Finite Group Databasebeta

← #124396

Subgroups — #124396

Order 512.

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

normal subgroups only

This group has 560 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#14243C211yesmaximal, characteristic
2#8700C211yesmaximal, characteristic
3C4×#323C211yesmaximal, characteristic
4#7131C212yesmaximal, complemented
5#7122C212yesmaximal, complemented
6C2×#393C411yescharacteristic
7C2×#391C411yescharacteristic
8C2×C4×C22⋊C4C2211yescharacteristic
9#1779—21no—
10C4×C22⋊C8—21no—
11C2×#323—21no—
12#1446C2211yescharacteristic
13#1442C2211yescharacteristic
14#1439C411yescharacteristic
15#1437—21no—
16#1434—21no—
17#1434—21no—
18C22⋊#95—21no—
19C4×#95C411yescharacteristic
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C23×Q8C2×C411yescharacteristic
24C24×C4D411yescharacteristic
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C22⋊C4—21no—
27C22×C22⋊C4—21no—
28C22×C22⋊C4C2×C411yescharacteristic
29C22×C22⋊C4D411yescharacteristic
30C22×C42—21no—
31C22×C42C2×C411yescharacteristic
32C22×C42D411yescharacteristic
33C2×C22⋊C8—22no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—41no—
36C2×C22⋊C8—41no—
37C2×C4×C8—41no—
38#393—22no—
39#393C814yescomplemented
40#391—41no—
41#391—41no—
42C4×C22⋊C4—41no—
43C4×C22⋊C4—41no—
44C2×#95—21no—
45C2×#95—21no—
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
60C22×Q8—21no—
61C22×Q8—41no—
62C22×Q8—41no—
63C22×Q8—81no—
64C22×Q8—81no—
65C22×Q8C2×C812yes—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—22no—
72C23×C4—21no—
73C23×C4—41no—
74C23×C4—42no—
75C23×C4—81no—
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22×C411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C22×C8—42no—
83C22×C8—42no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—81no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—24no—
89C2×C22⋊C4—41no—
90C2×C22⋊C4—41no—
91C2×C22⋊C4—41no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—81no—
95C2×C22⋊C4—81no—
96C2×C22⋊C4—81no—
97C2×C22⋊C4C2×C812yes—
98C2×C22⋊C4M1612yes—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—42no—
104C2×C42—42no—
105C2×C42—42no—
106C2×C42—81no—
107C2×C42—81no—
108C22⋊C8—24no—
109C22⋊C8—42no—
110C22⋊C8—42no—
111C22⋊C8—42no—
112C22⋊C8—82no—
113C22⋊C8—82no—
114C4×C8—82no—
115#95—24no—
116#95—21no—
117#95—41no—
118#95—41no—
119#95—41no—
120#95—42no—
121#95—42no—
122#95—42no—
123#95—41no—
124#95—41no—
125#95—42no—
126#95—81no—
127#95—81no—
128#95—81no—
129#95C2×C812yes—
130C24—21no—
131C24—21no—
132C24—21no—
133C24—21no—
134C24—21no—
135C24—21no—
136C24—41no—
137C24—41no—
138C24—41no—
139C24—41no—
140C24C22⋊C812yes—
141C24C2×C22⋊C411yescharacteristic
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8—82no—
145C2×Q8—82no—
146C2×Q8—84no—
147C2×Q8—82no—
148C2×Q8—82no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—41no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
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—42no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—81no—
206C22×C4—81no—
207C22×C4—84no—
208C22×C4—84no—
209C22×C4C22⋊C812yes—
210C22×C4#9911yescharacteristic
211C22×C4#9911yescharacteristic
212C22×C4#10412yes—
213C22×C4C22×C811yescharacteristic
214C22×C4C2×M1611yescharacteristic
215C22×C4#13512yes—
216C22×C4#13712yes—
217C2×C8—44no—
218C2×C8—82no—
219C2×C8—82no—
220C2×C8—82no—
221C2×C8—82no—
222C2×C8—82no—
223C4⋊C4—82no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—44no—
227C22⋊C4—84no—
228C22⋊C4—84no—
229C42—42no—
230C42—81no—
231C42—81no—
232C42—81no—
233C42—84no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—81no—
256C23—81no—
257C23—81no—
258C23—81no—
259C23—81no—
260C23—81no—
261C23C2×C22⋊C811yescharacteristic
262C23C2×#9911yescharacteristic
263C23#41311yescharacteristic
264C23#41611yescharacteristic
265C23#41911yescharacteristic
266C23C2×#10411yescharacteristic
267C23#42111yescharacteristic
268Q8—84no—
269Q8—84no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—41no—
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—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
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—42no—
309C2×C4—42no—
310C2×C4—84no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—84no—
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#35412yes—
335C8—84no—
336C8—162no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—44no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—82no—
354C22—82no—
355C22—82no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22#145911yescharacteristic
361C22#147611yescharacteristic
362C22#147911yescharacteristic
363C22#150811yescharacteristic
364C22#151611yescharacteristic
365C22#152911yescharacteristic
366C22#155411yescharacteristic
367C22#212111yescharacteristic
368C22#212211yescharacteristic
369C22C2×#35411yescharacteristic
370C22#213211yescharacteristic
371C4—22no—
372C4—42no—
373C4—42no—
374C4—42no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—42no—
379C4—82no—
380C4—82no—
381C4—82no—
382C4—161no—
383C2—21no—
384C2—21no—
385C2—21no—
386C2—21no—
387C2—82no—
388C2#758311yescharacteristic
389C2#1147311yescharacteristic
390C2#1147711yescharacteristic
391C2#1178711yescharacteristic
392C2#1179311yescharacteristic
393C2#1184511yescharacteristic
394C2#1185111yescharacteristic
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. #7131C2semidirect
5. #7122C2semidirect
39. #393C8semidirect

Maximal subgroups

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

subgroupindex
#142432
#87002
C4×#3232
#71312
#71222