Finite Group Databasebeta

← #10697387

Subgroups — #10697387

Order 768.

536 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 81 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×#3253C211yesmaximal, characteristic, complemented
2#83307C212yesmaximal, complemented
3#71812C212yesmaximal, complemented
4#70513C211yesmaximal, characteristic, complemented
5#67716C211yesmaximal, characteristic, complemented
6#10212—31nomaximal, Sylow
7C3×#532C2212yes—
8#5477C2211yescharacteristic
9C3×#476—22no—
10C3×#469—22no—
11C3×#459—22no—
12C3×#423C2211yescharacteristic
13#5238—24no—
14C2×#825—22no—
15#5083—22no—
16#5072—22no—
17#5058—22no—
18#5055—22no—
19#4932C2211yescharacteristic
20#4926C2212yes—
21#4537—24no—
22#3253D311yescharacteristic, complemented
23#3214—32no—
24#1963—31no—
25#1505—32no—
26#1307—31no—
27C2×C3×#41—22no—
28C2×#235—21no—
29C2×C3×D8—22no—
30C2×C3×M16D412yes—
31C3×C4⋊Q8D412yes—
32C3×#127D412yes—
33C3×#124—22no—
34C3×#121—22no—
35C3×#121—22no—
36C3×C4×D4—22no—
37C3×C4×D4—22no—
38C2×C3×C42C2311yescharacteristic
39C2×#215—24no—
40C2×#213—24no—
41#930—22no—
42C2×C4×Dic3—21no—
43#875—21no—
44#874D411yescharacteristic
45C4×D12—22no—
46#869D411yescharacteristic
47C3×#106—22no—
48C3×C4⋊C8—22no—
49C3×#102—24no—
50#835—24no—
51#832—24no—
52C3⋊#95—22no—
53#830D414yes—
54#825—44no—
55C4⋊(C3⋊C8)—22no—
56#803—22no—
57#532—31no—
58#532C2×D312yes—
59#496—62no—
60#489—62no—
61#476—22no—
62#469—22no—
63#461—62no—
64#459—22no—
65#459—62no—
66#449—64no—
67#432—32no—
68#423C2×D311yescharacteristic
69C2×#104—62no—
70#376—31no—
71#327—64no—
72C3×#41—84no—
73C2×C3×Q8—22no—
74C2×C3×D4—22no—
75C2×C3×D4—22no—
76C2×C3×D4—22no—
77C2×C3×D4—42no—
78C22×C3×C4—42no—
79C22×C3×C4C2×D412yes—
80C22×C3×C4C2×D411yescharacteristic
81C2×#67—41no—
82C22×Dic3—41no—
83#235—42no—
84#235—42no—
85C2×D12—22no—
86C2×D12—21no—
87C2×C4×D3—41no—
88C2×Dic6—21no—
89C3×D8—82no—
90C3×M16—42no—
91C2×C3×C8—22no—
92C3×C4⋊C4—22no—
93C3×C4⋊C4—22no—
94C3×C22⋊C4—42no—
95C3×C22⋊C4—42no—
96C3×C42C2×D411yescharacteristic
97C3×C42C2×D412yes—
98C3×C42#4111yescharacteristic
99#215—84no—
100#213—84no—
101(C2×D3)⋊C4—42no—
102C4⋊Dic3—22no—
103C4×Dic3—22no—
104C4×Dic3—41no—
105C2×C3⋊C8—24no—
106C2×#41—22no—
107C2×#41—61no—
108C2×SD16—64no—
109C2×D8—22no—
110C2×D8—64no—
111C2×M16#6712yes—
112C4⋊Q8—31no—
113C4⋊Q8#6712yes—
114#127—31no—
115#127#6712yes—
116#124—22no—
117#124—61no—
118#121—22no—
119#121—22no—
120#121—62no—
121C4×D4—22no—
122C4×D4—22no—
123C4×D4—62no—
124C2×C42—61no—
125C2×C42C22×D311yescharacteristic
126#106—22no—
127C4⋊C8—22no—
128C4⋊C8—62no—
129#104—124no—
130#103—64no—
131#102—24no—
132#102—64no—
133C22⋊C8—34no—
134#97—62no—
135#95—62no—
136C23×C3—42no—
137C23×C3#12711yescharacteristic
138C22×D3—41no—
139C3×Q8—44no—
140C3×D4—44no—
141C3×D4—42no—
142C3×D4—42no—
143C3×D4—82no—
144C3×D4—84no—
