Finite Group Databasebeta

← #280859

Subgroups — #280859

Order 512.

430 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 126 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#28297C211yesmaximal, characteristic
2#22809C211yesmaximal, characteristic
3#10639C212yesmaximal
4#10471C212yesmaximal
5#9669C211yesmaximal, characteristic
6#9627C211yesmaximal, characteristic
7#9178C212yesmaximal
8#8790C211yesmaximal, characteristic
9#8769C212yesmaximal
10#8677C211yesmaximal, characteristic
11#8252C211yesmaximal, characteristic
12#3078C2212yes—
13#2960C2211yescharacteristic
14C2×C4×Q16C2211yescharacteristic
15#2356C2212yes—
16#2064—22no—
17#1979—21no—
18#1969—21no—
19#1952—21no—
20#1927—22no—
21#1848C2211yescharacteristic
22#1846C2211yescharacteristic
23#1805C2212yes—
24#1801C2212yes—
25C4×#107—21no—
26C8×C4⋊C4C2211yescharacteristic
27#1571C2212yes—
28#1564C2211yescharacteristic
29#1542C2211yescharacteristic
30#1518C2212yes—
31#1503C2211yescharacteristic
32#1479C2212yes—
33#1461C2212yes—
34#1450C2211yescharacteristic
35#1303C2211yescharacteristic
36#1303C2212yes—
37#1301C2211yescharacteristic
38#1286C2212yes—
39#1286C2212yes—
40#1286C2212yes—
41C22×Q16D411yescharacteristic
42#533C2311yescharacteristic
43C2×C4×Q8C2312yes—
44#479—42no—
45#477—22no—
46#475—22no—
47#446—21no—
48#446—21no—
49C8×Q8—21no—
50C8×Q8—21no—
51C4×Q16—21no—
52C4×Q16—21no—
53C4×Q16—21no—
54C4×Q16—21no—
55#431C2311yescharacteristic
56#431C2311yescharacteristic
57#431C2312yes—
58C2×#108—21no—
59C2×#108—21no—
60#428—22no—
61C2×#107—21no—
62C2×#107D411yescharacteristic
63C2×#106—21no—
64#424—21no—
65#423C2312yes—
66#423C2311yescharacteristic
67C2×C4⋊C8C2311yescharacteristic
68#419—22no—
69C2×#103—22no—
70C2×#103D412yes—
71C2×#101—22no—
72#405—21no—
73C2×C4×C8—21no—
74C2×C4×C8C2311yescharacteristic
75#391—22no—
76#387—42no—
77#383—22no—
78#382C2312yes—
79C4×C4⋊C4—22no—
80C4×C4⋊C4C2311yescharacteristic
81#376—22no—
82#343—22no—
83#340—22no—
84#336—21no—
85#335—21no—
86#335—42no—
87#334—21no—
88#333—21no—
89#332—21no—
90#328—21no—
91#328—21no—
92#328—42no—
93Q8⋊C8—42no—
94#324—42no—
95#322—21no—
96C82—21no—
97C22×Q8C2×D412yes—
98C2×Q16—21no—
99C2×Q16—21no—
100C2×Q16—22no—
101C2×Q16—22no—
102C2×M16—21no—
103C2×M16—22no—
104C22×C8—21no—
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C4⋊Q8#4111yescharacteristic
108C4⋊Q8#4111yescharacteristic
109#125—21no—
110C22⋊Q8—21no—
111C22⋊Q8—21no—
112C4×Q8—22no—
113C4×Q8—22no—
114C4×Q8—22no—
115C4×Q8—21no—
116C4×Q8—21no—
117C4×Q8—22no—
118#117—21no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4C2×D412yes—
123C2×C42—21no—
124C2×C42—22no—
125C2×C42—42no—
126C2×C42C2411yescharacteristic
127#108—42no—
128#108—81no—
129#107—21no—
130#107—21no—
131#107—42no—
132#107—42no—
133#106—42no—
134#106—81no—
135C4⋊C8—21no—
136C4⋊C8—21no—
137C4⋊C8—22no—
138C4⋊C8—21no—
139C4⋊C8—21no—
140C4⋊C8—22no—
141C4⋊C8—21no—
142C4⋊C8—21no—
143C4⋊C8—22no—
144C4⋊C8—41no—
145C4⋊C8—41no—
146#103—22no—
