Finite Group Databasebeta

← C3×#2373 #84237

Subgroups — C3×#2373 #84237

Order 384.

496 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#126C211yesmaximal, characteristic
2C3×#389C211yesmaximal, characteristic, complemented
3C3×#388C211yesmaximal, characteristic
4C3×#388C211yesmaximal, characteristic
5C3×#388C211yesmaximal, characteristic
6C3×#387C211yesmaximal, characteristic, complemented
7C3×#387C211yesmaximal, characteristic, complemented
8C3×#387C211yesmaximal, characteristic, complemented
9C3×#385C211yesmaximal, characteristic
10C3×C4×C4⋊C4C211yesmaximal, characteristic, complemented
11C3×C4×C4⋊C4C211yesmaximal, characteristic, complemented
12C3×C4×C4⋊C4C211yesmaximal, characteristic, complemented
13C3×C4×C22⋊C4C211yesmaximal, characteristic
14C3×C4×C22⋊C4C211yesmaximal, characteristic
15C3×#376C211yesmaximal, characteristic, complemented
16#2373C311yesmaximal, characteristic, direct product factor, Sylow
17C23×C3×C4C2211yescharacteristic
18C3×#126C418yescomplemented
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×C3×C4⋊C4C2211yescharacteristic
21C2×C3×C4⋊C4C2211yescharacteristic
22C2×C3×C4⋊C4C2211yescharacteristic
23C2×C3×C4⋊C4C2211yescharacteristic
24C2×C3×C4⋊C4C2211yescharacteristic
25C2×C3×C4⋊C4C2211yescharacteristic
26C2×C3×C4⋊C4C2211yescharacteristic
27C2×C3×C22⋊C4C2211yescharacteristic
28C2×C3×C22⋊C4C2211yescharacteristic
29C2×C3×C22⋊C4C2211yescharacteristic
30C2×C3×C22⋊C4C2211yescharacteristic
31C2×C3×C22⋊C4C2211yescharacteristic
32C2×C3×C22⋊C4C2211yescharacteristic
33C2×C3×C42C2211yescharacteristic
34C2×C3×C42C2211yescharacteristic
35C2×C3×C42C2211yescharacteristic
36C2×C3×C42C2211yescharacteristic
37C2×C3×C42C2211yescharacteristic
38C2×C3×C42C2211yescharacteristic
39C2×C3×C42C2211yescharacteristic
40C2×C3×C42C2211yescharacteristic
41C3×#95C2211yescharacteristic
42C3×#95C2211yescharacteristic
43C3×#95C2211yescharacteristic
44C3×#95C2211yescharacteristic
45C3×#95C2211yescharacteristic
46C3×#95C2211yescharacteristic
47C3×#95C2211yescharacteristic
48C3×#95C2211yescharacteristic
49C3×#95C2211yescharacteristic
50C3×#95C2211yescharacteristic
51C3×#95C2211yescharacteristic
52C3×#95C2211yescharacteristic
53C2×#126C2×C311yescharacteristic
54#389C2×C311yescharacteristic, complemented
55#388C2×C311yescharacteristic
56#388C2×C311yescharacteristic
57#388C2×C311yescharacteristic
58#387C2×C311yescharacteristic, complemented
59#387C2×C311yescharacteristic, complemented
60#387C2×C311yescharacteristic, complemented
61#385C2×C311yescharacteristic
62C4×C4⋊C4C2×C311yescharacteristic, complemented
63C4×C4⋊C4C2×C311yescharacteristic, complemented
64C4×C4⋊C4C2×C311yescharacteristic, complemented
65C4×C22⋊C4C2×C311yescharacteristic
66C4×C22⋊C4C2×C311yescharacteristic
67#376C2×C311yescharacteristic, complemented
68C24×C3C2311yescharacteristic
69C22×C3×C4—22no—
70C22×C3×C4—22no—
71C22×C3×C4—41no—
72C22×C3×C4C2311yescharacteristic
73C22×C3×C4C2311yescharacteristic
74C22×C3×C4C2311yescharacteristic
75C22×C3×C4C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C22×C3×C4C2311yescharacteristic
81C22×C3×C4C2311yescharacteristic
82C22×C3×C4C2311yescharacteristic
83C22×C3×C4C2311yescharacteristic
