Finite Group Databasebeta

← #10693059

Subgroups — #10693059

Order 768.

522 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 111 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#82499C211yesmaximal, characteristic
2#82436C211yesmaximal, characteristic, complemented
3C3×#1801C211yesmaximal, characteristic, complemented
4#70496C212yesmaximal
5#67713C212yesmaximal, complemented
6#9490—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C2×#946—21no—
9C2×C4×D12C2211yescharacteristic
10C3×#423C2212yes—
11C2×C3×#101—21no—
12#5024—24no—
13#4991C412yes—
14#4987—21no—
15#4981C412yes—
16#4940—22no—
17#4936C2211yescharacteristic
18#4926—22no—
19#4925C2212yes—
20C3×#343—21no—
21#4483—24no—
22#2983—31no—
23#2536—31no—
24#1801D311yescharacteristic, complemented
25#1486—32no—
26#1286—32no—
27C22×C3×Q8D411yescharacteristic
28C22×D12C2×C411yescharacteristic
29C22×C4×D3—21no—
30C2×C3×M16—22no—
31C3×C4×Q8—22no—
32C3×C4×Q8—22no—
33C2×C3×C4⋊C4C2×C412yes—
34C2×C3×C4⋊C4D411yescharacteristic
35C2×C3×C42D412yes—
36C2×C3×C42C2311yescharacteristic
37#946—44no—
38C2×(C2×D3)⋊C4—22no—
39C2×(C2×D3)⋊C4—22no—
40C2×(C2×D3)⋊C4—21no—
41C2×C4⋊Dic3—21no—
42C2×C4⋊Dic3C2×C411yescharacteristic
43C2×Dic3⋊C4—22no—
44C4×D12—22no—
45C4×D12—22no—
46C3×C4⋊C8—22no—
47C3×C4⋊C8—22no—
48C3×#101—42no—
49C3⋊#95—22no—
50C3⋊#95—21no—
51C3⋊#95C2×C412yes—
52#830—22no—
53C4⋊(C3⋊C8)—24no—
54C4⋊(C3⋊C8)—22no—
55#803—24no—
56C4×C3⋊C8D414yes—
57C2×C22⋊Q8—61no—
58C2×C4×Q8C2×D311yescharacteristic
59C2×C4×D4—31no—
60#446—64no—
61#432—62no—
62#431—32no—
63#423C2×D312yes—
64C2×#101—21no—
65#398—32no—
66#394—32no—
67#388—62no—
68#382—31no—
69#380—61no—
70#343—21no—
71D4⋊C8—64no—
72C23×D3—21no—
73C2×C3×Q8—22no—
74C2×C3×Q8—21no—
75C2×C3×Q8—22no—
76C2×C3×Q8—21no—
77C22×C3×C4C22⋊C412yes—
78C22×C3×C4C22⋊C412yes—
79C22×C3×C4C22×C411yescharacteristic
80C22×C3×C4C2×D411yescharacteristic
81C22×C3×C4C2×D411yescharacteristic
82C22×Dic3—22no—
83C22×Dic3—21no—
84C2×D12—22no—
85C2×D12—42no—
86C2×C4×D3—42no—
87C2×C4×D3—42no—
88C2×C4×D3—41no—
89C2×C4×D3—41no—
90C3×M16—42no—
91C3×M16—42no—
92C2×C3×C8—42no—
93C3×C4⋊C4—24no—
94C3×C4⋊C4—22no—
95C3×C42—22no—
96C3×C42—22no—
97C3×C42C2×D412yes—
98C3×C42#4112yes—
99(C2×D3)⋊C4—44no—
100(C2×D3)⋊C4—44no—
101(C2×D3)⋊C4—41no—
102(C2×D3)⋊C4—41no—
103C4⋊Dic3—22no—
104C4⋊Dic3—42no—
105Dic3⋊C4—44no—
106C2×C3⋊C8—24no—
107C22×Q8#6711yescharacteristic
108C22×D4—31no—
109C23×C4—61no—
110C2×M16—22no—
111C22⋊Q8—124no—
112C4×Q8—22no—
113C4×Q8—22no—
114C4×D4—62no—
115C4×D4—62no—
116C2×C4⋊C4—31no—
117C2×C4⋊C4—62no—
118C2×C4⋊C4—61no—
119C2×C4⋊C4C4×D312yes—
120C2×C4⋊C4#6711yescharacteristic
121C2×C22⋊C4—62no—
122C2×C22⋊C4—62no—
123C2×C22⋊C4—61no—
124C2×C42D1212yes—
125C2×C42C22×D311yescharacteristic
126C4⋊C8—22no—
127C4⋊C8—22no—
128C4⋊C8—64no—
129C4⋊C8—62no—
130#101—42no—
131C22⋊C8—62no—
132#97—64no—
133C4×C8—34no—
134#95—32no—
135#95—62no—
136#95—61no—
137C23×C3C2×C22⋊C411yescharacteristic
138C22×D3—22no—
139C22×D3—41no—
140C22×D3—41no—
141C22×D3—41no—
142C22×D3—41no—
143C22×D3—41no—
144C22×D3—41no—
145C3×Q8—42no—
146C3×Q8—42no—
147C2×C3×C4—22no—
148C2×C3×C4—22no—
149C2×C3×C4—22no—
150C2×C3×C4—22no—
151C2×C3×C4—41no—
