Finite Group Databasebeta

← C3×#2417 #84281

Subgroups — C3×#2417 #84281

Order 384.

648 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×C3×C22⋊Q8C211yesmaximal, characteristic, complemented
3C2×C3×C4×D4C211yesmaximal, characteristic
4C22×C3×C4⋊C4C211yesmaximal, characteristic, complemented
5C3×#398C212yesmaximal, complemented
6C3×#396C212yesmaximal, complemented
7C3×#391C211yesmaximal, characteristic, complemented
8C3×#389C211yesmaximal, characteristic, complemented
9C3×#386C212yesmaximal
10C3×#380C211yesmaximal, characteristic, complemented
11C3×#380C212yesmaximal, complemented
12#2417C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q8C2211yescharacteristic
14C22×C3×D4C2211yescharacteristic
15C23×C3×C4C2211yescharacteristic
16C23×C3×C4C2212yes—
17C23×C3×C4C2211yescharacteristic
18C3×C22⋊Q8—24no—
19C3×C22⋊Q8—24no—
20C3×C4×D4—24no—
21C2×C3×C4⋊C4—24no—
22C2×C3×C4⋊C4C2211yescharacteristic
23C2×C3×C4⋊C4C2211yescharacteristic
24C2×C3×C4⋊C4C2212yes—
25C2×C3×C4⋊C4C2212yes—
26C2×C3×C4⋊C4C2212yes—
27C2×C3×C4⋊C4C2211yescharacteristic
28C2×C3×C4⋊C4C2212yes—
29C2×C3×C4⋊C4C2211yescharacteristic
30C2×C3×C22⋊C4C2211yescharacteristic
31C2×C3×C22⋊C4C2212yes—
32C2×C3×C22⋊C4C2212yes—
33C2×C3×C22⋊C4C2211yescharacteristic
34C2×C3×C22⋊C4C2212yes—
35C2×C3×C42C2211yescharacteristic
36C3×#95C2212yes—
37C3×#95C2212yes—
38C3×#95C2212yes—
39C3×#95C2212yes—
40C2×C22⋊Q8C2×C311yescharacteristic, complemented
41C2×C22⋊Q8C2×C311yescharacteristic, complemented
42C2×C4×D4C2×C311yescharacteristic
43C22×C4⋊C4C2×C311yescharacteristic, complemented
44#398C2×C312yescomplemented
45#396C2×C312yescomplemented
46#391C2×C311yescharacteristic, complemented
47#389C2×C311yescharacteristic, complemented
48#386C2×C312yes—
49#380C2×C311yescharacteristic, complemented
50#380C2×C312yescomplemented
51C24×C3C2311yescharacteristic
52C24×C3C2311yescharacteristic
53C2×C3×Q8—22no—
54C2×C3×Q8—22no—
55C2×C3×Q8—22no—
56C2×C3×D4—22no—
57C2×C3×D4—22no—
58C2×C3×D4Q814yescomplemented
59C22×C3×C4—22no—
60C22×C3×C4—22no—
61C22×C3×C4—24no—
62C22×C3×C4—24no—
63C22×C3×C4—22no—
64C22×C3×C4—22no—
65C22×C3×C4—22no—
66C22×C3×C4—42no—
67C22×C3×C4D414yes—
68C22×C3×C4C2312yes—
69C22×C3×C4C2311yescharacteristic
70C22×C3×C4C2312yes—
71C22×C3×C4C2312yes—
72C22×C3×C4C2311yescharacteristic
73C22×C3×C4C2312yes—
74C22×C3×C4C2311yescharacteristic
75C22×C3×C4C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C3×C4⋊C4—24no—
78C3×C4⋊C4—22no—
79C3×C4⋊C4—22no—
80C3×C4⋊C4—22no—
81C3×C4⋊C4—24no—
82C3×C4⋊C4—24no—
83C3×C4⋊C4—42no—
84C3×C4⋊C4D414yescomplemented
85C3×C22⋊C4—22no—
86C3×C22⋊C4—24no—
87C3×C22⋊C4—22no—
88C3×C22⋊C4—24no—
89C3×C22⋊C4—42no—
90C3×C42—22no—
91C22×Q8C22×C311yescharacteristic
