Finite Group Databasebeta

← #145145

Subgroups — #145145

Order 512.

395 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 395 conjugacy classes of subgroups — too many to draw. Only its 78 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#28231C211yesmaximal, characteristic
2#12886C211yesmaximal, characteristic, complemented
3#12818C211yesmaximal, characteristic, complemented
4#11599C211yesmaximal, characteristic, complemented
5#11587C211yesmaximal, characteristic, complemented
6#8662C211yesmaximal, characteristic, complemented
7#7278C211yesmaximal, characteristic, complemented
8C2×#469—21no—
9C2×C4×D8C2211yescharacteristic
10C2×#436C2211yescharacteristic
11#2537—21no—
12#2150C2211yescharacteristic
13C2×#361—21no—
14C4×M32C2211yescharacteristic
15#2034—22no—
16#1848C2211yescharacteristic
17#1844C2211yescharacteristic
18#1809—21no—
19#1802—21no—
20#1564C2211yescharacteristic
21#1486—21no—
22#1387—21no—
23#1383—22no—
24#1333—22no—
25#1330—22no—
26#1301—21no—
27C22×D8D411yescharacteristic
28C22×M16—21no—
29C2×#121—22no—
30C2×#121—41no—
31C2×C4×D4—21no—
32C2×C4×D4D411yescharacteristic
33C2×M32—21no—
34C2×M32—21no—
35#469—42no—
36#469—42no—
37C4×D8—21no—
38C4×D8—21no—
39C4×D8—21no—
40C4×D8—21no—
41#436—21no—
42#436—21no—
43#436—21no—
44#436—21no—
45#431—21no—
46C2×#108—21no—
47C2×#108—21no—
48C2×#107D411yescharacteristic
49C2×#106—21no—
50C2×#106D412yes—
51#423—21no—
52C2×C4⋊C8D411yescharacteristic
53C2×#102—22no—
54C2×#102—21no—
55C2×C22⋊C8—21no—
56C2×C4×C8C2311yescharacteristic
57#394—42no—
58#384—21no—
59#380—22no—
60#380—41no—
61C4⋊C16—21no—
62C4⋊C16—21no—
63#361—42no—
64#346D412yescomplemented
65C4×C16—21no—
66#340—22no—
67#335—21no—
68#334—21no—
69D4⋊C8—42no—
70C22×D4—21no—
71C22×D4—42no—
72C22×D4SD1611yescharacteristic
73C23×C4—21no—
74C23×C4—41no—
75C2×D8—21no—
76C2×D8—21no—
77C2×D8—21no—
78C2×D8—21no—
79C2×D8—21no—
80C2×D8D812yes—
81C2×M16—41no—
82C2×M16—41no—
83C2×M16—41no—
84C2×M16—41no—
85C2×M16—41no—
86C2×M16—41no—
87C2×M16—41no—
88C22×C8—21no—
89C22×C8C2×D411yescharacteristic
90C22×C8C2×D411yescharacteristic
91#121—44no—
92#121—82no—
93#121—84no—
94C4×D4—21no—
95C4×D4—21no—
96C4×D4—21no—
97C4×D4—21no—
98C4×D4—41no—
99C4×D4—41no—
100C4×D4—41no—
101C4×D4—41no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4SD1611yescharacteristic
107C2×C22⋊C4—21no—
108C2×C22⋊C4—42no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—41no—
111C2×C42C2×D411yescharacteristic
112M32—41no—
113M32—41no—
114M32—41no—
115M32—41no—
116C2×C16—22no—
117C2×C16—41no—
118#108—42no—
119#108—41no—
120#108—41no—
121#107—21no—
122#107—21no—
123#106—41no—
124#106—41no—
125#106—42no—
126C4⋊C8—21no—
127C4⋊C8—21no—
128C4⋊C8—41no—
129C4⋊C8—41no—
130C4⋊C8—41no—
131#102—44no—
132#102—41no—
133#102—41no—
134C22⋊C8—41no—
135C22⋊C8—41no—
136C22⋊C8—41no—
137C4×C8—41no—
138C4×C8C2×D411yescharacteristic
139C4×C8#4111yescharacteristic
140C4×C8#4111yescharacteristic
141C4×C8#4111yescharacteristic
142#95—42no—
143C24—21no—
144C24—41no—
145C2×D4—22no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—42no—
149C2×D4—41no—
150C2×D4—41no—
151C2×D4—41no—
152C2×D4—41no—
153C2×D4—41no—
154C2×D4—84no—
155C2×D4—84no—
