Finite Group Databasebeta

← C3×#13327 #10631934

Subgroups — C3×#13327 #10631934

Order 768.

594 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 146 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×SD32C212yesmaximal
2C2×C3×#494C211yesmaximal, characteristic, complemented
3C2×C3×#358C212yesmaximal, complemented
4C3×#1970C211yesmaximal, characteristic
5C3×#1374C211yesmaximal, characteristic, complemented
6#13327C311yesmaximal, characteristic, direct product factor, Sylow
7C22×C3×Q16—21no—
8C22×C3×Q16C2212yes—
9C22×C3×D8C2211yescharacteristic
10C2×C3×C4⋊Q8—21no—
11C2×C3×SD32—24no—
12C2×C3×SD32—24no—
13C2×C3×SD32—24no—
14C22×C3×C16C2212yes—
15C3×#494—24no—
16C2×C3×#107C2211yescharacteristic
17C2×C3×#102—21no—
18C2×C3×C4×C8C2211yescharacteristic
19C3×#390—21no—
20C3×#358—24no—
21C22×SD32C2×C312yes—
22C2×#494C2×C311yescharacteristic, complemented
23C2×#358C2×C312yescomplemented
24#1970C2×C311yescharacteristic
25#1374C2×C311yescharacteristic, complemented
26C22×C3×Q8—22no—
27C22×C3×D4—21no—
28C2×C3×Q16—24no—
29C2×C3×Q16—22no—
30C2×C3×Q16—22no—
31C2×C3×Q16—22no—
32C2×C3×Q16—22no—
33C2×C3×Q16—22no—
34C2×C3×Q16—42no—
35C2×C3×Q16—42no—
36C2×C3×Q16D414yes—
37C2×C3×D8—22no—
38C2×C3×D8—22no—
39C2×C3×D8—21no—
40C2×C3×D8D412yes—
41C22×C3×C8D411yescharacteristic
42C22×C3×C8C2311yescharacteristic
43C3×C4⋊Q8—24no—
44C3×C4⋊Q8—42no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—42no—
47C2×C3×C22⋊C4—42no—
48C2×C3×C42D411yescharacteristic
49C3×SD32—216no—
50C2×C3×C16—22no—
51C2×C3×C16—22no—
52C2×C3×C16D414yes—
53C3×#107—22no—
54C3×#102—42no—
55C3×C4×C8—22no—
56C22×Q16—21no—
57C22×Q16C22×C312yes—
58C22×D8C22×C311yescharacteristic
59C2×C4⋊Q8—21no—
60C2×SD32—24no—
61C2×SD32—24no—
62C2×SD32—24no—
63C22×C16C22×C312yes—
64#494—24no—
65C2×#107C22×C311yescharacteristic
66C2×#102—21no—
67C2×C4×C8C22×C311yescharacteristic
68#390—21no—
69#358—24no—
70C24×C3—41no—
71C2×C3×Q8—24no—
72C2×C3×Q8—42no—
73C2×C3×Q8—42no—
74C2×C3×Q8—42no—
75C2×C3×Q8—42no—
76C2×C3×Q8—42no—
77C2×C3×D4—22no—
78C2×C3×D4—41no—
79C2×C3×D4—41no—
80C2×C3×D4—41no—
81C2×C3×D4—41no—
82C2×C3×D4—41no—
83C22×C3×C4—42no—
84C22×C3×C4—41no—
85C22×C3×C4D812yes—
86C22×C3×C4C2×D411yescharacteristic
87C3×Q16—24no—
88C3×Q16—24no—
89C3×Q16—48no—
90C3×Q16—42no—
91C3×Q16—42no—
92C3×D8—24no—
93C3×D8—42no—
94C2×C3×C8—21no—
95C2×C3×C8—21no—
96C2×C3×C8D812yes—
97C2×C3×C8C2×D411yescharacteristic
98C2×C3×C8C2×D412yes—
99C2×C3×C8C2×D412yes—
100C2×C3×C8#4111yescharacteristic
101C3×C4⋊C4—48no—
102C3×C4⋊C4—41no—
103C3×C4⋊C4—41no—
104C3×C22⋊C4—84no—
105C3×C42—21no—
106C3×C42—21no—
107C3×C16—24no—
108C22×Q8—22no—
109C22×D4—21no—
110C2×Q16—24no—
111C2×Q16—22no—
112C2×Q16—22no—
113C2×Q16—22no—
114C2×Q16—22no—
115C2×Q16—22no—
116C2×Q16—42no—
117C2×Q16—42no—
118C2×Q16C3×D414yes—
119C2×D8—22no—
120C2×D8—22no—
121C2×D8—21no—
122C2×D8C3×D412yes—
123C22×C8C3×D411yescharacteristic
124C22×C8C23×C311yescharacteristic
125C4⋊Q8—24no—
126C4⋊Q8—42no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—42no—
129C2×C22⋊C4—42no—
130C2×C42C3×D411yescharacteristic
131SD32—216no—
132C2×C16—22no—
133C2×C16—22no—
134C2×C16C3×D414yes—
135#107—22no—
136#102—42no—
137C4×C8—22no—
138C23×C3—42no—
139C23×C3—81no—
140C23×C3—81no—
141C23×C3—82no—
142C23×C3—82no—
143C23×C3C2×D811yescharacteristic
144C3×Q8—44no—
145C3×Q8—44no—
146C3×Q8—82no—
147C3×Q8—82no—
148C3×D4—42no—
149C3×D4—42no—
150C3×D4—81no—
151C3×D4—81no—
152C2×C3×C4—22no—