92C22×D4C22×C311yescharacteristic
93C23×C4C22×C311yescharacteristic
94C23×C4C22×C312yes—
95C23×C4C22×C311yescharacteristic
96C22⋊Q8—24no—
97C22⋊Q8—24no—
98C4×D4—24no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4C22×C311yescharacteristic
101C2×C4⋊C4C22×C312yes—
102C2×C4⋊C4C22×C311yescharacteristic
103C2×C4⋊C4C22×C311yescharacteristic
104C2×C4⋊C4C22×C312yes—
105C2×C4⋊C4C22×C312yes—
106C2×C4⋊C4C22×C312yes—
107C2×C4⋊C4C22×C311yescharacteristic
108C2×C22⋊C4C22×C311yescharacteristic
109C2×C22⋊C4C22×C311yescharacteristic
110C2×C22⋊C4C22×C312yes—
111C2×C22⋊C4C22×C312yes—
112C2×C22⋊C4C22×C312yes—
113C2×C42C22×C311yescharacteristic
114#95C22×C312yes—
115#95C22×C312yes—
116#95C22×C312yes—
117#95C22×C312yes—
118C23×C3—21no—
119C23×C3—21no—
120C23×C3—21no—
121C23×C3—21no—
122C23×C3—21no—
123C23×C3—21no—
124C23×C3—21no—
125C23×C3—21no—
126C23×C3—21no—
127C23×C3—21no—
128C23×C3C2×D412yes—
129C23×C3C2×Q812yes—
130C23×C3C2×Q812yes—
131C23×C3#4112yes—
132C23×C3C2411yescharacteristic
133C3×Q8—44no—
134C3×D4—28no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—22no—
139C2×C3×C4—21no—
140C2×C3×C4—22no—
141C2×C3×C4—28no—
142C2×C3×C4—21no—
143C2×C3×C4—22no—
144C2×C3×C4—21no—
145C2×C3×C4—22no—
146C2×C3×C4—22no—
147C2×C3×C4—21no—
148C2×C3×C4—22no—
149C2×C3×C4—21no—
150C2×C3×C4—22no—
151C2×C3×C4—22no—
152C2×C3×C4—22no—
153C2×C3×C4—22no—
154C2×C3×C4—21no—
155C2×C3×C4—21no—
156C2×C3×C4—22no—
157C2×C3×C4—48no—
158C2×C3×C4—44no—
159C2×C3×C4C2×D412yes—
160C2×C3×C4C2×D414yes—
161C2×C3×C4C2×D412yes—
162C2×C3×C4C2×D412yes—
163C2×C3×C4C2×Q812yes—
164C2×C3×C4#4112yes—
165C24C23×C311yescharacteristic
166C24C23×C311yescharacteristic
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—22no—
170C2×D4—22no—
171C2×D4—22no—
172C2×D4C3×Q814yescomplemented
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—42no—
181C22×C4C3×D414yes—
182C22×C4C23×C312yes—
183C22×C4C23×C311yescharacteristic
184C22×C4C23×C312yes—
185C22×C4C23×C312yes—
186C22×C4C23×C311yescharacteristic
187C22×C4C23×C312yes—
188C22×C4C23×C311yescharacteristic
189C22×C4C23×C311yescharacteristic
190C22×C4C23×C311yescharacteristic
191C4⋊C4—24no—
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—24no—
196C4⋊C4—24no—
197C4⋊C4—42no—
198C4⋊C4C3×D414yescomplemented
199C22⋊C4—22no—
200C22⋊C4—24no—
201C22⋊C4—22no—
202C22⋊C4—24no—
203C22⋊C4—42no—
204C42—22no—
205C22×C3—22no—
206C22×C3—22no—
207C22×C3—22no—
208C22×C3—22no—
209C22×C3—22no—
210C22×C3—22no—
211C22×C3—22no—
212C22×C3—22no—
213C22×C3—22no—
214C22×C3—41no—
215C22×C3—41no—
216C22×C3—41no—
217C22×C3—41no—
218C22×C3C22⋊Q814yescomplemented
219C22×C3C22×D411yescharacteristic
220C22×C3C22×D411yescharacteristic
221C22×C3C22×Q811yescharacteristic
222C22×C3C2×#4111yescharacteristic
