Finite Group Databasebeta

← #10697793

Subgroups — #10697793

Order 768.

574 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 139 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×C3×#494C211yesmaximal, characteristic
2C2×#5094C212yesmaximal
3C2×#4547C212yesmaximal
4#69788C211yesmaximal, characteristic
5#67654C211yesmaximal, characteristic
6#10276—31nomaximal, Sylow
7C22×C3×Q16—22no—
8C2×C3×C4⋊Q8C2212yes—
9C2×C4×Dic6C2211yescharacteristic
10C3×#494—24no—
11C2×C3×C4×C8C2211yescharacteristic
12#5094—24no—
13#5094—24no—
14C2×#810—22no—
15C2×#808—22no—
16C2×C4⋊(C3⋊C8)C2211yescharacteristic
17C2×C4⋊(C3⋊C8)C2212yes—
18#4550—22no—
19#4547—24no—
20C2×#494D311yescharacteristic
21C2×#475—32no—
22#1984—31no—
23C2×#332—32no—
24#1277—31no—
25C22×C3×Q8—22no—
26C22×Dic6D411yescharacteristic
27C2×C3×Q16—28no—
28C2×C3×Q16—44no—
29C2×C3×Q16—44no—
30C22×C3×C8D412yes—
31C3×C4⋊Q8—22no—
32C3×C4⋊Q8—22no—
33C3×C4⋊Q8—22no—
34C3×C4⋊Q8D414yes—
35C2×C3×C4⋊C4—22no—
36C2×C3×C4⋊C4—22no—
37C2×C3×C42C2311yescharacteristic
38C2×C4⋊Dic3D411yescharacteristic
39C2×Dic3⋊C4—21no—
40C2×C4×Dic3—21no—
41C22×C3⋊C8—22no—
42C22×C3⋊C8—21no—
43C4×Dic6—22no—
44C4×Dic6—22no—
45C3×C4×C8—22no—
46#810—28no—
47#808—28no—
48C4⋊(C3⋊C8)—22no—
49C4⋊(C3⋊C8)—22no—
50C4⋊(C3⋊C8)D414yes—
51C22×Q16—22no—
52C2×C4⋊Q8C2×D312yes—
53C2×C4×Q8—31no—
54#494—24no—
55#475—64no—
56#475—64no—
57C2×#106—62no—
58C2×C4⋊C8—32no—
59C2×C4⋊C8—31no—
60C2×#103—62no—
61C2×C4×C8C2×D311yescharacteristic
62#336—62no—
63#332—64no—
64C2×C3×Q8—24no—
65C2×C3×Q8—42no—
66C2×C3×Q8—44no—
67C2×C3×Q8—44no—
68C22×C3×C4—42no—
69C22×C3×C4C2×D411yescharacteristic
70C22×C3×C4C2×D411yescharacteristic
71C22×C3×C4#4111yescharacteristic
72C22×Dic3—21no—
73C22×Dic3—21no—
74C2×Dic6—22no—
75C2×Dic6—22no—
76C2×Dic6—21no—
77C2×Dic6Q1612yescomplemented
78C3×Q16—416no—
79C2×C3×C8—22no—
80C2×C3×C8—22no—
81C2×C3×C8SD1614yes—
82C3×C4⋊C4—24no—
83C3×C4⋊C4—24no—
84C3×C4⋊C4—42no—
85C3×C4⋊C4—42no—
86C3×C42C2×D412yes—
87C3×C42C2×D412yes—
88C4⋊Dic3—22no—
89Dic3⋊C4—42no—
90C4×Dic3—41no—
91C4×Dic3—41no—
92C2×C3⋊C8—24no—
93C2×C3⋊C8—42no—
94C2×C3⋊C8—42no—
95C2×C3⋊C8—41no—
96C2×C3⋊C8—42no—
97C22×Q8—22no—
98C22×Q8—31no—
99C2×Q16—28no—
100C2×Q16—44no—
101C2×Q16—44no—
102C22×C8—62no—
103C22×C8—61no—
104C22×C8#6712yes—
105C4⋊Q8—22no—
106C4⋊Q8—22no—
107C4⋊Q8—22no—
108C4⋊Q8#6714yes—
109C4×Q8—62no—
110C4×Q8—62no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—31no—
114C2×C4⋊C4—61no—
115C2×C42—61no—
116C2×C42C22×D311yescharacteristic
117#106—68no—
118C4⋊C8—34no—
119C4⋊C8—62no—
120C4⋊C8—62no—
121#103—68no—
122C4×C8—22no—
123C23×C3#12111yescharacteristic
124C3×Q8—48no—
125C3×Q8—82no—
126C3×Q8—82no—
127C2×C3×C4—22no—
128C2×C3×C4—44no—
129C2×C3×C4—84no—
130C2×C3×C4#12012yes—
131C2×C3×C4#12011yescharacteristic
132C2×C3×C4#12112yes—
