Finite Group Databasebeta

← #387967

Subgroups — #387967

Order 512.

1574 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2501C211yesmaximal, characteristic
2#21049C212yesmaximal, complemented
3#11354C212yesmaximal, complemented
4#11044C212yesmaximal, complemented
5#10967C211yesmaximal, characteristic
6#10967C211yesmaximal, characteristic
7#10962C212yesmaximal, complemented
8#10857C212yesmaximal, complemented
9#10727C212yesmaximal, complemented
10C22×#123C414yes—
11C22×#121C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C2×#423C2212yes—
14C22×#100C2211yescharacteristic
15C22×#100C2212yes—
16C22×#100C2211yescharacteristic
17C2×#407C2212yes—
18C2×#407C2212yes—
19C2×C4×M16C2212yes—
20#2501—24no—
21#2501—24no—
22C2×#399C414yes—
23C2×#390C2212yes—
24C2×#384C2212yes—
25C2×C4×C22⋊C4C2211yescharacteristic
26#1924—24no—
27#1923—24no—
28C2×#343C2212yes—
29C2×#343C2212yes—
30C2×#343C2211yescharacteristic
31C2×#343C2212yes—
32C2×#343C2212yes—
33C2×#343C2211yescharacteristic
34C2×#343C2212yes—
35#1728C2212yes—
36#1728C2212yes—
37C23×D4C2×C412yes—
38C23×D4C2311yescharacteristic
39C24×C4C2×C412yes—
40C24×C4C2311yescharacteristic
41C22×M16C2312yes—
42C22×M16C2312yes—
43C22×M16C2312yes—
44C22×M16C2312yes—
45C2×#123—24no—
46C2×#123—28no—
47C2×#123—24no—
48C2×#123—416no—
49C2×#121—24no—
50C2×#121—42no—
51C2×#121—41no—
52C2×#121—41no—
53C22×C4⋊C4—24no—
54C22×C4⋊C4—42no—
55C22×C4⋊C4—41no—
56C22×C4⋊C4—41no—
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C418yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C22⋊C4C2×C412yes—
62C22×C22⋊C4C2×C412yes—
63C22×C22⋊C4C2×C418yes—
64C22×C42C2312yes—
65#423—24no—
66#423—42no—
67C2×C4⋊C8—24no—
68C2×#100—24no—
69C2×#100—24no—
70C2×#100—41no—
71C2×#100—42no—
72C2×#100—42no—
73C2×#100—41no—
74C2×#100—42no—
75#407—24no—
76#407—24no—
77#407—42no—
78#407—42no—
79C2×C22⋊C8—24no—
80C2×C22⋊C8—24no—
81C4×M16—24no—
82C4×M16—42no—
83C2×#97—22no—
84C2×C4×C8—22no—
85#399—24no—
86#399—28no—
87#399—24no—
88#390—22no—
89#390—22no—
90#390—24no—
91#384—22no—
92#384—22no—
93#384—24no—
94C4×C22⋊C4D414yes—
95C4×C22⋊C4D414yes—
96C2×#95C2×C412yes—
97#343—42no—
98#343—42no—
99#343—42no—
100#343—42no—
101#343—41no—
102#343—42no—
103#343—41no—
104#336—22no—
105#336—24no—
106#336—22no—
107C25C22×C411yescharacteristic
108C25C22×C412yes—
109C22×D4—24no—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4—22no—
113C22×D4—22no—
114C22×D4—21no—
115C22×D4—21no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—42no—
119C22×D4—42no—
120C22×D4—48no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—24no—
129C23×C4—42no—
130C23×C4—42no—
131C23×C4—48no—
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C414yes—
134C23×C4C22×C414yes—
135C23×C4C22×C412yes—
136C23×C4C2×D414yes—
137C23×C4C2411yescharacteristic
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—22no—
151C2×M16—22no—
152C2×M16—22no—
153C2×M16—22no—
154C2×M16—22no—
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16#4114yes—
161C2×M16#4114yes—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166#123—416no—
167#123—432no—
168#123—416no—
169#121—48no—
170#121—48no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—48no—
187C2×C4⋊C4—48no—
188C2×C4⋊C4—416no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—28no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—416no—
196C2×C22⋊C4C22⋊C412yes—
197C2×C22⋊C4C22⋊C414yes—
198C2×C22⋊C4C22⋊C412yes—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D414yes—
206C4⋊C8—48no—
207#100—44no—
208#100—48no—
209#100—44no—
210C22⋊C8—48no—
211C22⋊C8—48no—
212#97—44no—
213C4×C8—44no—
214#95C22⋊C412yes—
215#95C22⋊C414yes—
216#95C22⋊C412yes—
217C24—21no—
218C24—21no—
219C24—24no—
220C24—22no—
221C24—22no—
222C24—22no—
223C24—24no—
224C24—22no—
225C24—22no—
226C24—24no—
227C24—42no—
