Finite Group Databasebeta

← #145778

Subgroups — #145778

Order 512.

487 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 487 conjugacy classes of subgroups — too many to draw. Only its 80 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#17076C211yesmaximal, characteristic
2#11342C211yesmaximal, characteristic
3#8856C212yesmaximal
4#7251C212yesmaximal
5#7189C211yesmaximal, characteristic
6C2×#428—21no—
7C2×#400—21no—
8#2316C2211yescharacteristic
9#1852—22no—
10C2×#343C2211yescharacteristic
11C2×#339—21no—
12C2×#324C2212yes—
13#1444—22no—
14#1442—22no—
15#1441C2212yes—
16C4⋊#95—21no—
17C4⋊#95C2211yescharacteristic
18#1294—22no—
19C22×M16—21no—
20C2×#117—21no—
21C22×C4⋊C4—21no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4D411yescharacteristic
25C22×C4⋊C4D412yes—
26C22×C4⋊C4C2311yescharacteristic
27C22×C42D411yescharacteristic
28#428—42no—
29#428—42no—
30C2×#107—41no—
31C2×#107—42no—
32C2×#106—41no—
33C2×C22⋊C8—22no—
34#400—42no—
35#400—42no—
36#389—41no—
37#389—41no—
38#380—22no—
39C2×#95—21no—
40C2×#95—22no—
41C2×#95—22no—
42C2×#95—22no—
43C2×#95—22no—
44C2×#95—22no—
45C2×#95D412yes—
46#343—21no—
47#343Q812yes—
48#340—42no—
49#339—42no—
50#324—22no—
51#324Q814yes—
52C23×C4—21no—
53C23×C4—21no—
54C23×C4—22no—
55C23×C4—22no—
56C23×C4C2×D411yescharacteristic
57C23×C4C2×D412yes—
58C2×M16—22no—
59C2×M16—22no—
60C2×M16—22no—
61C2×M16—41no—
62C2×M16—41no—
63C2×M16—41no—
64C22×C8—41no—
65C22×C8—42no—
66#117—41no—
67#117—41no—
68#117—41no—
69#117—41no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—82no—
84C2×C4⋊C4—81no—
85C2×C4⋊C4—81no—
86C2×C4⋊C4—81no—
87C2×C4⋊C4—81no—
88C2×C4⋊C4—81no—
89C2×C4⋊C4—81no—
90C2×C4⋊C4D812yes—
91C2×C4⋊C4C2×Q812yes—
92C2×C4⋊C4C2×Q811yescharacteristic
93C2×C4⋊C4#4111yescharacteristic
94C2×C22⋊C4—41no—
95C2×C42—21no—
96C2×C42—41no—
97C2×C42—41no—
98C2×C42—41no—
99C2×C42Q1612yes—
100#107—44no—
101#107—84no—
102#106—44no—
103C22⋊C8—24no—
104C22⋊C8—42no—
105#95—42no—
106#95—44no—
107#95—44no—
108#95—42no—
109#95—44no—
110#95—42no—
111#95—42no—
112#95—42no—
113#95—82no—
114C24#12011yescharacteristic
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4C22⋊Q811yescharacteristic
152C22×C4C22⋊Q812yes—
153C22×C4C4⋊Q811yescharacteristic
154C22×C4C4⋊Q812yes—
155C22×C4C2×D811yescharacteristic
156C22×C4C2×Q1611yescharacteristic
157C22×C4AGL(1,ℤ/8)11yescharacteristic
158C22×C4#13711yescharacteristic
159M16—24no—
160M16—42no—
161M16—42no—
162M16—42no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—44no—
167C2×C8—82no—
168C2×C8—82no—
169C4⋊C4—41no—
170C4⋊C4—42no—
171C4⋊C4—41no—
172C4⋊C4—42no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—44no—
178C4⋊C4—81no—
179C4⋊C4—82no—
180C4⋊C4—82no—
181C4⋊C4—82no—
182C4⋊C4—84no—
183C4⋊C4—84no—
184C4⋊C4—84no—
185C22⋊C4—81no—
186C22⋊C4—81no—
187C42—42no—
188C42—81no—
189C42—81no—
190C42—81no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23#39511yescharacteristic
196C23#44711yescharacteristic
197C23#45111yescharacteristic
198C23#45311yescharacteristic
199C23#45611yescharacteristic
200C23B(4,𝔽2)11yescharacteristic
201C23#45811yescharacteristic
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—24no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—44no—
228C2×C4—42no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—81no—
248C2×C4—82no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—82no—
253C2×C4#47411yescharacteristic
254C2×C4#47711yescharacteristic
255C2×C4#50011yescharacteristic
256C2×C4#50111yescharacteristic
257C8—44no—
258C8—84no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22#159511yescharacteristic
272C22#160511yescharacteristic
273C22#161011yescharacteristic
274C22#161411yescharacteristic
275C22#205711yescharacteristic
276C22#206011yescharacteristic
277C22#206211yescharacteristic
278C22#219811yescharacteristic
279C22#219911yescharacteristic
280C22#220311yescharacteristic
281C22#220411yescharacteristic
282C4—21no—
283C4—21no—
284C4—21no—
285C4—21no—
286C4—44no—
287C4—42no—
288C4—42no—
289C4—82no—
290C4—162no—
291C4—161no—
292C4—161no—
293C4—162no—
294C2—21no—
295C2—21no—
296C2—21no—
297C2—21no—
298C2#1056711yescharacteristic
299C2#1209711yescharacteristic
300C2#1210011yescharacteristic
301C2#1225711yescharacteristic
302C2#1226011yescharacteristic
303C2#1346311yescharacteristic
304C2#1346511yescharacteristic
subgroupG/H
center, socleC23B(4,𝔽2)
commutator, Frattini, cosocleC22×C4⋊C4C23
prime radicalC24#120
LCSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24 ◁ C22×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#170762
#113422
#88562
#72512
#71892