84C22×C3×C4C2311yescharacteristic
85C22×C3×C4C2311yescharacteristic
86C3×C4⋊C4—22no—
87C3×C4⋊C4—22no—
88C3×C4⋊C4—22no—
89C3×C4⋊C4—22no—
90C3×C4⋊C4—41no—
91C3×C4⋊C4C2×C414yescomplemented
92C3×C4⋊C4C2×C414yescomplemented
93C3×C4⋊C4C2×C414yescomplemented
94C3×C22⋊C4—22no—
95C3×C22⋊C4—22no—
96C3×C22⋊C4—41no—
97C3×C22⋊C4C2×C414yes—
98C3×C22⋊C4C2×C414yes—
99C3×C22⋊C4C2×C414yes—
100C3×C42—22no—
101C3×C42—22no—
102C3×C42—22no—
103C3×C42—22no—
104C3×C42—41no—
105C3×C42—41no—
106C3×C42—41no—
107C3×C42C2×C414yescomplemented
108C23×C4C22×C311yescharacteristic
109#126C3×C418yescomplemented
110C2×C4⋊C4C22×C311yescharacteristic
111C2×C4⋊C4C22×C311yescharacteristic
112C2×C4⋊C4C22×C311yescharacteristic
113C2×C4⋊C4C22×C311yescharacteristic
114C2×C4⋊C4C22×C311yescharacteristic
115C2×C4⋊C4C22×C311yescharacteristic
116C2×C4⋊C4C22×C311yescharacteristic
117C2×C4⋊C4C22×C311yescharacteristic
118C2×C22⋊C4C22×C311yescharacteristic
119C2×C22⋊C4C22×C311yescharacteristic
120C2×C22⋊C4C22×C311yescharacteristic
121C2×C22⋊C4C22×C311yescharacteristic
122C2×C22⋊C4C22×C311yescharacteristic
123C2×C22⋊C4C22×C311yescharacteristic
124C2×C42C22×C311yescharacteristic
125C2×C42C22×C311yescharacteristic
126C2×C42C22×C311yescharacteristic
127C2×C42C22×C311yescharacteristic
128C2×C42C22×C311yescharacteristic
129C2×C42C22×C311yescharacteristic
130C2×C42C22×C311yescharacteristic
131C2×C42C22×C311yescharacteristic
132#95C22×C311yescharacteristic
133#95C22×C311yescharacteristic
134#95C22×C311yescharacteristic
135#95C22×C311yescharacteristic
136#95C22×C311yescharacteristic
137#95C22×C311yescharacteristic
138#95C22×C311yescharacteristic
139#95C22×C311yescharacteristic
140#95C22×C311yescharacteristic
141#95C22×C311yescharacteristic
142#95C22×C311yescharacteristic
143#95C22×C311yescharacteristic
144C23×C3—21no—
145C23×C3—21no—
146C23×C3—21no—
147C23×C3—21no—
148C23×C3—21no—
149C23×C3—21no—
150C23×C3C22×C412yes—
151C23×C3C2411yescharacteristic
152C2×C3×C4—21no—
153C2×C3×C4—21no—
154C2×C3×C4—21no—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4—21no—
158C2×C3×C4—21no—
159C2×C3×C4—21no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—44no—
185C2×C3×C4C22×C412yes—
186C2×C3×C4C22×C412yes—
187C2×C3×C4C22×C412yes—
188C2×C3×C4C22×C412yes—
189C2×C3×C4C22×C412yes—
190C2×C3×C4C22×C412yes—
191C2×C3×C4#4112yes—
192C2×C3×C4#4112yes—
193C2×C3×C4#4112yes—
194C2×C3×C4#4112yes—
195C24C23×C311yescharacteristic
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—41no—
199C22×C4C23×C311yescharacteristic
200C22×C4C23×C311yescharacteristic
201C22×C4C23×C311yescharacteristic
202C22×C4C23×C311yescharacteristic
203C22×C4C23×C311yescharacteristic
204C22×C4C23×C311yescharacteristic
205C22×C4C23×C311yescharacteristic
206C22×C4C23×C311yescharacteristic
207C22×C4C23×C311yescharacteristic
208C22×C4C23×C311yescharacteristic
209C22×C4C23×C311yescharacteristic
210C22×C4C23×C311yescharacteristic
211C22×C4C23×C311yescharacteristic
212C22×C4C23×C311yescharacteristic
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—41no—
218C4⋊C4C2×C3×C414yescomplemented
219C4⋊C4C2×C3×C414yescomplemented
220C4⋊C4C2×C3×C414yescomplemented
221C22⋊C4—22no—
222C22⋊C4—22no—
223C22⋊C4—41no—
224C22⋊C4C2×C3×C414yes—
