Finite Group Databasebeta

← #10692786

Subgroups — #10692786

Order 768.

504 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 107 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×#2979C211yesmaximal, characteristic
2#83288C211yesmaximal, characteristic
3#71810C211yesmaximal, characteristic
4#70780C212yesmaximal, complemented
5#67743C212yesmaximal, complemented
6#9435—31nomaximal, Sylow
7C3×#532C2211yescharacteristic
8C2×#951—22no—
9C2×#869C2211yescharacteristic
10C3×#436—22no—
11C3×C8×D4—22no—
12C3×#431C2212yes—
13C3×#408—22no—
14#5092—21no—
15#5076—21no—
16#5060C414yes—
17#4924C2212yes—
18C4×#208C2211yescharacteristic
19#4546—21no—
20#4544—21no—
21#4482—22no—
22#4482—22no—
23#3152—31no—
24#2979D311yescharacteristic
25#1560—31no—
26#1549—32no—
27#1299—32no—
28C2×C3×#41D412yes—
29C22×Dic6—21no—
30C2×C3×M16—22no—
31C22×C3×C8—22no—
32C3×C4⋊Q8D411yescharacteristic
33C3×#127D411yescharacteristic
34C3×#124C2×C412yes—
35C3×#121—22no—
36C3×C4×D4—22no—
37C2×C3×C42C2311yescharacteristic
38#951—44no—
39#951—44no—
40C2×#216—24no—
41C2×#214—24no—
42C2×C4⋊Dic3—22no—
43C2×#208—21no—
44C2×#208—22no—
45#869—22no—
46#869—21no—
47#869C2×C412yes—
48C3×C4⋊C8D412yes—
49C3×C4⋊C8D412yes—
50C3×C22⋊C8—24no—
51C3×C22⋊C8—22no—
52C3×C4×C8—22no—
53C4⋊(C3⋊C8)—22no—
54C4⋊(C3⋊C8)—22no—
55#803C2×C412yes—
56C4×C3⋊C8—21no—
57C2×#137—62no—
58#532C2×D311yescharacteristic
59C2×C4⋊Q8—31no—
60#497—34no—
61#494—61no—
62#492—61no—
63#436—22no—
64C8×D4—22no—
65#431C2×D312yes—
66#423—32no—
67#408—22no—
68C4×M16—31no—
69#333—61no—
70#332—62no—
71#332—62no—
72#331—61no—
73C3×#41—42no—
74C3×#41—42no—
75C2×C3×Q8C22⋊C412yes—
76C2×C3×D4—22no—
77C2×C3×D4—22no—
78C2×C3×D4C22⋊C412yes—
79C22×C3×C4—22no—
80C22×C3×C4C2×D411yescharacteristic
81C22×C3×C4#4112yes—
82C22×Dic3—41no—
83C2×Dic6—22no—
84C2×Dic6—42no—
85C2×Dic6—42no—
86C2×Dic6—41no—
87C3×M16—82no—
88C2×C3×C8—22no—
89C2×C3×C8—22no—
90C2×C3×C8—82no—
91C3×C4⋊C4—22no—
92C3×C22⋊C4—24no—
93C3×C42C22×C411yescharacteristic
94C3×C42C2×D411yescharacteristic
95C3×C42C2×D412yes—
96#216—48no—
97#214—48no—
98C4⋊Dic3—24no—
99C4⋊Dic3—42no—
100#208—22no—
101#208—22no—
102#208—82no—
103C2×C3⋊C8—22no—
104C2×C3⋊C8—42no—
105C2×#41#6712yes—
106C22×Q8—61no—
107#137—124no—
108#137—124no—
109C2×Q16—64no—
110C2×SD16—64no—
111C2×M16—22no—
112C2×M16—61no—
113C2×M16—62no—
114C22×C8—22no—
115C4⋊Q8—32no—
116C4⋊Q8—61no—
117C4⋊Q8—62no—
118C4⋊Q8D1211yescharacteristic
119#127D1211yescharacteristic
120#124C4×D312yes—
121#121—22no—
122C4×D4—22no—
123C2×C4⋊C4—62no—
124C2×C42C22×D311yescharacteristic
125C4⋊C8—62no—
126C4⋊C8—62no—
127C4⋊C8#6712yes—
128C4⋊C8#6712yes—
129C22⋊C8—24no—
130C22⋊C8—22no—
131#97—32no—
132C4×C8—22no—
