Finite Group Databasebeta

← #10697541

Subgroups — #10697541

Order 768.

560 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 137 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×#2582C211yesmaximal, characteristic
2C2×#5021C212yesmaximal
3C2×#4535C212yesmaximal
4#69579C211yesmaximal, characteristic
5#67634C211yesmaximal, characteristic
6#10237—31nomaximal, Sylow
7C2×C3×C4×Q8C2212yes—
8C2×#869C2211yescharacteristic
9C3×#398—22no—
10C3×#391—22no—
11C3×#389—22no—
12C3×#384C2211yescharacteristic
13C2×#835—22no—
14#5021—24no—
15#5021—24no—
16C2×#810—22no—
17C2×#807—22no—
18C2×C4⋊(C3⋊C8)C2212yes—
19C2×C4×C3⋊C8C2211yescharacteristic
20#4535—24no—
21C2×#477—32no—
22#2582D311yescharacteristic
23#1956—31no—
24C2×Q8⋊C8—32no—
25#1296—31no—
26C22×C3×Q8D412yes—
27C22×Dic6—21no—
28C3×C4×Q8—22no—
29C3×C4×Q8—22no—
30C3×C4×Q8—22no—
31C3×C4×Q8—22no—
32C2×C3×C4⋊C4—22no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C4⋊C4—22no—
35C2×C3×C4⋊C4D412yes—
36C2×C3×C4⋊C4D412yes—
37C2×C3×C42—22no—
38C2×C3×C42C2311yescharacteristic
39C2×C4⋊Dic3—22no—
40C22×C3⋊C8—21no—
41C22×C3⋊C8—22no—
42#869—22no—
43#869—21no—
44#869D412yes—
45C3×#95—24no—
46C3×#95—22no—
47#835—28no—
48#810—44no—
49#807—28no—
50C4⋊(C3⋊C8)—22no—
51C4⋊(C3⋊C8)D414yes—
52C4×C3⋊C8—22no—
53C2×C4⋊Q8—31no—
54C2×C4×Q8C2×D312yes—
55#477—64no—
56#477—64no—
57C2×#107—62no—
58C2×C4⋊C8—32no—
59C2×#103—62no—
60C2×#103—62no—
61C2×C4×C8—31no—
62#398—22no—
63#391—22no—
64#389—22no—
65#384C2×D311yescharacteristic
66Q8⋊C8—64no—
67C2×C3×Q8—22no—
68C2×C3×Q8—22no—
69C2×C3×Q8—22no—
70C2×C3×Q8—22no—
71C2×C3×Q8—22no—
72C2×C3×Q8Q1614yes—
73C22×C3×C4—22no—
74C22×C3×C4—22no—
75C22×C3×C4—22no—
76C22×C3×C4C2×D411yescharacteristic
77C22×C3×C4C2×D412yes—
78C22×Dic3—41no—
79C2×Dic6—22no—
80C2×Dic6—42no—
81C2×Dic6—42no—
82C2×Dic6—41no—
83C3×C4⋊C4—22no—
84C3×C4⋊C4—22no—
85C3×C4⋊C4—42no—
86C3×C4⋊C4—42no—
87C3×C4⋊C4—42no—
88C3×C4⋊C4—42no—
89C3×C4⋊C4—42no—
90C3×C4⋊C4—82no—
91C3×C42—42no—
92C3×C42—42no—
93C3×C42C2×D411yescharacteristic
94C3×C42C2×D412yes—
95C3×C42#4111yescharacteristic
96C4⋊Dic3—24no—
97C4⋊Dic3—42no—
98C2×C3⋊C8—22no—
99C2×C3⋊C8—24no—
100C2×C3⋊C8—41no—
101C2×C3⋊C8—41no—
102C2×C3⋊C8—42no—
103C2×C3⋊C8—42no—
104C22×Q8—61no—
105C22×Q8#6712yes—
106C22×C8—61no—
107C22×C8—62no—
108C4⋊Q8—32no—
109C4⋊Q8—62no—
110C4⋊Q8—61no—
111C4×Q8—22no—
112C4×Q8—22no—
113C4×Q8—22no—
114C4×Q8—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—62no—
119C2×C4⋊C4#6712yes—
120C2×C4⋊C4#6712yes—
121C2×C42—22no—
122C2×C42C22×D311yescharacteristic
123#107—68no—
124C4⋊C8—34no—
125C4⋊C8—62no—
126#103—68no—
127#103—124no—
128C4×C8—62no—
129#95—24no—
130#95—22no—
131C23×C3#12011yescharacteristic
132C3×Q8—24no—
133C3×Q8—24no—
134C3×Q8—42no—
135C3×Q8—42no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—24no—
139C2×C3×C4—42no—
140C2×C3×C4—42no—
141C2×C3×C4—42no—
142C2×C3×C4—44no—
