Finite Group Databasebeta

← #127009

Subgroups — #127009

Order 512.

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

normal subgroups only

This group has 826 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×#1463C211yesmaximal, characteristic
2C22×#324C211yesmaximal, characteristic
3#13924C211yesmaximal, characteristic
4#7189C214yesmaximal, complemented
5C23×C4⋊C4C2211yescharacteristic
6C2×#407—21no—
7C22×C22⋊C8—21no—
8C2×#380C411yescharacteristic
9C2×C4×C22⋊C4C411yescharacteristic
10#1463—22no—
11#1463C414yescomplemented
12C2×#324—21no—
13C2×#324—22no—
14C2×#324—21no—
15C2×#324C414yescomplemented
16C2×#324C2212yes—
17C2×#324C2212yes—
18#1445—21no—
19#1445—21no—
20C4⋊#95C412yes—
21C4⋊#95C2212yes—
22C22⋊#95—21no—
23C22⋊#95—21no—
24C24×C4C2×C411yescharacteristic
25C24×C4D411yescharacteristic
26C24×C4D411yescharacteristic
27C22×M16—41no—
28C23×C8—41no—
29C22×C4⋊C4—41no—
30C22×C4⋊C4—41no—
31C22×C4⋊C4—42no—
32C22×C4⋊C4C2×C412yes—
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C4⋊C4C2×C412yes—
35C22×C4⋊C4C2×C412yes—
36C22×C4⋊C4D411yescharacteristic
37C22×C4⋊C4D411yescharacteristic
38C22×C4⋊C4D412yes—
39C22×C4⋊C4Q812yes—
40C22×C4⋊C4C2311yescharacteristic
41C22×C22⋊C4—21no—
42C22×C22⋊C4—21no—
43C22×C42C2×C412yes—
44#407—24no—
45#407—42no—
46C2×C22⋊C8—24no—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8—41no—
50C2×C22⋊C8—42no—
51C2×C22⋊C8—41no—
52#380—41no—
53#380—41no—
54C4×C22⋊C4—42no—
55C2×#95—21no—
56C2×#95—21no—
57C2×#95—22no—
58C2×#95—22no—
59C2×#95—22no—
60C2×#95—22no—
61#324—24no—
62#324—24no—
63#324—22no—
64#324—22no—
65#324C2×C414yes—
66#324C2×C414yes—
67C25C22⋊C411yescharacteristic
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—22no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—22no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—22no—
77C23×C4—24no—
78C23×C4—22no—
79C23×C4—22no—
80C23×C4—42no—
81C23×C4—82no—
82C23×C4—82no—
83C23×C4C4212yes—
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22⋊C412yes—
88C23×C4C22⋊C412yes—
89C23×C4C4⋊C412yes—
90C23×C4C4⋊C412yes—
91C23×C4C4⋊C412yes—
92C23×C4C22×C411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C2×M16—44no—
96C2×M16—84no—
97C22×C8—44no—
98C22×C8—42no—
99C22×C8—42no—
100C22×C8—84no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4C4212yes—
121C2×C4⋊C4C22⋊C412yes—
122C2×C4⋊C4C22⋊C412yes—
123C2×C4⋊C4C4⋊C412yes—
124C2×C4⋊C4C4⋊C412yes—
125C2×C4⋊C4C4⋊C412yes—
126C2×C4⋊C4C22×C411yescharacteristic
127C2×C4⋊C4C22×C411yescharacteristic
128C2×C4⋊C4C2×D411yescharacteristic
129C2×C4⋊C4C2×Q811yescharacteristic
130C2×C22⋊C4—24no—
131C2×C22⋊C4—42no—
132C2×C22⋊C4—42no—
133C2×C22⋊C4—42no—
134C2×C22⋊C4—42no—
135C2×C22⋊C4—41no—
136C2×C22⋊C4—41no—
137C2×C42—24no—
138C2×C42—42no—
139C2×C42—42no—
140C22⋊C8—24no—
141C22⋊C8—24no—
142C22⋊C8—44no—
143C22⋊C8—42no—
144C22⋊C8—42no—
145C22⋊C8—44no—
146#95—42no—
147#95—42no—
148#95—82no—
149#95—82no—
150#95—82no—
151#95—82no—
152C24—22no—
153C24—22no—
154C24—22no—
155C24—22no—
156C24—42no—
157C24#9512yes—
158C24#9911yescharacteristic
159C24#9912yes—
160C24#9911yescharacteristic
161C24C2×C22⋊C411yescharacteristic
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—48no—
195C22×C4—44no—
196C22×C4—44no—
197C22×C4—44no—
198C22×C4—44no—
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—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—82no—
217C22×C4—82no—
218C22×C4—82no—
219C22×C4—82no—
220C22×C4—82no—
221C22×C4—82no—
222C22×C4—84no—
223C22×C4—84no—
224C22×C4#9512yes—
225C22×C4#9512yes—
226C22×C4#9512yes—
227C22×C4C2×C4211yescharacteristic
228C22×C4C2×C22⋊C411yescharacteristic
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4C2×C4⋊C411yescharacteristic
231C22×C4C2×C4⋊C411yescharacteristic
232C22×C4C2×C4⋊C411yescharacteristic
233C22×C4#13114yes—
234M16—88no—
235C2×C8—44no—
236C2×C8—44no—
237C2×C8—88no—
238C2×C8—84no—
239C2×C8—84no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—84no—
251C22⋊C4—48no—
252C22⋊C4—82no—
253C22⋊C4—82no—
254C42—84no—
255C23—22no—
256C23—22no—
257C23—21no—
258C23—21no—
259C23—22no—
260C23—22no—
261C23—21no—
262C23—21no—
263C23—24no—
264C23—24no—
265C23—24no—
266C23—22no—
267C23—22no—
268C23—22no—
269C23—42no—
270C23—42no—
271C23—44no—
272C23—44no—
273C23—44no—
274C23—44no—
275C23#34212yes—
276C23#34212yes—
277C23#35512yes—
278C23#35612yes—
279C23C2×#9511yescharacteristic
280C23#40812yes—
281C23C2×#9911yescharacteristic
282C23C2×#9911yescharacteristic
283C23#41312yes—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
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—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4#40512yes—
342C2×C4#42412yes—
343C8—84no—
344C8—84no—
345C22—22no—
346C22—22no—
347C22—24no—
348C22—22no—
349C22—22no—
350C22—24no—
351C22—21no—
352C22—22no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—48no—
361C22—44no—
362C22—44no—
363C22—42no—
364C22—42no—
365C22#139414yes—
366C22#146812yes—
367C22#146812yes—
368C22#172911yescharacteristic
369C22C2×#34211yescharacteristic
370C22#174211yescharacteristic
371C22#212712yes—
372C22C2×#35511yescharacteristic
373C22C2×#35611yescharacteristic
374C4—22no—
375C4—22no—
376C4—41no—
377C4—41no—
378C4—42no—
379C4—42no—
380C4—41no—
381C4—41no—
382C4—42no—
383C4—84no—
384C4—162no—
385C2—22no—
386C2—24no—
387C2—21no—
388C2—21no—
389C2—42no—
390C2#791311yescharacteristic
391C2#1155312yes—
392C2#1155512yes—
393C2#1258211yescharacteristic
394C2C2×#139411yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ 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. #7189C2semidirect
11. #1463C4semidirect
15. C2×#324C4semidirect

Maximal subgroups

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

subgroupindex
C2×#14632
C22×#3242
#139242
#71892