Finite Group Databasebeta

← #241171

Subgroups — #241171

Order 512.

1338 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
1#35002C211yesmaximal, characteristic
2C2×#1852C212yesmaximal
3#14130C211yesmaximal, characteristic
4#10569C214yesmaximal, complemented
5#10567C214yesmaximal, complemented
6#8023C212yesmaximal
7#7907C211yesmaximal, characteristic
8C2×#543—21no—
9C23×C4⋊C4C2211yescharacteristic
10C22×#107—22no—
11C22×C22⋊C8C2212yes—
12#2857—21no—
13#2857—21no—
14#2659C2212yes—
15#2659C2212yes—
16#2549—24no—
17#2440—22no—
18#2440—22no—
19#2315—21no—
20C2×#389—21no—
21C2×#385—21no—
22C2×#380C2211yescharacteristic
23C2×#380C2212yes—
24C2×C4×C22⋊C4C2211yescharacteristic
25#1852—24no—
26#1852C2218yes—
27C2×#340—22no—
28#1444C2212yes—
29#1442C2212yes—
30C4⋊#95—21no—
31C22⋊#95—21no—
32#1294C2218yes—
33#1280C2212yes—
34#1280C2212yes—
35C24×C4—21no—
36C24×C4C2311yescharacteristic
37C23×C8—22no—
38#543—44no—
39#543—44no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C22⋊Q8—24no—
43C2×C22⋊Q8—24no—
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—44no—
52C22×C4⋊C4D412yes—
53C22×C4⋊C4D412yes—
54C22×C4⋊C4C2312yes—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—21no—
57C22×C22⋊C4D411yescharacteristic
58C22×C22⋊C4D411yescharacteristic
59C22×C42D411yescharacteristic
60C22×C42D411yescharacteristic
61C2×#107—28no—
62C2×#107—48no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—24no—
65C2×C22⋊C8Q818yescomplemented
66C2×C22⋊C8C2314yes—
67#398—24no—
68#398—24no—
69#395—24no—
70#395—24no—
71#391—22no—
72#391—22no—
73#389—22no—
74#389—22no—
75#389—22no—
76#389—22no—
77#389—44no—
78#385—24no—
79#385—24no—
80#385—44no—
81#384—21no—
82#384—21no—
83#380—22no—
84#380—22no—
85#380—22no—
86#380—22no—
87#380—22no—
88#380D414yes—
89#380Q814yescomplemented
90#380C2314yes—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95C2311yescharacteristic
98C2×#95C2311yescharacteristic
99C2×#95C2312yes—
100#340—28no—
101C25C2×D411yescharacteristic
102C22×Q8—22no—
103C22×Q8—22no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—42no—
109C23×C4—42no—
110C23×C4—44no—
111C23×C4—42no—
112C23×C4—42no—
113C23×C4—42no—
114C23×C4—82no—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D412yes—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D412yes—
122C23×C4C2×Q814yes—
123C23×C4C2×Q812yes—
124C23×C4#4112yes—
125C23×C4C2411yescharacteristic
126C22×C8—28no—
127C22×C8—24no—
128C22×C8—44no—
129C22×C8—44no—
130C4⋊Q8—82no—
131C4⋊Q8—82no—
132C22⋊Q8—416no—
133C22⋊Q8—416no—
134C22⋊Q8—44no—
135C22⋊Q8—44no—
136C22⋊Q8—44no—
137C22⋊Q8—44no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—48no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4C2×D414yes—
166C2×C4⋊C4C2×D414yes—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—42no—
178C2×C22⋊C4D814yes—
179C2×C22⋊C4Q1614yes—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—42no—
185C2×C42—42no—
186#107—416no—
187C22⋊C8—28no—
188C22⋊C8—24no—
189C22⋊C8C2×Q818yes—
190#95—24no—
191#95—24no—
192#95—24no—
193#95—24no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—44no—
198#95—44no—
199#95—82no—
200#95C2×Q814yes—
201C24—41no—
202C24—41no—
203C24—41no—
204C24—41no—
205C24#12012yes—
206C24C22⋊Q812yes—
207C24C4⋊Q812yes—
208C24C2×D812yes—
209C24C2×Q1612yes—
210C24AGL(1,ℤ/8)12yes—
211C24#13712yes—
212C24C22×D411yescharacteristic
213C2×Q8—44no—
214C2×Q8—44no—
215C2×Q8—82no—
216C2×Q8—82no—
