Finite Group Databasebeta

← C2×#64128⋊C3 #13153465

Subgroups — C2×#64128⋊C3 #13153465

Order 1728.

991 conjugacy classes of subgroups.

The subgroup lattice has 375 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88722C212yesmaximal, complemented
2C2×Q16×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3C2×#3755⋊C3C212yesmaximal, complemented
4#64128⋊C3C218yesmaximal, direct product factor
5C2×#3745⋊C3C211yesmaximal, characteristic, complemented
6C2×C8×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
7C2×#64128C311yesmaximal, characteristic, complemented
8C2×C3×#919—31nomaximal
9C2×Q8×H(3,𝔽3)C2212yes—
10#88722—24no—
11#88722C2218yescomplemented
12C2×C4⋊((C3⋊D3)⋊C3)—22no—
13C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
14C2×C4×(C3⋊D3)⋊C3—22no—
15C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
16Q16×H(3,𝔽3)C2214yescomplemented
17C2×C8×H(3,𝔽3)C2211yescharacteristic
18#3755⋊C3C2218yescomplemented
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#3745⋊C3C2214yescomplemented
21C8×(C3⋊D3)⋊C3C2214yescomplemented
22C2×#3832C2×C312yescomplemented
23C2×C3×#239—32no—
24C2×C32×Q16—21no—
25C2×C32×Q16D311yescharacteristic, complemented
26C2×C32×Q16C2×C311yescharacteristic, complemented
27C2×#3755C2×C312yescomplemented
28#64128C2×C318yescomplemented
29C2×#3745C2×C311yescharacteristic, complemented
30C2×C8×C3⋊D3C2×C311yescharacteristic, complemented
31C2×C3×#215—32no—
32C3×#919—38no—
33C2×C3×D24—31no—
34C2×C3×D3×C8—31no—
35C22×(C3⋊D3)⋊C3—22no—
36C22×(C3⋊D3)⋊C3D411yescharacteristic
37Q8×H(3,𝔽3)—22no—
38Q8×H(3,𝔽3)C2314yes—
39C2×C4×H(3,𝔽3)—22no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×(C3⋊Dic3)⋊C3D411yescharacteristic
42C4⋊((C3⋊D3)⋊C3)—28no—
43C4⋊((C3⋊D3)⋊C3)—22no—
44C4⋊((C3⋊D3)⋊C3)C2314yes—
45C4×(C3⋊D3)⋊C3—28no—
46C4×(C3⋊D3)⋊C3D412yes—
47C4×(C3⋊D3)⋊C3C2312yes—
48C8×H(3,𝔽3)C2312yes—
49(C3⋊(C3⋊C8))⋊C3C2312yes—
50C2×C3×#135—91no—
51C2×#919—31no—
52C2×#919—91no—
53C2×C32×Q8—22no—
54C2×C32×Q8C2×D312yes—
55C2×C32×Q8C22×C312yes—
56#3832—24no—
57#3832C22×C318yescomplemented
58C2×C4⋊(C3⋊D3)—22no—
59C2×C4⋊(C3⋊D3)C22×C312yes—
60C2×C4×C3⋊D3—22no—
61C2×C4×C3⋊D3C22×C311yescharacteristic
62C3×#239—38no—
63C3×#239—64no—
64C2×C3×D12—32no—
65C2×C3×D12—62no—
66C2×C3×C4×D3—31no—
67C2×C3×C4×D3—62no—
68C32×Q16—24no—
69C32×Q16C2×D314yescomplemented
70C32×Q16C22×C314yescomplemented
71C2×C32×C8—21no—
72C2×C32×C8C2×D311yescharacteristic
73C2×C32×C8C22×C311yescharacteristic
74#3755C22×C318yescomplemented
75C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
76#3745C22×C314yescomplemented
77C8×C3⋊D3C22×C314yescomplemented
78C3×#215—38no—
79C2×C3×C3⋊C8—31no—
80C3×D24—34no—
81C3×D3×C8—34no—
82C22×H(3,𝔽3)C2×D411yescharacteristic
83C2×(C3⋊D3)⋊C3—22no—
84C2×(C3⋊D3)⋊C3—24no—
85C2×(C3⋊D3)⋊C3—44no—
86C2×(C3⋊D3)⋊C3C2×D412yes—
87C4×H(3,𝔽3)—24no—
88C4×H(3,𝔽3)C2×D411yescharacteristic
89C4×H(3,𝔽3)C2411yescharacteristic
90(C3⋊Dic3)⋊C3C2×D412yes—
