Finite Group Databasebeta

← #10697686

Subgroups — #10697686

Order 768.

474 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 103 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×#3249C211yesmaximal, characteristic
2#83367C212yesmaximal
3#71826C212yesmaximal
4#70472C211yesmaximal, characteristic
5#67705C211yesmaximal, characteristic
6#10259—31nomaximal, Sylow
7C3×#533C2212yes—
8C2×C4×Dic6C2211yescharacteristic
9C3×#477—22no—
10C3×#467—22no—
11C3×#462—22no—
12C2×C3×C4⋊C8C2211yescharacteristic
13#5094—22no—
14#5091—22no—
15#5063—24no—
16#5004—22no—
17#4995—22no—
18#4924C2212yes—
19C2×C4×C3⋊C8C2211yescharacteristic
20#4550—22no—
21#4535—24no—
22#3249D311yescharacteristic
23#3078—32no—
24#1984—31no—
25#1493—32no—
26#1292—31no—
27C22×Dic6D411yescharacteristic
28C2×C3×Q16—22no—
29C22×C3×C8D412yes—
30C3×C4⋊Q8D412yes—
31C3×C4⋊Q8D412yes—
32C3×#125—22no—
33C3×C22⋊Q8—22no—
34C3×C22⋊Q8—22no—
35C3×C4×Q8—22no—
36C3×C4×Q8—22no—
37C3×#117—22no—
38C2×C3×C42C2311yescharacteristic
39C2×C4⋊Dic3D411yescharacteristic
40C2×Dic3⋊C4—21no—
41C2×C4×Dic3—21no—
42C2×#208—22no—
43C22×C3⋊C8—21no—
44C4×Dic6—22no—
45C4×Dic6—22no—
46C3×#107—22no—
47C3×C4⋊C8—22no—
48C3×#103—24no—
49#810—24no—
50#810—24no—
51#808—24no—
52#807—24no—
53C4⋊(C3⋊C8)—22no—
54C4⋊(C3⋊C8)D414yes—
55C4×C3⋊C8—22no—
56#533C2×D312yes—
57C2×C4×Q8—31no—
58#479—64no—
59#477—22no—
60#477—62no—
61#475—62no—
62#467—22no—
63#462—22no—
64#428—62no—
65#423—32no—
66C2×C4⋊C8C2×D311yescharacteristic
67#419—62no—
68C2×C4×C8—31no—
69#336—62no—
70Q8⋊C8—64no—
71C2×C3×Q8—22no—
72C2×C3×Q8—22no—
73C22×C3×C4C2×D411yescharacteristic
74C22×C3×C4C2×D411yescharacteristic
75C22×C3×C4#4111yescharacteristic
76C22×Dic3—21no—
77C22×Dic3—21no—
78C2×Dic6—22no—
79C2×Dic6—22no—
80C2×Dic6—21no—
81C2×Dic6Q1612yescomplemented
82C3×Q16—82no—
83C2×C3×C8—22no—
84C2×C3×C8—42no—
85C3×C4⋊C4—22no—
86C3×C4⋊C4—22no—
87C3×C4⋊C4—22no—
88C3×C4⋊C4—22no—
89C3×C4⋊C4—42no—
90C3×C4⋊C4—42no—
91C3×C22⋊C4—42no—
92C3×C42—42no—
93C3×C42C2×D412yes—
94C3×C42C2×D412yes—
95C4⋊Dic3—22no—
96Dic3⋊C4—42no—
97C4×Dic3—41no—
98C4×Dic3—41no—
99#208—42no—
100#208—42no—
101C2×C3⋊C8—22no—
102C2×C3⋊C8—24no—
103C2×C3⋊C8—41no—
104C2×C3⋊C8—41no—
105C22×Q8—31no—
106C2×Q16—22no—
107C2×M16—62no—
108C22×C8—61no—
109C22×C8#6712yes—
110C4⋊Q8#6712yes—
111C4⋊Q8#6712yes—
112#125—22no—
113C22⋊Q8—22no—
114C22⋊Q8—22no—
115C4×Q8—22no—
116C4×Q8—22no—
117C4×Q8—62no—
118C4×Q8—62no—
119#117—22no—
120C2×C4⋊C4—31no—
121C2×C4⋊C4—61no—
122C2×C42—61no—
123C2×C42C22×D311yescharacteristic
124#107—22no—
125#107—64no—
126#106—64no—
127C4⋊C8—22no—
128C4⋊C8—34no—
129C4⋊C8—62no—
130#103—24no—
131#103—64no—
132#103—64no—
133C4×C8—62no—
134C23×C3#12111yescharacteristic
135C3×Q8—44no—
136C3×Q8—42no—
137C3×Q8—42no—
138C2×C3×C4—21no—
139C2×C3×C4—21no—
