Finite Group Databasebeta

← C3×#2716 #84580

Subgroups — C3×#2716 #84580

Order 384.

450 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×C3×#126C211yesmaximal, characteristic
3C3×#400C211yesmaximal, characteristic, complemented
4C3×#398C211yesmaximal, characteristic, complemented
5C3×#397C211yesmaximal, characteristic
6C3×#396C211yesmaximal, characteristic
7C3×#393C211yesmaximal, characteristic
8C3×#393C211yesmaximal, characteristic
9C3×#391C211yesmaximal, characteristic, complemented
10C3×#391C211yesmaximal, characteristic, complemented
11C3×#388C211yesmaximal, characteristic
12C3×#388C211yesmaximal, characteristic
13C3×#387C211yesmaximal, characteristic, complemented
14C3×#387C211yesmaximal, characteristic, complemented
15C3×#387C211yesmaximal, characteristic, complemented
16#2716C311yesmaximal, characteristic, direct product factor, Sylow
17C22×C3×Q8C2211yescharacteristic
18C22×C3×Q8C2211yescharacteristic
19C3×C4⋊Q8—24no—
20C3×#126—24no—
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×C22⋊C4C2211yescharacteristic
35C2×C3×C22⋊C4C2211yescharacteristic
36C2×C3×C22⋊C4C2211yescharacteristic
37C2×C3×C42C2211yescharacteristic
38C2×C3×C42C2211yescharacteristic
39C2×C3×C42C2211yescharacteristic
40C3×#95C2211yescharacteristic
41C3×#95C2211yescharacteristic
42C3×#95C2211yescharacteristic
43C3×#95C2211yescharacteristic
44C3×#95C2211yescharacteristic
45C3×#95C2211yescharacteristic
46C3×#95C2211yescharacteristic
47C3×#95C2211yescharacteristic
48C3×#95C2211yescharacteristic
49C3×#95C2211yescharacteristic
50C3×#95C2211yescharacteristic
51C3×#95C2211yescharacteristic
52C3×#95C2211yescharacteristic
53C3×#95C2211yescharacteristic
54C2×C4⋊Q8C2×C311yescharacteristic, complemented
55C2×#126C2×C311yescharacteristic
56#400C2×C311yescharacteristic, complemented
57#398C2×C311yescharacteristic, complemented
58#397C2×C311yescharacteristic
59#396C2×C311yescharacteristic
60#393C2×C311yescharacteristic
61#393C2×C311yescharacteristic
62#391C2×C311yescharacteristic, complemented
63#391C2×C311yescharacteristic, complemented
64#388C2×C311yescharacteristic
65#388C2×C311yescharacteristic
66#387C2×C311yescharacteristic, complemented
67#387C2×C311yescharacteristic, complemented
68#387C2×C311yescharacteristic, complemented
69C24×C3C2311yescharacteristic
70C2×C3×Q8—22no—
71C2×C3×Q8—22no—
72C2×C3×Q8—22no—
73C2×C3×Q8—22no—
74C2×C3×Q8—41no—
75C2×C3×Q8—41no—
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
86C22×C3×C4C2311yescharacteristic
87C22×C3×C4C2311yescharacteristic
88C22×C3×C4C2311yescharacteristic
89C22×C3×C4C2311yescharacteristic
90C3×C4⋊C4—22no—
91C3×C4⋊C4—22no—
92C3×C4⋊C4—22no—
93C3×C4⋊C4—22no—
94C3×C4⋊C4—22no—
95C3×C4⋊C4—22no—
96C3×C4⋊C4—22no—
97C3×C4⋊C4—41no—
98C3×C4⋊C4D414yes—
99C3×C22⋊C4—22no—
100C3×C22⋊C4—22no—
101C3×C22⋊C4—22no—
102C3×C22⋊C4—22no—
103C3×C22⋊C4—22no—
104C3×C22⋊C4—41no—
105C3×C22⋊C4—41no—
106C3×C42—22no—
107C3×C42—22no—
108C3×C42—41no—
109C22×Q8C22×C311yescharacteristic
110C22×Q8C22×C311yescharacteristic
111C4⋊Q8—24no—
112#126—24no—
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×C4⋊C4C22×C311yescharacteristic
121C2×C4⋊C4C22×C311yescharacteristic
122C2×C22⋊C4C22×C311yescharacteristic
123C2×C22⋊C4C22×C311yescharacteristic
124C2×C22⋊C4C22×C311yescharacteristic
125C2×C22⋊C4C22×C311yescharacteristic
126C2×C22⋊C4C22×C311yescharacteristic
127C2×C22⋊C4C22×C311yescharacteristic