223C22×C3C2×#4111yescharacteristic
224C22×C3H(5,𝔽2)11yescharacteristic
225C22×C32−1+411yescharacteristic
226C3×C4—22no—
227C3×C4—22no—
228C3×C4—24no—
229C3×C4—41no—
230C3×C4—42no—
231C3×C4—41no—
232C3×C4—42no—
233C3×C4—42no—
234C3×C4#12114yes—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23C2×C3×D412yes—
246C23C2×C3×Q812yes—
247C23C2×C3×Q812yes—
248C23C3×#4112yes—
249C23C24×C311yescharacteristic
250Q8—44no—
251D4—28no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—28no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—48no—
275C2×C4—44no—
276C2×C4C2×C3×D412yes—
277C2×C4C2×C3×D414yes—
278C2×C4C2×C3×D412yes—
279C2×C4C2×C3×D412yes—
280C2×C4C2×C3×Q812yes—
281C2×C4C3×#4112yes—
282C2×C3—24no—
283C2×C3—42no—
284C2×C3C2×#12111yescharacteristic
285C2×C3C2×C22⋊Q811yescharacteristic
286C2×C3#53711yescharacteristic
287C2×C3#54611yescharacteristic
288C2×C3D4×Q811yescharacteristic
289C2×C3#55211yescharacteristic
290C2×C3#55411yescharacteristic
291C22—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22C3×C22⋊Q814yescomplemented
305C22C22×C3×D411yescharacteristic
306C22C22×C3×D411yescharacteristic
307C22C22×C3×Q811yescharacteristic
308C22C2×C3×#4111yescharacteristic
309C22C2×C3×#4111yescharacteristic
310C22C3×H(5,𝔽2)11yescharacteristic
311C22C3×2−1+411yescharacteristic
312C4—22no—
313C4—22no—
314C4—24no—
315C4—41no—
316C4—42no—
317C4—41no—
318C4—42no—
319C4—42no—
320C4C3×#12114yes—
321C3#241711yescharacteristic, direct product factor, Sylow
322C2—24no—
323C2—42no—
324C2C2×C3×#12111yescharacteristic
325C2C2×C3×C22⋊Q811yescharacteristic
326C2C3×#53711yescharacteristic
327C2C3×#54611yescharacteristic
328C2C3×D4×Q811yescharacteristic
329C2C3×#55211yescharacteristic
330C2C3×#55411yescharacteristic
subgroupG/H
center, socleC23×C3C24
commutator, Frattini, cosocleC23C24×C3
prime radicalC22×C3×D4C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×C22⋊Q8C2semidirect
2. C2×C3×C22⋊Q8C2semidirect
4. C22×C3×C4⋊C4C2semidirect
5. C3×#398C2semidirect
6. C3×#396C2semidirect
7. C3×#391C2semidirect
8. C3×#389C2semidirect
10. C3×#380C2semidirect
11. C3×#380C2semidirect
12. #2417C3direct (N × H)
40. C2×C22⋊Q8C2×C3semidirect
41. C2×C22⋊Q8C2×C3semidirect
43. C22×C4⋊C4C2×C3semidirect
44. #398C2×C3semidirect
45. #396C2×C3semidirect
46. #391C2×C3semidirect
47. #389C2×C3semidirect
49. #380C2×C3semidirect
50. #380C2×C3semidirect
58. C2×C3×D4Q8semidirect
84. C3×C4⋊C4D4semidirect
172. C2×D4C3×Q8semidirect
198. C4⋊C4C3×D4semidirect
218. C22×C3C22⋊Q8semidirect
304. C22C3×C22⋊Q8semidirect
321. C3#2417direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊Q82
C2×C3×C4×D42
C22×C3×C4⋊C42
C3×#3982
C3×#3962
C3×#3912
C3×#3892
C3×#3862
C3×#3802
#24173