Finite Group Databasebeta

← #480679

Subgroups — #480679

Order 512.

1166 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1166 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×#2504C211yesmaximal, characteristic
2C2×#1463C212yesmaximal, complemented
3#11553C218yesmaximal, complemented
4#11511C212yesmaximal, complemented
5#11505C212yesmaximal, complemented
6C22×#121C411yescharacteristic
7C23×C4⋊C4C2211yescharacteristic
8C2×#407—22no—
9#2504—22no—
10#2504—22no—
11#2504C414yes—
12#2504C2214yes—
13C2×#397—22no—
14C2×#390C411yescharacteristic
15C2×#390C2211yescharacteristic
16C2×#386—22no—
17C2×#382C411yescharacteristic
18C2×#382C2211yescharacteristic
19C2×C4×C22⋊C4C411yescharacteristic
20#1463—22no—
21#1463—22no—
22#1463C2214yes—
23#1463C2214yescomplemented
24C2×#324—22no—
25C2×#324C2214yescomplemented
26#1324C2218yescomplemented
27#1318C2218yescomplemented
28C23×D4—21no—
29C23×D4C2×C411yescharacteristic
30C24×C4C2×C411yescharacteristic
31C24×C4D412yes—
32C22×M16—42no—
33C2×#121—22no—
34C2×#121—42no—
35C2×#121—41no—
36C2×#121—41no—
37C2×#121C2×C414yes—
38C22×C4⋊C4—42no—
39C22×C4⋊C4—41no—
40C22×C4⋊C4—41no—
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C412yes—
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4D412yes—
45C22×C4⋊C4D412yes—
46C22×C4⋊C4D412yes—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2311yescharacteristic
49C22×C22⋊C4—21no—
50C22×C22⋊C4—21no—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—22no—
53C22×C42C2×C411yescharacteristic
54C22×C42C2×C411yescharacteristic
55#407—24no—
56#407—24no—
57#407—42no—
58#407—42no—
59C2×C22⋊C8—24no—
60C2×C22⋊C8—42no—
61#397—24no—
62#397—24no—
63#397—24no—
64#397—24no—
65#390—21no—
66#390—21no—
67#390—22no—
68#390C2×C412yes—
69#390C2×C414yes—
70#390C2312yes—
71#386—24no—
72#386—24no—
73#386—24no—
74#386—24no—
75#382—21no—
76#382—21no—
77#382—22no—
78#382C2×C412yes—
79#382C2×C414yes—
80#382C2312yes—
81C4×C22⋊C4—22no—
82C4×C22⋊C4C2×C414yes—
83C2×#95—21no—
84C2×#95—22no—
85C2×#95—22no—
86#324—24no—
87#324—24no—
88#324—24no—
89#324C2318yes—
90C25—21no—
91C25C22⋊C411yescharacteristic
92C22×D4—24no—
93C22×D4—21no—
94C22×D4—42no—
95C22×D4—41no—
96C22×D4—41no—
97C22×D4—42no—
98C22×D4—48no—
99C22×D4—42no—
100C22×D4—41no—
101C22×D4—41no—
102C22×D4C22×C412yes—
103C23×C4—22no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—42no—
114C23×C4—42no—
115C23×C4—84no—
116C23×C4C22⋊C412yes—
117C23×C4C22⋊C412yes—
118C23×C4C22⋊C411yescharacteristic
119C23×C4C22⋊C412yes—
120C23×C4C22⋊C412yes—
121C23×C4C22×C411yescharacteristic
122C23×C4C22×C412yes—
123C23×C4C2×D412yes—
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C2×M16—44no—
127C2×M16—44no—
128C2×M16—88no—
129C22×C8—44no—
130#121—44no—
131#121—44no—
132#121—44no—
133#121—44no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4C22⋊C411yescharacteristic
152C2×C4⋊C4C22⋊C411yescharacteristic
153C2×C4⋊C4C22⋊C412yes—
154C2×C4⋊C4C22⋊C412yes—
155C2×C4⋊C4C22×C412yes—
156C2×C4⋊C4C22×C411yescharacteristic
157C2×C4⋊C4C22×C412yes—
158C2×C4⋊C4C2×D412yes—
159C2×C4⋊C4C2×D412yes—
160C2×C4⋊C4C2×D412yes—
161C2×C4⋊C4C2411yescharacteristic
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—42no—
169C2×C22⋊C4—42no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—84no—
