Finite Group Databasebeta

← C2×#64264⋊C3 #13164416

Subgroups — C2×#64264⋊C3 #13164416

Order 1728.

630 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 71 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×Q8⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#64264⋊C3C214yesmaximal, direct product factor
3C2×(C3⋊GL(2,𝔽3))⋊C3C211yesmaximal, characteristic, complemented
4C2×(C3⋊#226)⋊C3C211yesmaximal, characteristic, complemented
5C2×#64264C311yesmaximal, characteristic, complemented
6C2×C3×#942—31nomaximal
7C2×#638⋊C3—41nomaximal
8C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
9C2×Q8⋊H(3,𝔽3)C2212yescomplemented
10(C3⋊GL(2,𝔽3))⋊C3C2212yescomplemented
11(C3⋊#226)⋊C3C2212yescomplemented
12C22×C32×Q8D311yescharacteristic, complemented
13C2×C3×#235—31no—
14C22×C3×SL(2,𝔽3)—21no—
15C22×C3×SL(2,𝔽3)C2×C311yescharacteristic, complemented
16#64264C2×C314yescomplemented
17C2×C3⋊GL(2,𝔽3)C2×C311yescharacteristic, complemented
18C2×C3⋊#226C2×C311yescharacteristic, complemented
19C2×C3×#216—31no—
20C2×C3×#216—31no—
21C3×#942—34no—
22C3×#942—34no—
23C2×C3×#215—31no—
24C2×C3×#215—31no—
25C2×C3×#208—31no—
26C23×H(3,𝔽3)—41no—
27C22×(C3⋊D3)⋊C3—41no—
28#638⋊C3—44no—
29C2×(C3⋊Dic3)⋊C3—41no—
30Q8⋊H(3,𝔽3)C2311yescharacteristic
31C2×#983—91no—
32C2×C3×#137—91no—
33C2×#942—31no—
34C2×C32×Q8—31no—
35C2×C32×Q8—32no—
36C2×C32×Q8C2×D311yescharacteristic
37C2×C32×Q8C2×D312yescomplemented
38C22×C32×C4—31no—
39C2×#638—41no—
40C2×C3×#67—61no—
41C3×#235—34no—
42C3×#235—62no—
43C2×C3×D12—31no—
44C2×C3×C4×D3—61no—
45C2×C3×Dic6—31no—
46C2×C3×SL(2,𝔽3)—21no—
47C2×C3×SL(2,𝔽3)—22no—
48C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
49C2×C3×SL(2,𝔽3)C22×C312yescomplemented
50C3⋊GL(2,𝔽3)C22×C312yescomplemented
51C3⋊#226C22×C312yescomplemented
52C3×#216—32no—
53C3×#216—34no—
54C3×#216—32no—
55C3×#215—32no—
56C3×#215—34no—
57C3×#215—32no—
58C3×#208—34no—
59C2×C3×C3⋊C8—31no—
60C2×C3×C3⋊C8—31no—
61C22×H(3,𝔽3)—41no—
62C22×H(3,𝔽3)—42no—
63C22×H(3,𝔽3)—82no—
64C2×(C3⋊D3)⋊C3—42no—
65C2×(C3⋊D3)⋊C3—82no—
66(C3⋊Dic3)⋊C3—42no—
67C2×C3×#41—91no—
68C22×C3×Q8—31no—
69C22×C3×Q8C3×D311yescharacteristic
70C2×#235—31no—
71C22×SL(2,𝔽3)—31no—
72C22×SL(2,𝔽3)—61no—
73#983—94no—
74C2×GL(2,𝔽3)—91no—
75C2×#226—91no—
76C3×#137—94no—
77C3×#137—94no—
78C2×C3×Q16—91no—
79C2×C3×Q16—91no—
80C2×C3×SD16—91no—
81C2×C3×SD16—91no—
82C2×C3×M16—91no—
83C2×#216—31no—
84C2×#216—31no—
85#942—34no—
86#942—34no—
87C2×#215—31no—
88C2×#215—31no—
89C2×#208—31no—
90C23×C32—41no—
91C23×C32—81no—
92C23×C32S411yescharacteristic
93C22×C3⋊D3—41no—
94C22×C3×D3—61no—
95C32×Q8—31no—
96C32×Q8—32no—
97C32×Q8—61no—
98C32×Q8C22×D311yescharacteristic
99C2×C32×C4—31no—
100C2×C32×C4—32no—
101C2×C32×C4—31no—
102C2×C32×C4—32no—
103#638—44no—
104C2×C3⋊Dic3—41no—
105C3×#67—64no—
106C2×C3×Dic3—61no—
107C3×D12—32no—
108C3×D12—61no—
