Finite Group Databasebeta

← C3×#10590397 #13157415

Subgroups — C3×#10590397 #13157415

Order 1728.

898 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 898 conjugacy classes of subgroups — too many to draw. Only its 116 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×#64308C211yesmaximal, characteristic, complemented
2C3×#64083C211yesmaximal, characteristic, complemented
3C3×#63825C212yesmaximal, complemented
4C32×#805C212yesmaximal, complemented
5C3×#63559C211yesmaximal, characteristic, complemented
6C3×#4822—32nomaximal
7#10590397C311yesmaximal, characteristic, direct product factor
8C2×C3×C4⋊(C3⋊D3)—22no—
9C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
11C32×#235C2212yes—
12C3×#3811—22no—
13C3×D3×D12—22no—
14C3×#3799—22no—
15C33×C42C2211yescharacteristic, complemented
16C32×#208C2212yescomplemented
17C3×#3717—22no—
18C3×#3714—22no—
19C3×#1009—32no—
20#64308C2×C311yescharacteristic, complemented
21#64083C2×C311yescharacteristic, complemented
22C3×#949—32no—
23C3×#908—32no—
24C3×#874—32no—
25C3×#874—61no—
26#63825C2×C312yescomplemented
27C32×#104—32no—
28C3×#823—32no—
29C3×#805—22no—
30C3×#805D312yescomplemented
31C3×#805C2×C312yescomplemented
32#63559C2×C311yescharacteristic, complemented
33C22×C3×C3⋊D3—21no—
34C22×C3×C3⋊D3D412yes—
35C2×C3×D32—22no—
36C2×C33×C4—21no—
37C2×C33×C4C2311yescharacteristic
38C3×C4⋊(C3⋊D3)—22no—
39C3×C4⋊(C3⋊D3)—22no—
40C3×C4⋊(C3⋊D3)—44no—
41C32×#67—22no—
42C32×D12D412yescomplemented
43C32×C4×D3—22no—
44C32×Dic6D412yescomplemented
45C3×#626—24no—
46C3×#624—22no—
47C32×C3⋊C8—22no—
48C3×#453—91no—
49#4822—32no—
50C32×#41—32no—
51C2×C4⋊(C3⋊D3)—22no—
52C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
53C2×C4⋊(C3⋊D3)C22×C311yescharacteristic, complemented
54C3×#239—62no—
55C3×D3×D4—62no—
56C3×D3×D4—64no—
57C3×#235—22no—
58C3×#235—62no—
59C3×#235C2×D312yes—
60C3×#235C22×C312yes—
61C2×C3×D12—32no—
62C2×C3×D12—32no—
63C2×C3×D12—62no—
64C2×C3×D12—64no—
65C2×C3×D12—61no—
66C2×C3×D12—61no—
67C2×C3×D12—64no—
68#3811—22no—
69D3×D12—22no—
70#3799—22no—
71C32×M16—32no—
72C32×C42—21no—
73C32×C42—22no—
74C32×C42—41no—
75C32×C42C2×D312yescomplemented
76C32×C42C22×C311yescharacteristic, complemented
77C3×#215—62no—
78C3×#213—62no—
79C3×#208—22no—
80C3×#208C2×D312yescomplemented
81C3×#208C22×C312yescomplemented
82C3×D24—62no—
83C3×#204—62no—
84#3717—22no—
85#3714—22no—
86C22×C33C2×D411yescharacteristic
87C2×C3×C3⋊D3—22no—
88C2×C3×C3⋊D3—42no—
89C2×C3×C3⋊D3—42no—
90C2×C3×C3⋊D3—41no—
91C2×C32×D3—22no—