133C2×C3×C4#12711yescharacteristic
134C2×C3×C4C2×SD1612yes—
135C2×C3×C4C2×Q1611yescharacteristic
136C2×C3×C4#13511yescharacteristic
137C2×C3×C4AGL(1,ℤ/8)12yes—
138C2×C3×C4#13712yes—
139C2×Dic3—22no—
140C2×Dic3—41no—
141C2×Dic3—41no—
142C2×Dic3—41no—
143C2×Dic3—42no—
144Dic6—24no—
145Dic6—42no—
146C3×C8—24no—
147C3⋊C8—44no—
148C3⋊C8—81no—
149C2×Q8—24no—
150C2×Q8—32no—
151C2×Q8—42no—
152C2×Q8—44no—
153C2×Q8—44no—
154C2×Q8—61no—
155C2×Q8—62no—
156C2×Q8—62no—
157C22×C4—42no—
158C22×C4—61no—
159C22×C4—61no—
160C22×C4D3×D411yescharacteristic
161C22×C4#23711yescharacteristic
162C22×C4C2×#6711yescharacteristic
163Q16—416no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—64no—
167C2×C8—122no—
168C2×C8—122no—
169C2×C8—122no—
170C2×C8—121no—
171C2×C8#21414yes—
172C4⋊C4—24no—
173C4⋊C4—24no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—62no—
177C4⋊C4—122no—
178C42—121no—
179C42—121no—
180C42D3×D412yes—
181C42C2×#6712yes—
182C22×C3#39211yescharacteristic
183C22×C3#46012yes—
184C22×C3#46711yescharacteristic
185C22×C3#47011yescharacteristic
186C22×C3#47312yes—
187C3×C4—22no—
188C3×C4—84no—
189C3×C4#45012yes—
190C3×C4#45112yes—
191C3×C4#45212yes—
192C3×C4#49211yescharacteristic
193C3×C4#49611yescharacteristic
194Dic3—42no—
195Dic3—81no—
196C23#93811yescharacteristic
197Q8—48no—
198Q8—64no—
199Q8—82no—
200Q8—82no—
201Q8—122no—
202C2×C4—22no—
203C2×C4—44no—
204C2×C4—62no—
205C2×C4—84no—
206C2×C4—121no—
207C2×C4—121no—
208C2×C4—122no—
209C2×C4—121no—
210C2×C4D3×Q1611yescharacteristic
211C2×C4#91812yes—
212C2×C4#91911yescharacteristic
213C2×C4#93212yes—
214C2×C4C2×#21412yes—
215C2×C4#93712yes—
216C2×C4#93912yes—
217C2×C4#94011yescharacteristic
218C2×C4#95311yescharacteristic
219C8—24no—
220C8—124no—
221C8—241no—
222C2×C3#167012yes—
223C2×C3#169012yes—
224C2×C3#200111yescharacteristic
225C2×C3#201411yescharacteristic
226C2×C3#202011yescharacteristic
227C22#507512yes—
228C22#518811yescharacteristic
229C22#518911yescharacteristic
230C22#519212yes—
231C22#522311yescharacteristic
232C4—22no—
233C4—84no—
234C4—122no—
235C4—241no—
236C4#507411yescharacteristic
237C4#507611yescharacteristic
238C4#518612yes—
239C4#518712yes—
240C4#521812yes—
241C3#1027611yescharacteristic, complemented, Sylow
242C2#7130812yes—
243C2#7130912yes—
244C2#7176811yescharacteristic
245C2#7207111yescharacteristic
246C2#7207411yescharacteristic
subgroupG/H
centerC23#938
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#494C2
socle, prime radicalC23×C3#121
hypercenterC2×#494D3
hypocenterC3#10276
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#494

Complements

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

NH ≅ G/Ntype
77. C2×Dic6Q16semidirect
241. C3#10276semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4942
C2×#50942
C2×#45472
#697882
#676542
#102763