Finite Group Databasebeta

← C3×#11558 #10630165

Subgroups — C3×#11558 #10630165

Order 768.

446 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2486C211yesmaximal, characteristic
2C3×#1466C211yesmaximal, characteristic
3C3×#1459C211yesmaximal, characteristic
4C3×#1317C214yesmaximal, complemented
5#11558C311yesmaximal, characteristic, direct product factor, Sylow
6C2×C3×#123C412yes—
7C2×C3×C4×D4C2211yescharacteristic
8C22×C3×C4⋊C4C411yescharacteristic
9C3×#432—21no—
10C2×C3×C22⋊C8—21no—
11C3×#398—21no—
12C3×#397—21no—
13C3×#389—21no—
14C3×#381C2212yes—
15C3×#380—21no—
16C3×#380C411yescharacteristic
17C3×#324—21no—
18C3×#323—21no—
19C3×#323C2212yes—
20C3×#323C2212yes—
21#2486C2×C311yescharacteristic
22#1466C2×C311yescharacteristic
23#1459C2×C311yescharacteristic
24#1317C2×C314yescomplemented
25C22×C3×D4C2×C411yescharacteristic
26C23×C3×C4C2×C412yes—
27C23×C3×C4C2×C411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C22×C3×C8—41no—
30C3×#123—44no—
31C3×C4×D4—41no—
32C3×C4×D4—41no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C4⋊C4—21no—
35C2×C3×C4⋊C4—21no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4—41no—
38C2×C3×C4⋊C4—41no—
39C2×C3×C4⋊C4D411yescharacteristic
40C2×C3×C22⋊C4—22no—
41C2×C3×C22⋊C4C2×C412yes—
42C2×C3×C22⋊C4D411yescharacteristic
43C2×C3×C22⋊C4C2311yescharacteristic
44C2×C3×C42D411yescharacteristic
45C3×C4⋊C8—41no—
46C3×C22⋊C8—22no—
47C3×C22⋊C8—22no—
48C3×C22⋊C8—41no—
49C3×#97—41no—
50C3×#95—21no—
51C3×#95—22no—
52C3×#95—22no—
53C2×#123C3×C412yes—
54C2×C4×D4C22×C311yescharacteristic
55C22×C4⋊C4C3×C411yescharacteristic
56#432—21no—
57C2×C22⋊C8—21no—
58#398—21no—
59#397—21no—
60#389—21no—
61#381C22×C312yes—
62#380—21no—
63#380C3×C411yescharacteristic
64#324—21no—
65#323—21no—
66#323C22×C312yes—
67#323C22×C312yes—
68C24×C3C22⋊C411yescharacteristic
69C24×C3C22×C411yescharacteristic
70C2×C3×D4—22no—
71C2×C3×D4—22no—
72C2×C3×D4—41no—
73C22×C3×C4—21no—
74C22×C3×C4—21no—
75C22×C3×C4—21no—
76C22×C3×C4—21no—
77C22×C3×C4—22no—
78C22×C3×C4—22no—
79C22×C3×C4—24no—
80C22×C3×C4—42no—
81C22×C3×C4—42no—
82C22×C3×C4—42no—
83C22×C3×C4—42no—
84C22×C3×C4C22⋊C411yescharacteristic
85C22×C3×C4C22⋊C411yescharacteristic
86C22×C3×C4C22⋊C411yescharacteristic
87C22×C3×C4C2×D411yescharacteristic
88C22×C3×C4C2×D411yescharacteristic
89C2×C3×C8—42no—
90C2×C3×C8—42no—
91C2×C3×C8—82no—
92C3×C4⋊C4—41no—
93C3×C4⋊C4—42no—
94C3×C4⋊C4—42no—
95C3×C4⋊C4—44no—
96C3×C4⋊C4—41no—
97C3×C4⋊C4—41no—
98C3×C22⋊C4—41no—
99C3×C22⋊C4—41no—
100C3×C22⋊C4—44no—
101C3×C22⋊C4—42no—
102C3×C42—21no—
103C3×C42—21no—
104C22×D4C2×C3×C411yescharacteristic
105C23×C4C2×C3×C412yes—
106C23×C4C2×C3×C411yescharacteristic
107C23×C4C3×D411yescharacteristic
108C22×C8—41no—
109#123—44no—
110C4×D4—41no—
111C4×D4—41no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4C3×D411yescharacteristic
119C2×C22⋊C4—22no—
120C2×C22⋊C4C2×C3×C412yes—
121C2×C22⋊C4C3×D411yescharacteristic
122C2×C22⋊C4C23×C311yescharacteristic
123C2×C42C3×D411yescharacteristic
124C4⋊C8—41no—
125C22⋊C8—22no—
126C22⋊C8—22no—
127C22⋊C8—41no—
128#97—41no—
129#95—21no—
130#95—22no—
131#95—22no—
132C23×C3—21no—
133C23×C3—21no—
134C23×C3—21no—
135C23×C3—21no—
136C23×C3—41no—
