Finite Group Databasebeta

← #374151

Subgroups — #374151

Order 512.

1984 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2501C211yesmaximal, characteristic
2#34178C211yesmaximal, characteristic, complemented
3#20973C212yesmaximal
4C22×#343C211yesmaximal, characteristic, complemented
5#11210C212yesmaximal, complemented
6#11054C212yesmaximal, complemented
7#10958C212yesmaximal, complemented
8#10891C212yesmaximal, complemented
9#10732C212yesmaximal, complemented
10C23×M16C2212yes—
11C22×#123—22no—
12C22×#121C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C2×#418C2214yes—
15#2501—24no—
16#2501C418yescomplemented
17C2×#399—22no—
18#2345—24no—
19#2344—24no—
20#2316C2212yes—
21#2313—22no—
22C2×#390C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×#384C2211yescharacteristic
26#2280—22no—
27#2280C2212yes—
28#2274C2212yes—
29C2×C4×C22⋊C4C2211yescharacteristic
30#1895—24no—
31#1893—24no—
32C2×#343—24no—
33C2×#343C2212yes—
34C2×#343C2212yes—
35#1736C2214yes—
36C2×#339C2214yes—
37C2×#339C2214yes—
38C23×D4—21no—
39C23×D4C2311yescharacteristic
40C24×C4D412yes—
41C24×C4C2311yescharacteristic
42C22×M16—22no—
43C22×M16—24no—
44C22×M16—28no—
45C22×M16—24no—
46C22×M16—44no—
47C22×M16C2314yes—
48C23×C8—22no—
49C2×#123—216no—
50C2×#123—416no—
51C2×#121—22no—
52C2×#121—22no—
53C2×#121—22no—
54C2×#121—24no—
55C2×#121C2×C414yes—
56C2×#117—24no—
57C2×#117C2314yes—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—24no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4C2×C414yes—
64C22×C4⋊C4D412yes—
65C22×C4⋊C4D414yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75#418—28no—
76#418—44no—
77C2×#102—28no—
78#399—216no—
79#390—22no—
80#390—22no—
81#390C2×C414yes—
82#390C2×C414yes—
83#389—28no—
84#387—216no—
85#384—22no—
86#384—22no—
87#384C2×C414yes—
88#384C2×C414yes—
89#381—24no—
90#381—22no—
91#380—24no—
92#380—24no—
93#380D414yes—
94#380D414yes—
95#379—22no—
96C4×C4⋊C4—28no—
97C4×C22⋊C4—24no—
98C4×C22⋊C4—24no—
99C4×C22⋊C4—22no—
100C4×C22⋊C4—22no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4C2×C414yes—
103C2×#95—21no—
104#343—24no—
105#343—24no—
106#340—28no—
107#339—28no—
108#339—44no—
109C25—21no—
110C25C2×D411yescharacteristic
111C22×D4—24no—
112C22×D4—22no—
113C22×D4—21no—
114C22×D4—22no—
115C22×D4—24no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—48no—
119C22×D4—41no—
120C22×D4—41no—
121C22×D4—42no—
122C22×D4C22×C412yes—
123C23×C4—24no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—24no—
131C23×C4—21no—
132C23×C4—48no—
133C23×C4C22⋊C418yes—
134C23×C4C22×C412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D412yes—
137C23×C4C2×D412yes—
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4#4114yes—
142C23×C4C2411yescharacteristic
143C2×M16—24no—
144C2×M16—216no—
145C2×M16—24no—
146C2×M16—24no—
147C2×M16—24no—
148C2×M16—24no—
149C2×M16—48no—
150C2×M16—416no—
151C2×M16—48no—
152C2×M16—44no—
153C2×M16#4118yes—
154C22×C8—24no—
155C22×C8—48no—
156C22×C8—44no—
157#123—464no—
158#121—28no—
159#121—28no—
160#121—44no—
161#121—44no—
162#117—24no—
163#117—24no—
164#117—24no—
165#117—24no—
166#117—88no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—48no—
184C2×C4⋊C4—48no—
185C2×C4⋊C4C22⋊C418yes—
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C22×C412yes—
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C2×D418yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D412yes—
192C2×C22⋊C4—28no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—48no—
201C2×C22⋊C4—48no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C42—24no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—24no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—28no—
211C2×C42C22×C412yes—
212C2×C42C22×C412yes—
213#102—416no—
214#95—28no—
215#95—28no—
216#95—24no—
217#95—24no—
218#95—24no—
219C24—22no—
220C24—22no—
221C24—24no—
222C24—22no—
223C24—42no—
224C24—44no—
225C24—44no—
226C24—41no—
227C24—41no—
228C24—42no—
229C24—42no—
230C24C2×C22⋊C412yes—
231C24#12011yescharacteristic
232C24#12011yescharacteristic
