Finite Group Databasebeta

← #136157

Subgroups — #136157

Order 512.

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

normal subgroups only

This group has 340 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#22298C211yesmaximal, characteristic
2#11721C211yesmaximal, characteristic
3#11297C211yesmaximal, characteristic
4#9033C211yesmaximal, characteristic
5#8861C211yesmaximal, characteristic
6#7253C211yesmaximal, characteristic
7#7190C211yesmaximal, characteristic
8#2905C2211yescharacteristic
9#2396C2211yescharacteristic
10#1985—21no—
11#1939—21no—
12#1917—21no—
13#1916—21no—
14#1850—21no—
15#1798C2211yescharacteristic
16#1753—21no—
17#1742C2211yescharacteristic
18#1738—21no—
19C2×#341—21no—
20#1729C2211yescharacteristic
21#1571—21no—
22#1487—21no—
23#1458C2211yescharacteristic
24#1458C2211yescharacteristic
25#1354C412yes—
26#1353C412yes—
27#1303—21no—
28#1301—21no—
29C22×M16—21no—
30C22×M16—21no—
31C22×M16D411yescharacteristic
32C23×C8D411yescharacteristic
33C2×C4×Q8D411yescharacteristic
34C2×#117C2311yescharacteristic
35#431—21no—
36#431—21no—
37#424—42no—
38#423—21no—
39#423—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8D411yescharacteristic
43C2×C4⋊C8D411yescharacteristic
44C2×C22⋊C8—21no—
45C2×#97—21no—
46C2×C4×C8—21no—
47#400C2×C411yescharacteristic
48#399Q811yescharacteristic
49#398—21no—
50#396—21no—
51#395C2×C411yescharacteristic
52#393Q811yescharacteristic
53C4×C4⋊C4—21no—
54C4×C4⋊C4D411yescharacteristic
55C4×C22⋊C4—21no—
56#343—21no—
57#343C2×C412yes—
58#341—42no—
59#341—42no—
60#336—21no—
61#336—21no—
62#336—21no—
63#336—21no—
64#336—21no—
65#336—21no—
66#336—21no—
67#335—42no—
68Q8⋊C8—42no—
69#324—21no—
70#324C2×C412yes—
71#323—21no—
72#323—21no—
73C22×Q8C4⋊C411yescharacteristic
74C23×C4C2×D411yescharacteristic
75C2×M16—21no—
76C2×M16—22no—
77C2×M16—22no—
78C2×M16—41no—
79C2×M16—41no—
80C2×M16—41no—
81C2×M16—41no—
82C2×M16—41no—
83C2×M16—41no—
84C2×M16—41no—
85C2×M16—41no—
86C2×M16—41no—
87C2×M16—41no—
88C2×M16—41no—
89C2×M16—41no—
90C2×M16Q1612yes—
91C22×C8—21no—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—41no—
98C22×C8—41no—
99C22×C8—41no—
100C22×C8—41no—
101C4×Q8—41no—
102C4×Q8—41no—
103#117—41no—
104#117—41no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4C4⋊C411yescharacteristic
110C2×C4⋊C4C22×C411yescharacteristic
111C2×C4⋊C4#4111yescharacteristic
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4C2×Q811yescharacteristic
115C2×C22⋊C4#4111yescharacteristic
116C2×C42—21no—
117C2×C42—21no—
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C4⋊C8—22no—
121C4⋊C8—22no—
122C4⋊C8—41no—
123C4⋊C8—41no—
124C4⋊C8—41no—
125C4⋊C8—41no—
126C4⋊C8—41no—
127C4⋊C8—41no—
128C4⋊C8—81no—
129C4⋊C8—81no—
130C22⋊C8—22no—
131C22⋊C8—41no—
132C22⋊C8—41no—
133C22⋊C8—41no—
134#97—81no—
135C4×C8—41no—
136C4×C8—41no—
137C4×C8—81no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95C4⋊C411yescharacteristic
143#95C4⋊C411yescharacteristic
144C24C22⋊Q811yescharacteristic
145C2×Q8—41no—
146C2×Q8—81no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4C2×C4⋊C411yescharacteristic
156C22×C4C4×D411yescharacteristic
157C22×C4C4×D411yescharacteristic
158C22×C4#12011yescharacteristic
159C22×C4C22⋊Q811yescharacteristic
160C22×C4#12311yescharacteristic
161C22×C4C2×Q1611yescharacteristic
162C22×C4#13511yescharacteristic
163C22×C4AGL(1,ℤ/8)11yescharacteristic
164C22×C4#13711yescharacteristic
165M16—42no—
166M16—42no—
167M16—42no—
168M16—42no—
169M16—42no—
170M16—81no—
171M16—81no—
172M16—81no—
173M16—81no—
174M16—81no—
175C2×C8—22no—
176C2×C8—41no—
177C2×C8—41no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—42no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—81no—
192C2×C8—81no—
193C2×C8—81no—
194C2×C8—81no—
195C2×C8—81no—
196C2×C8—81no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—81no—
203C4⋊C4—81no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C22⋊C4—42no—
207C22⋊C4—81no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—41no—
213C42—41no—
214C42—41no—
215C42—41no—
216C23—21no—
217C23—21no—
218C23—41no—
219C23—41no—
220C23#38511yescharacteristic
221C23#47711yescharacteristic
222C23#47911yescharacteristic
223Q8—42no—
224Q8—81no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4C4×Q1611yescharacteristic
256C2×C4#44011yescharacteristic
257C2×C4#44911yescharacteristic
258C2×C4#45111yescharacteristic
259C2×C4#45411yescharacteristic
260C2×C4#45511yescharacteristic
261C2×C4B(4,𝔽2)11yescharacteristic
262C2×C4#45811yescharacteristic
263C8—42no—
264C8—82no—
265C8—82no—
266C8—81no—
267C8—82no—
268C8—81no—
269C8—81no—
270C8—161no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—81no—
280C22#186011yescharacteristic
281C22#186311yescharacteristic
282C22#186711yescharacteristic
283C4—21no—
284C4—21no—
285C4—42no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—81no—
291C4—81no—
292C4—81no—
293C4#159711yescharacteristic
294C4#160211yescharacteristic
295C4#161211yescharacteristic
296C4#161711yescharacteristic
297C2—21no—
298C2—21no—
299C2—41no—
300C2—41no—
301C2#886711yescharacteristic
302C2#1296311yescharacteristic
303C2#1297511yescharacteristic
subgroupG/H
centerC2×C4B(4,𝔽2)
commutatorC2×C4⋊C4C22×C4
Frattini, cosocleC2×#117C23
socleC22#1867
prime radicalC24C22⋊Q8
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#222982
#117212
#112972
#90332
#88612
#72532
#71902