Finite Group Databasebeta

← #100162

Subgroups — #100162

Order 512.

606 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 128 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#20533C211yesmaximal, characteristic
2#14223C211yesmaximal, characteristic
3#8706C212yesmaximal
4#8560C211yesmaximal, characteristic
5#7024C212yesmaximal
6C2×#431—21no—
7C2×#398C411yescharacteristic
8C2×#387—21no—
9C2×C4×C4⋊C4C2211yescharacteristic
10C2×C4×C4⋊C4C2211yescharacteristic
11#1797C2212yes—
12C8×C4⋊C4—24no—
13C4×C22⋊C8—21no—
14C4×C22⋊C8C2212yes—
15C2×#324—21no—
16#1445C411yescharacteristic
17#1442C411yescharacteristic
18#1441C411yescharacteristic
19#1436—21no—
20#1434—21no—
21#1434—21no—
22#1434C2211yescharacteristic
23#1280—21no—
24#1279—24no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4D411yescharacteristic
30C22×C42—21no—
31C22×C42D411yescharacteristic
32C22×C42D411yescharacteristic
33C22×C42C2311yescharacteristic
34#431—24no—
35#431—24no—
36C2×C4⋊C8—22no—
37C2×C4⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×C4×C8—22no—
41C2×C4×C8—22no—
42C2×C4×C8D414yes—
43#398—22no—
44#398C814yescomplemented
45#387—42no—
46#387—42no—
47C4×C4⋊C4—22no—
48C4×C4⋊C4—22no—
49C4×C4⋊C4—22no—
50C4×C4⋊C4—22no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95C2×C411yescharacteristic
62C2×#95C2×C411yescharacteristic
63C2×#95C2×C411yescharacteristic
64C2×#95D411yescharacteristic
65#336—22no—
66#336—22no—
67#336—22no—
68#324—42no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22⋊C411yescharacteristic
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22×C411yescharacteristic
78C23×C4C2×D411yescharacteristic
79C23×C4C2×D411yescharacteristic
80C22×C8—24no—
81C22×C8—42no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4C2×C812yes—
95C2×C4⋊C4C2×C812yes—
96C2×C4⋊C4M1612yes—
97C2×C4⋊C4D812yes—
98C2×C4⋊C4SD1612yes—
99C2×C42—22no—
100C2×C42—22no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—42no—
106C2×C42—42no—
107C2×C42—42no—
108C2×C42—41no—
109C2×C42—41no—
110C2×C42—42no—
111C2×C42C2×D412yes—
112C2×C42#4112yes—
113C4⋊C8—28no—
114C4⋊C8—44no—
115C22⋊C8—24no—
116C22⋊C8—24no—
117C22⋊C8—42no—
118C22⋊C8—42no—
119C4×C8—28no—
120C4×C8—44no—
121C4×C8—82no—
122#95—24no—
123#95—22no—
124#95—22no—
125#95—21no—
126#95—22no—
127#95—42no—
128#95—42no—
129#95—42no—
130#95—42no—
131#95—42no—
132#95—42no—
133#95—42no—
134#95—41no—
135#95—41no—
136#95—41no—
137#95—41no—
138#95C2×C812yes—
139C24C2×C22⋊C411yescharacteristic
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4C22⋊C812yes—
193C22×C4C22⋊C812yes—
194C22×C4#10212yes—
195C22×C4#10212yes—
196C22×C4#10412yes—
197C22×C4#10412yes—
198C22×C4C4×D412yes—
199C22×C4#12011yescharacteristic
200C22×C4#12112yes—
201C22×C4#12311yescharacteristic
202C22×C4C22×C811yescharacteristic
203C22×C4C2×M1611yescharacteristic
204C22×C4#13112yes—
205C22×C4C2×D811yescharacteristic
206C22×C4C2×SD1611yescharacteristic
207C22×C4#13512yes—
208C2×C8—28no—
209C2×C8—44no—
210C2×C8—44no—
211C2×C8—82no—
212C2×C8—82no—
213C2×C8—82no—
214C4⋊C4—24no—
215C4⋊C4—24no—
216C4⋊C4—24no—
217C4⋊C4—24no—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—42no—
221C4⋊C4—48no—
222C4⋊C4—42no—
223C42—22no—
224C42—22no—
225C42—24no—
226C42—24no—
227C42—42no—
228C42—42no—
229C42—81no—
230C42—82no—
231C42—82no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23#38611yescharacteristic
237C23C2×C22⋊C811yescharacteristic
238C23#40811yescharacteristic
239C23C2×#10211yescharacteristic
240C23#41611yescharacteristic
241C23C2×#10411yescharacteristic
242C23C2×#10411yescharacteristic
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4D4⋊C814yes—
310C2×C4C8×D412yes—
311C2×C4#43512yes—
312C2×C4#45812yes—
313C2×C4#46512yes—
314C2×C4#46612yes—
315C8—48no—
316C8—162no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22C2×D4⋊C811yescharacteristic
330C22#147611yescharacteristic
331C22#149211yescharacteristic
332C22#149411yescharacteristic
333C22#186911yescharacteristic
334C22#189011yescharacteristic
335C22#189111yescharacteristic
336C22#189311yescharacteristic
337C22#189611yescharacteristic
338C22#190911yescharacteristic
339C22#191211yescharacteristic
340C4—22no—
341C4—22no—
342C4—22no—
343C4—24no—
344C4—22no—
345C4—22no—
346C4—42no—
347C4—41no—
348C4—42no—
349C4—42no—
350C4—82no—
351C4—82no—
352C4—82no—
353C4—161no—
354C2—21no—
355C2—21no—
356C2—21no—
357C2—21no—
358C2#896611yescharacteristic
359C2#907211yescharacteristic
360C2#910011yescharacteristic
361C2#947011yescharacteristic
362C2#947811yescharacteristic
363C2#948611yescharacteristic
364C2#949111yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
44. #398C8semidirect

Maximal subgroups

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

subgroupindex
#205332
#142232
#87062
#85602
#70242