91C2×C3×#41—92no—
92C2×#239—32no—
93C2×#239—92no—
94C3×#135—98no—
95C2×C3×Q16—31no—
96C2×C3×Q16—31no—
97C2×C3×Q16—61no—
98C2×C3×Q16C3×D311yescharacteristic
99C2×C3×SD16—92no—
100C2×C3×D8—91no—
101C22×C3×C8—91no—
102C2×#215—32no—
103C2×#215—92no—
104#919—38no—
105#919—98no—
106C2×D24—31no—
107C2×D24—91no—
108C2×D3×C8—31no—
109C2×D3×C8—91no—
110C22×C3⋊D3—22no—
111C22×C3⋊D3C3×D411yescharacteristic
112C22×C3×D3—31no—
113C22×C3×D3—62no—
114C32×Q8—22no—
115C32×Q8—22no—
116C32×Q8—24no—
117C32×Q8—42no—
118C32×Q8C22×D314yes—
119C32×Q8C23×C314yes—
120C2×C32×C4—22no—
121C2×C32×C4—22no—
122C2×C32×C4—21no—
123C2×C32×C4—42no—
124C2×C32×C4C22×D311yescharacteristic
125C2×C32×C4C23×C311yescharacteristic
126C2×C3⋊Dic3C3×D411yescharacteristic
127C4⋊(C3⋊D3)—28no—
128C4⋊(C3⋊D3)—22no—
129C4⋊(C3⋊D3)C23×C314yes—
130C4×C3⋊D3—28no—
131C4×C3⋊D3C3×D412yes—
132C4×C3⋊D3C23×C312yes—
133C2×C3×Dic3—31no—
134C3×D12—34no—
135C3×D12—68no—
136C3×D12—62no—
137C3×C4×D3—32no—
138C3×C4×D3—32no—
139C3×C4×D3—68no—
140C32×C8—22no—
141C32×C8C22×D312yes—
142C32×C8C23×C312yes—
143C3⋊(C3⋊C8)C23×C312yes—
144C3×C3⋊C8—32no—
145C2×#135—91noSylow
146C2×H(3,𝔽3)#13512yescomplemented
147C2×H(3,𝔽3)C22×D411yescharacteristic
148(C3⋊D3)⋊C3—22no—
149(C3⋊D3)⋊C3—44no—
150C3×#41—98no—
151C3×#41—184no—
152C2×C3×Q8—32no—
153C2×C3×Q8—32no—
154C2×C3×Q8—62no—
155C2×C3×Q8C2×C3×D312yes—
156C2×C3×D4—92no—
157C2×C3×D4—182no—
158C22×C3×C4—91no—
159C22×C3×C4—182no—
160#239—38no—
161#239—64no—
162#239—98no—
163#239—184no—
164C2×D12—32no—
165C2×D12—62no—
166C2×D12—92no—
167C2×D12—182no—
168C2×C4×D3—31no—
169C2×C4×D3—62no—
170C2×C4×D3—91no—
171C2×C4×D3—182no—
172C3×Q16—34no—
173C3×Q16—34no—
174C3×Q16—64no—
175C3×Q16C2×C3×D314yes—
176C3×SD16—98no—
177C3×D8—94no—
178C2×C3×C8—31no—
179C2×C3×C8—31no—
180C2×C3×C8—61no—
181C2×C3×C8—94no—
182C2×C3×C8—91no—
183C2×C3×C8C2×C3×D311yescharacteristic
184#215—38no—
185#215—98no—
186C2×C3⋊C8—31no—
187C2×C3⋊C8—91no—
188D24—34no—
189D24—94no—
190D3×C8—34no—
191D3×C8—94no—
192C22×C32—21no—
193C22×C32D3×D411yescharacteristic
194C22×C32C2×C3×D411yescharacteristic
195C2×C3⋊D3—22no—
196C2×C3⋊D3—24no—
197C2×C3⋊D3—44no—
198C2×C3⋊D3C2×C3×D412yes—
199C2×C3×D3—32no—
200C2×C3×D3—62no—
201C2×C3×D3—64no—
202C2×C3×D3—124no—
203C32×C4—24no—
204C32×C4—24no—
205C32×C4—21no—
206C32×C4—21no—
207C32×C4—44no—
208C32×C4D3×D411yescharacteristic
209C32×C4C2×C3×D411yescharacteristic
210C32×C4C23×D311yescharacteristic
211C32×C4C24×C311yescharacteristic
212C3⋊Dic3C2×C3×D412yes—
213C3×Dic3—32no—
214C2×#41—92no—
215#135—98no—
216C2×Q16(C3⋊D3)⋊C311yescharacteristic, complemented
217C2×SD16—92no—
218C2×D8—91no—
219C22×C8—91no—
220H(3,𝔽3)C2×#13511yescharacteristic, complemented, Sylow
221C23×C3—91no—
222C23×C3—182no—
223C22×D3—31no—
224C22×D3—62no—
225C22×D3—91no—
226C22×D3—182no—
227C3×Q8—22no—
228C3×Q8—34no—
229C3×Q8—34no—
230C3×Q8—62no—
231C3×Q8—62no—
232C3×Q8—64no—
233C3×Q8—122no—
234C3×Q8C22×C3×D314yes—