109C3×C4×D3—64no—
110C3×Dic6—32no—
111C3×Dic6—61no—
112C3×SL(2,𝔽3)—21no—
113C3×SL(2,𝔽3)C23×C311yescharacteristic
114C3×C3⋊C8—32no—
115C3×C3⋊C8—32no—
116C2×#137—91noSylow
117C2×H(3,𝔽3)—42no—
118C2×H(3,𝔽3)—41no—
119C2×H(3,𝔽3)—82no—
120(C3⋊D3)⋊C3—82no—
121C3×#41—94no—
122C3×#41—182no—
123C2×C3×Q8—31no—
124C2×C3×Q8—32no—
125C2×C3×Q8—31no—
126C2×C3×Q8—32no—
127C2×C3×Q8—31no—
128C2×C3×Q8—32no—
129C2×C3×Q8—61no—
130C2×C3×Q8—62no—
131C2×C3×Q8—91no—
132C2×C3×Q8C2×C3×D311yescharacteristic
133C2×C3×Q8C2×C3×D312yes—
134C2×C3×D4—91no—
135C2×C3×D4—181no—
136C22×C3×C4—31no—
137C22×C3×C4—31no—
138C22×C3×C4—61no—
139C22×C3×C4—181no—
140C2×#67—61no—
141C2×#67—361no—
142#235—34no—
143#235—62no—
144C2×D12—31no—
145C2×C4×D3—61no—
146C2×Dic6—31no—
147C2×SL(2,𝔽3)—31no—
148C2×SL(2,𝔽3)—32no—
149C2×SL(2,𝔽3)—61no—
150C2×SL(2,𝔽3)—62no—
151GL(2,𝔽3)—92no—
152#226—92no—
153C3×Q16—92no—
154C3×Q16—94no—
155C3×Q16—92no—
156C3×SD16—92no—
157C3×SD16—94no—
158C3×SD16—92no—
159C3×M16—94no—
160C2×C3×C8—91no—
161C2×C3×C8—91no—
162#216—32no—
163#216—34no—
164#216—32no—
165#215—34no—
166#215—32no—
167#215—32no—
168#208—34no—
169C2×C3⋊C8—31no—
170C2×C3⋊C8—31no—
171C22×C32—22no—
172C22×C32—41no—
173C22×C32—42no—
174C22×C32—82no—
175C22×C32—84no—
176C22×C32—82no—
177C22×C32—81no—
178C22×C32C2×S412yes—
179C22×C32C2×S411yescharacteristic
180C2×C3⋊D3—42no—
181C2×C3⋊D3—82no—
182C2×C3×D3—62no—
183C2×C3×D3—122no—
184C32×C4—31no—
185C32×C4—31no—
186C32×C4—32no—
187C3⋊Dic3—42no—
188C3×Dic3—62no—
189C2×#41—91no—
190C22×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
191#137—94no—
192#137—94no—
193C2×Q16—91no—
194C2×Q16—91no—
195C2×SD16—91no—
196C2×SD16—91no—
197C2×M16—91no—
198H(3,𝔽3)—41noSylow
199C23×C3—31no—
200C23×C3—121no—
201C23×C3—181no—
202C23×C3—241no—
203C23×C3C3×S411yescharacteristic
204C22×D3—61no—
205C22×D3—361no—
206C3×Q8—31no—
207C3×Q8—32no—
208C3×Q8—31no—
209C3×Q8—32no—
210C3×Q8—31no—
211C3×Q8—61no—
212C3×Q8—62no—
213C3×Q8—61no—
214C3×Q8—62no—
215C3×Q8—61no—
216C3×Q8—92no—
217C3×Q8—181no—
218C3×Q8C22×C3×D311yescharacteristic
219C3×D4—92no—
220C3×D4—181no—
221C3×D4—184no—
222C2×C3×C4—31no—
223C2×C3×C4—32no—
224C2×C3×C4—31no—
225C2×C3×C4—32no—
226C2×C3×C4—31no—
227C2×C3×C4—32no—
228C2×C3×C4—31no—
229C2×C3×C4—32no—
230C2×C3×C4—62no—
231C2×C3×C4—62no—
232C2×C3×C4—61no—
233C2×C3×C4—61no—
234C2×C3×C4—184no—
235C2×C3×C4—181no—
236#67—64no—
237#67—364no—
238C2×Dic3—61no—
239C2×Dic3—361no—
240D12—32no—
241D12—61no—
242C4×D3—64no—
243Dic6—32no—
244Dic6—61no—
245SL(2,𝔽3)—31no—
246SL(2,𝔽3)—61no—
247C3×C8—92no—
248C3×C8—92no—
249C3⋊C8—32no—
250C3⋊C8—32no—
251C2×C32—22no—
252C2×C32—41no—
253C2×C32—42no—
254C2×C32—82no—
255C2×C32—84no—
256C2×C32—81no—
257C2×C32—82no—
258C2×C32#98312yescomplemented
259C2×C32C22×S411yescharacteristic
260C3⋊D3—82no—
261C3×D3—122no—
262#41—94no—
263#41—182no—
264C2×Q8—31no—
