Finite Group Databasebeta

← C2×#64319⋊C3 #13180892

Subgroups — C2×#64319⋊C3 #13180892

Order 1728.

1007 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 1007 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
1C2×C3⋊D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C2×C3×C3⋊(#41⋊C3)C212yesmaximal, complemented
3#64319⋊C3C214yesmaximal, direct product factor
4C2×#64319C311yesmaximal, characteristic, complemented
5C2×D3×#41⋊C3—32nomaximal
6C2×C32×SL(2,𝔽3)C2211yescharacteristic, complemented
7C2×C3×#624—41nomaximal
8C3⋊D3×SL(2,𝔽3)C2212yescomplemented
9C3×C3⋊(#41⋊C3)C2214yescomplemented
10C2×Q8×C3⋊D3C2×C311yescharacteristic
11C2×C3×#239C2×C312yescomplemented
12C2×C3×#41⋊C3—32no—
13#64319—34no—
14#64319C2×C314yescomplemented
15C2×#3802—31no—
16C2×C4×D32—31no—
17C2×#3795—32no—
18D3×#41⋊C3—38no—
19C2×D3×SL(2,𝔽3)—32no—
20C2×D3×SL(2,𝔽3)—61no—
21C2×C3⋊(#41⋊C3)—22no—
22C2×C3⋊(#41⋊C3)D312yescomplemented
23C22×C3×C3⋊D3—41no—
24C2×C32×Dic3—42no—
25C32×SL(2,𝔽3)C2311yescharacteristic, complemented
26C3×#624—44no—
27C2×D3×#41—32no—
28C22×#41⋊C3—91no—
29C22×D32—31no—
30C2×C32×Q8C22×C311yescharacteristic
31Q8×C3⋊D3—32no—
32Q8×C3⋊D3C22×C312yes—
33C2×C4×C3⋊D3—31no—
34C2×#634—31no—
35C3×#239—34no—
36C3×#239C22×C314yescomplemented
37C2×C3×D12—32no—
38C2×C3×C4×D3—32no—
39C2×C3×C4×D3—122no—
40C3×#41⋊C3—34no—
41C2×C3×SL(2,𝔽3)—22no—
42C2×C3×SL(2,𝔽3)—21no—
43C2×C3×SL(2,𝔽3)—41no—
44C2×C3×SL(2,𝔽3)C2×D312yescomplemented
45C2×#625—32no—
46C2×#624A411yescharacteristic
47C2×D3×Dic3—32no—
48#3802—34no—
49#3802—34no—
50C4×D32—34no—
51C4×D32—34no—
52#3795—38no—
53#3795—38no—
54D3×SL(2,𝔽3)—34no—
55D3×SL(2,𝔽3)—62no—
56C3⋊(#41⋊C3)—24no—
57C3⋊(#41⋊C3)C2×D314yescomplemented
58C22×C33—41no—
59C2×C3×C3⋊D3—44no—
60C2×C3×C3⋊D3—42no—
61C32×Dic3—44no—
62C2×C3×#41—32no—
63D3×#41—38no—
64D3×#41—98no—
65C2×#239C3×D312yescomplemented
66C2×D3×Q8—32no—
67C2×D3×Q8—61no—
68C2×#237—92no—
69C2×D3×D4—92no—
70C2×#235—92no—
71C22×C4×D3—92no—
72C2×#41⋊C3—32no—
73C2×#41⋊C3—62no—
74C2×#41⋊C3—94no—
75C22×SL(2,𝔽3)—91no—
76C22×C3⋊D3C2×A411yescharacteristic
77C22×C3×D3—32no—
78C22×C3×D3—122no—
79C22×C3×D3—241no—
80C2×D32—34no—
81C2×D32—64no—
82C32×Q8—31no—
83C32×Q8C23×C311yescharacteristic
84C2×C32×C4—31no—
85C2×C32×C4—122no—
86C2×C3⋊Dic3—31no—
87C4×C3⋊D3—32no—
88C4×C3⋊D3—32no—
89#634—32no—
90#634—32no—
91C2×C3×Dic3—42no—
92C2×C3×Dic3—82no—
93C2×C3×Dic3C2×A412yes—
94C3×D12—34no—
95C3×D12—34no—
96C3×C4×D3—34no—
97C3×C4×D3—34no—
98C3×C4×D3—128no—
99C3×SL(2,𝔽3)—22no—
100C3×SL(2,𝔽3)—21no—
101C3×SL(2,𝔽3)—41no—
102C3×SL(2,𝔽3)C22×D312yescomplemented
103#625—38no—
104#624C2×A414yes—
105D3×Dic3—38no—
