Finite Group Databasebeta

← #127657

Subgroups — #127657

Order 512.

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

normal subgroups only

This group has 894 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×#1821C211yesmaximal, characteristic
2C22×#324C211yesmaximal, characteristic
3#13870C211yesmaximal, characteristic
4#7251C214yesmaximal, complemented
5C22×#117C411yescharacteristic
6C23×C4⋊C4C2211yescharacteristic
7C2×#407—21no—
8C22×C22⋊C8—21no—
9C2×#381—21no—
10C2×#381C411yescharacteristic
11#1821—22no—
12#1821C414yescomplemented
13C2×#343—21no—
14C2×#343C2212yes—
15C2×#324—21no—
16C2×#324—22no—
17C2×#324C414yescomplemented
18C2×#324C2212yes—
19#1441C412yes—
20#1441C2212yes—
21#1436—21no—
22#1436—21no—
23C22⋊#95—21no—
24C24×C4C2×C411yescharacteristic
25C24×C4D411yescharacteristic
26C24×C4D411yescharacteristic
27C22×M16—22no—
28C23×C8—41no—
29C2×#117—44no—
30C2×#117—41no—
31C2×#117—41no—
32C22×C4⋊C4—41no—
33C22×C4⋊C4—41no—
34C22×C4⋊C4—42no—
35C22×C4⋊C4C2×C412yes—
36C22×C4⋊C4C2×C411yescharacteristic
37C22×C4⋊C4C2×C412yes—
38C22×C4⋊C4C2×C412yes—
39C22×C4⋊C4D411yescharacteristic
40C22×C4⋊C4D411yescharacteristic
41C22×C4⋊C4D412yes—
42C22×C4⋊C4Q812yes—
43C22×C4⋊C4C2311yescharacteristic
44C22×C22⋊C4—21no—
45C22×C22⋊C4—21no—
46C22×C42—21no—
47#407—24no—
48#407—42no—
49C2×C22⋊C8—24no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—41no—
52C2×C22⋊C8—42no—
53C2×C22⋊C8—41no—
54C2×C22⋊C8—41no—
55#381—44no—
56#381—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—21no—
72C23×C4—21no—
73C23×C4—22no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—22no—
77C23×C4—22no—
78C23×C4—21no—
79C23×C4—24no—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—42no—
83C23×C4—42no—
84C23×C4—42no—
85C23×C4—42no—
86C23×C4—42no—
87C23×C4C4212yes—
88C23×C4C22⋊C411yescharacteristic
89C23×C4C22⋊C412yes—
90C23×C4C22⋊C411yescharacteristic
91C23×C4C22⋊C411yescharacteristic
92C23×C4C22⋊C412yes—
93C23×C4C4⋊C412yes—
94C23×C4C4⋊C412yes—
95C23×C4C4⋊C412yes—
96C23×C4C22×C411yescharacteristic
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C2×M16—24no—
100C2×M16—42no—
101C2×M16—44no—
102C2×M16—44no—
103C22×C8—44no—
104C22×C8—42no—
105C22×C8—42no—
106C22×C8—82no—
107C22×C8—82no—
108#117—48no—
109#117—44no—
110#117—44no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—44no—
129C2×C4⋊C4C4212yes—
130C2×C4⋊C4C22⋊C412yes—
131C2×C4⋊C4C22⋊C412yes—
132C2×C4⋊C4C4⋊C412yes—
133C2×C4⋊C4C4⋊C412yes—
134C2×C4⋊C4C4⋊C412yes—
135C2×C4⋊C4C22×C411yescharacteristic
136C2×C4⋊C4C22×C411yescharacteristic
137C2×C4⋊C4C2×D411yescharacteristic
138C2×C4⋊C4C2×Q811yescharacteristic
139C2×C22⋊C4—24no—
140C2×C22⋊C4—42no—
141C2×C22⋊C4—41no—
142C2×C22⋊C4—41no—
143C2×C22⋊C4—42no—
144C2×C22⋊C4—42no—
145C2×C22⋊C4—42no—
146C2×C42—42no—
147C2×C42—44no—
148C22⋊C8—24no—
149C22⋊C8—44no—
150C22⋊C8—42no—
151C22⋊C8—44no—
152C22⋊C8—44no—
153#95—42no—
154#95—44no—
155#95—42no—
156#95—44no—
157#95—44no—
158#95—44no—
159C24—22no—
160C24—22no—
161C24—22no—
162C24—22no—
163C24—42no—
164C24#9512yes—
165C24#9911yescharacteristic
166C24#9912yes—
167C24#9911yescharacteristic
168C24C2×C22⋊C411yescharacteristic
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
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—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—48no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—48no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—44no—
214C22×C4—41no—
215C22×C4—44no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—81no—
237C22×C4—81no—
238C22×C4—84no—
239C22×C4#9512yes—
240C22×C4#9512yes—
241C22×C4#9512yes—
242C22×C4C2×C4211yescharacteristic
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C2×C4⋊C411yescharacteristic
246C22×C4C2×C4⋊C411yescharacteristic
247C22×C4C2×C4⋊C411yescharacteristic
248C22×C4#13514yes—
249M16—44no—
250M16—44no—
251M16—84no—
252C2×C8—44no—
253C2×C8—44no—
254C2×C8—84no—
255C2×C8—82no—
256C2×C8—82no—
257C2×C8—88no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C22⋊C4—44no—
271C22⋊C4—44no—
272C22⋊C4—84no—
273C42—48no—
274C23—21no—
275C23—21no—
276C23—22no—
277C23—22no—
278C23—22no—
279C23—21no—
280C23—21no—
281C23—22no—
282C23—24no—
283C23—24no—
284C23—22no—
285C23—22no—
286C23—24no—
287C23—22no—
288C23—42no—
289C23—42no—
290C23—44no—
291C23—44no—
292C23—44no—
293C23—44no—
294C23#34212yes—
295C23#34212yes—
296C23AΣL(1,𝔽16)12yes—
297C23#35212yes—
298C23C2×#9511yescharacteristic
299C23C2×#9911yescharacteristic
300C23C2×#9911yescharacteristic
301C23#41612yes—
302C23#42112yes—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—81no—
350C2×C4—82no—
351C2×C4—81no—
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—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4#41612yes—
365C2×C4#42712yes—
366C8—84no—
367C8—84no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—22no—
372C22—21no—
373C22—22no—
374C22—22no—
375C22—24no—
376C22—22no—
377C22—22no—
378C22—24no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—48no—
384C22—44no—
385C22—44no—
386C22—42no—
387C22—42no—
388C22#139014yes—
389C22#151612yes—
390C22#151612yes—
391C22#173311yescharacteristic
392C22#173511yescharacteristic
393C22C2×#34211yescharacteristic
394C22C2×AΣL(1,𝔽16)11yescharacteristic
395C22C2×#35211yescharacteristic
396C22#213312yes—
397C4—22no—
398C4—22no—
399C4—41no—
400C4—42no—
401C4—41no—
402C4—41no—
403C4—42no—
404C4—41no—
405C4—42no—
406C4—84no—
407C4—162no—
408C2—22no—
409C2—24no—
410C2—21no—
411C2—21no—
412C2—42no—
413C2#802411yescharacteristic
414C2#1184812yes—
415C2#1185012yes—
416C2C2×#139011yescharacteristic
417C2#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
12. #1821C4semidirect
17. C2×#324C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18212
C22×#3242
#138702
#72512