Finite Group Databasebeta

← #10694780

Subgroups — #10694780

Order 768.

410 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 131 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#71790C211yesmaximal, characteristic
3#71790C211yesmaximal, characteristic
4#71775C211yesmaximal, characteristic
5#71774C211yesmaximal, characteristic
6#69731C211yesmaximal, characteristic
7C8×#208C211yesmaximal, characteristic
8#9795—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C3×#487—21no—
12C3×#487—21no—
13C3×#487—22no—
14C2×C3×#103C414yes—
15C2×C3×C4×C8C2211yescharacteristic
16#4925C2211yescharacteristic
17#4924C2211yescharacteristic
18#4924C2211yescharacteristic
19C4×#208C2211yescharacteristic
20#4547—22no—
21#4546—22no—
22#4530—22no—
23#4530—22no—
24#4458—22no—
25#4454—21no—
26C8×C3⋊C8—21no—
27C2×#487D311yescharacteristic
28#1997—31no—
29#1997—31no—
30#1567—31no—
31#1549—31no—
32#1548—31no—
33C8×M16—31no—
34C22×C3×Q8—21no—
35C22×C3×Q8C2×C412yes—
36C22×C3×C8C2×C412yes—
37C3×C4⋊Q8—21no—
38C3×C4⋊Q8—21no—
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4C2×C412yes—
49C2×C3×C42C2311yescharacteristic
50C2×#208—21no—
51C2×#208—21no—
52C2×#208D412yes—
53C3×#103—24no—
54C3×#103—24no—
55C3×C4×C8D411yescharacteristic
56C3×C4×C8D411yescharacteristic
57C3×C4×C8D412yes—
58#830—21no—
59C4⋊(C3⋊C8)—21no—
60C4⋊(C3⋊C8)—21no—
61C4⋊(C3⋊C8)—21no—
62C4⋊(C3⋊C8)—21no—
63C4⋊(C3⋊C8)D412yes—
64#803—21no—
65C4×C3⋊C8—21no—
66C4×C3⋊C8—21no—
67C2×C4⋊Q8C2×D311yescharacteristic
68C2×C4⋊Q8C2×D311yescharacteristic
69#487—21no—
70#487—21no—
71#487—22no—
72#431—31no—
73#423—31no—
74#423—31no—
75C2×#103Dic314yes—
76C4×M16—31no—
77C2×C4×C8C2×D311yescharacteristic
78#343—62no—
79#343—62no—
80#335—62no—
81#333—62no—
82#332—62no—
83#322—61no—
84C82—61no—
85C2×C3×Q8—22no—
86C2×C3×Q8—22no—
87C2×C3×Q8—44no—
88C2×C3×Q8—41no—
89C2×C3×Q8—41no—
90C2×C3×Q8—82no—
91C22×C3×C4—22no—
92C22×C3×C4—22no—
93C22×C3×C4C22×C411yescharacteristic
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4#4111yescharacteristic
96C2×C3×C8—42no—
97C2×C3×C8C22⋊C412yes—
98C2×C3×C8C22⋊C412yes—
99C3×C4⋊C4—22no—
100C3×C4⋊C4—22no—
101C3×C4⋊C4—41no—
102C3×C4⋊C4—41no—
103C3×C4⋊C4—41no—
104C3×C4⋊C4—41no—
105C3×C4⋊C4—41no—
106C3×C4⋊C4—41no—
107C3×C42C2×D411yescharacteristic
108C3×C42C2×D411yescharacteristic
109C3×C42C2×D411yescharacteristic
110C3×C42#4111yescharacteristic
111#208—22no—
112#208—42no—
113#208—42no—
114#208Q1614yes—
115C2×C3⋊C8—22no—
116C2×C3⋊C8—22no—
117C2×C3⋊C8—41no—
118C2×C3⋊C8—41no—
119C22×Q8—21no—
120C22×Q8C2×Dic312yes—
121C2×M16—32no—
122C2×M16—61no—
123C2×M16—61no—
124C22×C8C2×Dic312yes—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4C2×Dic312yes—
137C2×C42C22×D311yescharacteristic
138C4⋊C8—32no—
139C4⋊C8—61no—
140C4⋊C8—61no—
141C4⋊C8—61no—
142C4⋊C8—61no—
143#103—24no—
144#103—24no—
145C22⋊C8—61no—
146#97—61no—
147C4×C8—61no—
