Finite Group Databasebeta

← #10697681

Subgroups — #10697681

Order 768.

526 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 103 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×#3249C211yesmaximal, characteristic, complemented
2#83350C212yesmaximal
3#71826C212yesmaximal, complemented
4#70470C211yesmaximal, characteristic
5#67705C211yesmaximal, characteristic, complemented
6#10258—31nomaximal, Sylow
7C3×#533C2212yes—
8C2×C4×D12C2211yescharacteristic
9C3×#477—22no—
10C3×#467—22no—
11C3×#462—22no—
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×(C2×D3)⋊C8—22no—
14#5087—22no—
15#5084—22no—
16#5066—24no—
17#5001—22no—
18#4995—22no—
19#4924C2212yes—
20C2×C4×C3⋊C8C2211yescharacteristic
21#4535—24no—
22#3249D311yescharacteristic, complemented
23#3077—32no—
24#1954—31no—
25#1493—32no—
26#1292—31no—
27C22×D12D411yescharacteristic
28C22×C4×D3—21no—
29C2×C3×Q16—22no—
30C22×C3×C8D412yes—
31C3×C4⋊Q8D412yes—
32C3×C4⋊Q8D412yes—
33C3×#125—22no—
34C3×C22⋊Q8—22no—
35C3×C22⋊Q8—22no—
36C3×C4×Q8—22no—
37C3×C4×Q8—22no—
38C3×#117—22no—
39C2×C3×C42C2311yescharacteristic
40C2×(C2×D3)⋊C4—21no—
41C2×C4⋊Dic3D411yescharacteristic
42C2×#208—22no—
43C22×C3⋊C8—21no—
44C4×D12—22no—
45C4×D12—22no—
46C3×#107—22no—
47C3×C4⋊C8—22no—
48C3×#103—24no—
49(C2×D3)⋊C8—44no—
50#809—24no—
51#809—24no—
52#808—24no—
53#807—24no—
54C4⋊(C3⋊C8)—22no—
55C4⋊(C3⋊C8)D414yes—
56C4×C3⋊C8—22no—
57#533C2×D312yes—
58C2×C4×D4—31no—
59#478—64no—
60#477—22no—
61#476—62no—
62#474—62no—
63#467—22no—
64#462—22no—
65#428—62no—
66#423—32no—
67C2×C4⋊C8C2×D311yescharacteristic
68#418—62no—
69C2×C22⋊C8—62no—
70C2×C4×C8—31no—
71Q8⋊C8—64no—
72C23×D3—21no—
73C2×C3×Q8—22no—
74C2×C3×Q8—22no—
75C22×C3×C4C2×D411yescharacteristic
76C22×C3×C4C2×D411yescharacteristic
77C22×C3×C4#4111yescharacteristic
78C22×Dic3—21no—
79C2×D12—22no—
80C2×D12—22no—
81C2×D12—21no—
82C2×D12Q1612yescomplemented
83C2×C4×D3—42no—
84C2×C4×D3—42no—
85C2×C4×D3—41no—
86C2×C4×D3—41no—
87C3×Q16—82no—
88C2×C3×C8—22no—
89C2×C3×C8—42no—
90C3×C4⋊C4—22no—
91C3×C4⋊C4—22no—
92C3×C4⋊C4—22no—
93C3×C4⋊C4—22no—
94C3×C4⋊C4—42no—
95C3×C4⋊C4—42no—
96C3×C22⋊C4—42no—
97C3×C42—42no—
98C3×C42C2×D412yes—
99C3×C42C2×D412yes—
100(C2×D3)⋊C4—42no—
101C4⋊Dic3—22no—
102#208—42no—
103#208—42no—
104C2×C3⋊C8—22no—
105C2×C3⋊C8—24no—
106C2×C3⋊C8—41no—
107C2×C3⋊C8—41no—
108C22×D4—31no—
109C23×C4—61no—
110C2×Q16—22no—
111C2×M16—62no—
112C22×C8—61no—
113C22×C8#6712yes—
114C4⋊Q8#6712yes—
115C4⋊Q8#6712yes—
116#125—22no—
117C22⋊Q8—22no—
118C22⋊Q8—22no—
119C4×Q8—22no—
120C4×Q8—22no—
121C4×D4—62no—
122C4×D4—62no—
123#117—22no—
124C2×C4⋊C4—31no—
125C2×C22⋊C4—61no—
126C2×C42C22×D311yescharacteristic
127#107—22no—
128#107—64no—
129#106—64no—
130C4⋊C8—22no—
131C4⋊C8—34no—
132C4⋊C8—62no—
133#103—24no—
134#102—64no—
135#102—64no—
136C22⋊C8—124no—
137C4×C8—62no—
138C23×C3#12111yescharacteristic
139C22×D3—22no—
140C22×D3—42no—
141C22×D3—42no—
142C22×D3—41no—
143C22×D3—41no—
144C3×Q8—44no—
145C3×Q8—42no—
146C3×Q8—42no—
147C2×C3×C4—21no—
148C2×C3×C4—21no—
149C2×C3×C4—41no—
150C2×C3×C4—42no—
151C2×C3×C4—42no—
