Finite Group Databasebeta

← #10707448

Subgroups — #10707448

Order 768.

398 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 59 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#76602C211yesmaximal, characteristic, complemented
2C3×#2007C211yesmaximal, characteristic
3#71875C212yesmaximal
4#67643C211yesmaximal, characteristic, complemented
5#67496C212yesmaximal, complemented
6#12131—31nomaximal, Sylow
7C2×C3×#137—21no—
8C3×#537C2211yescharacteristic
9C2×C3×#104—21no—
10C3×#391C2211yescharacteristic
11#5229—21no—
12#5034—22no—
13#4986C2212yes—
14#4928—21no—
15#4661—22no—
16#4655—22no—
17C3×#343C2211yescharacteristic
18#4537—22no—
19#4528C2212yes—
20#2809—31no—
21#2007D311yescharacteristic
22#1616—32no—
23#1353—32no—
24#1319—31no—
25C2×C3×#41—21no—
26C22×C3×Q8D411yescharacteristic
27C3×#137—42no—
28C3×#137—42no—
29C2×C3×Q16—41no—
30C2×C3×SD16—41no—
31C2×C3×M16—21no—
32C3×C22⋊Q8—41no—
33C3×#121—22no—
34C3×#121—41no—
35C2×C3×C4⋊C4C2311yescharacteristic
36C2×C3×C42D411yescharacteristic
37C2×C4⋊Dic3—21no—
38C2×C4⋊Dic3—21no—
39C2×C4⋊Dic3—21no—
40C2×C4⋊Dic3D412yes—
41C2×Dic3⋊C4—22no—
42C2×C4×Dic3—21no—
43C3×#104—42no—
44C3×#95—22no—
45#832—42no—
46C3⋊#95—22no—
47C3⋊#95—22no—
48C3⋊#95—22no—
49C3⋊#95D412yes—
50#830—22no—
51#808—42no—
52C2×#137—21no—
53#537C2×D311yescharacteristic
54#481—62no—
55C2×#104—21no—
56#400—32no—
57#398—62no—
58#391—61no—
59#391C2×D311yescharacteristic
60#389—61no—
61#387—62no—
62#343C2×D311yescharacteristic
63#327—62no—
64#324—32no—
65C3×#41—41no—
66C3×#41—41no—
67C3×#41—41no—
68C3×#41—41no—
69C2×C3×Q8—21no—
70C2×C3×Q8—21no—
71C2×C3×Q8—41no—
72C2×C3×Q8—41no—
73C2×C3×Q8SD1612yes—
74C2×C3×D4—21no—
75C2×C3×D4—22no—
76C2×C3×D4—41no—
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C22×C3×C4—41no—
80C22×C3×C4C2×D412yes—
81C22×C3×C4C2×D411yescharacteristic
82C22×Dic3—22no—
83C22×Dic3—22no—
84C3×Q16—44no—
85C3×SD16—44no—
86C3×M16—22no—
87C3×M16—41no—
88C2×C3×C8—41no—
89C3×C4⋊C4—22no—
90C3×C4⋊C4—41no—
91C3×C22⋊C4—42no—
92C3×C42—41no—
93C4⋊Dic3—42no—
94C4⋊Dic3—42no—
95C4⋊Dic3—81no—
96C4⋊Dic3—81no—
97Dic3⋊C4—44no—
98C4×Dic3—81no—
99C2×C3⋊C8—42no—
100C2×#41—21no—
101C22×Q8#6711yescharacteristic
102#137—42no—
103#137—42no—
104C2×Q16—41no—
105C2×SD16—41no—
106C2×M16—21no—
107C22⋊Q8—41no—
108#121—22no—
109#121—41no—
110C2×C4⋊C4—32no—
111C2×C4⋊C4—62no—
112C2×C4⋊C4—61no—
113C2×C4⋊C4—61no—
114C2×C4⋊C4—61no—
115C2×C4⋊C4C22×D311yescharacteristic
116C2×C42—61no—
117C2×C42#6711yescharacteristic
118#106—122no—
119#104—42no—
120#102—122no—
121C22⋊C8—62no—
122#95—22no—
123#95—32no—
124#95—62no—
125#95—62no—
126#95—62no—
127C23×C3—41no—
128C23×C3#12011yescharacteristic
129C3×Q8—42no—
130C3×Q8—41no—
131C3×Q8—41no—
132C3×Q8—81no—
133C3×D4—41no—
134C3×D4—41no—
135C3×D4—82no—
136C3×D4—81no—
137C3×D4—81no—
138C2×C3×C4—21no—
139C2×C3×C4—21no—
