Finite Group Databasebeta

← #480674

Subgroups — #480674

Order 512.

1362 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1362 conjugacy classes of subgroups — too many to draw. Only its 202 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×#2501C211yesmaximal, characteristic
2C2×#1463C212yesmaximal, complemented
3#11549C218yesmaximal, complemented
4#11512C212yesmaximal, complemented
5#11505C212yesmaximal, complemented
6C22×#123—22no—
7C22×#121C411yescharacteristic
8C23×C4⋊C4C2211yescharacteristic
9C2×#407—22no—
10#2501—22no—
11#2501—22no—
12#2501C414yes—
13#2501C2214yes—
14C2×#399—22no—
15C2×#390C411yescharacteristic
16C2×#390C2211yescharacteristic
17C2×#384C411yescharacteristic
18C2×#384C2211yescharacteristic
19C2×C4×C22⋊C4C411yescharacteristic
20#1463—22no—
21#1463—22no—
22#1463C2214yes—
23#1463C2214yescomplemented
24C2×#324—22no—
25C2×#324C2214yescomplemented
26#1325C2218yescomplemented
27#1318C2218yescomplemented
28C23×D4—21no—
29C23×D4C2×C411yescharacteristic
30C24×C4C2×C411yescharacteristic
31C24×C4D412yes—
32C22×M16—42no—
33C2×#123—24no—
34C2×#123—24no—
35C2×#123—24no—
36C2×#123—24no—
37C2×#123—416no—
38C2×#121—22no—
39C2×#121—42no—
40C2×#121—41no—
41C2×#121—41no—
42C2×#121C2×C414yes—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—41no—
46C22×C4⋊C4—42no—
47C22×C4⋊C4—41no—
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C412yes—
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4D412yes—
52C22×C4⋊C4D412yes—
53C22×C4⋊C4D412yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C22⋊C4—21no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—22no—
60C22×C42C2×C411yescharacteristic
61C22×C42C2×C411yescharacteristic
62#407—24no—
63#407—24no—
64#407—42no—
65#407—42no—
66C2×C22⋊C8—24no—
67C2×C22⋊C8—42no—
68#399—24no—
69#399—24no—
70#399—24no—
71#399—24no—
72#390—21no—
73#390—21no—
74#390—22no—
75#390C2×C412yes—
76#390C2×C414yes—
77#390C2312yes—
78#384—21no—
79#384—21no—
80#384—22no—
81#384C2×C412yes—
82#384C2×C414yes—
83#384C2312yes—
84C4×C22⋊C4—22no—
85C4×C22⋊C4C2×C414yes—
86C2×#95—21no—
87#324—24no—
88#324—24no—
89#324—24no—
90#324C2318yes—
91C25—21no—
92C25C22⋊C411yescharacteristic
93C22×D4—24no—
94C22×D4—21no—
95C22×D4—42no—
96C22×D4—41no—
97C22×D4—41no—
98C22×D4—42no—
99C22×D4—48no—
100C22×D4—42no—
101C22×D4—41no—
102C22×D4—41no—
103C22×D4C22×C412yes—
104C23×C4—22no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—22no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—21no—
114C23×C4—42no—
115C23×C4—42no—
116C23×C4—48no—
117C23×C4C22⋊C412yes—
118C23×C4C22⋊C412yes—
119C23×C4C22⋊C411yescharacteristic
120C23×C4C22⋊C412yes—
121C23×C4C22⋊C412yes—
122C23×C4C22×C411yescharacteristic
123C23×C4C22×C412yes—
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4C2×D412yes—
127C2×M16—44no—
128C2×M16—44no—
129C2×M16—88no—
130C22×C8—44no—
131#123—416no—
132#123—416no—
133#123—416no—
134#123—416no—
135#121—44no—
136#121—44no—
137#121—44no—
138#121—44no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—48no—
161C2×C4⋊C4—48no—
162C2×C4⋊C4C22⋊C411yescharacteristic
163C2×C4⋊C4C22⋊C411yescharacteristic
164C2×C4⋊C4C22⋊C412yes—
165C2×C4⋊C4C22⋊C412yes—
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C22×C411yescharacteristic
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C2×D412yes—
170C2×C4⋊C4C2×D412yes—
171C2×C4⋊C4C2×D412yes—
172C2×C4⋊C4C2411yescharacteristic
173C2×C22⋊C4—28no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—48no—
