Finite Group Databasebeta

← #127601

Subgroups — #127601

Order 512.

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

normal subgroups only

This group has 936 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×#1740C211yesmaximal, characteristic
2C2×#1464C211yesmaximal, characteristic
3#13938C211yesmaximal, characteristic
4#7250C214yesmaximal
5C22×#117—21no—
6C22×#117C2211yescharacteristic
7C2×#432—21no—
8C2×#383C411yescharacteristic, complemented
9C2×C4×C4⋊C4C411yescharacteristic, complemented
10#1740—22no—
11#1740C414yescomplemented
12C2×#342—21no—
13C2×#342C2212yes—
14#1464—22no—
15#1464C414yescomplemented
16C2×#324—21no—
17C2×#323C2212yes—
18#1441C412yescomplemented
19#1439C2212yes—
20#1437—21no—
21#1436—21no—
22#1434—21no—
23C22⋊#95—21no—
24C24×C4D411yescharacteristic
25C2×#117—28no—
26C2×#117—24no—
27C2×#117—41no—
28C2×#117—41no—
29C2×#117—42no—
30C2×#117—42no—
31C2×#117—44no—
32C2×#117C2×C412yes—
33C2×#117C2×C412yes—
34C2×#117D412yes—
35C2×#117Q812yes—
36C22×C4⋊C4—41no—
37C22×C4⋊C4C2×C412yes—
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4D411yescharacteristic
40C22×C22⋊C4—22no—
41C22×C22⋊C4—22no—
42C22×C22⋊C4D411yescharacteristic
43C22×C22⋊C4C2311yescharacteristic
44C22×C42—21no—
45C22×C42—41no—
46C22×C42C2×C411yescharacteristic
47C22×C42D411yescharacteristic
48#432—24no—
49#432—42no—
50C2×C4⋊C8—41no—
51C2×C22⋊C8—22no—
52C2×#97—41no—
53#383—42no—
54C4×C4⋊C4—41no—
55C4×C4⋊C4—41no—
56C2×#95—22no—
57C2×#95—22no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95—22no—
62C2×#95C2×C412yes—
63#342—24no—
64#342—22no—
65#342C2×C414yes—
66#324—24no—
67#323—22no—
68#323C2×C414yes—
69C25C2×D411yescharacteristic
70C23×C4—22no—
71C23×C4—22no—
72C23×C4—21no—
73C23×C4—22no—
74C23×C4—21no—
75C23×C4—24no—
76C23×C4—22no—
77C23×C4—22no—
78C23×C4—42no—
79C23×C4C22⋊C412yes—
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22⋊C411yescharacteristic
84C23×C4C4⋊C412yes—
85C23×C4C22×C411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C22×C8—42no—
88#117—216no—
89#117—42no—
90#117—42no—
91#117—42no—
92#117—42no—
93#117—42no—
94#117—42no—
95#117—42no—
96#117—42no—
97#117—416no—
98#117—48no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—48no—
105C2×C4⋊C4—44no—
106C2×C4⋊C4—44no—
107C2×C4⋊C4C4212yes—
108C2×C4⋊C4C22⋊C412yes—
109C2×C4⋊C4C4⋊C412yes—
110C2×C4⋊C4C4⋊C412yes—
111C2×C22⋊C4—28no—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—41no—
114C2×C22⋊C4—41no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—44no—
119C2×C22⋊C4—44no—
120C2×C22⋊C4—42no—
121C2×C22⋊C4—42no—
122C2×C22⋊C4—82no—
123C2×C22⋊C4C22⋊C412yes—
124C2×C22⋊C4C4⋊C412yes—
125C2×C22⋊C4C22×C411yescharacteristic
126C2×C22⋊C4C22×C411yescharacteristic
127C2×C22⋊C4C2×D411yescharacteristic
128C2×C22⋊C4C2×Q811yescharacteristic
129C2×C42—21no—
130C2×C42—21no—
131C2×C42—22no—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42—22no—
135C2×C42—48no—
136C2×C42—44no—
137C2×C42—42no—
138C2×C42—82no—
139C2×C42C4212yes—
140C2×C42C22⋊C412yes—
141C2×C42C4⋊C412yes—
142C2×C42C4⋊C412yes—
143C4⋊C8—44no—
144C22⋊C8—24no—
145C22⋊C8—42no—
146#97—44no—
147#95—42no—
148#95—44no—
149#95—42no—
150#95—41no—
151#95—41no—
152#95—42no—
153#95—44no—
154#95—82no—
155C24—21no—
156C24—21no—
157C24—22no—
158C24—42no—
159C24—41no—
160C24—41no—
161C24C2×C22⋊C411yescharacteristic
162C24C2×C22⋊C411yescharacteristic
