Finite Group Databasebeta

← C3×#2015 #72730

Subgroups — C3×#2015 #72730

Order 384.

390 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×Q16C211yesmaximal, characteristic, complemented
2C22×C3×SD16C211yesmaximal, characteristic
3C2×C3×#124C211yesmaximal, characteristic
4C2×C3×#103C211yesmaximal, characteristic, complemented
5C2×C3×#102C211yesmaximal, characteristic
6C3×#391C211yesmaximal, characteristic, complemented
7C3×#336C211yesmaximal, characteristic, complemented
8#2015C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×Q8C2211yescharacteristic
10C22×C3×Q8C2211yescharacteristic
11C22×C3×D4C2211yescharacteristic
12C2×C3×Q16—22no—
13C2×C3×Q16—22no—
14C2×C3×Q16—22no—
15C2×C3×SD16—22no—
16C2×C3×SD16—22no—
17C2×C3×SD16—22no—
18C22×C3×C8C2211yescharacteristic
19C22×C3×C8C2211yescharacteristic
20C3×#124—22no—
21C3×#124—22no—
22C2×C3×C4⋊C4C2211yescharacteristic
23C2×C3×C22⋊C4—22no—
24C2×C3×C42C2211yescharacteristic
25C3×#103—22no—
26C3×#102—22no—
27C3×#95—22no—
28C22×Q16C2×C311yescharacteristic, complemented
29C22×SD16C2×C311yescharacteristic
30C2×#124C2×C311yescharacteristic
31C2×#103C2×C311yescharacteristic, complemented
32C2×#102C2×C311yescharacteristic
33#391C2×C311yescharacteristic, complemented
34#336C2×C311yescharacteristic, complemented
35C24×C3—21no—
36C2×C3×Q8—21no—
37C2×C3×Q8—21no—
38C2×C3×Q8—21no—
39C2×C3×Q8—21no—
40C2×C3×Q8—21no—
41C2×C3×Q8—21no—
42C2×C3×Q8—21no—
43C2×C3×Q8—21no—
44C2×C3×Q8—21no—
45C2×C3×Q8—21no—
46C2×C3×Q8D412yes—
47C2×C3×Q8D412yes—
48C2×C3×D4—21no—
49C2×C3×D4—21no—
50C2×C3×D4—21no—
51C2×C3×D4—21no—
52C2×C3×D4—21no—
53C2×C3×D4D412yes—
54C22×C3×C4—21no—
55C22×C3×C4—21no—
56C22×C3×C4—21no—
57C22×C3×C4D412yes—
58C22×C3×C4C2311yescharacteristic
59C3×Q16—28no—
60C3×SD16—28no—
61C2×C3×C8—21no—
62C2×C3×C8—21no—
63C2×C3×C8—21no—
64C2×C3×C8—21no—
65C2×C3×C8D412yes—
66C2×C3×C8D412yes—
67C3×C4⋊C4—21no—
68C3×C4⋊C4—21no—
69C3×C22⋊C4—28no—
70C3×C42—21no—
71C3×C42—21no—
72C22×Q8C22×C311yescharacteristic
73C22×Q8C22×C311yescharacteristic
74C22×D4C22×C311yescharacteristic
75C2×Q16—22no—
76C2×Q16—22no—
77C2×Q16—22no—
78C2×SD16—22no—
79C2×SD16—22no—
80C2×SD16—22no—
81C22×C8C22×C311yescharacteristic
82C22×C8C22×C311yescharacteristic
83#124—22no—
84#124—22no—
85C2×C4⋊C4C22×C311yescharacteristic
86C2×C22⋊C4—22no—
87C2×C42C22×C311yescharacteristic
88#103—22no—
89#102—22no—
90#95—22no—
91C23×C3—22no—
92C23×C3—41no—
93C23×C3—41no—
94C23×C3—42no—
95C23×C3—42no—
96C23×C3C2×D411yescharacteristic
97C3×Q8—22no—
98C3×Q8—22no—
99C3×Q8—22no—
100C3×Q8—22no—
101C3×Q8—41no—
102C3×Q8—41no—
103C3×Q8—41no—
104C3×Q8—41no—
105C3×D4—22no—
106C3×D4—22no—
107C3×D4—41no—
108C3×D4—41no—
109C2×C3×C4—22no—
110C2×C3×C4—22no—
111C2×C3×C4—22no—
112C2×C3×C4—22no—
113C2×C3×C4—22no—
114C2×C3×C4—41no—
115C2×C3×C4—41no—
116C2×C3×C4—41no—
117C2×C3×C4—41no—
118C2×C3×C4—41no—
119C2×C3×C4—42no—
120C2×C3×C4C2×D411yescharacteristic
