Finite Group Databasebeta

← #10696028

Subgroups — #10696028

Order 768.

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

normal subgroups only

This group has 734 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, complemented
2#82308C212yesmaximal, complemented
3#71598C212yesmaximal, complemented
4#69440C211yesmaximal, characteristic, complemented
5#67614C211yesmaximal, characteristic, complemented
6#10012—31nomaximal, Sylow
7C2×C3×C4×D4—22no—
8C2×C3×C4×D4C2212yes—
9#5477C2211yescharacteristic
10C3×#399—22no—
11C3×#392—22no—
12C3×#376C2211yescharacteristic
13#5217—22no—
14#5015—22no—
15C4×#213—24no—
16#5007—22no—
17#4967—22no—
18C2×#805—22no—
19#4925C2212yes—
20#4924C2211yescharacteristic
21D4⋊(C3⋊C8)—24no—
22#3100—32no—
23#2397D311yescharacteristic, complemented
24#1963—31no—
25#1486—32no—
26#1274—31no—
27C22×C3×D4—22no—
28C22×C3×D4D412yes—
29C23×C3×C4—22no—
30C2×#235—21no—
31C3×C4×D4—22no—
32C3×C4×D4—22no—
33C3×C4×D4—22no—
34C3×C4×D4—22no—
35C3×C4×D4—44no—
36C3×C4×D4—44no—
37C2×C3×C4⋊C4—22no—
38C2×C3×C4⋊C4D412yes—
39C2×C3×C22⋊C4—24no—
40C2×C3×C22⋊C4—22no—
41C2×C3×C42D412yes—
42C2×C3×C42C2311yescharacteristic
43C2×#213—24no—
44#930—22no—
45C2×#208—21no—
46#875—21no—
47#874D411yescharacteristic, complemented
48C4×D12—22no—
49#869D411yescharacteristic, complemented
50C3×#95—22no—
51#832—24no—
52#830—22no—
53#809—24no—
54#807—24no—
55#805—44no—
56C4⋊(C3⋊C8)—22no—
57C4⋊(C3⋊C8)—22no—
58C4×C3⋊C8D414yescomplemented
59#532—31no—
60C2×C4×D4—22no—
61C2×C4×D4C2×D312yes—
62#480—62no—
63#474—62no—
64#459—62no—
65#447—62no—
66C4×D8—64no—
67#431—32no—
68#423—31no—
69C2×#104—62no—
70#399—22no—
71#392—22no—
72#376C2×D311yescharacteristic
73D4⋊C8—64no—
74C24×C3—22no—
75C2×C3×D4—22no—
76C2×C3×D4—22no—
77C2×C3×D4—44no—
78C2×C3×D4—44no—
79C2×C3×D4—48no—
80C22×C3×C4—22no—
81C22×C3×C4—22no—
82C22×C3×C4—42no—
83C22×C3×C4—42no—
84C22×C3×C4—44no—
85C22×C3×C4—44no—
86C22×C3×C4C2×D412yes—
87C22×C3×C4C2×D411yescharacteristic
88C2×#67—41no—
89#235—42no—
90#235—42no—
91C2×D12—22no—
92C2×D12—21no—
93C2×C4×D3—41no—
94C2×Dic6—21no—
95C3×C4⋊C4—22no—
96C3×C4⋊C4—22no—
97C3×C4⋊C4—44no—
98C3×C22⋊C4—48no—
99C3×C22⋊C4—42no—
100C3×C22⋊C4—42no—
101C3×C42—22no—
102C3×C42D814yescomplemented
103C3×C42C2×D411yescharacteristic
104C3×C42C2×D412yes—
105C3×C42#4111yescharacteristic
106#213—48no—
107(C2×D3)⋊C4—42no—
108C4⋊Dic3—22no—
109#208—42no—
110C2×C3⋊C8—24no—
111C2×C3⋊C8—41no—
112C2×#41—61no—
113C22×D4—22no—
114C22×D4#6712yes—
115C23×C4—22no—
116C2×D8—64no—
117C2×M16—61no—
118C4⋊Q8—31no—
119#127—31no—
120#124—61no—
121#121—62no—
122C4×D4—22no—
123C4×D4—22no—
124C4×D4—22no—
125C4×D4—22no—
126C4×D4—44no—
127C4×D4—44no—
128C4×D4—62no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4#6712yes—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—22no—
133C2×C42#6712yes—
134C2×C42C22×D311yescharacteristic
135#107—64no—
136C4⋊C8—62no—
137C4⋊C8—62no—
138#104—124no—
139#102—64no—
140#102—64no—
141C22⋊C8—62no—
142C4×C8—34no—
143#95—22no—
144C23×C3—24no—
145C23×C3—44no—
146C23×C3—42no—
147C23×C3—42no—
148C23×C3—44no—
149C23×C3#12011yescharacteristic
150C22×D3—41no—
151C3×D4—24no—
152C3×D4—24no—
153C3×D4—42no—
