Finite Group Databasebeta

← #10694575

Subgroups — #10694575

Order 768.

408 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 115 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#78463C211yesmaximal, characteristic
2C3×#1452C211yesmaximal, characteristic, complemented
3#70793C212yesmaximal, complemented
4#70790C212yesmaximal
5#69766C211yesmaximal, characteristic, complemented
6#9758—31nomaximal, Sylow
7#5477C2211yescharacteristic
8#5470C2211yescharacteristic
9C3×#431C2212yes—
10C2×C3×C4×C8C2211yescharacteristic
11#5113—21no—
12#5113—21no—
13C2×#818—21no—
14C2×#817—21no—
15#4936C2212yes—
16#4697—22no—
17#4694C412yescomplemented
18#4681C412yescomplemented
19C3×#322—22no—
20#4470—24no—
21#4467—24no—
22#3131—31no—
23#1846—31no—
24#1518—32no—
25#1517—32no—
26#1452D311yescharacteristic, complemented
27C2×#235—21no—
28C22×C3×C8D411yescharacteristic
29C22×C3×C8D411yescharacteristic
30C2×C3×C42C2311yescharacteristic
31#930—22no—
32#926—21no—
33C2×C4⋊Dic3—22no—
34#924—22no—
35#875—21no—
36#874C2×C411yescharacteristic
37C4×D12—22no—
38#870—21no—
39#869C2×C411yescharacteristic
40#869C2×C412yes—
41C4×Dic6—22no—
42C3×C4⋊C8—22no—
43C3×C22⋊C8D414yes—
44C3×C4×C8—22no—
45C3×C4×C8C2×C412yes—
46C3×C4×C8D412yes—
47C3⋊#95—22no—
48C3⋊#95—22no—
49#821—22no—
50#821—22no—
51#818—42no—
52#817—42no—
53#816—22no—
54#816—22no—
55#533—31no—
56#532—31no—
57#488—62no—
58#487—32no—
59#486—32no—
60#431C2×D312yes—
61C2×#107—61no—
62C2×#106—61no—
63#416—61no—
64#416—61no—
65C2×C4×C8C2×D311yescharacteristic
66#382—32no—
67#328—64no—
68#327—64no—
69#322—22no—
70C22×C3×C4C2×D412yes—
71C22×C3×C4C2×D411yescharacteristic
72C2×#67—41no—
73C22×Dic3—42no—
74#235—41no—
75#235—41no—
76#235—41no—
77#235—41no—
78C2×D12—21no—
79C2×D12—22no—
80C2×C4×D3—41no—
81C2×Dic6—21no—
82C2×Dic6—22no—
83C2×C3×C8—21no—
84C2×C3×C8—21no—
85C2×C3×C8—21no—
86C2×C3×C8—21no—
87C2×C3×C8—24no—
88C2×C3×C8C22⋊C412yes—
89C2×C3×C8C22⋊C412yes—
90C3×C42C22×C411yescharacteristic
91C3×C42C2×D411yescharacteristic
92C3×C42#4112yes—
93#217—41no—
94(C2×D3)⋊C4—42no—
95C4⋊Dic3—22no—
96C4⋊Dic3—22no—
97C4⋊Dic3—22no—
98C4⋊Dic3—42no—
99C4⋊Dic3—42no—
100Dic3⋊C4—42no—
101C4×Dic3—41no—
102C2×#41—61no—
103C22×C8D1211yescharacteristic
104C22×C8D1211yescharacteristic
105C4⋊Q8—31no—
106C4⋊Q8—32no—
107#127—31no—
108#125—61no—
109#124—61no—
110C22⋊Q8—62no—
111#121—62no—
112C4×Q8—62no—
113C4×D4—62no—
114#117—61no—
115C2×C4⋊C4—62no—
116C2×C42C22×D311yescharacteristic
117#107—122no—
118#106—122no—
119C4⋊C8—22no—
120#103—62no—
121#103—62no—
122#102—62no—
123#102—62no—
124C22⋊C8D1214yes—
125C4×C8—22no—
126C4×C8C4×D312yes—
127C4×C8#6712yes—
128#95—62no—
129#95—62no—
130C23×C3#12711yescharacteristic
131C22×D3—41no—
132C2×C3×C4—42no—
133C2×C3×C4C2×C22⋊C411yescharacteristic
134C2×C3×C4C4×D412yes—
135C2×C3×C4#12112yes—
136C2×C3×C4#12411yescharacteristic
137C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
138C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
