Finite Group Databasebeta

← #207277

Subgroups — #207277

Order 512.

1124 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic
2#21934C211yesmaximal, characteristic
3#13807C211yesmaximal, characteristic
4#9912C212yesmaximal, complemented
5#9911C212yesmaximal, complemented
6#9909C212yesmaximal, complemented
7#9908C212yesmaximal, complemented
8#8019C212yesmaximal
9#8015C212yesmaximal
10C2×#543—21no—
11C2×#407C2212yes—
12#2857C2212yes—
13#2857C2212yes—
14#2659—21no—
15#2659—21no—
16#2549—24no—
17#2440C414yescomplemented
18#2440C414yescomplemented
19#2315—21no—
20C2×#381C2211yescharacteristic
21C2×#380C2212yes—
22C2×C4×C22⋊C4C2211yescharacteristic
23C2×C4×C22⋊C4C2211yescharacteristic
24#1851—24no—
25#1797C2212yes—
26#1797C2212yes—
27#1437—21no—
28#1437—21no—
29#1434—21no—
30#1434C2211yescharacteristic
31#1434C2211yescharacteristic
32C4×#95—21no—
33C4×#95C2211yescharacteristic
34C4×#95C2211yescharacteristic
35#1294C2214yes—
36#1294C2214yes—
37#1294C2214yes—
38#1294C2214yes—
39C24×C4C2311yescharacteristic
40C22×M16—22no—
41#543—44no—
42#543—44no—
43C2×C4⋊Q8C2×C412yes—
44C2×C4⋊Q8C2×C412yes—
45C2×C22⋊Q8—22no—
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8—24no—
48C2×C22⋊Q8—24no—
49C22×C4⋊C4C2×C412yes—
50C22×C4⋊C4C2×C412yes—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—21no—
53C22×C22⋊C4—21no—
54C22×C22⋊C4D411yescharacteristic
55C22×C22⋊C4D411yescharacteristic
56C22×C42—21no—
57C22×C42—21no—
58C22×C42—21no—
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63#407—24no—
64#407—24no—
65C2×C22⋊C8—22no—
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#389C2×C414yes—
76#389C2×C414yes—
77#385—24no—
78#385—24no—
79#384C2×C412yes—
80#384C2×C412yes—
81#381—22no—
82#381Q814yescomplemented
83#380—22no—
84#380—22no—
85#380—22no—
86#380—22no—
87#380C2×C414yes—
88#380C2314yes—
89C4×C22⋊C4—22no—
90C4×C22⋊C4C2×C414yes—
91C4×C22⋊C4D414yes—
92C4×C22⋊C4Q814yescomplemented
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95D411yescharacteristic
102C2×#95D411yescharacteristic
103C2×#95C2311yescharacteristic
104C2×#95C2311yescharacteristic
105C2×#95C2311yescharacteristic
106C2×#95C2311yescharacteristic
107#336—22no—
108#336—22no—
109C25C2×D411yescharacteristic
110C22×Q8—22no—
111C22×Q8—22no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—81no—
119C23×C4—81no—
120C23×C4C22×C412yes—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D412yes—
127C23×C4C2×Q812yes—
128C23×C4C2×Q812yes—
129C23×C4#4112yes—
130C23×C4#4112yes—
131C23×C4C2411yescharacteristic
132C2×M16—44no—
133C2×M16—82no—
134C2×M16—82no—
135C22×C8—24no—
136C4⋊Q8—82no—
137C4⋊Q8—82no—
138C22⋊Q8—416no—
139C22⋊Q8—416no—
140C22⋊Q8—44no—
141C22⋊Q8—44no—
142C22⋊Q8—44no—
143C22⋊Q8—44no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—82no—
155C2×C4⋊C4—82no—
156C2×C4⋊C4C22×C414yes—
157C2×C4⋊C4C22×C414yes—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—42no—
167C2×C22⋊C4—42no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—81no—
171C2×C22⋊C4—81no—
172C2×C22⋊C4C22⋊C414yes—
173C2×C22⋊C4C22⋊C414yes—
174C2×C42—24no—
175C2×C42—42no—
176C2×C42—42no—
177C2×C42—42no—
178C2×C42—42no—
179C2×C42—81no—
180C2×C42—81no—
181C2×C42—81no—
182C2×C42—81no—
183C2×C42—81no—
184C2×C42—81no—
185C2×C42C22×C412yes—
186C2×C42C22×C412yes—
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189C4⋊C8—82no—
190C4⋊C8—82no—
191C22⋊C8—24no—
192C22⋊C8—24no—
193C22⋊C8—44no—
194#95—24no—
195#95—24no—
196#95—24no—
197#95—24no—
198#95—24no—