228C24—44no—
229C24—44no—
230C24C2×C22⋊C412yes—
231C24C23×C411yescharacteristic
232C2×D4—48no—
233C2×D4—48no—
234C2×D4—48no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—41no—
246C2×D4—42no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—42no—
250C2×D4—41no—
251C2×D4—88no—
252C2×D4—88no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—28no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—44no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—41no—
283C22×C4—48no—
284C22×C4—48no—
285C22×C4—48no—
286C22×C4—48no—
287C22×C4—44no—
288C22×C4—88no—
289C22×C4—88no—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C414yes—
292C22×C4C2×C22⋊C414yes—
293C22×C4C22×D412yes—
294C22×C4C2×#4112yes—
295C22×C4C2×#4112yes—
296C22×C4H(5,𝔽2)12yes—
297C22×C4H(5,𝔽2)11yescharacteristic
298C22×C4H(5,𝔽2)11yescharacteristic
299C22×C4H(5,𝔽2)12yes—
300C22×C42−1+412yes—
301M16—24no—
302M16—24no—
303M16—24no—
304M16—24no—
305M16—42no—
306M16—42no—
307M16—42no—
308M16—42no—
309M16—42no—
310M16—42no—
311M16—42no—
312M16—42no—
313M16—42no—
314M16—42no—
315M16—42no—
316M16—42no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C4⋊C4—48no—
330C4⋊C4—416no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—416no—
338C22⋊C4—24no—
339C22⋊C4—28no—
340C22⋊C4—48no—
341C22⋊C4—416no—
342C22⋊C4—416no—
343C22⋊C4#10018yescomplemented
344C42—22no—
345C42—22no—
346C42—24no—
347C42—22no—
348C42—24no—
349C42—22no—
350C23—22no—
351C23—22no—
352C23—24no—
353C23—24no—
354C23—22no—
355C23—24no—
356C23—22no—
357C23—24no—
358C23—48no—
359C23—48no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—41no—
364C23—41no—
365C23—42no—
366C23—44no—
367C23—44no—
368C23—44no—
369C23—44no—
370C23—42no—
371C23—42no—
372C23—82no—
373C23—82no—
374C23—82no—
375C23—82no—
376C23—88no—
377C23—88no—
378C23C2×#10014yes—
379C23C22×C22⋊C411yescharacteristic
380C23C4×#4112yes—
381C23#51811yescharacteristic
382C23#51811yescharacteristic
383C23#52012yes—
384D4—816no—
385D4—82no—
386D4—82no—
387D4—82no—
388D4—82no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—48no—
397C2×C4—48no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—48no—
408C2×C4—44no—
409C2×C4—816no—
410C2×C4—84no—
411C2×C4—84no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4C2×#10018yes—
415C2×C4#53512yes—
416C2×C4#53512yes—
417C2×C4#53812yes—
418C2×C4#53912yes—
419C2×C4#54012yes—
420C2×C4#54212yes—
421C2×C4#54612yes—
422C2×C4#54612yes—
423C2×C4#54712yes—
424C2×C4#54712yes—
425C2×C4#55111yescharacteristic
426C2×C4#55312yes—
427C2×C4#55911yescharacteristic
428C2×C4#56011yescharacteristic
429C2×C4#56212yes—
430C2×C4#56311yescharacteristic
431C8—44no—
432C8—44no—
433C8—82no—
434C8—82no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—24no—
440C22—24no—
441C22—24no—
442C22—28no—
443C22—48no—
444C22—44no—
445C22—44no—
446C22—42no—
447C22—42no—
448C22—44no—
449C22—82no—
450C22—82no—
451C22—82no—
452C22—88no—
453C22—164no—
454C22—162no—
455C22—162no—
456C22#230911yescharacteristic
457C22#233512yes—
458C22#233912yes—
459C22#237411yescharacteristic
460C22#237611yescharacteristic
461C22C22×#10012yes—
462C22#288912yes—
463C4—28no—
464C4—42no—
465C4—44no—
466C4—84no—
467C4—82no—
468C4#270312yes—
469C4#270612yes—
470C4#271012yes—
471C4#271812yes—
472C2—22no—
473C2—21no—
474C2—21no—
475C2—44no—
476C2—162no—
477C2#1450611yescharacteristic
478C2#2179711yescharacteristic
479C2#2180211yescharacteristic
480C2#2247914yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21049C2semidirect
3. #11354C2semidirect
4. #11044C2semidirect
7. #10962C2semidirect
8. #10857C2semidirect
9. #10727C2semidirect
343. C22⋊C4#100semidirect

Maximal subgroups

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

subgroupindex
C2×#25012
#210492
#113542
#110442
#109672
#109622
#108572
#107272