Finite Group Databasebeta

← C2×C3×#63689 #13164408

Subgroups — C2×C3×#63689 #13164408

Order 1728.

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

normal subgroups only

This group has 587 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
1C22×C3×Q8⋊C9C211yesmaximal, characteristic, complemented
2C3×#63689C214yesmaximal, direct product factor
3C2×C3×#3689C211yesmaximal, characteristic, complemented
4C2×C3×#3688C211yesmaximal, characteristic, complemented
5C2×C3×#942—31nomaximal
6C2×#63689C311yesmaximal, characteristic, direct product factor
7C2×C3×#611—41nomaximal
8C2×C3×Q8⋊C9C2211yescharacteristic
9C2×C3×Q8⋊C9C2212yescomplemented
10C3×#3689C2212yescomplemented
11C3×#3688C2212yescomplemented
12C22×C32×Q8D311yescharacteristic
13C2×C3×#235—31no—
14C2×C3×#216—31no—
15C2×C3×#216—31no—
16C3×#942—34no—
17C3×#942—34no—
18C2×C3×#215—31no—
19C2×C3×#215—31no—
20C2×C3×#208—31no—
21C22×Q8⋊C9—21no—
22C22×Q8⋊C9C2×C311yescharacteristic, complemented
23#63689C2×C314yesdirect product factor
24C2×#3689C2×C311yescharacteristic, complemented
25C2×#3688C2×C311yescharacteristic, complemented
26C23×C3×C9—41no—
27C22×C3×D9—41no—
28C3×#611—44no—
29C2×C3×Dic9—41no—
30C3×Q8⋊C9C2311yescharacteristic
31C2×C3×#137—91no—
32C2×#942—31no—
33C2×C32×Q8—31no—
34C2×C32×Q8—32no—
35C2×C32×Q8C2×D311yescharacteristic
36C2×C32×Q8C2×D312yes—
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—34no—
46C3×#216—32no—
47C3×#215—34no—
48C3×#215—32no—
49C3×#215—32no—
50C3×#208—34no—
51C2×C3×C3⋊C8—31no—
52C2×C3×C3⋊C8—31no—
53C2×#611—41no—
54C2×Q8⋊C9—21no—
55C2×Q8⋊C9—22no—
56C2×Q8⋊C9C22×C311yescharacteristic
57C2×Q8⋊C9C22×C312yescomplemented
58#3689C22×C312yescomplemented
59#3688C22×C312yescomplemented
60C22×C3×C9—41no—
61C22×C3×C9—42no—
62C22×C3×C9—82no—
63C2×C3×D9—42no—
64C2×C3×D9—82no—
65C3×Dic9—42no—
66C2×C3×#41—91no—
67C22×C3×Q8—21no—
68C22×C3×Q8D911yescharacteristic, complemented
69C22×C3×Q8C3×D311yescharacteristic
70C2×#235—31no—
71C3×#137—94no—
72C3×#137—94no—
73C2×C3×Q16—91no—
74C2×C3×Q16—91no—
75C2×C3×SD16—91no—
76C2×C3×SD16—91no—
77C2×C3×M16—91no—
78C2×#216—31no—
79C2×#216—31no—
80#942—34no—
81#942—34no—
82C2×#215—31no—
83C2×#215—31no—
84C2×#208—31no—
85C23×C32S411yescharacteristic
86C22×C3×D3—61no—
87C32×Q8—31no—
88C32×Q8—32no—
89C32×Q8—61no—
90C32×Q8C22×D311yescharacteristic
91C2×C32×C4—31no—
92C2×C32×C4—32no—
93C2×C32×C4—31no—
94C2×C32×C4—32no—
95C3×#67—64no—
96C2×C3×Dic3—61no—
97C3×D12—32no—
98C3×D12—61no—
99C3×C4×D3—64no—
100C3×Dic6—32no—
101C3×Dic6—61no—
102C23×C9—41no—
103C23×C9—81no—
104C22×D9—41no—
105C3×C3⋊C8—32no—
106C3×C3⋊C8—32no—
107#611—44no—
108C2×Dic9—41no—
109Q8⋊C9—21no—
