Finite Group Databasebeta

← #374155

Subgroups — #374155

Order 512.

1788 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2504C211yesmaximal, characteristic
2#34178C211yesmaximal, characteristic, complemented
3#20973C212yesmaximal
4C22×#343C211yesmaximal, characteristic, complemented
5#11211C212yesmaximal, complemented
6#11054C212yesmaximal, complemented
7#10958C212yesmaximal, complemented
8#10892C212yesmaximal, complemented
9#10732C212yesmaximal, complemented
10C23×M16C2212yes—
11C22×#121C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C2×#418C2214yes—
14#2504—24no—
15#2504C418yescomplemented
16C2×#397—22no—
17#2345—24no—
18#2344—24no—
19#2316C2212yes—
20#2313—22no—
21C2×#390C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#386—22no—
24C2×#382C2211yescharacteristic
25C2×#382C2211yescharacteristic
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×#121—22no—
50C2×#121—22no—
51C2×#121—22no—
52C2×#121—24no—
53C2×#121C2×C414yes—
54C2×#117—24no—
55C2×#117C2314yes—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—24no—
59C22×C4⋊C4C2×C414yes—
60C22×C4⋊C4D412yes—
61C22×C4⋊C4D414yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71#418—28no—
72#418—44no—
73C2×#102—28no—
74#397—216no—
75#390—22no—
76#390—22no—
77#390C2×C414yes—
78#390C2×C414yes—
79#389—28no—
80#387—216no—
81#386—216no—
82#382—22no—
83#382—22no—
84#382C2×C414yes—
85#382C2×C414yes—
86#381—24no—
87#381—22no—
88#380—24no—
89#380—24no—
90#380D414yes—
91#380D414yes—
92#379—22no—
93C4×C4⋊C4—28no—
94C4×C22⋊C4—24no—
95C4×C22⋊C4—24no—
96C4×C22⋊C4—22no—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—22no—
99C4×C22⋊C4C2×C414yes—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—22no—
103#343—24no—
104#343—24no—
105#340—28no—
106#339—28no—
107#339—44no—
108C25—21no—
109C25C2×D411yescharacteristic
110C22×D4—24no—
111C22×D4—22no—
112C22×D4—21no—
113C22×D4—22no—
114C22×D4—24no—
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—48no—
118C22×D4—41no—
119C22×D4—41no—
120C22×D4—42no—
121C22×D4C22×C412yes—
122C23×C4—24no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—24no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—84no—
132C23×C4C22⋊C418yes—
133C23×C4C22×C412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2×D412yes—
137C23×C4C2×D412yes—
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C23×C4#4114yes—
141C23×C4C2411yescharacteristic
142C2×M16—24no—
143C2×M16—216no—
144C2×M16—24no—
145C2×M16—24no—
146C2×M16—24no—
147C2×M16—24no—
148C2×M16—48no—
149C2×M16—416no—
150C2×M16—48no—
151C2×M16—44no—
152C2×M16#4118yes—
153C22×C8—24no—
154C22×C8—48no—
155C22×C8—44no—
156#121—28no—
157#121—28no—
158#121—44no—
159#121—44no—
160#117—24no—
161#117—24no—
162#117—24no—
163#117—24no—
164#117—88no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4C22⋊C418yes—
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C22×C412yes—
182C2×C4⋊C4C22×C412yes—
183C2×C4⋊C4C2×D418yes—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D412yes—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—28no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—84no—
197C2×C22⋊C4—84no—
198C2×C42—24no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—24no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—28no—
205C2×C42C22×C412yes—
206C2×C42C22×C412yes—
207#102—416no—
208#95—28no—
209#95—28no—
210#95—24no—
211#95—28no—
212#95—28no—
213#95—24no—
214#95—24no—
215C24—22no—
216C24—22no—
217C24—24no—
218C24—22no—
219C24—42no—
220C24—44no—
221C24—44no—
222C24—41no—
223C24—41no—
224C24—42no—
225C24—42no—
226C24C2×C22⋊C412yes—
227C24#12011yescharacteristic
228C24#12011yescharacteristic
229C24#12312yes—
230C24C22×D411yescharacteristic
231C2×D4—24no—
232C2×D4—24no—
233C2×D4—24no—
234C2×D4—416no—
235C2×D4—44no—
