Finite Group Databasebeta

← #190519

Subgroups — #190519

Order 512.

1138 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2318C211yesmaximal, characteristic
2#21906C211yesmaximal, characteristic, complemented
3#21902C211yesmaximal, characteristic, complemented
4#8698C212yesmaximal, complemented
5#8562C212yesmaximal, complemented
6#8561C212yesmaximal, complemented
7#8301C212yesmaximal
8#8155C212yesmaximal, complemented
9#8154C212yesmaximal, complemented
10C22×C4×D4C2211yescharacteristic
11#2318—24no—
12#2318—24no—
13C2×#388C2211yescharacteristic
14C2×#388C2211yescharacteristic
15C2×#387C2211yescharacteristic
16C2×#387C2211yescharacteristic
17C2×#386C411yescharacteristic
18C2×#386C411yescharacteristic
19C2×#385C411yescharacteristic
20C2×#385C411yescharacteristic
21C2×#383C411yescharacteristic
22C2×#382C411yescharacteristic
23C2×#381C411yescharacteristic
24C2×#380C411yescharacteristic
25C2×C4×C22⋊C4C2211yescharacteristic
26C22×#95C2211yescharacteristic
27#1924—24no—
28#1923—24no—
29#1797C2212yes—
30#1797C2212yes—
31#1779C2212yes—
32#1779C2212yes—
33#1749C2212yes—
34C4×C22⋊C8C2212yes—
35C2×#324C2214yes—
36C2×#323C2214yes—
37#1280C2214yes—
38#1280C2214yes—
39C23×D4C2×C411yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2×C411yescharacteristic
42C24×C4C2×C411yescharacteristic
43C24×C4C2311yescharacteristic
44C2×C4×D4—22no—
45C2×C4×D4—22no—
46C2×C4×D4—24no—
47C2×C4×D4—24no—
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C22⋊C4C2×C411yescharacteristic
54C22×C22⋊C4C2×C411yescharacteristic
55C22×C22⋊C4C2×C411yescharacteristic
56C22×C22⋊C4C2×C411yescharacteristic
57C22×C22⋊C4C2×C411yescharacteristic
58C22×C22⋊C4C2×C411yescharacteristic
59C22×C22⋊C4C2×C411yescharacteristic
60C22×C22⋊C4C2311yescharacteristic
61C22×C42C2×C411yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8C2314yes—
71C2×C22⋊C8C2314yes—
72C2×#97—22no—
73C2×C4×C8—22no—
74#388—22no—
75#388—22no—
76#388—22no—
77#388—22no—
78#387—22no—
79#387—22no—
80#387—22no—
81#387—22no—
82#386—24no—
83#386—24no—
84#385—24no—
85#385—24no—
86#383—24no—
87#382—24no—
88#381—24no—
89#380—24no—
90C4×C22⋊C4D414yes—
91C4×C22⋊C4D414yes—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95C2×C411yescharacteristic
95C2×#95C2×C411yescharacteristic
96C2×#95C2×C411yescharacteristic
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×#95C2311yescharacteristic
107C2×#95C2311yescharacteristic
108#336—22no—
109#336—22no—
110#336—22no—
111#336—22no—
112#324—28no—
113#323—28no—
114C25C22×C411yescharacteristic
115C25C22×C411yescharacteristic
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—24no—
119C22×D4—22no—
120C22×D4—42no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—24no—
129C23×C4—42no—
130C23×C4—81no—
131C23×C4—81no—
132C23×C4—81no—
133C23×C4—81no—
134C23×C4—81no—
135C23×C4—81no—
136C23×C4—81no—
137C23×C4—81no—
138C23×C4C22×C411yescharacteristic
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C22×C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C411yescharacteristic
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C23×C4C2411yescharacteristic
153C22×C8—24no—
154C22×C8—24no—
155C4×D4—216no—
156C4×D4—216no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—81no—
165C2×C4⋊C4—81no—
166C2×C4⋊C4—81no—
167C2×C4⋊C4—81no—
168C2×C4⋊C4#4114yes—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4—81no—
182C2×C22⋊C4—81no—
183C2×C22⋊C4—81no—
184C2×C22⋊C4—81no—
185C2×C22⋊C4C22⋊C414yes—
186C2×C22⋊C4C22⋊C414yes—
187C2×C22⋊C4#4114yes—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—81no—
192C2×C42—81no—
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C4⋊C8—82no—
198C4⋊C8—82no—
199C22⋊C8—24no—
200C22⋊C8—24no—
201C22⋊C8—24no—
202C22⋊C8—24no—
