Finite Group Databasebeta

← #10696151

Subgroups — #10696151

Order 768.

636 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 105 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×#3150C211yesmaximal, characteristic, complemented
2#83298C212yesmaximal, complemented
3#71800C212yesmaximal, complemented
4#69837C211yesmaximal, characteristic, complemented
5#67660C211yesmaximal, characteristic, complemented
6#10032—31nomaximal, Sylow
7C2×C3×#135—22no—
8C3×#532C2212yes—
9C2×C4×D12C2211yescharacteristic
10C3×#497—22no—
11C3×#496—22no—
12C2×C3×#97C2211yescharacteristic
13C2×(C2×D3)⋊C8—22no—
14#5087—22no—
15#5072—22no—
16#5036—22no—
17#5033—22no—
18C4×#213—24no—
19#4925C2212yes—
20C2×C4⋊(C3⋊C8)C2211yescharacteristic
21D4⋊(C3⋊C8)—24no—
22#3150D311yescharacteristic, complemented
23#3100—32no—
24#1955—31no—
25#1480—32no—
26#1275—31no—
27C2×C3×#41—22no—
28C22×D12D411yescharacteristic
29C22×C4×D3—21no—
30C3×#135—44no—
31C3×#135—44no—
32C2×C3×Q16—22no—
33C2×C3×SD16—24no—
34C2×C3×D8—22no—
35C22×C3×C8D412yes—
36C3×C4⋊Q8D412yes—
37C3×#127D412yes—
38C3×#124—22no—
39C3×#121—22no—
40C3×#121—22no—
41C3×C4×D4—22no—
42C3×C4×D4—22no—
43C2×C3×C42C2311yescharacteristic
44C2×#213—24no—
45C2×(C2×D3)⋊C4—21no—
46C2×C4⋊Dic3D411yescharacteristic
47C22×C3⋊C8—21no—
48C4×D12—22no—
49C4×D12—22no—
50C3×#97—22no—
51#832—24no—
52#830—22no—
53(C2×D3)⋊C8—44no—
54#809—24no—
55#807—24no—
56C4⋊(C3⋊C8)—22no—
57C4⋊(C3⋊C8)—22no—
58C4×C3⋊C8D414yes—
59C2×#135—22no—
60#532C2×D312yes—
61C2×C4×D4—31no—
62#497—22no—
63#496—22no—
64#480—62no—
65#474—62no—
66#459—62no—
67#447—62no—
68C4×D8—64no—
69#431—32no—
70C2×C4⋊C8—31no—
71C2×C22⋊C8—62no—
72C2×#97C2×D311yescharacteristic
73D4⋊C8—64no—
74C23×D3—21no—
75C3×#41—44no—
76C3×#41—44no—
77C2×C3×Q8—22no—
78C2×C3×D4—22no—
79C2×C3×D4—22no—
80C2×C3×D4—22no—
81C2×C3×D4—42no—
82C22×C3×C4—42no—
83C22×C3×C4C2×D411yescharacteristic
84C22×C3×C4C2×D411yescharacteristic
85C22×C3×C4#4111yescharacteristic
86C22×Dic3—21no—
87C2×D12—22no—
88C2×D12—42no—
89C2×C4×D3—41no—
90C2×C4×D3—41no—
91C2×C4×D3—42no—
92C2×C4×D3—42no—
93C3×Q16—44no—
94C3×SD16—48no—
95C3×D8—44no—
96C2×C3×C8—22no—
97C2×C3×C8—22no—
98C2×C3×C8D814yescomplemented
99C3×C4⋊C4—22no—
100C3×C4⋊C4—22no—
101C3×C22⋊C4—42no—
102C3×C22⋊C4—42no—
103C3×C42C2×D412yes—
104C3×C42C2×D412yes—
105#213—48no—
106(C2×D3)⋊C4—42no—
107C4⋊Dic3—22no—
108C2×C3⋊C8—24no—
109C2×C3⋊C8—41no—
110C2×C3⋊C8—42no—
111C2×#41—22no—
112C22×D4—31no—
113C23×C4—61no—
114#135—44no—
115#135—44no—
116C2×Q16—22no—
117C2×SD16—24no—
118C2×D8—22no—
119C2×D8—64no—
120C22×C8—61no—
121C22×C8#6712yes—
122C4⋊Q8#6712yes—
123#127#6712yes—
124#124—22no—
125#121—22no—
126#121—22no—
127C4×D4—22no—
128C4×D4—22no—
129C4×D4—62no—
130C4×D4—62no—
131C2×C4⋊C4—31no—
132C2×C22⋊C4—61no—
133C2×C42C22×D311yescharacteristic
134#107—64no—
135C4⋊C8—62no—
136C4⋊C8—62no—
137#102—64no—
138#102—64no—
139C22⋊C8—62no—
140C22⋊C8—124no—
141#97—22no—
142C4×C8—34no—
143C23×C3—42no—
144C23×C3#12111yescharacteristic
145C22×D3—22no—
146C22×D3—41no—
147C22×D3—42no—
148C22×D3—41no—
149C22×D3—42no—
150C3×Q8—44no—
151C3×D4—44no—
152C3×D4—44no—
153C3×D4—44no—
154C3×D4—84no—
155C2×C3×C4—41no—
156C2×C3×C4—41no—
157C2×C3×C4—42no—