153C2×C3×C4—22no—
154C2×C3×C4—44no—
155C2×C3×C4—82no—
156C2×C3×C4—82no—
157C2×C3×C4—81no—
158C2×C3×C4—82no—
159C2×C3×C4SD3214yescomplemented
160C2×C3×C4#12011yescharacteristic
161C2×C3×C4#12111yescharacteristic
162C2×C3×C4#12711yescharacteristic
163C2×C3×C4C2×D811yescharacteristic
164C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
165C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
166C3×C8—22no—
167C3×C8#12012yes—
168C3×C8#12112yes—
169C24—41no—
170C2×Q8—24no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×D4—22no—
177C2×D4—41no—
178C2×D4—41no—
179C2×D4—41no—
180C2×D4—41no—
181C2×D4—41no—
182C22×C4—42no—
183C22×C4—41no—
184C22×C4C3×D812yes—
185C22×C4C2×C3×D411yescharacteristic
186Q16—24no—
187Q16—24no—
188Q16—48no—
189Q16—42no—
190Q16—42no—
191D8—24no—
192D8—42no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8C3×D812yes—
196C2×C8C2×C3×D411yescharacteristic
197C2×C8C2×C3×D412yes—
198C2×C8C2×C3×D412yes—
199C2×C8C3×#4111yescharacteristic
200C4⋊C4—48no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C22⋊C4—84no—
204C42—21no—
205C42—21no—
206C16—24no—
207C22×C3—84no—
208C22×C3—82no—
209C22×C3—162no—
210C22×C3—161no—
211C22×C3—161no—
212C22×C3#44711yescharacteristic
213C22×C3#45911yescharacteristic
214C22×C3#49311yescharacteristic
215C22×C3C2×SD3212yes—
216C22×C3#51012yes—
217C3×C4—24no—
218C3×C4—84no—
219C3×C4—161no—
220C3×C4#39211yescharacteristic
221C3×C4#44711yescharacteristic
222C3×C4#45911yescharacteristic
223C3×C4#49611yescharacteristic
224C23—42no—
225C23—82no—
226C23—82no—
227C23—81no—
228C23—81no—
229C23C2×C3×D811yescharacteristic
230Q8—44no—
231Q8—44no—
232Q8—82no—
233Q8—82no—
234D4—42no—
235D4—42no—
236D4—81no—
237D4—81no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—44no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—81no—
245C2×C4C3×SD3214yescomplemented
246C2×C4C3×#12011yescharacteristic
247C2×C4C3×#12111yescharacteristic
248C2×C4C3×#12711yescharacteristic
249C2×C4C2×C3×D811yescharacteristic
250C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
251C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
252C8—22no—
253C8C3×#12012yes—
254C8C3×#12112yes—
255C2×C3—162no—
256C2×C3#201111yescharacteristic
257C2×C3#218512yes—
258C2×C3#220812yes—
259C2×C3#224811yescharacteristic
260C2×C3#225111yescharacteristic
261C22—82no—
262C22—84no—
263C22—161no—
264C22—161no—
265C22—162no—
266C22C3×#44711yescharacteristic
267C22C3×#45911yescharacteristic
268C22C3×#49311yescharacteristic
269C22C2×C3×SD3212yes—
270C22C3×#51012yes—
271C4—24no—
272C4—84no—
273C4—161no—
274C4C3×#39211yescharacteristic
275C4C3×#44711yescharacteristic
276C4C3×#45911yescharacteristic
277C4C3×#49611yescharacteristic
278C3#1332711yescharacteristic, direct product factor, Sylow
279C2—162no—
280C2C3×#201111yescharacteristic
281C2C3×#218512yes—
282C2C3×#220812yes—
283C2C3×#224811yescharacteristic
284C2C3×#225111yescharacteristic
subgroupG/H
center, socleC23×C3C2×D8
commutator, Frattini, cosocleC22×C8C23×C3
prime radicalC22×C3×D8C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#494C2semidirect
3. C2×C3×#358C2semidirect
5. C3×#1374C2semidirect
6. #13327C3direct (N × H)
22. C2×#494C2×C3semidirect
23. C2×#358C2×C3semidirect
25. #1374C2×C3semidirect
159. C2×C3×C4SD32semidirect
245. C2×C4C3×SD32semidirect
278. C3#13327direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×SD322
C2×C3×#4942
C2×C3×#3582
C3×#19702
C3×#13742
#133273