236C2×D4—44no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—44no—
242C2×D4—44no—
243C2×D4—44no—
244C2×D4—44no—
245C2×D4—88no—
246C2×D4—84no—
247C2×D4—84no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—28no—
260C22×C4—28no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—28no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—28no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—28no—
271C22×C4—48no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—44no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—88no—
284C22×C4—84no—
285C22×C4—88no—
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C414yes—
288C22×C4C2×C22⋊C414yes—
289C22×C4C4×D414yes—
290C22×C4C4×D414yes—
291C22×C4#12012yes—
292C22×C4#12114yes—
293C22×C4#12114yes—
294C22×C4#12312yes—
295C22×C4C23×C411yescharacteristic
296C22×C4C22×D412yes—
297C22×C4C22×D411yescharacteristic
298C22×C4C2×#4112yes—
299C22×C4C2×#4114yes—
300C22×C4H(5,𝔽2)12yes—
301C22×C42−1+412yes—
302M16—28no—
303M16—28no—
304M16—416no—
305M16—44no—
306M16—44no—
307C2×C8—28no—
308C2×C8—416no—
309C2×C8—44no—
310C2×C8—44no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—216no—
316C4⋊C4—24no—
317C4⋊C4—28no—
318C4⋊C4—28no—
319C4⋊C4—42no—
320C4⋊C4—88no—
321C22⋊C4—48no—
322C22⋊C4—48no—
323C22⋊C4—44no—
324C22⋊C4—44no—
325C22⋊C4—48no—
326C22⋊C4—48no—
327C22⋊C4—88no—
328C22⋊C4—88no—
329C42—24no—
330C42—24no—
331C42—42no—
332C42—41no—
333C42—41no—
334C42—44no—
335C42—44no—
336C42—88no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—24no—
343C23—24no—
344C23—22no—
345C23—24no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—42no—
350C23—42no—
351C23—48no—
352C23—48no—
353C23—44no—
354C23—44no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—82no—
359C23—88no—
360C23—88no—
361C23—82no—
362C23—82no—
363C23—82no—
364C23—82no—
365C23—82no—
366C23#38612yes—
367C23#38612yes—
368C23#45414yes—
369C23C22×C22⋊C411yescharacteristic
370C23C2×#12011yescharacteristic
371C23C2×#12311yescharacteristic
372C23#53212yes—
373C23#53712yes—
374D4—48no—
375D4—88no—
376D4—88no—
377D4—82no—
378D4—82no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—22no—
383C2×C4—28no—
384C2×C4—28no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—416no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—48no—
403C2×C4—48no—
404C2×C4—48no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—88no—
410C2×C4—84no—
411C2×C4—84no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4#38614yes—
415C2×C4C2×C4×D412yes—
416C2×C4C4×#4112yes—
417C2×C4#52012yes—
418C2×C4C2×#12112yes—
419C2×C4#52512yes—
420C2×C4#53511yescharacteristic
421C2×C4#53611yescharacteristic
422C2×C4#54614yes—
423C2×C4#54714yes—
424C2×C4#55114yes—
425C2×C4#55214yes—
426C8—48no—
427C22—22no—
428C22—24no—
429C22—24no—
430C22—28no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—44no—
440C22—44no—
441C22—48no—
442C22—44no—
443C22—84no—
444C22—84no—
445C22—82no—
446C22—82no—
447C22—82no—
448C22—164no—
449C22—164no—
450C22#188814yes—
451C22C2×#38611yescharacteristic
452C22#229711yescharacteristic
453C22#231111yescharacteristic
454C22#243012yes—
455C22#249812yes—
456C22C2×#45412yes—
457C22#301912yes—
458C4—24no—
459C4—44no—
460C4—44no—
461C4—82no—
462C4—82no—
463C4—88no—
464C4—82no—
465C4#235914yes—
466C4#241112yes—
467C4#241312yes—
468C2—21no—
469C2—21no—
470C2—22no—
471C2—24no—
472C2—42no—
473C2—162no—
474C2#1413511yescharacteristic
475C2C2×#188811yescharacteristic
476C2#2095311yescharacteristic
477C2#2552114yes—
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. #11211C2semidirect
6. #11054C2semidirect
7. #10958C2semidirect
8. #10892C2semidirect
9. #10732C2semidirect
15. #2504C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25042
#341782
#209732
C22×#3432
#112112
#110542
#109582
#108922
#107322