225C22⋊C4C2×C3×C414yes—
226C22⋊C4C2×C3×C414yes—
227C42—22no—
228C42—22no—
229C42—22no—
230C42—22no—
231C42—41no—
232C42—41no—
233C42—41no—
234C42C2×C3×C414yescomplemented
235C22×C3—22no—
236C22×C3—22no—
237C22×C3—22no—
238C22×C3—41no—
239C22×C3—41no—
240C22×C3—41no—
241C22×C3—41no—
242C22×C3C23×C411yescharacteristic
243C22×C3C2×#4111yescharacteristic
244C22×C3C2×#4111yescharacteristic
245C22×C3C2×#4111yescharacteristic
246C22×C3C2×#4111yescharacteristic
247C22×C3H(5,𝔽2)11yescharacteristic
248C22×C32−1+411yescharacteristic
249C3×C4—22no—
250C3×C4—22no—
251C3×C4—22no—
252C3×C4—22no—
253C3×C4—22no—
254C3×C4—22no—
255C3×C4—22no—
256C3×C4—22no—
257C3×C4—22no—
258C3×C4—22no—
259C3×C4—41no—
260C3×C4—41no—
261C3×C4—41no—
262C3×C4—41no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23C22×C3×C412yes—
270C23C24×C311yescharacteristic
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—44no—
304C2×C4C22×C3×C412yes—
305C2×C4C22×C3×C412yes—
306C2×C4C22×C3×C412yes—
307C2×C4C22×C3×C412yes—
308C2×C4C22×C3×C412yes—
309C2×C4C22×C3×C412yes—
310C2×C4C3×#4112yes—
311C2×C4C3×#4112yes—
312C2×C4C3×#4112yes—
313C2×C4C3×#4112yes—
314C2×C3—42no—
315C2×C3C4×#4111yescharacteristic
316C2×C3C4×#4111yescharacteristic
317C2×C3#52011yescharacteristic
318C2×C3#55111yescharacteristic
319C2×C3#55211yescharacteristic
320C2×C3#55311yescharacteristic
321C2×C3#55611yescharacteristic
322C22—22no—
323C22—22no—
324C22—22no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22C23×C3×C411yescharacteristic
330C22C2×C3×#4111yescharacteristic
331C22C2×C3×#4111yescharacteristic
332C22C2×C3×#4111yescharacteristic
333C22C2×C3×#4111yescharacteristic
334C22C3×H(5,𝔽2)11yescharacteristic
335C22C3×2−1+411yescharacteristic
336C4—22no—
337C4—22no—
338C4—22no—
339C4—22no—
340C4—22no—
341C4—22no—
342C4—22no—
343C4—22no—
344C4—22no—
345C4—22no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C3#237311yescharacteristic, direct product factor, Sylow
351C2—42no—
352C2C3×C4×#4111yescharacteristic
353C2C3×C4×#4111yescharacteristic
354C2C3×#52011yescharacteristic
355C2C3×#55111yescharacteristic
356C2C3×#55211yescharacteristic
357C2C3×#55311yescharacteristic
358C2C3×#55611yescharacteristic
subgroupG/H
center, socleC23×C3C24
commutatorC22C23×C3×C4
Frattini, cosocleC23C24×C3
prime radicalC24×C3C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#389C2semidirect
6. C3×#387C2semidirect
7. C3×#387C2semidirect
8. C3×#387C2semidirect
10. C3×C4×C4⋊C4C2semidirect
11. C3×C4×C4⋊C4C2semidirect
12. C3×C4×C4⋊C4C2semidirect
15. C3×#376C2semidirect
16. #2373C3direct (N × H)
18. C3×#126C4semidirect
54. #389C2×C3semidirect
58. #387C2×C3semidirect
59. #387C2×C3semidirect
60. #387C2×C3semidirect
62. C4×C4⋊C4C2×C3semidirect
63. C4×C4⋊C4C2×C3semidirect
64. C4×C4⋊C4C2×C3semidirect
67. #376C2×C3semidirect
91. C3×C4⋊C4C2×C4semidirect
92. C3×C4⋊C4C2×C4semidirect
93. C3×C4⋊C4C2×C4semidirect
107. C3×C42C2×C4semidirect
109. #126C3×C4semidirect
218. C4⋊C4C2×C3×C4semidirect
219. C4⋊C4C2×C3×C4semidirect
220. C4⋊C4C2×C3×C4semidirect
234. C42C2×C3×C4semidirect
350. C3#2373direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#1262
C3×#3892
C3×#3882
C3×#3872
C3×#3852
C3×C4×C4⋊C42
C3×C4×C22⋊C42
C3×#3762
#23733