Finite Group Databasebeta

← C2×C3×#63691 #13164421

Subgroups — C2×C3×#63691 #13164421

Order 1728.

575 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 575 conjugacy classes of subgroups — too many to draw. Only its 84 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
1C2×C3×#41⋊C9C211yesmaximal, characteristic, complemented
2C3×#63691C214yesmaximal, direct product factor
3C2×C3×#3689C211yesmaximal, characteristic, complemented
4C2×C3×#3688C211yesmaximal, characteristic, complemented
5C2×C3×#950—31nomaximal
6C2×#63691C311yesmaximal, characteristic, direct product factor
7C2×C3×C4×D9—41nomaximal
8C3×#41⋊C9C2212yescomplemented
9C2×C3×Q8⋊C9C2211yescharacteristic
10C3×#3689C2212yescomplemented
11C3×#3688C2212yescomplemented
12C2×C32×#41D311yescharacteristic
13C2×C3×#235—31no—
14C3×#950—34no—
15C3×#950—34no—
16C2×C3×#216—31no—
17C2×C3×#215—31no—
18C2×C3×#214—31no—
19C2×C3×#213—31no—
20C22×C3×C3⋊C8—31no—
21C2×#41⋊C9—21no—
22C2×#41⋊C9C2×C311yescharacteristic, complemented
23#63691C2×C314yesdirect product factor
24C2×#3689C2×C311yescharacteristic, complemented
25C2×#3688C2×C311yescharacteristic, complemented
26C22×C3×D9—41no—
27C2×C3×C4×C9—41no—
28C2×C3×Dic9—41no—
29C3×C4×D9—44no—
30C3×Q8⋊C9C2311yescharacteristic
31C2×C3×#135—91no—
32C2×#950—31no—
33C32×#41—32no—
34C32×#41C2×D312yes—
35C2×C32×Q8C2×D311yescharacteristic
36C2×C32×D4—31no—
37C22×C32×C4—31no—
38C2×C3×#67—61no—
39C3×#235—34no—
40C3×#235—62no—
41C2×C3×D12—31no—
42C2×C3×C4×D3—61no—
43C2×C3×Dic6—31no—
44C3×#216—32no—
45C3×#216—32no—
46C3×#215—32no—
47C3×#215—32no—
48C3×#214—34no—
49C3×#213—34no—
50C2×C3×C3⋊C8—31no—
51C2×C3×C3⋊C8—31no—
52C2×C3×C3⋊C8—34no—
53C2×C4×D9—41no—
54#41⋊C9—22no—
55#41⋊C9C22×C312yescomplemented
56C2×Q8⋊C9—21no—
57C2×Q8⋊C9C22×C311yescharacteristic
58#3689C22×C312yescomplemented
59#3688C22×C312yescomplemented
60C22×C3×C9—41no—
61C2×C3×D9—42no—
62C2×C3×D9—44no—
63C3×C4×C9—42no—
64C3×Dic9—42no—
65C2×C3×#41—21no—
66C2×C3×#41—91no—
67C2×C3×#41D911yescharacteristic, complemented
68C2×C3×#41C3×D311yescharacteristic
69C2×#235—31no—
70C3×#135—94no—
71C3×#135—94no—
72C2×C3×Q16—91no—
73C2×C3×SD16—91no—
74C2×C3×SD16—91no—
75C2×C3×D8—91no—
76C22×C3×C8—91no—
77#950—34no—
78#950—34no—
79C2×#216—31no—
80C2×#215—31no—
81C2×#214—31no—
82C2×#213—31no—
83C22×C3⋊C8—31no—
84C23×C32—31no—
85C22×C3×D3—61no—
86C32×Q8—31no—
87C32×Q8C22×D311yescharacteristic
88C32×D4—32no—
89C32×D4—61no—
90C2×C32×C4—31no—
91C2×C32×C4—32no—
92C2×C32×C4—32no—
93C2×C32×C4S411yescharacteristic
94C3×#67—64no—
95C2×C3×Dic3—61no—
96C3×D12—32no—
97C3×D12—61no—
98C3×C4×D3—64no—
99C3×Dic6—32no—
100C3×Dic6—61no—
101C22×D9—41no—
