Finite Group Databasebeta

← #206930

Subgroups — #206930

Order 512.

1132 conjugacy classes of subgroups.

The subgroup lattice has 507 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2329C211yesmaximal, characteristic
2#21934C211yesmaximal, characteristic
3#20524C211yesmaximal, characteristic
4#9883C212yesmaximal, complemented
5#9881C212yesmaximal, complemented
6#9880C212yesmaximal, complemented
7#9878C212yesmaximal, complemented
8#7911C212yesmaximal
9#7909C212yesmaximal
10C22×C22⋊Q8C411yescharacteristic
11C2×#407C2212yes—
12C2×#407C2212yes—
13#2329—24no—
14#2329—24no—
15C2×#391C411yescharacteristic
16C2×#391C2211yescharacteristic
17C2×#389C411yescharacteristic
18C2×#389C2211yescharacteristic
19C2×#387C411yescharacteristic
20C2×#387C411yescharacteristic
21C2×#387C2211yescharacteristic
22C2×#387C2211yescharacteristic
23C2×#385C411yescharacteristic
24C2×#385C411yescharacteristic
25C2×#380C411yescharacteristic
26C2×C4×C22⋊C4C2211yescharacteristic
27C2×C4×C22⋊C4C2211yescharacteristic
28C22×#95C2211yescharacteristic
29#1851—24no—
30#1797C2212yes—
31#1797C2212yes—
32#1753—24no—
33#1749C2212yes—
34C4×C22⋊C8C2212yes—
35#1280C2214yes—
36#1280C2214yes—
37#1280C2214yes—
38#1280C2214yes—
39C23×Q8C2×C411yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2×C411yescharacteristic
42C24×C4C2311yescharacteristic
43C22×M16—22no—
44C22×M16—22no—
45C2×C22⋊Q8—24no—
46C2×C22⋊Q8—24no—
47C2×C22⋊Q8—24no—
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C22⋊C4C2×C411yescharacteristic
58C22×C22⋊C4C2×C411yescharacteristic
59C22×C22⋊C4C2×C411yescharacteristic
60C22×C22⋊C4C2×C411yescharacteristic
61C22×C42C2×C411yescharacteristic
62C22×C42C2×C411yescharacteristic
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67#407—24no—
68#407—28no—
69#407Q818yescomplemented
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8C2314yes—
73C2×C22⋊C8C2314yes—
74C2×#97—22no—
75C2×C4×C8—22no—
76#391—22no—
77#391—22no—
78#391—24no—
79#389—22no—
80#389—22no—
81#389—24no—
82#387—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#387—24no—
87#387—24no—
88#385—24no—
89#385—24no—
90#380—24no—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4D414yes—
94C4×C22⋊C4D414yes—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95C2×C411yescharacteristic
98C2×#95C2×C411yescharacteristic
99C2×#95C2×C411yescharacteristic
100C2×#95C2×C411yescharacteristic
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C411yescharacteristic
105C2×#95C2×C411yescharacteristic
106C2×#95C2×C411yescharacteristic
107C2×#95C2311yescharacteristic
108C2×#95C2311yescharacteristic
109C2×#95C2311yescharacteristic
110C2×#95C2311yescharacteristic
111#336—22no—
112#336—22no—
113#336—22no—
114#336—22no—
115C25C22×C411yescharacteristic
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—81no—
119C22×Q8—81no—
120C22×Q8M1614yescomplemented
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—42no—
126C23×C4—81no—
127C23×C4—81no—
128C23×C4—81no—
129C23×C4—81no—
130C23×C4—81no—
131C23×C4—81no—
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C411yescharacteristic
134C23×C4C22×C411yescharacteristic
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C411yescharacteristic
138C23×C4C22×C411yescharacteristic
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C22×C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2×Q814yes—
148C23×C4#4114yes—
149C23×C4C2411yescharacteristic
150C2×M16—28no—
151C2×M16—44no—
152C2×M16—82no—
153C2×M16—82no—
154C2×M16—82no—
155C2×M16—82no—
156C22×C8—24no—
157C22×C8—24no—
158C22⋊Q8—232no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—81no—
174C2×C4⋊C4—81no—
175C2×C4⋊C4—81no—
176C2×C4⋊C4—81no—
177C2×C4⋊C4—81no—
178C2×C4⋊C4—81no—
179C2×C4⋊C4M1614yescomplemented
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4C22⋊C414yes—
