Finite Group Databasebeta

← C3×#12130 #10630737

Subgroups — C3×#12130 #10630737

Order 768.

442 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C3×#2811C211yesmaximal, characteristic, complemented
2C3×#2007C211yesmaximal, characteristic
3C3×#1604C212yesmaximal
4C3×#1354C212yesmaximal, complemented
5C3×#1319C211yesmaximal, characteristic, complemented
6#12130C311yesmaximal, characteristic, direct product factor, Sylow
7C2×C3×#137—21no—
8C3×#537C2211yescharacteristic
9C3×#448—22no—
10C2×C3×#104—21no—
11C3×#395—22no—
12C3×#395C2212yes—
13C3×#391—21no—
14C3×#391—21no—
15C3×#391—22no—
16C3×#391C2211yescharacteristic
17C3×#343C2211yescharacteristic
18C3×#327—22no—
19C3×#324C2212yes—
20#2811C2×C311yescharacteristic, complemented
21#2007C2×C311yescharacteristic
22#1604C2×C312yes—
23#1354C2×C312yescomplemented
24#1319C2×C311yescharacteristic, complemented
25C2×C3×#41—21no—
26C22×C3×Q8—21no—
27C22×C3×Q8D411yescharacteristic
28C22×C3×Q8D412yes—
29C3×#137—42no—
30C3×#137—42no—
31C2×C3×Q16—41no—
32C2×C3×SD16—41no—
33C2×C3×SD16—42no—
34C2×C3×M16—21no—
35C3×C22⋊Q8—41no—
36C3×#121—22no—
37C3×#121—41no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—22no—
41C2×C3×C4⋊C4C2311yescharacteristic
42C2×C3×C42—21no—
43C2×C3×C42D411yescharacteristic
44C3×#104—42no—
45C3×#103—42no—
46C3×C22⋊C8—22no—
47C3×#95—22no—
48C3×#95—22no—
49C3×#95—22no—
50C3×#95—22no—
51C3×#95D412yes—
52C2×#137—21no—
53#537C22×C311yescharacteristic
54#448—22no—
55C2×#104—21no—
56#395—22no—
57#395C22×C312yes—
58#391—21no—
59#391—21no—
60#391—22no—
61#391C22×C311yescharacteristic
62#343C22×C311yescharacteristic
63#327—22no—
64#324C22×C312yes—
65C3×#41—41no—
66C3×#41—41no—
67C3×#41—41no—
68C3×#41—41no—
69C2×C3×Q8—24no—
70C2×C3×Q8—21no—
71C2×C3×Q8—21no—
72C2×C3×Q8—42no—
73C2×C3×Q8—41no—
74C2×C3×Q8—41no—
75C2×C3×Q8—82no—
76C2×C3×Q8—81no—
77C2×C3×Q8—81no—
78C2×C3×Q8SD1612yescomplemented
79C2×C3×D4—21no—
80C2×C3×D4—22no—
81C2×C3×D4—41no—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—22no—
85C22×C3×C4—22no—
86C22×C3×C4—41no—
87C22×C3×C4C2×D411yescharacteristic
88C22×C3×C4C2×D412yes—
89C3×Q16—44no—
90C3×SD16—44no—
91C3×SD16—84no—
92C3×M16—22no—
93C3×M16—41no—
94C2×C3×C8—41no—
95C2×C3×C8—42no—
96C3×C4⋊C4—22no—
97C3×C4⋊C4—41no—
98C3×C4⋊C4—82no—
99C3×C4⋊C4—81no—
100C3×C4⋊C4—81no—
101C3×C22⋊C4—42no—
102C3×C42—41no—
103C3×C42—81no—
104C2×#41—21no—
105C22×Q8—21no—
106C22×Q8C3×D411yescharacteristic
107C22×Q8C3×D412yes—
108#137—42no—
109#137—42no—
110C2×Q16—41no—
111C2×SD16—41no—
112C2×SD16—42no—
113C2×M16—21no—
114C22⋊Q8—41no—
115#121—22no—
116#121—41no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4C23×C311yescharacteristic
121C2×C42—21no—
122C2×C42C3×D411yescharacteristic
123#104—42no—
124#103—42no—
125C22⋊C8—22no—
126#95—22no—
127#95—22no—
128#95—22no—
129#95—22no—
130#95C3×D412yes—
131C23×C3—41no—
132C23×C3#12011yescharacteristic
133C3×Q8—42no—
134C3×Q8—41no—
135C3×Q8—41no—
136C3×Q8—44no—
137C3×Q8—81no—
138C3×Q8—84no—
139C3×Q8—82no—
140C3×D4—41no—
141C3×D4—41no—
142C3×D4—82no—
