Finite Group Databasebeta

← C3×#2505 #84369

Subgroups — C3×#2505 #84369

Order 384.

456 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#400C211yesmaximal, characteristic, complemented
2C3×#400C211yesmaximal, characteristic, complemented
3C3×#400C211yesmaximal, characteristic, complemented
4C3×#398C211yesmaximal, characteristic, complemented
5C3×#389C211yesmaximal, characteristic, complemented
6C3×#387C211yesmaximal, characteristic, complemented
7C3×#387C211yesmaximal, characteristic, complemented
8C3×#385C211yesmaximal, characteristic
9C3×#385C211yesmaximal, characteristic
10C3×#385C211yesmaximal, characteristic
11C3×#385C211yesmaximal, characteristic
12C3×#382C211yesmaximal, characteristic, complemented
13C3×#380C211yesmaximal, characteristic
14C3×C4×C22⋊C4C211yesmaximal, characteristic
15C2×C3×#95C211yesmaximal, characteristic
16#2505C311yesmaximal, characteristic, direct product factor, Sylow
17C23×C3×C4C2211yescharacteristic
18C23×C3×C4C2211yescharacteristic
19C23×C3×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×C4⋊C4C2211yescharacteristic
28C2×C3×C4⋊C4C2211yescharacteristic
29C2×C3×C4⋊C4C2211yescharacteristic
30C2×C3×C22⋊C4C2211yescharacteristic
31C2×C3×C22⋊C4C2211yescharacteristic
32C2×C3×C22⋊C4C2211yescharacteristic
33C2×C3×C22⋊C4C2211yescharacteristic
34C2×C3×C42C2211yescharacteristic
35C2×C3×C42C2211yescharacteristic
36C3×#95C2211yescharacteristic
37C3×#95C2211yescharacteristic
38C3×#95C2211yescharacteristic
39C3×#95C2211yescharacteristic
40C3×#95C2211yescharacteristic
41C3×#95C2211yescharacteristic
42C3×#95C2211yescharacteristic
43C3×#95C2211yescharacteristic
44C3×#95C2211yescharacteristic
45C3×#95C2211yescharacteristic
46C3×#95C2211yescharacteristic
47C3×#95C2211yescharacteristic
48C3×#95C2211yescharacteristic
49C3×#95C2211yescharacteristic
50C3×#95C2211yescharacteristic
51C3×#95C2211yescharacteristic
52#400C2×C311yescharacteristic, complemented
53#400C2×C311yescharacteristic, complemented
54#400C2×C311yescharacteristic, complemented
55#398C2×C311yescharacteristic, complemented
56#389C2×C311yescharacteristic, complemented
57#387C2×C311yescharacteristic, complemented
58#387C2×C311yescharacteristic, complemented
59#385C2×C311yescharacteristic
60#385C2×C311yescharacteristic
61#385C2×C311yescharacteristic
62#385C2×C311yescharacteristic
63#382C2×C311yescharacteristic, complemented
64#380C2×C311yescharacteristic
65C4×C22⋊C4C2×C311yescharacteristic
66C2×#95C2×C311yescharacteristic
67C24×C3C2311yescharacteristic
68C22×C3×C4—22no—
69C22×C3×C4—22no—
70C22×C3×C4—22no—
71C22×C3×C4—22no—
72C22×C3×C4—22no—
73C22×C3×C4—41no—
74C22×C3×C4—41no—
75C22×C3×C4—41no—
76C22×C3×C4—41no—
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
86C22×C3×C4C2311yescharacteristic
87C22×C3×C4C2311yescharacteristic
88C22×C3×C4C2311yescharacteristic
89C22×C3×C4C2311yescharacteristic
90C22×C3×C4C2311yescharacteristic
91C3×C4⋊C4—22no—
92C3×C4⋊C4—22no—
93C3×C4⋊C4—22no—
94C3×C4⋊C4—41no—
95C3×C4⋊C4—41no—
96C3×C4⋊C4—41no—
97C3×C4⋊C4—41no—
98C3×C4⋊C4—41no—
99C3×C4⋊C4—41no—
100C3×C4⋊C4—41no—
101C3×C22⋊C4—22no—
102C3×C22⋊C4—22no—
103C3×C22⋊C4—22no—
104C3×C22⋊C4Q814yes—
105C3×C42—22no—
106C3×C42—41no—
107C23×C4C22×C311yescharacteristic
108C23×C4C22×C311yescharacteristic
109C23×C4C22×C311yescharacteristic
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×C4⋊C4C22×C311yescharacteristic
119C2×C4⋊C4C22×C311yescharacteristic
120C2×C22⋊C4C22×C311yescharacteristic
121C2×C22⋊C4C22×C311yescharacteristic
122C2×C22⋊C4C22×C311yescharacteristic
123C2×C22⋊C4C22×C311yescharacteristic
124C2×C42C22×C311yescharacteristic
125C2×C42C22×C311yescharacteristic
126#95C22×C311yescharacteristic
127#95C22×C311yescharacteristic
128#95C22×C311yescharacteristic
129#95C22×C311yescharacteristic
