Finite Group Databasebeta

← #145073

Subgroups — #145073

Order 512.

317 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 317 conjugacy classes of subgroups — too many to draw. Only its 74 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#28707C211yesmaximal, characteristic
2#13289C211yesmaximal, characteristic, complemented
3#13185C211yesmaximal, characteristic, complemented
4#11935C211yesmaximal, characteristic, complemented
5#11583C211yesmaximal, characteristic
6#8658C211yesmaximal, characteristic
7#7175C211yesmaximal, characteristic, complemented
8C2×#499C2211yescharacteristic
9C2×C8×D4C2211yescharacteristic
10#2562—21no—
11C2×#365—21no—
12#2150C2211yescharacteristic
13C4×M32C2211yescharacteristic
14#2076—21no—
15#2075—21no—
16#1989—21no—
17#1848C2211yescharacteristic
18#1807—21no—
19#1807—21no—
20C4×#106C2211yescharacteristic
21#1564C2211yescharacteristic
22#1486—21no—
23#1387—21no—
24#1387—21no—
25#1383—21no—
26#1363—22no—
27D4⋊C16—22no—
28#1301—21no—
29C23×C8—21no—
30C2×#125—21no—
31C2×C4×D4D411yescharacteristic
32C2×M32—21no—
33C2×M32—21no—
34#499—21no—
35#499—21no—
36#499—21no—
37#499—21no—
38C8×D4—21no—
39C8×D4—21no—
40C8×D4—21no—
41C8×D4—21no—
42#431—21no—
43C2×#108—21no—
44C2×#108—21no—
45C2×#107D411yescharacteristic
46C2×#107D411yescharacteristic
47C2×#106—21no—
48C2×#106D411yescharacteristic
49C2×#106D411yescharacteristic
50#423—21no—
51C2×C4⋊C8D411yescharacteristic
52C2×C22⋊C8—21no—
53C2×C4×C8C2311yescharacteristic
54#400—41no—
55#394—41no—
56#388—41no—
57#388—41no—
58#380—21no—
59#380—21no—
60#380—21no—
61#380—21no—
62C4×C4⋊C4—21no—
63#365—42no—
64C4⋊C16—21no—
65C4⋊C16D412yes—
66#346—21no—
67C4×C16—21no—
68#340—21no—
69#340—21no—
70#340—21no—
71#340—21no—
72#340—21no—
73#335—21no—
74#334—21no—
75D4⋊C8—42no—
76C22×D4SD1611yescharacteristic
77C23×C4—21no—
78C2×M16—41no—
79C22×C8—21no—
80C22×C8—41no—
81C22×C8—41no—
82C22×C8—41no—
83C22×C8—41no—
84C22×C8—41no—
85C22×C8—41no—
86C22×C8C2×D411yescharacteristic
87C22×C8C2×D411yescharacteristic
88#125—41no—
89#125—41no—
90#125—41no—
91#125—41no—
92C4×D4—21no—
93C4×D4—21no—
94C4×D4—21no—
95C4×D4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4SD1611yescharacteristic
103C2×C22⋊C4—21no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4—41no—
108C2×C42—41no—
109C2×C42C2×D411yescharacteristic
110M32—22no—
111M32—41no—
112M32—41no—
113M32—41no—
114C2×C16—22no—
115C2×C16—41no—
116#108—42no—
117#108—41no—
118#108—41no—
119#107—21no—
120#107—21no—
121#107—21no—
122#107—21no—
123#106—21no—
124#106—21no—
125#106—21no—
126#106—21no—
127#106—41no—
128#106—41no—
129C4⋊C8—21no—
130C4⋊C8—21no—
131C4⋊C8—41no—
132C4⋊C8—41no—
133C4⋊C8—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—41no—
143#95—41no—
144#95—41no—
145#95—41no—
146#95—41no—
147C24—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—42no—
151C22×C4—21no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4#12011yescharacteristic
163C22×C4C2×SD1611yescharacteristic
164C22×C4AGL(1,ℤ/8)11yescharacteristic
165M16—81no—
166M16—81no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—41no—
172C2×C8—81no—
173C2×C8—81no—
174C2×C8—81no—
175C2×C8—81no—
176C2×C8—81no—
177C2×C8—81no—
178C2×C8#12111yescharacteristic
179C2×C8#12111yescharacteristic
180C2×C8#12311yescharacteristic
181C2×C8#12411yescharacteristic
182C2×C8#13511yescharacteristic
183C2×C8#13511yescharacteristic
184C2×C8AGL(1,ℤ/8)11yescharacteristic
185C2×C8#13711yescharacteristic
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—81no—
197C4⋊C4—81no—
198C4⋊C4—81no—
199C4⋊C4—81no—
200C22⋊C4—41no—
201C22⋊C4—41no—
202C22⋊C4—82no—
203C22⋊C4—82no—
204C22⋊C4—161no—
205C22⋊C4—161no—
206C42—161no—
207C42#12111yescharacteristic
208C42#12311yescharacteristic
209C42#13511yescharacteristic
210C42AGL(1,ℤ/8)11yescharacteristic
211C16—42no—
212C16—42no—
213C23—22no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23#45011yescharacteristic
221D4—22no—
222D4—22no—
223D4—41no—
224D4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4#39411yescharacteristic
247C2×C4#44911yescharacteristic
248C2×C4#44911yescharacteristic
249C2×C4#45211yescharacteristic
250C2×C4#45411yescharacteristic
251C2×C4#45611yescharacteristic
252C2×C4#46511yescharacteristic
253C2×C4#46911yescharacteristic
254C2×C4#48111yescharacteristic
255C2×C4#48211yescharacteristic
256C8—21no—
257C8—21no—
258C8—81no—
259C8—161no—
260C8—161no—
261C8#46311yescharacteristic
262C8#46411yescharacteristic
263C8#48811yescharacteristic
264C8#48911yescharacteristic
265C22—21no—
266C22—21no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—81no—
271C22—81no—
272C22—81no—
273C22—81no—
274C22#163511yescharacteristic
275C22#165411yescharacteristic
276C22#203411yescharacteristic
277C4—21no—
278C4—21no—
279C4—21no—
280C4—21no—
281C4—81no—
282C4—161no—
283C4—161no—
284C4—161no—
285C4—161no—
286C4#162811yescharacteristic
287C4#165011yescharacteristic
288C4#203811yescharacteristic
289C4#204211yescharacteristic
290C2—21no—
291C2—21no—
292C2—82no—
293C2#1041711yescharacteristic
294C2#1339511yescharacteristic
295C2#1340011yescharacteristic
subgroupG/H
center, socleC22#2034
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC22×D4SD16
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13289C2semidirect
3. #13185C2semidirect
4. #11935C2semidirect
7. #7175C2semidirect

Maximal subgroups

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

subgroupindex
#287072
#132892
#131852
#119352
#115832
#86582
#71752