92C3×D32—82no—
93C33×C4—22no—
94C33×C4C2×D412yes—
95C32×Dic3—22no—
96C3×H(5,𝔽2)—91no—
97#1009—32no—
98C3×AGL(1,ℤ/8)—92no—
99C3×#127—91no—
100#949—32no—
101#908—32no—
102#874—32no—
103#874—61no—
104C3×#104—32no—
105C3×#104—32no—
106C3×#104—62no—
107C3×#100—91no—
108#823—32no—
109#805C3×D312yescomplemented
110C22×C3⋊D3—21no—
111C22×C3⋊D3C3×D412yes—
112C22×C3×D3—34no—
113C22×C3×D3—62no—
114C22×C3×D3—64no—
115C22×C3×D3—62no—
116C22×C3×D3—62no—
117C2×D32—22no—
118C32×Q8—32no—
119C32×D4—32no—
120C32×D4—62no—
121C2×C32×C4—21no—
122C2×C32×C4—22no—
123C2×C32×C4—22no—
124C2×C32×C4—21no—
125C2×C32×C4—22no—
126C2×C32×C4—41no—
127C2×C32×C4—42no—
128C2×C32×C4—42no—
129C2×C32×C4—62no—
130C2×C32×C4C22×D312yes—
131C2×C32×C4C23×C311yescharacteristic
132C4⋊(C3⋊D3)—22no—
133C4⋊(C3⋊D3)—22no—
134C4⋊(C3⋊D3)—44no—
135C3×#67—22no—
136C3×#67—22no—
137C3×#67—42no—
138C3×#67—64no—
139C3×D12—22no—
140C3×D12—64no—
141C3×D12—62no—
142C3×D12—62no—
143C3×D12—62no—
144C3×D12—62no—
145C3×D12—122no—
146C3×D12—122no—
147C3×D12—128no—
148C3×D12—128no—
149C3×D12D1212yescomplemented
150C3×D12C3×D412yescomplemented
151C3×C4×D3—22no—
152C3×C4×D3—22no—
153C3×C4×D3—42no—
154C3×C4×D3—64no—
155C3×Dic6—22no—
156C3×Dic6D1212yescomplemented
157C3×Dic6C3×D412yescomplemented
158#626—24no—
159#624—22no—
160C32×C8—62no—
161C3×C3⋊C8—22no—
162C3×C3⋊C8—22no—
163C3×C3⋊C8—42no—
164#453—91noSylow
165C2×C33—21no—
166C2×C33#12011yescharacteristic
167C3×C3⋊D3—42no—
168C3×C3⋊D3—81no—
169C32×D3—42no—
170C3×#41—32no—
171C3×#41—32no—
172C3×#41—62no—
173C3×#41—182no—
174C2×C3×D4—91no—
175C2×C3×D4—91no—
176C2×C3×D4—182no—
177C2×C3×D4—182no—
178C2×C3×D4—182no—
179#239—62no—
180D3×D4—62no—
181D3×D4—64no—
182#235—62no—
183#235C2×C3×D312yes—
184C2×D12—32no—
185C2×D12—32no—
186C2×D12—61no—
187C2×D12—61no—
188C2×D12—62no—
189C2×D12—64no—
190C2×D12—64no—
191C3×SD16—182no—
192C3×D8—182no—
193C3×M16—32no—
194C3×M16—32no—
195C3×M16—62no—
196C3×C42—21no—
197C3×C42—22no—
198C3×C42—41no—
199C3×C42D3211yescharacteristic, complemented
200C3×C42C2×C3×D312yescomplemented
201#215—62no—
202#213—62no—
203#208C2×C3×D312yescomplemented
204D24—62no—
205#204—62no—
206C22×C32—21no—
207C22×C32—22no—
208C22×C32—41no—
209C22×C32—62no—
210C22×C32D3×D412yes—
211C22×C32C2×C3×D411yescharacteristic
212C2×C3⋊D3—22no—
213C2×C3⋊D3—42no—
214C2×C3⋊D3—42no—
215C2×C3⋊D3—41no—
216C2×C3×D3—22no—
217C2×C3×D3—22no—
218C2×C3×D3—42no—
219C2×C3×D3—64no—
220C2×C3×D3—122no—
221C2×C3×D3—124no—
222C2×C3×D3—124no—
223C2×C3×D3—124no—
224C2×C3×D3—124no—
225C2×C3×D3—122no—
