Finite Group Databasebeta

← C22×(C2×D3)⋊#157 #13180996

Subgroups — C22×(C2×D3)⋊#157 #13180996

Order 1728.

3112 conjugacy classes of subgroups; only the 290 normal subgroups are shown.

normal subgroups only

This group has 3112 conjugacy classes of subgroups — too many to draw. Only its 290 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×D3×C3⋊D3C211yesmaximal, characteristic
2C23×C3⋊#157C211yesmaximal, characteristic, complemented
3C23×C3×#157C211yesmaximal, characteristic, complemented
4C2×(C2×D3)⋊#157C2112yesmaximal, direct product factor
5C22×C22⋊#157—31nomaximal
6C23×C3⋊(C3⋊D3)C411yescharacteristic, complemented
7C23×C3×C3⋊D3C2211yescharacteristic
8C23×C32×D3C411yescharacteristic, complemented
9C22×D3×C3⋊D3—24no—
10C22×D3×C3⋊D3—24no—
11C22×D3×C3⋊D3C416yescomplemented
12C22×D3×C3⋊D3C2216yes—
13C22×C3⋊#157—24no—
14C22×C3⋊#157C2216yescomplemented
15C22×C3×#157—24no—
16C22×C3×#157C2216yescomplemented
17(C2×D3)⋊#157C22116yesdirect product factor
18C24×C3⋊D3—31no—
19C23×D32—62no—
20C23×#157—31no—
21C23×#157D311yescharacteristic, complemented
22C2×C22⋊#157—312no—
23C23×C33C2×C411yescharacteristic
24C22×C3⋊(C3⋊D3)—24no—
25C22×C3⋊(C3⋊D3)C2×C416yescomplemented
26C22×C3×C3⋊D3C2×C413yes—
27C22×C3×C3⋊D3D414yes—
28C22×C3×C3⋊D3D414yes—
29C22×C3×C3⋊D3C2313yes—
30C22×C32×D3—24no—
31C22×C32×D3C2×C416yescomplemented
32C2×D3×C3⋊D3—224no—
33C2×D3×C3⋊D3—212no—
34C2×D3×C3⋊D3—212no—
35C2×D3×C3⋊D3C2×C4112yescomplemented
36C2×D3×C3⋊D3C2314yes—
37C2×C3⋊#157—212no—
38C2×C3⋊#157C2314yescomplemented
39C2×C3×#157—212no—
40C2×C3×#157C2314yescomplemented
41C22×(C2×D3)⋊C4—91nomaximal
42C24×C32—31no—
43C23×C3⋊D3—31no—
44C23×C3⋊D3—36no—
45C23×C3⋊D3—36no—
46C23×C3⋊D3—62no—
47C23×C3⋊D3—64no—
48C23×C3⋊D3—64no—
49C23×C3⋊D3—122no—
50C23×C3⋊D3C2×D311yescharacteristic
51C23×C3×D3—22no—
52C23×C3×D3—62no—
53C22×D32—612no—
54C22×D32—612no—
55C22×D32—616no—
56C22×D32—616no—
57C22×#157—24no—
58C22×#157—36no—
59C22×#157—64no—
60C22×#157C2×D316yescomplemented
61C22⋊#157—316no—
62C22×C33C22⋊C414yescomplemented
63C22×C33C22×C413yes—
64C2×C3⋊(C3⋊D3)—212no—
65C2×C3⋊(C3⋊D3)C22×C414yescomplemented
66C2×C3×C3⋊D3C22⋊C4112yescomplemented
67C2×C3×C3⋊D3C22×C413yes—
68C2×C3×C3⋊D3C2×D416yes—
69C2×C3×C3⋊D3C2×D416yes—
70C2×C3×C3⋊D3C2411yescharacteristic
71C2×C32×D3—212no—
72C2×C32×D3C22×C414yescomplemented
73D3×C3⋊D3—224no—
74D3×C3⋊D3—24no—
75D3×C3⋊D3—24no—
76C3⋊#157—24no—
77C3×#157—24no—
78C24×D3—91no—
79C24×D3—182no—
80C23×C3×C4—91no—
81C23×Dic3—91no—
82C2×(C2×D3)⋊C4—912no—
83C23×C32—22no—
84C23×C32—36no—
85C23×C32—42no—
86C23×C32—64no—
87C23×C32C4×D311yescharacteristic
88C22×C3⋊D3—36no—
89C22×C3⋊D3—312no—
90C22×C3⋊D3—34no—
91C22×C3⋊D3—612no—
92C22×C3⋊D3—64no—
93C22×C3⋊D3—624no—
94C22×C3⋊D3—612no—
95C22×C3⋊D3—612no—
96C22×C3⋊D3—616no—
97C22×C3⋊D3—1216no—
98C22×C3⋊D3—1212no—
99C22×C3⋊D3C4×D313yes—
100C22×C3⋊D3D1214yes—
101C22×C3⋊D3#6714yes—
102C22×C3⋊D3C22×D313yes—
103C22×C3×D3—216no—
104C22×C3×D3—212no—
105C22×C3×D3—66no—
106C22×C3×D3—68no—
107C22×C3×D3—68no—
108C22×C3×D3—66no—
109C2×D32—624no—