140C2×C3×C4—41no—
141C2×C3×C4—42no—
142C2×C3×C4—42no—
143C2×C3×C4#12012yes—
144C2×C3×C4#12011yescharacteristic
145C2×C3×C4#12112yes—
146C2×C3×C4#12711yescharacteristic
147C2×C3×C4C2×Q1611yescharacteristic
148C2×C3×C4#13511yescharacteristic
149C2×C3×C4#13712yes—
150C2×Dic3—22no—
151C2×Dic3—41no—
152C2×Dic3—41no—
153C2×Dic3—42no—
154C2×Dic3—41no—
155Dic6—24no—
156Dic6—42no—
157C3×C8—42no—
158C3⋊C8—42no—
159C3⋊C8—44no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—32no—
163C2×Q8—61no—
164C2×Q8—62no—
165C2×Q8—62no—
166C22×C4—61no—
167C22×C4—61no—
168C22×C4D3×D411yescharacteristic
169C22×C4#23711yescharacteristic
170C22×C4C2×#6711yescharacteristic
171Q16—82no—
172M16—122no—
173M16—122no—
174C2×C8—22no—
175C2×C8—42no—
176C2×C8—62no—
177C2×C8—64no—
178C2×C8—121no—
179C2×C8—121no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—62no—
187C4⋊C4—122no—
188C22⋊C4—42no—
189C42—42no—
190C42—121no—
191C42—121no—
192C42D3×D412yes—
193C42C2×#6712yes—
194C22×C3—21no—
195C22×C3—21no—
196C22×C3#39211yescharacteristic
197C22×C3#46711yescharacteristic
198C22×C3#47011yescharacteristic
199C3×C4—21no—
200C3×C4—22no—
201C3×C4—82no—
202C3×C4—82no—
203C3×C4#45112yes—
204C3×C4#45212yes—
205C3×C4#45511yescharacteristic
206C3×C4#45611yescharacteristic
207Dic3—42no—
208Dic3—81no—
209C23#93811yescharacteristic
210Q8—44no—
211Q8—42no—
212Q8—42no—
213Q8—64no—
214Q8—122no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—62no—
221C2×C4—121no—
222C2×C4—121no—
223C2×C4—122no—
224C2×C4—121no—
225C2×C4D3×Q1611yescharacteristic
226C2×C4#91812yes—
227C2×C4#91911yescharacteristic
228C2×C4#93712yes—
229C2×C4#93912yes—
230C2×C4#94011yescharacteristic
231C2×C4#95311yescharacteristic
232C8—42no—
233C8—122no—
234C8—124no—
235C2×C3—21no—
236C2×C3—21no—
237C2×C3#200111yescharacteristic
238C2×C3#200711yescharacteristic
239C2×C3#201711yescharacteristic
240C22—21no—
241C22—21no—
242C22#518811yescharacteristic
243C22#518911yescharacteristic
244C22#522311yescharacteristic
245C4—22no—
246C4—21no—
247C4—82no—
248C4—82no—
249C4—122no—
250C4—241no—
251C4#518612yes—
252C4#518712yes—
253C4#524211yescharacteristic
254C4#524611yescharacteristic
255C3#1025911yescharacteristic, complemented, Sylow
256C2—21no—
257C2—21no—
258C2#7188111yescharacteristic
259C2#7207111yescharacteristic
260C2#7212111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3249C2
socleC22×C3#392
prime radicalC23×C3#121
hypercenter#3249D3
hypocenterC3#10259
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3249

Complements

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

NH ≅ G/Ntype
81. C2×Dic6Q16semidirect
255. C3#10259semidirect

Maximal subgroups

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

subgroupindex
C3×#32492
#833672
#718262
#704722
#677052
#102593