163C24C2×C4⋊C411yescharacteristic
164C24AGL(1,ℤ/8)12yes—
165C24#13712yes—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—28no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—28no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—24no—
195C22×C4—41no—
196C22×C4—42no—
197C22×C4—41no—
198C22×C4—42no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—42no—
202C22×C4—44no—
203C22×C4—48no—
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—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—84no—
217C22×C4—82no—
218C22×C4—82no—
219C22×C4#9512yes—
220C22×C4#9512yes—
221C22×C4#9512yes—
222C22×C4#9512yes—
223C22×C4#9912yes—
224C22×C4#9911yescharacteristic
225C22×C4#9911yescharacteristic
226C22×C4#10414yes—
227C22×C4C2×C4211yescharacteristic
228C22×C4C2×C22⋊C411yescharacteristic
229C22×C4C2×C4⋊C411yescharacteristic
230C22×C4C2×C4⋊C411yescharacteristic
231C2×C8—44no—
232C2×C8—82no—
233C2×C8—82no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—416no—
239C4⋊C4—44no—
240C4⋊C4—44no—
241C22⋊C4—28no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—41no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—41no—
248C22⋊C4—48no—
249C22⋊C4—44no—
250C22⋊C4—84no—
251C22⋊C4—84no—
252C42—22no—
253C42—22no—
254C42—22no—
255C42—22no—
256C42—22no—
257C42—22no—
258C42—22no—
259C42—22no—
260C42—416no—
261C42—84no—
262C23—22no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—22no—
268C23—22no—
269C23—21no—
270C23—21no—
271C23—22no—
272C23—22no—
273C23—42no—
274C23—42no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—41no—
283C23—41no—
284C23—81no—
285C23—81no—
286C23—82no—
287C23—82no—
288C23C2×#9511yescharacteristic
289C23C2×#9911yescharacteristic
290C23C2×#9911yescharacteristic
291C23#41712yes—
292C23#41811yescharacteristic
293C23#41911yescharacteristic
294C23C2×#10412yes—
295C23#42812yes—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—28no—
301C2×C4—28no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—24no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—48no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—84no—
340C2×C4—84no—
341C2×C4—88no—
342C2×C4—88no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4#33914yes—
353C2×C4#34212yes—
354C2×C4#34212yes—
355C8—84no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—21no—
365C22—21no—
366C22—22no—
367C22—41no—
368C22—41no—
369C22—42no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—42no—
378C22—82no—
379C22—82no—
380C22—81no—
381C22—81no—
382C22#150412yes—
383C22#150412yes—
384C22C2×#33911yescharacteristic
385C22#173611yescharacteristic
386C22C2×#34211yescharacteristic
387C22#212012yes—
388C22#212811yescharacteristic
389C22#212911yescharacteristic
390C4—24no—
391C4—22no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—42no—
396C4—41no—
397C4—42no—
398C4—41no—
399C4—88no—
400C4—84no—
401C4—162no—
402C2—22no—
403C2—21no—
404C2—21no—
405C2—41no—
406C2—41no—
407C2—42no—
408C2#802011yescharacteristic
409C2#1175412yes—
410C2#1175612yes—
411C2#1257311yescharacteristic
412C2#1258311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. C2×#383C4semidirect
9. C2×C4×C4⋊C4C4semidirect
11. #1740C4semidirect
15. #1464C4semidirect
18. #1441C4semidirect

Maximal subgroups

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

subgroupindex
C2×#17402
C2×#14642
#139382
#72502