Finite Group Databasebeta

← #508514

Subgroups — #508514

Order 512.

500 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 134 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#33132C211yesmaximal, characteristic
2#20807C211yesmaximal, characteristic, complemented
3#13541C212yesmaximal, complemented
4#13535C212yesmaximal, complemented
5#13510C212yesmaximal, complemented
6#13487C212yesmaximal, complemented
7#13004C212yesmaximal
8#12928C212yesmaximal, complemented
9#12847C211yesmaximal, characteristic, complemented
10C2×#466C2211yescharacteristic
11C2×#430—21no—
12C22×#108C2212yes—
13#2905—21no—
14C2×#405C2211yescharacteristic
15#2624—21no—
16C2×#365C2214yes—
17C2×#361C2214yes—
18#2112C2212yes—
19#1972C2211yescharacteristic
20#1972C2211yescharacteristic
21#1944C2211yescharacteristic
22#1944C2211yescharacteristic
23#1934—21no—
24#1934—21no—
25#1895—21no—
26#1849C2212yes—
27#1849C2212yes—
28#1848—21no—
29#1848—21no—
30#1807C2211yescharacteristic
31#1387C2214yes—
32#1383C2214yes—
33#1377C2212yes—
34#1377C2212yes—
35C22×D8C2311yescharacteristic
36C22×M16—22no—
37C22×M16D411yescharacteristic
38C23×C8C2311yescharacteristic
39C2×#121—21no—
40C2×#117D411yescharacteristic
41C2×M32C2314yes—
42#466—24no—
43#430—44no—
44C2×#108—22no—
45C2×#108—22no—
46C2×#108—22no—
47C2×#108—44no—
48C2×#108C2314yes—
49C2×#107C2311yescharacteristic
50C2×#106C2311yescharacteristic
51C2×#106C2311yescharacteristic
52#424—44no—
53#423—22no—
54#423—22no—
55C2×#102—21no—
56C2×#102—21no—
57C2×#102—21no—
58#405—24no—
59C2×#97C2311yescharacteristic
60C2×#97C2311yescharacteristic
61C2×C4×C8D411yescharacteristic
62C2×C4×C8D411yescharacteristic
63#399—41no—
64#397—41no—
65#394—41no—
66#394—41no—
67#390—21no—
68#390—21no—
69#389—21no—
70#389—21no—
71#380—21no—
72#365—44no—
73#361—44no—
74C22⋊C16—24no—
75#340—21no—
76#340—21no—
77#340—21no—
78C22×D4—21no—
79C22×D4—41no—
80C23×C4C2×D411yescharacteristic
81C2×D8—22no—
82C2×D8—22no—
83C2×D8—22no—
84C2×M16—22no—
85C2×M16—22no—
86C2×M16—22no—
87C2×M16—24no—
88C2×M16—42no—
89C2×M16—44no—
90C2×M16—44no—
91C22×C8—22no—
92C22×C8—22no—
93C22×C8—42no—
94C22×C8C2×D411yescharacteristic
95C22×C8C2×D411yescharacteristic
96C22×C8C2×D411yescharacteristic
97C22×C8#4114yes—
98C22×C8C2411yescharacteristic
99#121—42no—
100#121—42no—
101#117—22no—
102#117—22no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4SD1611yescharacteristic
109C2×C4⋊C4SD1611yescharacteristic
110C2×C22⋊C4—41no—
111C2×C22⋊C4—41no—
112C2×C22⋊C4—41no—
113C2×C22⋊C4—41no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4SD1611yescharacteristic
117C2×C22⋊C4SD1611yescharacteristic
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D411yescharacteristic
120M32—24no—
121M32—24no—
122C2×C16—24no—
123#108—24no—
124#108—24no—
125#108—44no—
126#108—44no—
127#108—44no—
128#107—22no—
129#106—22no—
130#106—22no—
131C4⋊C8—44no—
132C4⋊C8—44no—
133#102—42no—
134#102—42no—
135#102—42no—
136#97—22no—
137#97—22no—
138C4×C8—22no—
139C4×C8—22no—
140#95—41no—
141#95—41no—
142#95—41no—
143C24—41no—
144C24C2×SD1611yescharacteristic
145C2×D4—41no—
146C2×D4—41no—
147C2×D4—41no—
148C2×D4—41no—
149C2×D4—41no—
150C2×D4—41no—
151C2×D4—82no—
152C2×D4—161no—
153C2×D4—161no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4#12312yes—
162C22×C4C2×SD1611yescharacteristic
163C22×C4C2×SD1611yescharacteristic
164C22×C4C2×SD1611yescharacteristic
165C22×C4C2×SD1611yescharacteristic
166C22×C4C2×SD1611yescharacteristic
167C22×C4AGL(1,ℤ/8)12yes—
168C22×C4C22×D411yescharacteristic
169D8—44no—
170M16—28no—
171M16—44no—
172M16—44no—
173M16—84no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—28no—
181C2×C8—44no—
182C2×C8—84no—
183C2×C8#12312yes—
184C2×C8AGL(1,ℤ/8)12yes—
185C2×C8#13712yes—
186C2×C8C2×#4112yes—
187C2×C8H(5,𝔽2)12yes—
188C2×C82−1+412yes—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—161no—
198C4⋊C4—161no—
199C22⋊C4—22no—
200C22⋊C4—22no—
201C22⋊C4—82no—
202C22⋊C4—161no—
203C22⋊C4—161no—
204C22⋊C4—161no—
205C22⋊C4—161no—
206C22⋊C4—161no—
207C42—21no—
208C42—21no—
209C42—21no—
210C42—21no—
211C16—44no—
212C23—21no—
213C23—21no—
214C23—41no—
215C23—41no—
216C23—81no—
217C23—81no—
218C23—81no—
219C23—81no—
220C23—81no—
221C23—81no—
222C23—81no—
223C23#48112yes—
224C23C22×SD1611yescharacteristic
225D4—81no—
226D4—81no—
227D4—81no—
228D4—81no—
229D4—164no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4#48112yes—
254C2×C4C2×#12311yescharacteristic
255C2×C4#53511yescharacteristic
256C2×C4#53611yescharacteristic
257C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
258C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
259C2×C4C2×#13711yescharacteristic
260C8—22no—
261C8—22no—
262C8—22no—
263C8—84no—
264C8#53914yes—
265C22—21no—
266C22—21no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—81no—
271C22—81no—
272C22—161no—
273C22—161no—
274C22—161no—
275C22—161no—
276C22—161no—
277C22—161no—
278C22—161no—
279C22C2×#48111yescharacteristic
280C22#320611yescharacteristic
281C22#322711yescharacteristic
282C4—22no—
283C4—41no—
284C4—41no—
285C4—41no—
286C4—41no—
287C4—161no—
288C4—161no—
289C4—161no—
290C4#262311yescharacteristic
291C4#309211yescharacteristic
292C4#321711yescharacteristic
293C4#323811yescharacteristic
294C2—21no—
295C2—21no—
296C2—42no—
297C2—321no—
298C2#2990711yescharacteristic
299C2#3365711yescharacteristic
300C2#3365911yescharacteristic
subgroupG/H
center, socleC22C2×#481
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20807C2semidirect
3. #13541C2semidirect
4. #13535C2semidirect
5. #13510C2semidirect
6. #13487C2semidirect
8. #12928C2semidirect
9. #12847C2semidirect

Maximal subgroups

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

subgroupindex
#331322
#208072
#135412
#135352
#135102
#134872
#130042
#129282
#128472