110Q8⋊C9C23×C311yescharacteristic
111C2×#137—91noSylow
112C2×C3×C9—42no—
113C2×C3×C9—41no—
114C2×C3×C9—82no—
115C3×D9—82no—
116C3×#41—94no—
117C3×#41—182no—
118C2×C3×Q8—21no—
119C2×C3×Q8—22no—
120C2×C3×Q8—31no—
121C2×C3×Q8—32no—
122C2×C3×Q8—31no—
123C2×C3×Q8—32no—
124C2×C3×Q8—61no—
125C2×C3×Q8—62no—
126C2×C3×Q8—91no—
127C2×C3×Q8C2×D911yescharacteristic
128C2×C3×Q8C2×D912yescomplemented
129C2×C3×Q8C2×C3×D311yescharacteristic
130C2×C3×Q8C2×C3×D312yes—
131C2×C3×D4—91no—
132C2×C3×D4—181no—
133C22×C3×C4—31no—
134C22×C3×C4—31no—
135C22×C3×C4—61no—
136C22×C3×C4—181no—
137C2×#67—61no—
138#235—34no—
139#235—62no—
140C2×D12—31no—
141C2×C4×D3—61no—
142C2×Dic6—31no—
143C3×Q16—92no—
144C3×Q16—94no—
145C3×Q16—92no—
146C3×SD16—92no—
147C3×SD16—94no—
148C3×SD16—92no—
149C3×M16—94no—
150C2×C3×C8—91no—
151C2×C3×C8—91no—
152#216—32no—
153#216—34no—
154#216—32no—
155#215—34no—
156#215—32no—
157#215—32no—
158#208—34no—
159C2×C3⋊C8—31no—
160C2×C3⋊C8—31no—
161C22×C32—22no—
162C22×C32C2×S412yes—
163C22×C32C2×S411yescharacteristic
164C2×C3×D3—62no—
165C2×C3×D3—122no—
166C32×C4—31no—
167C32×C4—31no—
168C32×C4—32no—
169C3×Dic3—62no—
170C22×C9—41no—
171C22×C9—42no—
172C22×C9—82no—
173C22×C9—84no—
174C22×C9—82no—
175C22×C9—81no—
176C2×D9—42no—
177C2×D9—82no—
178Dic9—42no—
179C2×#41—91no—
180C22×Q8C3×D911yescharacteristic, complemented
181#137—94no—
182#137—94no—
183C2×Q16—91no—
184C2×Q16—91no—
185C2×SD16—91no—
186C2×SD16—91no—
187C2×M16—91no—
188C3×C9—41noSylow
189C23×C3—21no—
190C23×C3—181no—
191C23×C3(C22⋊C9)⋊C211yescharacteristic
192C23×C3C3×S411yescharacteristic
193C22×D3—61no—
194C3×Q8—21no—
195C3×Q8—31no—
196C3×Q8—32no—
197C3×Q8—31no—
198C3×Q8—32no—
199C3×Q8—61no—
200C3×Q8—62no—
201C3×Q8—61no—
202C3×Q8—61no—
203C3×Q8—62no—
204C3×Q8—92no—
205C3×Q8—181no—
206C3×Q8C22×D911yescharacteristic
207C3×Q8C22×C3×D311yescharacteristic
208C3×D4—92no—
209C3×D4—181no—
210C3×D4—184no—
211C2×C3×C4—31no—
212C2×C3×C4—32no—
213C2×C3×C4—31no—
214C2×C3×C4—32no—
215C2×C3×C4—32no—
216C2×C3×C4—32no—
217C2×C3×C4—31no—
218C2×C3×C4—31no—
219C2×C3×C4—61no—
220C2×C3×C4—62no—
221C2×C3×C4—61no—
222C2×C3×C4—62no—
223C2×C3×C4—184no—
224C2×C3×C4—181no—
225#67—64no—
226C2×Dic3—61no—
227D12—32no—
228D12—61no—
229C4×D3—64no—
230Dic6—32no—
231Dic6—61no—
232C3×C8—92no—
233C3×C8—92no—
234C3⋊C8—32no—
235C3⋊C8—32no—
236C2×C32—22no—
237C2×C32#98312yes—
238C2×C32C22×S411yescharacteristic
239C3×D3—122no—
240C2×C9—41no—
241C2×C9—42no—
242C2×C9—82no—
243C2×C9—84no—
244C2×C9—81no—
245C2×C9—82no—
246D9—82no—
247#41—94no—
248#41—182no—
249C2×Q8—31no—
250C2×Q8—32no—
251C2×Q8—91no—
252C2×Q8C2×C3×D911yescharacteristic