265C2×Q8—32no—
266C2×Q8—91no—
267C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
268C2×Q8C2×(C3⋊D3)⋊C312yescomplemented
269C2×D4—91no—
270C2×D4—181no—
271C22×C4—31no—
272C22×C4—181no—
273Q16—92no—
274Q16—94no—
275Q16—92no—
276SD16—92no—
277SD16—94no—
278SD16—92no—
279M16—94no—
280C2×C8—91no—
281C2×C8—91no—
282C22×C3—22no—
283C22×C3—32no—
284C22×C3—31no—
285C22×C3—62no—
286C22×C3—122no—
287C22×C3—121no—
288C22×C3—182no—
289C22×C3—242no—
290C22×C3—244no—
291C22×C3—242no—
292C22×C3—241no—
293C22×C3—362no—
294C22×C3C2×C3×S412yes—
295C22×C3C2×C3×S411yescharacteristic
296C2×D3—62no—
297C2×D3—122no—
298C2×D3—362no—
299C2×D3—722no—
300C3×C4—31no—
301C3×C4—32no—
302C3×C4—31no—
303C3×C4—31no—
304C3×C4—31no—
305C3×C4—32no—
306C3×C4—61no—
307C3×C4—62no—
308C3×C4—61no—
309C3×C4—182no—
310Dic3—62no—
311Dic3—362no—
312C32—41no—
313C32—81no—
314C32C2×#98311yescharacteristic, complemented
315C23—181no—
316C23(C3⋊S4)⋊C311yescharacteristic
317Q8—31no—
318Q8—32no—
319Q8—61no—
320Q8—92no—
321Q8—181no—
322Q8C22×(C3⋊D3)⋊C311yescharacteristic
323D4—92no—
324D4—181no—
325D4—184no—
326C2×C4—32no—
327C2×C4—32no—
328C2×C4—31no—
329C2×C4—31no—
330C2×C4—184no—
331C2×C4—181no—
332C8—92no—
333C8—92no—
334C2×C3—22no—
335C2×C3—31no—
336C2×C3—32no—
337C2×C3—62no—
338C2×C3—121no—
339C2×C3—122no—
340C2×C3—242no—
341C2×C3—244no—
342C2×C3—241no—
343C2×C3—242no—
344C2×C3—362no—
345C2×C3C3×#98312yes—
346C2×C3C22×C3×S411yescharacteristic
347D3—122no—
348D3—722no—
349C22—22no—
350C22—182no—
351C22—362no—
352C22C2×(C3⋊S4)⋊C312yes—
353C22C2×(C3⋊S4)⋊C311yescharacteristic
354C4—31no—
355C4—32no—
356C4—31no—
357C4—182no—
358C3—31no—
359C3—121no—
360C3—241no—
361C3C2×C3×#98311yescharacteristic
362C2—22no—
363C2—362no—
364C2#64264⋊C312yesdirect product factor
365C2C22×(C3⋊S4)⋊C311yescharacteristic
subgroupG/H
centerC22C2×(C3⋊S4)⋊C3
commutator, cosocle, hypocenterC3×SL(2,𝔽3)C23×C3
FrattiniC2×C3C22×C3×S4
FittingC22×C32×Q8D3
socleC22×C3C2×C3×S4
hypercenterC23(C3⋊S4)⋊C3
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×Q8⋊H(3,𝔽3)C2semidirect
2. #64264⋊C3C2direct (N × H)
3. C2×(C3⋊GL(2,𝔽3))⋊C3C2semidirect
4. C2×(C3⋊#226)⋊C3C2semidirect
5. C2×#64264C3semidirect
9. C2×Q8⋊H(3,𝔽3)C22semidirect
10. (C3⋊GL(2,𝔽3))⋊C3C22semidirect
11. (C3⋊#226)⋊C3C22semidirect
12. C22×C32×Q8D3semidirect
15. C22×C3×SL(2,𝔽3)C2×C3semidirect
16. #64264C2×C3semidirect
17. C2×C3⋊GL(2,𝔽3)C2×C3semidirect
18. C2×C3⋊#226C2×C3semidirect
37. C2×C32×Q8C2×D3semidirect
49. C2×C3×SL(2,𝔽3)C22×C3semidirect
50. C3⋊GL(2,𝔽3)C22×C3semidirect
51. C3⋊#226C22×C3semidirect
190. C22×Q8(C3⋊D3)⋊C3semidirect
258. C2×C32#983semidirect
268. C2×Q8C2×(C3⋊D3)⋊C3semidirect
314. C32C2×#983semidirect
364. C2#64264⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q8⋊H(3,𝔽3)2
#64264⋊C32
C2×(C3⋊GL(2,𝔽3))⋊C32
C2×(C3⋊#226)⋊C32
C2×#642643
C2×C3×#9423
C2×#638⋊C34