Finite Group Databasebeta

← #10697543

Subgroups — #10697543

Order 768.

560 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 133 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×#4850C212yesmaximal
2#74343C211yesmaximal, characteristic
3C2×#4484C212yesmaximal
4#69579C211yesmaximal, characteristic
5C3×#1296C211yesmaximal, characteristic
6#10237—31nomaximal, Sylow
7C2×#869C2211yescharacteristic
8C2×C4×Dic6C2212yes—
9C2×C3×C4⋊C8C2212yes—
10C3×#384C2211yescharacteristic
11C2×#818—22no—
12C2×#816—22no—
13#4984—22no—
14#4981—22no—
15#4973—22no—
16C2×#810—22no—
17C2×C4×C3⋊C8C2211yescharacteristic
18#4850—24no—
19#4850—24no—
20#4484—24no—
21C2×#477—32no—
22#2582—31no—
23#1956—31no—
24C2×Q8⋊C8—32no—
25#1296D311yescharacteristic
26C22×Dic6—21no—
27C22×Dic6D412yes—
28C22×C3×C8—22no—
29C2×C3×C4⋊C4—22no—
30C2×C3×C4⋊C4D412yes—
31C2×C3×C42C2311yescharacteristic
32C2×C4⋊Dic3—22no—
33C2×C4⋊Dic3—22no—
34C2×C4⋊Dic3D412yes—
35C2×Dic3⋊C4—22no—
36C2×C4×Dic3—22no—
37C22×C3⋊C8—21no—
38#869—22no—
39#869—21no—
40#869D412yes—
41C4×Dic6—22no—
42C4×Dic6—22no—
43C4×Dic6—22no—
44C4×Dic6—22no—
45C3×C4⋊C8—22no—
46C3×C4⋊C8D414yes—
47C3⋊#95—22no—
48C3⋊#95—24no—
49#818—28no—
50#816—28no—
51#810—44no—
52C4×C3⋊C8—22no—
53C2×C4⋊Q8—31no—
54C2×C4×Q8—32no—
55#477—64no—
56#477—64no—
57C2×#107—62no—
58C2×C4⋊C8C2×D312yes—
59C2×#103—62no—
60C2×#103—62no—
61C2×C4×C8—31no—
62#398—62no—
63#391—62no—
64#389—62no—
65#384C2×D311yescharacteristic
66Q8⋊C8—64no—
67C22×C3×C4—22no—
68C22×C3×C4C2×D411yescharacteristic
69C22×C3×C4C2×D412yes—
70C22×Dic3—22no—
71C22×Dic3—22no—
72C22×Dic3—41no—
73C2×Dic6—22no—
74C2×Dic6—22no—
75C2×Dic6—22no—
76C2×Dic6—22no—
77C2×Dic6—22no—
78C2×Dic6—22no—
79C2×Dic6—42no—
80C2×Dic6—42no—
81C2×Dic6—41no—
82C2×Dic6Q1614yes—
83C2×C3×C8—24no—
84C2×C3×C8—42no—
85C2×C3×C8—42no—
86C3×C4⋊C4—42no—
87C3×C4⋊C4—42no—
88C3×C4⋊C4—42no—
89C3×C42C2×D412yes—
90C3×C42C2×D411yescharacteristic
91C3×C42#4111yescharacteristic
92C4⋊Dic3—24no—
93C4⋊Dic3—22no—
94C4⋊Dic3—22no—
95C4⋊Dic3—42no—
96C4⋊Dic3—82no—
97Dic3⋊C4—42no—
98Dic3⋊C4—42no—
99C4×Dic3—42no—
100C4×Dic3—42no—
101C2×C3⋊C8—22no—
102C2×C3⋊C8—41no—
103C2×C3⋊C8—41no—
104C22×Q8—32no—
105C22×Q8—61no—
106C22×C8—22no—
107C22×C8—61no—
108C4⋊Q8—32no—
109C4⋊Q8—62no—
110C4⋊Q8—61no—
111C4×Q8—62no—
112C4×Q8—62no—
113C4×Q8—62no—
114C4×Q8—62no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—32no—
117C2×C4⋊C4—62no—
118C2×C4⋊C4—62no—
119C2×C4⋊C4—62no—
120C2×C4⋊C4D1212yes—
121C2×C42—62no—
122C2×C42C22×D311yescharacteristic
123#107—68no—
124C4⋊C8—22no—
125C4⋊C8D1214yes—
126#103—68no—
127#103—124no—
128C4×C8—62no—
129#95—64no—
130#95—62no—
131C23×C3#12011yescharacteristic
132C2×C3×C4—21no—
133C2×C3×C4—21no—
134C2×C3×C4—44no—
135C2×C3×C4—42no—
136C2×C3×C4#12012yes—
137C2×C3×C4#12111yescharacteristic
