Finite Group Databasebeta

← #10694668

Subgroups — #10694668

Order 768.

406 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 127 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×#487C211yesmaximal, characteristic
2#71775C211yesmaximal, characteristic
3#71774C211yesmaximal, characteristic
4#71773C211yesmaximal, characteristic
5#71773C211yesmaximal, characteristic
6#69792C211yesmaximal, characteristic
7#69726C211yesmaximal, characteristic
8#9775—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C3×#487—22no—
12C3×#487C414yes—
13C2×C3×#103—22no—
14C2×C3×C4×C8C2211yescharacteristic
15C2×#819—22no—
16#4925C2211yescharacteristic
17#4925C2211yescharacteristic
18#4924C2211yescharacteristic
19#4924C2211yescharacteristic
20#4547—22no—
21#4547—22no—
22#4547—22no—
23#4546—22no—
24#4454—22no—
25C2×#487D311yescharacteristic
26#1848—31no—
27#1549—31no—
28#1548—31no—
29#1546—31no—
30#1546—31no—
31#1452—31no—
32C22×C3×Q8—21no—
33C22×C3×Q8—21no—
34C22×C3×C8D412yes—
35C3×C4⋊Q8—21no—
36C3×C4⋊Q8—21no—
37C3×C4⋊Q8—21no—
38C3×C4⋊Q8—21no—
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8C2×C412yes—
42C3×C4⋊Q8C2×C412yes—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C42C2311yescharacteristic
48C2×#208—21no—
49C2×#208—21no—
50C3×#103—28no—
51C3×C4×C8C2×C412yes—
52C3×C4×C8D412yes—
53#830—21no—
54#830—21no—
55#819—44no—
56C4⋊(C3⋊C8)—21no—
57C4⋊(C3⋊C8)—21no—
58C4⋊(C3⋊C8)—21no—
59C4⋊(C3⋊C8)—21no—
60C4⋊(C3⋊C8)D412yes—
61C4⋊(C3⋊C8)D412yes—
62C4×C3⋊C8—21no—
63C4×C3⋊C8—21no—
64C2×C4⋊Q8C2×D311yescharacteristic
65C2×C4⋊Q8C2×D311yescharacteristic
66#487—22no—
67#487Dic314yes—
68#431—31no—
69#431—31no—
70C2×#108—62no—
71#423—31no—
72#423—31no—
73C2×#103—22no—
74C2×C4×C8C2×D311yescharacteristic
75#333—62no—
76#332—62no—
77#332—62no—
78#332—62no—
79#322—62no—
80C2×C3×Q8—22no—
81C2×C3×Q8—22no—
82C2×C3×Q8—41no—
83C2×C3×Q8—41no—
84C2×C3×Q8—42no—
85C2×C3×Q8—42no—
86C2×C3×Q8—82no—
87C22×C3×C4—41no—
88C22×C3×C4—41no—
89C22×C3×C4C2×D411yescharacteristic
90C22×C3×C4#4111yescharacteristic
91C22×C3×C4#4111yescharacteristic
92C2×C3×C8—22no—
93C2×C3×C8—22no—
94C2×C3×C8C22⋊C414yes—
95C3×C4⋊C4—22no—
96C3×C4⋊C4—22no—
97C3×C4⋊C4—22no—
98C3×C4⋊C4—22no—
99C3×C4⋊C4—41no—
100C3×C4⋊C4—41no—
101C3×C4⋊C4—41no—
102C3×C4⋊C4—41no—
103C3×C42C22×C411yescharacteristic
104C3×C42C2×D411yescharacteristic
105C3×C42C2×D411yescharacteristic
106C3×C42C2×D411yescharacteristic
107#208—42no—
108#208—42no—
109C2×C3⋊C8—22no—
110C2×C3⋊C8—22no—
111C2×C3⋊C8—41no—
112C2×C3⋊C8—41no—
113C22×Q8—21no—
114C22×Q8—21no—
115C2×M16—61no—
116C2×M16—61no—
117C22×C8#6712yes—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8C2×Dic312yes—
125C4⋊Q8C2×Dic312yes—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C42C22×D311yescharacteristic
131#108—124no—
132C4⋊C8—32no—
133C4⋊C8—32no—
134C4⋊C8—61no—
135C4⋊C8—61no—
136C4⋊C8—61no—