183C2×C22⋊C4—48no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—42no—
187C2×C42—42no—
188C2×C42C22×C412yes—
189C2×C42C22×C412yes—
190C22⋊C8—28no—
191C22⋊C8—44no—
192C22⋊C8—44no—
193C22⋊C8—44no—
194#95—24no—
195C24—24no—
196C24—22no—
197C24—41no—
198C24—41no—
199C24—42no—
200C24—44no—
201C24—42no—
202C24—42no—
203C24—44no—
204C24—41no—
205C24—41no—
206C24#9912yes—
207C24#9911yescharacteristic
208C24#9911yescharacteristic
209C24C2×C22⋊C411yescharacteristic
210C24C2×C22⋊C412yes—
211C2×D4—24no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—48no—
217C2×D4—48no—
218C2×D4—44no—
219C2×D4—44no—
220C2×D4—42no—
221C2×D4—84no—
222C2×D4—84no—
223C2×D4—88no—
224C2×D4—84no—
225C2×D4—84no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—44no—
260C22×C4—48no—
261C22×C4—44no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—48no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—84no—
277C22×C4—84no—
278C22×C4—84no—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C412yes—
284C22×C4C23×C411yescharacteristic
285C22×C4C22×D412yes—
286M16—88no—
287M16—88no—
288C2×C8—48no—
289C2×C8—88no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—41no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—41no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—48no—
303C4⋊C4—48no—
304C4⋊C4—48no—
305C4⋊C4—48no—
306C22⋊C4—216no—
307C22⋊C4—48no—
308C22⋊C4—48no—
309C22⋊C4—48no—
310C22⋊C4—48no—
311C22⋊C4—48no—
312C42—24no—
313C42—42no—
314C42—42no—
315C42—42no—
316C23—21no—
317C23—21no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—24no—
323C23—24no—
324C23—22no—
325C23—24no—
326C23—41no—
327C23—42no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—42no—
332C23—48no—
333C23—48no—
334C23—44no—
335C23—44no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—82no—
340C23—82no—
341C23—82no—
342C23—88no—
343C23—88no—
344C23—82no—
345C23—82no—
346C23—82no—
347C23#35314yes—
348C23C2×#9912yes—
349C23C2×#9911yescharacteristic
350C23C2×#9912yes—
351C23C2×#9911yescharacteristic
352C23C22×C22⋊C411yescharacteristic
353D4—84no—
354D4—84no—
355D4—88no—
356D4—88no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—48no—
377C2×C4—48no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4#41312yes—
405C2×C4#56812yes—
406C8—88no—
407C22—24no—
408C22—24no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—22no—
414C22—21no—
415C22—21no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—44no—
421C22—44no—
422C22—44no—
423C22—48no—
424C22—44no—
425C22—84no—
426C22—84no—
427C22—82no—
428C22—82no—
429C22—82no—
430C22—164no—
431C22—164no—
432C22C2×#35312yes—
433C22C2×#35314yes—
434C22#213312yes—
435C22#287911yescharacteristic
436C22C22×#9911yescharacteristic
437C22C2×#41311yescharacteristic
438C4—22no—
439C4—22no—
440C4—44no—
441C4—44no—
442C4—42no—
443C4—42no—
444C4—42no—
445C4—42no—
446C4—42no—
447C4—82no—
448C2—21no—
449C2—21no—
450C2—22no—
451C2—22no—
452C2—24no—
453C2—42no—
454C2—162no—
455C2#1739211yescharacteristic
456C2C22×#35311yescharacteristic
457C2C2×#213311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×#121C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1463C2semidirect
3. #11549C2semidirect
4. #11512C2semidirect
5. #11505C2semidirect
23. #1463C22semidirect
25. C2×#324C22semidirect
26. #1325C22semidirect
27. #1318C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25012
C2×#14632
#115492
#115122
#115052