Finite Group Databasebeta

← C22×C3×#157⋊D3 #72252978

Subgroups — C22×C3×#157⋊D3 #72252978

Order 2592.

1946 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 1946 conjugacy classes of subgroups — too many to draw. Only its 230 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
1C22×C3×#1113C212yesmaximal, complemented
2C22×C32×#157C211yesmaximal, characteristic, complemented
3C2×C3×#157⋊D3C2112yesmaximal, direct product factor
4C22×#157⋊D3C311yesmaximal, characteristic, direct product factor
5C22×C3×GO+(4,𝔽2)—31nomaximal
6C22×C32×C3⋊D3—22no—
7C22×C32×C3⋊D3C2211yescharacteristic
8C2×C3×#1113—26no—
9C2×C3×#1113C22112yescomplemented
10C2×C32×#157C2216yescomplemented
11C3×#157⋊D3C22116yesdirect product factor
12C22×#1113C2×C312yescomplemented
13C22×C3×D32—32no—
14C22×C3×D32—62no—
15C22×C3×#157—21no—
16C22×C3×#157D311yescharacteristic, complemented
17C22×C3×#157C2×C311yescharacteristic, complemented
18C2×#157⋊D3C2×C3112yesdirect product factor
19C2×C3×GO+(4,𝔽2)—312no—
20C22×C34D411yescharacteristic, complemented
21C2×C32×C3⋊D3—212no—
22C2×C32×C3⋊D3D413yescomplemented
23C2×C32×C3⋊D3C2313yes—
24C3×#1113—212no—
25C3×#1113C2318yescomplemented
26C32×#157C2314yescomplemented
27C22×GO+(4,𝔽2)—31no—
28C22×C3×D12—91nomaximal
29C22×C3×C3⋊D3—22no—
30C22×C3×C3⋊D3—21no—
31C22×C3×C3⋊D3—22no—
32C22×C3×C3⋊D3—42no—
33C22×C3×C3⋊D3C2×D311yescharacteristic
34C22×C3×C3⋊D3C22×C311yescharacteristic
35C22×C32×D3—62no—
36C22×C32×D3—62no—
37C22×C32×D3—62no—
38C22×C32×D3—122no—
39C2×#1113—26no—
40C2×#1113C22×C3112yescomplemented
41C2×C3×D32—312no—
42C2×C3×D32—612no—
43C2×C3×D32—612no—
44C2×C3×D32—66no—
45C2×C3×#157—26no—
46C2×C3×#157C2×D316yescomplemented
47C2×C3×#157C22×C316yescomplemented
48#157⋊D3C22×C3116yesdirect product factor
49C3×GO+(4,𝔽2)—316no—
50C2×C34C2×D413yescomplemented
51C32×C3⋊D3—28no—
52C32×C3⋊D3C2×D413yescomplemented
53C32×C3⋊D3C2411yescharacteristic
54C23×C3×D3—92no—
55C23×C3×D3—182no—
56C22×D32—32no—
57C22×D32—62no—
58C22×#157C3×D311yescharacteristic, complemented
59C2×GO+(4,𝔽2)—312no—
60C22×C32×C4—91no—
61C2×C3×D12—912no—
62C22×C33—22no—
63C22×C33—21no—
64C22×C33—42no—
65C22×C33—42no—
66C22×C33—82no—
67C22×C33D1211yescharacteristic, complemented
68C22×C33C3×D411yescharacteristic, complemented
69C2×C3×C3⋊D3—212no—
70C2×C3×C3⋊D3—23no—
71C2×C3×C3⋊D3—23no—
72C2×C3×C3⋊D3—212no—
73C2×C3×C3⋊D3—412no—
74C2×C3×C3⋊D3D1213yescomplemented
75C2×C3×C3⋊D3C3×D413yescomplemented
76C2×C3×C3⋊D3C22×D313yes—
