22⋊C4C2×C3×C3⋊D3semidirect
358. C32#4741semidirect
359. C32C3×B(4,𝔽2)semidirect
365. C23C3×C4⋊(C3⋊D3)semidirect
380. C2×C4C3×C4⋊(C3⋊D3)semidirect
415. C3C3×#4741semidirect
416. C3#10592440direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643612
C3×#641462
C3×#640912
C33×#992
C3×#636332
#105924403
C3×#47413