Finite Group Databasebeta

← C3×#2720 #84584

Subgroups — C3×#2720 #84584

Order 384.

472 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#125C211yesmaximal, characteristic, complemented
2C2×C3×C22⋊Q8C211yesmaximal, characteristic
3C3×#400C211yesmaximal, characteristic, complemented
4C3×#399C211yesmaximal, characteristic
5C3×#398C211yesmaximal, characteristic, complemented
6C3×#396C211yesmaximal, characteristic
7C3×#395C211yesmaximal, characteristic, complemented
8C3×#395C211yesmaximal, characteristic, complemented
9C3×#389C211yesmaximal, characteristic, complemented
10C3×#388C211yesmaximal, characteristic
11C3×#388C211yesmaximal, characteristic
12C3×#387C211yesmaximal, characteristic, complemented
13C3×#385C211yesmaximal, characteristic
14C3×#385C211yesmaximal, characteristic
15C3×#384C211yesmaximal, characteristic, complemented
16#2720C311yesmaximal, characteristic, direct product factor, Sylow
17C22×C3×Q8C2211yescharacteristic
18C23×C3×C4C2211yescharacteristic
19C3×#125—24no—
20C3×C22⋊Q8—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×C4⋊C4C2211yescharacteristic
31C2×C3×C4⋊C4C2211yescharacteristic
32C2×C3×C4⋊C4C2211yescharacteristic
33C2×C3×C4⋊C4C2211yescharacteristic
34C2×C3×C4⋊C4C2211yescharacteristic
35C2×C3×C4⋊C4C2211yescharacteristic
36C2×C3×C22⋊C4C2211yescharacteristic
37C2×C3×C22⋊C4C2211yescharacteristic
38C2×C3×C22⋊C4C2211yescharacteristic
39C2×C3×C22⋊C4C2211yescharacteristic
40C2×C3×C22⋊C4C2211yescharacteristic
41C2×C3×C22⋊C4C2211yescharacteristic
42C2×C3×C42C2211yescharacteristic
43C2×C3×C42C2211yescharacteristic
44C3×#95C2211yescharacteristic
45C3×#95C2211yescharacteristic
46C3×#95C2211yescharacteristic
47C3×#95C2211yescharacteristic
48C3×#95C2211yescharacteristic
49C3×#95C2211yescharacteristic
50C3×#95C2211yescharacteristic
51C3×#95C2211yescharacteristic
52C3×#95C2211yescharacteristic
53C3×#95C2211yescharacteristic
54C2×#125C2×C311yescharacteristic, complemented
55C2×C22⋊Q8C2×C311yescharacteristic
56#400C2×C311yescharacteristic, complemented
57#399C2×C311yescharacteristic
58#398C2×C311yescharacteristic, complemented
59#396C2×C311yescharacteristic
60#395C2×C311yescharacteristic, complemented
61#395C2×C311yescharacteristic, complemented
62#389C2×C311yescharacteristic, complemented
63#388C2×C311yescharacteristic
64#388C2×C311yescharacteristic
65#387C2×C311yescharacteristic, complemented
66#385C2×C311yescharacteristic
67#385C2×C311yescharacteristic
68#384C2×C311yescharacteristic, complemented
69C24×C3C2311yescharacteristic
70C2×C3×Q8—22no—
71C2×C3×Q8—41no—
72C2×C3×Q8—41no—
73C22×C3×C4—22no—
74C22×C3×C4—41no—
75C22×C3×C4—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—22no—
98C3×C4⋊C4—41no—
99C3×C4⋊C4—41no—
100C3×C4⋊C4—41no—
101C3×C4⋊C4—41no—
102C3×C4⋊C4—41no—
103C3×C4⋊C4—41no—
104C3×C4⋊C4D414yes—
105C3×C22⋊C4—22no—
106C3×C22⋊C4—22no—
107C3×C22⋊C4—22no—
108C3×C22⋊C4—41no—
109C3×C22⋊C4—41no—
110C3×C22⋊C4Q814yescomplemented
111C3×C42—22no—
112C3×C42—22no—
113C22×Q8C22×C311yescharacteristic
114C23×C4C22×C311yescharacteristic
115#125—24no—
116C22⋊Q8—24no—
117C2×C4⋊C4C22×C311yescharacteristic
118C2×C4⋊C4C22×C311yescharacteristic
119C2×C4⋊C4C22×C311yescharacteristic
120C2×C4⋊C4C22×C311yescharacteristic
121C2×C4⋊C4C22×C311yescharacteristic
122C2×C4⋊C4C22×C311yescharacteristic
123C2×C4⋊C4C22×C311yescharacteristic
124C2×C4⋊C4C22×C311yescharacteristic
125C2×C4⋊C4C22×C311yescharacteristic
126C2×C4⋊C4C22×C311yescharacteristic
127C2×C4⋊C4C22×C311yescharacteristic
128C2×C4⋊C4C22×C311yescharacteristic
129C2×C4⋊C4C22×C311yescharacteristic
130C2×C4⋊C4C22×C311yescharacteristic
131C2×C4⋊C4C22×C311yescharacteristic
132C2×C22⋊C4C22×C311yescharacteristic
133C2×C22⋊C4C22×C311yescharacteristic
134C2×C22⋊C4C22×C311yescharacteristic
135C2×C22⋊C4C22×C311yescharacteristic
136C2×C22⋊C4C22×C311yescharacteristic
137C2×C22⋊C4C22×C311yescharacteristic
138C2×C42C22×C311yescharacteristic
139C2×C42C22×C311yescharacteristic