130#95C22×C311yescharacteristic
131#95C22×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
142C23×C3—21no—
143C23×C3—21no—
144C23×C3—21no—
145C23×C3—21no—
146C23×C3C2×Q812yes—
147C23×C3#4112yes—
148C23×C3#4112yes—
149C23×C3C2411yescharacteristic
150C2×C3×C4—21no—
151C2×C3×C4—21no—
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—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—44no—
189C2×C3×C4—44no—
190C2×C3×C4—44no—
191C2×C3×C4C2×Q812yes—
192C2×C3×C4C2×Q812yes—
193C2×C3×C4#4112yes—
194C2×C3×C4#4112yes—
195C24C23×C311yescharacteristic
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4C23×C311yescharacteristic
206C22×C4C23×C311yescharacteristic
207C22×C4C23×C311yescharacteristic
208C22×C4C23×C311yescharacteristic
209C22×C4C23×C311yescharacteristic
210C22×C4C23×C311yescharacteristic
211C22×C4C23×C311yescharacteristic
212C22×C4C23×C311yescharacteristic
213C22×C4C23×C311yescharacteristic
214C22×C4C23×C311yescharacteristic
215C22×C4C23×C311yescharacteristic
216C22×C4C23×C311yescharacteristic
217C22×C4C23×C311yescharacteristic
218C22×C4C23×C311yescharacteristic
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—22no—
232C22⋊C4C3×Q814yes—
233C42—22no—
234C42—41no—
235C22×C3—22no—
236C22×C3—22no—
237C22×C3—22no—
238C22×C3—22no—
239C22×C3—22no—
240C22×C3—22no—
241C22×C3#12514yes—
242C22×C3C22×Q811yescharacteristic
243C22×C3C2×#4111yescharacteristic
244C22×C3C2×#4111yescharacteristic
245C22×C3C2×#4111yescharacteristic
246C22×C3C2×#4111yescharacteristic
247C22×C3H(5,𝔽2)11yescharacteristic
248C22×C3H(5,𝔽2)11yescharacteristic
249C3×C4—22no—
250C3×C4—22no—
251C3×C4—22no—
252C3×C4—22no—
253C3×C4—41no—
254C3×C4—41no—
255C3×C4—41no—
256C3×C4—41no—
257C3×C4—41no—
258C3×C4—41no—
259C3×C4—41no—
260C3×C4—41no—
261C3×C4—41no—
262C3×C4—41no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23C2×C3×Q812yes—
268C23C3×#4112yes—
269C23C3×#4112yes—
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—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4C2×C3×Q812yes—
313C2×C4C2×C3×Q812yes—
314C2×C4C3×#4112yes—
315C2×C4C3×#4112yes—
316C2×C3—24no—
317C2×C3C2×#12511yescharacteristic
318C2×C3#52911yescharacteristic
319C2×C3#53811yescharacteristic
320C2×C3#55111yescharacteristic
321C2×C3#55111yescharacteristic
322C2×C3#55411yescharacteristic
323C2×C3#55411yescharacteristic
324C22—22no—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22C3×#12514yes—
331C22C22×C3×Q811yescharacteristic
332C22C2×C3×#4111yescharacteristic
333C22C2×C3×#4111yescharacteristic
334C22C2×C3×#4111yescharacteristic
335C22C2×C3×#4111yescharacteristic
336C22C3×H(5,𝔽2)11yescharacteristic
337C22C3×H(5,𝔽2)11yescharacteristic
338C4—22no—
339C4—22no—
340C4—22no—
341C4—22no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—41no—
352C3#250511yescharacteristic, direct product factor, Sylow
353C2—24no—
354C2C2×C3×#12511yescharacteristic
355C2C3×#52911yescharacteristic
356C2C3×#53811yescharacteristic
357C2C3×#55111yescharacteristic
358C2C3×#55111yescharacteristic
359C2C3×#55411yescharacteristic
360C2C3×#55411yescharacteristic
subgroupG/H
center, socleC23×C3C24
commutator, Frattini, cosocleC23C24×C3
prime radicalC24×C3C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#400C2semidirect
2. C3×#400C2semidirect
3. C3×#400C2semidirect
4. C3×#398C2semidirect
5. C3×#389C2semidirect
6. C3×#387C2semidirect
7. C3×#387C2semidirect
12. C3×#382C2semidirect
16. #2505C3direct (N × H)
52. #400C2×C3semidirect
53. #400C2×C3semidirect
54. #400C2×C3semidirect
55. #398C2×C3semidirect
56. #389C2×C3semidirect
57. #387C2×C3semidirect
58. #387C2×C3semidirect
63. #382C2×C3semidirect
352. C3#2505direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#4002
C3×#3982
C3×#3892
C3×#3872
C3×#3852
C3×#3822
C3×#3802
C3×C4×C22⋊C42
C2×C3×#952
#25053