154C3×D4—42no—
155C3×D4—416no—
156C2×C3×C4—24no—
157C2×C3×C4—42no—
158C2×C3×C4—44no—
159C2×C3×C4—42no—
160C2×C3×C4—44no—
161C2×C3×C4—84no—
162C2×C3×C4—82no—
163C2×C3×C4—82no—
164C2×C3×C4#12012yes—
165C2×C3×C4#12111yescharacteristic
166C2×C3×C4#12112yes—
167C2×C3×C4#12711yescharacteristic
168C2×C3×C4C2×D812yes—
169C2×C3×C4AGL(1,ℤ/8)12yes—
170#67—82no—
171C2×Dic3—41no—
172D12—44no—
173D12—42no—
174C4×D3—81no—
175C4×D3—81no—
176Dic6—42no—
177C3⋊C8—44no—
178C3⋊C8—81no—
179C24—22no—
180#41—122no—
181#41—122no—
182C2×Q8—61no—
183C2×D4—22no—
184C2×D4—22no—
185C2×D4—44no—
186C2×D4—44no—
187C2×D4—48no—
188C2×D4—62no—
189C2×D4—61no—
190C2×D4—121no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—121no—
198C22×C4C2×#6712yes—
199C22×C4C2×#6711yescharacteristic
200D8—128no—
201M16—122no—
202C2×C8—64no—
203C2×C8—121no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—44no—
207C4⋊C4—62no—
208C22⋊C4—48no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—122no—
212C42—22no—
213C42#21314yescomplemented
214C42D3×D411yescharacteristic
215C42D3×D412yes—
216C42#23711yescharacteristic
217C22×C3—22no—
218C22×C3—48no—
219C22×C3—44no—
220C22×C3—84no—
221C22×C3—82no—
222C22×C3—82no—
223C22×C3#39211yescharacteristic
224C22×C3#44712yes—
225C2×D3—82no—
226C2×D3—81no—
227C3×C4—22no—
228C3×C4—22no—
229C3×C4—44no—
230C3×C4—82no—
231C3×C4#45912yes—
232C3×C4#49311yescharacteristic
233C3×C4#49611yescharacteristic
234Dic3—81no—
235C23—24no—
236C23—44no—
237C23—42no—
238C23—44no—
239C23—42no—
240C23—121no—
241C23#95311yescharacteristic
242Q8—122no—
243D4—24no—
244D4—24no—
245D4—42no—
246D4—42no—
247D4—416no—
248D4—124no—
249D4—122no—
250D4—242no—
251C2×C4—24no—
252C2×C4—42no—
253C2×C4—44no—
254C2×C4—44no—
255C2×C4—42no—
256C2×C4—84no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—121no—
260C2×C4—241no—
261C2×C4—241no—
262C2×C4C2×#21312yes—
263C2×C4#93212yes—
264C2×C4#93712yes—
265C2×C4#93811yescharacteristic
266C2×C4#93912yes—
267C2×C4#94011yescharacteristic
268C8—124no—
269C8—241no—
270C2×C3—22no—
271C2×C3—84no—
272C2×C3#201012yes—
273C2×C3#201111yescharacteristic
274D3—161no—
275C22—22no—
276C22—48no—
277C22—44no—
278C22—84no—
279C22—82no—
280C22—82no—
281C22—242no—
282C22—241no—
283C22#521712yes—
284C22#522311yescharacteristic
285C4—22no—
286C4—22no—
287C4—44no—
288C4—82no—
289C4—241no—
290C4#507212yes—
291C4#507311yescharacteristic
292C4#507411yescharacteristic
293C3#1001211yescharacteristic, complemented, Sylow
294C2—22no—
295C2—84no—
296C2—481no—
297C2#7171512yes—
298C2#7176711yescharacteristic
subgroupG/H
centerC2×C4#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2397C2
socleC22×C3#392
hypercenter#2397D3
hypocenterC3#10012
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
1. C3×#2397C2semidirect
2. #82308C2semidirect
3. #71598C2semidirect
4. #69440C2semidirect
5. #67614C2semidirect
23. #2397D3semidirect
47. #874D4semidirect
49. #869D4semidirect
58. C4×C3⋊C8D4semidirect
102. C3×C42D8semidirect
213. C42#213semidirect
293. C3#10012semidirect

Maximal subgroups

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

subgroupindex
C3×#23972
#823082
#715982
#694402
#676142
#100123