138C2×C3×C4#12112yes—
139C2×C3×C4#12711yescharacteristic
140C2×C3×C4C2×Q1612yes—
141C2×C3×C4#13512yes—
142C2×C3×C4#13712yes—
143C2×C3×C4#13712yes—
144C2×Dic3—24no—
145C2×Dic3—42no—
146C2×Dic3—42no—
147C2×Dic3—42no—
148C2×Dic3—42no—
149C2×Dic3—42no—
150C2×Dic3—42no—
151C2×Dic3—81no—
152C2×Dic3—81no—
153Dic6—24no—
154Dic6—24no—
155Dic6—42no—
156Dic6—42no—
157Dic6—44no—
158Dic6—82no—
159C3×C8—44no—
160C3⋊C8—42no—
161C2×Q8—34no—
162C2×Q8—62no—
163C2×Q8—62no—
164C2×Q8—62no—
165C2×Q8—62no—
166C2×Q8—62no—
167C2×Q8—62no—
168C2×Q8—122no—
169C2×Q8—122no—
170C2×Q8—121no—
171C22×C4—22no—
172C22×C4—62no—
173C22×C4—62no—
174C22×C4—121no—
175C22×C4C2×D1211yescharacteristic
176C22×C4D3×D412yes—
177C2×C8—24no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—62no—
181C2×C8—121no—
182C2×C8—121no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—64no—
187C4⋊C4—62no—
188C4⋊C4—62no—
189C4⋊C4—122no—
190C4⋊C4—122no—
191C4⋊C4—122no—
192C4⋊C4—242no—
193C42—122no—
194C42—122no—
195C42C2×D1212yes—
196C42D3×D411yescharacteristic
197C42#23911yescharacteristic
198C22×C3#39211yescharacteristic
199C22×C3#45112yes—
200C22×C3#45212yes—
201C22×C3#45311yescharacteristic
202C22×C3#45411yescharacteristic
203C3×C4—21no—
204C3×C4—21no—
205C3×C4—82no—
206C3×C4#45112yes—
207C3×C4#45212yes—
208C3×C4#46712yes—
209C3×C4#47012yes—
210Dic3—44no—
211Dic3—82no—
212Dic3—82no—
213C23#88211yescharacteristic
214Q8—64no—
215Q8—64no—
216Q8—124no—
217Q8—122no—
218Q8—122no—
219Q8—242no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—42no—
223C2×C4—44no—
224C2×C4—64no—
225C2×C4—122no—
226C2×C4—122no—
227C2×C4—122no—
228C2×C4—122no—
229C2×C4—122no—
230C2×C4—122no—
231C2×C4—241no—
232C2×C4—241no—
233C2×C4#87411yescharacteristic
234C2×C4#88212yes—
235C2×C4#89511yescharacteristic
236C2×C4#89512yes—
237C2×C4#91212yes—
238C2×C4#91512yes—
239C2×C4#91512yes—
240C2×C4D3×Q1612yes—
241C8—44no—
242C8—122no—
243C2×C3#163011yescharacteristic
244C2×C3#163311yescharacteristic
245C2×C3#163612yes—
246C2×C3#200112yes—
247C2×C3#200611yescharacteristic
248C22#467111yescharacteristic
249C22#474711yescharacteristic
250C22#475311yescharacteristic
251C22#478312yes—
252C22#480912yes—
253C4—21no—
254C4—21no—
255C4—82no—
256C4—124no—
257C4—242no—
258C4—242no—
259C4#478312yes—
260C4#480912yes—
261C4#486712yes—
262C4#488712yes—
263C3#1023711yescharacteristic, complemented, Sylow
264C2#6840411yescharacteristic
265C2#6845812yes—
266C2#6850511yescharacteristic
267C2#6888812yes—
268C2#6900811yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1296C2
socle, prime radicalC23×C3#120
hypercenter#1296D3
hypocenterC3#10237
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

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
C2×#48502
#743432
C2×#44842
#695792
C3×#12962
#102373