137C4⋊C8—61no—
138#103—28no—
139C22⋊C8—61no—
140C22⋊C8—61no—
141C4×C8—61no—
142C4×C8—61no—
143C4×C8C2×Dic312yes—
144C4×C8#6712yes—
145C23×C3#12411yescharacteristic
146C3×Q8—44no—
147C3×Q8—82no—
148C3×Q8—82no—
149C3×Q8—81no—
150C3×Q8—81no—
151C2×C3×C4—22no—
152C2×C3×C4—22no—
153C2×C3×C4—42no—
154C2×C3×C4—42no—
155C2×C3×C4—82no—
156C2×C3×C4—82no—
157C2×C3×C4C2×C22⋊C411yescharacteristic
158C2×C3×C4C4×D411yescharacteristic
159C2×C3×C4C4×D411yescharacteristic
160C2×C3×C4#12111yescharacteristic
161C2×C3×C4#12111yescharacteristic
162C2×C3×C4#12711yescharacteristic
163C2×C3×C4AGL(1,ℤ/8)12yes—
164C2×C3×C4#13712yes—
165C3×C8—24no—
166C3⋊C8—42no—
167C3⋊C8—42no—
168C3⋊C8—81no—
169C3⋊C8—81no—
170C2×Q8—22no—
171C2×Q8—22no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—82no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4#23711yescharacteristic
180C22×C4#23711yescharacteristic
181C22×C4C2×#6711yescharacteristic
182M16—122no—
183M16—122no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—62no—
187C2×C8—62no—
188C2×C8—121no—
189C2×C8—121no—
190C2×C8#21714yes—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C42D3×D411yescharacteristic
200C42D3×D411yescharacteristic
201C42C22×Dic311yescharacteristic
202C42C2×#6711yescharacteristic
203C22×C3—21no—
204C22×C3—21no—
205C22×C3#39011yescharacteristic
206C22×C3#48912yes—
207C3×C4—21no—
208C3×C4—21no—
209C3×C4—82no—
210C3×C4—82no—
211C3×C4#41712yes—
212C3×C4#47112yes—
213C3×C4#47312yes—
214C3×C4#49611yescharacteristic
215C3×C4#49711yescharacteristic
216C23#93611yescharacteristic
217Q8—44no—
218Q8—82no—
219Q8—82no—
220Q8—81no—
221Q8—81no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4D4×Dic311yescharacteristic
229C2×C4D4×Dic311yescharacteristic
230C2×C4#93811yescharacteristic
231C2×C4#93811yescharacteristic
232C2×C4#94011yescharacteristic
233C2×C4#94912yes—
234C2×C4#95112yes—
235C2×C4C2×#21711yescharacteristic
236C8—24no—
237C8—122no—
238C8—122no—
239C8—241no—
240C8—241no—
241C2×C3—22no—
242C2×C3#195911yescharacteristic
243C2×C3#197711yescharacteristic
244C2×C3#197911yescharacteristic
245C22—21no—
246C22—21no—
247C22#505512yes—
248C22#522111yescharacteristic
249C4—21no—
250C4—21no—
251C4—82no—
252C4—82no—
253C4#505811yescharacteristic
254C4#506011yescharacteristic
255C4#517712yes—
256C4#519212yes—
257C4#523212yes—
258C3#977511yescharacteristic, complemented, Sylow
259C2—22no—
260C2#7172611yescharacteristic
261C2#7205211yescharacteristic
262C2#7206911yescharacteristic
subgroupG/H
centerC22#5221
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×#487C2
socleC22×C3#390
prime radicalC23×C3#124
cosocleC2×C3×C42C23
hypercenterC2×#487D3
hypocenterC3#9775
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
258. C3#9775semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4872
#717752
#717742
#717732
#697922
#697262
#97753