Finite Group Databasebeta

← C3×#10592521 #13162860

Subgroups — C3×#10592521 #13162860

Order 1728.

1184 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64366C211yesmaximal, characteristic, complemented
2C3×#64158C211yesmaximal, characteristic, complemented
3C3×#64117C211yesmaximal, characteristic, complemented
4C3×#64083C211yesmaximal, characteristic, complemented
5C33×#104C211yesmaximal, characteristic, complemented
6C3×#63650C211yesmaximal, characteristic, complemented
7C3×#63632C211yesmaximal, characteristic, complemented
8#10592521C311yesmaximal, characteristic, direct product factor
9C3×#4822—34nomaximal
10C33×#41C2211yescharacteristic
11C3×#3832—21no—
12C3×D4×C3⋊D3—22no—
13C3×D4×C3⋊D3—21no—
14C3×#3828—21no—
15C3×#3828C2211yescharacteristic
16C2×C3×C4⋊(C3⋊D3)—21no—
17C2×C3×C4⋊(C3⋊D3)—22no—
18C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
19C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
20C33×M16C2211yescharacteristic, complemented
21C33×C42C2211yescharacteristic, complemented
22C3×#3755—21no—
23C3×#3753—21no—
24C3×#3748C2211yescharacteristic, complemented
25C3×#3745—21no—
26C3×#3744—21no—
27#64366C2×C311yescharacteristic, complemented
28C3×#1009—34no—
29#64158C2×C311yescharacteristic, complemented
30#64117C2×C311yescharacteristic, complemented
31#64083C2×C311yescharacteristic, complemented
32C3×#949—34no—
33C3×#908—34no—
34C3×#874—34no—
35C32×#104—24no—
36C32×#104D314yescomplemented
37C32×#104C2×C311yescharacteristic, complemented
38#63650C2×C311yescharacteristic, complemented
39#63632C2×C311yescharacteristic, complemented
40C3×#823—34no—
41C3×#805—34no—
42C22×C3×C3⋊D3—22no—
43C22×C3×C3⋊D3—21no—
44C22×C3×C3⋊D3D412yes—
45C33×Q8D411yescharacteristic, complemented
46C33×D4—21no—
47C33×D4D411yescharacteristic, complemented
48C2×C33×C4—21no—
49C2×C33×C4—21no—
50C2×C33×C4C2311yescharacteristic
51C3×#638—22no—
52C3×#638—21no—
53C3×C4⋊(C3⋊D3)—22no—
54C3×C4⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)—21no—
57C3×C4⋊(C3⋊D3)—21no—
58C3×C4⋊(C3⋊D3)—44no—
59C3×C4⋊(C3⋊D3)D411yescharacteristic, complemented
60C3×C4×C3⋊D3—22no—
61C3×C4×C3⋊D3—21no—
62C3×#634D411yescharacteristic, complemented
63C33×C8—21no—
64C3×C3⋊(C3⋊C8)—21no—
65C3×#453—91no—
66#4822—34no—
67C32×#41—24no—
68C32×#41C2×D314yes—
69C32×#41C22×C311yescharacteristic
70#3832—21no—
71D4×C3⋊D3—22no—
72D4×C3⋊D3—21no—
73#3828—21no—
74#3828C22×C311yescharacteristic
75C2×C4⋊(C3⋊D3)—21no—
76C2×C4⋊(C3⋊D3)—22no—
77C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
78C2×C4⋊(C3⋊D3)C22×C311yescharacteristic, complemented
79C3×#239—64no—
80C3×D3×D4—68no—
81C3×D3×D4—64no—
82C3×#235—34no—
83C3×#235—64no—
84C2×C3×D12—34no—
85C2×C3×D12—34no—
86C2×C3×D12—64no—
87C2×C3×D12—68no—
88C32×M16—24no—
89C32×M16C2×D314yescomplemented
90C32×M16C22×C311yescharacteristic, complemented
91C32×C42—24no—
92C32×C42C2×D314yescomplemented
93C32×C42C22×C311yescharacteristic, complemented
94#3755—21no—
95#3753—21no—
96#3748C22×C311yescharacteristic, complemented
97#3745—21no—
98#3744—21no—
99C3×#215—64no—
100C3×#213—64no—
101C3×#208—34no—
102C3×D24—64no—
103C3×#204—64no—
104C22×C33—21no—
105C22×C33C2×D411yescharacteristic
106C2×C3×C3⋊D3—22no—
107C2×C3×C3⋊D3—21no—
108C2×C3×C3⋊D3—42no—
109C2×C3×C3⋊D3—42no—
110C2×C3×C3⋊D3—41no—
111C2×C3×C3⋊D3—82no—
112C33×C4—22no—
113C33×C4—21no—
114C33×C4C2×D411yescharacteristic
115C33×C4C2×D411yescharacteristic
116C3×C3⋊Dic3—21no—
