Finite Group Databasebeta

← #10708552

Subgroups — #10708552

Order 768.

262 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 53 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
1#80898C211yesmaximal, characteristic
2#71090C211yesmaximal, characteristic
3#69117C211yesmaximal, characteristic
4#68718C211yesmaximal, characteristic
5C3×#1342C211yesmaximal, characteristic
6#67533C211yesmaximal, characteristic
7#67484C211yesmaximal, characteristic
8#12305—31nomaximal, Sylow
9#5625—21no—
10C3×#470C2211yescharacteristic
11#5182—21no—
12#5137C2211yescharacteristic
13#5125C2211yescharacteristic
14#5119C2211yescharacteristic
15#5051—21no—
16#4857—21no—
17#4799—21no—
18#4795—21no—
19C3×#349C2211yescharacteristic
20C3×#338C2211yescharacteristic
21#4565—21no—
22#4559—21no—
23#4558—21no—
24#4553C2211yescharacteristic
25#4517—21no—
26#4513—21no—
27#4487—21no—
28#3261—31no—
29#2239—31no—
30#2239—31no—
31#2103—31no—
32#1342—31no—
33#1342D311yescharacteristic
34#1328—31no—
35C2×C3×Q16D411yescharacteristic
36C2×C3×M16—21no—
37C2×C3×M16C2311yescharacteristic
38C3×C22⋊Q8—21no—
39Q8×Dic3—21no—
40#944—21no—
41C2×#216—21no—
42#926D411yescharacteristic
43#924—21no—
44C2×#208—21no—
45C22×C3⋊C8D411yescharacteristic
46C2×Dic12D411yescharacteristic
47#890—21no—
48#890—41no—
49#889—41no—
50#887—21no—
51#879—41no—
52#879—41no—
53C3×M32—21no—
54C3×#106D411yescharacteristic
55C3×#103—21no—
56#835—21no—
57#819—21no—
58#819—21no—
59#817D411yescharacteristic
60#816—21no—
61Dic3⋊C8—21no—
62Dic3⋊C8—21no—
63#812—21no—
64#810—21no—
65#807—21no—
66#541—61no—
67#479—61no—
68#479—61no—
69#470—31no—
70#470C2×D311yescharacteristic
71#467—61no—
72#462—61no—
73#462—61no—
74#430—31no—
75#424—31no—
76#365—61no—
77#365—61no—
78#362—61no—
79#362—61no—
80#349—31no—
81#349C2×D311yescharacteristic
82#344—61no—
83#341—61no—
84#338C2×D311yescharacteristic
85C2×C3×Q8—21no—
86C22×C3×C4C2×D411yescharacteristic
87C2×Dic6—21no—
88C3×Q16—41no—
89C3×M16—41no—
90C3×M16—41no—
91C3×M16—41no—
92C2×C3×C8—41no—
93C2×C3×C8C2×D411yescharacteristic
94C2×C3×C8C2×D411yescharacteristic
95C3×C4⋊C4—21no—
96C3×C4⋊C4—41no—
97C3×C22⋊C4—41no—
98#217—41no—
99#217SD1611yescharacteristic
100#217SD1611yescharacteristic
101#216—81no—
102C4⋊Dic3—21no—
103C4⋊Dic3—21no—
104C4⋊Dic3—41no—
105C4⋊Dic3—41no—
106Dic3⋊C4—41no—
107Dic3⋊C4—41no—
108Dic3⋊C4—41no—
109Dic3⋊C4—41no—
110Dic3⋊C4—41no—
111C4×Dic3—21no—
112C4×Dic3—41no—
113C4×Dic3—41no—
114#208—41no—
115#208—41no—
116C2×C3⋊C8—21no—
117C2×C3⋊C8—41no—
118C2×C3⋊C8—41no—
119Dic12—41no—
120C3×C16—41no—
121C3⋊C16—41no—
122C2×Q16—31no—
123C2×Q16—61no—
124C2×Q16D1211yescharacteristic
125C2×M16—21no—
126C2×M16—61no—
127C2×M16C22×D311yescharacteristic
128C22×C8—31no—
129C4⋊Q8—61no—
130#126—121no—
131#126—121no—
132#125—61no—
