Finite Group Databasebeta

← #10692610

Subgroups — #10692610

Order 768.

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

normal subgroups only

This group has 706 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#82378C211yesmaximal, characteristic
2#82304C211yesmaximal, characteristic, complemented
3C3×#1800C211yesmaximal, characteristic, complemented
4#70502C212yesmaximal, complemented
5#67715C212yesmaximal, complemented
6#9405—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#940—21no—
9C2×#930—22no—
10C2×#874C2211yescharacteristic
11C3×#423C2212yes—
12C2×C3×#100—21no—
13#5207—22no—
14#5012—24no—
15#4988—21no—
16#4960C414yes—
17C4×C4⋊Dic3C2211yescharacteristic
18#4925C2212yes—
19C3×#343—21no—
20#4485—24no—
21#2975—31no—
22#2541—31no—
23#1800D311yescharacteristic, complemented
24#1545—32no—
25#1303—32no—
26C22×C3×D4D411yescharacteristic
27C23×C3×C4D412yes—
28C22×#67—24no—
29C22×D12—21no—
30C22×D12—21no—
31C22×D12C2×C412yes—
32C2×C3×M16—22no—
33C3×C4×D4—22no—
34C3×C4×D4—22no—
35C2×C3×C4⋊C4D411yescharacteristic
36C2×C3×C22⋊C4C2×C412yes—
37C2×C3×C42C2311yescharacteristic
38#940—44no—
39#930—48no—
40C2×(C2×D3)⋊C4—24no—
41C2×C4⋊Dic3—21no—
42C2×C4×Dic3—21no—
43C2×#208—24no—
44#874—22no—
45#874—22no—
46C3×C4⋊C8—22no—
47C3×C4⋊C8—22no—
48C3×#100—42no—
49C3⋊#95C2×C412yes—
50#830—24no—
51#830—22no—
52C4⋊(C3⋊C8)D414yescomplemented
53C4×C3⋊C8—22no—
54C2×#127—31no—
55C2×#127—61no—
56C2×#121—62no—
57C2×C4×D4C2×D311yescharacteristic
58#436—64no—
59#431—32no—
60#423C2×D312yes—
61C2×#100—21no—
62#407—62no—
63#392—34no—
64#390—61no—
65C4×C4⋊C4—31no—
66#343—21no—
67#331—64no—
68C24×C3C22⋊C412yes—
69C23×D3—22no—
70C2×C3×D4—22no—
71C2×C3×D4—21no—
72C2×C3×D4—22no—
73C2×C3×D4—21no—
74C22×C3×C4—22no—
75C22×C3×C4—22no—
76C22×C3×C4—44no—
77C22×C3×C4C22⋊C412yes—
78C22×C3×C4C22×C411yescharacteristic
79C22×C3×C4C2×D411yescharacteristic
80C22×C3×C4C2×D411yescharacteristic
81C2×#67—48no—
82C2×#67—48no—
83C2×#67—84no—
84C22×Dic3—22no—
85C2×D12—24no—
86C2×D12—44no—
87C2×D12—41no—
88C2×D12—42no—
89C2×D12—41no—
90C2×D12—42no—
91C2×D12—41no—
92C2×D12—41no—
93C2×D12—82no—
94C3×M16—42no—
95C3×M16—42no—
96C2×C3×C8—42no—
97C3×C4⋊C4—22no—
98C3×C22⋊C4—24no—
99C3×C42C2×D412yes—
100C3×C42#4112yes—
101(C2×D3)⋊C4—48no—
102C4⋊Dic3—24no—
103C4×Dic3—42no—
104#208—84no—
105C2×C3⋊C8—24no—
106C22×D4—32no—
107C22×D4—64no—
108C22×D4—61no—
109C22×D4—61no—
110C22×D4#6711yescharacteristic
111C23×C4D1212yes—
112C2×M16—22no—
113C2×M16—64no—
114#127—62no—
115#127—62no—
116#127—124no—
117#121—128no—
118C4×D4—22no—
119C4×D4—22no—
120C2×C4⋊C4—61no—
121C2×C4⋊C4#6711yescharacteristic
122C2×C22⋊C4—64no—
123C2×C22⋊C4C4×D312yes—
124C2×C42—61no—
125C2×C42C22×D311yescharacteristic
126C4⋊C8—22no—
127C4⋊C8—22no—
128C4⋊C8—34no—
129#100—42no—
130C22⋊C8—64no—
131C22⋊C8—62no—
132C4×C8—62no—
133#95—32no—
134C23×C3—22no—
135C23×C3—22no—
136C23×C3—22no—
137C23×C3—42no—
138C23×C3—42no—
139C23×C3C2×C22⋊C411yescharacteristic
140C22×D3—44no—
141C22×D3—44no—
142C22×D3—42no—
143C22×D3—42no—
144C22×D3—42no—
145C3×D4—42no—
146C3×D4—42no—
147C2×C3×C4—22no—
148C2×C3×C4—22no—
149C2×C3×C4—41no—
150C2×C3×C4—44no—
151C2×C3×C4—42no—
152C2×C3×C4—82no—
153C2×C3×C4C4×D412yes—
154C2×C3×C4#12011yescharacteristic
155C2×C3×C4#12112yes—