102C3×C3⋊C8—32no—
103C3×C3⋊C8—32no—
104C2×C4×C9—41no—
105C2×C4×C9—81no—
106C2×Dic9—41no—
107C4×D9—44no—
108Q8⋊C9—21no—
109Q8⋊C9C23×C311yescharacteristic
110C2×#135—91noSylow
111C2×C3×C9—42no—
112C2×C3×C9—41no—
113C3×D9—44no—
114C3×#41—22no—
115C3×#41—32no—
116C3×#41—32no—
117C3×#41—62no—
118C3×#41—94no—
119C3×#41—182no—
120C3×#41C2×D912yescomplemented
121C3×#41C2×C3×D312yes—
122C2×C3×Q8—21no—
123C2×C3×Q8—91no—
124C2×C3×Q8C2×D911yescharacteristic
125C2×C3×Q8C2×C3×D311yescharacteristic
126C2×C3×D4—31no—
127C2×C3×D4—31no—
128C2×C3×D4—61no—
129C2×C3×D4—91no—
130C2×C3×D4—181no—
131C22×C3×C4—31no—
132C22×C3×C4—31no—
133C22×C3×C4—61no—
134C22×C3×C4—181no—
135C2×#67—61no—
136#235—34no—
137#235—62no—
138C2×D12—31no—
139C2×C4×D3—61no—
140C2×Dic6—31no—
141C3×Q16—92no—
142C3×Q16—92no—
143C3×SD16—94no—
144C3×SD16—92no—
145C3×SD16—92no—
146C3×D8—94no—
147C2×C3×C8—91no—
148C2×C3×C8—91no—
149C2×C3×C8—94no—
150#216—32no—
151#216—32no—
152#215—32no—
153#215—32no—
154#214—34no—
155#213—34no—
156C2×C3⋊C8—31no—
157C2×C3⋊C8—31no—
158C2×C3⋊C8—34no—
159C22×C32—32no—
160C22×C32—62no—
161C22×C32C2×S411yescharacteristic
162C2×C3×D3—62no—
163C2×C3×D3—122no—
164C32×C4—31no—
165C32×C4—31no—
166C32×C4C2×S412yes—
167C3×Dic3—62no—
168C22×C9—41no—
169C22×C9—81no—
170C2×D9—42no—
171C2×D9—44no—
172C4×C9—42no—
173C4×C9—82no—
174Dic9—42no—
175C2×#41—91no—
176C2×#41C3×D911yescharacteristic, complemented
177#135—94no—
178#135—94no—
179C2×Q16—91no—
180C2×SD16—91no—
181C2×SD16—91no—
182C2×D8—91no—
183C22×C8—91no—
184C3×C9—41noSylow
185C23×C3—31no—
186C23×C3—31no—
187C23×C3—61no—
188C23×C3—181no—
189C22×D3—61no—
190C3×Q8—21no—
191C3×Q8—31no—
192C3×Q8—31no—
193C3×Q8—61no—
194C3×Q8—92no—
195C3×Q8—181no—
196C3×Q8C22×D911yescharacteristic
197C3×Q8C22×C3×D311yescharacteristic
198C3×D4—32no—
199C3×D4—32no—
200C3×D4—62no—
201C3×D4—61no—
202C3×D4—61no—
203C3×D4—62no—
204C3×D4—92no—
205C3×D4—181no—
206C3×D4—184no—
207C2×C3×C4—21no—
208C2×C3×C4—31no—
209C2×C3×C4—31no—
210C2×C3×C4—32no—
211C2×C3×C4—32no—
212C2×C3×C4—32no—
213C2×C3×C4—32no—
214C2×C3×C4—62no—
215C2×C3×C4—62no—
216C2×C3×C4—61no—
217C2×C3×C4—181no—
218C2×C3×C4—184no—
219C2×C3×C4(C22⋊C9)⋊C211yescharacteristic
220C2×C3×C4C3×S411yescharacteristic
221#67—64no—
222C2×Dic3—61no—
223D12—32no—
224D12—61no—
225C4×D3—64no—
226Dic6—32no—
227Dic6—61no—
228C3×C8—92no—
229C3×C8—92no—
230C3⋊C8—32no—
231C3⋊C8—32no—
232C2×C32—62no—
233C2×C32#98512yes—
234C2×C32C22×S411yescharacteristic
235C3×D3—122no—
236C2×C9—41no—
237C2×C9—42no—
238C2×C9—81no—
239C2×C9—82no—
240D9—44no—
241#41—32no—
242#41—94no—
243#41—182no—
244#41C2×C3×D912yescomplemented
245C2×Q8—91no—