121C2×C3×C4C2×D411yescharacteristic
122C2×C3×C4C2×D411yescharacteristic
123C2×C3×C4C2×D411yescharacteristic
124C2×C3×C4C2×D411yescharacteristic
125C2×C3×C4#4111yescharacteristic
126C3×C8—22no—
127C3×C8—22no—
128C24—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8C3×D412yes—
140C2×Q8C3×D412yes—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4C3×D412yes—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4C3×D412yes—
151C22×C4C23×C311yescharacteristic
152Q16—28no—
153SD16—28no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8C3×D412yes—
159C2×C8C3×D412yes—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C22⋊C4—28no—
163C42—21no—
164C42—21no—
165C22×C3—42no—
166C22×C3—44no—
167C22×C3—82no—
168C22×C3—81no—
169C22×C3—81no—
170C22×C3#12011yescharacteristic
171C22×C3#12111yescharacteristic
172C22×C3#12711yescharacteristic
173C22×C3#13512yes—
174C22×C3#13712yes—
175C3×C4—42no—
176C3×C4—42no—
177C3×C4—42no—
178C3×C4—81no—
179C3×C4#12011yescharacteristic
180C3×C4#12011yescharacteristic
181C3×C4#12111yescharacteristic
182C3×C4#12111yescharacteristic
183C23—22no—
184C23—41no—
185C23—41no—
186C23—42no—
187C23—42no—
188C23C2×C3×D411yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—22no—
192Q8—22no—
193Q8—41no—
194Q8—41no—
195Q8—41no—
196Q8—41no—
197D4—22no—
198D4—22no—
199D4—41no—
200D4—41no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4C2×C3×D411yescharacteristic
213C2×C4C2×C3×D411yescharacteristic
214C2×C4C2×C3×D411yescharacteristic
215C2×C4C2×C3×D411yescharacteristic
216C2×C4C2×C3×D411yescharacteristic
217C2×C4C3×#4111yescharacteristic
218C8—22no—
219C8—22no—
220C2×C3—82no—
221C2×C3#39211yescharacteristic
222C2×C3#45212yes—
223C2×C3#46412yes—
224C2×C3#49511yescharacteristic
225C2×C3#49711yescharacteristic
226C22—42no—
227C22—44no—
228C22—82no—
229C22—81no—
230C22—81no—
231C22C3×#12011yescharacteristic
232C22C3×#12111yescharacteristic
233C22C3×#12711yescharacteristic
234C22C3×#13512yes—
235C22C3×#13712yes—
236C4—42no—
237C4—42no—
238C4—42no—
239C4—81no—
240C4C3×#12011yescharacteristic
241C4C3×#12011yescharacteristic
242C4C3×#12111yescharacteristic
243C4C3×#12111yescharacteristic
244C3#201511yescharacteristic, direct product factor, Sylow
245C2—82no—
246C2C3×#39211yescharacteristic
247C2C3×#45212yes—
248C2C3×#46412yes—
249C2C3×#49511yescharacteristic
250C2C3×#49711yescharacteristic
subgroupG/H
center, socleC23×C3C2×D4
commutator, Frattini, cosocleC22×C4C23×C3
prime radicalC22×C3×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C3×Q16C2semidirect
4. C2×C3×#103C2semidirect
6. C3×#391C2semidirect
7. C3×#336C2semidirect
8. #2015C3direct (N × H)
28. C22×Q16C2×C3semidirect
31. C2×#103C2×C3semidirect
33. #391C2×C3semidirect
34. #336C2×C3semidirect
244. C3#2015direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×Q162
C22×C3×SD162
C2×C3×#1242
C2×C3×#1032
C2×C3×#1022
C3×#3912
C3×#3362
#20153