156C2×C3×C4#12311yescharacteristic
157C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
158C2×C3×C4AGL(1,ℤ/8)12yes—
159C2×C3×C4#13711yescharacteristic
160#67—816no—
161C2×Dic3—24no—
162C2×Dic3—42no—
163C2×Dic3—42no—
164D12—84no—
165D12—82no—
166D12—82no—
167D12—82no—
168D12—82no—
169C3×C8—82no—
170C3⋊C8—44no—
171C24—62no—
172C24(C2×D3)⋊C412yes—
173C2×D4—21no—
174C2×D4—22no—
175C2×D4—21no—
176C2×D4—22no—
177C2×D4—64no—
178C2×D4—124no—
179C2×D4—128no—
180C2×D4—128no—
181C2×D4—121no—
182C2×D4—121no—
183C2×D4—121no—
184C2×D4—122no—
185C2×D4—121no—
186C2×D4—122no—
187C2×D4—242no—
188C2×D4—244no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—44no—
192C22×C4—62no—
193C22×C4(C2×D3)⋊C412yes—
194C22×C4C2×C4×D311yescharacteristic
195C22×C4C2×D1211yescharacteristic
196C22×C4C2×#6711yescharacteristic
197M16—42no—
198M16—42no—
199M16—244no—
200C2×C8—42no—
201C2×C8—64no—
202C4⋊C4—22no—
203C4⋊C4—64no—
204C22⋊C4—24no—
205C22⋊C4—128no—
206C42—122no—
207C42D3×D412yes—
208C42#23712yes—
209C22×C3—21no—
210C22×C3—21no—
211C22×C3—44no—
212C22×C3—42no—
213C22×C3—42no—
214C22×C3—42no—
215C22×C3—82no—
216C22×C3#38611yescharacteristic
217C22×C3#41711yescharacteristic
218C22×C3#41811yescharacteristic
219C2×D3—84no—
220C2×D3—84no—
221C2×D3—82no—
222C2×D3—82no—
223C2×D3—82no—
224C3×C4—21no—
225C3×C4—22no—
226C3×C4—42no—
227C3×C4#45312yes—
228C3×C4#46812yes—
229C3×C4#46912yes—
230Dic3—44no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—42no—
235C23—42no—
236C23—124no—
237C23—124no—
238C23—122no—
239C23—122no—
240C23—122no—
241C23C2×(C2×D3)⋊C411yescharacteristic
242D4—42no—
243D4—42no—
244D4—244no—
245D4—242no—
246D4—2416no—
247D4—242no—
248D4—242no—
249D4—242no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—44no—
254C2×C4—42no—
255C2×C4—64no—
256C2×C4—82no—
257C2×C4—122no—
258C2×C4—122no—
259C2×C4#67⋊C412yes—
260C2×C4#88211yescharacteristic
261C2×C4#88611yescharacteristic
262C2×C4#93211yescharacteristic
263C2×C4#93912yes—
264C2×C4#94211yescharacteristic
265C2×C4#94912yes—
266C8—82no—
267C8—124no—
268C2×C3—21no—
269C2×C3—21no—
270C2×C3—82no—
271C2×C3#188711yescharacteristic
272C2×C3#189511yescharacteristic
273C2×C3#191411yescharacteristic
274D3—162no—
275C22—21no—
276C22—21no—
277C22—44no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—82no—
282C22—244no—
283C22—244no—
284C22—242no—
285C22—242no—
286C22—242no—
287C22#495811yescharacteristic
288C22#496611yescharacteristic
289C22#500111yescharacteristic
290C4—21no—
291C4—22no—
292C4—42no—
293C4—124no—
294C4#474712yes—
295C4#503912yes—
296C4#504912yes—
297C3#940511yescharacteristic, complemented, Sylow
298C2—21no—
299C2—21no—
300C2—82no—
301C2—482no—
302C2#6951711yescharacteristic
303C2#7007011yescharacteristic
304C2#7013711yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1800C2
socleC22×C3#386
prime radical#82378C2
cosocleC2×C3×C42C23
hypercenter#1800D3
hypocenterC3#9405
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
2. #82304C2semidirect
3. C3×#1800C2semidirect
4. #70502C2semidirect
5. #67715C2semidirect
23. #1800D3semidirect
52. C4⋊(C3⋊C8)D4semidirect
297. C3#9405semidirect

Maximal subgroups

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

subgroupindex
#823782
#823042
C3×#18002
#705022
#677152
#94053