117C3×H(5,𝔽2)—91no—
118#1009—34no—
119C3×AGL(1,ℤ/8)—91no—
120C3×AGL(1,ℤ/8)—91no—
121C3×#127—91no—
122#949—34no—
123#908—34no—
124#874—34no—
125C3×#104—24no—
126C3×#104—91no—
127C3×#104C3×D314yescomplemented
128C3×#104C3⋊D311yescharacteristic, complemented
129C3×#100—91no—
130#823—34no—
131#805—34no—
132C22×C3⋊D3—22no—
133C22×C3⋊D3—21no—
134C22×C3⋊D3C3×D412yes—
135C22×C3×D3—38no—
136C22×C3×D3—68no—
137C22×C3×D3—64no—
138C32×Q8—24no—
139C32×Q8D1214yescomplemented
140C32×Q8C3×D411yescharacteristic, complemented
141C32×D4—21no—
142C32×D4—24no—
143C32×D4—24no—
144C32×D4—44no—
145C32×D4D1214yescomplemented
146C32×D4C3×D411yescharacteristic, complemented
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—24no—
150C2×C32×C4—24no—
151C2×C32×C4—24no—
152C2×C32×C4—44no—
153C2×C32×C4—44no—
154C2×C32×C4C22×D314yes—
155C2×C32×C4C23×C311yescharacteristic
156#638—22no—
157#638—21no—
158C4⋊(C3⋊D3)—22no—
159C4⋊(C3⋊D3)—21no—
160C4⋊(C3⋊D3)—21no—
161C4⋊(C3⋊D3)—21no—
162C4⋊(C3⋊D3)—21no—
163C4⋊(C3⋊D3)—44no—
164C4⋊(C3⋊D3)C3×D411yescharacteristic, complemented
165C4×C3⋊D3—22no—
166C4×C3⋊D3—21no—
167#634C3×D411yescharacteristic, complemented
168C3×#67—68no—
169C3×#67—64no—
170C3×D12—34no—
171C3×D12—68no—
172C3×D12—64no—
173C3×D12—64no—
174C3×D12—64no—
175C3×D12—64no—
176C3×D12—1216no—
177C3×C4×D3—68no—
178C3×C4×D3—64no—
179C3×Dic6—34no—
180C32×C8—21no—
181C32×C8—24no—
182C32×C8—44no—
183C3⋊(C3⋊C8)—21no—
184C3×C3⋊C8—64no—
185#453—91noSylow
186C2×C33—21no—
187C2×C33—41no—
188C2×C33#12011yescharacteristic
189C3×C3⋊D3—42no—
190C3×C3⋊D3—41no—
191C3×C3⋊D3—81no—
192C3×#41—24no—
193C3×#41—91no—
194C3×#41—181no—
195C3×#41—181no—
196C3×#41C2×C3×D314yes—
197C3×#41C2×C3⋊D311yescharacteristic
198C2×C3×D4—91no—
199C2×C3×D4—91no—
200C2×C3×D4—181no—
201C2×C3×D4—182no—
202C2×C3×D4—182no—
203C2×C3×D4—181no—
204#239—64no—
205D3×D4—68no—
206D3×D4—64no—
207#235—34no—
208#235—64no—
209C2×D12—34no—
210C2×D12—34no—
211C2×D12—64no—
212C2×D12—68no—
213C3×SD16—181no—
214C3×SD16—181no—
215C3×D8—181no—
216C3×D8—181no—
217C3×M16—24no—
218C3×M16—91no—
219C3×M16C2×C3×D314yescomplemented
220C3×M16C2×C3⋊D311yescharacteristic, complemented
221C3×C42—24no—
222C3×C42C2×C3×D314yescomplemented
223C3×C42C2×C3⋊D311yescharacteristic, complemented
224#215—64no—
225#213—64no—
226#208—34no—
227D24—64no—
228#204—64no—
229C22×C32—21no—
230C22×C32—24no—
231C22×C32—24no—
232C22×C32—44no—
233C22×C32D3×D414yes—
234C22×C32C2×C3×D411yescharacteristic
235C2×C3⋊D3—22no—
236C2×C3⋊D3—21no—
237C2×C3⋊D3—42no—
238C2×C3⋊D3—42no—
239C2×C3⋊D3—41no—
240C2×C3⋊D3—82no—
241C2×C3×D3—64no—
242C2×C3×D3—68no—
243C2×C3×D3—128no—
244C2×C3×D3—128no—
245C2×C3×D3—124no—
246C2×C3×D3—248no—
247C32×C4—22no—
248C32×C4—21no—
249C32×C4—24no—
250C32×C4—24no—
251C32×C4—28no—
252C32×C4—24no—
253C32×C4—44no—
254C32×C4—48no—
255C32×C4C2×D1214yes—
256C32×C4D3×D414yes—
257C32×C4C2×C3×D411yescharacteristic
258C32×C4C2×C3×D411yescharacteristic
259C3⋊Dic3—21no—
260C3×Dic3—64no—
261H(5,𝔽2)—91no—
262AGL(1,ℤ/8)—91no—
263AGL(1,ℤ/8)—91no—
264#127—91no—
265#104—91no—
266#104C3×C3⋊D311yescharacteristic, complemented
267#100—91no—
268C33#45311yescharacteristic, complemented, Sylow
269C23×C3—92no—
270C23×C3—182no—
271C23×C3—181no—
272C22×D3—38no—
273C22×D3—68no—
274C22×D3—64no—
275C3×Q8—24no—
276C3×Q8—91no—
277C3×Q8C3×D1214yescomplemented
278C3×Q8C4⋊(C3⋊D3)11yescharacteristic, complemented
279C3×D4—24no—
280C3×D4—21no—
