Finite Group Databasebeta

← C22⋊C4×#1113 #13178280

Subgroups — C22⋊C4×#1113 #13178280

Order 1728.

3126 conjugacy classes of subgroups; only the 327 normal subgroups are shown.

normal subgroups only

This group has 3126 conjugacy classes of subgroups — too many to draw. Only its 327 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
1C23×#1113C211yesmaximal, characteristic
2C2×C4×#1113C212yesmaximal, complemented
3C3×C22⋊C4×C3⋊D3C213yesmaximal, complemented
4#11723723C213yesmaximal, complemented
5#11723700C216yesmaximal, complemented
6D32×C22⋊C4—33nomaximal
7C23×C3×C3⋊D3C2213yes—
8C22×#1113—28no—
9C22×#1113C418yescomplemented
10C22×#1113C2213yes—
11C22×#1113C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2216yescomplemented
13C2×#6294C2216yescomplemented
14C4×#1113—28no—
15C33×C22⋊C4C2211yescharacteristic, complemented
16C3×#3757C2213yescomplemented
17C3×#3752C2216yescomplemented
18#88814C2216yescomplemented
19C23×D32—33no—
20C2×C4×D32—36no—
21C22⋊C4×C3⋊D3D313yescomplemented
22C3×D3×C22⋊C4—36no—
23C3×D3×C22⋊C4—63no—
24#63974—33no—
25D3×#217—36no—
26#63921—36no—
27D3×(C2×D3)⋊C4—312no—
28C23×C33C2311yescharacteristic
29C22×C3×C3⋊D3—212no—
30C22×C3×C3⋊D3C2×C4112yescomplemented
31C22×C3×C3⋊D3C2313yes—
32C22×C3×C3⋊D3C2313yes—
33C2×#1113—212no—
34C2×#1113—232no—
35C2×#1113C2×C414yes—
36C2×#1113C2×C4112yescomplemented
37C2×#1113D418yes—
38C2×C33×C4C2312yescomplemented
39C2×C3×C3⋊Dic3C2316yescomplemented
40C3×C4×C3⋊D3—212no—
41#6294—212no—
42C2×D3×C22⋊C4—93no—
43C23×C3⋊D3C2×D313yes—
44C23×C3×D3—36no—
45C23×C3×D3—63no—
46C22×D32—33no—
47C22×D32—324no—
48C22×D32—36no—
49C22×D32—33no—
50C22×D32—624no—
51C2×C4×C3⋊D3C2×D316yescomplemented
52C2×C3×C4×D3—312no—
53C2×C3×C4×D3—66no—
54C2×#624—36no—
55C2×D3×Dic3—312no—
56C4×D32—624no—
57C32×C22⋊C4—23no—
58C32×C22⋊C4—41no—
59C32×C22⋊C4C2×D313yescomplemented
60#3757C2×D313yescomplemented
61#3752C2×D316yescomplemented
62C3×#217—36no—
63C3×#217—63no—
64C3×(C2×D3)⋊C4—312no—
65C3×(C2×D3)⋊C4—66no—
66#3722—36no—
67(C2×D3)⋊Dic3—312no—
68C22×C33—22no—
69C22×C33C22×C412yescomplemented
70C22×C33C2411yescharacteristic
71C2×C3×C3⋊D3—26no—
72C2×C3×C3⋊D3—224no—
73C2×C3×C3⋊D3C22×C416yes—
74C2×C3×C3⋊D3C22×C416yescomplemented
75C2×C3×C3⋊D3C2×D4112yes—
76#1113—224no—
77#1113C22⋊C4116yesdirect product factor
78C33×C4—22no—
79C3×C3⋊Dic3—26no—
80C24×D3—93no—
81C22×C4×D3—96no—
82C2×C3×C22⋊C4—93no—
83C2×#217—93no—
84C2×(C2×D3)⋊C4—96no—
85D3×C22⋊C4—36no—
86D3×C22⋊C4—63no—
87D3×C22⋊C4—912no—
88D3×C22⋊C4—96no—
89C23×C32—23no—
90C23×C32—41no—
91C23×C32C22×D313yes—
92C22×C3⋊D3—212no—
93C22×C3⋊D3C4×D3112yescomplemented
94C22×C3⋊D3C22×D313yes—
95C22×C3⋊D3C22×D313yes—
96C22×C3×D3—36no—
97C22×C3×D3—324no—
98C22×C3×D3—36no—
99C22×C3×D3—63no—
100C22×C3×D3—63no—
101C22×C3×D3—612no—
102C22×C3×D3—624no—
103C22×C3×D3—1212no—
104C2×D32—312no—
105C2×D32—324no—
106C2×D32—312no—
107C2×D32—324no—
