Finite Group Databasebeta

← C2×#11722520 #13164767

Subgroups — C2×#11722520 #13164767

Order 1728.

3271 conjugacy classes of subgroups; only the 563 normal subgroups are shown.

normal subgroups only

This group has 3271 conjugacy classes of subgroups — too many to draw. Only its 563 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×Q8×#272C211yesmaximal, characteristic, complemented
2C2×#88722C212yesmaximal, complemented
3#11722520C2116yesmaximal, direct product factor
4C2×#88655C213yesmaximal, complemented
5C2×C4×((C3⋊D3)⋊C3)⋊C2C213yesmaximal, complemented
6C2×#88648C216yesmaximal, complemented
7C2×D3×#239—32nomaximal
8C22×((C3⋊D3)⋊C3)⋊C2C2213yes—
9C2×Q8×H(3,𝔽3)C2211yescharacteristic, complemented
10Q8×#272C2218yescomplemented
11C2×C4×#272C2213yes—
12C2×#6229C2213yescomplemented
13#88722C22116yescomplemented
14C2×C4⋊((C3⋊D3)⋊C3)C2216yescomplemented
15C2×C4×(C3⋊D3)⋊C3C2216yescomplemented
16C2×#6199C2211yescharacteristic
17C2×#6198C2216yescomplemented
18C2×((C3⋊D3)⋊C3)⋊C4C2216yescomplemented
19#88655C22124yescomplemented
20C4×((C3⋊D3)⋊C3)⋊C2C22124yescomplemented
21#88648C22148yescomplemented
22C2×#3832D312yescomplemented
23C2×C3×#239—32no—
24C2×C3×D3×Q8—32no—
25C2×C3×D3×Q8—61no—
26D3×#239—332no—
27C2×D3×D12—36no—
28C2×C4×D32—36no—
29C2×#3799—36no—
30C2×#3796—36no—
31C22×#272C2311yescharacteristic
32C22×(C3⋊D3)⋊C3C2316yes—
33C2×((C3⋊D3)⋊C3)⋊C2—212no—
34C2×((C3⋊D3)⋊C3)⋊C2C23112yes—
35Q8×H(3,𝔽3)C2314yescomplemented
36C2×C4×H(3,𝔽3)C2313yes—
37C2×#1084C2313yes—
38C4×#272C23112yes—
39#6229C23112yescomplemented
40C2×(C3⋊Dic3)⋊C3C2312yes—
41C4⋊((C3⋊D3)⋊C3)C23124yescomplemented
42C4×(C3⋊D3)⋊C3C23124yescomplemented
43#6199C2314yes—
44#6198C23124yescomplemented
45((C3⋊D3)⋊C3)⋊C4C23124yescomplemented
46C2×D3×#41—92no—
47C22×#239—91no—
48C22×D32—36no—
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×D3×Q8—316no—
56C3×D3×Q8—68no—
57C2×C3×D12—36no—
58C2×C3×C4×D3—36no—
59C2×C3×C4×D3—36no—
60C2×C3×C4×D3—63no—
61C2×C3×Dic6—36no—
62C2×C3×Dic6—63no—
63C2×#626—312no—
64C2×#624—36no—
65C2×D3×Dic3—36no—
66C2×D3×Dic3—32no—
67D3×D12—348no—
68C4×D32—348no—
69#3799—348no—
70#3796—348no—
71C22×H(3,𝔽3)C2411yescharacteristic
72C2×#272#4114yescomplemented
73C2×#272C2412yes—
74C2×(C3⋊D3)⋊C3—212no—
75C2×(C3⋊D3)⋊C3C24112yes—
76((C3⋊D3)⋊C3)⋊C2—224no—
77C4×H(3,𝔽3)C2416yes—
78#1084C2416yes—
79(C3⋊Dic3)⋊C3C2414yes—
80C2×C3×#41—92no—
81C22×C3×Q8—91no—
82D3×#41—932no—
83C2×#239—32no—
84C2×#239—92no—
85C2×#239—96no—
86C2×#239—916no—
87C2×D3×Q8—32no—
88C2×D3×Q8—61no—
89C2×#237—96no—
90C2×D3×D4—96no—
91C2×#235—96no—
92C22×D12—93no—
93C22×C4×D3—93no—
94C22×C4×D3—96no—
95C22×C3⋊D3C22×D316yes—
96C22×C3×D3—32no—
97C22×C3×D3—36no—
98C22×C3×D3—61no—
99C2×D32—324no—