133#125—121no—
134#125—121no—
135C22⋊Q8—21no—
136C22⋊Q8—61no—
137C4×Q8—61no—
138C4×Q8—61no—
139#117—31no—
140M32—21no—
141M32—61no—
142#108—61no—
143#108—61no—
144#107—61no—
145#106—31no—
146#106D1211yescharacteristic
147C4⋊C8—61no—
148C4⋊C8—61no—
149#103—21no—
150#103—61no—
151#103—61no—
152#103—61no—
153C23×C3C2×SD1611yescharacteristic
154C3×Q8—41no—
155C3×Q8—41no—
156C2×C3×C4—21no—
157C2×C3×C4—21no—
158C2×C3×C4—41no—
159C2×C3×C4—41no—
160C2×C3×C4#12011yescharacteristic
161C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
162C2×Dic3—21no—
163C2×Dic3—21no—
164C2×Dic3—41no—
165C2×Dic3—41no—
166Dic6—41no—
167Dic6—41no—
168C3×C8—21no—
169C3×C8—21no—
170C3×C8—81no—
171C3⋊C8—41no—
172C3⋊C8—81no—
173C2×Q8—21no—
174C2×Q8—61no—
175C22×C4D3×D411yescharacteristic
176Q16—41no—
177Q16—121no—
178Q16—241no—
179M16—41no—
180M16—41no—
181M16—41no—
182M16—121no—
183M16—121no—
184C2×C8—41no—
185C2×C8—61no—
186C2×C8—121no—
187C2×C8—121no—
188C2×C8C2×D1211yescharacteristic
189C2×C8D3×D411yescharacteristic
190C4⋊C4—21no—
191C4⋊C4—41no—
192C4⋊C4—61no—
193C4⋊C4—61no—
194C4⋊C4—121no—
195C4⋊C4—121no—
196C4⋊C4—121no—
197C4⋊C4—121no—
198C4⋊C4—121no—
199C4⋊C4—121no—
200C4⋊C4—121no—
201C22⋊C4—31no—
202C22⋊C4—31no—
203C22⋊C4—41no—
204C22⋊C4—121no—
205C42—61no—
206C42—121no—
207C42—121no—
208C16—41no—
209C16—121no—
210C22×C3—21no—
211C22×C3—41no—
212C22×C3#45011yescharacteristic
213C3×C4—21no—
214C3×C4—81no—
215C3×C4—81no—
216C3×C4#45411yescharacteristic
217C3×C4#45811yescharacteristic
218Dic3—41no—
219Dic3—41no—
220Dic3—81no—
221Dic3—81no—
222C23D3×SD1611yescharacteristic
223Q8—41no—
224Q8—41no—
225Q8—121no—
226Q8—121no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—61no—
232C2×C4—61no—
233C2×C4—121no—
234C2×C4—121no—
235C2×C4#88211yescharacteristic
236C2×C4#91111yescharacteristic
237C8—21no—
238C8—21no—
239C8—81no—
240C8—121no—
241C8—241no—
242C2×C3—21no—
243C2×C3—41no—
244C2×C3#161611yescharacteristic
245C22—21no—
246C22—41no—
247C22#477611yescharacteristic
248C4—21no—
249C4—81no—
250C4—81no—
251C4—121no—
252C4—121no—
253C4—241no—
254C4—241no—
255C4#474211yescharacteristic
256C4#482711yescharacteristic
257C3#1230511yescharacteristic, complemented, Sylow
258C2—21no—
259C2—41no—
260C2#6829911yescharacteristic
subgroupG/H
centerC2#68299
commutator, cosocleC2×C3×M16C23
FrattiniC2×M16C22×D3
FittingC3×#1342C2
socleC2×C3#1616
prime radicalC23×C3C2×SD16
hypercenter#1342D3
hypocenterC3#12305
LCSG ▷ C2×C3×M16 ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #1342

Complements

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

NH ≅ G/Ntype
257. C3#12305semidirect

Maximal subgroups

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

subgroupindex
#808982
#710902
#691172
#687182
C3×#13422
#675332
#674842
#123053