281C3×D4—24no—
282C3×D4—44no—
283C3×D4—91no—
284C3×D4—182no—
285C3×D4—182no—
286C3×D4—181no—
287C3×D4—181no—
288C3×D4—181no—
289C3×D4—181no—
290C3×D4—181no—
291C3×D4—364no—
292C3×D4C3×D1214yescomplemented
293C3×D4C4⋊(C3⋊D3)11yescharacteristic, complemented
294C2×C3×C4—24no—
295C2×C3×C4—21no—
296C2×C3×C4—24no—
297C2×C3×C4—21no—
298C2×C3×C4—24no—
299C2×C3×C4—44no—
300C2×C3×C4—44no—
301C2×C3×C4—182no—
302C2×C3×C4—181no—
303C2×C3×C4C22×C3×D314yes—
304C2×C3×C4C22×C3⋊D311yescharacteristic
305#67—68no—
306#67—64no—
307D12—34no—
308D12—68no—
309D12—64no—
310D12—64no—
311D12—64no—
312D12—64no—
313D12—1216no—
314C4×D3—68no—
315C4×D3—64no—
316Dic6—34no—
317C3×C8—24no—
318C3×C8—21no—
319C3×C8—44no—
320C3×C8—181no—
321C3⋊C8—64no—
322C2×C32—21no—
323C2×C32—24no—
324C2×C32—24no—
325C2×C32—44no—
326C2×C32—41no—
327C2×C32—44no—
328C2×C32—84no—
329C2×C32#88214yes—
330C2×C32C3×#12011yescharacteristic
331C3⋊D3—42no—
332C3⋊D3—41no—
333C3⋊D3—81no—
334C3×D3—128no—
335C3×D3—124no—
336C3×D3—244no—
337#41—91no—
338#41—181no—
339#41—181no—
340#41C2×C3×C3⋊D311yescharacteristic
341C2×D4—91no—
342C2×D4—91no—
343C2×D4—181no—
344C2×D4—182no—
345C2×D4—182no—
346C2×D4—181no—
347SD16—181no—
348SD16—181no—
349D8—181no—
350D8—181no—
351M16—91no—
352M16C2×C3×C3⋊D311yescharacteristic, complemented
353C42C2×C3×C3⋊D311yescharacteristic, complemented
354C22×C3—24no—
355C22×C3—21no—
356C22×C3—24no—
357C22×C3—44no—
358C22×C3—181no—
359C22×C3—182no—
360C22×C3—362no—
361C22×C3—362no—
362C22×C3—361no—
363C22×C3—722no—
364C22×C3C3×D3×D414yes—
365C22×C3D4×C3⋊D311yescharacteristic
366C2×D3—64no—
367C2×D3—68no—
368C2×D3—128no—
369C2×D3—128no—
370C2×D3—124no—
371C2×D3—248no—
372C3×C4—28no—
373C3×C4—22no—
374C3×C4—24no—
375C3×C4—21no—
376C3×C4—24no—
377C3×C4—24no—
378C3×C4—44no—
379C3×C4—48no—
380C3×C4—181no—
381C3×C4C2×C3×D1214yes—
382C3×C4C3×D3×D414yes—
383C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
384C3×C4D4×C3⋊D311yescharacteristic
385Dic3—64no—
386C32—24no—
387C32#482214yescomplemented
388C32C3×#45311yescharacteristic, complemented
389C23—92no—
390C23—182no—
391C23—181no—
392Q8—91no—
393Q8C3×C4⋊(C3⋊D3)11yescharacteristic, complemented
394D4—21no—
395D4—91no—
396D4—182no—
397D4—182no—
398D4—181no—
399D4—181no—
400D4—181no—
401D4—181no—
402D4—181no—
403D4—364no—
404D4C3×C4⋊(C3⋊D3)11yescharacteristic, complemented
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—182no—
408C2×C4—181no—
409C2×C4C22×C3×C3⋊D311yescharacteristic
410C8—21no—
411C8—181no—
412C2×C3—21no—
413C2×C3—24no—
414C2×C3—24no—
415C2×C3—44no—
416C2×C3—41no—
417C2×C3—44no—
418C2×C3—84no—
419C2×C3—362no—
420C2×C3—361no—
421C2×C3—721no—
422C2×C3C3×#88214yes—
423C2×C3#6409111yescharacteristic
424D3—128no—
425D3—124no—
426D3—244no—
427C22—21no—
428C22—181no—
429C22—182no—
430C22—362no—
431C22—362no—
432C22—361no—
433C22—722no—
434C22C3×D4×C3⋊D311yescharacteristic
435C4—22no—
436C4—21no—
437C4—181no—
438C4C2×C3×C4⋊(C3⋊D3)11yescharacteristic
439C4C3×D4×C3⋊D311yescharacteristic
440C3—24no—
441C3C3×#482214yescomplemented
442C3#1059252111yescharacteristic, direct product factor
443C2—21no—
444C2—41no—
445C2—362no—
446C2—361no—
447C2—721no—
448C2C3×#6409111yescharacteristic
subgroupG/H
centerC2×C3#64091
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#104C2
socleC2×C33#120
hypercenterC3×#104C3⋊D3
hypocenterC32C3×#453
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#104

