Finite Group Databasebeta

← C3×#2005 #72720

Subgroups — C3×#2005 #72720

Order 384.

362 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 92 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
1C2×C3×#137C211yesmaximal, characteristic
2C3×#536C211yesmaximal, characteristic
3C2×C3×C4⋊Q8C211yesmaximal, characteristic, complemented
4C2×C3×#104C211yesmaximal, characteristic
5C3×#419C211yesmaximal, characteristic, complemented
6C2×C3×#101C211yesmaximal, characteristic, complemented
7C3×#339C211yesmaximal, characteristic, complemented
8#2005C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×#41C2211yescharacteristic
10C22×C3×Q8C2211yescharacteristic
11C22×C3×Q8C2211yescharacteristic
12C3×#137—22no—
13C3×#137—22no—
14C2×C3×Q16—21no—
15C2×C3×SD16—21no—
16C2×C3×M16C2211yescharacteristic
17C2×C3×M16C2211yescharacteristic
18C3×C4⋊Q8—22no—
19C3×C4⋊Q8—21no—
20C3×C4⋊Q8—22no—
21C3×#124—21no—
22C3×#123—22no—
23C3×C22⋊Q8—22no—
24C3×#117C2211yescharacteristic
25C2×C3×C4⋊C4—22no—
26C2×C3×C42C2211yescharacteristic
27C3×#104—22no—
28C3×#103—22no—
29C3×#101—22no—
30C2×#137C2×C311yescharacteristic
31#536C2×C311yescharacteristic
32C2×C4⋊Q8C2×C311yescharacteristic, complemented
33C2×#104C2×C311yescharacteristic
34#419C2×C311yescharacteristic, complemented
35C2×#101C2×C311yescharacteristic, complemented
36#339C2×C311yescharacteristic, complemented
37C3×#41—21no—
38C3×#41—21no—
39C3×#41—21no—
40C3×#41—21no—
41C2×C3×Q8—21no—
42C2×C3×Q8—21no—
43C2×C3×Q8—21no—
44C2×C3×Q8—21no—
45C2×C3×Q8—21no—
46C2×C3×Q8—21no—
47C2×C3×Q8—21no—
48C2×C3×Q8—21no—
49C2×C3×Q8—21no—
50C2×C3×Q8—21no—
51C2×C3×Q8—21no—
52C2×C3×Q8D411yescharacteristic
53C2×C3×Q8D412yes—
54C2×C3×D4—21no—
55C2×C3×D4D411yescharacteristic
56C22×C3×C4—21no—
57C22×C3×C4—21no—
58C22×C3×C4—21no—
59C22×C3×C4D412yes—
60C22×C3×C4C2311yescharacteristic
61C3×Q16—24no—
62C3×SD16—24no—
63C3×M16—21no—
64C3×M16—21no—
65C3×M16—21no—
66C3×M16D412yes—
67C2×C3×C8—21no—
68C2×C3×C8—21no—
69C3×C4⋊C4—28no—
70C3×C4⋊C4—24no—
71C3×C4⋊C4D412yes—
72C3×C22⋊C4—24no—
73C3×C22⋊C4—21no—
74C3×C42—21no—
75C3×C42—21no—
76C3×C42D412yes—
77C2×#41C22×C311yescharacteristic
78C22×Q8C22×C311yescharacteristic
79C22×Q8C22×C311yescharacteristic
80#137—22no—
81#137—22no—
82C2×Q16—21no—
83C2×SD16—21no—
84C2×M16C22×C311yescharacteristic
85C2×M16C22×C311yescharacteristic
86C4⋊Q8—22no—
87C4⋊Q8—21no—
88C4⋊Q8—22no—
89#124—21no—
90#123—22no—
91C22⋊Q8—22no—
92#117C22×C311yescharacteristic
93C2×C4⋊C4—22no—
94C2×C42C22×C311yescharacteristic
95#104—22no—
96#103—22no—
97#101—22no—
98C23×C3—21no—
99C23×C3C2×D411yescharacteristic
100C3×Q8—22no—
101C3×Q8—21no—
102C3×Q8—22no—
103C3×Q8—21no—
104C3×Q8—41no—
105C3×Q8—41no—
106C3×Q8—41no—
107C3×Q8—41no—
108C3×Q8—41no—
109C3×Q8—41no—
110C3×D4—21no—
111C3×D4—21no—
112C3×D4—41no—
113C3×D4—41no—
114C2×C3×C4—24no—
115C2×C3×C4—22no—
116C2×C3×C4—22no—
117C2×C3×C4—22no—
118C2×C3×C4—22no—
