Finite Group Databasebeta

← C3×#10590344 #13157358

Subgroups — C3×#10590344 #13157358

Order 1728.

786 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 156 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×C4×D12C212yesmaximal, complemented
2C2×C3×#3713C211yesmaximal, characteristic
3C3×#63630C211yesmaximal, characteristic, complemented
4C3×#63574C211yesmaximal, characteristic, complemented
5C3×#63562C212yesmaximal, complemented
6C3×#5014—32nomaximal
7#10590344C311yesmaximal, characteristic, direct product factor
8C2×C32×D12C2212yes—
9C2×C32×C4×D3—22no—
10C33×C42C2211yescharacteristic
11C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
12C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
13C32×(C2×D3)⋊C4—22no—
14C32×C4⋊Dic3C2212yes—
15C3×#3713C414yescomplemented
16C32×C4×D4—32no—
17C2×C3×#213—32no—
18C3×C4×D12—22no—
19C3×C4×D12D312yescomplemented
20C3×C4×D12C2×C312yescomplemented
21C2×#3713C2×C311yescharacteristic
22#63630C2×C311yescharacteristic, complemented
23C3×#832—32no—
24C3×#809—32no—
25C3×#808—32no—
26C3×#803—32no—
27C3×#803—61no—
28#63574C2×C311yescharacteristic, complemented
29#63562C2×C312yescomplemented
30C22×C32×D3—22no—
31C2×C33×C4D412yes—
32C2×C33×C4C2311yescharacteristic
33C2×C32×Dic3—22no—
34C32×D12—22no—
35C32×D12C2×C414yescomplemented
36C32×C4×D3—44no—
37C3×C3⋊(C3⋊C8)—21no—
38C3×C3⋊(C3⋊C8)C2×C412yescomplemented
39C3×#442—91no—
40#5014—32no—
41C2×C32×D4—32no—
42C22×C32×C4—62no—
43C2×C3×D12—22no—
44C2×C3×D12C2×D312yes—
45C2×C3×D12C22×C312yes—
46C2×C3×C4×D3—24no—
47C2×C3×C4×D3—22no—
48C2×C3×C4×D3—22no—
49C32×C4⋊C4—32no—
50C32×C22⋊C4—62no—
51C32×C42—21no—
52C32×C42—22no—
53C32×C42—41no—
54C32×C42C2×D312yes—
55C32×C42C22×C311yescharacteristic
56C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
57C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
58C3×#213—38no—
59C3×(C2×D3)⋊C4—24no—
60C3×(C2×D3)⋊C4—22no—
61C3×(C2×D3)⋊C4—22no—
62C3×C4⋊Dic3—22no—
63C3×C4⋊Dic3C2×D312yes—
64C3×C4⋊Dic3C22×C312yes—
65C2×C3×C3⋊C8—32no—
66C2×C3×C3⋊C8—32no—
67C2×C3×C3⋊C8—61no—
68C2×C3×C3⋊C8—61no—
69#3713C3×C414yescomplemented
70C22×C33C2×D411yescharacteristic
71C2×C32×D3—24no—
72C2×C32×D3—42no—
73C2×C32×D3—42no—
74C33×C4—22no—
75C33×C4C22×C411yescharacteristic
76C33×C4#4111yescharacteristic
77C32×Dic3—42no—
78C2×C3×D8—91no—
79C3×C4×D4—32no—
80C3×C4×D4—32no—
81C3×C4×D4—62no—
82C2×#213—32no—
83C4×D12C3×D312yescomplemented
84C3×#106—91no—
85C3×#102—92no—
86C3×#97—91no—
87#832—32no—
88#809—32no—
89#808—32no—
90#803—32no—
91#803—61no—
92C23×C32—62no—
93C22×C3×D3—24no—
94C22×C3×D3—22no—
95C22×C3×D3—22no—
96C32×D4—34no—
97C32×D4—62no—
98C2×C32×C4—22no—
99C2×C32×C4—22no—
100C2×C32×C4—22no—
101C2×C32×C4—21no—
102C2×C32×C4—22no—
103C2×C32×C4—42no—
104C2×C32×C4—41no—
105C2×C32×C4—62no—
106C2×C32×C4—124no—
107C2×C32×C4#6712yes—
108C2×C32×C4#6712yes—
109C2×C32×C4C3×D412yes—
