Finite Group Databasebeta

← #145146

Subgroups — #145146

Order 512.

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

normal subgroups only

This group has 353 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#28232C211yesmaximal, characteristic
2#12886C211yesmaximal, characteristic, complemented
3#12819C211yesmaximal, characteristic, complemented
4#11600C211yesmaximal, characteristic, complemented
5#11587C211yesmaximal, characteristic
6#8662C211yesmaximal, characteristic
7#7278C211yesmaximal, characteristic, complemented
8C2×#470—21no—
9C2×C4×Q16C2211yescharacteristic
10C2×#436C2211yescharacteristic
11#2538—21no—
12#2150C2211yescharacteristic
13C2×#362—21no—
14C4×M32C2211yescharacteristic
15#2035—22no—
16#1848C2211yescharacteristic
17#1844C2211yescharacteristic
18#1809—21no—
19#1803—21no—
20#1564C2211yescharacteristic
21#1486—21no—
22#1387—21no—
23#1383—22no—
24#1334—22no—
25#1330—22no—
26#1301—21no—
27C22×Q16D411yescharacteristic
28C22×M16—21no—
29C2×C22⋊Q8—22no—
30C2×C4×Q8—21no—
31C2×C4×D4D411yescharacteristic
32C2×M32—21no—
33C2×M32—21no—
34#470—42no—
35#470—42no—
36C4×Q16—21no—
37C4×Q16—21no—
38C4×Q16—21no—
39C4×Q16—21no—
40#436—21no—
41#436—21no—
42#436—21no—
43#436—21no—
44#431—21no—
45C2×#108—21no—
46C2×#108—21no—
47C2×#107D411yescharacteristic
48C2×#106—21no—
49C2×#106D412yes—
50#423—21no—
51C2×C4⋊C8D411yescharacteristic
52C2×#103—22no—
53C2×#103—21no—
54C2×C22⋊C8—21no—
55C2×C4×C8C2311yescharacteristic
56#396—42no—
57#390—41no—
58#389—41no—
59#384—21no—
60#380—22no—
61C4⋊C16—21no—
62C4⋊C16—21no—
63#362—42no—
64#346D412yes—
65C4×C16—21no—
66#340—22no—
67#335—21no—
68#334—21no—
69D4⋊C8—42no—
70C22×Q8—21no—
71C22×D4SD1611yescharacteristic
72C23×C4—21no—
73C2×Q16—21no—
74C2×Q16—21no—
75C2×Q16—21no—
76C2×Q16—21no—
77C2×Q16—21no—
78C2×Q16D812yes—
79C2×M16—41no—
80C2×M16—41no—
81C2×M16—41no—
82C2×M16—41no—
83C2×M16—41no—
84C2×M16—41no—
85C2×M16—41no—
86C22×C8—21no—
87C22×C8C2×D411yescharacteristic
88C22×C8C2×D411yescharacteristic
89C22⋊Q8—44no—
90C22⋊Q8—82no—
91C4×Q8—41no—
92C4×Q8—41no—
93C4×Q8—41no—
94C4×Q8—41no—
95C4×D4—21no—
96C4×D4—21no—
97C4×D4—21no—
98C4×D4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4SD1611yescharacteristic
106C2×C22⋊C4—21no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—42no—
109C2×C42—41no—
110C2×C42C2×D411yescharacteristic
111M32—41no—
112M32—41no—
113M32—41no—
114M32—41no—
115C2×C16—22no—
116C2×C16—41no—
117#108—42no—
118#108—41no—
119#108—41no—
120#107—21no—
121#107—21no—
122#106—41no—
123#106—41no—
124#106—42no—
125C4⋊C8—21no—
126C4⋊C8—21no—
127C4⋊C8—41no—
128C4⋊C8—41no—
129C4⋊C8—41no—
130#103—44no—
131#103—41no—
132#103—41no—
133C22⋊C8—41no—
134C22⋊C8—41no—
135C22⋊C8—41no—
136C4×C8—41no—
137C4×C8C2×D411yescharacteristic
138C4×C8#4111yescharacteristic
139C4×C8#4111yescharacteristic
140C4×C8#4111yescharacteristic
141#95—42no—
142C24—21no—
143C2×Q8—22no—
144C2×Q8—41no—
145C2×Q8—41no—
146C2×Q8—41no—
147C2×Q8—41no—
148C2×Q8—41no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—42no—
152C22×C4—21no—
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—42no—
162C22×C4#12011yescharacteristic
163C22×C4C2×SD1611yescharacteristic
164C22×C4AGL(1,ℤ/8)11yescharacteristic
165Q16—22no—
166Q16—22no—
167Q16—41no—
168Q16—41no—
169M16—81no—
170M16—81no—
171M16—81no—
172M16—81no—
173M16—81no—
174M16—81no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—81no—
181C2×C8—81no—
182C2×C8#12111yescharacteristic
183C2×C8#12111yescharacteristic
184C2×C8#12311yescharacteristic
185C2×C8#12411yescharacteristic
186C2×C8C2×D811yescharacteristic
187C2×C8#13511yescharacteristic
188C2×C8AGL(1,ℤ/8)11yescharacteristic
189C2×C8#13711yescharacteristic
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—41no—
193C4⋊C4—42no—
194C4⋊C4—41no—
195C4⋊C4—42no—
196C4⋊C4—82no—
197C4⋊C4—84no—
198C4⋊C4—81no—
199C4⋊C4—81no—
200C4⋊C4—161no—
201C22⋊C4—41no—
202C22⋊C4—41no—
203C22⋊C4—84no—
204C22⋊C4—162no—
205C42—81no—
206C42—81no—
207C42#12111yescharacteristic
208C42#12311yescharacteristic
209C42#13511yescharacteristic
210C42AGL(1,ℤ/8)11yescharacteristic
211C16—42no—
212C16—81no—
213C23—42no—
214C23—42no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23#45011yescharacteristic
219Q8—42no—
220Q8—42no—
221Q8—81no—
222Q8—81no—
223D4—42no—
224D4—81no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4—81no—
241C2×C4—82no—
242C2×C4—81no—
243C2×C4#39411yescharacteristic
244C2×C4#44711yescharacteristic
245C2×C4#44911yescharacteristic
246C2×C4#45211yescharacteristic
247C2×C4#45411yescharacteristic
248C2×C4#45511yescharacteristic
249C2×C4#46511yescharacteristic
250C2×C4#46911yescharacteristic
251C2×C4#48111yescharacteristic
252C2×C4#48211yescharacteristic
253C8—21no—
254C8—21no—
255C8—81no—
256C8—161no—
257C8—161no—
258C8#46611yescharacteristic
259C8#46811yescharacteristic
260C8#48011yescharacteristic
261C8#48211yescharacteristic
262C22—21no—
263C22—21no—
264C22—82no—
265C22—82no—
266C22—81no—
267C22—81no—
268C22—81no—
269C22#163511yescharacteristic
270C22#163911yescharacteristic
271C22#203411yescharacteristic
272C4—21no—
273C4—21no—
274C4—21no—
275C4—81no—
276C4—82no—
277C4—162no—
278C4—161no—
279C4#162511yescharacteristic
280C4#164611yescharacteristic
281C4#203311yescharacteristic
282C4#204011yescharacteristic
283C4#223912yes—
284C2—21no—
285C2—21no—
286C2—161no—
287C2#1043111yescharacteristic
288C2#1339911yescharacteristic
289C2#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×#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. #12819C2semidirect
4. #11600C2semidirect
7. #7278C2semidirect

Maximal subgroups

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

subgroupindex
#282322
#128862
#128192
#116002
#115872
#86622
#72782