77C2×C3×C3⋊D3C23×C313yes—
78C2×C32×D3—66no—
79C2×C32×D3—66no—
80C2×C32×D3—612no—
81C2×C32×D3—612no—
82C2×C32×D3—1212no—
83#1113—212no—
84#1113C23×C318yescomplemented
85C3×D32—38no—
86C3×D32—68no—
87C3×D32—624no—
88C3×D32—612no—
89C3×#157—24no—
90C3×#157C22×D314yescomplemented
91C3×#157C23×C314yescomplemented
92C22×C3×D4—271no—
93C22×D12—91no—
94C34C22×D411yescharacteristic, complemented, Sylow
95C23×C32—91no—
96C23×C32—182no—
97C22×C3⋊D3—22no—
98C22×C3⋊D3C2×C3×D311yescharacteristic
99C22×C3×D3—62no—
100C22×C3×D3—62no—
101C22×C3×D3—62no—
102C22×C3×D3—62no—
103C22×C3×D3—62no—
104C22×C3×D3—62no—
105C22×C3×D3—912no—
106C22×C3×D3—122no—
107C22×C3×D3—122no—
108C22×C3×D3—122no—
109C22×C3×D3—122no—
110C22×C3×D3—122no—
111C22×C3×D3—124no—
112C22×C3×D3—1812no—
113C22×C3×D3—186no—
114C22×C3×D3—1812no—
115C2×D32—312no—
116C2×D32—612no—
117C2×D32—612no—
118C2×D32—66no—
119C2×#157C2×C3×D316yescomplemented
120GO+(4,𝔽2)—316no—
121C2×C32×C4—96no—
122C3×D12—916no—
123C2×C33—26no—
124C2×C33—23no—
125C2×C33—46no—
126C2×C33—46no—
127C2×C33—86no—
128C2×C33C2×D1213yescomplemented
129C2×C33C2×C3×D413yescomplemented
130C3×C3⋊D3—28no—
131C3×C3⋊D3—28no—
132C3×C3⋊D3—21no—
133C3×C3⋊D3—23no—
134C3×C3⋊D3—48no—
135C3×C3⋊D3C2×D1213yescomplemented
136C3×C3⋊D3C2×C3×D413yescomplemented
137C3×C3⋊D3C23×D311yescharacteristic
138C3×C3⋊D3C24×C311yescharacteristic
139C32×D3—66no—
140C32×D3—62no—
141C32×D3—68no—
142C32×D3—68no—
143C32×D3—128no—
144C24×C3—272no—
145C23×D3—92no—
146C23×D3—182no—
147C2×C3×D4—2712no—
148C22×C3×C4—91no—
149C22×C3×C4—91no—
150C22×C3×C4—181no—
151C2×D12—912no—
152C22×C32—22no—
153C22×C32—22no—
154C22×C32—42no—
155C22×C32—42no—
156C22×C32—42no—
157C22×C32—44no—
158C22×C32—42no—
159C22×C32—42no—
160C22×C32—82no—
161C22×C32—82no—
162C22×C32—84no—
163C22×C32—93no—
164C22×C32—93no—
165C22×C32—1812no—
166C22×C32C3×D1211yescharacteristic, complemented
167C22×C32GO+(4,𝔽2)11yescharacteristic, complemented
168C2×C3⋊D3—212no—
169C2×C3⋊D3C3×D1213yescomplemented
170C2×C3⋊D3C22×C3×D313yes—
171C2×C3×D3—66no—
172C2×C3×D3—66no—
173C2×C3×D3—66no—
174C2×C3×D3—66no—
175C2×C3×D3—612no—
176C2×C3×D3—612no—
177C2×C3×D3—612no—
178C2×C3×D3—612no—
179C2×C3×D3—98no—
180C2×C3×D3—126no—
181C2×C3×D3—126no—
182C2×C3×D3—1212no—
183C2×C3×D3—1212no—
184C2×C3×D3—1212no—
185C2×C3×D3—1212no—
186C2×C3×D3—1224no—
187C2×C3×D3—188no—
188C2×C3×D3—1812no—
189C2×C3×D3—1824no—
190D32—38no—
191D32—68no—
192D32—624no—
193D32—612no—
