Finite Group Databasebeta

← #137236

Subgroups — #137236

Order 512.

397 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 397 conjugacy classes of subgroups — too many to draw. Only its 90 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#23666C211yesmaximal, characteristic
2#21969C211yesmaximal, characteristic
3#12869C212yesmaximal
4C2×#1322C211yesmaximal, characteristic
5#7260C212yesmaximal
6#2920—21no—
7C2×#427C2211yescharacteristic
8C22×C4⋊C8C2211yescharacteristic
9#2296—21no—
10C2×C22⋊C16C2212yes—
11#2088—21no—
12#2088—21no—
13#2087C411yescharacteristic
14#2087C411yescharacteristic
15#2086C411yescharacteristic
16#2086C411yescharacteristic
17#1950—24no—
18#1810—21no—
19#1810C2212yes—
20#1735—21no—
21#1735C2211yescharacteristic
22#1382—24no—
23#1322—21no—
24#1322—21no—
25C23×C8D411yescharacteristic
26C23×C8C2311yescharacteristic
27C2×#117—21no—
28C2×#117—21no—
29C22×C42D411yescharacteristic
30C22×C16—22no—
31#432—41no—
32#432—41no—
33#431—41no—
34#431—41no—
35C2×#108—22no—
36C2×#108D414yes—
37#427—22no—
38#427—22no—
39C2×#107C2×C411yescharacteristic
40C2×#107C2×C411yescharacteristic
41C2×#106—21no—
42#423—22no—
43C2×C4⋊C8—22no—
44C2×C4⋊C8—22no—
45C2×C4⋊C8C2×C411yescharacteristic
46C2×C4⋊C8C2×C411yescharacteristic
47C2×C4⋊C8D411yescharacteristic
48C2×C4⋊C8D411yescharacteristic
49#407—22no—
50C4×M16—22no—
51#389—21no—
52#389—21no—
53#389—21no—
54#389—21no—
55#385—41no—
56#385—41no—
57C4×C4⋊C4—41no—
58C4×C4⋊C4—41no—
59C22⋊C16—22no—
60C22⋊C16—22no—
61#341—22no—
62#341—22no—
63#340—21no—
64#340—21no—
65#340—21no—
66#340—21no—
67#340—21no—
68#340C2×C411yescharacteristic
69#340C2×C411yescharacteristic
70C23×C4SD1611yescharacteristic
71C23×C4SD1611yescharacteristic
72C23×C4C2×D411yescharacteristic
73C2×M16—24no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—42no—
77C22×C8—42no—
78C22×C8C22⋊C411yescharacteristic
79C22×C8C22⋊C411yescharacteristic
80C22×C8C22×C411yescharacteristic
81C22×C8C2×D412yes—
82C22×C8C2×D411yescharacteristic
83C22×C8#4112yes—
84#117—24no—
85#117—42no—
86#117—41no—
87#117—41no—
88#117—41no—
89#117—41no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—41no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—41no—
105C2×C42—41no—
106C2×C42C22⋊C411yescharacteristic
107C2×C42C22⋊C411yescharacteristic
108C2×C42D811yescharacteristic
109C2×C42D811yescharacteristic
110C2×C16—44no—
111C2×C16—42no—
112#108—44no—
113#108—82no—
114#107—22no—
115#107—22no—
116#106—42no—
117C4⋊C8—22no—
118C4⋊C8—22no—
119C4⋊C8—22no—
120C4⋊C8—22no—
121C4⋊C8—42no—
122C4⋊C8—42no—
123C22⋊C8—42no—
124C22⋊C8—42no—
125#97—42no—
126C4×C8—42no—
127#95—41no—
128#95—41no—
129#95—41no—
130#95—41no—
131C24C2×SD1611yescharacteristic
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—81no—
145C22×C4#10211yescharacteristic
146C22×C4#10211yescharacteristic
147C22×C4#10211yescharacteristic
148C22×C4#10211yescharacteristic
149C22×C4C2×C22⋊C411yescharacteristic
150C22×C4#12011yescharacteristic
151C22×C4#12311yescharacteristic
152C22×C4C2×D811yescharacteristic
153C22×C4AGL(1,ℤ/8)11yescharacteristic
154C22×C4#13711yescharacteristic
155M16—44no—
156M16—44no—
157C2×C8—24no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—42no—
163C2×C8—42no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8—44no—
167C2×C8—81no—
168C2×C8C4×D412yes—
169C2×C8#12112yes—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—82no—
179C4⋊C4—82no—
180C4⋊C4—82no—
181C4⋊C4—82no—
182C22⋊C4—44no—
183C22⋊C4—81no—
184C22⋊C4—81no—
185C42—21no—
186C42—21no—
187C42—21no—
188C42—21no—
189C42—21no—
190C42—21no—
191C42—21no—
192C42—21no—
193C42—44no—
194C42—81no—
195C42—81no—
196C16—82no—
197C23—22no—
198C23—21no—
199C23—21no—
200C23—41no—
201C23C2×#10211yescharacteristic
202C23#44811yescharacteristic
203C23#48111yescharacteristic
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
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#35912yes—
237C2×C4#38611yescharacteristic
238C2×C4#41711yescharacteristic
239C2×C4#44711yescharacteristic
240C2×C4#45612yes—
241C2×C4#45812yes—
242C2×C4#48011yescharacteristic
243C8—22no—
244C8—22no—
245C8—41no—
246C8—41no—
247C8—84no—
248C22—21no—
249C22—21no—
250C22—21no—
251C22—21no—
252C22—21no—
253C22—21no—
254C22—21no—
255C22—42no—
256C22—42no—
257C22#161112yes—
258C22#187411yescharacteristic
259C22C2×#35911yescharacteristic
260C22#214311yescharacteristic
261C4—21no—
262C4—21no—
263C4—22no—
264C4—41no—
265C4—41no—
266C4—41no—
267C4—82no—
268C4—82no—
269C4—161no—
270C4—161no—
271C4#161012yes—
272C4#188611yescharacteristic
273C4#189011yescharacteristic
274C2—21no—
275C2—21no—
276C2—22no—
277C2—41no—
278C2#902711yescharacteristic
279C2#1299011yescharacteristic
280C2#1299811yescharacteristic
subgroupG/H
centerC2×C4#447
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
socleC22#1874
prime radicalC24C2×SD16
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#236662
#219692
#128692
C2×#13222
#72602