Finite Group Databasebeta

← #10693024

Subgroups — #10693024

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#82388C211yesmaximal, characteristic
2#82301C211yesmaximal, characteristic
3C3×#1800C211yesmaximal, characteristic
4#70500C212yesmaximal, complemented
5#67713C212yesmaximal, complemented
6#9483—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×C4×Dic6C2211yescharacteristic
9C3×#423C2212yes—
10C2×C3×#100—21no—
11#5221—21no—
12#5210—22no—
13C2×#830—22no—
14D4×C3⋊C8—24no—
15#4972—21no—
16#4949C412yes—
17#4947C412yes—
18#4936C2211yescharacteristic
19#4925C2212yes—
20C3×#343—21no—
21#4484—24no—
22#2972—31no—
23#2580—31no—
24#1800D311yescharacteristic
25#1487—32no—
26#1286—32no—
27C22×C3×D4D411yescharacteristic
28C23×C3×C4D412yes—
29C22×Dic6C2×C411yescharacteristic
30C2×C3×M16—22no—
31C3×C4×D4—22no—
32C3×C4×D4—22no—
33C2×C3×C4⋊C4D411yescharacteristic
34C2×C3×C22⋊C4C2×C412yes—
35C2×C3×C42C2311yescharacteristic
36C2×#217—22no—
37C2×#217—22no—
38C2×C4⋊Dic3—21no—
39C2×C4⋊Dic3C2×C411yescharacteristic
40C2×Dic3⋊C4—21no—
41C2×Dic3⋊C4—22no—
42C2×C4×Dic3—21no—
43C22×C3⋊C8—24no—
44C4×Dic6—22no—
45C4×Dic6—22no—
46C3×C4⋊C8—22no—
47C3×C4⋊C8—22no—
48C3×#100—42no—
49C3⋊#95—22no—
50C3⋊#95—21no—
51C3⋊#95C2×C412yes—
52#830—24no—
53#830—22no—
54C4⋊(C3⋊C8)—22no—
55C4×C3⋊C8D414yes—
56C2×C4×Q8—31no—
57C2×C4×D4C2×D311yescharacteristic
58C8×D4—64no—
59#431—32no—
60#423C2×D312yes—
61C2×#100—21no—
62C2×C22⋊C8—62no—
63#399—32no—
64#393—32no—
65#390—61no—
66#389—61no—
67#385—62no—
68#382—31no—
69#343—21no—
70Q8⋊C8—64no—
71C24×C3C22⋊C412yes—
72C2×C3×D4—22no—
73C2×C3×D4—22no—
74C2×C3×D4SD1612yes—
75C2×C3×D4Q1612yes—
76C22×C3×C4—22no—
77C22×C3×C4—22no—
78C22×C3×C4—82no—
79C22×C3×C4C22⋊C412yes—
80C22×C3×C4C22×C411yescharacteristic
81C22×C3×C4C2×D411yescharacteristic
82C22×C3×C4C2×D411yescharacteristic
83C22×Dic3—21no—
84C22×Dic3—22no—
85C22×Dic3—21no—
86C2×Dic6—22no—
87C2×Dic6—42no—
88C3×M16—42no—
89C3×M16—42no—
90C2×C3×C8—42no—
91C3×C4⋊C4—22no—
92C3×C22⋊C4—24no—
93C3×C42C2×D412yes—
94C3×C42#4112yes—
95#217—44no—
96#217—82no—
97C4⋊Dic3—22no—
98C4⋊Dic3—42no—
99Dic3⋊C4—41no—
100Dic3⋊C4—41no—
101Dic3⋊C4—82no—
102C4×Dic3—42no—
103C2×C3⋊C8—24no—
104C2×C3⋊C8—84no—
105C22×Q8—31no—
106C22×D4#6711yescharacteristic
107C23×C4D1212yes—
108C2×M16—22no—
109C22×C8—64no—
110C4×Q8—62no—
111C4×Q8—62no—
112C4×D4—22no—
113C4×D4—22no—
114C2×C4⋊C4—31no—
115C2×C4⋊C4—61no—
116C2×C4⋊C4—61no—
117C2×C4⋊C4—62no—
118C2×C4⋊C4#6711yescharacteristic
119C2×C22⋊C4—62no—
120C2×C22⋊C4—62no—
121C2×C22⋊C4C4×D312yes—
122C2×C42—61no—
123C2×C42C22×D311yescharacteristic
124C4⋊C8—22no—
125C4⋊C8—22no—
126C4⋊C8—62no—
127#100—42no—
128C22⋊C8—64no—
129C22⋊C8—62no—
130C4×C8—34no—
131#95—32no—
132#95—62no—
133#95—61no—
134C23×C3—22no—
135C23×C3—22no—
136C23×C3—42no—
137C23×C3—42no—
138C23×C3#10314yes—
139C23×C3C2×C22⋊C411yescharacteristic
140C3×D4—24no—
141C3×D4—24no—
142C2×C3×C4—22no—
143C2×C3×C4—41no—
144C2×C3×C4—41no—
145C2×C3×C4—44no—
146C2×C3×C4—42no—
147C2×C3×C4—82no—
148C2×C3×C4C4×D412yes—
149C2×C3×C4#12011yescharacteristic