152C2×C3×C4#12012yes—
153C2×C3×C4#12011yescharacteristic
154C2×C3×C4#12112yes—
155C2×C3×C4#12711yescharacteristic
156C2×C3×C4C2×Q1611yescharacteristic
157C2×C3×C4#13511yescharacteristic
158C2×C3×C4#13712yes—
159C2×Dic3—42no—
160C2×Dic3—41no—
161D12—24no—
162D12—42no—
163C4×D3—82no—
164C4×D3—81no—
165C4×D3—81no—
166C3×C8—42no—
167C3⋊C8—42no—
168C3⋊C8—44no—
169C24—61no—
170C2×Q8—22no—
171C2×Q8—22no—
172C2×D4—32no—
173C2×D4—62no—
174C2×D4—61no—
175C2×D4—62no—
176C22×C4—61no—
177C22×C4—122no—
178C22×C4—122no—
179C22×C4—121no—
180C22×C4—121no—
181C22×C4D3×D411yescharacteristic
182C22×C4#23711yescharacteristic
183C22×C4C2×#6711yescharacteristic
184Q16—82no—
185M16—122no—
186M16—122no—
187C2×C8—22no—
188C2×C8—42no—
189C2×C8—62no—
190C2×C8—64no—
191C2×C8—121no—
192C2×C8—121no—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—62no—
200C22⋊C4—42no—
201C22⋊C4—122no—
202C42—42no—
203C42D3×D412yes—
204C42C2×#6712yes—
205C22×C3—21no—
206C22×C3—21no—
207C22×C3#39211yescharacteristic
208C22×C3#46711yescharacteristic
209C22×C3#47011yescharacteristic
210C2×D3—44no—
211C2×D3—42no—
212C2×D3—82no—
213C2×D3—81no—
214C2×D3—81no—
215C3×C4—21no—
216C3×C4—22no—
217C3×C4—82no—
218C3×C4—82no—
219C3×C4#45112yes—
220C3×C4#45212yes—
221C3×C4#45311yescharacteristic
222C3×C4#45411yescharacteristic
223Dic3—81no—
224C23—62no—
225C23—122no—
226C23—122no—
227C23—121no—
228C23—121no—
229C23#93811yescharacteristic
230Q8—44no—
231Q8—42no—
232Q8—42no—
233D4—64no—
234D4—122no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—122no—
241C2×C4—121no—
242C2×C4—242no—
243C2×C4—241no—
244C2×C4—241no—
245C2×C4D3×Q1611yescharacteristic
246C2×C4#91812yes—
247C2×C4#91911yescharacteristic
248C2×C4#93712yes—
249C2×C4#93912yes—
250C2×C4#94011yescharacteristic
251C2×C4#95311yescharacteristic
252C8—42no—
253C8—122no—
254C8—124no—
255C2×C3—21no—
256C2×C3—21no—
257C2×C3#200111yescharacteristic
258C2×C3#200611yescharacteristic
259C2×C3#201711yescharacteristic
260D3—82no—
261C22—21no—
262C22—21no—
263C22—124no—
264C22—122no—
265C22—242no—
266C22—241no—
267C22—241no—
268C22#518811yescharacteristic
269C22#518911yescharacteristic
270C22#522311yescharacteristic
271C4—22no—
272C4—21no—
273C4—82no—
274C4—82no—
275C4—241no—
276C4#518612yes—
277C4#518712yes—
278C4#524111yescharacteristic
279C4#524511yescharacteristic
280C3#1025811yescharacteristic, complemented, Sylow
281C2—21no—
282C2—21no—
283C2—242no—
284C2#7187911yescharacteristic
285C2#7207111yescharacteristic
286C2#7212111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3249C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#3249D3
hypocenterC3#10258
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3249

Complements

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

NH ≅ G/Ntype
1. C3×#3249C2semidirect
3. #71826C2semidirect
5. #67705C2semidirect
22. #3249D3semidirect
82. C2×D12Q16semidirect
280. C3#10258semidirect

Maximal subgroups

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

subgroupindex
C3×#32492
#833502
#718262
#704702
#677052
#102583