Finite Group Databasebeta

← #124388

Subgroups — #124388

Order 512.

558 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 92 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#14229C211yesmaximal, characteristic
2#8700C211yesmaximal, characteristic
3C4×#323C211yesmaximal, characteristic
4#7122C212yesmaximal, complemented
5#7104C212yesmaximal, complemented
6C2×#397C411yescharacteristic
7C2×#389C411yescharacteristic
8C2×#388—21no—
9C2×C4×C22⋊C4C2211yescharacteristic
10#1779—21no—
11C4×C22⋊C8—21no—
12C2×#323—21no—
13#1445C2211yescharacteristic
14#1442C2211yescharacteristic
15#1439C411yescharacteristic
16#1436—21no—
17#1434—21no—
18C22⋊#95—21no—
19C4×#95C411yescharacteristic
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C24×C4D411yescharacteristic
24C22×C4⋊C4—21no—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C22⋊C4—21no—
28C22×C22⋊C4—21no—
29C22×C22⋊C4C2×C411yescharacteristic
30C22×C22⋊C4D411yescharacteristic
31C22×C42—21no—
32C22×C42C2×C411yescharacteristic
33C22×C42D411yescharacteristic
34C2×C22⋊C8—22no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—41no—
37C2×C22⋊C8—41no—
38C2×C4×C8—41no—
39#397—22no—
40#397C814yescomplemented
41#389—41no—
42#389—41no—
43#388—44no—
44C4×C22⋊C4—41no—
45C4×C22⋊C4—41no—
46C2×#95—21no—
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—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—22no—
67C23×C4—21no—
68C23×C4—41no—
69C23×C4—42no—
70C23×C4—81no—
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22×C411yescharacteristic
75C23×C4C2×D411yescharacteristic
76C23×C4C2×D411yescharacteristic
77C22×C8—42no—
78C22×C8—42no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—81no—
87C2×C4⋊C4C2×C812yes—
88C2×C22⋊C4—24no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—41no—
92C2×C22⋊C4—41no—
93C2×C22⋊C4—41no—
94C2×C22⋊C4—42no—
95C2×C22⋊C4—42no—
96C2×C22⋊C4—42no—
97C2×C22⋊C4—42no—
98C2×C22⋊C4—81no—
99C2×C22⋊C4C2×C812yes—
100C2×C22⋊C4M1612yes—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—42no—
106C2×C42—42no—
107C2×C42—42no—
108C2×C42—81no—
109C2×C42—81no—
110C22⋊C8—24no—
111C22⋊C8—42no—
112C22⋊C8—42no—
113C22⋊C8—42no—
114C22⋊C8—82no—
115C22⋊C8—82no—
116C4×C8—82no—
117#95—24no—
118#95—21no—
119#95—41no—
120#95—41no—
121#95—41no—
122#95—42no—
123#95—42no—
124#95—41no—
125#95—41no—
126#95—42no—
127#95—42no—
128#95—42no—
129#95—81no—
130#95C2×C812yes—
131C24—21no—
132C24—21no—
133C24—21no—
134C24—21no—
135C24—21no—
136C24—21no—
137C24—41no—
138C24—41no—
139C24—41no—
140C24—41no—
141C24C22⋊C812yes—
142C24C2×C22⋊C411yescharacteristic
143C22×C4—21no—
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—21no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—41no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—41no—
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—42no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—81no—
200C22×C4—81no—
201C22×C4—84no—
202C22×C4—84no—
203C22×C4C22⋊C812yes—
204C22×C4#9911yescharacteristic
205C22×C4#9911yescharacteristic
206C22×C4#10412yes—
207C22×C4C22×C811yescharacteristic
208C22×C4C2×M1611yescharacteristic
209C22×C4#13512yes—
210C22×C4AGL(1,ℤ/8)12yes—
211C2×C8—44no—
212C2×C8—82no—
213C2×C8—82no—
214C2×C8—82no—
215C2×C8—82no—
216C2×C8—82no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—84no—
220C4⋊C4—82no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—44no—
224C22⋊C4—48no—
225C22⋊C4—84no—
226C42—42no—
227C42—81no—
228C42—81no—
229C42—81no—
230C42—84no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—81no—
253C23—81no—
254C23—81no—
255C23—81no—
256C23—81no—
257C23—81no—
258C23C2×C22⋊C811yescharacteristic
259C23C2×#9911yescharacteristic
260C23#41311yescharacteristic
261C23#41611yescharacteristic
262C23#41811yescharacteristic
263C23C2×#10411yescharacteristic
264C23#42111yescharacteristic
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—84no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—84no—
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#35412yes—
330C8—84no—
331C8—162no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—44no—
345C22—42no—
346C22—42no—
347C22—42no—
348C22—82no—
349C22—82no—
350C22—82no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22#145911yescharacteristic
356C22#147611yescharacteristic
357C22#147811yescharacteristic
358C22#150711yescharacteristic
359C22#151611yescharacteristic
360C22#152911yescharacteristic
361C22#155311yescharacteristic
362C22#212111yescharacteristic
363C22#212311yescharacteristic
364C22C2×#35411yescharacteristic
365C22#213211yescharacteristic
366C4—22no—
367C4—42no—
368C4—42no—
369C4—42no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—42no—
374C4—82no—
375C4—82no—
376C4—82no—
377C4—161no—
378C2—21no—
379C2—21no—
380C2—21no—
381C2—21no—
382C2—82no—
383C2#758211yescharacteristic
384C2#1147411yescharacteristic
385C2#1147711yescharacteristic
386C2#1177711yescharacteristic
387C2#1178311yescharacteristic
388C2#1184511yescharacteristic
389C2#1185311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C22×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. #7122C2semidirect
5. #7104C2semidirect
40. #397C8semidirect

Maximal subgroups

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

subgroupindex
#142292
#87002
C4×#3232
#71222
#71042