139C2×C3×C4#13711yescharacteristic
140C2×C3×C4#13711yescharacteristic
141#67—81no—
142#67—81no—
143C2×Dic3—42no—
144C2×Dic3—41no—
145C2×Dic3—84no—
146C2×Dic3—82no—
147D12—41no—
148D12—41no—
149D12—82no—
150C4×D3—81no—
151C4×D3—81no—
152Dic6—41no—
153Dic6—41no—
154Dic6—82no—
155C3×C8—22no—
156C3×C8—22no—
157C3×C8—44no—
158#41—121no—
159#41—121no—
160#41—121no—
161#41—121no—
162C2×Q8—62no—
163C2×Q8—61no—
164C2×D4—62no—
165C2×D4—61no—
166C2×D4—121no—
167C22×C4—121no—
168C22×C4—122no—
169C22×C4C2×D1212yes—
170C22×C4C2×D1211yescharacteristic
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—24no—
176C2×C8(C2×D3)⋊C412yes—
177C2×C8(C2×D3)⋊C412yes—
178C4⋊C4—62no—
179C4⋊C4—62no—
180C4⋊C4—62no—
181C4⋊C4—122no—
182C4⋊C4—122no—
183C4⋊C4—122no—
184C22⋊C4—121no—
185C22⋊C4—122no—
186C42—121no—
187C42C2×C4×D311yescharacteristic
188C42#23512yes—
189C42C2×#6711yescharacteristic
190C22×C3—22no—
191C22×C3#39011yescharacteristic
192C22×C3#49611yescharacteristic
193C22×C3#49711yescharacteristic
194C2×D3—81no—
195C2×D3—82no—
196C3×C4—22no—
197C3×C4—22no—
198C3×C4#41711yescharacteristic
199C3×C4#41711yescharacteristic
200C3×C4#48911yescharacteristic
201C3×C4#48911yescharacteristic
202Dic3—81no—
203Dic3—82no—
204Dic3—162no—
205C23—121no—
206C23#87411yescharacteristic
207Q8—121no—
208Q8—121no—
209Q8—242no—
210D4—121no—
211D4—121no—
212D4—242no—
213D4—241no—
214D4—241no—
215C2×C4—42no—
216C2×C4—122no—
217C2×C4—121no—
218C2×C4—241no—
219C2×C4—241no—
220C2×C4—244no—
221C2×C4—242no—
222C2×C4C4×D1212yes—
223C2×C4#87511yescharacteristic
224C2×C4#90811yescharacteristic
225C2×C4#90811yescharacteristic
226C2×C4#90911yescharacteristic
227C2×C4#90911yescharacteristic
228C2×C4C2×(C2×D3)⋊C411yescharacteristic
229C2×C4#93012yes—
230C8—22no—
231C8—22no—
232C8—44no—
233C2×C3—21no—
234C2×C3—21no—
235C2×C3#195911yescharacteristic
236C2×C3#197612yes—
237D3—161no—
238C22—22no—
239C22—242no—
240C22—241no—
241C22#471211yescharacteristic
242C22#471511yescharacteristic
243C22#493911yescharacteristic
244C4—22no—
245C4—22no—
246C4—242no—
247C4—241no—
248C4—482no—
249C4#471411yescharacteristic
250C4#471411yescharacteristic
251C4#513311yescharacteristic
252C4#513311yescharacteristic
253C3#975811yescharacteristic, complemented, Sylow
254C2—21no—
255C2—21no—
256C2—481no—
257C2#6982111yescharacteristic
258C2#6989712yes—
subgroupG/H
centerC22#4939
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1452C2
socleC22×C3#390
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1452D3
hypocenterC3#9758
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1452

Complements

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

NH ≅ G/Ntype
2. C3×#1452C2semidirect
3. #70793C2semidirect
5. #69766C2semidirect
17. #4694C4semidirect
18. #4681C4semidirect
26. #1452D3semidirect
253. C3#9758semidirect

Maximal subgroups

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

subgroupindex
#784632
C3×#14522
#707932
#707902
#697662
#97583