217C2×Q8—82no—
218C2×Q8—82no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—44no—
236C22×C4—48no—
237C22×C4—44no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—48no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—88no—
270C22×C4—88no—
271C22×C4—84no—
272C22×C4—84no—
273C22×C4#12011yescharacteristic
274C22×C4#12011yescharacteristic
275C22×C4C22⋊Q814yes—
276C22×C4C22⋊Q812yes—
277C22×C4C22⋊Q814yes—
278C22×C4C4⋊Q814yes—
279C22×C4C4⋊Q812yes—
280C22×C4C4⋊Q814yes—
281C22×C4C2×D812yes—
282C22×C4C2×D812yes—
283C22×C4C2×Q1612yes—
284C22×C4C2×Q1612yes—
285C22×C4C22×D412yes—
286C22×C4C22×Q812yes—
287C22×C4C22×Q811yescharacteristic
288C22×C4C2×#4111yescharacteristic
289C2×C8—28no—
290C2×C8—44no—
291C2×C8—44no—
292C2×C8—416no—
293C4⋊C4—28no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—48no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4—44no—
301C4⋊C4—48no—
302C4⋊C4—48no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4—82no—
309C4⋊C4—82no—
310C4⋊C4—88no—
311C22⋊C4—28no—
312C22⋊C4—28no—
313C22⋊C4—44no—
314C22⋊C4—44no—
315C22⋊C4—48no—
316C22⋊C4—48no—
317C42—42no—
318C42—42no—
319C42—81no—
320C42—81no—
321C23—21no—
322C23—22no—
323C23—21no—
324C23—22no—
325C23—21no—
326C23—22no—
327C23—21no—
328C23—22no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23#39514yes—
342C23#44714yes—
343C23#45114yes—
344C23#47414yes—
345C23#47714yes—
346C23#50014yes—
347C23#50114yes—
348C23C2×#12011yescharacteristic
349C23C2×C22⋊Q811yescharacteristic
350C23C2×C4⋊Q811yescharacteristic
351C23C22×D811yescharacteristic
352C23C22×Q1611yescharacteristic
353C23C2×AGL(1,ℤ/8)11yescharacteristic
354C23C2×#13711yescharacteristic
355Q8—164no—
356Q8—164no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—28no—
363C2×C4—28no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—816no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—164no—
412C2×C4—164no—
413C2×C4#39512yes—
414C2×C4#39512yes—
415C2×C4B(4,𝔽2)12yes—
416C2×C4#45812yes—
417C2×C4C2×C22⋊Q812yes—
418C2×C4C2×C4⋊Q812yes—
419C8—48no—
420C22—21no—
421C22—21no—
422C22—24no—
423C22—21no—
424C22—21no—
425C22—24no—
426C22—21no—
427C22—21no—
428C22—24no—
429C22—21no—
430C22—21no—
431C22—24no—
432C22—21no—
433C22—21no—
434C22—24no—
435C22—21no—
436C22—21no—
437C22—24no—
438C22—44no—
439C22—44no—
440C22—44no—
441C22—44no—
442C22#205718yes—
443C22C2×#39511yescharacteristic
444C22C2×#44711yescharacteristic
445C22C2×#45111yescharacteristic
446C22#302111yescharacteristic
447C22#302211yescharacteristic
448C22C2×B(4,𝔽2)11yescharacteristic
449C22C2×#45811yescharacteristic
450C22C2×#47411yescharacteristic
451C22C2×#47711yescharacteristic
452C22C2×#50011yescharacteristic
453C22C2×#50111yescharacteristic
454C4—22no—
455C4—22no—
456C4—22no—
457C4—42no—
458C4—42no—
459C4—42no—
460C4—42no—
461C4—84no—
462C4—84no—
463C4—84no—
464C4—84no—
465C4#206214yes—
466C2—21no—
467C2—21no—
468C2—28no—
469C2—21no—
470C2—21no—
471C2#1819411yescharacteristic
472C2#1819711yescharacteristic
473C2#1820211yescharacteristic
474C2#1820511yescharacteristic
475C2C2×#205711yescharacteristic
476C2#2356211yescharacteristic
477C2C2×#206211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10569C2semidirect
5. #10567C2semidirect
65. C2×C22⋊C8Q8semidirect
89. #380Q8semidirect

Maximal subgroups

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

subgroupindex
#350022
C2×#18522
#141302
#105692
#105672
#80232
#79072