Finite Group Databasebeta

← #127604

Subgroups — #127604

Order 512.

1062 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1062 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#13870C211yesmaximal, characteristic
4#7251C214yesmaximal, complemented
5C23×M16—21no—
6C22×#117C411yescharacteristic
7C23×C4⋊C4C2211yescharacteristic
8C2×#407—21no—
9C2×#381—21no—
10C2×#381C411yescharacteristic
11C2×#343—21no—
12C2×#343—22no—
13C2×#343C414yescomplemented
14C2×#343C2212yes—
15#1463—22no—
16#1463C414yescomplemented
17C2×#324—21no—
18C2×#324C2212yes—
19#1441C412yes—
20#1441C2212yes—
21#1436—21no—
22#1436—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—
35C2×#117—44no—
36C2×#117—41no—
37C2×#117—41no—
38C22×C4⋊C4—41no—
39C22×C4⋊C4—41no—
40C22×C4⋊C4—42no—
41C22×C4⋊C4C2×C412yes—
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4C2×C412yes—
44C22×C4⋊C4C2×C412yes—
45C22×C4⋊C4D411yescharacteristic
46C22×C4⋊C4D411yescharacteristic
47C22×C4⋊C4D412yes—
48C22×C4⋊C4Q812yes—
49C22×C4⋊C4C2311yescharacteristic
50C22×C22⋊C4—21no—
51C22×C22⋊C4—21no—
52C22×C42—21no—
53#407—24no—
54#407—42no—
55C2×C22⋊C8—22no—
56C2×C22⋊C8—41no—
57#381—44no—
58#381—42no—
59C2×#95—22no—
60C2×#95—21no—
61C2×#95—22no—
62C2×#95—22no—
63C2×#95—22no—
64C2×#95C2×C412yes—
65#343—24no—
66#343—22no—
67#343C2×C414yes—
68#324—24no—
69#324—22no—
70#324C2×C414yes—
71C25C22⋊C411yescharacteristic
72C23×C4—21no—
73C23×C4—22no—
74C23×C4—21no—
75C23×C4—24no—
76C23×C4—22no—
77C23×C4—22no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—22no—
82C23×C4—21no—
83C23×C4—22no—
84C23×C4—22no—
85C23×C4—22no—
86C23×C4—42no—
87C23×C4—42no—
88C23×C4—42no—
89C23×C4—42no—
90C23×C4C4212yes—
91C23×C4C22⋊C412yes—
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C412yes—
96C23×C4C4⋊C412yes—
97C23×C4C4⋊C412yes—
98C23×C4C4⋊C412yes—
99C23×C4C22×C411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C2×M16—28no—
103C2×M16—216no—
104C2×M16—24no—
105C2×M16—416no—
106C2×M16—48no—
107C2×M16—44no—
108C2×M16—42no—
109C2×M16—48no—
110C2×M16—44no—
111C2×M16—44no—
112C2×M16—82no—
113C2×M16—82no—
114C22×C8—48no—
115C22×C8—44no—
116C22×C8—42no—
117C22×C8—42no—
118#117—48no—
119#117—44no—
120#117—44no—
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—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4C4212yes—
140C2×C4⋊C4C22⋊C412yes—
141C2×C4⋊C4C22⋊C412yes—
142C2×C4⋊C4C4⋊C412yes—
143C2×C4⋊C4C4⋊C412yes—
144C2×C4⋊C4C4⋊C412yes—
145C2×C4⋊C4C22×C411yescharacteristic
146C2×C4⋊C4C22×C411yescharacteristic
147C2×C4⋊C4C2×D411yescharacteristic
148C2×C4⋊C4C2×Q811yescharacteristic
149C2×C22⋊C4—24no—
150C2×C22⋊C4—42no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4—41no—
153C2×C22⋊C4—42no—
154C2×C22⋊C4—42no—
155C2×C22⋊C4—42no—
156C2×C42—42no—
157C2×C42—44no—
158C22⋊C8—24no—
159C22⋊C8—42no—
160C22⋊C8—44no—
161#95—42no—
162#95—42no—
163#95—44no—
164#95—44no—
165#95—44no—
166#95—42no—
167#95—42no—
168C24—22no—
169C24—22no—
170C24—22no—
171C24—22no—
172C24—42no—
173C24#9512yes—
174C24#9911yescharacteristic
175C24#9912yes—
176C24#9911yescharacteristic
177C24C2×C22⋊C411yescharacteristic
178C22×C4—24no—
179C22×C4—28no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—28no—
183C22×C4—24no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—24no—
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—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—24no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—48no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—44no—
226C22×C4—41no—
227C22×C4—44no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—81no—
249C22×C4—81no—
250C22×C4—84no—
251C22×C4#9512yes—
252C22×C4#9512yes—
253C22×C4#9512yes—
254C22×C4#10414yes—
255C22×C4C2×C4211yescharacteristic
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C4⋊C411yescharacteristic
259C22×C4C2×C4⋊C411yescharacteristic
260C22×C4C2×C4⋊C411yescharacteristic
261C22×C4AGL(1,ℤ/8)12yes—
262C22×C4#13712yes—
263M16—28no—
264M16—416no—
265M16—48no—
266M16—44no—
267M16—88no—
268C2×C8—416no—
269C2×C8—44no—
270C2×C8—48no—
271C2×C8—44no—
272C2×C8—82no—
273C2×C8—82no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C22⋊C4—44no—
287C22⋊C4—44no—
288C22⋊C4—84no—
289C42—48no—
290C23—21no—
291C23—21no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—21no—
296C23—21no—
297C23—22no—
298C23—24no—
299C23—24no—
300C23—22no—
301C23—22no—
302C23—24no—
303C23—22no—
304C23—42no—
305C23—42no—
306C23—44no—
307C23—44no—
308C23—44no—
309C23—44no—
310C23#34212yes—
311C23#34212yes—
312C23AΣL(1,𝔽16)12yes—
313C23#35212yes—
314C23C2×#9511yescharacteristic
315C23C2×#9911yescharacteristic
316C23C2×#9911yescharacteristic
317C23#41712yes—
318C23C2×#10412yes—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—28no—
324C2×C4—28no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—81no—
369C2×C4—82no—
370C2×C4—81no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4#33914yes—
384C2×C4#41811yescharacteristic
385C2×C4#41911yescharacteristic
386C2×C4#42812yes—
387C8—48no—
388C8—84no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—22no—
393C22—21no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—24no—
398C22—22no—
399C22—22no—
400C22—24no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—48no—
405C22—44no—
406C22—44no—
407C22—42no—
408C22—42no—
409C22#139014yes—
410C22#150412yes—
411C22#150412yes—
412C22C2×#33911yescharacteristic
413C22#173611yescharacteristic
414C22C2×#34211yescharacteristic
415C22C2×AΣL(1,𝔽16)11yescharacteristic
416C22C2×#35211yescharacteristic
417C22#213312yes—
418C4—21no—
419C4—24no—
420C4—21no—
421C4—22no—
422C4—41no—
423C4—42no—
424C4—41no—
425C4—41no—
426C4—42no—
427C4—41no—
428C4—84no—
429C4—162no—
430C2—22no—
431C2—24no—
432C2—21no—
433C2—21no—
434C2—42no—
435C2#802011yescharacteristic
436C2#1176212yes—
437C2#1176412yes—
438C2C2×#139011yescharacteristic
439C2#1258611yescharacteristic
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
13. C2×#343C4semidirect
16. #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
#138702
#72512