Finite Group Databasebeta

← #360818

Subgroups — #360818

Order 512.

1724 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34181C211yesmaximal, characteristic
2C2×C4×#99C211yesmaximal, characteristic
3#13966C212yesmaximal, complemented
4#13960C211yesmaximal, characteristic
5#11287C214yesmaximal, complemented
6#10854C212yesmaximal, complemented
7#10745C214yesmaximal
8C2×#518C2212yes—
9C22×C4×D4C2211yescharacteristic
10C22×#99C2212yes—
11#2349—24no—
12D4×C4⋊C4—24no—
13#2316—21no—
14#2315—21no—
15#2315C2212yes—
16C2×#386C2212yes—
17#2285—21no—
18C2×#380—21no—
19C2×#380C2211yescharacteristic
20C2×#380C2212yes—
21C2×C4×C22⋊C4C2212yes—
22C22×#95C2211yescharacteristic
23C4×#99C418yescomplemented
24C2×#342C2218yes—
25C2×#342C2212yes—
26#1442—24no—
27#1441—21no—
28#1439—21no—
29#1434—21no—
30#1434—24no—
31C22⋊#95—21no—
32C22⋊#95C2212yes—
33C22⋊#95C2218yes—
34C23×D4C2311yescharacteristic
35C24×C4D411yescharacteristic
36C24×C4C2312yes—
37C24×C4C2311yescharacteristic
38#518—28no—
39#518—44no—
40C2×C4×D4—22no—
41C2×C4×D4—22no—
42C2×C4×D4—24no—
43C2×C4×D4—28no—
44C2×C4×D4C2×C414yes—
45C2×C4×D4Q814yes—
46C2×#117—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4D411yescharacteristic
50C22×C4⋊C4D414yes—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—22no—
53C22×C22⋊C4D411yescharacteristic
54C22×C22⋊C4C2312yes—
55C22×C22⋊C4C2314yes—
56C22×C22⋊C4C2314yes—
57C22×C22⋊C4C2311yescharacteristic
58C22×C42—22no—
59C22×C42D411yescharacteristic
60C2×#99—24no—
61C2×#99—24no—
62C2×#99C2×C418yes—
63#389—28no—
64#386—28no—
65#385—28no—
66#385—28no—
67#384—22no—
68#382—22no—
69#380—22no—
70#380—22no—
71#380—22no—
72#380—22no—
73#380—22no—
74#380—22no—
75#380—22no—
76#380—22no—
77#380—24no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—22no—
80C4×C22⋊C4—24no—
81C4×C22⋊C4C2×C418yescomplemented
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—22no—
87C2×#95—28no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—28no—
94C2×#95D414yes—
95#342—48no—
96#342C2×C418yescomplemented
97C25C2×D411yescharacteristic
98C25C2411yescharacteristic
99C22×D4—22no—
100C22×D4—22no—
101C22×D4—22no—
102C22×D4—22no—
103C22×D4C22×C412yes—
104C22×D4C2×Q812yes—
105C22×D4#4114yes—
106C22×D4#4114yes—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—24no—
112C23×C4—24no—
113C23×C4—24no—
114C23×C4—22no—
115C23×C4—24no—
116C23×C4—24no—
117C23×C4—44no—
118C23×C4—82no—
119C23×C4—82no—
120C23×C4—82no—
121C23×C4C4⋊C414yes—
122C23×C4C22×C412yes—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D418yes—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D412yes—
130C23×C4C2×D412yes—
131C23×C4C2×Q812yes—
132C23×C4#4114yes—
133C4×D4—28no—
134C4×D4—28no—
135C4×D4—28no—
136C4×D4—28no—
137C4×D4—432no—
138#117—44no—
139#117—44no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—84no—
149C2×C4⋊C4—84no—
150C2×C4⋊C4—82no—
151C2×C4⋊C4C4⋊C414yes—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—28no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—28no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—44no—
173C2×C22⋊C4—82no—
174C2×C22⋊C4C4⋊C414yes—
175C2×C22⋊C4C22×C418yes—
176C2×C22⋊C4C22×C412yes—
177C2×C22⋊C4C2×Q812yes—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—22no—
181C2×C42—24no—
182C2×C42—28no—
183C2×C42—84no—
184C2×C42C4⋊C414yes—
185#99—216no—
186#95—24no—
187#95—24no—
188#95—28no—
189#95—24no—
190#95—24no—
191#95—416no—
192#95—44no—
193#95—41no—
194#95—42no—
195#95—41no—
196#95—42no—
197#95—44no—
198#95—82no—
199#95—88no—
200C24—21no—
201C24—21no—
202C24—21no—
203C24—21no—
204C24—42no—
205C24—42no—
206C24—42no—
207C24—42no—
208C24C2×C4⋊C412yes—
209C24#12412yes—
210C24#12412yes—
211C24C23×C411yescharacteristic
212C24C22×D411yescharacteristic
213C24C22×D412yes—
214C24C22×Q811yescharacteristic
215C24C2×#4112yes—
216C24C2×#4112yes—
217C24C2×#4112yes—
218C24H(5,𝔽2)12yes—
219C242−1+412yes—
220C2×D4—24no—
221C2×D4—24no—
222C2×D4—24no—
223C2×D4—24no—
224C2×D4—28no—
225C2×D4—28no—
226C2×D4—28no—
227C2×D4—28no—
228C2×D4#11718yes—