233C24#12312yes—
234C24C22×D411yescharacteristic
235C2×D4—24no—
236C2×D4—24no—
237C2×D4—24no—
238C2×D4—416no—
239C2×D4—44no—
240C2×D4—44no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—44no—
248C2×D4—44no—
249C2×D4—88no—
250C2×D4—84no—
251C2×D4—84no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—28no—
264C22×C4—28no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—28no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—28no—
275C22×C4—22no—
276C22×C4—28no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—48no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—416no—
285C22×C4—44no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—88no—
289C22×C4—84no—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C414yes—
292C22×C4C2×C22⋊C414yes—
293C22×C4C4×D414yes—
294C22×C4C4×D414yes—
295C22×C4#12012yes—
296C22×C4#12114yes—
297C22×C4#12114yes—
298C22×C4#12312yes—
299C22×C4C23×C411yescharacteristic
300C22×C4C22×D412yes—
301C22×C4C22×D411yescharacteristic
302C22×C4C2×#4112yes—
303C22×C4C2×#4114yes—
304C22×C4H(5,𝔽2)12yes—
305C22×C42−1+412yes—
306M16—28no—
307M16—28no—
308M16—416no—
309M16—44no—
310M16—44no—
311C2×C8—28no—
312C2×C8—416no—
313C2×C8—44no—
314C2×C8—44no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—216no—
320C4⋊C4—24no—
321C4⋊C4—28no—
322C4⋊C4—28no—
323C4⋊C4—416no—
324C4⋊C4—416no—
325C4⋊C4—42no—
326C4⋊C4—88no—
327C22⋊C4—216no—
328C22⋊C4—416no—
329C22⋊C4—416no—
330C22⋊C4—48no—
331C22⋊C4—44no—
332C22⋊C4—44no—
333C22⋊C4—48no—
334C22⋊C4—48no—
335C42—24no—
336C42—24no—
337C42—42no—
338C42—41no—
339C42—41no—
340C42—44no—
341C42—44no—
342C42—88no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—24no—
349C23—24no—
350C23—22no—
351C23—24no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—42no—
356C23—42no—
357C23—48no—
358C23—48no—
359C23—44no—
360C23—44no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—82no—
365C23—88no—
366C23—88no—
367C23—82no—
368C23—82no—
369C23—82no—
370C23—82no—
371C23—82no—
372C23#38612yes—
373C23#38612yes—
374C23#45314yes—
375C23C22×C22⋊C411yescharacteristic
376C23C2×#12011yescharacteristic
377C23C2×#12311yescharacteristic
378C23#53212yes—
379C23#53712yes—
380D4—48no—
381D4—88no—
382D4—88no—
383D4—82no—
384D4—82no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—28no—
390C2×C4—28no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—48no—
400C2×C4—48no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—416no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—48no—
412C2×C4—48no—
413C2×C4—48no—
414C2×C4—42no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—88no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4#38614yes—
421C2×C4C2×C4×D412yes—
422C2×C4C4×#4112yes—
423C2×C4#52012yes—
424C2×C4C2×#12112yes—
425C2×C4#52512yes—
426C2×C4#53511yescharacteristic
427C2×C4#53611yescharacteristic
428C2×C4#54614yes—
429C2×C4#54714yes—
430C2×C4#55114yes—
431C2×C4#55214yes—
432C8—48no—
433C22—22no—
434C22—24no—
435C22—24no—
436C22—28no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—44no—
446C22—44no—
447C22—48no—
448C22—44no—
449C22—84no—
450C22—84no—
451C22—82no—
452C22—82no—
453C22—82no—
454C22—164no—
455C22—164no—
456C22#188714yes—
457C22C2×#38611yescharacteristic
458C22#229711yescharacteristic
459C22#231111yescharacteristic
460C22#243012yes—
461C22#249812yes—
462C22C2×#45312yes—
463C22#302012yes—
464C4—24no—
465C4—44no—
466C4—44no—
467C4—44no—
468C4—44no—
469C4—88no—
470C4—82no—
471C4#235914yes—
472C4#241112yes—
473C4#241312yes—
474C2—21no—
475C2—21no—
476C2—22no—
477C2—24no—
478C2—42no—
479C2—162no—
480C2#1413511yescharacteristic
481C2C2×#188711yescharacteristic
482C2#2095411yescharacteristic
483C2#2551714yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34178C2semidirect
4. C22×#343C2semidirect
5. #11210C2semidirect
6. #11054C2semidirect
7. #10958C2semidirect
8. #10891C2semidirect
9. #10732C2semidirect
16. #2501C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25012
#341782
#209732
C22×#3432
#112102
#110542
#109582
#108912
#107322