Finite Group Databasebeta

← #126384

Subgroups — #126384

Order 512.

322 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 66 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#31190C211yesmaximal, characteristic
2#13538C211yesmaximal, characteristic, complemented
3#13481C211yesmaximal, characteristic, complemented
4#10616C211yesmaximal, characteristic
5#7121C211yesmaximal, characteristic, complemented
6#7113C211yesmaximal, characteristic, complemented
7#7102C211yesmaximal, characteristic
8C2×#484—21no—
9#2417—21no—
10C2×#358—21no—
11#2068—21no—
12#2036—21no—
13#2001—21no—
14#1969C2211yescharacteristic
15#1954C2211yescharacteristic
16#1942C2211yescharacteristic
17#1852—21no—
18#1845C2211yescharacteristic
19#1802—21no—
20#1380—21no—
21#1380—21no—
22#1380—21no—
23#1374C2211yescharacteristic
24#1374C2211yescharacteristic
25#1296—21no—
26#1277C2211yescharacteristic
27C22×Q16D411yescharacteristic
28C2×C22⋊Q8—21no—
29C2×C22⋊Q8—21no—
30C2×C4×D4D411yescharacteristic
31C22×C4⋊C4—21no—
32C22×C16—21no—
33C22×C16—21no—
34#484—42no—
35#484—42no—
36C2×#107—21no—
37C2×#107—21no—
38C2×#107—21no—
39C2×#107D411yescharacteristic
40C2×#107D411yescharacteristic
41C2×#107D411yescharacteristic
42C2×C4⋊C8—21no—
43C2×C4⋊C8D411yescharacteristic
44C2×#103—21no—
45C2×#103—21no—
46C2×#103—21no—
47C2×#102—41no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C4×C8C2311yescharacteristic
51#398—41no—
52#396—41no—
53#391—21no—
54#389—21no—
55#386—41no—
56#384—21no—
57#380—21no—
58#380—41no—
59#358—42no—
60#340—21no—
61#340—21no—
62#340—41no—
63C22×Q8—21no—
64C22×D4D811yescharacteristic
65C23×C4—21no—
66C23×C4—41no—
67C2×Q16—21no—
68C2×Q16—21no—
69C2×Q16—41no—
70C2×Q16—41no—
71C22×C8—21no—
72C22×C8—41no—
73C22×C8C2×D411yescharacteristic
74C22×C8C2×D411yescharacteristic
75C22⋊Q8—42no—
76C22⋊Q8—81no—
77C22⋊Q8—81no—
78C22⋊Q8—81no—
79C4×D4—41no—
80C4×D4—41no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—81no—
93C2×C4⋊C4D811yescharacteristic
94C2×C22⋊C4—21no—
95C2×C22⋊C4—41no—
96C2×C22⋊C4—41no—
97C2×C22⋊C4—41no—
98C2×C42C2×D411yescharacteristic
99C2×C16—41no—
100C2×C16—41no—
101C2×C16—41no—
102C2×C16—41no—
103C2×C16—41no—
104C2×C16—41no—
105#107—21no—
106#107—42no—
107#107—41no—
108#107—41no—
109#107—42no—
110#107—41no—
111#107—41no—
112#107SD1612yes—
113C4⋊C8—41no—
114C4⋊C8—41no—
115C4⋊C8—41no—
116#103—42no—
117#103—41no—
118#103—41no—
119#103—81no—
120#102—82no—
121C22⋊C8—41no—
122C22⋊C8—41no—
123C22⋊C8—41no—
124C22⋊C8—41no—
125C4×C8—41no—
126#95—41no—
127#95—41no—
128#95—41no—
129#95—41no—
130C24—21no—
131C2×Q8—41no—
132C2×Q8—41no—
133C2×Q8—41no—
134C2×Q8—41no—
135C2×Q8—41no—
136C2×Q8—41no—
137C2×D4—21no—
138C2×D4—41no—
139C2×D4—41no—
140C2×D4Q3212yes—
141C22×C4—21no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—82no—
153C22×C4—82no—
154C22×C4—81no—
155C22×C4—161no—
156C22×C4#12011yescharacteristic
157C22×C4C2×D811yescharacteristic
158C22×C4AGL(1,ℤ/8)11yescharacteristic
159Q16—42no—
160Q16—81no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—41no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—81no—
169C2×C8—81no—
170C2×C8—81no—
171C2×C8C2×SD1611yescharacteristic
172C2×C8#13511yescharacteristic
173C2×C8AGL(1,ℤ/8)11yescharacteristic
174C2×C8AGL(1,ℤ/8)11yescharacteristic
175C4⋊C4—22no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C4⋊C4—81no—
183C4⋊C4—81no—
184C4⋊C4—82no—
185C4⋊C4—82no—
186C4⋊C4—82no—
187C4⋊C4—81no—
188C4⋊C4—81no—
189C4⋊C4—161no—
190C4⋊C4—161no—
191C22⋊C4—81no—
192C22⋊C4—82no—
193C22⋊C4—82no—
194C22⋊C4—161no—
195C42—21no—
196C42—21no—
197C16—81no—
198C16—81no—
199C23—22no—
200C23—41no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23#44711yescharacteristic
207Q8—81no—
208Q8—81no—
209Q8—81no—
210Q8—81no—
211D4—42no—
212D4—81no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—41no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—164no—
238C2×C4—161no—
239C2×C4—161no—
240C2×C4—161no—
241C2×C4#44911yescharacteristic
242C2×C4#45011yescharacteristic
243C2×C4#45411yescharacteristic
244C2×C4#45511yescharacteristic
245C2×C4#45511yescharacteristic
246C2×C4#45811yescharacteristic
247C2×C4C2×Q3211yescharacteristic
248C2×C4#51011yescharacteristic
249C8—21no—
250C8—21no—
251C8—21no—
252C8—21no—
253C8—81no—
254C8—161no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22—81no—
259C22—81no—
260C22—81no—
261C22—81no—
262C22#160711yescharacteristic
263C22#162511yescharacteristic
264C22#163911yescharacteristic
265C22#218811yescharacteristic
266C22#219011yescharacteristic
267C22#219311yescharacteristic
268C22#219411yescharacteristic
269C4—21no—
270C4—21no—
271C4—41no—
272C4—82no—
273C4—81no—
274C4—161no—
275C4—161no—
276C4—161no—
277C4—321no—
278C4#161211yescharacteristic
279C4#161611yescharacteristic
280C4#164211yescharacteristic
281C4#166211yescharacteristic
282C2—82no—
283C2#779511yescharacteristic
284C2#1228211yescharacteristic
285C2#1228511yescharacteristic
286C2#1234611yescharacteristic
287C2#1234911yescharacteristic
288C2#1237411yescharacteristic
289C2#1237811yescharacteristic
subgroupG/H
center, socleC23#447
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC22×D4D8
LCSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1954 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13538C2semidirect
3. #13481C2semidirect
5. #7121C2semidirect
6. #7113C2semidirect

Maximal subgroups

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

subgroupindex
#311902
#135382
#134812
#106162
#71212
#71132
#71022