147#103—22no—
148#103—42no—
149#103—42no—
150#103—82no—
151#101—82no—
152C22⋊C8—21no—
153C22⋊C8—21no—
154C22⋊C8—22no—
155#97—41no—
156C4×C8—21no—
157C4×C8—21no—
158C4×C8—22no—
159C4×C8—41no—
160C4×C8—42no—
161C4×C8#4111yescharacteristic
162C4×C8#4111yescharacteristic
163C4×C8#4111yescharacteristic
164C4×C8#4111yescharacteristic
165#95—22no—
166#95—22no—
167#95—42no—
168#95—42no—
169#95—42no—
170#95#4111yescharacteristic
171#95#4111yescharacteristic
172C2×Q8—22no—
173C2×Q8—21no—
174C2×Q8—22no—
175C2×Q8—21no—
176C2×Q8—22no—
177C2×Q8—22no—
178C2×Q8—22no—
179C2×Q8—22no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—42no—
185C22×C4C22×D411yescharacteristic
186C22×C4C2×#4111yescharacteristic
187C22×C4H(5,𝔽2)11yescharacteristic
188Q16—42no—
189Q16—42no—
190M16—42no—
191M16—41no—
192M16—42no—
193M16—41no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—41no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8#12311yescharacteristic
205C2×C8#12311yescharacteristic
206C2×C8#12311yescharacteristic
207C2×C8#12311yescharacteristic
208C2×C8#13512yes—
209C2×C8#13712yes—
210C4⋊C4—22no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—22no—
215C4⋊C4—21no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—41no—
219C4⋊C4—42no—
220C4⋊C4—41no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C22⋊C4—41no—
224C42—41no—
225C42—42no—
226C42—42no—
227C42—41no—
228C42—41no—
229C42—84no—
230C42C2×#4111yescharacteristic
231C42C2×#4111yescharacteristic
232C42C2×#4111yescharacteristic
233C42H(5,𝔽2)11yescharacteristic
234C23#54611yescharacteristic
235Q8—42no—
236Q8—42no—
237Q8—42no—
238Q8—42no—
239Q8—81no—
240Q8—81no—
241C2×C4—22no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4C2×#12311yescharacteristic
260C2×C4#52911yescharacteristic
261C2×C4#54011yescharacteristic
262C2×C4#54611yescharacteristic
263C2×C4#55311yescharacteristic
264C2×C4#55511yescharacteristic
265C2×C4C2×#13511yescharacteristic
266C2×C4C2×#13711yescharacteristic
267C2×C4#57611yescharacteristic
268C2×C4#57811yescharacteristic
269C8—21no—
270C8—21no—
271C8—82no—
272C8—81no—
273C8—81no—
274C8—81no—
275C8—81no—
276C8—82no—
277C8#48512yes—
278C8#48512yes—
279C22—21no—
280C22—21no—
281C22#242011yescharacteristic
282C22#330211yescharacteristic
283C22#333311yescharacteristic
284C4—21no—
285C4—21no—
286C4—21no—
287C4—21no—
288C4—42no—
289C4—81no—
290C4—82no—
291C4—82no—
292C4—81no—
293C4—81no—
294C4—162no—
295C4C2×#48511yescharacteristic
296C4#309811yescharacteristic
297C4#330511yescharacteristic
298C4#332811yescharacteristic
299C2—21no—
300C2—21no—
301C2#2497011yescharacteristic
302C2#2498511yescharacteristic
303C2#2498911yescharacteristic
subgroupG/H
center, socleC22#2420
commutator, Frattini, cosocleC2×C42C24
prime radicalC23#546
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ 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
#282972
#228092
#106392
#104712
#96692
#96272
#91782
#87902
#87692
#86772
#82522