150C2×C3×C4#12112yes—
151C2×C3×C4#12311yescharacteristic
152C2×C3×C4C2×SD1611yescharacteristic
153C2×C3×C4C2×Q1611yescharacteristic
154C2×C3×C4#13512yes—
155C2×Dic3—22no—
156C2×Dic3—41no—
157C2×Dic3—41no—
158C2×Dic3—41no—
159C2×Dic3—42no—
160C2×Dic3—42no—
161C2×Dic3—41no—
162C2×Dic3—41no—
163C2×Dic3—42no—
164Dic6—24no—
165Dic6—42no—
166C3×C8—82no—
167C3⋊C8—44no—
168C24(C2×D3)⋊C412yes—
169C2×Q8—62no—
170C2×Q8—122no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4#21412yes—
174C2×D4#21612yes—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—61no—
178C22×C4—62no—
179C22×C4—61no—
180C22×C4—82no—
181C22×C4(C2×D3)⋊C412yes—
182C22×C4C2×C4×D311yescharacteristic
183C22×C4C2×D1211yescharacteristic
184C22×C4C2×#6711yescharacteristic
185M16—42no—
186M16—42no—
187C2×C8—42no—
188C2×C8—64no—
189C2×C8—244no—
190C4⋊C4—22no—
191C4⋊C4—62no—
192C4⋊C4—121no—
193C4⋊C4—121no—
194C4⋊C4—122no—
195C4⋊C4—242no—
196C22⋊C4—24no—
197C22⋊C4—124no—
198C22⋊C4—242no—
199C42—122no—
200C42D3×D412yes—
201C42#23712yes—
202C22×C3—21no—
203C22×C3—21no—
204C22×C3—24no—
205C22×C3—44no—
206C22×C3—42no—
207C22×C3—42no—
208C22×C3—82no—
209C22×C3#38611yescharacteristic
210C22×C3C2×#10311yescharacteristic
211C22×C3#41611yescharacteristic
212C3×C4—22no—
213C3×C4—21no—
214C3×C4—21no—
215C3×C4—42no—
216C3×C4#46512yes—
217C3×C4#46712yes—
218Dic3—42no—
219Dic3—82no—
220Dic3—81no—
221C23—22no—
222C23—22no—
223C23—42no—
224C23—42no—
225C23#81014yes—
226C23C2×(C2×D3)⋊C411yescharacteristic
227Q8—64no—
228Q8—122no—
229D4—24no—
230D4—24no—
231C2×C4—22no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—44no—
235C2×C4—42no—
236C2×C4—62no—
237C2×C4—82no—
238C2×C4—121no—
239C2×C4—121no—
240C2×C4—122no—
241C2×C4—121no—
242C2×C4—122no—
243C2×C4—121no—
244C2×C4—121no—
245C2×C4—122no—
246C2×C4#67⋊C412yes—
247C2×C4#88211yescharacteristic
248C2×C4#88611yescharacteristic
249C2×C4C2×#21411yescharacteristic
250C2×C4#93912yes—
251C2×C4C2×#21611yescharacteristic
252C2×C4#95012yes—
253C8—82no—
254C8—124no—
255C2×C3—21no—
256C2×C3—21no—
257C2×C3—44no—
258C2×C3#189111yescharacteristic
259C2×C3#189211yescharacteristic
260C2×C3#191111yescharacteristic
261C22—21no—
262C22—21no—
263C22—24no—
264C22—44no—
265C22—42no—
266C22—42no—
267C22—82no—
268C22#495811yescharacteristic
269C22C2×#81011yescharacteristic
270C22#500211yescharacteristic
271C4—21no—
272C4—21no—
273C4—22no—
274C4—42no—
275C4—122no—
276C4—242no—
277C4—241no—
278C4#504112yes—
279C4#505112yes—
280C3#948311yescharacteristic, complemented, Sylow
281C2—21no—
282C2—21no—
283C2—44no—
284C2#6951411yescharacteristic
285C2#7007111yescharacteristic
286C2#7013611yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1800C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenter#1800D3
hypocenterC3#9483
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
4. #70500C2semidirect
5. #67713C2semidirect
280. C3#9483semidirect

Maximal subgroups

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

subgroupindex
#823882
#823012
C3×#18002
#705002
#677132
#94833