137C23×C3—41no—
138C23×C3—41no—
139C23×C3—41no—
140C23×C3#10012yes—
141C23×C3C2×C22⋊C411yescharacteristic
142C23×C3#13112yes—
143C3×D4—44no—
144C2×C3×C4—22no—
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4—21no—
148C2×C3×C4—21no—
149C2×C3×C4—21no—
150C2×C3×C4—21no—
151C2×C3×C4—42no—
152C2×C3×C4—42no—
153C2×C3×C4—42no—
154C2×C3×C4—41no—
155C2×C3×C4—41no—
156C2×C3×C4—41no—
157C2×C3×C4—42no—
158C2×C3×C4—42no—
159C2×C3×C4—42no—
160C2×C3×C4—42no—
161C2×C3×C4—42no—
162C2×C3×C4—42no—
163C2×C3×C4—81no—
164C2×C3×C4—84no—
165C2×C3×C4#9911yescharacteristic
166C2×C3×C4#9911yescharacteristic
167C3×C8—82no—
168C3×C8—82no—
169C24C3×C22⋊C411yescharacteristic
170C24C22×C3×C411yescharacteristic
171C2×D4—22no—
172C2×D4—22no—
173C2×D4—41no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—24no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4C3×C22⋊C411yescharacteristic
186C22×C4C3×C22⋊C411yescharacteristic
187C22×C4C3×C22⋊C411yescharacteristic
188C22×C4C2×C3×D411yescharacteristic
189C22×C4C2×C3×D411yescharacteristic
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—82no—
193C4⋊C4—41no—
194C4⋊C4—42no—
195C4⋊C4—44no—
196C4⋊C4—42no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C22⋊C4—41no—
200C22⋊C4—41no—
201C22⋊C4—44no—
202C22⋊C4—42no—
203C42—21no—
204C42—21no—
205C22×C3—21no—
206C22×C3—21no—
207C22×C3—22no—
208C22×C3—22no—
209C22×C3—42no—
210C22×C3—42no—
211C22×C3—42no—
212C22×C3—41no—
213C22×C3—41no—
214C22×C3—41no—
215C22×C3—41no—
216C22×C3—81no—
217C22×C3#40811yescharacteristic
218C22×C3C2×#9911yescharacteristic
219C22×C3C2×#10011yescharacteristic
220C3×C4—21no—
221C3×C4—41no—
222C3×C4—41no—
223C3×C4—41no—
224C3×C4—41no—
225C3×C4—82no—
226C3×C4—82no—
227C3×C4#35212yes—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23C3×#10012yes—
237C23C2×C3×C22⋊C411yescharacteristic
238C23C3×#13112yes—
239D4—44no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—81no—
260C2×C4—84no—
261C2×C4C3×#9911yescharacteristic
262C2×C4C3×#9911yescharacteristic
263C8—82no—
264C8—82no—
265C2×C3—21no—
266C2×C3—21no—
267C2×C3—42no—
268C2×C3—42no—
269C2×C3#147111yescharacteristic
270C2×C3C2×#35211yescharacteristic
271C2×C3#212311yescharacteristic
272C22—21no—
273C22—21no—
274C22—22no—
275C22—22no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—81no—
284C22C3×#40811yescharacteristic
285C22C2×C3×#9911yescharacteristic
286C22C2×C3×#10011yescharacteristic
287C4—21no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—82no—
293C4—82no—
294C4C3×#35212yes—
295C3#1155811yescharacteristic, direct product factor, Sylow
296C2—21no—
297C2—21no—
298C2—42no—
299C2—42no—
300C2C3×#147111yescharacteristic
301C2C2×C3×#35211yescharacteristic
302C2C3×#212311yescharacteristic
subgroupG/H
center, socleC22×C3C2×#99
commutatorC24C22×C3×C4
Frattini, cosocleC2×C22⋊C4C23×C3
prime radicalC22×C3×D4C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×C4×D4 ◁ G

Complements

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

NH ≅ G/Ntype
4. C3×#1317C2semidirect
5. #11558C3direct (N × H)
24. #1317C2×C3semidirect
295. C3#11558direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#24862
C3×#14662
C3×#14592
C3×#13172
#115583