Finite Group Databasebeta

← #125370

Subgroups — #125370

Order 512.

580 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 90 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
1#15165C211yesmaximal, characteristic
2C4×#323C211yesmaximal, characteristic
3#7911C211yesmaximal, characteristic
4#7123C212yesmaximal, complemented
5#7122C212yesmaximal, complemented
6C2×#407—21no—
7C2×#396—21no—
8C2×#389C411yescharacteristic
9C2×C4×C22⋊C4C2211yescharacteristic
10C4×C22⋊C8—21no—
11C2×#323—21no—
12#1444C2211yescharacteristic
13#1442C2211yescharacteristic
14#1441—21no—
15#1439C411yescharacteristic
16#1439C411yescharacteristic
17#1436—21no—
18C4⋊#95C411yescharacteristic
19C22⋊#95—21no—
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C24×C4D411yescharacteristic
24C22×M16—41no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C22⋊C4—21no—
31C22×C22⋊C4—21no—
32C22×C22⋊C4C2×C411yescharacteristic
33C22×C22⋊C4D411yescharacteristic
34C22×C42C2×C411yescharacteristic
35C22×C42D411yescharacteristic
36#407—24no—
37#407—42no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—41no—
41C2×C4×C8—41no—
42#396—44no—
43#389—41no—
44#389—41no—
45C4×C22⋊C4—41no—
46C4×C22⋊C4—41no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95C2×C411yescharacteristic
55C2×#95C2×C411yescharacteristic
56C2×#95D411yescharacteristic
57C2×#95C2311yescharacteristic
58#336—41no—
59#323—24no—
60C25C22⋊C411yescharacteristic
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—22no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—81no—
69C23×C4—81no—
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4SD1612yes—
74C23×C4Q1612yes—
75C23×C4C22×C411yescharacteristic
76C23×C4C2×D411yescharacteristic
77C23×C4C2×D411yescharacteristic
78C2×M16—44no—
79C2×M16—82no—
80C2×M16—82no—
81C22×C8—42no—
82C22×C8—42no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—81no—
94C2×C4⋊C4—81no—
95C2×C4⋊C4—81no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—41no—
101C2×C22⋊C4—41no—
102C2×C22⋊C4—42no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—81no—
107C2×C22⋊C4—81no—
108C2×C42—21no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—81no—
113C2×C42—81no—
114C22⋊C8—24no—
115C22⋊C8—24no—
116C22⋊C8—44no—
117C22⋊C8—42no—
118C22⋊C8—42no—
119C4×C8—82no—
120#95—22no—
121#95—22no—
122#95—41no—
123#95—41no—
124#95—42no—
125#95—41no—
126#95—41no—
127#95—42no—
128#95—42no—
129#95—81no—
130#95—81no—
131#95—81no—
132C24—21no—
133C24—21no—
134C24—21no—
135C24—21no—
136C24—21no—
137C24—21no—
138C24—41no—
139C24—41no—
140C24—41no—
141C24—41no—
142C24#10312yes—
143C24C2×C22⋊C411yescharacteristic
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—81no—
201C22×C4—81no—
202C22×C4—82no—
203C22×C4—82no—
204C22×C4—84no—
205C22×C4#9911yescharacteristic
206C22×C4#9911yescharacteristic
207C22×C4#10112yes—
208C22×C4#10312yes—
209C22×C4#10412yes—
210C22×C4#10412yes—
211C22×C4#13112yes—
212C22×C4C2×SD1611yescharacteristic
213C22×C4C2×Q1611yescharacteristic
214C22×C4AGL(1,ℤ/8)12yes—
215M16—88no—
216C2×C8—44no—
217C2×C8—44no—
218C2×C8—82no—
219C2×C8—82no—
220C2×C8—82no—
221C2×C8—82no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—84no—
225C4⋊C4—84no—
226C4⋊C4—84no—
227C4⋊C4—82no—
228C22⋊C4—44no—
229C22⋊C4—44no—
230C22⋊C4—84no—
231C22⋊C4—84no—
232C42—42no—
233C42—81no—
234C42—81no—
235C42—81no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—81no—
258C23—81no—
259C23—81no—
260C23—81no—
261C23—81no—
262C23—81no—
263C23#40811yescharacteristic
264C23C2×#9911yescharacteristic
265C23C2×#10111yescharacteristic
266C23C2×#10311yescharacteristic
267C23#41811yescharacteristic
268C23C2×#10411yescharacteristic
269C23C2×#10411yescharacteristic
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4#32812yes—
336C2×C4#32812yes—
337C8—84no—
338C8—84no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—44no—
352C22—42no—
353C22—42no—
354C22—42no—
355C22—82no—
356C22—82no—
357C22—82no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22#147211yescharacteristic
363C22#148411yescharacteristic
364C22#149311yescharacteristic
365C22C2×#32811yescharacteristic
366C22#150711yescharacteristic
367C22#152611yescharacteristic
368C22#154911yescharacteristic
369C22#212011yescharacteristic
370C22#212111yescharacteristic
371C22#212711yescharacteristic
372C22#213211yescharacteristic
373C4—22no—
374C4—22no—
375C4—42no—
376C4—42no—
377C4—42no—
378C4—41no—
379C4—41no—
380C4—42no—
381C4—82no—
382C4—82no—
383C4—161no—
384C4—161no—
385C2—21no—
386C2—21no—
387C2—21no—
388C2—21no—
389C2—82no—
390C2#768811yescharacteristic
391C2#1157211yescharacteristic
392C2#1157711yescharacteristic
393C2#1171611yescharacteristic
394C2#1172111yescharacteristic
395C2#1177411yescharacteristic
396C2#1178311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7123C2semidirect
5. #7122C2semidirect

Maximal subgroups

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

subgroupindex
#151652
C4×#3232
#79112
#71232
#71222