189C2×C22⋊C4C22⋊C414yes—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—81no—
193C2×C42—81no—
194C2×C42—81no—
195C2×C42—81no—
196C2×C42C2×D412yes—
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C4⋊C8—82no—
201C4⋊C8—82no—
202C22⋊C8—28no—
203C22⋊C8—24no—
204C22⋊C8—28no—
205C22⋊C8—44no—
206C22⋊C8C2×Q818yes—
207#97—82no—
208C4×C8—82no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—21no—
225#95C22⋊C414yes—
226#95C22⋊C414yes—
227C24—21no—
228C24—21no—
229C24—22no—
230C24—41no—
231C24—41no—
232C24—41no—
233C24—41no—
234C24C2×C22⋊C412yes—
235C24C4×Q814yes—
236C24C23×C411yescharacteristic
237C2×Q8—28no—
238C2×Q8—82no—
239C2×Q8—82no—
240C2×Q8—84no—
241C2×Q8—84no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—28no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
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—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—44no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—84no—
297C22×C4—84no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4—84no—
303C22×C4—84no—
304C22×C4—88no—
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C412yes—
310C22×C4C4×Q814yes—
311C22×C4C22⋊Q814yes—
312C22×C4C22⋊Q814yes—
313C22×C4#12414yes—
314C22×C4C4⋊Q814yes—
315C22×C4C2×M1612yes—
316C22×C4C2×M1612yes—
317C22×C4C2×M1612yes—
318C22×C4C2×M1612yes—
319C22×C4C2×M1612yes—
320C22×C4C2×M1612yes—
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×Q812yes—
324C22×C4C2×#4112yes—
325M16—88no—
326M16—88no—
327C2×C8—28no—
328C2×C8—44no—
329C2×C8—44no—
330C2×C8—44no—
331C2×C8—44no—
332C2×C8—44no—
333C4⋊C4—28no—
334C4⋊C4—28no—
335C4⋊C4—28no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C4⋊C4—82no—
339C4⋊C4—82no—
340C4⋊C4—82no—
341C4⋊C4—82no—
342C22⋊C4—28no—
343C22⋊C4—28no—
344C22⋊C4—44no—
345C22⋊C4—44no—
346C42—42no—
347C42—42no—
348C42—42no—
349C42—42no—
350C42—82no—
351C42—82no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—24no—
363C23—24no—
364C23—44no—
365C23—44no—
366C23—42no—
367C23—42no—
368C23—44no—
369C23—42no—
370C23—42no—
371C23—44no—
372C23#39114yes—
373C23C22×C22⋊C411yescharacteristic
374C23C2×C4×Q812yes—
375C23C22×M1611yescharacteristic
376C23C22×M1611yescharacteristic
377C23#56812yes—
378Q8—88no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—88no—
442C2×C4—88no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4#39114yes—
448C2×C4#40714yes—
449C2×C4C2×C22⋊Q812yes—
450C2×C4C2×#12411yescharacteristic
451C2×C4C2×C4⋊Q811yescharacteristic
452C8—48no—
453C8—84no—
454C22—22no—
455C22—22no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—22no—
461C22—22no—
462C22—24no—
463C22—24no—
464C22—24no—
465C22—48no—
466C22—48no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22C2×#39111yescharacteristic
472C22C2×#40711yescharacteristic
473C22#287911yescharacteristic
474C22#288311yescharacteristic
475C22#288311yescharacteristic
476C22#288811yescharacteristic
477C22#288911yescharacteristic
478C22Q8×M1612yes—
479C22Q8×M1612yes—
480C4—22no—
481C4—21no—
482C4—21no—
483C4—24no—
484C4—24no—
485C4—42no—
486C4—42no—
487C4—42no—
488C4—42no—
489C4—42no—
490C4—42no—
491C4—42no—
492C4—81no—
493C4—81no—
494C4—81no—
495C4—81no—
496C2—22no—
497C2—22no—
498C2—44no—
499C2#1736711yescharacteristic
500C2#1737011yescharacteristic
501C2#1740511yescharacteristic
502C2#1740611yescharacteristic
503C2#2130311yescharacteristic
504C2#2133311yescharacteristic
505C2#2133811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ 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. #9883C2semidirect
5. #9881C2semidirect
6. #9880C2semidirect
7. #9878C2semidirect
69. #407Q8semidirect
120. C22×Q8M16semidirect
179. C2×C4⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×#23292
#219342
#205242
#98832
#98812
#98802
#98782
#79112
#79092