194#157C22×C3×D314yescomplemented
195C32×C4—94no—
196C22×D4—271noSylow
197C33—22no—
198C33—21no—
199C33—42no—
200C33—42no—
201C33—82no—
202C33C22×D1211yescharacteristic, complemented
203C33C22×C3×D411yescharacteristic, complemented
204C23×C3—91no—
205C23×C3—91no—
206C23×C3—181no—
207C23×C3—182no—
208C23×C3—182no—
209C23×C3—2712no—
210C23×C3—362no—
211C23×C3—546no—
212C22×D3—62no—
213C22×D3—62no—
214C22×D3—62no—
215C22×D3—912no—
216C22×D3—122no—
217C22×D3—1812no—
218C22×D3—186no—
219C22×D3—1812no—
220C3×D4—2716no—
221C2×C3×C4—96no—
222C2×C3×C4—96no—
223C2×C3×C4—186no—
224D12—916no—
225C2×C32—26no—
226C2×C32—26no—
227C2×C32—46no—
228C2×C32—46no—
229C2×C32—46no—
230C2×C32—412no—
231C2×C32—46no—
232C2×C32—46no—
233C2×C32—86no—
234C2×C32—86no—
235C2×C32—812no—
236C2×C32—91no—
237C2×C32—93no—
238C2×C32—188no—
239C2×C32C2×C3×D1213yescomplemented
240C2×C32C2×GO+(4,𝔽2)13yescomplemented
241C3⋊D3—28no—
242C3⋊D3C2×C3×D1213yescomplemented
243C3⋊D3C23×C3×D311yescharacteristic
244C3×D3—66no—
245C3×D3—66no—
246C3×D3—62no—
247C3×D3—62no—
248C3×D3—68no—
249C3×D3—68no—
250C3×D3—68no—
251C3×D3—68no—
252C3×D3—122no—
253C3×D3—126no—
254C3×D3—128no—
255C3×D3—128no—
256C3×D3—128no—
257C3×D3—128no—
258C3×D3—1216no—
259C24—272no—
260C2×D4—2712no—
261C22×C4—91no—
262C22×C3—22no—
263C22×C3—21no—
264C22×C3—42no—
265C22×C3—42no—
266C22×C3—82no—
267C22×C3—93no—
268C22×C3—93no—
269C22×C3—93no—
270C22×C3—93no—
271C22×C3—183no—
272C22×C3—183no—
273C22×C3—1812no—
274C22×C3—1812no—
275C22×C3—278no—
276C22×C3—3612no—
277C22×C3—5412no—
278C22×C3C3×GO+(4,𝔽2)11yescharacteristic, complemented
279C22×C3#157⋊D311yescharacteristic, direct product factor
280C2×D3—66no—
281C2×D3—66no—
282C2×D3—612no—
283C2×D3—612no—
284C2×D3—98no—
285C2×D3—1212no—
286C2×D3—188no—
287C2×D3—1812no—
288C2×D3—1824no—
289C3×C4—94no—
290C3×C4—94no—
291C3×C4—184no—
292C32—22no—
293C32—22no—
294C32—42no—
295C32—42no—
296C32—42no—
297C32—44no—
298C32—42no—
299C32—42no—
300C32—82no—
301C32—82no—
302C32—84no—
303C32C22×C3×D1211yescharacteristic, complemented
304C32C22×GO+(4,𝔽2)11yescharacteristic, complemented
305C23—91no—
306C23—182no—
307C23—2712no—
308C23—546no—
309D4—2716no—
310C2×C4—96no—
311C2×C3—26no—
312C2×C3—23no—
313C2×C3—46no—
314C2×C3—46no—
315C2×C3—86no—
316C2×C3—91no—
317C2×C3—93no—
318C2×C3—91no—
319C2×C3—93no—
320C2×C3—181no—
321C2×C3—183no—
322C2×C3—188no—
323C2×C3—188no—
324C2×C3—368no—
325C2×C3C2×C3×GO+(4,𝔽2)13yescomplemented
