Finite Group Databasebeta

← #125239

Subgroups — #125239

Order 512.

558 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 78 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#14969C211yesmaximal, characteristic
2#8700C211yesmaximal, characteristic
3C4×#323C211yesmaximal, characteristic
4#7123C212yesmaximal, complemented
5#7123C212yesmaximal, complemented
6C2×#389C411yescharacteristic
7C2×#387—21no—
8C2×#385—21no—
9C2×C4×C22⋊C4C2211yescharacteristic
10#1779—21no—
11C4×C22⋊C8—21no—
12C2×#323—21no—
13#1445—21no—
14#1444C2211yescharacteristic
15#1444C2211yescharacteristic
16#1439—21no—
17#1439C411yescharacteristic
18#1439C411yescharacteristic
19#1434C411yescharacteristic
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C24×C4D411yescharacteristic
24C22×C4⋊C4—21no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C22⋊C4—21no—
30C22×C22⋊C4—21no—
31C22×C22⋊C4C2×C411yescharacteristic
32C22×C22⋊C4D411yescharacteristic
33C22×C42C2×C411yescharacteristic
34C22×C42D411yescharacteristic
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—41no—
38C2×C22⋊C8—41no—
39C2×C4×C8—41no—
40#389—41no—
41#389—41no—
42#387—44no—
43#385—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×#95C2×C411yescharacteristic
54C2×#95C2×C411yescharacteristic
55C2×#95D411yescharacteristic
56C2×#95C2311yescharacteristic
57#336—41no—
58#323—24no—
59C25C22⋊C411yescharacteristic
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—22no—
66C23×C4—21no—
67C23×C4—41no—
68C23×C4—42no—
69C23×C4—81no—
70C23×C4C22⋊C411yescharacteristic
71C23×C4C22⋊C411yescharacteristic
72C23×C4C22⋊C411yescharacteristic
73C23×C4C22×C411yescharacteristic
74C23×C4C2×D411yescharacteristic
75C23×C4C2×D411yescharacteristic
76C22×C8—42no—
77C22×C8—42no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—81no—
91C2×C4⋊C4—81no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—41no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—42no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4—42no—
103C2×C22⋊C4—81no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—42no—
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—22no—
119#95—22no—
120#95—41no—
121#95—41no—
122#95—42no—
123#95—42no—
124#95—42no—
125#95—42no—
126#95—42no—
127#95—81no—
128C24—21no—
129C24—21no—
130C24—21no—
131C24—21no—
132C24—21no—
133C24—21no—
134C24—41no—
135C24—41no—
136C24—41no—
137C24—41no—
138C24#10412yes—
139C24C2×C22⋊C411yescharacteristic
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—41no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—41no—
162C22×C4—41no—
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—42no—
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—81no—
198C22×C4—81no—
199C22×C4—84no—
200C22×C4—84no—
201C22×C4#9911yescharacteristic
202C22×C4#9911yescharacteristic
203C22×C4#10112yes—
204C22×C4#13112yes—
205C22×C4#13512yes—
206C22×C4#13512yes—
207C2×C8—44no—
208C2×C8—82no—
209C2×C8—82no—
210C2×C8—82no—
211C2×C8—82no—
212C2×C8—82no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—48no—
216C4⋊C4—82no—
217C4⋊C4—84no—
218C4⋊C4—82no—
219C4⋊C4—82no—
220C22⋊C4—44no—
221C22⋊C4—44no—
222C22⋊C4—48no—
223C22⋊C4—84no—
224C42—42no—
225C42—81no—
226C42—81no—
227C42—81no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—81no—
250C23—81no—
251C23—81no—
252C23—81no—
253C23—81no—
254C23—81no—
255C23#40811yescharacteristic
256C23C2×#9911yescharacteristic
257C23C2×#10111yescharacteristic
258C23#41611yescharacteristic
259C23#41611yescharacteristic
260C23C2×#10411yescharacteristic
261C23#42111yescharacteristic
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—84no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—84no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
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#35312yes—
325C8—84no—
326C8—162no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—44no—
340C22—42no—
341C22—42no—
342C22—42no—
343C22—82no—
344C22—82no—
345C22—82no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22#147211yescharacteristic
351C22#148211yescharacteristic
352C22#149411yescharacteristic
353C22#150111yescharacteristic
354C22#151611yescharacteristic
355C22#152811yescharacteristic
356C22#154411yescharacteristic
357C22#212011yescharacteristic
358C22#212111yescharacteristic
359C22C2×#35311yescharacteristic
360C22#213511yescharacteristic
361C4—22no—
362C4—42no—
363C4—42no—
364C4—42no—
365C4—41no—
366C4—42no—
367C4—82no—
368C4—82no—
369C4—82no—
370C4—81no—
371C4—161no—
372C2—21no—
373C2—21no—
374C2—21no—
375C2—21no—
376C2—82no—
377C2#767111yescharacteristic
378C2#1157011yescharacteristic
379C2#1157711yescharacteristic
380C2#1172611yescharacteristic
381C2#1173911yescharacteristic
382C2#1184311yescharacteristic
383C2#1184911yescharacteristic
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. #7123C2semidirect

Maximal subgroups

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

subgroupindex
#149692
#87002
C4×#3232
#71232