119C2×C3×C4—22no—
120C2×C3×C4—21no—
121C2×C3×C4—22no—
122C2×C3×C4—41no—
123C2×C3×C4—41no—
124C2×C3×C4—41no—
125C2×C3×C4—41no—
126C2×C3×C4C2×D411yescharacteristic
127C2×C3×C4C2×D411yescharacteristic
128C2×C3×C4C2×D411yescharacteristic
129C2×C3×C4C2×D411yescharacteristic
130C2×C3×C4C2×D411yescharacteristic
131C2×C3×C4#4111yescharacteristic
132C3×C8—22no—
133C3×C8—41no—
134#41—21no—
135#41—21no—
136#41—21no—
137#41—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8C3×D411yescharacteristic
150C2×Q8C3×D412yes—
151C2×D4—21no—
152C2×D4C3×D411yescharacteristic
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4C3×D412yes—
157C22×C4C23×C311yescharacteristic
158Q16—24no—
159SD16—24no—
160M16—21no—
161M16—21no—
162M16—21no—
163M16C3×D412yes—
164C2×C8—21no—
165C2×C8—21no—
166C4⋊C4—28no—
167C4⋊C4—24no—
168C4⋊C4C3×D412yes—
169C22⋊C4—24no—
170C22⋊C4—21no—
171C42—21no—
172C42—21no—
173C42C3×D412yes—
174C22×C3—22no—
175C22×C3—41no—
176C22×C3—42no—
177C22×C3#12011yescharacteristic
178C22×C3#12011yescharacteristic
179C22×C3#12111yescharacteristic
180C3×C4—24no—
181C3×C4—42no—
182C3×C4—41no—
183C3×C4—42no—
184C3×C4—42no—
185C3×C4#12011yescharacteristic
186C3×C4#12111yescharacteristic
187C3×C4#12111yescharacteristic
188C3×C4#12711yescharacteristic
189C23—21no—
190C23C2×C3×D411yescharacteristic
191Q8—22no—
192Q8—21no—
193Q8—22no—
194Q8—21no—
195Q8—41no—
196Q8—41no—
197Q8—41no—
198Q8—41no—
199Q8—41no—
200Q8—41no—
201D4—21no—
202D4—21no—
203D4—41no—
204D4—41no—
205C2×C4—24no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4C2×C3×D411yescharacteristic
218C2×C4C2×C3×D411yescharacteristic
219C2×C4C2×C3×D411yescharacteristic
220C2×C4C2×C3×D411yescharacteristic
221C2×C4C2×C3×D411yescharacteristic
222C2×C4C3×#4111yescharacteristic
223C8—22no—
224C8—41no—
225C2×C3—21no—
226C2×C3—21no—
227C2×C3—81no—
228C2×C3#39211yescharacteristic
229C2×C3#45612yes—
230C22—22no—
231C22—41no—
232C22—42no—
233C22C3×#12011yescharacteristic
234C22C3×#12011yescharacteristic
235C22C3×#12111yescharacteristic
236C4—24no—
237C4—42no—
238C4—41no—
239C4—42no—
240C4—42no—
241C4C3×#12011yescharacteristic
242C4C3×#12111yescharacteristic
243C4C3×#12111yescharacteristic
244C4C3×#12711yescharacteristic
245C3#200511yescharacteristic, direct product factor, Sylow
246C2—21no—
247C2—21no—
248C2—81no—
249C2C3×#39211yescharacteristic
250C2C3×#45612yes—
subgroupG/H
center, socleC22×C3#120
commutator, Frattini, cosocleC22×C4C23×C3
prime radicalC2×C3×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C3×C4⋊Q8C2semidirect
5. C3×#419C2semidirect
6. C2×C3×#101C2semidirect
7. C3×#339C2semidirect
8. #2005C3direct (N × H)
32. C2×C4⋊Q8C2×C3semidirect
34. #419C2×C3semidirect
35. C2×#101C2×C3semidirect
36. #339C2×C3semidirect
245. C3#2005direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#1372
C3×#5362
C2×C3×C4⋊Q82
C2×C3×#1042
C3×#4192
C2×C3×#1012
C3×#3392
#20053