Finite Group Databasebeta

← C3×#2000 #72715

Subgroups — C3×#2000 #72715

Order 384.

444 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×SD16C211yesmaximal, characteristic, complemented
2C2×C3×C22⋊Q8C211yesmaximal, characteristic, complemented
3C2×C3×#121C211yesmaximal, characteristic
4C2×C3×#103C211yesmaximal, characteristic, complemented
5C2×C3×#102C211yesmaximal, characteristic, complemented
6C2×C3×C22⋊C8C211yesmaximal, characteristic, complemented
7C3×#340C211yesmaximal, characteristic, complemented
8#2000C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×Q8C2211yescharacteristic
10C22×C3×D4—21no—
11C22×C3×D4C2211yescharacteristic
12C23×C3×C4C2211yescharacteristic
13C2×C3×SD16—22no—
14C2×C3×SD16—22no—
15C2×C3×SD16—22no—
16C22×C3×C8C2211yescharacteristic
17C22×C3×C8C2211yescharacteristic
18C3×C22⋊Q8—22no—
19C3×C22⋊Q8—22no—
20C3×#121—22no—
21C3×#121—22no—
22C2×C3×C4⋊C4—21no—
23C2×C3×C4⋊C4C2211yescharacteristic
24C2×C3×C4⋊C4C2211yescharacteristic
25C2×C3×C22⋊C4—21no—
26C2×C3×C22⋊C4—21no—
27C3×#103—22no—
28C3×#102—22no—
29C3×C22⋊C8—22no—
30C22×SD16C2×C311yescharacteristic, complemented
31C2×C22⋊Q8C2×C311yescharacteristic, complemented
32C2×#121C2×C311yescharacteristic
33C2×#103C2×C311yescharacteristic, complemented
34C2×#102C2×C311yescharacteristic, complemented
35C2×C22⋊C8C2×C311yescharacteristic, complemented
36#340C2×C311yescharacteristic, complemented
37C24×C3—21no—
38C24×C3D411yescharacteristic
39C2×C3×Q8—21no—
40C2×C3×Q8—21no—
41C2×C3×Q8—21no—
42C2×C3×Q8—21no—
43C2×C3×Q8—21no—
44C2×C3×Q8D412yescomplemented
45C2×C3×D4—24no—
46C2×C3×D4—21no—
47C2×C3×D4—21no—
48C2×C3×D4—21no—
49C2×C3×D4—21no—
50C2×C3×D4—21no—
51C2×C3×D4—42no—
52C2×C3×D4—42no—
53C2×C3×D4D412yes—
54C22×C3×C4—21no—
55C22×C3×C4—21no—
56C22×C3×C4—21no—
57C22×C3×C4—21no—
58C22×C3×C4—21no—
59C22×C3×C4—21no—
60C22×C3×C4—21no—
61C22×C3×C4—21no—
62C22×C3×C4—21no—
63C22×C3×C4D411yescharacteristic
64C22×C3×C4C2311yescharacteristic
65C3×SD16—28no—
66C2×C3×C8—21no—
67C2×C3×C8—21no—
68C2×C3×C8—21no—
69C2×C3×C8—21no—
70C2×C3×C8—21no—
71C2×C3×C8D412yescomplemented
72C3×C4⋊C4—24no—
73C3×C4⋊C4—21no—
74C3×C4⋊C4—21no—
75C3×C4⋊C4D412yes—
76C3×C4⋊C4D412yescomplemented
77C3×C22⋊C4—24no—
78C3×C22⋊C4—24no—
79C22×Q8C22×C311yescharacteristic
80C22×D4—21no—
81C22×D4C22×C311yescharacteristic
82C23×C4C22×C311yescharacteristic
83C2×SD16—22no—
84C2×SD16—22no—
85C2×SD16—22no—
86C22×C8C22×C311yescharacteristic
87C22×C8C22×C311yescharacteristic
88C22⋊Q8—22no—
89C22⋊Q8—22no—
90#121—22no—
91#121—22no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4C22×C311yescharacteristic
94C2×C4⋊C4C22×C311yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97#103—22no—
98#102—22no—
99C22⋊C8—22no—
100C23×C3—21no—
101C23×C3—21no—
102C23×C3—21no—
103C23×C3—21no—
104C23×C3—21no—
105C23×C3—21no—
106C23×C3—22no—
107C23×C3—41no—
108C23×C3—41no—
109C23×C3—41no—
110C23×C3—41no—
111C23×C3—41no—
112C23×C3—41no—
113C23×C3SD1612yescomplemented
114C23×C3C2×D411yescharacteristic
115C3×Q8—22no—
116C3×Q8—22no—
117C3×Q8—41no—
118C3×Q8—41no—
119C3×D4—22no—
120C3×D4—22no—
121C3×D4—48no—
122C3×D4—41no—
123C3×D4—41no—
124C2×C3×C4—21no—
125C2×C3×C4—22no—
126C2×C3×C4—22no—
127C2×C3×C4—22no—
128C2×C3×C4—21no—
129C2×C3×C4—21no—
130C2×C3×C4—41no—
131C2×C3×C4—41no—
132C2×C3×C4—41no—
133C2×C3×C4—41no—
134C2×C3×C4—41no—
135C2×C3×C4—41no—
136C2×C3×C4—41no—
137C2×C3×C4—41no—
138C2×C3×C4—41no—
139C2×C3×C4—41no—
140C2×C3×C4C2×D411yescharacteristic
141C2×C3×C4C2×D411yescharacteristic
142C2×C3×C4C2×D411yescharacteristic
143C2×C3×C4C2×D411yescharacteristic
144C2×C3×C4C2×D411yescharacteristic
145C2×C3×C4#4111yescharacteristic
146C3×C8—22no—
147C3×C8—41no—
148C24—21no—
149C24C3×D411yescharacteristic