226C2×C3×D3—122no—
227C2×C3×D3—244no—
228D32—82no—
229C32×C4—22no—
230C32×C4—24no—
231C32×C4—24no—
232C32×C4—22no—
233C32×C4—22no—
234C32×C4—22no—
235C32×C4—42no—
236C32×C4—44no—
237C32×C4—44no—
238C32×C4—62no—
239C32×C4C2×D1212yes—
240C32×C4D3×D412yes—
241C32×C4C2×C3×D412yes—
242C3×Dic3—22no—
243C3×Dic3—22no—
244C3×Dic3—42no—
245H(5,𝔽2)—91no—
246AGL(1,ℤ/8)—92no—
247#127—91no—
248#104—32no—
249#100—91no—
250C33#45311yescharacteristic, complemented, Sylow
251C23×C3—92no—
252C23×C3—182no—
253C23×C3—181no—
254C22×D3—34no—
255C22×D3—62no—
256C22×D3—64no—
257C22×D3—62no—
258C22×D3—62no—
259C3×Q8—32no—
260C3×Q8—32no—
261C3×Q8—62no—
262C3×D4—32no—
263C3×D4—32no—
264C3×D4—62no—
265C3×D4—62no—
266C3×D4—62no—
267C3×D4—122no—
268C3×D4—184no—
269C3×D4—182no—
270C3×D4—182no—
271C3×D4—364no—
272C2×C3×C4—22no—
273C2×C3×C4—21no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—21no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—41no—
280C2×C3×C4—62no—
281C2×C3×C4—62no—
282C2×C3×C4—122no—
283C2×C3×C4—182no—
284C2×C3×C4C2×D3211yescharacteristic
285C2×C3×C4C22×C3×D312yes—
286#67—22no—
287#67—64no—
288D12—64no—
289D12—62no—
290D12—62no—
291D12—62no—
292D12—62no—
293D12—122no—
294D12—122no—
295D12—128no—
296D12—128no—
297D12C3×D1212yescomplemented
298C4×D3—22no—
299C4×D3—64no—
300Dic6C3×D1212yescomplemented
301C3×C8—62no—
302C3×C8—62no—
303C3×C8—122no—
304C3⋊C8—22no—
305C2×C32—21no—
306C2×C32—22no—
307C2×C32—22no—
308C2×C32—22no—
309C2×C32—21no—
310C2×C32—41no—
311C2×C32—42no—
312C2×C32—42no—
313C2×C32—122no—
314C2×C32#88212yes—
315C2×C32C3×#12011yescharacteristic
316C3⋊D3—42no—
317C3⋊D3—81no—
318C3×D3—42no—
319C3×D3—42no—
320C3×D3—82no—
321C3×D3—124no—
322C3×D3—242no—
323C3×D3—242no—
324C3×D3—242no—
325#41—32no—
326#41—182no—
327C2×D4—91no—
328C2×D4—91no—
329C2×D4—182no—
330C2×D4—182no—
331C2×D4—182no—
332SD16—182no—
333D8—182no—
334M16—32no—
335C42C3×D3211yescharacteristic, complemented
336C22×C3—21no—
337C22×C3—22no—
338C22×C3—41no—
339C22×C3—62no—
340C22×C3—62no—
341C22×C3—122no—
342C22×C3—182no—
343C22×C3—362no—
344C22×C3—362no—
345C22×C3—361no—
346C22×C3—722no—
347C22×C3#381111yescharacteristic
348C22×C3C3×D3×D412yes—
349C2×D3—22no—
350C2×D3—64no—
351C2×D3—122no—
352C2×D3—124no—
353C2×D3—124no—
354C2×D3—124no—
355C2×D3—124no—
356C2×D3—122no—
357C2×D3—122no—
358C2×D3—244no—
359C3×C4—24no—
360C3×C4—24no—
361C3×C4—22no—
362C3×C4—22no—
363C3×C4—22no—
364C3×C4—22no—
365C3×C4—44no—
366C3×C4—44no—
367C3×C4—42no—
368C3×C4—62no—
369C3×C4—62no—
