Finite Group Databasebeta

← C2×#11722521 #13164768

Subgroups — C2×#11722521 #13164768

Order 1728.

3307 conjugacy classes of subgroups; only the 555 normal subgroups are shown.

normal subgroups only

This group has 3307 conjugacy classes of subgroups — too many to draw. Only its 555 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×#88748C211yesmaximal, characteristic, complemented
2C2×#88722C212yesmaximal, complemented
3#11722521C2116yesmaximal, direct product factor
4C2×#88655C213yesmaximal, complemented
5C2×#88654C216yesmaximal, complemented
6C2×#88650C213yesmaximal, complemented
7C2×#64320—32nomaximal
8C22×((C3⋊D3)⋊C3)⋊C2C2216yescomplemented
9C2×Q8×H(3,𝔽3)C2211yescharacteristic, complemented
10#88748C2218yescomplemented
11C2×C4⋊#272C2213yescomplemented
12C2×C4×#272C2213yescomplemented
13#88722C22116yescomplemented
14C2×C4⋊((C3⋊D3)⋊C3)C2216yescomplemented
15C2×C4×(C3⋊D3)⋊C3C2216yescomplemented
16C2×#6200C2216yescomplemented
17C2×#6198C2213yescomplemented
18C2×#6196C2211yescharacteristic, complemented
19#88655C22124yescomplemented
20#88654C22148yescomplemented
21#88650C22124yescomplemented
22C2×#3832D312yescomplemented
23C2×C3×#239—32no—
24C2×C3×#239—32no—
25C2×C3×#239—61no—
26#64320—332no—
27C2×#3802—36no—
28C2×D3×D12—36no—
29C2×D3×D12—36no—
30C2×#3798—36no—
31C22×#272C2313yes—
32C22×(C3⋊D3)⋊C3C2316yes—
33C2×((C3⋊D3)⋊C3)⋊C2—412no—
34C2×((C3⋊D3)⋊C3)⋊C2C23124yescomplemented
35Q8×H(3,𝔽3)C2314yescomplemented
36C2×C4×H(3,𝔽3)C2313yes—
37C2×#1084C2311yescharacteristic
38C4⋊#272C23112yescomplemented
39C4×#272C23112yescomplemented
40C2×(C3⋊Dic3)⋊C3C2312yes—
41C4⋊((C3⋊D3)⋊C3)C23124yescomplemented
42C4×(C3⋊D3)⋊C3C23124yescomplemented
43#6200C23124yescomplemented
44#6198C23112yescomplemented
45#6196C2314yescomplemented
46C2×#1009—91no—
47C2×#1009—92no—
48C22×D32—312no—
49C2×C32×Q8—21no—
50C2×C32×Q8C2×D312yescomplemented
51#3832C2×D3116yescomplemented
52C2×C4⋊(C3⋊D3)C2×D316yescomplemented
53C2×C4×C3⋊D3C2×D316yescomplemented
54C3×#239—316no—
55C3×#239—316no—
56C3×#239—68no—
57C2×C3×D12—36no—
58C2×C3×D12—36no—
59C2×C3×D12—63no—
60C2×C3×C4×D3—36no—
61C2×C3×C4×D3—36no—
62C2×C3×C4×D3—63no—
63C2×#627—32no—
64C2×#626—36no—
65C2×#626—36no—
66C2×#625—36no—
67#3802—348no—
68D3×D12—348no—
69D3×D12—348no—
70#3798—348no—
71C22×H(3,𝔽3)C2411yescharacteristic
72C2×#272—26no—
73C2×#272C2416yes—
74C2×(C3⋊D3)⋊C3—212no—
75C2×(C3⋊D3)⋊C3C24112yes—
76((C3⋊D3)⋊C3)⋊C2—424no—
77C4×H(3,𝔽3)C2416yes—
78#1084C2412yes—
79(C3⋊Dic3)⋊C3C2414yes—
80C2×C3×#41—92no—
81C2×C3×#41—91no—
82#1009—916no—
83#1009—932no—
84C2×#239—32no—
85C2×#239—32no—
86C2×#239—61no—
87C2×#239—92no—
88C2×D3×D4—96no—
89C2×D3×D4—96no—
90C2×D3×D4—96no—
91C2×#235—93no—
92C2×#235—96no—
93C22×D12—93no—
94C22×D12—96no—
95C22×C3⋊D3C22×D316yes—
96C22×C3×D3—36no—
97C22×C3×D3—36no—
98C22×C3×D3—63no—
99C2×D32—348no—
100C2×D32—1224no—
101C32×Q8—24no—