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8C3×D412yescomplemented
156C2×D4—24no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4C3×D412yes—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4C3×D411yescharacteristic
175C22×C4C23×C311yescharacteristic
176SD16—28no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8C3×D412yescomplemented
183C4⋊C4—24no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4C3×D412yes—
187C4⋊C4C3×D412yescomplemented
188C22⋊C4—24no—
189C22⋊C4—24no—
190C22×C3—22no—
191C22×C3—22no—
192C22×C3—22no—
193C22×C3—41no—
194C22×C3—41no—
195C22×C3—41no—
196C22×C3—41no—
197C22×C3—42no—
198C22×C3—42no—
199C22×C3—42no—
200C22×C3—81no—
201C22×C3—81no—
202C22×C3—81no—
203C22×C3—81no—
204C22×C3#12011yescharacteristic
205C22×C3#12011yescharacteristic
206C22×C3#12111yescharacteristic
207C22×C3C2×SD1611yescharacteristic
208C22×C3#13511yescharacteristic
209C22×C3AGL(1,ℤ/8)11yescharacteristic
210C22×C3#13711yescharacteristic
211C3×C4—41no—
212C3×C4—42no—
213C3×C4—42no—
214C3×C4—42no—
215C3×C4#12011yescharacteristic
216C3×C4#12111yescharacteristic
217C3×C4#12111yescharacteristic
218C3×C4#12711yescharacteristic
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—22no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23C3×SD1612yescomplemented
233C23C2×C3×D411yescharacteristic
234Q8—22no—
235Q8—22no—
236Q8—41no—
237Q8—41no—
238D4—22no—
239D4—22no—
240D4—48no—
241D4—41no—
242D4—41no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4C2×C3×D411yescharacteristic
260C2×C4C2×C3×D411yescharacteristic
261C2×C4C2×C3×D411yescharacteristic
262C2×C4C2×C3×D411yescharacteristic
263C2×C4C2×C3×D411yescharacteristic
264C2×C4C3×#4111yescharacteristic
265C8—22no—
266C8—41no—
267C2×C3—42no—
268C2×C3—82no—
269C2×C3#39211yescharacteristic
270C2×C3#44811yescharacteristic
271C2×C3#44911yescharacteristic
272C2×C3#45011yescharacteristic
273C2×C3#45211yescharacteristic
274C2×C3#46511yescharacteristic
275C2×C3#46811yescharacteristic
276C22—22no—
277C22—22no—
278C22—22no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—81no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22C3×#12011yescharacteristic
291C22C3×#12011yescharacteristic
292C22C3×#12111yescharacteristic
293C22C2×C3×SD1611yescharacteristic
294C22C3×#13511yescharacteristic
295C22C3×AGL(1,ℤ/8)11yescharacteristic
296C22C3×#13711yescharacteristic
297C4—41no—
298C4—42no—
299C4—42no—
300C4—42no—
301C4C3×#12011yescharacteristic
302C4C3×#12111yescharacteristic
303C4C3×#12111yescharacteristic
304C4C3×#12711yescharacteristic
305C3#200011yescharacteristic, direct product factor, Sylow
306C2—42no—
307C2—82no—
308C2C3×#39211yescharacteristic
309C2C3×#44811yescharacteristic
310C2C3×#44911yescharacteristic
311C2C3×#45011yescharacteristic
312C2C3×#45211yescharacteristic
313C2C3×#46511yescharacteristic
314C2C3×#46811yescharacteristic
subgroupG/H
center, socleC23×C3C2×D4
commutator, Frattini, cosocleC22×C4C23×C3
prime radicalC2×C3×#121C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C3×SD16C2semidirect
2. C2×C3×C22⋊Q8C2semidirect
4. C2×C3×#103C2semidirect
5. C2×C3×#102C2semidirect
6. C2×C3×C22⋊C8C2semidirect
7. C3×#340C2semidirect
8. #2000C3direct (N × H)
30. C22×SD16C2×C3semidirect
31. C2×C22⋊Q8C2×C3semidirect
33. C2×#103C2×C3semidirect
34. C2×#102C2×C3semidirect
35. C2×C22⋊C8C2×C3semidirect
36. #340C2×C3semidirect
44. C2×C3×Q8D4semidirect
71. C2×C3×C8D4semidirect
76. C3×C4⋊C4D4semidirect
113. C23×C3SD16semidirect
155. C2×Q8C3×D4semidirect
182. C2×C8C3×D4semidirect
187. C4⋊C4C3×D4semidirect
232. C23C3×SD16semidirect
305. C3#2000direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×SD162
C2×C3×C22⋊Q82
C2×C3×#1212
C2×C3×#1032
C2×C3×#1022
C2×C3×C22⋊C82
C3×#3402
#20003