143C3×D4—81no—
144C3×D4—81no—
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4—22no—
148C2×C3×C4—24no—
149C2×C3×C4—22no—
150C2×C3×C4—22no—
151C2×C3×C4—41no—
152C2×C3×C4—42no—
153C2×C3×C4—41no—
154C2×C3×C4—41no—
155C2×C3×C4—42no—
156C2×C3×C4—41no—
157C2×C3×C4—42no—
158C2×C3×C4—42no—
159C2×C3×C4—42no—
160C2×C3×C4—41no—
161C2×C3×C4—42no—
162C2×C3×C4—81no—
163C2×C3×C4—81no—
164C2×C3×C4C2×SD1611yescharacteristic
165C2×C3×C4#13711yescharacteristic
166C3×C8—42no—
167C3×C8—82no—
168#41—41no—
169#41—41no—
170#41—41no—
171#41—41no—
172C2×Q8—24no—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×Q8—42no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—82no—
179C2×Q8—81no—
180C2×Q8—81no—
181C2×Q8C3×SD1612yescomplemented
182C2×D4—21no—
183C2×D4—22no—
184C2×D4—41no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—41no—
190C22×C4C2×C3×D411yescharacteristic
191C22×C4C2×C3×D412yes—
192Q16—44no—
193SD16—44no—
194SD16—84no—
195M16—22no—
196M16—41no—
197C2×C8—41no—
198C2×C8—42no—
199C4⋊C4—22no—
200C4⋊C4—41no—
201C4⋊C4—82no—
202C4⋊C4—81no—
203C4⋊C4—81no—
204C22⋊C4—42no—
205C42—41no—
206C42—81no—
207C22×C3—21no—
208C22×C3—21no—
209C22×C3—81no—
210C22×C3—81no—
211C22×C3—81no—
212C22×C3#44811yescharacteristic
213C22×C3#45311yescharacteristic
214C22×C3B(4,𝔽2)11yescharacteristic
215C3×C4—21no—
216C3×C4—21no—
217C3×C4—42no—
218C3×C4—44no—
219C3×C4—42no—
220C3×C4—42no—
221C3×C4—81no—
222C3×C4—82no—
223C23—41no—
224C23C3×#12011yescharacteristic
225Q8—42no—
226Q8—41no—
227Q8—41no—
228Q8—44no—
229Q8—81no—
230Q8—84no—
231Q8—82no—
232D4—41no—
233D4—41no—
234D4—82no—
235D4—81no—
236D4—81no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—24no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4C2×C3×SD1611yescharacteristic
257C2×C4C3×#13711yescharacteristic
258C8—42no—
259C8—82no—
260C2×C3—21no—
261C2×C3—21no—
262C2×C3—161no—
263C2×C3#159611yescharacteristic
264C2×C3#219911yescharacteristic
265C2×C3#220511yescharacteristic
266C22—21no—
267C22—21no—
268C22—81no—
269C22—81no—
270C22—81no—
271C22C3×#44811yescharacteristic
272C22C3×#45311yescharacteristic
273C22C3×B(4,𝔽2)11yescharacteristic
274C4—21no—
275C4—21no—
276C4—42no—
277C4—44no—
278C4—42no—
279C4—42no—
280C4—81no—
281C4—82no—
282C3#1213011yescharacteristic, direct product factor, Sylow
283C2—21no—
284C2—21no—
285C2—161no—
286C2C3×#159611yescharacteristic
287C2C3×#219911yescharacteristic
288C2C3×#220511yescharacteristic
subgroupG/H
center, socleC22×C3B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C3
prime radicalC3×#537C22
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3 ◁ C2×C3×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2811C2semidirect
4. C3×#1354C2semidirect
5. C3×#1319C2semidirect
6. #12130C3direct (N × H)
20. #2811C2×C3semidirect
23. #1354C2×C3semidirect
24. #1319C2×C3semidirect
78. C2×C3×Q8SD16semidirect
181. C2×Q8C3×SD16semidirect
282. C3#12130direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#28112
C3×#20072
C3×#16042
C3×#13542
C3×#13192
#121303