172C2×C22⋊C4—84no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—42no—
176C2×C42—42no—
177C2×C42C22×C412yes—
178C2×C42C22×C412yes—
179C22⋊C8—28no—
180C22⋊C8—44no—
181C22⋊C8—44no—
182C22⋊C8—44no—
183#95—24no—
184#95—24no—
185#95—24no—
186#95—24no—
187#95—24no—
188C24—24no—
189C24—22no—
190C24—41no—
191C24—41no—
192C24—42no—
193C24—44no—
194C24—42no—
195C24—42no—
196C24—44no—
197C24—41no—
198C24—41no—
199C24#9912yes—
200C24#9911yescharacteristic
201C24#9911yescharacteristic
202C24C2×C22⋊C411yescharacteristic
203C24C2×C22⋊C412yes—
204C2×D4—24no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—416no—
210C2×D4—44no—
211C2×D4—44no—
212C2×D4—42no—
213C2×D4—84no—
214C2×D4—84no—
215C2×D4—88no—
216C2×D4—84no—
217C2×D4—84no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4—44no—
250C22×C4—48no—
251C22×C4—44no—
252C22×C4—48no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—44no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—84no—
269C22×C4—84no—
270C22×C4—84no—
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C23×C411yescharacteristic
277C22×C4C22×D412yes—
278M16—88no—
279M16—88no—
280C2×C8—48no—
281C2×C8—88no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—41no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C22⋊C4—48no—
295C22⋊C4—48no—
296C22⋊C4—84no—
297C22⋊C4—84no—
298C22⋊C4—84no—
299C22⋊C4—84no—
300C42—24no—
301C42—42no—
302C42—42no—
303C42—42no—
304C23—21no—
305C23—21no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—24no—
311C23—24no—
312C23—24no—
313C23—22no—
314C23—41no—
315C23—42no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—42no—
320C23—48no—
321C23—48no—
322C23—44no—
323C23—44no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—82no—
328C23—82no—
329C23—82no—
330C23—88no—
331C23—88no—
332C23—82no—
333C23—82no—
334C23—82no—
335C23#35514yes—
336C23C2×#9912yes—
337C23C2×#9911yescharacteristic
338C23C2×#9911yescharacteristic
339C23C2×#9912yes—
340C23C22×C22⋊C411yescharacteristic
341D4—84no—
342D4—84no—
343D4—88no—
344D4—88no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—84no—
389C2×C4—84no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4#41312yes—
393C2×C4#56812yes—
394C8—88no—
395C22—24no—
396C22—24no—
397C22—21no—
398C22—22no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—44no—
409C22—44no—
410C22—44no—
411C22—48no—
412C22—44no—
413C22—84no—
414C22—84no—
415C22—82no—
416C22—82no—
417C22—82no—
418C22—164no—
419C22—164no—
420C22#212712yes—
421C22C2×#35512yes—
422C22C2×#35514yes—
423C22#287911yescharacteristic
424C22C22×#9911yescharacteristic
425C22C2×#41311yescharacteristic
426C4—22no—
427C4—22no—
428C4—42no—
429C4—42no—
430C4—42no—
431C4—42no—
432C4—42no—
433C4—82no—
434C4—82no—
435C4—82no—
436C2—21no—
437C2—21no—
438C2—22no—
439C2—22no—
440C2—24no—
441C2—42no—
442C2—162no—
443C2#1739211yescharacteristic
444C2C2×#212711yescharacteristic
445C2C22×#35511yescharacteristic
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. #11553C2semidirect
4. #11511C2semidirect
5. #11505C2semidirect
23. #1463C22semidirect
25. C2×#324C22semidirect
26. #1324C22semidirect
27. #1318C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25042
C2×#14632
#115532
#115112
#115052