140C2×C3×C4—22no—
141C2×C3×C4—22no—
142C2×C3×C4—22no—
143C2×C3×C4—41no—
144C2×C3×C4—41no—
145C2×C3×C4—41no—
146C2×C3×C4—41no—
147C2×C3×C4—42no—
148C2×C3×C4—41no—
149C2×C3×C4—81no—
150C2×C3×C4—81no—
151C2×C3×C4C2×SD1611yescharacteristic
152C2×C3×C4#13711yescharacteristic
153C2×Dic3—42no—
154C2×Dic3—42no—
155C2×Dic3—42no—
156C2×Dic3—42no—
157C2×Dic3—42no—
158C2×Dic3—42no—
159C3×C8—42no—
160C3⋊C8—82no—
161#41—41no—
162#41—41no—
163#41—41no—
164#41—41no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—41no—
168C2×Q8—41no—
169C2×Q8#21412yes—
170C2×D4—21no—
171C2×D4—22no—
172C2×D4—41no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—41no—
176C22×C4—62no—
177C22×C4—62no—
178C22×C4D3×D412yes—
179C22×C4C2×#6711yescharacteristic
180Q16—44no—
181SD16—44no—
182M16—22no—
183M16—41no—
184C2×C8—41no—
185C2×C8—122no—
186C4⋊C4—22no—
187C4⋊C4—41no—
188C4⋊C4—122no—
189C4⋊C4—124no—
190C4⋊C4—122no—
191C4⋊C4—241no—
192C4⋊C4—241no—
193C22⋊C4—42no—
194C42—41no—
195C42—241no—
196C22×C3—21no—
197C22×C3—21no—
198C22×C3—81no—
199C22×C3—81no—
200C22×C3—81no—
201C22×C3#44811yescharacteristic
202C22×C3#45311yescharacteristic
203C22×C3B(4,𝔽2)11yescharacteristic
204C3×C4—21no—
205C3×C4—21no—
206C3×C4—42no—
207C3×C4—42no—
208C3×C4—42no—
209C3×C4—81no—
210Dic3—82no—
211Dic3—82no—
212C23—41no—
213C23#93711yescharacteristic
214Q8—42no—
215Q8—41no—
216Q8—41no—
217Q8—81no—
218D4—41no—
219D4—41no—
220D4—82no—
221D4—81no—
222D4—81no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—122no—
237C2×C4—122no—
238C2×C4—122no—
239C2×C4—122no—
240C2×C4—122no—
241C2×C4—122no—
242C2×C4C2×#21411yescharacteristic
243C2×C4#95111yescharacteristic
244C8—42no—
245C8—242no—
246C2×C3—21no—
247C2×C3—21no—
248C2×C3—161no—
249C2×C3#159611yescharacteristic
250C2×C3#220111yescharacteristic
251C2×C3#220411yescharacteristic
252C22—21no—
253C22—21no—
254C22—81no—
255C22—81no—
256C22—81no—
257C22#503211yescharacteristic
258C22#504011yescharacteristic
259C22#507711yescharacteristic
260C4—21no—
261C4—21no—
262C4—42no—
263C4—42no—
264C4—42no—
265C4—81no—
266C4—242no—
267C4—242no—
268C3#1213111yescharacteristic, complemented, Sylow
269C2—21no—
270C2—21no—
271C2—161no—
272C2#6969811yescharacteristic
273C2#7099211yescharacteristic
274C2#7099411yescharacteristic
subgroupG/H
centerC22#5032
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#2007C2
socleC22×C3B(4,𝔽2)
prime radicalC3×#537C22
hypercenter#2007D3
hypocenterC3#12131
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2007

Complements

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

NH ≅ G/Ntype
1. #76602C2semidirect
4. #67643C2semidirect
5. #67496C2semidirect
268. C3#12131semidirect

Maximal subgroups

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

subgroupindex
#766022
C3×#20072
#718752
#676432
#674962
#121313