Finite Group Databasebeta

← C3×#10590988 #13157997

Subgroups — C3×#10590988 #13157997

Order 1728.

606 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 104 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×#64116C211yesmaximal, characteristic, complemented
2C3×#63830C212yesmaximal, complemented
3C2×C3×#3713C211yesmaximal, characteristic, complemented
4C32×#823C212yesmaximal, complemented
5C3×#63578C211yesmaximal, characteristic, complemented
6C3×#4749—32nomaximal
7#10590988C311yesmaximal, characteristic, direct product factor
8C3×#3828C2211yescharacteristic
9C2×C32×D12C2212yes—
10C33×M16C2211yescharacteristic
11C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
12C3×#3743—21no—
13C3×C8×C3⋊D3—21no—
14C32×#208C2212yes—
15C3×#3716—22no—
16C3×#3714—22no—
17C3×#3713—21no—
18C3×#3713—21no—
19#64116C2×C311yescharacteristic, complemented
20C3×#932—32no—
21C2×C3×#213—32no—
22C3×#908—32no—
23C3×#907—32no—
24C3×#907—61no—
25#63830C2×C312yescomplemented
26C2×#3713C2×C311yescharacteristic, complemented
27C32×#100—32no—
28C3×#823—22no—
29C3×#823D312yescomplemented
30C3×#823C2×C312yescomplemented
31C3×#822—32no—
32#63578C2×C311yescharacteristic, complemented
33C22×C32×D3—22no—
34C2×C33×C4C2311yescharacteristic
35C3×#638—21no—
36C3×C4⋊(C3⋊D3)D411yescharacteristic
37C3×C4×C3⋊D3—21no—
38C3×#634D411yescharacteristic
39C32×D12—22no—
40C32×D12—22no—
41C33×C8—21no—
42C3×C3⋊(C3⋊C8)D411yescharacteristic
43C3×C3⋊(C3⋊C8)D411yescharacteristic
44C32×C3⋊C8—22no—
45C3×#472—91no—
46#4749—32no—
47C2×C32×D4—32no—
48#3828C22×C311yescharacteristic
49C3×#235—32no—
50C3×#235—61no—
51C2×C3×D12—22no—
52C2×C3×D12C2×D312yes—
53C2×C3×D12C22×C312yes—
54C32×M16—21no—
55C32×M16—22no—
56C32×M16—32no—
57C32×M16—41no—
58C32×M16C2×D312yes—
59C32×M16C22×C311yescharacteristic
60C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
61#3743—21no—
62C8×C3⋊D3—21no—
63C3×#214—62no—
64C3×#213—62no—
65C3×#213—62no—
66C3×#213—62no—
67C3×#208—22no—
68C3×#208C2×D312yes—
69C3×#208C22×C312yes—
70C2×C3×C3⋊C8—32no—
71C2×C3×C3⋊C8—61no—
72C3×D24—62no—
73C3×#204—62no—
74C3×ΓL(1,𝔽25)—62no—
75C3×ΓL(1,𝔽25)—62no—
76C3×D3×C8—62no—
77C3×D3×C8—62no—
78#3716—22no—
79#3714—22no—
80#3713—21no—
81#3713—21no—
82C22×C33#4111yescharacteristic
83C2×C3×C3⋊D3—21no—
84C2×C32×D3—42no—
85C2×C32×D3—82no—
86C33×C4C2×D411yescharacteristic
87C33×C4C2×D411yescharacteristic
88C3×C3⋊Dic3—21no—
89C3×AGL(1,ℤ/8)—92no—
90C2×C3×D8—91no—
91C3×#131—91no—
92#932—32no—
93C2×#213—32no—
94#908—32no—
95#907—32no—
96#907—61no—
97C3×#108—91no—
98C3×#100—32no—
99C3×#100—32no—
100C3×#100—62no—
101#823C3×D312yescomplemented
102#822—32no—
103C23×C32—62no—
104C22×C3×D3—24no—
105C22×C3×D3—22no—
106C22×C3×D3—22no—
107C32×D4—62no—
108C32×D4—62no—
109C2×C32×C4—21no—
110C2×C32×C4—22no—
111C2×C32×C4—41no—
112C2×C32×C4C22×D312yes—
113C2×C32×C4C23×C311yescharacteristic
