Finite Group Databasebeta

← #10693626

Subgroups — #10693626

Order 768.

338 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 107 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×C4×C4⋊C8C211yesmaximal, characteristic
2#70437C211yesmaximal, characteristic
3#70429C211yesmaximal, characteristic
4#70421C211yesmaximal, characteristic
5#69396C211yesmaximal, characteristic
6#67707C211yesmaximal, characteristic
7#67611C211yesmaximal, characteristic
8#9579—31nomaximal, Sylow
9#5470C2211yescharacteristic
10C2×C3×C4⋊C8C2211yescharacteristic
11C2×C3×C4⋊C8C2211yescharacteristic
12C2×C3×C4×C8—21no—
13C3×C43C2211yescharacteristic
14#4925C2211yescharacteristic
15#4925C2211yescharacteristic
16#4924C2211yescharacteristic
17C3×#336—21no—
18#4523—21no—
19#4523—21no—
20#4486C412yescomplemented
21#4484—22no—
22#4481C412yescomplemented
23#4480—22no—
24#1994—31no—
25C4×C4⋊C8D311yescharacteristic
26#1570—31no—
27#1542—31no—
28#1481—31no—
29#1301—31no—
30#1276—31no—
31C22×C3×C8—21no—
32C22×C3×C8—21no—
33C2×C3×C42—21no—
34C2×C3×C42—21no—
35C2×C3×C42D411yescharacteristic
36C2×C3×C42D411yescharacteristic
37C2×C3×C42C2311yescharacteristic
38#926—21no—
39#924—21no—
40#924—21no—
41C2×#208—21no—
42#870—21no—
43#869C2×C411yescharacteristic
44#869C2×C411yescharacteristic
45C4×Dic6—21no—
46C4×Dic6—21no—
47C3×C4⋊C8—21no—
48C3×C4⋊C8—21no—
49C3×C4⋊C8C2×C412yes—
50C3×C4⋊C8D412yes—
51C3×C4×C8—42no—
52#830—21no—
53#830—21no—
54C4⋊(C3⋊C8)—21no—
55C4⋊(C3⋊C8)—21no—
56C4⋊(C3⋊C8)—21no—
57C4⋊(C3⋊C8)C2×C412yes—
58C4×C3⋊C8—21no—
59C4×C3⋊C8—21no—
60#533—31no—
61#431—31no—
62#431—31no—
63#423—31no—
64C2×C4⋊C8C2×D311yescharacteristic
65C2×C4⋊C8C2×D311yescharacteristic
66C2×C4×C8—21no—
67C43C2×D311yescharacteristic
68#339—61no—
69#339—61no—
70#336—21no—
71#334—62no—
72#333—32no—
73#332—32no—
74Q8⋊C8—62no—
75C22×C3×C4—21no—
76C22×C3×C4—21no—
77C22×C3×C4C2×D411yescharacteristic
78C22×C3×C4#4111yescharacteristic
79C22×C3×C4#4111yescharacteristic
80C2×Dic6—21no—
81C2×Dic6—21no—
82C2×C3×C8—22no—
83C2×C3×C8—22no—
84C2×C3×C8—41no—
85C2×C3×C8—41no—
86C2×C3×C8—41no—
87C2×C3×C8—41no—
88C3×C42—24no—
89C3×C42—21no—
90C3×C42—21no—
91C3×C42—21no—
92C3×C42—42no—
93C3×C42SD1612yes—
94C3×C42C22×C411yescharacteristic
95C3×C42C2×D411yescharacteristic
96C3×C42#4111yescharacteristic
97C3×C42#4111yescharacteristic
98#217—41no—
99C4⋊Dic3—21no—
100C4⋊Dic3—21no—
101C4⋊Dic3—21no—
102C4⋊Dic3—21no—
103Dic3⋊C4—41no—
104Dic3⋊C4—41no—
105C4×Dic3—41no—
106#208—41no—
107#208—41no—
108C2×C3⋊C8—22no—
109C2×C3⋊C8—41no—
110C2×C3⋊C8—41no—
111C2×M16—61no—
112C22×C8—21no—
113C22×C8—21no—
114C4⋊Q8—31no—
115C4⋊Q8—31no—
116#125—61no—
117C22⋊Q8—61no—
118C22⋊Q8—61no—
119C4×Q8—61no—
120C4×Q8—61no—
121#117—61no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42#6711yescharacteristic
125C2×C42#6711yescharacteristic
126C2×C42C22×D311yescharacteristic
127C4⋊C8—21no—
128C4⋊C8—21no—
129C4⋊C8—32no—
130C4⋊C8—61no—
131C4⋊C8—61no—
132C4⋊C8—61no—
133C4⋊C8C4×D312yes—
134C4⋊C8D1212yes—
135C22⋊C8—61no—
136C22⋊C8—61no—
137C4×C8—42no—
138C4×C8—61no—
139C4×C8—61no—
140C23×C3#12311yescharacteristic
141C2×C3×C4—22no—
142C2×C3×C4—22no—
143C2×C3×C4—22no—
144C2×C3×C4—22no—
145C2×C3×C4—22no—
146C2×C3×C4—41no—
147C2×C3×C4—41no—
148C2×C3×C4—41no—