100C2×D32—624no—
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—36no—
110C2×C3×Dic3—63no—
111C3×D12—324no—
112C3×C4×D3—324no—
113C3×C4×D3—324no—
114C3×C4×D3—612no—
115C3×Dic6—324no—
116C3×Dic6—612no—
117#626—348no—
118#624—324no—
119D3×Dic3—324no—
120D3×Dic3—38no—
121C22×#41—271noSylow
122C2×H(3,𝔽3)C2×#4112yescomplemented
123C2×H(3,𝔽3)C2511yescharacteristic
124#272C2×#4114yescomplemented
125(C3⋊D3)⋊C3—212no—
126C23×D3—93no—
127C23×D3—96no—
128C3×#41—916no—
129C2×C3×Q8—32no—
130C2×C3×Q8—61no—
131C2×C3×Q8—98no—
132C2×C3×Q8—93no—
133C2×C3×Q8D3211yescharacteristic
134C2×C3×D4—96no—
135C22×C3×C4—96no—
136C22×C3×C4—93no—
137C2×#67—912no—
138C22×Dic3—91no—
139C22×Dic3—96no—
140#239—316no—
141#239—916no—
142#239—948no—
143D3×Q8—316no—
144D3×Q8—68no—
145#237—948no—
146D3×D4—948no—
147#235—948no—
148C2×D12—36no—
149C2×D12—96no—
150C2×D12—924no—
151C2×D12—96no—
152C2×C4×D3—36no—
153C2×C4×D3—36no—
154C2×C4×D3—63no—
155C2×C4×D3—96no—
156C2×C4×D3—924no—
157C2×C4×D3—948no—
158C2×C4×D3—96no—
159C2×C4×D3—96no—
160C2×C4×D3—92no—
161C2×Dic6—36no—
162C2×Dic6—63no—
163C22×C32—21no—
164C22×C32C23×D312yes—
165C2×C3⋊D3—212no—
166C2×C3⋊D3C23×D3112yes—
167C2×C3×D3—38no—
168C2×C3×D3—34no—
169C2×C3×D3—312no—
170C2×C3×D3—64no—
171C2×C3×D3—62no—
172C2×C3×D3—612no—
173D32—648no—
174C32×C4—26no—
175C32×C4C23×D3112yes—
176C3⋊Dic3C23×D314yes—
177C3×Dic3—34no—
178C3×Dic3—312no—
179C3×Dic3—66no—
180C2×#41—92no—
181C2×#41—276no—
182C2×#41—2716no—
183C22×Q8—91no—
184C22×D4—273no—
185C23×C4—273no—
186H(3,𝔽3)C22×#4111yescharacteristic, complemented, Sylow
187C23×C3—96no—
188C23×C3—91no—
189C22×D3—32no—
190C22×D3—36no—
191C22×D3—61no—
192C22×D3—96no—
193C22×D3—912no—
194C22×D3—924no—
195C22×D3—1824no—
196C22×D3—1812no—
197C3×Q8—38no—
198C3×Q8—64no—
199C3×Q8—912no—
200C3×Q8C2×D3214yes—
201C3×D4—924no—
202C2×C3×C4—36no—
203C2×C3×C4—63no—
204C2×C3×C4—924no—
205C2×C3×C4—92no—
206C2×C3×C4—93no—
207C2×C3×C4—912no—
208C2×C3×C4C2×D3213yes—
209#67—948no—
210C2×Dic3—36no—
211C2×Dic3—32no—
212C2×Dic3—63no—
213C2×Dic3—92no—
214C2×Dic3—94no—
215C2×Dic3—924no—
216D12—324no—
217D12—924no—
218D12—924no—
219C4×D3—324no—
220C4×D3—324no—
221C4×D3—612no—
222C4×D3—924no—
223C4×D3—924no—
224C4×D3—924no—
225C4×D3—98no—
226Dic6—324no—
227Dic6—612no—
228C2×C32—22no—
229C2×C32—21no—
230C2×C32D3×#4114yescomplemented
231C2×C32C24×D312yes—
232C3⋊D3—212no—
233C3×D3—38no—
234C3×D3—64no—
235C3×D3—612no—
236C24—273no—
237#41—916no—
238#41—2748no—
239C2×Q8—98no—
240C2×Q8—93no—
241C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
242C2×D4—96no—
243C2×D4—2724no—
244C2×D4—276no—
245C22×C4—93no—
246C22×C4—96no—
247C22×C4—2724no—
248C22×C4—271no—
249C22×C4—276no—
250C22×C3—32no—
251C22×C3—61no—