114#638—21no—
115C4⋊(C3⋊D3)C3×D411yescharacteristic
116C4×C3⋊D3—21no—
117#634C3×D411yescharacteristic
118C3×#67—62no—
119C3×#67—62no—
120C3×D12—22no—
121C3×D12—22no—
122C3×D12—22no—
123C3×D12—22no—
124C3×D12—32no—
125C3×D12—42no—
126C3×D12—42no—
127C3×D12—61no—
128C3×C4×D3—62no—
129C3×C4×D3—62no—
130C3×Dic6—32no—
131C3×Dic6—61no—
132C32×C8—21no—
133C32×C8—22no—
134C32×C8—22no—
135C32×C8—42no—
136C32×C8—42no—
137C32×C8—62no—
138C3⋊(C3⋊C8)C3×D411yescharacteristic
139C3⋊(C3⋊C8)C3×D411yescharacteristic
140C3×C3⋊C8—22no—
141C3×C3⋊C8—22no—
142C3×C3⋊C8—32no—
143C3×C3⋊C8—32no—
144C3×C3⋊C8—42no—
145C3×C3⋊C8—61no—
146C3×C3⋊C8—61no—
147#472—91noSylow
148C2×C33—21no—
149C2×C33#12111yescharacteristic
150C3×C3⋊D3—41no—
151C32×D3—82no—
152C3×#41—91no—
153C2×C3×D4—32no—
154C2×C3×D4—32no—
155C2×C3×D4—62no—
156#235—32no—
157#235—61no—
158C2×D12C2×C3×D312yes—
159C3×SD16—182no—
160C3×D8—182no—
161C3×D8—181no—
162C3×D8—181no—
163C3×M16—21no—
164C3×M16—22no—
165C3×M16—32no—
166C3×M16—32no—
167C3×M16—41no—
168C3×M16—62no—
169C3×M16—181no—
170C3×M16D3211yescharacteristic
171C3×M16C2×C3×D312yes—
172C2×C3×C8—91no—
173C2×C3×C8—181no—
174#214—62no—
175#213—62no—
176#213—62no—
177#213—62no—
178#208C2×C3×D312yes—
179C2×C3⋊C8—32no—
180C2×C3⋊C8—61no—
181D24—62no—
182#204—62no—
183ΓL(1,𝔽25)—62no—
184ΓL(1,𝔽25)—62no—
185D3×C8—62no—
186D3×C8—62no—
187C22×C32—21no—
188C22×C32—22no—
189C22×C32—41no—
190C22×C32—122no—
191C22×C32—242no—
192C22×C32#23512yes—
193C22×C32C3×#4111yescharacteristic
194C2×C3⋊D3—21no—
195C2×C3×D3—44no—
196C2×C3×D3—42no—
197C2×C3×D3—42no—
198C2×C3×D3—62no—
199C2×C3×D3—62no—
200C2×C3×D3—82no—
201C2×C3×D3—82no—
202C2×C3×D3—84no—
203C32×C4—21no—
204C32×C4—21no—
205C32×C4—22no—
206C32×C4—22no—
207C32×C4—41no—
208C32×C4—41no—
209C32×C4D3×D412yes—
210C32×C4D3×D412yes—
211C32×C4C2×C3×D411yescharacteristic
212C32×C4C2×C3×D411yescharacteristic
213C3⋊Dic3—21no—
214C3×Dic3—62no—
215C3×Dic3—62no—
216AGL(1,ℤ/8)—92no—
217C2×D8—91no—
218#131—91no—
219#108—91no—
220#100—32no—
221C33#47211yescharacteristic, complemented, Sylow
222C23×C3—64no—
223C23×C3—62no—
224C23×C3—62no—
225C22×D3—22no—
226C3×Q8—91no—
227C3×D4—62no—
228C3×D4—62no—
229C3×D4—62no—
230C3×D4—62no—
231C3×D4—91no—
232C3×D4—122no—
233C3×D4—122no—
234C3×D4—181no—
235C2×C3×C4—22no—
236C2×C3×C4—21no—
237C2×C3×C4—41no—
238C2×C3×C4—181no—
239C2×C3×C4C2×D3211yescharacteristic
240C2×C3×C4C22×C3×D312yes—
241#67—62no—
242#67—62no—
243D12—22no—
244D12—22no—
245D12—32no—
246D12—61no—
247C4×D3—62no—
248C4×D3—62no—
249Dic6—32no—
250Dic6—61no—
251C3×C8—22no—
252C3×C8—21no—
253C3×C8—22no—
254C3×C8—42no—
255C3×C8—42no—
256C3×C8—62no—
257C3×C8—62no—
258C3×C8—91no—
259C3×C8—91no—
260C3×C8—122no—
261C3⋊C8—22no—
262C3⋊C8—32no—
263C3⋊C8—32no—
264C3⋊C8—61no—
265C3⋊C8—61no—
266C2×C32—22no—
267C2×C32—21no—
268C2×C32—22no—