229C2×D4#12518yes—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—28no—
241C22×C4—24no—
242C22×C4—28no—
243C22×C4—28no—
244C22×C4—216no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—44no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—48no—
253C22×C4—48no—
254C22×C4—48no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—48no—
265C22×C4—48no—
266C22×C4—48no—
267C22×C4—48no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4—84no—
279C22×C4—84no—
280C22×C4—88no—
281C22×C4—88no—
282C22×C4—84no—
283C22×C4—84no—
284C22×C4—84no—
285C22×C4C2×C4⋊C412yes—
286C22×C4C2×C4⋊C412yes—
287C22×C4C2×C4⋊C412yes—
288C22×C4C2×C4⋊C412yes—
289C22×C4C2×C4⋊C412yes—
290C22×C4C4×D414yes—
291C22×C4C4×D414yes—
292C22×C4#12011yescharacteristic
293C22×C4#12012yes—
294C22×C4#12011yescharacteristic
295C22×C4C22⋊Q814yes—
296C22×C4C22⋊Q814yes—
297C22×C4#12312yes—
298C22×C4#12312yes—
299C22×C4#12412yes—
300C22×C4#12412yes—
301C4⋊C4—28no—
302C4⋊C4—48no—
303C4⋊C4—48no—
304C4⋊C4—84no—
305C4⋊C4—84no—
306C4⋊C4—84no—
307C4⋊C4—84no—
308C22⋊C4—24no—
309C22⋊C4—24no—
310C22⋊C4—28no—
311C22⋊C4—28no—
312C22⋊C4—28no—
313C22⋊C4—48no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C22⋊C4—48no—
317C22⋊C4—44no—
318C22⋊C4—44no—
319C22⋊C4—48no—
320C22⋊C4—88no—
321C42—24no—
322C42—24no—
323C42—48no—
324C42—88no—
325C23—24no—
326C23—22no—
327C23—24no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—41no—
336C23—41no—
337C23—44no—
338C23—42no—
339C23—44no—
340C23—41no—
341C23—41no—
342C23—44no—
343C23—42no—
344C23—44no—
345C23—44no—
346C23—42no—
347C23—42no—
348C23—44no—
349C23—42no—
350C23—42no—
351C23—81no—
352C23—82no—
353C23—82no—
354C23—81no—
355C23#38214yes—
356C23C22×C4⋊C411yescharacteristic
357C23C2×#11712yes—
358C23C2×C4×D412yes—
359C23#52012yes—
360C23C2×#12011yescharacteristic
361C23C2×C22⋊Q812yes—
362C23C2×#12311yescharacteristic
363C23C2×#12411yescharacteristic
364C23C2×#12411yescharacteristic
365C23C2×#12512yes—
366C23#53511yescharacteristic
367C23#53611yescharacteristic
368C23#54412yes—
369C23#54614yes—
370C23#54814yes—
371C23#55214yes—
372C23#55314yes—
373D4—216no—
374D4—216no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—48no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—48no—
400C2×C4—44no—
401C2×C4—48no—
402C2×C4—416no—
403C2×C4—44no—
404C2×C4—48no—
405C2×C4—48no—
406C2×C4—48no—
407C2×C4—44no—
408C2×C4—48no—
409C2×C4—84no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—88no—
413C2×C4—88no—
414C2×C4—82no—
415C2×C4—88no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—816no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—168no—
428C2×C4#38214yes—
429C2×C4#38512yes—
430C2×C4#38512yes—
431C2×C4#38514yes—
432C22—28no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—28no—
437C22—22no—
438C22—22no—
439C22—21no—
440C22—21no—
441C22—22no—
442C22—22no—
443C22—21no—
444C22—21no—
445C22—22no—
446C22—24no—
447C22—42no—
448C22—42no—
449C22—42no—
450C22—42no—
451C22—44no—
452C22—44no—
453C22—44no—
454C22—44no—
455C22—44no—
456C22—42no—
457C22—42no—
458C22—82no—
459C22—82no—
460C22—82no—
461C22C2×#38211yescharacteristic
462C22C2×#38511yescharacteristic
463C22#231711yescharacteristic
464C22#235214yes—
465C22#243511yescharacteristic
466C22#243611yescharacteristic
467C22#244214yes—
468C22#250311yescharacteristic
469C22#250411yescharacteristic
470C22#302512yes—
471C22#302612yes—
472C4—24no—
473C4—22no—
474C4—22no—
475C4—24no—
476C4—42no—
477C4—42no—
478C4—88no—
479C4—88no—
480C4—84no—
481C4—162no—
482C2—24no—
483C2—21no—
484C2—21no—
485C2—21no—
486C2—21no—
487C2—24no—
488C2—44no—
489C2#1388111yescharacteristic
490C2#2088811yescharacteristic
491C2#2088911yescharacteristic
492C2#2590112yes—
493C2#2594912yes—
subgroupG/H
center, socleC23C2×#120
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13966C2semidirect
5. #11287C2semidirect
6. #10854C2semidirect
23. C4×#99C4semidirect
81. C4×C22⋊C4C2×C4semidirect
96. #342C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341812
C2×C4×#992
#139662
#139602
#112872
#108542
#107452