Finite Group Databasebeta

← #127607

Subgroups — #127607

Order 512.

896 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 896 conjugacy classes of subgroups — too many to draw. Only its 154 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
1C2×#1742C211yesmaximal, characteristic
2C2×#1464C211yesmaximal, characteristic
3#13957C211yesmaximal, characteristic
4#7251C214yesmaximal, complemented
5C23×M16—21no—
6C22×#117C2211yescharacteristic
7C2×#432—21no—
8C2×#382C411yescharacteristic
9C2×#376C411yescharacteristic
10#1742—22no—
11#1742C414yescomplemented
12C2×#343—21no—
13C2×#343C2212yes—
14#1464—22no—
15#1464C414yescomplemented
16C2×#324C2212yes—
17C2×#323—21no—
18#1441C2212yes—
19#1439C412yes—
20#1437—21no—
21#1436—21no—
22#1434—21no—
23C22⋊#95—21no—
24C24×C4D411yescharacteristic
25C22×M16—28no—
26C22×M16—24no—
27C22×M16—22no—
28C22×M16—41no—
29C22×M16—42no—
30C22×M16—44no—
31C23×C8—41no—
32C2×#117—42no—
33C2×#117—41no—
34C2×#117—41no—
35C2×#117C2×C412yes—
36C2×#117C2×C412yes—
37C2×#117D412yes—
38C2×#117Q812yes—
39C22×C4⋊C4C2×C412yes—
40C22×C4⋊C4D411yescharacteristic
41C22×C4⋊C4C2311yescharacteristic
42C22×C22⋊C4—22no—
43C22×C22⋊C4C2×C411yescharacteristic
44C22×C22⋊C4D411yescharacteristic
45C22×C42—21no—
46C22×C42C2×C411yescharacteristic
47C22×C42D411yescharacteristic
48#432—24no—
49#432—42no—
50C2×C4⋊C8—41no—
51C2×C22⋊C8—22no—
52C2×#97—41no—
53#382—41no—
54#382—41no—
55#376—42no—
56C2×#95—22no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95—22no—
61C2×#95—22no—
62C2×#95C2×C412yes—
63#343—24no—
64#343—22no—
65#343C2×C414yes—
66#324—22no—
67#324C2×C414yes—
68#323—24no—
69C25C22⋊C411yescharacteristic
70C23×C4—24no—
71C23×C4—21no—
72C23×C4—22no—
73C23×C4—21no—
74C23×C4—22no—
75C23×C4—22no—
76C23×C4—22no—
77C23×C4—22no—
78C23×C4C22⋊C412yes—
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C4⋊C412yes—
83C23×C4C22×C411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C2×M16—216no—
87C2×M16—28no—
88C2×M16—24no—
89C2×M16—416no—
90C2×M16—48no—
91C2×M16—42no—
92C2×M16—44no—
93C2×M16—48no—
94C2×M16—44no—
95C22×C8—48no—
96C22×C8—42no—
97C22×C8—44no—
98C22×C8—42no—
99#117—42no—
100#117—42no—
101#117—42no—
102#117—42no—
103#117—42no—
104#117—42no—
105#117—42no—
106#117—42no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—82no—
113C2×C4⋊C4C22⋊C412yes—
114C2×C4⋊C4C4⋊C412yes—
115C2×C4⋊C4C22×C411yescharacteristic
116C2×C4⋊C4C22×C411yescharacteristic
117C2×C4⋊C4C2×D411yescharacteristic
118C2×C4⋊C4C2×Q811yescharacteristic
119C2×C22⋊C4—41no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—41no—
122C2×C22⋊C4—41no—
123C2×C22⋊C4—44no—
124C2×C22⋊C4—44no—
125C2×C22⋊C4—82no—
126C2×C22⋊C4C4212yes—
127C2×C22⋊C4C22⋊C412yes—
128C2×C22⋊C4C4⋊C412yes—
129C2×C22⋊C4C4⋊C412yes—
130C2×C42—22no—
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42—21no—
135C2×C42—22no—
136C2×C42—42no—
137C2×C42—44no—
138C2×C42C4212yes—
139C2×C42C22⋊C412yes—
140C2×C42C4⋊C412yes—
141C2×C42C4⋊C412yes—
142C4⋊C8—44no—
143C22⋊C8—24no—