143C2×C3×C4—42no—
144C2×C3×C4—42no—
145C2×C3×C4—42no—
146C2×C3×C4#12012yes—
147C2×C3×C4#12111yescharacteristic
148C2×C3×C4#12112yes—
149C2×C3×C4#12711yescharacteristic
150C2×C3×C4C2×Q1612yes—
151C2×C3×C4#13512yes—
152C2×C3×C4#13712yes—
153C2×C3×C4#13712yes—
154C2×Dic3—42no—
155C2×Dic3—81no—
156C2×Dic3—81no—
157Dic6—44no—
158Dic6—82no—
159C3⋊C8—42no—
160C3⋊C8—44no—
161C2×Q8—22no—
162C2×Q8—22no—
163C2×Q8—22no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—62no—
167C2×Q8—122no—
168C2×Q8—122no—
169C2×Q8—121no—
170C2×Q8#21614yes—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—121no—
175C22×C4C2×#6711yescharacteristic
176C22×C4C2×#6712yes—
177C2×C8—62no—
178C2×C8—64no—
179C2×C8—121no—
180C2×C8—121no—
181C2×C8—122no—
182C2×C8—122no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—64no—
191C4⋊C4—82no—
192C4⋊C4—122no—
193C42—42no—
194C42—42no—
195C42D3×D411yescharacteristic
196C42D3×D412yes—
197C42#23711yescharacteristic
198C22×C3#39211yescharacteristic
199C22×C3#45112yes—
200C22×C3#45212yes—
201C22×C3#45311yescharacteristic
202C22×C3#45411yescharacteristic
203C3×C4—21no—
204C3×C4—21no—
205C3×C4—44no—
206C3×C4—82no—
207C3×C4—82no—
208C3×C4#45112yes—
209C3×C4#45212yes—
210C3×C4#46712yes—
211C3×C4#47012yes—
212Dic3—82no—
213C23#95311yescharacteristic
214Q8—24no—
215Q8—24no—
216Q8—42no—
217Q8—42no—
218Q8—124no—
219Q8—242no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—24no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—44no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—122no—
231C2×C4—241no—
232C2×C4—241no—
233C2×C4#93712yes—
234C2×C4#93811yescharacteristic
235C2×C4#93912yes—
236C2×C4#94011yescharacteristic
237C2×C4#94212yes—
238C2×C4C2×#21612yes—
239C2×C4#95012yes—
240C2×C4#95112yes—
241C8—122no—
242C8—124no—
243C2×C3#163011yescharacteristic
244C2×C3#163311yescharacteristic
245C2×C3#163612yes—
246C2×C3#200112yes—
247C2×C3#200611yescharacteristic
248C22#522311yescharacteristic
249C22#522812yes—
250C22#523912yes—
251C22#524111yescharacteristic
252C22#524511yescharacteristic
253C4—21no—
254C4—21no—
255C4—44no—
256C4—82no—
257C4—82no—
258C4—242no—
259C4#504712yes—
260C4#505012yes—
261C4#505112yes—
262C4#505212yes—
263C3#1023711yescharacteristic, complemented, Sylow
264C2#7170012yes—
265C2#7187911yescharacteristic
266C2#7193612yes—
267C2#7194011yescharacteristic
268C2#7194411yescharacteristic
subgroupG/H
centerC23#953
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2582C2
socle, prime radicalC23×C3#120
hypercenter#2582D3
hypocenterC3#10237
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2582

Complements

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

NH ≅ G/Ntype
263. C3#10237semidirect

Maximal subgroups

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

subgroupindex
C3×#25822
C2×#50212
C2×#45352
#695792
#676342
#102373