110C2×D32—68no—
111C2×D32—696no—
112C2×D32—648no—
113C2×D32—648no—
114C2×#157—212no—
115C2×#157—34no—
116C2×#157—612no—
117C2×#157C22×D314yescomplemented
118C22×C22⋊C4—271noSylow
119C2×C33C2×C22⋊C416yescomplemented
120C2×C33C23×C411yescharacteristic
121C3⋊(C3⋊D3)—24no—
122C3×C3⋊D3C2×C22⋊C416yescomplemented
123C3×C3⋊D3C22×D411yescharacteristic
124C3×C3⋊D3C22×D411yescharacteristic
125C32×D3—24no—
126C24×C3—62no—
127C24×C3—91no—
128C23×D3—62no—
129C23×D3—91no—
130C23×D3—96no—
131C23×D3—96no—
132C23×D3—1812no—
133C23×D3—1812no—
134C23×D3—182no—
135C23×D3—184no—
136C23×D3—184no—
137C23×D3—1816no—
138C23×D3—1816no—
139C23×D3—362no—
140C23×D3#15711yescharacteristic, complemented
141C22×C3×C4—96no—
142C22×C3×C4—184no—
143C22×Dic3—96no—
144C22×Dic3—184no—
145(C2×D3)⋊C4—916no—
146C22×C32—26no—
147C22×C32—28no—
148C22×C32—34no—
149C22×C32—48no—
150C22×C32—46no—
151C22×C32—612no—
152C22×C32(C2×D3)⋊C414yescomplemented
153C22×C32C2×C4×D313yes—
154C2×C3⋊D3—34no—
155C2×C3⋊D3—68no—
156C2×C3⋊D3—612no—
157C2×C3⋊D3—624no—
158C2×C3⋊D3—64no—
159C2×C3⋊D3—64no—
160C2×C3⋊D3—648no—
161C2×C3⋊D3—1248no—
162C2×C3⋊D3—128no—
163C2×C3⋊D3(C2×D3)⋊C4112yescomplemented
164C2×C3⋊D3C2×C4×D313yes—
165C2×C3⋊D3C2×D1216yes—
166C2×C3⋊D3C2×#6716yes—
167C2×C3⋊D3C23×D311yescharacteristic
168C2×C3×D3—248no—
169C2×C3×D3—28no—
170C2×C3×D3—624no—
171C2×C3×D3—66no—
172C2×C3×D3—612no—
173C2×C3×D3—612no—
174C2×C3×D3—62no—
175D32—696no—
176D32—616no—
177D32—616no—
178#157—24no—
179#157—64no—
180C25—271no—
181C23×C4—91no—
182C23×C4—271no—
183C2×C22⋊C4—2712no—
184C33C22×C22⋊C411yescharacteristic, complemented, Sylow
185C23×C3—22no—
186C23×C3—42no—
187C23×C3—612no—
188C23×C3—616no—
189C23×C3—93no—
190C23×C3—93no—
191C23×C3—94no—
192C23×C3—94no—
193C23×C3C2×#15711yescharacteristic
194C22×D3—24no—
195C22×D3—66no—
196C22×D3—68no—
197C22×D3—68no—
198C22×D3—66no—
199C22×D3—96no—
200C22×D3—94no—
201C22×D3—912no—
202C22×D3—188no—
203C22×D3—1824no—
204C22×D3—1812no—
205C22×D3—1812no—
206C22×D3—1812no—
207C22×D3—1824no—
208C22×D3—1896no—
209C22×D3—1848no—
210C22×D3—1848no—
211C22×D3—1816no—
212C22×D3—184no—
213C22×D3—3612no—
214C22×D3—3616no—
215C22×D3C2×#15716yescomplemented
216C2×C3×C4—94no—
217C2×C3×C4—1812no—
218C2×Dic3—94no—
219C2×Dic3—1812no—
220C2×C32—212no—
221C2×C32—22no—
222C2×C32—412no—
223C2×C32—42no—
224C2×C32—64no—
225C2×C32C2×(C2×D3)⋊C416yescomplemented
226C2×C32C22×C4×D311yescharacteristic
227C3⋊D3—64no—
228C3⋊D3—616no—
229C3⋊D3—1216no—
230C3⋊D3C2×(C2×D3)⋊C416yescomplemented
231C3⋊D3C22×D1211yescharacteristic
232C3⋊D3C22×#6711yescharacteristic
233C3×D3—216no—
234C3×D3—612no—
235C3×D3—62no—
236C3×D3—62no—
237C24—31no—
238C24—91no—
239C24—276no—
240C24—276no—
241C24—271no—
242C24—544no—
243C24—544no—
244C22×C4—96no—
245C22×C4—184no—
246C22×C4—276no—
247C22×C4—544no—
248C22⋊C4—2716no—
249C22×C3—26no—
250C22×C3—28no—
251C22×C3—46no—
252C22×C3—48no—
253C22×C3—68no—
254C22×C3—648no—
255C22×C3—96no—
256C22×C3—96no—
257C22×C3—912no—
258C22×C3—91no—
259C22×C3—93no—
260C22×C3C22⋊#15714yescomplemented