144C22⋊C8—42no—
145#97—44no—
146#95—42no—
147#95—44no—
148#95—42no—
149#95—42no—
150#95—41no—
151#95—41no—
152#95—42no—
153#95—42no—
154#95—82no—
155C24—21no—
156C24—21no—
157C24—22no—
158C24—42no—
159C24—41no—
160C24—41no—
161C24#9512yes—
162C24#10414yes—
163C24C2×C22⋊C411yescharacteristic
164C22×C4—24no—
165C22×C4—28no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—28no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—24no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—24no—
193C22×C4—42no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—42no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4#9512yes—
213C22×C4#9512yes—
214C22×C4#9512yes—
215C22×C4#9912yes—
216C22×C4#9911yescharacteristic
217C22×C4#9911yescharacteristic
218C22×C4C2×C4211yescharacteristic
219C22×C4C2×C22⋊C411yescharacteristic
220C22×C4C2×C22⋊C411yescharacteristic
221C22×C4C2×C4⋊C411yescharacteristic
222C22×C4C2×C4⋊C411yescharacteristic
223C22×C4C2×C4⋊C411yescharacteristic
224C22×C4AGL(1,ℤ/8)12yes—
225C22×C4#13712yes—
226M16—28no—
227M16—416no—
228M16—48no—
229M16—44no—
230C2×C8—416no—
231C2×C8—44no—
232C2×C8—48no—
233C2×C8—44no—
234C2×C8—82no—
235C2×C8—82no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—84no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—88no—
248C42—22no—
249C42—22no—
250C42—22no—
251C42—22no—
252C42—22no—
253C42—22no—
254C42—22no—
255C42—22no—
256C42—48no—
257C23—21no—
258C23—21no—
259C23—22no—
260C23—22no—
261C23—24no—
262C23—22no—
263C23—24no—
264C23—44no—
265C23—42no—
266C23—42no—
267C23—44no—
268C23—41no—
269C23—41no—
270C23—42no—
271C23—81no—
272C23—81no—
273C23—82no—
274C23—82no—
275C23#33914yes—
276C23C2×#9511yescharacteristic
277C23C2×#9911yescharacteristic
278C23C2×#9911yescharacteristic
279C23#41712yes—
280C23C2×#10412yes—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—28no—
286C2×C4—28no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4#34212yes—
337C2×C4#34212yes—
338C2×C4#41811yescharacteristic
339C2×C4#41911yescharacteristic
340C2×C4#42812yes—
341C8—48no—
342C8—84no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—24no—
352C22—44no—
353C22—42no—
354C22—42no—
355C22—44no—
356C22—44no—
357C22—84no—
358C22—82no—
359C22#150412yes—
360C22#150412yes—
361C22C2×#33911yescharacteristic
362C22#173611yescharacteristic
363C22C2×#34211yescharacteristic
364C22#212211yescharacteristic
365C22#212311yescharacteristic
366C22#213212yes—
367C4—21no—
368C4—22no—
369C4—22no—
370C4—21no—
371C4—22no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—42no—
377C4—42no—
378C4—84no—
379C4—162no—
380C2—22no—
381C2—21no—
382C2—21no—
383C2—44no—
384C2#802011yescharacteristic
385C2#1176512yes—
386C2#1176712yes—
387C2#1257511yescharacteristic
388C2#1258511yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7251C2semidirect
11. #1742C4semidirect
15. #1464C4semidirect

Maximal subgroups

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

subgroupindex
C2×#17422
C2×#14642
#139572
#72512