152C2×C3×C4—41no—
153C2×C3×C4—42no—
154C2×C3×C4C4×D412yes—
155C2×C3×C4#12011yescharacteristic
156C2×C3×C4#12112yes—
157C2×C3×C4#12311yescharacteristic
158C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
159C2×C3×C4AGL(1,ℤ/8)12yes—
160C2×C3×C4#13711yescharacteristic
161C2×Dic3—42no—
162C2×Dic3—42no—
163C2×Dic3—41no—
164C2×Dic3—41no—
165C2×Dic3—41no—
166C2×Dic3—42no—
167D12—24no—
168D12—42no—
169C4×D3—82no—
170C4×D3—82no—
171C3×C8—82no—
172C3⋊C8—44no—
173C24—61no—
174C2×Q8—21no—
175C2×Q8—22no—
176C2×Q8—21no—
177C2×Q8—22no—
178C2×D4—62no—
179C2×D4—122no—
180C22×C4—62no—
181C22×C4—61no—
182C22×C4—121no—
183C22×C4—122no—
184C22×C4—121no—
185C22×C4—122no—
186C22×C4(C2×D3)⋊C412yes—
187C22×C4(C2×D3)⋊C412yes—
188C22×C4C2×C4×D311yescharacteristic
189C22×C4C2×D1211yescharacteristic
190C22×C4C2×#6711yescharacteristic
191M16—42no—
192M16—42no—
193C2×C8—42no—
194C2×C8—64no—
195C4⋊C4—24no—
196C4⋊C4—22no—
197C4⋊C4—62no—
198C4⋊C4—124no—
199C4⋊C4—122no—
200C22⋊C4—124no—
201C22⋊C4—124no—
202C22⋊C4—121no—
203C22⋊C4—121no—
204C42—22no—
205C42—22no—
206C42D3×D412yes—
207C42#23712yes—
208C22×C3—21no—
209C22×C3—21no—
210C22×C3#38611yescharacteristic
211C22×C3#41711yescharacteristic
212C22×C3#41811yescharacteristic
213C2×D3—42no—
214C2×D3—42no—
215C2×D3—42no—
216C2×D3—81no—
217C2×D3—81no—
218C2×D3—81no—
219C2×D3—81no—
220C3×C4—22no—
221C3×C4—21no—
222C3×C4—21no—
223C3×C4—42no—
224C3×C4—42no—
225C3×C4#46812yes—
226C3×C4#46912yes—
227Dic3—82no—
228Dic3—81no—
229C23—62no—
230C23—121no—
231C23—121no—
232C23—121no—
233C23—121no—
234C23—121no—
235C23—121no—
236C23C2×(C2×D3)⋊C411yescharacteristic
237Q8—42no—
238Q8—42no—
239D4—64no—
240D4—122no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—122no—
249C2×C4—122no—
250C2×C4—121no—
251C2×C4—121no—
252C2×C4—122no—
253C2×C4—121no—
254C2×C4—242no—
255C2×C4—242no—
256C2×C4#67⋊C412yes—
257C2×C4#88211yescharacteristic
258C2×C4#88611yescharacteristic
259C2×C4#93211yescharacteristic
260C2×C4#93912yes—
261C2×C4#94211yescharacteristic
262C2×C4#94912yes—
263C8—82no—
264C8—124no—
265C2×C3—21no—
266C2×C3—21no—
267C2×C3#189111yescharacteristic
268C2×C3#189511yescharacteristic
269C2×C3#191211yescharacteristic
270D3—82no—
271C22—21no—
272C22—21no—
273C22—122no—
274C22—122no—
275C22—122no—
276C22—241no—
277C22—241no—
278C22—241no—
279C22—241no—
280C22#495811yescharacteristic
281C22#496611yescharacteristic
282C22#500111yescharacteristic
283C4—21no—
284C4—21no—
285C4—22no—
286C4—42no—
287C4—42no—
288C4—242no—
289C4—241no—
290C4#503912yes—
291C4#504912yes—
292C3#949011yescharacteristic, complemented, Sylow
293C2—21no—
294C2—21no—
295C2—242no—
296C2#6951711yescharacteristic
297C2#7010511yescharacteristic
298C2#7013611yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1801C2
socleC22×C3#386
prime radicalC22×D12C2×C4
cosocleC2×C3×C42C23
hypercenter#1801D3
hypocenterC3#9490
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
2. #82436C2semidirect
3. C3×#1801C2semidirect
5. #67713C2semidirect
24. #1801D3semidirect
292. C3#9490semidirect

Maximal subgroups

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

subgroupindex
#824992
#824362
C3×#18012
#704962
#677132
#94903