235C3×D4—94no—
236C3×D4—182no—
237C3×D4—188no—
238C2×C3×C4—22no—
239C2×C3×C4—31no—
240C2×C3×C4—31no—
241C2×C3×C4—62no—
242C2×C3×C4—62no—
243C2×C3×C4—61no—
244C2×C3×C4—92no—
245C2×C3×C4—92no—
246C2×C3×C4—91no—
247C2×C3×C4—122no—
248C2×C3×C4—188no—
249C2×C3×C4C22×C3×D311yescharacteristic
250C2×Dic3—31no—
251C2×Dic3—91no—
252D12—34no—
253D12—68no—
254D12—62no—
255D12—94no—
256D12—182no—
257D12—188no—
258C4×D3—32no—
259C4×D3—32no—
260C4×D3—68no—
261C4×D3—92no—
262C4×D3—92no—
263C4×D3—188no—
264C3×C8—32no—
265C3×C8—32no—
266C3×C8—62no—
267C3×C8—92no—
268C3×C8C22×C3×D312yes—
269C3⋊C8—32no—
270C3⋊C8—92no—
271C2×C32—22no—
272C2×C32—21no—
273C2×C32#91912yescomplemented
274C2×C32C3×#13512yescomplemented
275C2×C32C2×D3×D411yescharacteristic
276C2×C32C22×C3×D411yescharacteristic
277C3⋊D3—22no—
278C3⋊D3—44no—
279C3×D3—62no—
280C3×D3—124no—
281#41—98no—
282#41—184no—
283C2×Q8C2×(C3⋊D3)⋊C312yes—
284C2×D4—92no—
285C2×D4—182no—
286C22×C4—91no—
287C22×C4—182no—
288Q16C2×(C3⋊D3)⋊C314yescomplemented
289SD16—98no—
290D8—94no—
291C2×C8—91no—
292C2×C8—94no—
293C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
294C22×C3—31no—
295C22×C3—31no—
296C22×C3—61no—
297C22×C3—92no—
298C22×C3—182no—
299C22×C3—184no—
300C22×C3—364no—
301C22×C3C3×D3×D411yescharacteristic
302C2×D3—32no—
303C2×D3—62no—
304C2×D3—64no—
305C2×D3—92no—
306C2×D3—124no—
307C2×D3—182no—
308C2×D3—184no—
309C2×D3—364no—
310C3×C4—24no—
311C3×C4—31no—
312C3×C4—31no—
313C3×C4—31no—
314C3×C4—31no—
315C3×C4—64no—
316C3×C4—64no—
317C3×C4—61no—
318C3×C4—61no—
319C3×C4—92no—
320C3×C4—124no—
321C3×C4C3×D3×D411yescharacteristic
322C3×C4C23×C3×D311yescharacteristic
323Dic3—32no—
324Dic3—92no—
325C32—21no—
326C32C2×#91911yescharacteristic, complemented
327C32C2×C3×#13511yescharacteristic, complemented
328C23—91no—
329C23—182no—
330Q8—22no—
331Q8C22×(C3⋊D3)⋊C314yes—
332D4—94no—
333D4—188no—
334D4—182no—
335C2×C4—22no—
336C2×C4—91no—
337C2×C4—92no—
338C2×C4—92no—
339C2×C4—188no—
340C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
341C8—92no—
342C8C22×(C3⋊D3)⋊C312yes—
343C2×C3—32no—
344C2×C3—32no—
345C2×C3—31no—
346C2×C3—31no—
347C2×C3—62no—
348C2×C3—61no—
349C2×C3—182no—
350C2×C3—364no—
351C2×C3C3×#91912yes—
352C2×C3C2×C3×D3×D411yescharacteristic
353D3—62no—
354D3—124no—
355D3—182no—
356D3—364no—
357C22—92no—
358C22—182no—
359C22—184no—
360C22—364no—
361C22D4×(C3⋊D3)⋊C311yescharacteristic
362C4—24no—
363C4—92no—
364C4D4×(C3⋊D3)⋊C311yescharacteristic
365C4C23×(C3⋊D3)⋊C311yescharacteristic
366C3—31no—
367C3—31no—
368C3—61no—
369C3C2×C3×#91911yescharacteristic
370C2—182no—
371C2—364no—
372C2#64128⋊C312yesdirect product factor
373C2C2×D4×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutator, cosocleC32×C4C24×C3
FrattiniC3×C4C23×C3×D3
FittingC2×Q16×H(3,𝔽3)C2
socleC22×C3C3×D3×D4
hypercenterC2×Q16(C3⋊D3)⋊C3
hypocenterC32C2×C3×#135
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q16