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4—42no—
148C2×C3×C4—42no—
149C2×C3×C4—84no—
150C2×C3×C4#12012yes—
151C2×C3×C4#12011yescharacteristic
152C2×C3×C4#12111yescharacteristic
153C2×C3×C4#12112yes—
154#67—82no—
155C2×Dic3—42no—
156C2×Dic3—41no—
157C2×Dic3—82no—
158D12—42no—
159D12—42no—
160D12—42no—
161C4×D3—81no—
162C4×D3—81no—
163Dic6—42no—
164C3×C8—42no—
165C3⋊C8—44no—
166#41—84no—
167#41—122no—
168#41—122no—
169C2×Q8—22no—
170C2×Q8—61no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4—22no—
174C2×D4—42no—
175C2×D4—62no—
176C2×D4—61no—
177C2×D4—121no—
178C22×C4—42no—
179C22×C4—121no—
180C22×C4—121no—
181C22×C4D3×D412yes—
182C22×C4C2×#6711yescharacteristic
183SD16—244no—
184D8—82no—
185D8—244no—
186M16—42no—
187C2×C8—22no—
188C2×C8—64no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—62no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—122no—
195C42—62no—
196C42—121no—
197C42D3×D411yescharacteristic
198C42#23711yescharacteristic
199C42C2×#6712yes—
200C22×C3—21no—
201C22×C3—22no—
202C22×C3—82no—
203C22×C3—82no—
204C22×C3—82no—
205C22×C3#39211yescharacteristic
206C2×D3—81no—
207C2×D3—82no—
208C3×C4—21no—
209C3×C4—22no—
210C3×C4—22no—
211C3×C4—82no—
212C3×C4#45311yescharacteristic
213C3×C4#45411yescharacteristic
214Dic3—82no—
215Dic3—81no—
216C23—42no—
217C23—121no—
218C23#94011yescharacteristic
219Q8—44no—
220Q8—122no—
221D4—44no—
222D4—42no—
223D4—42no—
224D4—82no—
225D4—84no—
226D4—122no—
227D4—122no—
228D4—122no—
229D4—242no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—84no—
235C2×C4—122no—
236C2×C4—121no—
237C2×C4—242no—
238C2×C4—241no—
239C2×C4—241no—
240C2×C4#93712yes—
241C2×C4#93811yescharacteristic
242C2×C4#93912yes—
243C2×C4#95311yescharacteristic
244C8—42no—
245C8—124no—
246C2×C3—41no—
247C2×C3—162no—
248C2×C3#200611yescharacteristic
249C2×C3#200812yes—
250D3—161no—
251C22—22no—
252C22—21no—
253C22—82no—
254C22—82no—
255C22—82no—
256C22—242no—
257C22—241no—
258C22#522311yescharacteristic
259C4—22no—
260C4—22no—
261C4—21no—
262C4—82no—
263C4—242no—
264C4—241no—
265C4#524111yescharacteristic
266C4#524511yescharacteristic
267C3#1021211yescharacteristic, complemented, Sylow
268C2—41no—
269C2—162no—
270C2—481no—
271C2#7187911yescharacteristic
272C2#7209912yes—
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3253C2
socleC22×C3#392
hypercenter#3253D3
hypocenterC3#10212
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3253

Complements

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

NH ≅ G/Ntype
1. C3×#3253C2semidirect
2. #83307C2semidirect
3. #71812C2semidirect
4. #70513C2semidirect
5. #67716C2semidirect
22. #3253D3semidirect
267. C3#10212semidirect

Maximal subgroups

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

subgroupindex
C3×#32532
#833072
#718122
#705132
#677162
#102123