Finite Group Databasebeta

← #10693039

Subgroups — #10693039

Order 768.

542 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 127 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#82435C211yesmaximal, characteristic, complemented
3C3×#1801C211yesmaximal, characteristic, complemented
4#70496C212yesmaximal
5#67713C212yesmaximal, complemented
6#9486—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C2×#946—21no—
9C2×C4×D12C2211yescharacteristic
10C3×#423C2212yes—
11C2×C3×#101—21no—
12Q8×C3⋊C8—24no—
13#4991C412yes—
14#4987—21no—
15#4981C412yes—
16#4940—22no—
17#4936C2211yescharacteristic
18#4925—22no—
19#4925C2212yes—
20C3×#343—21no—
21#4483—24no—
22#2982—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—42no—
38#946—42no—
39C2×(C2×D3)⋊C4—22no—
40C2×(C2×D3)⋊C4—22no—
41C2×(C2×D3)⋊C4—21no—
42C2×C4⋊Dic3—21no—
43C2×C4⋊Dic3C2×C411yescharacteristic
44C2×Dic3⋊C4—22no—
45C4×D12—22no—
46C4×D12—22no—
47C3×C4⋊C8—22no—
48C3×C4⋊C8—22no—
49C3×#101—42no—
50C3⋊#95—22no—
51C3⋊#95—21no—
52C3⋊#95C2×C412yes—
53#830—22no—
54C4⋊(C3⋊C8)—24no—
55C4⋊(C3⋊C8)—22no—
56C4×C3⋊C8—24no—
57C4×C3⋊C8D414yes—
58C2×C22⋊Q8—61no—
59C2×C4×Q8C2×D311yescharacteristic
60C2×C4×D4—31no—
61C8×Q8—64no—
62#431—32no—
63#431—62no—
64#423C2×D312yes—
65C2×#101—21no—
66#398—32no—
67#394—32no—
68#388—62no—
69#382—31no—
70#380—61no—
71#343—21no—
72D4⋊C8—64no—
73C23×D3—21no—
74C2×C3×Q8—22no—
75C2×C3×Q8—22no—
76C2×C3×Q8D812yes—
77C2×C3×Q8SD1612yes—
78C22×C3×C4C22⋊C412yes—
79C22×C3×C4C22⋊C412yes—
80C22×C3×C4C22×C411yescharacteristic
81C22×C3×C4C2×D411yescharacteristic
82C22×C3×C4C2×D411yescharacteristic
83C22×Dic3—22no—
84C22×Dic3—21no—
85C2×D12—22no—
86C2×D12—42no—
87C2×C4×D3—42no—
88C2×C4×D3—41no—
89C2×C4×D3—41no—
90C2×C4×D3—42no—
91C3×M16—42no—
92C3×M16—42no—
93C2×C3×C8—42no—
94C3×C4⋊C4—24no—
95C3×C4⋊C4—22no—
96C3×C42—22no—
97C3×C42—22no—
98C3×C42C2×D412yes—
99C3×C42#4112yes—
100(C2×D3)⋊C4—44no—
101(C2×D3)⋊C4—44no—
102(C2×D3)⋊C4—41no—
103(C2×D3)⋊C4—41no—
104C4⋊Dic3—22no—
105C4⋊Dic3—41no—
106C4⋊Dic3—41no—
107Dic3⋊C4—44no—
108C2×C3⋊C8—24no—
109C22×Q8#6711yescharacteristic
110C22×D4—31no—
111C23×C4—61no—
112C2×M16—22no—
113C22⋊Q8—122no—
114C22⋊Q8—122no—
115C4×Q8—22no—
116C4×Q8—22no—
117C4×D4—62no—
118C4×D4—62no—
119C2×C4⋊C4—31no—
120C2×C4⋊C4—62no—
121C2×C4⋊C4—61no—
122C2×C4⋊C4C4×D312yes—
123C2×C4⋊C4#6711yescharacteristic
124C2×C22⋊C4—62no—
125C2×C22⋊C4—62no—
126C2×C22⋊C4—61no—
127C2×C42D1212yes—
128C2×C42C22×D311yescharacteristic
129C4⋊C8—22no—
130C4⋊C8—22no—
131C4⋊C8—64no—
132C4⋊C8—62no—
133#101—42no—
134C22⋊C8—62no—
135C4×C8—34no—
136C4×C8—64no—
137#95—32no—
138#95—62no—
139#95—61no—
140C23×C3C2×C22⋊C411yescharacteristic
141C22×D3—22no—
142C22×D3—41no—
143C22×D3—41no—
144C22×D3—41no—
145C22×D3—41no—
146C22×D3—41no—
147C22×D3—41no—
148C3×Q8—24no—
149C3×Q8—24no—
150C2×C3×C4—22no—
151C2×C3×C4—22no—
152C2×C3×C4—22no—
153C2×C3×C4—41no—
154C2×C3×C4—41no—
155C2×C3×C4—42no—
156C2×C3×C4#10214yes—
157C2×C3×C4C4×D412yes—
158C2×C3×C4#12011yescharacteristic