253C2×Q8C2×C3×D912yescomplemented
254C2×D4—91no—
255C2×D4—181no—
256C22×C4—31no—
257C22×C4—181no—
258Q16—92no—
259Q16—94no—
260Q16—92no—
261SD16—92no—
262SD16—94no—
263SD16—92no—
264M16—94no—
265C2×C8—91no—
266C2×C8—91no—
267C22×C3—24no—
268C22×C3—22no—
269C22×C3—22no—
270C22×C3—22no—
271C22×C3—21no—
272C22×C3—182no—
273C22×C3—362no—
274C22×C3C2×(C22⋊C9)⋊C212yes—
275C22×C3C2×(C22⋊C9)⋊C211yescharacteristic
276C22×C3C2×C3×S412yes—
277C22×C3C2×C3×S411yescharacteristic
278C2×D3—62no—
279C2×D3—122no—
280C3×C4—31no—
281C3×C4—31no—
282C3×C4—32no—
283C3×C4—31no—
284C3×C4—32no—
285C3×C4—31no—
286C3×C4—61no—
287C3×C4—62no—
288C3×C4—61no—
289C3×C4—182no—
290Dic3—62no—
291C32C2×#98311yescharacteristic
292C9—41no—
293C9—81no—
294C23—181no—
295C23C3×(C22⋊C9)⋊C211yescharacteristic
296Q8—31no—
297Q8—32no—
298Q8—61no—
299Q8—92no—
300Q8—181no—
301Q8C22×C3×D911yescharacteristic
302D4—92no—
303D4—181no—
304D4—184no—
305C2×C4—31no—
306C2×C4—32no—
307C2×C4—31no—
308C2×C4—32no—
309C2×C4—184no—
310C2×C4—181no—
311C8—92no—
312C8—92no—
313C2×C3—24no—
314C2×C3—22no—
315C2×C3—22no—
316C2×C3—21no—
317C2×C3—22no—
318C2×C3—362no—
319C2×C3#6368912yesdirect product factor
320C2×C3C22×(C22⋊C9)⋊C211yescharacteristic
321C2×C3C3×#98312yes—
322C2×C3C22×C3×S411yescharacteristic
323D3—122no—
324C22—22no—
325C22—182no—
326C22—362no—
327C22C2×C3×(C22⋊C9)⋊C212yes—
328C22C2×C3×(C22⋊C9)⋊C211yescharacteristic
329C4—31no—
330C4—32no—
331C4—31no—
332C4—182no—
333C3—21no—
334C3C2×#6368911yescharacteristic, direct product factor
335C3C2×C3×#98311yescharacteristic
336C2—22no—
337C2—362no—
338C2C3×#6368912yesdirect product factor
339C2C22×C3×(C22⋊C9)⋊C211yescharacteristic
subgroupG/H
centerC22×C3C2×(C22⋊C9)⋊C2
commutator, cosocle, hypocenterQ8⋊C9C23×C3
FrattiniC2×C3C22×C3×S4
FittingC22×C32×Q8D3
socleC22×C32C2×S4
hypercenterC23×C3(C22⋊C9)⋊C2
LCSG ▷ Q8⋊C9
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C22×C3×Q8⋊C9C2semidirect
2. C3×#63689C2direct (N × H)
3. C2×C3×#3689C2semidirect
4. C2×C3×#3688C2semidirect
6. C2×#63689C3direct (N × H)
9. C2×C3×Q8⋊C9C22semidirect
10. C3×#3689C22semidirect
11. C3×#3688C22semidirect
22. C22×Q8⋊C9C2×C3semidirect
23. #63689C2×C3direct (N × H)
24. C2×#3689C2×C3semidirect
25. C2×#3688C2×C3semidirect
57. C2×Q8⋊C9C22×C3semidirect
58. #3689C22×C3semidirect
59. #3688C22×C3semidirect
68. C22×C3×Q8D9semidirect
128. C2×C3×Q8C2×D9semidirect
180. C22×Q8C3×D9semidirect
253. C2×Q8C2×C3×D9semidirect
319. C2×C3#63689direct (N × H)
334. C3C2×#63689direct (N × H)
338. C2C3×#63689direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×Q8⋊C92
C3×#636892
C2×C3×#36892
C2×C3×#36882
C2×C3×#9423
C2×#636893
C2×C3×#6114