156C2×D4—162no—
157C22×C4—21no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—42no—
166C22×C4—82no—
167C22×C4—82no—
168C22×C4—81no—
169C22×C4—81no—
170C22×C4#12011yescharacteristic
171C22×C4C2×SD1611yescharacteristic
172C22×C4AGL(1,ℤ/8)11yescharacteristic
173D8—22no—
174D8—22no—
175D8—41no—
176D8—41no—
177M16—81no—
178M16—81no—
179M16—81no—
180M16—81no—
181M16—81no—
182M16—81no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—81no—
189C2×C8—81no—
190C2×C8#12111yescharacteristic
191C2×C8#12111yescharacteristic
192C2×C8#12311yescharacteristic
193C2×C8#12411yescharacteristic
194C2×C8C2×D811yescharacteristic
195C2×C8#13511yescharacteristic
196C2×C8AGL(1,ℤ/8)11yescharacteristic
197C2×C8#13711yescharacteristic
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—41no—
201C4⋊C4—42no—
202C4⋊C4—41no—
203C4⋊C4—42no—
204C4⋊C4—82no—
205C4⋊C4—161no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C22⋊C4—84no—
209C22⋊C4—84no—
210C22⋊C4—81no—
211C22⋊C4—81no—
212C42#12111yescharacteristic
213C42#12311yescharacteristic
214C42#13511yescharacteristic
215C42AGL(1,ℤ/8)11yescharacteristic
216C16—42no—
217C16—81no—
218C23—42no—
219C23—42no—
220C23—41no—
221C23—42no—
222C23—41no—
223C23—41no—
224C23—81no—
225C23—81no—
226C23—81no—
227C23—81no—
228C23—81no—
229C23—81no—
230C23#45011yescharacteristic
231D4—42no—
232D4—42no—
233D4—42no—
234D4—81no—
235D4—81no—
236D4—81no—
237D4—168no—
238C2×C4—22no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—81no—
251C2×C4—82no—
252C2×C4—81no—
253C2×C4—162no—
254C2×C4—162no—
255C2×C4#39411yescharacteristic
256C2×C4#44711yescharacteristic
257C2×C4#44911yescharacteristic
258C2×C4#45211yescharacteristic
259C2×C4#45411yescharacteristic
260C2×C4#45511yescharacteristic
261C2×C4#46511yescharacteristic
262C2×C4#46911yescharacteristic
263C2×C4#48111yescharacteristic
264C2×C4#48211yescharacteristic
265C8—21no—
266C8—21no—
267C8—81no—
268C8—161no—
269C8—161no—
270C8#46611yescharacteristic
271C8#46811yescharacteristic
272C8#48011yescharacteristic
273C8#48211yescharacteristic
274C22—21no—
275C22—21no—
276C22—82no—
277C22—82no—
278C22—82no—
279C22—82no—
280C22—81no—
281C22—81no—
282C22—81no—
283C22—82no—
284C22—161no—
285C22—161no—
286C22—161no—
287C22—161no—
288C22#163511yescharacteristic
289C22#163911yescharacteristic
290C22#203411yescharacteristic
291C4—21no—
292C4—21no—
293C4—21no—
294C4—81no—
295C4—162no—
296C4—161no—
297C4#162511yescharacteristic
298C4#164611yescharacteristic
299C4#203311yescharacteristic
300C4#204011yescharacteristic
301C4#223812yes—
302C2—21no—
303C2—21no—
304C2—161no—
305C2—162no—
306C2#1043111yescharacteristic
307C2#1339811yescharacteristic
308C2#1340011yescharacteristic
subgroupG/H
center, socleC22#2034
commutator, Frattini, cosocleC2×C4×C8C23
prime radical#28231C2
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#436 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12886C2semidirect
3. #12818C2semidirect
4. #11599C2semidirect
5. #11587C2semidirect
6. #8662C2semidirect
7. #7278C2semidirect
64. #346D4semidirect

Maximal subgroups

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

subgroupindex
#282312
#128862
#128182
#115992
#115872
#86622
#72782