370C3×C4—122no—
371C3×C4D3×D1212yes—
372C3×C4C2×C3×D1212yes—
373C3×C4C3×D3×D412yes—
374Dic3—22no—
375C32—22no—
376C32—21no—
377C32—41no—
378C32#482212yescomplemented
379C32C3×#45311yescharacteristic, complemented
380C23—92no—
381C23—182no—
382C23—181no—
383Q8—32no—
384D4—32no—
385D4—62no—
386D4—184no—
387D4—182no—
388D4—182no—
389D4—364no—
390C2×C4—21no—
391C2×C4—62no—
392C2×C4—182no—
393C2×C4C2×C3×D3211yescharacteristic
394C8—62no—
395C2×C3—22no—
396C2×C3—22no—
397C2×C3—21no—
398C2×C3—22no—
399C2×C3—21no—
400C2×C3—42no—
401C2×C3—42no—
402C2×C3—41no—
403C2×C3—122no—
404C2×C3—122no—
405C2×C3—242no—
406C2×C3—362no—
407C2×C3—721no—
408C2×C3#6392311yescharacteristic
409C2×C3C3×#88212yes—
410D3—42no—
411D3—124no—
412D3—242no—
413D3—242no—
414D3—242no—
415C22—62no—
416C22—182no—
417C22—362no—
418C22—362no—
419C22—361no—
420C22—722no—
421C22C3×#381111yescharacteristic
422C4—22no—
423C4—62no—
424C4C3×D3×D1212yes—
425C3—21no—
426C3—22no—
427C3—41no—
428C3#1059039711yescharacteristic, direct product factor
429C3C3×#482212yescomplemented
430C2—21no—
431C2—122no—
432C2—362no—
433C2—721no—
434C2C3×#6392311yescharacteristic
subgroupG/H
centerC2×C3#63923
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC2×C33#120
hypercenterC3×C42D32
hypocenterC32C3×#453
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
1. C3×#64308C2semidirect
2. C3×#64083C2semidirect
3. C3×#63825C2semidirect
4. C32×#805C2semidirect
5. C3×#63559C2semidirect
7. #10590397C3direct (N × H)
10. C2×C3×C4⋊(C3⋊D3)C22semidirect
15. C33×C42C22semidirect
16. C32×#208C22semidirect
20. #64308C2×C3semidirect
21. #64083C2×C3semidirect
26. #63825C2×C3semidirect
30. C3×#805D3semidirect
31. C3×#805C2×C3semidirect
32. #63559C2×C3semidirect
42. C32×D12D4semidirect
44. C32×Dic6D4semidirect
53. C2×C4⋊(C3⋊D3)C22×C3semidirect
75. C32×C42C2×D3semidirect
76. C32×C42C22×C3semidirect
80. C3×#208C2×D3semidirect
81. C3×#208C22×C3semidirect
109. #805C3×D3semidirect
149. C3×D12D12semidirect
150. C3×D12C3×D4semidirect
156. C3×Dic6D12semidirect
157. C3×Dic6C3×D4semidirect
199. C3×C42D32semidirect
200. C3×C42C2×C3×D3semidirect
203. #208C2×C3×D3semidirect
250. C33#453semidirect
297. D12C3×D12semidirect
300. Dic6C3×D12semidirect
335. C42C3×D32semidirect
378. C32#4822semidirect
379. C32C3×#453semidirect
428. C3#10590397direct (N × H)
429. C3C3×#4822semidirect

Maximal subgroups

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

subgroupindex
C3×#643082
C3×#640832
C3×#638252
C32×#8052
C3×#635592
C3×#48223
#105903973