Finite Group Databasebeta

← #142181

Subgroups — #142181

Order 512.

574 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 94 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#15071C211yesmaximal, characteristic
2C4×#343C211yesmaximal, characteristic
3#8560C211yesmaximal, characteristic
4#7251C212yesmaximal
5#7251C212yesmaximal
6C22×#125C411yescharacteristic
7C2×C4×M16—21no—
8C2×#387—21no—
9C2×C4×C4⋊C4C2211yescharacteristic
10#1797—21no—
11C2×#343—21no—
12C2×#343C2212yes—
13C2×#324C2212yes—
14#1444—21no—
15#1444C411yescharacteristic
16#1442—21no—
17#1442C411yescharacteristic
18#1441C2211yescharacteristic
19#1441C2211yescharacteristic
20#1436—21no—
21#1434C411yescharacteristic
22#1280—21no—
23C22×M16—22no—
24C2×#125—41no—
25C2×#125—41no—
26C2×#125—44no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4D411yescharacteristic
33C22×C4⋊C4C2311yescharacteristic
34C22×C42C2×C411yescharacteristic
35C22×C42D411yescharacteristic
36C22×C42D411yescharacteristic
37C2×C4⋊C8—41no—
38C2×C22⋊C8—22no—
39C4×M16—24no—
40C4×M16—42no—
41C2×#97—41no—
42C2×C4×C8—41no—
43#387—44no—
44C4×C4⋊C4—41no—
45C4×C4⋊C4—41no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95C2×C411yescharacteristic
56C2×#95C2×C411yescharacteristic
57C2×#95C2×C411yescharacteristic
58C2×#95D411yescharacteristic
59#343—24no—
60#343—22no—
61#343Q814yes—
62#336—41no—
63#324—24no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4C22⋊C411yescharacteristic
69C23×C4C22⋊C411yescharacteristic
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22×C411yescharacteristic
73C23×C4C2×D411yescharacteristic
74C23×C4C2×D411yescharacteristic
75C2×M16—24no—
76C2×M16—24no—
77C2×M16—24no—
78C2×M16—42no—
79C2×M16—42no—
80C2×M16—42no—
81C22×C8—42no—
82C22×C8—42no—
83#125—44no—
84#125—44no—
85#125—44no—
86#125—44no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4C2×Q812yes—
107C2×C4⋊C4#4112yes—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—22no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—21no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—42no—
118C2×C42—42no—
119C4⋊C8—82no—
120C22⋊C8—42no—
121C22⋊C8—42no—
122#97—44no—
123C4×C8—44no—
124#95—41no—
125#95—41no—
126#95—41no—
127#95—41no—
128#95—42no—
129#95—42no—
130#95—42no—
131#95—42no—
132#95—42no—
133#95—42no—
134#95—42no—
135#95—42no—
136#95—41no—
137#95—41no—
138C24C2×C22⋊C411yescharacteristic
139C22×C4—21no—
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—21no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
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×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4#9911yescharacteristic
197C22×C4#9911yescharacteristic
198C22×C4#10412yes—
199C22×C4#10412yes—
200C22×C4C4×Q812yes—
201C22×C4C22⋊Q812yes—
202C22×C4#12411yescharacteristic
203C22×C4C4⋊Q811yescharacteristic
204C22×C4#13512yes—
205C22×C4AGL(1,ℤ/8)11yescharacteristic
206C22×C4#13711yescharacteristic
207M16—28no—
208M16—44no—
209M16—44no—
210M16—44no—
211C2×C8—44no—
212C2×C8—44no—
213C2×C8—44no—
214C2×C8—82no—
215C2×C8—82no—
216C2×C8—82no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—44no—
220C4⋊C4—44no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—48no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—48no—
227C4⋊C4—44no—
228C42—22no—
229C42—22no—
230C42—22no—
231C42—22no—
232C42—42no—
233C42—42no—
234C42—42no—
235C42—81no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23#39111yescharacteristic
241C23C2×#9911yescharacteristic
242C23#41011yescharacteristic
243C23#41611yescharacteristic
244C23#41711yescharacteristic
245C23C2×#10411yescharacteristic
246C23C2×#10411yescharacteristic
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
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×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4AΣL(1,𝔽16)12yes—
318C2×C4#47812yes—
319C2×C4#47912yes—
320C8—48no—
321C8—162no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22#150011yescharacteristic
335C22#150111yescharacteristic
336C22#150411yescharacteristic
337C22#150911yescharacteristic
338C22#198911yescharacteristic
339C22#199411yescharacteristic
340C22#199611yescharacteristic
341C22C2×AΣL(1,𝔽16)11yescharacteristic
342C22#212311yescharacteristic
343C22#213211yescharacteristic
344C22#213311yescharacteristic
345C4—22no—
346C4—22no—
347C4—22no—
348C4—42no—
349C4—41no—
350C4—42no—
351C4—41no—
352C4—41no—
353C4—42no—
354C4—82no—
355C4—81no—
356C4—82no—
357C4—82no—
358C4—161no—
359C2—21no—
360C2—21no—
361C2—21no—
362C2—21no—
363C2#991011yescharacteristic
364C2#1173311yescharacteristic
365C2#1174011yescharacteristic
366C2#1176211yescharacteristic
367C2#1176711yescharacteristic
368C2#1329311yescharacteristic
369C2#1330311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#150712
C4×#3432
#85602
#72512