199#95—21no—
200#95—21no—
201#95—21no—
202#95—21no—
203#95—42no—
204#95—42no—
205#95—42no—
206#95—42no—
207#95—81no—
208#95—81no—
209#95C22⋊C414yes—
210#95C22⋊C414yes—
211#95C2×Q812yes—
212#95C2×Q812yes—
213#95C2×Q812yes—
214#95C2×Q812yes—
215C24—21no—
216C24—21no—
217C24—21no—
218C24—21no—
219C24—41no—
220C24—41no—
221C24—41no—
222C24—41no—
223C24C2×C22⋊C412yes—
224C24C22⋊Q812yes—
225C24C22⋊Q812yes—
226C24C22×D411yescharacteristic
227C2×Q8—44no—
228C2×Q8—44no—
229C2×Q8—82no—
230C2×Q8—82no—
231C2×Q8—82no—
232C2×Q8—82no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C412yes—
298C22×C4C4×Q814yes—
299C22×C4C4×Q814yes—
300C22×C4C22⋊Q812yes—
301C22×C4C22⋊Q812yes—
302C22×C4#12414yes—
303C22×C4C4⋊Q814yes—
304C22×C4AGL(1,ℤ/8)12yes—
305C22×C4AGL(1,ℤ/8)12yes—
306C22×C4#13712yes—
307C22×C4#13712yes—
308C22×C4C23×C411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C22×Q811yescharacteristic
311C22×C4C22×Q811yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C22×C4C2×#4111yescharacteristic
314M16—88no—
315C2×C8—44no—
316C2×C8—44no—
317C2×C8—44no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—48no—
322C4⋊C4—44no—
323C4⋊C4—48no—
324C4⋊C4—82no—
325C4⋊C4—82no—
326C4⋊C4—82no—
327C4⋊C4—82no—
328C4⋊C4—82no—
329C4⋊C4—82no—
330C22⋊C4—28no—
331C22⋊C4—28no—
332C22⋊C4—48no—
333C22⋊C4—48no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C22⋊C4—84no—
337C42—42no—
338C42—42no—
339C42—42no—
340C42—42no—
341C42—84no—
342C42—84no—
343C42—84no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—81no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23—81no—
370C23#39114yes—
371C23C22×C22⋊C411yescharacteristic
372C23C2×C22⋊Q811yescharacteristic
373C23C2×C22⋊Q811yescharacteristic
374C23C2×AGL(1,ℤ/8)11yescharacteristic
375C23C2×AGL(1,ℤ/8)11yescharacteristic
376C23C2×#13711yescharacteristic
377C23C2×#13711yescharacteristic
378Q8—164no—
379Q8—164no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—164no—
449C2×C4—164no—
450C2×C4#39114yes—
451C2×C4#41814yes—
452C2×C4#41914yes—
453C2×C4C2×C4×Q811yescharacteristic
454C2×C4C2×C4×Q811yescharacteristic
455C2×C4C2×#12411yescharacteristic
456C2×C4C2×C4⋊Q811yescharacteristic
457C8—84no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—24no—
471C22—24no—
472C22—24no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22—82no—
478C22—82no—
479C22—82no—
480C22—82no—
481C22C2×#39111yescharacteristic
482C22#288311yescharacteristic
483C22#288311yescharacteristic
484C22C2×#41811yescharacteristic
485C22C2×#41911yescharacteristic
486C22#290111yescharacteristic
487C22#290111yescharacteristic
488C22#307911yescharacteristic
489C22#307911yescharacteristic
490C22#307911yescharacteristic
491C22#307911yescharacteristic
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—24no—
497C4—24no—
498C4—24no—
499C4—44no—
500C4—42no—
501C4—42no—
502C4—82no—
503C4—82no—
504C4—82no—
505C4—84no—
506C4—84no—
507C2—21no—
508C2—21no—
509C2—21no—
510C2—21no—
511C2—44no—
512C2#1762011yescharacteristic
513C2#1762111yescharacteristic
514C2#1762611yescharacteristic
515C2#1762711yescharacteristic
516C2#2131311yescharacteristic
517C2#2133011yescharacteristic
518C2#2133311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9912C2semidirect
5. #9911C2semidirect
6. #9909C2semidirect
7. #9908C2semidirect
17. #2440C4semidirect
18. #2440C4semidirect
82. #381Q8semidirect
92. C4×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#350022
#219342
#138072
#99122
#99112
#99092
#99082
#80192
#80152