106C22×#41—91noSylow
107C2×C33—42no—
108C2×C33—41no—
109C3×C3⋊D3—44no—
110C23×D3—92no—
111C3×#41—34no—
112C3×#41—94no—
113C2×C3×Q8—21no—
114C2×C3×Q8C2×C3×D312yes—
115C2×C3×D4—92no—
116C22×C3×C4—92no—
117C22×C3×C4—361no—
118C2×#67—94no—
119C22×Dic3—92no—
120#239—34no—
121#239C2×C3×D314yescomplemented
122D3×Q8—34no—
123D3×Q8—62no—
124D3×Q8—94no—
125D3×Q8—182no—
126#237—98no—
127#237—98no—
128D3×D4—98no—
129D3×D4—98no—
130#235—98no—
131#235—98no—
132C2×D12—32no—
133C2×C4×D3—32no—
134C2×C4×D3—32no—
135C2×C4×D3—98no—
136C2×C4×D3—98no—
137C2×C4×D3—92no—
138C2×C4×D3—92no—
139C2×C4×D3—181no—
140C2×Dic6—92no—
141C2×Dic6—181no—
142#41⋊C3—34no—
143#41⋊C3—64no—
144C2×SL(2,𝔽3)—22no—
145C2×SL(2,𝔽3)—41no—
146C2×SL(2,𝔽3)—92no—
147C2×SL(2,𝔽3)D3211yescharacteristic, complemented
148C22×C32—42no—
149C22×C32—81no—
150C22×C32—82no—
151C22×C32—161no—
152C22×C32C22×A411yescharacteristic
153C2×C3⋊D3#41⋊C314yescomplemented
154C2×C3⋊D3C22×A412yes—
155C2×C3×D3—34no—
156C2×C3×D3—64no—
157C2×C3×D3—124no—
158C2×C3×D3—128no—
159C2×C3×D3—244no—
160C2×C3×D3—242no—
161D32—68no—
162C32×C4—31no—
163C32×C4—31no—
164C32×C4—124no—
165C3⋊Dic3—32no—
166C3×Dic3—44no—
167C3×Dic3—84no—
168C3×Dic3C22×A414yes—
169C2×#41—32no—
170C2×#41—94no—
171C2×#41—272no—
172C2×#41—274no—
173C22×Q8—91no—
174C22×D4—271no—
175C23×C4—271no—
176C33—41noSylow
177C23×C3—92no—
178C23×C3—361no—
179C22×D3—32no—
180C22×D3—32no—
181C22×D3—61no—
182C22×D3—98no—
183C22×D3—188no—
184C3×Q8—21no—
185C3×Q8—32no—
186C3×Q8—61no—
187C3×Q8C22×C3×D312yes—
188C3×D4—94no—
189C3×D4—94no—
190C2×C3×C4—32no—
191C2×C3×C4—32no—
192C2×C3×C4—61no—
193C2×C3×C4—94no—
194C2×C3×C4—94no—
195C2×C3×C4—122no—
196C2×C3×C4—242no—
197C2×C3×C4—364no—
198#67—916no—
199C2×Dic3—92no—
200C2×Dic3—98no—
201C2×Dic3—181no—
202C2×Dic3D3×A412yes—
203D12—34no—
204D12—34no—
205C4×D3—38no—
206C4×D3—34no—
207C4×D3—34no—
208C4×D3—98no—
209C4×D3—94no—
210C4×D3—94no—
211C4×D3—182no—
212C4×D3—182no—
213Dic6—94no—
214Dic6—94no—
215Dic6—182no—
216Dic6—182no—
217SL(2,𝔽3)—22no—
218SL(2,𝔽3)—41no—
219SL(2,𝔽3)C2×D3211yescharacteristic, complemented
220C2×C32—44no—
221C2×C32—42no—
222C2×C32—81no—
223C2×C32—82no—
224C2×C32—84no—
225C2×C32—82no—
226C2×C32—161no—
227C2×C32—162no—
228C2×C32C2×#41⋊C312yescomplemented
229C2×C32C23×A411yescharacteristic
230C3⋊D3C2×#41⋊C314yescomplemented
231C3×D3—64no—
232C3×D3—128no—
233C3×D3—244no—
234C24—271no—
235#41—34no—
236#41—94no—
237#41—278no—
238#41—278no—
239C2×Q8—92no—
240C2×Q8—271no—
241C2×Q8—272no—
242C2×Q8C3×D3211yescharacteristic
243C2×D4—92no—
244C2×D4—274no—
245C2×D4—274no—
246C2×D4—272no—
247C22×C4—92no—
248C22×C4—91no—