140#95C22×C311yescharacteristic
141#95C22×C311yescharacteristic
142#95C22×C311yescharacteristic
143#95C22×C311yescharacteristic
144#95C22×C311yescharacteristic
145#95C22×C311yescharacteristic
146#95C22×C311yescharacteristic
147#95C22×C311yescharacteristic
148#95C22×C311yescharacteristic
149#95C22×C311yescharacteristic
150C23×C3—21no—
151C23×C3—21no—
152C23×C3—21no—
153C23×C3—21no—
154C23×C3—21no—
155C23×C3—21no—
156C23×C3C2×Q812yes—
157C23×C3C2411yescharacteristic
158C3×Q8—44no—
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×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—44no—
196C2×C3×C4C2×D412yes—
197C2×C3×C4C2×D412yes—
198C2×C3×C4C2×D412yes—
199C2×C3×C4C2×Q812yes—
200C2×C3×C4C2×Q812yes—
201C2×C3×C4#4112yes—
202C24C23×C311yescharacteristic
203C2×Q8—22no—
204C2×Q8—41no—
205C2×Q8—41no—
206C22×C4—22no—
207C22×C4—41no—
208C22×C4—41no—
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
219C22×C4C23×C311yescharacteristic
220C22×C4C23×C311yescharacteristic
221C22×C4C23×C311yescharacteristic
222C22×C4C23×C311yescharacteristic
223C4⋊C4—22no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4C3×D414yes—
238C22⋊C4—22no—
239C22⋊C4—22no—
240C22⋊C4—22no—
241C22⋊C4—41no—
242C22⋊C4—41no—
243C22⋊C4C3×Q814yescomplemented
244C42—22no—
245C42—22no—
246C22×C3—22no—
247C22×C3—22no—
248C22×C3—22no—
249C22×C3—41no—
250C22×C3—41no—
251C22×C3—41no—
252C22×C3—41no—
253C22×C3C22×D411yescharacteristic
254C22×C3C22×Q811yescharacteristic
255C22×C3C2×#4111yescharacteristic
256C22×C3H(5,𝔽2)11yescharacteristic
257C22×C3H(5,𝔽2)11yescharacteristic
258C22×C32−1+411yescharacteristic
259C22×C32−1+411yescharacteristic
260C3×C4—22no—
261C3×C4—22no—
262C3×C4—22no—
263C3×C4—22no—
264C3×C4—22no—
265C3×C4—22no—
266C3×C4—41no—
267C3×C4—41no—
268C3×C4—41no—
269C3×C4—41no—
270C3×C4—41no—
271C3×C4—41no—
272C3×C4—41no—
273C3×C4—41no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23C2×C3×Q812yes—
281C23C24×C311yescharacteristic
282Q8—44no—
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×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—44no—
320C2×C4C2×C3×D412yes—
321C2×C4C2×C3×D412yes—
322C2×C4C2×C3×D412yes—
323C2×C4C2×C3×Q812yes—
324C2×C4C2×C3×Q812yes—
325C2×C4C3×#4112yes—
326C2×C3—42no—
327C2×C3#53511yescharacteristic
328C2×C3#53911yescharacteristic
329C2×C3#54411yescharacteristic
330C2×C3#54711yescharacteristic
331C2×C3D4×Q811yescharacteristic
332C2×C3#55411yescharacteristic
333C2×C3#56311yescharacteristic
334C22—22no—
335C22—22no—
336C22—22no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22C22×C3×D411yescharacteristic
342C22C22×C3×Q811yescharacteristic
343C22C2×C3×#4111yescharacteristic
344C22C3×H(5,𝔽2)11yescharacteristic
345C22C3×H(5,𝔽2)11yescharacteristic
346C22C3×2−1+411yescharacteristic
347C22C3×2−1+411yescharacteristic
348C4—22no—
349C4—22no—
350C4—22no—
351C4—22no—
352C4—22no—
353C4—22no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—41no—
362C3#272011yescharacteristic, direct product factor, Sylow
363C2—42no—
364C2C3×#53511yescharacteristic
365C2C3×#53911yescharacteristic
366C2C3×#54411yescharacteristic
367C2C3×#54711yescharacteristic
368C2C3×D4×Q811yescharacteristic
369C2C3×#55411yescharacteristic
370C2C3×#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×#125C2semidirect
3. C3×#400C2semidirect
5. C3×#398C2semidirect
7. C3×#395C2semidirect
8. C3×#395C2semidirect
9. C3×#389C2semidirect
12. C3×#387C2semidirect
15. C3×#384C2semidirect
16. #2720C3direct (N × H)
54. C2×#125C2×C3semidirect
56. #400C2×C3semidirect
58. #398C2×C3semidirect
60. #395C2×C3semidirect
61. #395C2×C3semidirect
62. #389C2×C3semidirect
65. #387C2×C3semidirect
68. #384C2×C3semidirect
110. C3×C22⋊C4Q8semidirect
243. C22⋊C4C3×Q8semidirect
362. C3#2720direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#1252
C2×C3×C22⋊Q82
C3×#4002
C3×#3992
C3×#3982
C3×#3962
C3×#3952
C3×#3892
C3×#3882
C3×#3872
C3×#3852
C3×#3842
#27203