148C4×C8—61no—
149C4×C8#6711yescharacteristic
150C4×C8#6711yescharacteristic
151C4×C8#6712yes—
152C23×C3C4×D411yescharacteristic
153C3×Q8—44no—
154C3×Q8—82no—
155C3×Q8—82no—
156C3×Q8—82no—
157C2×C3×C4—22no—
158C2×C3×C4—22no—
159C2×C3×C4—42no—
160C2×C3×C4—42no—
161C2×C3×C4—82no—
162C2×C3×C4—82no—
163C2×C3×C4C2×C22⋊C411yescharacteristic
164C2×C3×C4C4×D411yescharacteristic
165C2×C3×C4#12111yescharacteristic
166C2×C3×C4#12111yescharacteristic
167C2×C3×C4#12411yescharacteristic
168C2×C3×C4#12711yescharacteristic
169C2×C3×C4C2×Q1612yes—
170C2×C3×C4#13512yes—
171C3×C8—22no—
172C3×C8—22no—
173C3⋊C8—24no—
174C3⋊C8—42no—
175C3⋊C8—81no—
176C3⋊C8—81no—
177C2×Q8—22no—
178C2×Q8—22no—
179C2×Q8—44no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8—82no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4D3×D411yescharacteristic
186C22×C4#23711yescharacteristic
187C22×C4C22×Dic311yescharacteristic
188M16—34no—
189M16—62no—
190M16—122no—
191M16—122no—
192C2×C8—42no—
193C2×C8—62no—
194C2×C8—62no—
195C2×C8—121no—
196C2×C8—121no—
197C2×C8#21712yes—
198C2×C8#21712yes—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C42D3×D411yescharacteristic
208C42#23711yescharacteristic
209C42C2×#6711yescharacteristic
210C42C2×#6711yescharacteristic
211C22×C3—21no—
212C22×C3—21no—
213C22×C3#39011yescharacteristic
214C22×C3C4×Q1612yes—
215C3×C4—21no—
216C3×C4—21no—
217C3×C4—82no—
218C3×C4—82no—
219C3×C4#41312yes—
220C3×C4#46712yes—
221C3×C4#47112yes—
222C3×C4#49411yescharacteristic
223C3×C4#49511yescharacteristic
224C23D4×Dic311yescharacteristic
225Q8—44no—
226Q8—82no—
227Q8—82no—
228Q8—82no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4D3×Q1612yes—
236C2×C4#91912yes—
237C2×C4D4×Dic311yescharacteristic
238C2×C4#93611yescharacteristic
239C2×C4#93811yescharacteristic
240C2×C4#93811yescharacteristic
241C2×C4#94011yescharacteristic
242C2×C4C2×#21711yescharacteristic
243C8—22no—
244C8—22no—
245C8—64no—
246C8—122no—
247C8—241no—
248C8—241no—
249C2×C3—22no—
250C2×C3#196711yescharacteristic
251C2×C3#196911yescharacteristic
252C2×C3#197711yescharacteristic
253C22—21no—
254C22—21no—
255C22Dic3×Q1612yes—
256C22#522111yescharacteristic
257C4—21no—
258C4—21no—
259C4—82no—
260C4—82no—
261C4#517712yes—
262C4#518311yescharacteristic
263C4#518812yes—
264C4#519111yescharacteristic
265C4#523712yes—
266C3#979511yescharacteristic, complemented, Sylow
267C2—22no—
268C2#7172711yescharacteristic
269C2#7205211yescharacteristic
270C2#7206611yescharacteristic
subgroupG/H
centerC22#5221
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×#487C2
socleC22×C3#390
prime radicalC23×C3C4×D4
cosocleC2×C3×C42C23
hypercenterC2×#487D3
hypocenterC3#9795
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
266. C3#9795semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4872
#717902
#717752
#717742
#697312
C8×#2082
#97953