133C4×C8—61no—
134C23×C3—22no—
135C23×C3#12311yescharacteristic
136C3×Q8—42no—
137C3×D4—42no—
138C3×D4—42no—
139C3×D4—42no—
140C3×D4—82no—
141C2×C3×C4—21no—
142C2×C3×C4—21no—
143C2×C3×C4—22no—
144C2×C3×C4—22no—
145C2×C3×C4—22no—
146C2×C3×C4—42no—
147C2×C3×C4—42no—
148C2×C3×C4C2×C22⋊C411yescharacteristic
149C2×C3×C4C4×D412yes—
150C2×C3×C4#12011yescharacteristic
151C2×C3×C4#12112yes—
152C2×Dic3—42no—
153C2×Dic3—82no—
154Dic6—44no—
155Dic6—81no—
156Dic6—81no—
157C3×C8—42no—
158C3×C8—42no—
159C3⋊C8—24no—
160C3⋊C8—82no—
161#41—42no—
162#41—42no—
163C2×Q8—62no—
164C2×Q8—122no—
165C2×Q8—122no—
166C2×Q8—121no—
167C2×Q8(C2×D3)⋊C412yes—
168C2×D4—22no—
169C2×D4—22no—
170C2×D4(C2×D3)⋊C412yes—
171C22×C4—22no—
172C22×C4—121no—
173C22×C4#23512yes—
174C22×C4C2×#6711yescharacteristic
175Q16—128no—
176SD16—128no—
177M16—62no—
178M16—62no—
179M16—82no—
180M16—242no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—62no—
184C2×C8—82no—
185C2×C8—122no—
186C4⋊C4—22no—
187C4⋊C4—64no—
188C4⋊C4—122no—
189C22⋊C4—24no—
190C42C2×C4×D311yescharacteristic
191C42C2×D1211yescharacteristic
192C42C2×#6712yes—
193C22×C3—21no—
194C22×C3—22no—
195C22×C3—42no—
196C22×C3—44no—
197C22×C3#38611yescharacteristic
198C3×C4—21no—
199C3×C4—22no—
200C3×C4—42no—
201C3×C4#45311yescharacteristic
202C3×C4#45311yescharacteristic
203C3×C4#47112yes—
204C3×C4#47312yes—
205Dic3—82no—
206C23—22no—
207C23#92911yescharacteristic
208Q8—42no—
209Q8—124no—
210Q8—241no—
211Q8—241no—
212D4—42no—
213D4—42no—
214D4—42no—
215D4—82no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—122no—
224C2×C4—242no—
225C2×C4C2×(C2×D3)⋊C411yescharacteristic
226C2×C4C4×#6712yes—
227C2×C4#93012yes—
228C2×C4#95311yescharacteristic
229C8—42no—
230C8—42no—
231C8—64no—
232C8—242no—
233C2×C3—41no—
234C2×C3—82no—
235C2×C3#188711yescharacteristic
236C2×C3#190812yes—
237C22—22no—
238C22—21no—
239C22—44no—
240C22—42no—
241C22#521311yescharacteristic
242C4—22no—
243C4—21no—
244C4—42no—
245C4—242no—
246C4#514112yes—
247C4#514312yes—
248C4#524111yescharacteristic
249C4#524111yescharacteristic
250C3#943511yescharacteristic, complemented, Sylow
251C2—41no—
252C2—82no—
253C2#7165011yescharacteristic
254C2#7201912yes—
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2979C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#2979D3
hypocenterC3#9435
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2979

Complements

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

NH ≅ G/Ntype
4. #70780C2semidirect
5. #67743C2semidirect
250. C3#9435semidirect

Maximal subgroups

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

subgroupindex
C3×#29792
#832882
#718102
#707802
#677432
#94353