Complements

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

NH ≅ G/Ntype
1. C2×#88722C2semidirect
2. C2×Q16×H(3,𝔽3)C2semidirect
3. C2×#3755⋊C3C2semidirect
4. #64128⋊C3C2direct (N × H)
5. C2×#3745⋊C3C2semidirect
6. C2×C8×(C3⋊D3)⋊C3C2semidirect
7. C2×#64128C3semidirect
11. #88722C22semidirect
16. Q16×H(3,𝔽3)C22semidirect
18. #3755⋊C3C22semidirect
20. #3745⋊C3C22semidirect
21. C8×(C3⋊D3)⋊C3C22semidirect
22. C2×#3832C2×C3semidirect
25. C2×C32×Q16D3semidirect
26. C2×C32×Q16C2×C3semidirect
27. C2×#3755C2×C3semidirect
28. #64128C2×C3semidirect
29. C2×#3745C2×C3semidirect
30. C2×C8×C3⋊D3C2×C3semidirect
57. #3832C22×C3semidirect
69. C32×Q16C2×D3semidirect
70. C32×Q16C22×C3semidirect
74. #3755C22×C3semidirect
76. #3745C22×C3semidirect
77. C8×C3⋊D3C22×C3semidirect
146. C2×H(3,𝔽3)#135semidirect
216. C2×Q16(C3⋊D3)⋊C3semidirect
220. H(3,𝔽3)C2×#135semidirect
273. C2×C32#919semidirect
274. C2×C32C3×#135semidirect
288. Q16C2×(C3⋊D3)⋊C3semidirect
326. C32C2×#919semidirect
327. C32C2×C3×#135semidirect
372. C2#64128⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#887222
C2×Q16×H(3,𝔽3)2
C2×#3755⋊C32
#64128⋊C32
C2×#3745⋊C32
C2×C8×(C3⋊D3)⋊C32
C2×#641283
C2×C3×#9193