Finite Group Databasebeta

← #326554

Subgroups — #326554

Order 512.

417 conjugacy classes of subgroups.

The subgroup lattice has 309 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21733C211yesmaximal, characteristic
2#21676C211yesmaximal, characteristic
3#10669C212yesmaximal
4#10373C212yesmaximal
5#9749C211yesmaximal, characteristic
6#9749C211yesmaximal, characteristic
7#9628C211yesmaximal, characteristic
8#9627C211yesmaximal, characteristic
9#8881C212yesmaximal
10#8879C212yesmaximal
11#8780C211yesmaximal, characteristic
12#3160C2211yescharacteristic
13#2961C2212yes—
14#2402C2211yescharacteristic
15#1995C2212yes—
16#1946—21no—
17#1945—21no—
18#1940C2211yescharacteristic
19#1940C2211yescharacteristic
20#1917—21no—
21#1917—21no—
22#1848C2211yescharacteristic
23#1832C2211yescharacteristic
24#1773—21no—
25#1772—21no—
26C8×C4⋊C4—21no—
27C8×C4⋊C4—21no—
28C8×C4⋊C4C2211yescharacteristic
29C8×C4⋊C4C2211yescharacteristic
30#1726C2211yescharacteristic
31#1571C2212yes—
32#1570C2212yes—
33#1564C2211yescharacteristic
34#1564C2211yescharacteristic
35#1503C2212yes—
36#1503C2212yes—
37#1481C2212yes—
38#1481C2212yes—
39#1303C2212yes—
40#1303C2212yes—
41#1281C2212yes—
42#1274C2212yes—
43#533C2312yes—
44#500Q811yescharacteristic
45#500Q811yescharacteristic
46#499—21no—
47#499—21no—
48#498Q812yes—
49#446—42no—
50#446—42no—
51#431C2312yes—
52#431C2312yes—
53C2×#108—21no—
54C2×#108—21no—
55#427—21no—
56#427—21no—
57#424—22no—
58#423C2312yes—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8C2311yescharacteristic
62C2×C4⋊C8C2311yescharacteristic
63#405—22no—
64C2×#97D411yescharacteristic
65C2×#97D411yescharacteristic
66C2×#97C2311yescharacteristic
67C2×C4×C8—21no—
68C2×C4×C8—21no—
69C2×C4×C8D411yescharacteristic
70C2×C4×C8D411yescharacteristic
71C2×C4×C8C2311yescharacteristic
72#400—21no—
73#400—21no—
74#398—21no—
75#398—21no—
76C4×C4⋊C4—21no—
77C4×C4⋊C4—21no—
78C4×C4⋊C4C2311yescharacteristic
79C4×C4⋊C4C2311yescharacteristic
80#376C2311yescharacteristic
81#339—22no—
82#339—22no—
83#336—21no—
84#336—21no—
85#336—21no—
86#336—21no—
87#336—21no—
88#336—21no—
89#335—21no—
90#335—21no—
91#335—42no—
92#334—21no—
93#334—21no—
94#334—42no—
95#328—22no—
96#328—22no—
97#328—22no—
98#328—22no—
99Q8⋊C8—42no—
100Q8⋊C8—42no—
101C2×M16—22no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8C2×D411yescharacteristic
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C22×C8C2×D411yescharacteristic
108C4⋊Q8C2×Q812yes—
109C4⋊Q8C2×Q812yes—
110#125—22no—
111C22⋊Q8—22no—
112C22⋊Q8—22no—
113C4×Q8—22no—
114C4×Q8—22no—
115#117—22no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42C2×D411yescharacteristic
125C2×C42C2×D411yescharacteristic
126C2×C42C2411yescharacteristic
127#108—42no—
128#108—42no—
129#107—21no—
130#107—21no—
131#107—21no—
132#107—21no—
133#106—22no—
134#106—22no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—21no—
138C4⋊C8—21no—
139C4⋊C8—21no—
140C4⋊C8—22no—
141C4⋊C8—21no—
142C4⋊C8—22no—
143C4⋊C8—42no—
144C4⋊C8—44no—
145C4⋊C8—42no—
146C22⋊C8—22no—
147C22⋊C8—22no—
148#97—22no—
149#97—21no—
150#97—21no—
151#97—41no—
152#97—42no—
153C4×C8—22no—
154C4×C8—22no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8—42no—
158C4×C8—42no—
159C4×C8—42no—
160C4×C8C2×Q811yescharacteristic
161C4×C8C2×Q811yescharacteristic
162C4×C8#4111yescharacteristic
163C4×C8#4111yescharacteristic
164#95—21no—
165#95—21no—
166#95—21no—
167#95—21no—
168#95—21no—
169#95—21no—
170#95#4111yescharacteristic
171#95#4111yescharacteristic
172#95#4111yescharacteristic
173#95#4111yescharacteristic
174C2×Q8—22no—
175C2×Q8—22no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4C22×D411yescharacteristic
183C22×C4C2×#4111yescharacteristic
184C22×C4C2×#4111yescharacteristic
185M16—42no—
186M16—42no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—42no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8C22⋊Q811yescharacteristic
204C2×C8C22⋊Q811yescharacteristic
205C2×C8C22⋊Q811yescharacteristic
206C2×C8C22⋊Q811yescharacteristic
207C2×C8#13512yes—
208C2×C8#13712yes—
209C2×C8#13712yes—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C22⋊C4—42no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—42no—
228C42—41no—
229C42—41no—
230C42—41no—
231C42—41no—
232C42C22×Q811yescharacteristic
233C42C2×#4111yescharacteristic
234C42H(5,𝔽2)11yescharacteristic
235C42H(5,𝔽2)11yescharacteristic
236C23#52511yescharacteristic
237Q8—82no—
238Q8—82no—
239C2×C4—22no—
240C2×C4—22no—
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×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4C2×C22⋊Q811yescharacteristic
262C2×C4#53511yescharacteristic
263C2×C4#55411yescharacteristic
264C2×C4#55411yescharacteristic
265C2×C4#55911yescharacteristic
266C2×C4#55911yescharacteristic
267C2×C4C2×#13511yescharacteristic
268C2×C4C2×#13711yescharacteristic
269C2×C4C2×#13711yescharacteristic
270C2×C4#57511yescharacteristic
271C8—21no—
272C8—21no—
273C8—22no—
274C8—41no—
275C8—82no—
276C8—82no—
277C8—81no—
278C8—82no—
279C8—81no—
280C8#47912yes—
281C8#47912yes—
282C22—21no—
283C22—21no—
284C22#258011yescharacteristic
285C22#310711yescharacteristic
286C22#311611yescharacteristic
287C4—21no—
288C4—21no—
289C4—21no—
290C4—21no—
291C4—42no—
292C4—42no—
293C4—81no—
294C4—81no—
295C4—81no—
296C4—81no—
297C4—82no—
298C4—82no—
299C4C2×#47911yescharacteristic
300C4#307811yescharacteristic
301C4#321211yescharacteristic
302C4#321811yescharacteristic
303C2—21no—
304C2—21no—
305C2#2925711yescharacteristic
306C2#2927011yescharacteristic
307C2#2927011yescharacteristic
subgroupG/H
centerC2×C4#535
commutator, Frattini, cosocleC2×C42C24
socleC22#2580
prime radicalC23#525
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#217332
#216762
#106692
#103732
#97492
#96282
#96272
#88812
#88792
#87802