252C22×C3—912no—
253C22×C3—94no—
254C22×C3—92no—
255C22×C3—1812no—
256C22×C3C22×D3211yescharacteristic
257C2×D3—34no—
258C2×D3—38no—
259C2×D3—312no—
260C2×D3—62no—
261C2×D3—64no—
262C2×D3—612no—
263C2×D3—912no—
264C2×D3—1812no—
265C2×D3—1848no—
266C2×D3—1824no—
267C3×C4—312no—
268C3×C4—66no—
269C3×C4—94no—
270C3×C4—96no—
271C3×C4C22×D3216yes—
272Dic3—312no—
273Dic3—34no—
274Dic3—66no—
275Dic3—94no—
276C32—21no—
277C32C2×D3×#4112yescomplemented
278C23—91no—
279C23—96no—
280C23—2712no—
281C23—5412no—
282Q8—912no—
283Q8C2×((C3⋊D3)⋊C3)⋊C214yescomplemented
284D4—924no—
285D4—2724no—
286C2×C4—92no—
287C2×C4—93no—
288C2×C4—912no—
289C2×C4—924no—
290C2×C4—274no—
291C2×C4—2724no—
292C2×C4C2×((C3⋊D3)⋊C3)⋊C213yes—
293C2×C3—34no—
294C2×C3—32no—
295C2×C3—62no—
296C2×C3—61no—
297C2×C3—94no—
298C2×C3—1812no—
299C2×C3#6431912yes—
300C2×C3C23×D3211yescharacteristic
301D3—38no—
302D3—64no—
303D3—612no—
304D3—1812no—
305C22—94no—
306C22—92no—
307C22—912no—
308C22—1812no—
309C22—5424no—
310C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
311C4—94no—
312C4—96no—
313C4C22×((C3⋊D3)⋊C3)⋊C216yes—
314C3—32no—
315C3—61no—
316C3C2×#6431911yescharacteristic
317C2—94no—
318C2—1812no—
319C2#1172252012yesdirect 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)C22×#41
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×Q8×#272C2semidirect
2. C2×#88722C2semidirect
3. #11722520C2direct (N × H)
4. C2×#88655C2semidirect
5. C2×C4×((C3⋊D3)⋊C3)⋊C2C2semidirect
6. C2×#88648C2semidirect
9. C2×Q8×H(3,𝔽3)C22semidirect
10. Q8×#272C22semidirect
12. C2×#6229C22semidirect
13. #88722C22semidirect
14. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
15. C2×C4×(C3⋊D3)⋊C3C22semidirect
17. C2×#6198C22semidirect
18. C2×((C3⋊D3)⋊C3)⋊C4C22semidirect
19. #88655C22semidirect
20. C4×((C3⋊D3)⋊C3)⋊C2C22semidirect
21. #88648C22semidirect
22. C2×#3832D3semidirect
35. Q8×H(3,𝔽3)C23semidirect
39. #6229C23semidirect
41. C4⋊((C3⋊D3)⋊C3)C23semidirect
42. C4×(C3⋊D3)⋊C3C23semidirect
44. #6198C23semidirect
45. ((C3⋊D3)⋊C3)⋊C4C23semidirect
50. C2×C32×Q8C2×D3semidirect
51. #3832C2×D3semidirect
52. C2×C4⋊(C3⋊D3)C2×D3semidirect
53. C2×C4×C3⋊D3C2×D3semidirect
72. C2×#272#41semidirect
102. C32×Q8C22×D3semidirect
106. C4⋊(C3⋊D3)C22×D3semidirect
107. C4×C3⋊D3C22×D3semidirect
122. C2×H(3,𝔽3)C2×#41semidirect
124. #272C2×#41semidirect
186. H(3,𝔽3)C22×#41semidirect
230. C2×C32D3×#41semidirect
241. C2×Q8((C3⋊D3)⋊C3)⋊C2semidirect
277. C32C2×D3×#41semidirect
283. Q8C2×((C3⋊D3)⋊C3)⋊C2semidirect
319. C2#11722520direct (N × H)

Maximal subgroups

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

subgroupindex
C2×Q8×#2722
C2×#887222
#117225202
C2×#886552
C2×C4×((C3⋊D3)⋊C3)⋊C22
C2×#886482
C2×D3×#2393