108C2×D32—696no—
109C2×D32—624no—
110C2×D32—612no—
111C2×C32×C4—26no—
112C2×C32×C4—42no—
113C2×C32×C4C22×D316yescomplemented
114C2×C3⋊Dic3C22×D316yescomplemented
115C4×C3⋊D3—212no—
116C2×C3×Dic3—312no—
117C2×C3×Dic3—66no—
118C3×C4×D3—624no—
119C3×C4×D3—1212no—
120#624—612no—
121D3×Dic3—624no—
122C22×C22⋊C4—271noSylow
123C2×C33—22no—
124C2×C33C23×C411yescharacteristic
125C2×C33C22×D412yes—
126C3×C3⋊D3—26no—
127C3×C3⋊D3C2×C22⋊C4112yescomplemented
128C24×C3—93no—
129C23×D3—36no—
130C23×D3—63no—
131C23×D3—93no—
132C23×D3—96no—
133C23×D3—93no—
134C23×D3—924no—
135C23×D3—1824no—
136C22×C3×C4—96no—
137C22×Dic3—96no—
138C2×C4×D3—312no—
139C2×C4×D3—66no—
140C2×C4×D3—912no—
141C2×C4×D3—1824no—
142C3×C22⋊C4—23no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—93no—
145C3×C22⋊C4—96no—
146C3×C22⋊C4D3213yescomplemented
147#217—36no—
148#217—63no—
149#217—96no—
150(C2×D3)⋊C4—312no—
151(C2×D3)⋊C4—66no—
152(C2×D3)⋊C4—912no—
153C22×C32—26no—
154C22×C32—23no—
155C22×C32—26no—
156C22×C32—41no—
157C22×C32—42no—
158C22×C32—46no—
159C22×C32—82no—
160C22×C32C2×C4×D316yescomplemented
161C22×C32C23×D313yes—
162C2×C3⋊D3—26no—
163C2×C3⋊D3—224no—
164C2×C3⋊D3C2×C4×D316yes—
165C2×C3⋊D3C2×C4×D316yescomplemented
166C2×C3⋊D3D3×D4112yes—
167C2×C3×D3—312no—
168C2×C3×D3—324no—
169C2×C3×D3—312no—
170C2×C3×D3—66no—
171C2×C3×D3—612no—
172C2×C3×D3—66no—
173C2×C3×D3—648no—
174C2×C3×D3—612no—
175C2×C3×D3—1224no—
176C2×C3×D3—126no—
177D32—348no—
178D32—648no—
179D32—624no—
180C32×C4—26no—
181C32×C4—46no—
182C32×C4—82no—
183C3⋊Dic3—26no—
184C3×Dic3—612no—
185C3×Dic3—126no—
186C25—271no—
187C23×C4—272no—
188C2×C22⋊C4—93no—
189C2×C22⋊C4—276no—
190C2×C22⋊C4—273no—
191C33C22×C22⋊C411yescharacteristic, complemented, Sylow
192C23×C3—23no—
193C23×C3—41no—
194C23×C3—93no—
195C23×C3—93no—
196C23×C3—912no—
197C23×C3—1812no—
198C23×C3C2×D3213yes—
199C22×D3—36no—
200C22×D3—36no—
201C22×D3—324no—
202C22×D3—63no—
203C22×D3—612no—
204C22×D3—63no—
205C22×D3—624no—
206C22×D3—924no—
207C22×D3—912no—
208C22×D3—924no—
209C22×D3—912no—
210C22×D3—1212no—
211C22×D3—1824no—
212C22×D3—1896no—
213C22×D3—1812no—
214C2×C3×C4—26no—
215C2×C3×C4—42no—
216C2×C3×C4—96no—
217C2×C3×C4—1812no—
218C2×C3×C4C2×D3216yescomplemented
219C2×Dic3—312no—
220C2×Dic3—66no—
221C2×Dic3—1812no—
222C4×D3—624no—
223C4×D3—1212no—
224C4×D3—1824no—
225C2×C32—26no—
226C2×C32—23no—
227C2×C32—26no—
228C2×C32—41no—
229C2×C32—42no—
230C2×C32—46no—
231C2×C32—82no—
232C2×C32C22×C4×D313yes—
233C2×C32C2×D3×D416yes—
234C3⋊D3—26no—
235C3⋊D3D3×C22⋊C4112yescomplemented
236C3×D3—324no—
237C3×D3—612no—
238C3×D3—612no—
239C3×D3—126no—
240C24—93no—
241C24—271no—
242C24—273no—
243C24—278no—
244C24—548no—
245C22×C4—96no—
246C22×C4—276no—
247C22×C4—548no—
248C22⋊C4—93no—
249C22⋊C4—96no—
250C22⋊C4—276no—
251C22⋊C4#111311yescharacteristic, direct product factor
252C22×C3—26no—
253C22×C3—23no—
254C22×C3—26no—