249C22×C4—271no—
250C22×C4—272no—
251C22×C4—274no—
252C22×C4—274no—
253C22×C3—21no—
254C22×C3—41no—
255C22×C3—82no—
256C22×C3—94no—
257C22×C3—161no—
258C22×C3—184no—
259C22×C3—362no—
260C22×C3—364no—
261C22×C3C2×D3×A412yes—
262C2×D3—38no—
263C2×D3—34no—
264C2×D3—34no—
265C2×D3—62no—
266C2×D3—64no—
267C2×D3—64no—
268C2×D3—1816no—
269C3×C4—32no—
270C3×C4—32no—
271C3×C4—34no—
272C3×C4—61no—
273C3×C4—61no—
274C3×C4—124no—
275C3×C4—244no—
276Dic3—94no—
277Dic3—182no—
278Dic3C2×D3×A414yes—
279C32—42no—
280C32—81no—
281C32—82no—
282C32—161no—
283C32C22×#41⋊C311yescharacteristic, complemented
284C23—91no—
285C23—92no—
286C23—274no—
287C23—544no—
288Q8—31no—
289Q8—272no—
290Q8—272no—
291Q8C2×C3×D3211yescharacteristic
292D4—94no—
293D4—94no—
294D4—278no—
295C2×C4—31no—
296C2×C4—32no—
297C2×C4—94no—
298C2×C4—94no—
299C2×C4—94no—
300C2×C4—272no—
301C2×C4—272no—
302C2×C4—278no—
303C2×C4—271no—
304C2×C3—21no—
305C2×C3—22no—
306C2×C3—41no—
307C2×C3—42no—
308C2×C3—82no—
309C2×C3—84no—
310C2×C3—161no—
311C2×C3—162no—
312C2×C3—184no—
313C2×C3—364no—
314C2×C3D3×#41⋊C314yescomplemented
315C2×C3C22×D3×A412yes—
316D3—38no—
317D3—64no—
318D3—64no—
319C22—92no—
320C22—94no—
321C22—94no—
322C22—184no—
323C22—548no—
324C22D32×A411yescharacteristic
325C4—31no—
326C4—31no—
327C4—34no—
328C4—272no—
329C3—21no—
330C3—41no—
331C3—82no—
332C3—161no—
333C3C2×D3×#41⋊C312yescomplemented
334C2—94no—
335C2—184no—
336C2#64319⋊C312yesdirect product factor
337C2C2×D32×A411yescharacteristic
subgroupG/H
center, hypercenterC22D32×A4
commutator, cosocle, hypocenterC32×Q8C23×C3
FrattiniC2C2×D32×A4
FittingC2×C32×Q8C22×C3
socleC22×C32C22×A4
LCSG ▷ C32×Q8
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×C3⋊D3×SL(2,𝔽3)C2semidirect
2. C2×C3×C3⋊(#41⋊C3)C2semidirect
3. #64319⋊C3C2direct (N × H)
4. C2×#64319C3semidirect
6. C2×C32×SL(2,𝔽3)C22semidirect
8. C3⋊D3×SL(2,𝔽3)C22semidirect
9. C3×C3⋊(#41⋊C3)C22semidirect
11. C2×C3×#239C2×C3semidirect
14. #64319C2×C3semidirect
22. C2×C3⋊(#41⋊C3)D3semidirect
25. C32×SL(2,𝔽3)C23semidirect
36. C3×#239C22×C3semidirect
44. C2×C3×SL(2,𝔽3)C2×D3semidirect
57. C3⋊(#41⋊C3)C2×D3semidirect
65. C2×#239C3×D3semidirect
102. C3×SL(2,𝔽3)C22×D3semidirect
121. #239C2×C3×D3semidirect
147. C2×SL(2,𝔽3)D32semidirect
153. C2×C3⋊D3#41⋊C3semidirect
219. SL(2,𝔽3)C2×D32semidirect
228. C2×C32C2×#41⋊C3semidirect
230. C3⋊D3C2×#41⋊C3semidirect
283. C32C22×#41⋊C3semidirect
314. C2×C3D3×#41⋊C3semidirect
333. C3C2×D3×#41⋊C3semidirect
336. C2#64319⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×SL(2,𝔽3)2
C2×C3×C3⋊(#41⋊C3)2
#64319⋊C32
C2×#643193
C2×D3×#41⋊C33
C2×C3×#6244