269C2×C32—21no—
270C2×C32—41no—
271C2×C32—42no—
272C2×C32—41no—
273C2×C32—81no—
274C2×C32—242no—
275C2×C32#88412yes—
276C2×C32C3×#12111yescharacteristic
277C3⋊D3—41no—
278C3×D3—84no—
279C3×D3—82no—
280C3×D3—82no—
281C3×D3—122no—
282C3×D3—122no—
283#41—91no—
284C2×D4—32no—
285SD16—182no—
286D8—182no—
287D8—181no—
288D8—181no—
289M16—32no—
290M16—181no—
291M16C3×D3211yescharacteristic
292C2×C8—91no—
293C2×C8—181no—
294C22×C3—21no—
295C22×C3—22no—
296C22×C3—41no—
297C22×C3—124no—
298C22×C3—122no—
299C22×C3—122no—
300C22×C3—181no—
301C22×C3—242no—
302C22×C3—242no—
303C22×C3—244no—
304C22×C3#379811yescharacteristic
305C22×C3C3×#23512yes—
306C2×D3—42no—
307C2×D3—62no—
308C2×D3—62no—
309C2×D3—82no—
310C3×C4—22no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—22no—
314C3×C4—41no—
315C3×C4—41no—
316C3×C4—181no—
317C3×C4#380211yescharacteristic
318C3×C4#381111yescharacteristic
319C3×C4C3×D3×D412yes—
320C3×C4C3×D3×D412yes—
321Dic3—62no—
322Dic3—62no—
323C32—21no—
324C32—22no—
325C32—41no—
326C32#474912yescomplemented
327C32C3×#47211yescharacteristic, complemented
328C23—62no—
329Q8—91no—
330D4—62no—
331D4—62no—
332D4—91no—
333D4—181no—
334C2×C4—181no—
335C2×C4C2×C3×D3211yescharacteristic
336C8—21no—
337C8—62no—
338C8—91no—
339C8—91no—
340C2×C3—21no—
341C2×C3—22no—
342C2×C3—21no—
343C2×C3—22no—
344C2×C3—42no—
345C2×C3—41no—
346C2×C3—41no—
347C2×C3—81no—
348C2×C3—244no—
349C2×C3—242no—
350C2×C3—242no—
351C2×C3—361no—
352C2×C3#6391211yescharacteristic
353C2×C3C3×#88412yes—
354D3—82no—
355D3—122no—
356D3—122no—
357C22—122no—
358C22—181no—
359C22—242no—
360C22C3×#379811yescharacteristic
361C4—181no—
362C4C3×#380211yescharacteristic
363C4C3×#381111yescharacteristic
364C3—21no—
365C3—22no—
366C3—41no—
367C3#1059098811yescharacteristic, direct product factor
368C3C3×#474912yescomplemented
369C2—21no—
370C2—242no—
371C2—361no—
372C2C3×#6391211yescharacteristic
subgroupG/H
centerC2×C3#63912
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×M16C22
socleC2×C33#121
hypercenterC3×M16D32
hypocenterC32C3×#472
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×M16

Complements

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

NH ≅ G/Ntype
1. C3×#64116C2semidirect
2. C3×#63830C2semidirect
3. C2×C3×#3713C2semidirect
4. C32×#823C2semidirect
5. C3×#63578C2semidirect
7. #10590988C3direct (N × H)
19. #64116C2×C3semidirect
25. #63830C2×C3semidirect
26. C2×#3713C2×C3semidirect
29. C3×#823D3semidirect
30. C3×#823C2×C3semidirect
32. #63578C2×C3semidirect
101. #823C3×D3semidirect
221. C33#472semidirect
326. C32#4749semidirect
327. C32C3×#472semidirect
367. C3#10590988direct (N × H)
368. C3C3×#4749semidirect

Maximal subgroups

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

subgroupindex
C3×#641162
C3×#638302
C2×C3×#37132
C32×#8232
C3×#635782
C3×#47493
#105909883