Finite Group Databasebeta

← #10696045

Subgroups — #10696045

Order 768.

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

normal subgroups only

This group has 696 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×#2397C211yesmaximal, characteristic
2#82314C212yesmaximal, complemented
3#71598C212yesmaximal, complemented
4#69441C211yesmaximal, characteristic, complemented
5#67614C211yesmaximal, characteristic, complemented
6#10015—31nomaximal, Sylow
7C2×C3×C4×D4—22no—
8C2×C3×C4×D4C2212yes—
9#5470C2211yescharacteristic
10C3×#399—22no—
11C3×#392—22no—
12C3×#376C2211yescharacteristic
13#5218—22no—
14#5019—22no—
15C4×#214—24no—
16#5008—22no—
17#4971—22no—
18#4925C2212yes—
19#4924C2211yescharacteristic
20D4⋊(C3⋊C8)—24no—
21#4523—22no—
22#3101—32no—
23#2397D311yescharacteristic
24#1995—31no—
25#1486—32no—
26#1274—31no—
27C22×C3×D4—22no—
28C22×C3×D4D412yes—
29C23×C3×C4—22no—
30C3×C4×D4—22no—
31C3×C4×D4—22no—
32C3×C4×D4—22no—
33C3×C4×D4—22no—
34C3×C4×D4—44no—
35C3×C4×D4—44no—
36C2×C3×C4⋊C4—22no—
37C2×C3×C4⋊C4D412yes—
38C2×C3×C22⋊C4—24no—
39C2×C3×C22⋊C4—22no—
40C2×C3×C42D412yes—
41C2×C3×C42C2311yescharacteristic
42C2×#214—24no—
43#926—21no—
44#924—22no—
45C2×#208—21no—
46#870—21no—
47#869D412yescomplemented
48C4×Dic6—22no—
49C3×#95—22no—
50#832—24no—
51#830—22no—
52#810—24no—
53#808—24no—
54C4⋊(C3⋊C8)—22no—
55C4⋊(C3⋊C8)—22no—
56C4×C3⋊C8D414yescomplemented
57#533—31no—
58C2×C4×D4—22no—
59C2×C4×D4C2×D312yes—
60#483—62no—
61#476—62no—
62#461—62no—
63#450—62no—
64C4×SD16—64no—
65#431—32no—
66#423—31no—
67#399—22no—
68#392—22no—
69#376C2×D311yescharacteristic
70#339—62no—
71D4⋊C8—64no—
72C24×C3—22no—
73C2×C3×D4—22no—
74C2×C3×D4—22no—
75C2×C3×D4—44no—
76C2×C3×D4—44no—
77C2×C3×D4—48no—
78C22×C3×C4—22no—
79C22×C3×C4—22no—
80C22×C3×C4—42no—
81C22×C3×C4—42no—
82C22×C3×C4—44no—
83C22×C3×C4—44no—
84C22×C3×C4C2×D412yes—
85C22×C3×C4C2×D411yescharacteristic
86C2×Dic6—22no—
87C3×C4⋊C4—22no—
88C3×C4⋊C4—22no—
89C3×C4⋊C4—44no—
90C3×C22⋊C4—48no—
91C3×C22⋊C4—42no—
92C3×C22⋊C4—42no—
93C3×C42—22no—
94C3×C42SD1614yescomplemented
95C3×C42C2×D411yescharacteristic
96C3×C42C2×D412yes—
97C3×C42#4111yescharacteristic
98#217—41no—
99#214—48no—
100C4⋊Dic3—22no—
101C4⋊Dic3—22no—
102Dic3⋊C4—42no—
103C4×Dic3—41no—
104#208—42no—
105C2×C3⋊C8—24no—
106C2×C3⋊C8—41no—
107C22×D4—22no—
108C22×D4#6712yes—
109C23×C4—22no—
110C2×SD16—64no—
111C2×M16—61no—
112C4⋊Q8—32no—
113#125—61no—
114C22⋊Q8—62no—
115C4×Q8—62no—
116C4×D4—22no—
117C4×D4—22no—
118C4×D4—22no—
119C4×D4—22no—
120C4×D4—44no—
121C4×D4—44no—
122#117—61no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4#6712yes—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—22no—
127C2×C42#6712yes—
128C2×C42C22×D311yescharacteristic
129#106—64no—
130C4⋊C8—62no—
131C4⋊C8—62no—
132#103—64no—
133#102—64no—
134C22⋊C8—62no—
135C4×C8—34no—
136#95—22no—
137C23×C3—24no—
138C23×C3—44no—
139C23×C3—42no—
140C23×C3—42no—
141C23×C3—44no—
142C23×C3#12011yescharacteristic
143C3×D4—24no—
144C3×D4—24no—