102C32×Q8C22×D318yescomplemented
103C2×C32×C4—23no—
104C2×C32×C4C22×D316yes—
105C2×C3⋊Dic3C22×D312yes—
106C4⋊(C3⋊D3)C22×D3124yescomplemented
107C4×C3⋊D3C22×D3124yescomplemented
108C2×C3×Dic3—32no—
109C2×C3×Dic3—32no—
110C2×C3×Dic3—61no—
111C3×D12—324no—
112C3×D12—324no—
113C3×D12—612no—
114C3×C4×D3—324no—
115C3×C4×D3—324no—
116C3×C4×D3—612no—
117#627—38no—
118#626—324no—
119#626—324no—
120#625—324no—
121C2×H(5,𝔽2)—271noSylow
122C2×H(3,𝔽3)H(5,𝔽2)12yescomplemented
123C2×H(3,𝔽3)C2511yescharacteristic
124#272—26no—
125(C3⋊D3)⋊C3—212no—
126C23×D3—96no—
127C23×D3—912no—
128C3×#41—916no—
129C3×#41—98no—
130C2×C3×Q8—32no—
131C2×C3×Q8—61no—
132C2×C3×Q8D3211yescharacteristic
133C2×C3×D4—96no—
134C2×C3×D4—93no—
135C22×C3×C4—96no—
136C22×C3×C4—93no—
137C2×#67—96no—
138C2×#67—96no—
139C2×#67—96no—
140#239—316no—
141#239—316no—
142#239—68no—
143#239—916no—
144D3×D4—948no—
145D3×D4—948no—
146D3×D4—948no—
147#235—924no—
148#235—948no—
149C2×D12—36no—
150C2×D12—36no—
151C2×D12—63no—
152C2×D12—96no—
153C2×D12—924no—
154C2×D12—948no—
155C2×D12—93no—
156C2×D12—96no—
157C2×C4×D3—36no—
158C2×C4×D3—36no—
159C2×C4×D3—63no—
160C2×C4×D3—96no—
161C2×Dic6—91no—
162C2×Dic6—92no—
163C22×C32—21no—
164C22×C32C23×D312yes—
165C2×C3⋊D3—212no—
166C2×C3⋊D3C23×D3112yes—
167C2×C3×D3—312no—
168C2×C3×D3—312no—
169C2×C3×D3—66no—
170C2×C3×D3—612no—
171C2×C3×D3—612no—
172C2×C3×D3—126no—
173D32—1248no—
174C32×C4—26no—
175C32×C4C23×D3112yes—
176C3⋊Dic3C23×D314yes—
177C3×Dic3—34no—
178C3×Dic3—34no—
179C3×Dic3—62no—
180H(5,𝔽2)—2716no—
181C2×#41—91no—
182C2×#41—92no—
183C2×#41—273no—
184C22×D4—273no—
185C22×D4—276no—
186H(3,𝔽3)C2×H(5,𝔽2)11yescharacteristic, complemented, Sylow
187C23×C3—96no—
188C23×C3—93no—
189C22×D3—36no—
190C22×D3—36no—
191C22×D3—63no—
192C22×D3—96no—
193C22×D3—924no—
194C22×D3—948no—
195C22×D3—3612no—
196C22×D3—3624no—
197C3×Q8—38no—
198C3×Q8—64no—
199C3×Q8C2×D3214yes—
200C3×D4—924no—
201C3×D4—912no—
202C2×C3×C4—36no—
203C2×C3×C4—63no—
204C2×C3×C4—924no—
205C2×C3×C4—92no—
206C2×C3×C4—91no—
207C2×C3×C4—912no—
208C2×C3×C4C2×D3213yes—
209#67—924no—
210#67—924no—
211#67—924no—
212C2×Dic3—32no—
213C2×Dic3—32no—
214C2×Dic3—61no—
215C2×Dic3—92no—
216D12—324no—
217D12—324no—
218D12—612no—
219D12—924no—
220D12—912no—
221D12—924no—
222C4×D3—324no—
223C4×D3—324no—
224C4×D3—612no—
225C4×D3—924no—
226Dic6—94no—
227Dic6—98no—
228C2×C32—22no—
229C2×C32—21no—
230C2×C32#100914yescomplemented
231C2×C32C24×D312yes—
232C3⋊D3—212no—
233C3×D3—612no—
234C3×D3—612no—
235C3×D3—126no—
236C24—276no—
237#41—98no—
238#41—916no—
239#41—2724no—
240C2×Q8—271no—
241C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
242C2×D4—93no—
243C2×D4—96no—
244C2×D4—2724no—
245C2×D4—2748no—
246C2×D4—273no—
247C2×D4—276no—
248C22×C4—93no—
249C22×C4—96no—
250C22×C3—32no—
251C22×C3—61no—
252C22×C3—912no—
253C22×C3—96no—
254C22×C3—1812no—
