Finite Group Databasebeta

← #136233

Subgroups — #136233

Order 512.

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

normal subgroups only

This group has 362 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#11343C211yesmaximal, characteristic
4#9033C211yesmaximal, characteristic
5#8862C211yesmaximal, characteristic
6#7253C211yesmaximal, characteristic
7#7190C211yesmaximal, characteristic
8C2×#430—21no—
9#2905C2211yescharacteristic
10#2396C2211yescharacteristic
11#1985—21no—
12#1939—21no—
13#1917—21no—
14#1916—21no—
15#1850—21no—
16#1798C2211yescharacteristic
17#1753—21no—
18#1742C2211yescharacteristic
19C2×#341—21no—
20#1729C2211yescharacteristic
21#1570—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#430—42no—
38#430—42no—
39C2×#108—42no—
40#424—42no—
41#423—21no—
42#423—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8D411yescharacteristic
46C2×C4⋊C8D411yescharacteristic
47C2×C22⋊C8—21no—
48C2×#97—21no—
49C2×C4×C8—21no—
50#400C2×C411yescharacteristic
51#399Q811yescharacteristic
52#398—21no—
53#396—21no—
54#395C2×C411yescharacteristic
55#393Q811yescharacteristic
56C4×C4⋊C4—21no—
57C4×C4⋊C4D411yescharacteristic
58C4×C22⋊C4—21no—
59#343—21no—
60#343C2×C412yes—
61#341—42no—
62#336—21no—
63#336—21no—
64#336—21no—
65#336—21no—
66#336—21no—
67#336—21no—
68#336—21no—
69#334—42no—
70Q8⋊C8—42no—
71#324—21no—
72#324C2×C412yes—
73#323—21no—
74#323—21no—
75C22×Q8C4⋊C411yescharacteristic
76C23×C4C2×D411yescharacteristic
77C2×M16—21no—
78C2×M16—22no—
79C2×M16—22no—
80C2×M16—22no—
81C2×M16—22no—
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×M16—41no—
91C2×M16—41no—
92C2×M16SD1612yes—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8—41no—
100C22×C8—41no—
101C22×C8—41no—
102C22×C8—41no—
103C4×Q8—41no—
104C4×Q8—41no—
105#117—41no—
106#117—41no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4C4⋊C411yescharacteristic
112C2×C4⋊C4C22×C411yescharacteristic
113C2×C4⋊C4#4111yescharacteristic
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4C2×Q811yescharacteristic
117C2×C22⋊C4#4111yescharacteristic
118C2×C42—21no—
119C2×C42—21no—
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D411yescharacteristic
122#108—48no—
123C4⋊C8—22no—
124C4⋊C8—22no—
125C4⋊C8—41no—
126C4⋊C8—41no—
127C4⋊C8—41no—
128C4⋊C8—41no—
129C4⋊C8—41no—
130C4⋊C8—41no—
131C4⋊C8—81no—
132C4⋊C8—81no—
133C22⋊C8—22no—
134C22⋊C8—41no—
135C22⋊C8—41no—
136C22⋊C8—41no—
137#97—81no—
138C4×C8—41no—
139C4×C8—41no—
140C4×C8—81no—
141#95—21no—
142#95—21no—
143#95—21no—
144#95—21no—
145#95C4⋊C411yescharacteristic
146#95C4⋊C411yescharacteristic
147C24C22⋊Q811yescharacteristic
148C2×Q8—41no—
149C2×Q8—81no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4C2×C4⋊C411yescharacteristic
159C22×C4C4×D411yescharacteristic
160C22×C4C4×D411yescharacteristic
161C22×C4#12011yescharacteristic
162C22×C4C22⋊Q811yescharacteristic
163C22×C4#12311yescharacteristic
164C22×C4C2×SD1611yescharacteristic
165C22×C4#13511yescharacteristic
166C22×C4#13711yescharacteristic
167C22×C4#13711yescharacteristic
168M16—24no—
169M16—42no—
170M16—42no—
171M16—42no—
172M16—42no—
173M16—42no—
174M16—42no—
175M16—81no—
176M16—81no—
177M16—81no—
178C2×C8—22no—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8—42no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—42no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—81no—
197C2×C8—81no—
198C2×C8—81no—
199C2×C8—81no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—81no—
206C4⋊C4—81no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—42no—
210C22⋊C4—81no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—41no—
216C42—41no—
217C42—41no—
218C42—41no—
219C23—21no—
220C23—21no—
221C23—41no—
222C23—41no—
223C23#38511yescharacteristic
224C23#47511yescharacteristic
225C23#47911yescharacteristic
226Q8—42no—
227Q8—81no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
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×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4C4×SD1611yescharacteristic
259C2×C4#44111yescharacteristic
260C2×C4#44811yescharacteristic
261C2×C4#45211yescharacteristic
262C2×C4#45311yescharacteristic
263C2×C4#45611yescharacteristic
264C2×C4B(4,𝔽2)11yescharacteristic
265C2×C4#45811yescharacteristic
266C8—44no—
267C8—42no—
268C8—82no—
269C8—82no—
270C8—81no—
271C8—81no—
272C8—81no—
273C8—161no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—81no—
283C22#185911yescharacteristic
284C22#186511yescharacteristic
285C22#186711yescharacteristic
286C4—21no—
287C4—21no—
288C4—42no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—81no—
294C4—81no—
295C4—81no—
296C4#159611yescharacteristic
297C4#160311yescharacteristic
298C4#161111yescharacteristic
299C4#161811yescharacteristic
300C2—21no—
301C2—21no—
302C2—41no—
303C2—41no—
304C2#887111yescharacteristic
305C2#1296311yescharacteristic
306C2#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
#113432
#90332
#88622
#72532
#71902