246C2×Q8C2×C3×D911yescharacteristic
247C2×D4—31no—
248C2×D4—91no—
249C2×D4—181no—
250C22×C4—31no—
251C22×C4—181no—
252Q16—92no—
253Q16—92no—
254SD16—94no—
255SD16—92no—
256SD16—92no—
257D8—94no—
258C2×C8—91no—
259C2×C8—91no—
260C2×C8—94no—
261C22×C3—21no—
262C22×C3—32no—
263C22×C3—32no—
264C22×C3—62no—
265C22×C3—62no—
266C22×C3—62no—
267C22×C3—122no—
268C22×C3—182no—
269C22×C3—362no—
270C22×C3C2×(C22⋊C9)⋊C211yescharacteristic
271C22×C3C2×C3×S411yescharacteristic
272C2×D3—62no—
273C2×D3—122no—
274C3×C4—22no—
275C3×C4—31no—
276C3×C4—31no—
277C3×C4—31no—
278C3×C4—31no—
279C3×C4—61no—
280C3×C4—61no—
281C3×C4—182no—
282C3×C4C2×(C22⋊C9)⋊C212yes—
283C3×C4C2×C3×S412yes—
284Dic3—62no—
285C32C2×#98511yescharacteristic
286C9—41no—
287C9—81no—
288C23—31no—
289C23—181no—
290Q8—31no—
291Q8—92no—
292Q8—181no—
293Q8C22×C3×D911yescharacteristic
294D4—32no—
295D4—61no—
296D4—92no—
297D4—181no—
298D4—184no—
299C2×C4—31no—
300C2×C4—32no—
301C2×C4—32no—
302C2×C4—181no—
303C2×C4—184no—
304C2×C4C3×(C22⋊C9)⋊C211yescharacteristic
305C8—92no—
306C8—92no—
307C2×C3—21no—
308C2×C3—22no—
309C2×C3—62no—
310C2×C3—62no—
311C2×C3—122no—
312C2×C3—362no—
313C2×C3#6369112yesdirect product factor
314C2×C3C22×(C22⋊C9)⋊C211yescharacteristic
315C2×C3C3×#98512yes—
316C2×C3C22×C3×S411yescharacteristic
317D3—122no—
318C22—32no—
319C22—62no—
320C22—182no—
321C22—362no—
322C22C2×C3×(C22⋊C9)⋊C211yescharacteristic
323C4—31no—
324C4—31no—
325C4—182no—
326C4C2×C3×(C22⋊C9)⋊C212yes—
327C3—21no—
328C3C2×#6369111yescharacteristic, direct product factor
329C3C2×C3×#98511yescharacteristic
330C2—62no—
331C2—362no—
332C2C3×#6369112yesdirect product factor
333C2C22×C3×(C22⋊C9)⋊C211yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4(C22⋊C9)⋊C2
commutator, cosocle, hypocenterQ8⋊C9C23×C3
FrattiniC2×C3C22×C3×S4
FittingC2×C32×#41D3
socleC22×C32C2×S4
LCSG ▷ Q8⋊C9
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#41⋊C9C2semidirect
2. C3×#63691C2direct (N × H)
3. C2×C3×#3689C2semidirect
4. C2×C3×#3688C2semidirect
6. C2×#63691C3direct (N × H)
8. C3×#41⋊C9C22semidirect
10. C3×#3689C22semidirect
11. C3×#3688C22semidirect
22. C2×#41⋊C9C2×C3semidirect
23. #63691C2×C3direct (N × H)
24. C2×#3689C2×C3semidirect
25. C2×#3688C2×C3semidirect
55. #41⋊C9C22×C3semidirect
58. #3689C22×C3semidirect
59. #3688C22×C3semidirect
67. C2×C3×#41D9semidirect
120. C3×#41C2×D9semidirect
176. C2×#41C3×D9semidirect
244. #41C2×C3×D9semidirect
313. C2×C3#63691direct (N × H)
328. C3C2×#63691direct (N × H)
332. C2C3×#63691direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#41⋊C92
C3×#636912
C2×C3×#36892
C2×C3×#36882
C2×C3×#9503
C2×#636913
C2×C3×C4×D94