326C2×C3C2×#157⋊D313yesdirect product factor
327D3—62no—
328D3—66no—
329D3—68no—
330D3—68no—
331D3—128no—
332C22—93no—
333C22—93no—
334C22—1812no—
335C22—278no—
336C22—5412no—
337C22C3×#157⋊D311yescharacteristic, direct product factor
338C4—94no—
339C3—22no—
340C3—21no—
341C3—42no—
342C3—42no—
343C3—82no—
344C3C22×C3×GO+(4,𝔽2)11yescharacteristic, complemented
345C3C22×#157⋊D311yescharacteristic, direct product factor
346C2—91no—
347C2—93no—
348C2—188no—
349C2C2×C3×#157⋊D313yesdirect product factor
subgroupG/H
center, hypercenterC22×C3#157⋊D3
commutator, cosocleC3×C3⋊D3C24×C3
FrattiniC1G
Fitting, socleC22×C34D4
hypocenterC33C22×C3×D4
LCSG ▷ C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C22×C3×#1113C2semidirect
2. C22×C32×#157C2semidirect
3. C2×C3×#157⋊D3C2direct (N × H)
4. C22×#157⋊D3C3direct (N × H)
9. C2×C3×#1113C22semidirect
10. C2×C32×#157C22semidirect
11. C3×#157⋊D3C22direct (N × H)
12. C22×#1113C2×C3semidirect
16. C22×C3×#157D3semidirect
17. C22×C3×#157C2×C3semidirect
18. C2×#157⋊D3C2×C3direct (N × H)
20. C22×C34D4semidirect
22. C2×C32×C3⋊D3D4semidirect
25. C3×#1113C23semidirect
26. C32×#157C23semidirect
40. C2×#1113C22×C3semidirect
46. C2×C3×#157C2×D3semidirect
47. C2×C3×#157C22×C3semidirect
48. #157⋊D3C22×C3direct (N × H)
50. C2×C34C2×D4semidirect
52. C32×C3⋊D3C2×D4semidirect
58. C22×#157C3×D3semidirect
67. C22×C33D12semidirect
68. C22×C33C3×D4semidirect
74. C2×C3×C3⋊D3D12semidirect
75. C2×C3×C3⋊D3C3×D4semidirect
84. #1113C23×C3semidirect
90. C3×#157C22×D3semidirect
91. C3×#157C23×C3semidirect
94. C34C22×D4semidirect
119. C2×#157C2×C3×D3semidirect
128. C2×C33C2×D12semidirect
129. C2×C33C2×C3×D4semidirect
135. C3×C3⋊D3C2×D12semidirect
136. C3×C3⋊D3C2×C3×D4semidirect
166. C22×C32C3×D12semidirect
167. C22×C32GO+(4,𝔽2)semidirect
169. C2×C3⋊D3C3×D12semidirect
194. #157C22×C3×D3semidirect
202. C33C22×D12semidirect
203. C33C22×C3×D4semidirect
239. C2×C32C2×C3×D12semidirect
240. C2×C32C2×GO+(4,𝔽2)semidirect
242. C3⋊D3C2×C3×D12semidirect
278. C22×C3C3×GO+(4,𝔽2)semidirect
279. C22×C3#157⋊D3direct (N × H)
303. C32C22×C3×D12semidirect
304. C32C22×GO+(4,𝔽2)semidirect
325. C2×C3C2×C3×GO+(4,𝔽2)semidirect
326. C2×C3C2×#157⋊D3direct (N × H)
337. C22C3×#157⋊D3direct (N × H)
344. C3C22×C3×GO+(4,𝔽2)semidirect
345. C3C22×#157⋊D3direct (N × H)
349. C2C2×C3×#157⋊D3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#11132
C22×C32×#1572
C2×C3×#157⋊D32
C22×#157⋊D33
C22×C3×GO+(4,𝔽2)3
C22×C3×D129