261C22×C3C22×#15713yes—
262C2×D3—212no—
263C2×D3—624no—
264C2×D3—66no—
265C2×D3—612no—
266C2×D3—612no—
267C2×D3—62no—
268C2×D3—94no—
269C2×D3—188no—
270C2×D3—184no—
271C2×D3—184no—
272C2×D3—1824no—
273C2×D3—1896no—
274C2×D3—1816no—
275C2×D3—1816no—
276C2×D3—1848no—
277C2×D3—1812no—
278C2×D3—368no—
279C2×D3—3648no—
280C2×D3C22×#15714yescomplemented
281C3×C4—184no—
282Dic3—184no—
283C32—22no—
284C32—42no—
285C32C22×(C2×D3)⋊C411yescharacteristic, complemented
286C23—36no—
287C23—64no—
288C23—93no—
289C23—93no—
290C23—94no—
291C23—94no—
292C23—274no—
293C23—2712no—
294C23—276no—
295C23—5412no—
296C23—5412no—
297C23—5424no—
298C23—544no—
299C23D3×#15711yescharacteristic
300C2×C4—94no—
301C2×C4—1812no—
302C2×C4—274no—
303C2×C4—5412no—
304C2×C3—22no—
305C2×C3—212no—
306C2×C3—42no—
307C2×C3—412no—
308C2×C3—616no—
309C2×C3—91no—
310C2×C3—91no—
311C2×C3—96no—
312C2×C3C2×C22⋊#15716yescomplemented
313C2×C3C23×#15711yescharacteristic
314D3—24no—
315D3—612no—
316D3—62no—
317D3—62no—
318D3—1816no—
319D3—184no—
320D3—3616no—
321C22—34no—
322C22—612no—
323C22—91no—
324C22—93no—
325C22—96no—
326C22—96no—
327C22—912no—
328C22—274no—
329C22—544no—
330C22—544no—
331C22—5424no—
332C22—5412no—
333C22(C2×D3)⋊#15714yesdirect product factor
334C22C2×D3×#15713yes—
335C4—184no—
336C4—544no—
337C3—22no—
338C3—42no—
339C3C22×C22⋊#15711yescharacteristic, complemented
340C2—64no—
341C2—91no—
342C2—91no—
343C2—96no—
344C2—544no—
345C2C2×(C2×D3)⋊#15716yesdirect product factor
346C2C22×D3×#15711yescharacteristic
subgroupG/H
center, hypercenterC23D3×#157
commutatorC2×C33C23×C4
FrattiniC2C22×D3×#157
Fitting, socleC23×C33C2×C4
prime radicalC23×D3×C3⋊D3C2
cosocleC2×C3×C3⋊D3C24
hypocenterC33C22×C22⋊C4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C23×C3⋊#157C2semidirect
3. C23×C3×#157C2semidirect
4. C2×(C2×D3)⋊#157C2direct (N × H)
6. C23×C3⋊(C3⋊D3)C4semidirect
8. C23×C32×D3C4semidirect
11. C22×D3×C3⋊D3C4semidirect
14. C22×C3⋊#157C22semidirect
16. C22×C3×#157C22semidirect
17. (C2×D3)⋊#157C22direct (N × H)
21. C23×#157D3semidirect
25. C22×C3⋊(C3⋊D3)C2×C4semidirect
31. C22×C32×D3C2×C4semidirect
35. C2×D3×C3⋊D3C2×C4semidirect
38. C2×C3⋊#157C23semidirect
40. C2×C3×#157C23semidirect
60. C22×#157C2×D3semidirect
62. C22×C33C22⋊C4semidirect
65. C2×C3⋊(C3⋊D3)C22×C4semidirect
66. C2×C3×C3⋊D3C22⋊C4semidirect
72. C2×C32×D3C22×C4semidirect
117. C2×#157C22×D3semidirect
119. C2×C33C2×C22⋊C4semidirect
122. C3×C3⋊D3C2×C22⋊C4semidirect
140. C23×D3#157semidirect
152. C22×C32(C2×D3)⋊C4semidirect
163. C2×C3⋊D3(C2×D3)⋊C4semidirect
184. C33C22×C22⋊C4semidirect
215. C22×D3C2×#157semidirect
225. C2×C32C2×(C2×D3)⋊C4semidirect
230. C3⋊D3C2×(C2×D3)⋊C4semidirect
260. C22×C3C22⋊#157semidirect
280. C2×D3C22×#157semidirect
285. C32C22×(C2×D3)⋊C4semidirect
312. C2×C3C2×C22⋊#157semidirect
333. C22(C2×D3)⋊#157direct (N × H)
339. C3C22×C22⋊#157semidirect
345. C2C2×(C2×D3)⋊#157direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D3×C3⋊D32
C23×C3⋊#1572
C23×C3×#1572
C2×(C2×D3)⋊#1572
C22×C22⋊#1573
C22×(C2×D3)⋊C49