110C2×C32×C4C22×D312yes—
111C2×C32×C4C23×C311yescharacteristic
112C2×C3×Dic3—24no—
113C2×C3×Dic3—22no—
114C2×C3×Dic3—22no—
115C3×D12—24no—
116C3×D12—22no—
117C3×D12—22no—
118C3×D12—24no—
119C3×D12C4×D314yescomplemented
120C3×D12C2×C3×C414yescomplemented
121C3×C4×D3—48no—
122C3×C4×D3—44no—
123C3×C4×D3—44no—
124C3⋊(C3⋊C8)—21no—
125C3⋊(C3⋊C8)C2×C3×C412yescomplemented
126C3×C3⋊C8—34no—
127C3×C3⋊C8—62no—
128C3×C3⋊C8—62no—
129C3×C3⋊C8—62no—
130#442—91noSylow
131C2×C33C4×D411yescharacteristic
132C2×C33AGL(1,ℤ/8)12yes—
133C32×D3—44no—
134C2×C3×D4—32no—
135C2×C3×D4—32no—
136C2×C3×D4—62no—
137C22×C3×C4—64no—
138C22×C3×C4—62no—
139C22×C3×C4—62no—
140C2×D12C2×C3×D312yes—
141C2×C4×D3—22no—
142C3×D8—94no—
143C2×C3×C8—91no—
144C2×C3×C8—91no—
145C3×C4⋊C4—32no—
146C3×C4⋊C4—32no—
147C3×C4⋊C4—62no—
148C3×C22⋊C4—64no—
149C3×C22⋊C4—62no—
150C3×C22⋊C4—62no—
151C3×C42—21no—
152C3×C42—22no—
153C3×C42—41no—
154C3×C42D3211yescharacteristic
155C3×C42C2×C3×D312yes—
156#213—38no—
157(C2×D3)⋊C4—22no—
158C4⋊Dic3C2×C3×D312yes—
159C2×C3⋊C8—32no—
160C2×C3⋊C8—32no—
161C2×C3⋊C8—61no—
162C2×C3⋊C8—61no—
163C22×C32—21no—
164C22×C32—22no—
165C22×C32—41no—
166C22×C32—64no—
167C22×C32—122no—
168C22×C32—122no—
169C22×C32C2×#6712yes—
170C22×C32C2×C3×D411yescharacteristic
171C2×C3×D3—28no—
172C2×C3×D3—24no—
173C2×C3×D3—24no—
174C2×C3×D3—44no—
175C2×C3×D3—44no—
176C2×C3×D3—42no—
177C2×C3×D3—42no—
178C2×C3×D3—42no—
179C2×C3×D3—42no—
180C32×C4—22no—
181C32×C4—24no—
182C32×C4—22no—
183C32×C4—22no—
184C32×C4—21no—
185C32×C4—21no—
186C32×C4—22no—
187C32×C4—22no—
188C32×C4—41no—
189C32×C4—41no—
190C32×C4—42no—
191C32×C4—42no—
192C32×C4—44no—
193C32×C4—122no—
194C32×C4C2×C4×D312yes—
195C32×C4#23512yes—
196C32×C4C22×C3×C411yescharacteristic
197C32×C4C3×#4111yescharacteristic
198C3×Dic3—44no—
199C3×Dic3—42no—
200C3×Dic3—42no—
201C2×D8—91no—
202C4×D4—32no—
203#106—91no—
204#102—92no—
205#97—91no—
206C33#44211yescharacteristic, complemented, Sylow
207C23×C3—64no—
208C23×C3—62no—
209C23×C3—62no—
210C22×D3—22no—
211C3×D4—34no—
212C3×D4—34no—
213C3×D4—64no—
214C3×D4—64no—
215C3×D4—62no—
216C3×D4—62no—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—22no—
220C2×C3×C4—22no—
221C2×C3×C4—21no—
222C2×C3×C4—42no—
223C2×C3×C4—41no—
224C2×C3×C4—64no—
225C2×C3×C4—62no—
226C2×C3×C4—62no—
227C2×C3×C4—128no—
228C2×C3×C4—124no—
229C2×C3×C4—124no—
230C2×C3×C4#62512yes—
231C2×C3×C4C3×#6712yes—
232C2×C3×C4C3×#6712yes—
233C2×C3×C4C2×D3211yescharacteristic
234C2×C3×C4C22×C3×D312yes—
235C2×Dic3—22no—
236D12—22no—
237D12C3×C4×D314yescomplemented
238C4×D3—44no—
239C3×C8—92no—
240C3×C8—181no—
241C3⋊C8—34no—
242C3⋊C8—62no—
243C3⋊C8—62no—
244C3⋊C8—62no—
245C2×C32—22no—
246C2×C32—22no—
247C2×C32—22no—
248C2×C32—22no—
249C2×C32—21no—
250C2×C32—41no—
251C2×C32—42no—
252C2×C32—124no—
253C2×C32C4×#6712yes—
254C2×C32#93212yes—
255C2×C32#94912yes—
256C2×C32C3×C4×D411yescharacteristic