203C22⋊C8—82no—
204C22⋊C8—82no—
205C22⋊C8—82no—
206C22⋊C8—82no—
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—22no—
218#95—22no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95C22⋊C414yes—
224#95C22⋊C414yes—
225C24—21no—
226C24—21no—
227C24—22no—
228C24—21no—
229C24—21no—
230C24—22no—
231C24—22no—
232C24—41no—
233C24—41no—
234C24—41no—
235C24—41no—
236C24—41no—
237C24—41no—
238C24—41no—
239C24—41no—
240C24C2×C22⋊C412yes—
241C24#11712yes—
242C24#13114yes—
243C24C23×C411yescharacteristic
244C2×D4—28no—
245C2×D4—48no—
246C2×D4—48no—
247C2×D4—44no—
248C2×D4—44no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—28no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—82no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4—82no—
310C22×C4—84no—
311C22×C4—84no—
312C22×C4—84no—
313C22×C4—84no—
314C22×C4—88no—
315C22×C4—88no—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C412yes—
320C22×C4C2×C22⋊C412yes—
321C22×C4#11712yes—
322C22×C4#11712yes—
323C22×C4#11712yes—
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C2×#4112yes—
327C22×C4C2×#4112yes—
328C2×C8—44no—
329C2×C8—44no—
330C2×C8—44no—
331C2×C8—44no—
332C2×C8—44no—
333C2×C8—44no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C4⋊C4—82no—
339C4⋊C4—82no—
340C22⋊C4—28no—
341C22⋊C4—28no—
342C22⋊C4—24no—
343C22⋊C4—24no—
344C22⋊C4—44no—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C22⋊C4—82no—
348C22⋊C4—82no—
349C42—28no—
350C42—42no—
351C42—42no—
352C42—42no—
353C42—42no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—24no—
364C23—24no—
365C23—22no—
366C23—22no—
367C23—24no—
368C23—44no—
369C23—44no—
370C23—42no—
371C23—42no—
372C23—44no—
373C23—42no—
374C23—42no—
375C23—44no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—42no—
387C23—82no—
388C23—82no—
389C23—82no—
390C23—82no—
391C23#40814yes—
392C23C22×C22⋊C411yescharacteristic
393C23C2×#11711yescharacteristic
394C23C2×#11711yescharacteristic
395C23C2×#13112yes—
396C23#56812yes—
397D4—416no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—42no—
422C2×C4—42no—
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—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—84no—
449C2×C4—84no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—88no—
455C2×C4—88no—
456C2×C4—88no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4#52512yes—
462C2×C4#52512yes—
463C8—84no—
464C8—84no—
465C22—22no—
466C22—22no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—22no—
472C22—22no—
473C22—24no—
474C22—24no—
475C22—24no—
476C22—48no—
477C22—48no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—84no—
486C22—84no—
487C22—81no—
488C22—81no—
489C22—81no—
490C22—81no—
491C22#228011yescharacteristic
492C22C2×#40811yescharacteristic
493C22#287911yescharacteristic
494C22#288311yescharacteristic
495C22#288311yescharacteristic
496C22#288811yescharacteristic
497C22#288911yescharacteristic
498C22#292412yes—
499C22#292412yes—
500C4—22no—
501C4—22no—
502C4—42no—
503C4—42no—
504C4—42no—
505C4—42no—
506C4—42no—
507C4—42no—
508C4—42no—
509C4—42no—
510C4—81no—
511C4—81no—
512C4—81no—
513C4—81no—
514C2—22no—
515C2—22no—
516C2—44no—
517C2—82no—
518C2#1738611yescharacteristic
519C2#1738711yescharacteristic
520C2#1740511yescharacteristic
521C2#1740611yescharacteristic
522C2#2071711yescharacteristic
523C2#2072411yescharacteristic
524C2#2073111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×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
2. #21906C2semidirect
3. #21902C2semidirect
4. #8698C2semidirect
5. #8562C2semidirect
6. #8561C2semidirect
8. #8155C2semidirect
9. #8154C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23182
#219062
#219022
#86982
#85622
#85612
#83012
#81552
#81542