255C22×C3—186no—
256C22×C3C22×D3211yescharacteristic
257C2×D3—312no—
258C2×D3—312no—
259C2×D3—66no—
260C2×D3—612no—
261C2×D3—612no—
262C2×D3—912no—
263C2×D3—126no—
264C2×D3—1812no—
265C2×D3—3624no—
266C2×D3—3648no—
267C3×C4—312no—
268C3×C4—66no—
269C3×C4—94no—
270C3×C4—92no—
271C3×C4C22×D3216yes—
272Dic3—34no—
273Dic3—34no—
274Dic3—62no—
275Dic3—94no—
276C32—21no—
277C32C2×#100912yescomplemented
278C23—93no—
279C23—96no—
280C23—2724no—
281C23—10812no—
282Q8—274no—
283Q8C2×((C3⋊D3)⋊C3)⋊C214yescomplemented
284D4—912no—
285D4—924no—
286D4—2712no—
287D4—2724no—
288C2×C4—91no—
289C2×C4—92no—
290C2×C4—912no—
291C2×C4—924no—
292C2×C4C2×((C3⋊D3)⋊C3)⋊C213yes—
293C2×C3—34no—
294C2×C3—32no—
295C2×C3—62no—
296C2×C3—61no—
297C2×C3—1812no—
298C2×C3—186no—
299C2×C3#6432012yes—
300C2×C3C23×D3211yescharacteristic
301D3—612no—
302D3—612no—
303D3—126no—
304D3—1812no—
305C22—96no—
306C22—912no—
307C22—186no—
308C22—1812no—
309C22—10824no—
310C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
311C4—92no—
312C4—94no—
313C4C22×((C3⋊D3)⋊C3)⋊C216yes—
314C3—32no—
315C3—61no—
316C3C2×#6432011yescharacteristic
317C2—186no—
318C2—1812no—
319C2#1172252112yesdirect product factor
320C2C23×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC2×H(3,𝔽3)C25
FrattiniC2×C3C23×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3C22×D32
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×H(5,𝔽2)
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#88748C2semidirect
2. C2×#88722C2semidirect
3. #11722521C2direct (N × H)
4. C2×#88655C2semidirect
5. C2×#88654C2semidirect
6. C2×#88650C2semidirect
8. C22×((C3⋊D3)⋊C3)⋊C2C22semidirect
9. C2×Q8×H(3,𝔽3)C22semidirect
10. #88748C22semidirect
11. C2×C4⋊#272C22semidirect
12. C2×C4×#272C22semidirect
13. #88722C22semidirect
14. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
15. C2×C4×(C3⋊D3)⋊C3C22semidirect
16. C2×#6200C22semidirect
17. C2×#6198C22semidirect
18. C2×#6196C22semidirect
19. #88655C22semidirect
20. #88654C22semidirect
21. #88650C22semidirect
22. C2×#3832D3semidirect
34. C2×((C3⋊D3)⋊C3)⋊C2C23semidirect
35. Q8×H(3,𝔽3)C23semidirect
38. C4⋊#272C23semidirect
39. C4×#272C23semidirect
41. C4⋊((C3⋊D3)⋊C3)C23semidirect
42. C4×(C3⋊D3)⋊C3C23semidirect
43. #6200C23semidirect
44. #6198C23semidirect
45. #6196C23semidirect
50. C2×C32×Q8C2×D3semidirect
51. #3832C2×D3semidirect
52. C2×C4⋊(C3⋊D3)C2×D3semidirect
53. C2×C4×C3⋊D3C2×D3semidirect
102. C32×Q8C22×D3semidirect
106. C4⋊(C3⋊D3)C22×D3semidirect
107. C4×C3⋊D3C22×D3semidirect
122. C2×H(3,𝔽3)H(5,𝔽2)semidirect
186. H(3,𝔽3)C2×H(5,𝔽2)semidirect
230. C2×C32#1009semidirect
241. C2×Q8((C3⋊D3)⋊C3)⋊C2semidirect
277. C32C2×#1009semidirect
283. Q8C2×((C3⋊D3)⋊C3)⋊C2semidirect
319. C2#11722521direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#887482
C2×#887222
#117225212
C2×#886552
C2×#886542
C2×#886502
C2×#643203