257C2×C32C3×AGL(1,ℤ/8)12yes—
258C3×D3—48no—
259C3×D3—44no—
260C3×D3—44no—
261C2×D4—32no—
262C22×C4—62no—
263D8—94no—
264C2×C8—91no—
265C2×C8—91no—
266C4⋊C4—32no—
267C22⋊C4—62no—
268C42C3×D3211yescharacteristic
269C22×C3—22no—
270C22×C3—21no—
271C22×C3—41no—
272C22×C3—68no—
273C22×C3—64no—
274C22×C3—64no—
275C22×C3—124no—
276C22×C3—124no—
277C22×C3—122no—
278C22×C3—122no—
279C22×C3—122no—
280C22×C3—122no—
281C22×C3C2×#62511yescharacteristic
282C22×C3C2×C3×#6712yes—
283C2×D3—24no—
284C2×D3—42no—
285C2×D3—42no—
286C3×C4—24no—
287C3×C4—22no—
288C3×C4—22no—
289C3×C4—22no—
290C3×C4—21no—
291C3×C4—22no—
292C3×C4—21no—
293C3×C4—22no—
294C3×C4—42no—
295C3×C4—42no—
296C3×C4—44no—
297C3×C4—41no—
298C3×C4—41no—
299C3×C4—124no—
300C3×C4—122no—
301C3×C4—122no—
302C3×C4#379811yescharacteristic
303C3×C4C4×D3211yescharacteristic
304C3×C4C2×C3×C4×D312yes—
305C3×C4C3×#23512yes—
306Dic3—42no—
307C32—22no—
308C32—21no—
309C32—41no—
310C32#501412yescomplemented
311C32C3×#44211yescharacteristic, complemented
312C23—62no—
313D4—34no—
314D4—62no—
315C2×C4—62no—
316C2×C4—124no—
317C2×C4C3×#62512yes—
318C2×C4C2×C3×D3211yescharacteristic
319C8—92no—
320C8—181no—
321C2×C3—22no—
322C2×C3—22no—
323C2×C3—22no—
324C2×C3—22no—
325C2×C3—21no—
326C2×C3—42no—
327C2×C3—41no—
328C2×C3—128no—
329C2×C3—124no—
330C2×C3—124no—
331C2×C3#6382312yes—
332C2×C3C4×#62511yescharacteristic
333C2×C3C3×C4×#6712yes—
334C2×C3C3×#93212yes—
335C2×C3C3×#94912yes—
336D3—44no—
337C22—64no—
338C22—122no—
339C22—122no—
340C22C2×C3×#62511yescharacteristic
341C4—22no—
342C4—122no—
343C4C3×#379811yescharacteristic
344C4C3×C4×D3211yescharacteristic
345C3—22no—
346C3—21no—
347C3—41no—
348C3#1059034411yescharacteristic, direct product factor
349C3C3×#501412yescomplemented
350C2—124no—
351C2C3×#6382312yes—
352C2C3×C4×#62511yescharacteristic
subgroupG/H
centerC22×C3C2×#625
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC2×C3×#3713C2
cosocleC2×C32×C4C23×C3
hypercenterC3×C42D32
hypocenterC32C3×#442
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
1. C32×C4×D12C2semidirect
3. C3×#63630C2semidirect
4. C3×#63574C2semidirect
5. C3×#63562C2semidirect
7. #10590344C3direct (N × H)
15. C3×#3713C4semidirect
19. C3×C4×D12D3semidirect
20. C3×C4×D12C2×C3semidirect
22. #63630C2×C3semidirect
28. #63574C2×C3semidirect
29. #63562C2×C3semidirect
35. C32×D12C2×C4semidirect
38. C3×C3⋊(C3⋊C8)C2×C4semidirect
69. #3713C3×C4semidirect
83. C4×D12C3×D3semidirect
119. C3×D12C4×D3semidirect
120. C3×D12C2×C3×C4semidirect
125. C3⋊(C3⋊C8)C2×C3×C4semidirect
206. C33#442semidirect
237. D12C3×C4×D3semidirect
310. C32#5014semidirect
311. C32C3×#442semidirect
348. C3#10590344direct (N × H)
349. C3C3×#5014semidirect

Maximal subgroups

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

subgroupindex
C32×C4×D122
C2×C3×#37132
C3×#636302
C3×#635742
C3×#635622
C3×#50143
#105903443