255C22×C3—46no—
256C22×C3—41no—
257C22×C3—42no—
258C22×C3—82no—
259C22×C3—96no—
260C22×C3—912no—
261C22×C3—96no—
262C22×C3—1824no—
263C22×C3—186no—
264C22×C3C4×D3216yescomplemented
265C22×C3C22×D3213yes—
266C2×D3—312no—
267C2×D3—324no—
268C2×D3—312no—
269C2×D3—612no—
270C2×D3—66no—
271C2×D3—66no—
272C2×D3—648no—
273C2×D3—612no—
274C2×D3—948no—
275C2×D3—1224no—
276C2×D3—126no—
277C2×D3—1848no—
278C2×D3—1824no—
279C3×C4—26no—
280C3×C4—46no—
281C3×C4—82no—
282C3×C4—186no—
283Dic3—612no—
284Dic3—126no—
285C32—23no—
286C32—41no—
287C32C2×D3×C22⋊C413yescomplemented
288C23—93no—
289C23—93no—
290C23—912no—
291C23—1812no—
292C23—278no—
293C23—274no—
294C23—2712no—
295C23—5412no—
296C23—5432no—
297C23C2×#111311yescharacteristic
298C2×C4—96no—
299C2×C4—1812no—
300C2×C4—5412no—
301C2×C4C2×#111312yescomplemented
302C2×C3—26no—
303C2×C3—23no—
304C2×C3—26no—
305C2×C3—46no—
306C2×C3—41no—
307C2×C3—42no—
308C2×C3—82no—
309C2×C3—912no—
310C2×C3—186no—
311C2×C3C2×C4×D3213yes—
312C2×C3D32×D416yes—
313D3—324no—
314D3—612no—
315D3—612no—
316D3—126no—
317C22—22no—
318C22—96no—
319C22—912no—
320C22—96no—
321C22—1824no—
322C22—186no—
323C22—2716no—
324C22—5424no—
325C22C4×#111312yescomplemented
326C22C22×#111311yescharacteristic
327C4—22no—
328C4—186no—
329C3—23no—
330C3—41no—
331C3D32×C22⋊C413yescomplemented
332C2—22no—
333C2—912no—
334C2—186no—
335C2C2×C4×#111311yescharacteristic
336C2D4×#111312yes—
subgroupG/H
center, FrattiniC22C22×#1113
commutatorC2×C33C23×C4
FittingC33×C22⋊C4C22
socle, cosocleC22×C33C24
prime radicalC23×#1113C2
hypercenterC22⋊C4#1113
hypocenterC33C22×C22⋊C4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×#1113C2semidirect
3. C3×C22⋊C4×C3⋊D3C2semidirect
4. #11723723C2semidirect
5. #11723700C2semidirect
9. C22×#1113C4semidirect
12. C2×C3×C4×C3⋊D3C22semidirect
13. C2×#6294C22semidirect
15. C33×C22⋊C4C22semidirect
16. C3×#3757C22semidirect
17. C3×#3752C22semidirect
18. #88814C22semidirect
21. C22⋊C4×C3⋊D3D3semidirect
30. C22×C3×C3⋊D3C2×C4semidirect
36. C2×#1113C2×C4semidirect
38. C2×C33×C4C23semidirect
39. C2×C3×C3⋊Dic3C23semidirect
51. C2×C4×C3⋊D3C2×D3semidirect
59. C32×C22⋊C4C2×D3semidirect
60. #3757C2×D3semidirect
61. #3752C2×D3semidirect
69. C22×C33C22×C4semidirect
74. C2×C3×C3⋊D3C22×C4semidirect
77. #1113C22⋊C4direct (N × H)
93. C22×C3⋊D3C4×D3semidirect
113. C2×C32×C4C22×D3semidirect
114. C2×C3⋊Dic3C22×D3semidirect
127. C3×C3⋊D3C2×C22⋊C4semidirect
146. C3×C22⋊C4D32semidirect
160. C22×C32C2×C4×D3semidirect
165. C2×C3⋊D3C2×C4×D3semidirect
191. C33C22×C22⋊C4semidirect
218. C2×C3×C4C2×D32semidirect
235. C3⋊D3D3×C22⋊C4semidirect
251. C22⋊C4#1113direct (N × H)
264. C22×C3C4×D32semidirect
287. C32C2×D3×C22⋊C4semidirect
301. C2×C4C2×#1113semidirect
325. C22C4×#1113semidirect
331. C3D32×C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C23×#11132
C2×C4×#11132
C3×C22⋊C4×C3⋊D32
#117237232
#117237002
D32×C22⋊C43