Complements

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

NH ≅ G/Ntype
1. C3×#64366C2semidirect
2. C3×#64158C2semidirect
3. C3×#64117C2semidirect
4. C3×#64083C2semidirect
5. C33×#104C2semidirect
6. C3×#63650C2semidirect
7. C3×#63632C2semidirect
8. #10592521C3direct (N × H)
19. C2×C3×C4⋊(C3⋊D3)C22semidirect
20. C33×M16C22semidirect
21. C33×C42C22semidirect
24. C3×#3748C22semidirect
27. #64366C2×C3semidirect
29. #64158C2×C3semidirect
30. #64117C2×C3semidirect
31. #64083C2×C3semidirect
36. C32×#104D3semidirect
37. C32×#104C2×C3semidirect
38. #63650C2×C3semidirect
39. #63632C2×C3semidirect
45. C33×Q8D4semidirect
47. C33×D4D4semidirect
59. C3×C4⋊(C3⋊D3)D4semidirect
62. C3×#634D4semidirect
78. C2×C4⋊(C3⋊D3)C22×C3semidirect
89. C32×M16C2×D3semidirect
90. C32×M16C22×C3semidirect
92. C32×C42C2×D3semidirect
93. C32×C42C22×C3semidirect
96. #3748C22×C3semidirect
127. C3×#104C3×D3semidirect
128. C3×#104C3⋊D3semidirect
139. C32×Q8D12semidirect
140. C32×Q8C3×D4semidirect
145. C32×D4D12semidirect
146. C32×D4C3×D4semidirect
164. C4⋊(C3⋊D3)C3×D4semidirect
167. #634C3×D4semidirect
219. C3×M16C2×C3×D3semidirect
220. C3×M16C2×C3⋊D3semidirect
222. C3×C42C2×C3×D3semidirect
223. C3×C42C2×C3⋊D3semidirect
266. #104C3×C3⋊D3semidirect
268. C33#453semidirect
277. C3×Q8C3×D12semidirect
278. C3×Q8C4⋊(C3⋊D3)semidirect
292. C3×D4C3×D12semidirect
293. C3×D4C4⋊(C3⋊D3)semidirect
352. M16C2×C3×C3⋊D3semidirect
353. C42C2×C3×C3⋊D3semidirect
387. C32#4822semidirect
388. C32C3×#453semidirect
393. Q8C3×C4⋊(C3⋊D3)semidirect
404. D4C3×C4⋊(C3⋊D3)semidirect
441. C3C3×#4822semidirect
442. C3#10592521direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643662
C3×#641582
C3×#641172
C3×#640832
C33×#1042
C3×#636502
C3×#636322
#105925213
C3×#48223