Finite Group Databasebeta

← #127606

Subgroups — #127606

Order 512.

994 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 162 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
1C22×#343C211yesmaximal, characteristic
2C2×#1463C211yesmaximal, characteristic
3#13923C211yesmaximal, characteristic
4#7251C214yesmaximal, complemented
5C23×M16—21no—
6C23×C4⋊C4C2211yescharacteristic
7C2×#407—21no—
8C2×#380C411yescharacteristic
9C2×C4×C22⋊C4C411yescharacteristic
10C2×#343—21no—
11C2×#343—22no—
12C2×#343C414yescomplemented
13C2×#343C2212yes—
14#1463—22no—
15#1463C414yescomplemented
16C2×#324—21no—
17C2×#324C2212yes—
18#1441C412yes—
19#1441C2212yes—
20#1436—21no—
21#1436—21no—
22C22⋊#95—21no—
23C22⋊#95—21no—
24C24×C4C2×C411yescharacteristic
25C24×C4D411yescharacteristic
26C24×C4D411yescharacteristic
27C22×M16—28no—
28C22×M16—24no—
29C22×M16—22no—
30C22×M16—42no—
31C22×M16—41no—
32C22×M16—44no—
33C22×M16—41no—
34C23×C8—41no—
35C22×C4⋊C4—41no—
36C22×C4⋊C4—41no—
37C22×C4⋊C4—42no—
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4C2×C412yes—
40C22×C4⋊C4C2×C412yes—
41C22×C4⋊C4C2×C412yes—
42C22×C4⋊C4D411yescharacteristic
43C22×C4⋊C4D411yescharacteristic
44C22×C4⋊C4D412yes—
45C22×C4⋊C4Q812yes—
46C22×C4⋊C4C2311yescharacteristic
47C22×C22⋊C4—21no—
48C22×C22⋊C4—21no—
49C22×C42—21no—
50#407—24no—
51#407—42no—
52C2×C22⋊C8—22no—
53C2×C22⋊C8—41no—
54#380—41no—
55#380—41no—
56C4×C22⋊C4—42no—
57C2×#95—22no—
58C2×#95—21no—
59C2×#95—22no—
60C2×#95—22no—
61C2×#95—22no—
62C2×#95C2×C412yes—
63#343—24no—
64#343—22no—
65#343C2×C414yes—
66#324—24no—
67#324—22no—
68#324C2×C414yes—
69C25C22⋊C411yescharacteristic
70C23×C4—21no—
71C23×C4—22no—
72C23×C4—21no—
73C23×C4—24no—
74C23×C4—22no—
75C23×C4—22no—
76C23×C4—21no—
77C23×C4—22no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4—22no—
84C23×C4—82no—
85C23×C4—82no—
86C23×C4C4212yes—
87C23×C4C22⋊C412yes—
88C23×C4C22⋊C411yescharacteristic
89C23×C4C22⋊C411yescharacteristic
90C23×C4C22⋊C411yescharacteristic
91C23×C4C22⋊C412yes—
92C23×C4C4⋊C412yes—
93C23×C4C4⋊C412yes—
94C23×C4C4⋊C412yes—
95C23×C4C22×C411yescharacteristic
96C23×C4C2×D411yescharacteristic
97C23×C4C2×D411yescharacteristic
98C2×M16—28no—
99C2×M16—216no—
100C2×M16—24no—
101C2×M16—416no—
102C2×M16—48no—
103C2×M16—44no—
104C2×M16—42no—
105C2×M16—48no—
106C2×M16—44no—
107C2×M16—44no—
108C2×M16—84no—
109C22×C8—48no—
110C22×C8—44no—
111C22×C8—42no—
112C22×C8—42no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—82no—
131C2×C4⋊C4C4212yes—
132C2×C4⋊C4C22⋊C412yes—
133C2×C4⋊C4C22⋊C412yes—
134C2×C4⋊C4C4⋊C412yes—
135C2×C4⋊C4C4⋊C412yes—
136C2×C4⋊C4C4⋊C412yes—
137C2×C4⋊C4C22×C411yescharacteristic
138C2×C4⋊C4C22×C411yescharacteristic
139C2×C4⋊C4C2×D411yescharacteristic
140C2×C4⋊C4C2×Q811yescharacteristic
141C2×C22⋊C4—24no—
142C2×C22⋊C4—42no—
143C2×C22⋊C4—42no—
144C2×C22⋊C4—42no—
145C2×C22⋊C4—42no—
146C2×C22⋊C4—41no—
147C2×C22⋊C4—41no—
148C2×C42—42no—
149C2×C42—82no—
150C22⋊C8—24no—
151C22⋊C8—42no—
152C22⋊C8—44no—
153#95—42no—
154#95—42no—
155#95—44no—
156#95—42no—
157#95—42no—
158#95—82no—
159#95—82no—
160C24—22no—
161C24—22no—