145C3×D4—42no—
146C3×D4—42no—
147C3×D4—416no—
148C2×C3×C4—24no—
149C2×C3×C4—42no—
150C2×C3×C4—44no—
151C2×C3×C4—42no—
152C2×C3×C4—44no—
153C2×C3×C4—84no—
154C2×C3×C4—82no—
155C2×C3×C4—82no—
156C2×C3×C4#12012yes—
157C2×C3×C4#12111yescharacteristic
158C2×C3×C4#12112yes—
159C2×C3×C4#12711yescharacteristic
160C2×C3×C4C2×SD1612yes—
161C2×C3×C4AGL(1,ℤ/8)12yes—
162C2×Dic3—42no—
163Dic6—44no—
164C3⋊C8—44no—
165C3⋊C8—81no—
166C24—22no—
167C2×Q8—62no—
168C2×D4—22no—
169C2×D4—22no—
170C2×D4—44no—
171C2×D4—44no—
172C2×D4—48no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—44no—
176C22×C4—44no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4C2×#6712yes—
180C22×C4C2×#6711yescharacteristic
181SD16—128no—
182M16—122no—
183C2×C8—64no—
184C2×C8—121no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—44no—
188C4⋊C4—62no—
189C4⋊C4—62no—
190C4⋊C4—122no—
191C22⋊C4—48no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—121no—
195C42—22no—
196C42—121no—
197C42#21414yescomplemented
198C42D3×D411yescharacteristic
199C42D3×D412yes—
200C42#23711yescharacteristic
201C22×C3—22no—
202C22×C3—48no—
203C22×C3—44no—
204C22×C3—84no—
205C22×C3—82no—
206C22×C3—82no—
207C22×C3#39211yescharacteristic
208C22×C3#45012yes—
209C3×C4—22no—
210C3×C4—22no—
211C3×C4—44no—
212C3×C4—82no—
213C3×C4#46012yes—
214C3×C4#49211yescharacteristic
215C3×C4#49611yescharacteristic
216Dic3—82no—
217C23—24no—
218C23—44no—
219C23—42no—
220C23—44no—
221C23—42no—
222C23#95311yescharacteristic
223Q8—124no—
224D4—24no—
225D4—24no—
226D4—42no—
227D4—42no—
228D4—416no—
229C2×C4—24no—
230C2×C4—42no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—42no—
234C2×C4—82no—
235C2×C4—84no—
236C2×C4—82no—
237C2×C4—122no—
238C2×C4#93212yes—
239C2×C4C2×#21412yes—
240C2×C4#93712yes—
241C2×C4#93811yescharacteristic
242C2×C4#93912yes—
243C2×C4#94011yescharacteristic
244C8—124no—
245C8—241no—
246C2×C3—22no—
247C2×C3—84no—
248C2×C3#201012yes—
249C2×C3#201411yescharacteristic
250C22—22no—
251C22—48no—
252C22—44no—
253C22—84no—
254C22—82no—
255C22—82no—
256C22#521812yes—
257C22#522311yescharacteristic
258C4—22no—
259C4—22no—
260C4—44no—
261C4—82no—
262C4—242no—
263C4#507411yescharacteristic
264C4#507512yes—
265C4#507611yescharacteristic
266C3#1001511yescharacteristic, complemented, Sylow
267C2—22no—
268C2—84no—
269C2#7171512yes—
270C2#7176811yescharacteristic
subgroupG/H
centerC2×C4#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#2397C2
socleC22×C3#392
hypercenter#2397D3
hypocenterC3#10015
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2397

Complements

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

NH ≅ G/Ntype
2. #82314C2semidirect
3. #71598C2semidirect
4. #69441C2semidirect
5. #67614C2semidirect
47. #869D4semidirect
56. C4×C3⋊C8D4semidirect
94. C3×C42SD16semidirect
197. C42#214semidirect
266. C3#10015semidirect

Maximal subgroups

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

subgroupindex
C3×#23972
#823142
#715982
#694412
#676142
#100153