128C2×C22⋊C4C22×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
144#95C22×C311yescharacteristic
145#95C22×C311yescharacteristic
146C23×C3—21no—
147C23×C3—21no—
148C23×C3—21no—
149C23×C3—21no—
150C23×C3—21no—
151C23×C3—21no—
152C23×C3—21no—
153C23×C3C2411yescharacteristic
154C3×Q8—44no—
155C3×Q8—44no—
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—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4C2×D412yes—
194C2×C3×C4C2×D412yes—
195C2×C3×C4C2×D412yes—
196C2×C3×C4#4112yes—
197C2×C3×C4#4112yes—
198C24C23×C311yescharacteristic
199C2×Q8—22no—
200C2×Q8—22no—
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—41no—
204C2×Q8—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—22no—
223C4⋊C4—22no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—41no—
227C4⋊C4C3×D414yes—
228C22⋊C4—22no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—41no—
234C22⋊C4—41no—
235C42—22no—
236C42—22no—
237C42—41no—
238C22×C3—41no—
239C22×C3—41no—
240C22×C3—41no—
241C22×C3—41no—
242C22×C3—41no—
243C22×C3—41no—
244C22×C3—41no—
245C22×C3C22×D411yescharacteristic
246C22×C3C2×#4111yescharacteristic
247C22×C3C2×#4111yescharacteristic
248C22×C3H(5,𝔽2)11yescharacteristic
249C22×C32−1+411yescharacteristic
250C22×C32−1+411yescharacteristic
251C22×C32−1+411yescharacteristic
252C3×C4—22no—
253C3×C4—22no—
254C3×C4—22no—
255C3×C4—22no—
256C3×C4—22no—
257C3×C4—41no—
258C3×C4—41no—
259C3×C4—41no—
260C3×C4—41no—
261C3×C4—41no—
262C3×C4—41no—
263C3×C4—41no—
264C3×C4—41no—
265C3×C4—41no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23C24×C311yescharacteristic
274Q8—44no—
275Q8—44no—
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—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4C2×C3×D412yes—
314C2×C4C2×C3×D412yes—
315C2×C4C2×C3×D412yes—
316C2×C4C3×#4112yes—
317C2×C4C3×#4112yes—
318C2×C3—81no—
319C2×C3#53611yescharacteristic
320C2×C3#54111yescharacteristic
321C2×C3#54211yescharacteristic
322C2×C3#54611yescharacteristic
323C2×C3#54811yescharacteristic
324C2×C3#55611yescharacteristic
325C2×C3#56311yescharacteristic
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22C22×C3×D411yescharacteristic
334C22C2×C3×#4111yescharacteristic
335C22C2×C3×#4111yescharacteristic
336C22C3×H(5,𝔽2)11yescharacteristic
337C22C3×2−1+411yescharacteristic
338C22C3×2−1+411yescharacteristic
339C22C3×2−1+411yescharacteristic
340C4—22no—
341C4—22no—
342C4—22no—
343C4—22no—
344C4—22no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—41no—
354C3#271611yescharacteristic, direct product factor, Sylow
355C2—81no—
356C2C3×#53611yescharacteristic
357C2C3×#54111yescharacteristic
358C2C3×#54211yescharacteristic
359C2C3×#54611yescharacteristic
360C2C3×#54811yescharacteristic
361C2C3×#55611yescharacteristic
362C2C3×#56311yescharacteristic
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. C2×C3×C4⋊Q8C2semidirect
3. C3×#400C2semidirect
4. C3×#398C2semidirect
9. C3×#391C2semidirect
10. C3×#391C2semidirect
13. C3×#387C2semidirect
14. C3×#387C2semidirect
15. C3×#387C2semidirect
16. #2716C3direct (N × H)
54. C2×C4⋊Q8C2×C3semidirect
56. #400C2×C3semidirect
57. #398C2×C3semidirect
62. #391C2×C3semidirect
63. #391C2×C3semidirect
66. #387C2×C3semidirect
67. #387C2×C3semidirect
68. #387C2×C3semidirect
354. C3#2716direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊Q82
C2×C3×#1262
C3×#4002
C3×#3982
C3×#3972
C3×#3962
C3×#3932
C3×#3912
C3×#3882
C3×#3872
#27163