162C24—22no—
163C24—22no—
164C24—42no—
165C24#9512yes—
166C24#9911yescharacteristic
167C24#9912yes—
168C24#9911yescharacteristic
169C24C2×C22⋊C411yescharacteristic
170C22×C4—24no—
171C22×C4—28no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—28no—
175C22×C4—24no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—24no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—24no—
208C22×C4—44no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—82no—
227C22×C4—82no—
228C22×C4—82no—
229C22×C4—82no—
230C22×C4—82no—
231C22×C4—84no—
232C22×C4—84no—
233C22×C4—82no—
234C22×C4—82no—
235C22×C4#9512yes—
236C22×C4#9512yes—
237C22×C4#9512yes—
238C22×C4#10414yes—
239C22×C4C2×C4211yescharacteristic
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×C4⋊C411yescharacteristic
243C22×C4C2×C4⋊C411yescharacteristic
244C22×C4C2×C4⋊C411yescharacteristic
245C22×C4AGL(1,ℤ/8)12yes—
246C22×C4#13712yes—
247M16—28no—
248M16—416no—
249M16—48no—
250M16—44no—
251M16—88no—
252C2×C8—416no—
253C2×C8—44no—
254C2×C8—48no—
255C2×C8—44no—
256C2×C8—84no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—84no—
268C22⋊C4—48no—
269C22⋊C4—82no—
270C22⋊C4—82no—
271C42—84no—
272C23—22no—
273C23—22no—
274C23—21no—
275C23—21no—
276C23—22no—
277C23—22no—
278C23—21no—
279C23—21no—
280C23—24no—
281C23—24no—
282C23—22no—
283C23—22no—
284C23—24no—
285C23—22no—
286C23—42no—
287C23—42no—
288C23—44no—
289C23—44no—
290C23—44no—
291C23—44no—
292C23#34212yes—
293C23#34212yes—
294C23#35512yes—
295C23#35612yes—
296C23C2×#9511yescharacteristic
297C23C2×#9911yescharacteristic
298C23C2×#9911yescharacteristic
299C23#41712yes—
300C23C2×#10412yes—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—28no—
306C2×C4—28no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—81no—
343C2×C4—81no—
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—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4#33914yes—
362C2×C4#41811yescharacteristic
363C2×C4#41911yescharacteristic
364C2×C4#42812yes—
365C8—48no—
366C8—84no—
367C22—21no—
368C22—21no—
369C22—22no—
370C22—21no—
371C22—21no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—24no—
376C22—22no—
377C22—22no—
378C22—24no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—48no—
383C22—44no—
384C22—44no—
385C22—42no—
386C22—42no—
387C22#139414yes—
388C22#150412yes—
389C22#150412yes—
390C22C2×#33911yescharacteristic
391C22#173611yescharacteristic
392C22C2×#34211yescharacteristic
393C22#212112yes—
394C22C2×#35511yescharacteristic
395C22C2×#35611yescharacteristic
396C4—21no—
397C4—24no—
398C4—21no—
399C4—22no—
400C4—41no—
401C4—42no—
402C4—41no—
403C4—42no—
404C4—41no—
405C4—41no—
406C4—84no—
407C4—162no—
408C2—22no—
409C2—24no—
410C2—21no—
411C2—21no—
412C2—42no—
413C2#802011yescharacteristic
414C2#1176612yes—
415C2#1176812yes—
416C2#1257411yescharacteristic
417C2C2×#139411yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7251C2semidirect
12. C2×#343C4semidirect
15. #1463C4semidirect

Maximal subgroups

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

subgroupindex
C22×#3432
C2×#14632
#139232
#72512