149C2×C3×C4#11711yescharacteristic
150C2×C3×C4C4×D411yescharacteristic
151C2×C3×C4C4×D411yescharacteristic
152C2×C3×C4#12111yescharacteristic
153C2×C3×C4#12411yescharacteristic
154C2×C3×C4#12611yescharacteristic
155C2×C3×C4C2×SD1611yescharacteristic
156C2×C3×C4#13511yescharacteristic
157C2×C3×C4#13711yescharacteristic
158C2×C3×C4#13711yescharacteristic
159C2×Dic3—41no—
160C2×Dic3—41no—
161Dic6—42no—
162Dic6—81no—
163C3×C8—42no—
164C3×C8—42no—
165C3⋊C8—42no—
166C3⋊C8—81no—
167C3⋊C8—81no—
168C2×Q8—61no—
169C2×Q8—61no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4#23511yescharacteristic
173C22×C4#23511yescharacteristic
174C22×C4C2×#6711yescharacteristic
175M16—121no—
176M16—121no—
177C2×C8—22no—
178C2×C8—22no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—62no—
184C2×C8—121no—
185C2×C8—121no—
186C4⋊C4—61no—
187C4⋊C4—61no—
188C4⋊C4—61no—
189C4⋊C4—61no—
190C4⋊C4—121no—
191C4⋊C4—121no—
192C22⋊C4—121no—
193C42—24no—
194C42—21no—
195C42—21no—
196C42—21no—
197C42—42no—
198C42—121no—
199C42#21412yes—
200C42C2×C4×D311yescharacteristic
201C42C2×D1211yescharacteristic
202C42#23511yescharacteristic
203C42#23511yescharacteristic
204C22×C3—21no—
205C22×C3—21no—
206C22×C3#38811yescharacteristic
207C22×C3#48311yescharacteristic
208C22×C3#48511yescharacteristic
209C3×C4—24no—
210C3×C4—21no—
211C3×C4—21no—
212C3×C4—21no—
213C3×C4—42no—
214C3×C4C4×SD1611yescharacteristic
215C3×C4#44111yescharacteristic
216C3×C4#44312yes—
217C3×C4#46111yescharacteristic
218C3×C4#46411yescharacteristic
219Dic3—81no—
220Dic3—81no—
221C23#92911yescharacteristic
222Q8—122no—
223Q8—241no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—121no—
233C2×C4—121no—
234C2×C4#87211yescharacteristic
235C2×C4C4×D1211yescharacteristic
236C2×C4#87511yescharacteristic
237C2×C4#87611yescharacteristic
238C2×C4C4×#6711yescharacteristic
239C2×C4#93011yescharacteristic
240C2×C4C2×#21411yescharacteristic
241C2×C4#94211yescharacteristic
242C2×C4#95011yescharacteristic
243C2×C4#95111yescharacteristic
244C8—42no—
245C8—42no—
246C8—122no—
247C8—241no—
248C8—241no—
249C2×C3—21no—
250C2×C3—21no—
251C2×C3#192711yescharacteristic
252C2×C3#193211yescharacteristic
253C2×C3#193711yescharacteristic
254C22—21no—
255C22—21no—
256C22#494011yescharacteristic
257C22#497111yescharacteristic
258C22#499611yescharacteristic
259C4—24no—
260C4—21no—
261C4—21no—
262C4—21no—
263C4—42no—
264C4—241no—
265C4—241no—
266C4#470012yes—
267C4C4×#21411yescharacteristic
268C4#501911yescharacteristic
269C4#502811yescharacteristic
270C4#503011yescharacteristic
271C3#957911yescharacteristic, complemented, Sylow
272C2—21no—
273C2—21no—
274C2#6946411yescharacteristic
275C2#6979311yescharacteristic
276C2#6987411yescharacteristic
subgroupG/H
centerC2×C4#930
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×C4×C4⋊C8C2
socleC22×C3#388
prime radicalC23×C3#123
cosocleC2×C3×C42C23
hypercenterC4×C4⋊C8D3
hypocenterC3#9579
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×C4⋊C8

Complements

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

NH ≅ G/Ntype
20. #4486C4semidirect
22. #4481C4semidirect
271. C3#9579semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C4⋊C82
#704372
#704292
#704212
#693962
#677072
#676112
#95793