158C2×C3×C4—82no—
159C2×C3×C4—82no—
160C2×C3×C4#12012yes—
161C2×C3×C4#12011yescharacteristic
162C2×C3×C4#12112yes—
163C2×C3×C4#12711yescharacteristic
164C2×C3×C4C2×D812yes—
165C2×C3×C4AGL(1,ℤ/8)12yes—
166C2×Dic3—41no—
167C2×Dic3—42no—
168D12—24no—
169D12—42no—
170C4×D3—82no—
171C4×D3—81no—
172C4×D3—81no—
173C3×C8—24no—
174C3⋊C8—44no—
175C3⋊C8—81no—
176C24—61no—
177#41—44no—
178#41—44no—
179C2×Q8—22no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—42no—
184C2×D4—62no—
185C2×D4—122no—
186C22×C4—42no—
187C22×C4—61no—
188C22×C4—122no—
189C22×C4—122no—
190C22×C4—121no—
191C22×C4—121no—
192C22×C4D3×D411yescharacteristic
193C22×C4#23711yescharacteristic
194C22×C4C2×#6711yescharacteristic
195Q16—44no—
196SD16—48no—
197D8—44no—
198D8—128no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—64no—
202C2×C8—122no—
203C2×C8—121no—
204C2×C8#21314yescomplemented
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—62no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—122no—
211C42D3×D412yes—
212C42C2×#6712yes—
213C22×C3—21no—
214C22×C3—21no—
215C22×C3—82no—
216C22×C3—82no—
217C22×C3—82no—
218C22×C3#39211yescharacteristic
219C22×C3#45912yes—
220C2×D3—44no—
221C2×D3—42no—
222C2×D3—82no—
223C2×D3—81no—
224C2×D3—81no—
225C3×C4—22no—
226C3×C4—22no—
227C3×C4—82no—
228C3×C4#44712yes—
229C3×C4#49311yescharacteristic
230C3×C4#49611yescharacteristic
231Dic3—81no—
232C23—42no—
233C23—62no—
234C23—121no—
235C23—122no—
236C23—121no—
237C23—122no—
238C23#93811yescharacteristic
239Q8—44no—
240D4—44no—
241D4—44no—
242D4—44no—
243D4—64no—
244D4—84no—
245D4—122no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—121no—
252C2×C4—122no—
253C2×C4—242no—
254C2×C4—241no—
255C2×C4—241no—
256C2×C4C2×#21312yes—
257C2×C4#93212yes—
258C2×C4#93712yes—
259C2×C4#93912yes—
260C2×C4#94011yescharacteristic
261C2×C4#95311yescharacteristic
262C8—24no—
263C8—124no—
264C8—241no—
265C2×C3—22no—
266C2×C3—162no—
267C2×C3#201011yescharacteristic
268C2×C3#201111yescharacteristic
269C2×C3#201711yescharacteristic
270D3—82no—
271C22—21no—
272C22—21no—
273C22—82no—
274C22—82no—
275C22—82no—
276C22—124no—
277C22—122no—
278C22—242no—
279C22—241no—
280C22—241no—
281C22#507212yes—
282C22#522311yescharacteristic
283C4—22no—
284C4—22no—
285C4—82no—
286C4—241no—
287C4#507311yescharacteristic
288C4#507411yescharacteristic
289C4#521712yes—
290C3#1003211yescharacteristic, complemented, Sylow
291C2—22no—
292C2—162no—
293C2—242no—
294C2#7176711yescharacteristic
295C2#7211811yescharacteristic
296C2#7212111yescharacteristic
subgroupG/H
centerC2×C4#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3150C2
socleC22×C3#392
hypercenter#3150D3
hypocenterC3#10032
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3150

Complements

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

NH ≅ G/Ntype
1. C3×#3150C2semidirect
2. #83298C2semidirect
3. #71800C2semidirect
4. #69837C2semidirect
5. #67660C2semidirect
22. #3150D3semidirect
98. C2×C3×C8D8semidirect
204. C2×C8#213semidirect
290. C3#10032semidirect

Maximal subgroups

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

subgroupindex
C3×#31502
#832982
#718002
#698372
#676602
#100323