159C2×C3×C4#12112yes—
160C2×C3×C4#12311yescharacteristic
161C2×C3×C4C2×D811yescharacteristic
162C2×C3×C4C2×SD1611yescharacteristic
163C2×C3×C4#13512yes—
164C2×Dic3—42no—
165C2×Dic3—42no—
166C2×Dic3—41no—
167C2×Dic3—41no—
168C2×Dic3—42no—
169C2×Dic3—41no—
170D12—24no—
171D12—42no—
172C4×D3—82no—
173C4×D3—82no—
174C3×C8—82no—
175C3⋊C8—44no—
176C24—61no—
177C2×Q8—22no—
178C2×Q8—22no—
179C2×Q8#21312yes—
180C2×Q8#21512yes—
181C2×D4—62no—
182C2×D4—122no—
183C22×C4—62no—
184C22×C4—61no—
185C22×C4—121no—
186C22×C4—122no—
187C22×C4—121no—
188C22×C4—122no—
189C22×C4(C2×D3)⋊C412yes—
190C22×C4(C2×D3)⋊C412yes—
191C22×C4C2×C4×D311yescharacteristic
192C22×C4C2×D1211yescharacteristic
193C22×C4C2×#6711yescharacteristic
194M16—42no—
195M16—42no—
196C2×C8—42no—
197C2×C8—64no—
198C4⋊C4—24no—
199C4⋊C4—22no—
200C4⋊C4—62no—
201C4⋊C4—124no—
202C4⋊C4—121no—
203C4⋊C4—121no—
204C22⋊C4—124no—
205C22⋊C4—124no—
206C22⋊C4—121no—
207C22⋊C4—121no—
208C42—22no—
209C42—22no—
210C42D3×D412yes—
211C42#23712yes—
212C22×C3—21no—
213C22×C3—21no—
214C22×C3#38611yescharacteristic
215C22×C3C2×#10211yescharacteristic
216C22×C3#41611yescharacteristic
217C2×D3—42no—
218C2×D3—42no—
219C2×D3—42no—
220C2×D3—81no—
221C2×D3—81no—
222C2×D3—81no—
223C2×D3—81no—
224C3×C4—22no—
225C3×C4—24no—
226C3×C4—21no—
227C3×C4—21no—
228C3×C4—42no—
229C3×C4#46512yes—
230C3×C4#46612yes—
231Dic3—82no—
232Dic3—81no—
233C23—62no—
234C23—121no—
235C23—121no—
236C23—121no—
237C23—121no—
238C23—121no—
239C23—121no—
240C23C2×(C2×D3)⋊C411yescharacteristic
241Q8—24no—
242Q8—24no—
243D4—64no—
244D4—122no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—122no—
252C2×C4—122no—
253C2×C4—121no—
254C2×C4—122no—
255C2×C4—121no—
256C2×C4—121no—
257C2×C4—242no—
258C2×C4—242no—
259C2×C4#80914yes—
260C2×C4#67⋊C412yes—
261C2×C4#88211yescharacteristic
262C2×C4#88611yescharacteristic
263C2×C4C2×#21311yescharacteristic
264C2×C4#93912yes—
265C2×C4C2×#21511yescharacteristic
266C2×C4#95012yes—
267C8—82no—
268C8—124no—
269C2×C3—21no—
270C2×C3—21no—
271C2×C3#189111yescharacteristic
272C2×C3#189311yescharacteristic
273C2×C3#191211yescharacteristic
274D3—82no—
275C22—21no—
276C22—21no—
277C22—122no—
278C22—122no—
279C22—122no—
280C22—241no—
281C22—241no—
282C22—241no—
283C22—241no—
284C22#495811yescharacteristic
285C22C2×#80911yescharacteristic
286C22#500211yescharacteristic
287C4—24no—
288C4—21no—
289C4—21no—
290C4—22no—
291C4—42no—
292C4—242no—
293C4—241no—
294C4#503812yes—
295C4#504812yes—
296C3#948611yescharacteristic, complemented, Sylow
297C2—21no—
298C2—21no—
299C2—242no—
300C2#6951511yescharacteristic
301C2#7010511yescharacteristic
302C2#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#9486
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. #82435C2semidirect
3. C3×#1801C2semidirect
5. #67713C2semidirect
24. #1801D3semidirect
296. C3#9486semidirect

Maximal subgroups

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

subgroupindex
#824992
#824352
C3×#18012
#704962
#677132
#94863