Finite Group Databasebeta

← #190520

Subgroups — #190520

Order 512.

1064 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2319C211yesmaximal, characteristic
2#21916C211yesmaximal, characteristic, complemented
3#21912C211yesmaximal, characteristic, complemented
4#8699C212yesmaximal, complemented
5#8561C212yesmaximal, complemented
6#8561C212yesmaximal, complemented
7#8302C212yesmaximal
8#8155C212yesmaximal, complemented
9#8155C212yesmaximal, complemented
10C22×C4×D4C2211yescharacteristic
11#2319—24no—
12#2319—24no—
13C2×#390C411yescharacteristic
14C2×#390C411yescharacteristic
15C2×#389C411yescharacteristic
16C2×#389C411yescharacteristic
17C2×#388C2211yescharacteristic
18C2×#388C2211yescharacteristic
19C2×#388C2211yescharacteristic
20C2×#388C2211yescharacteristic
21C2×#381C411yescharacteristic
22C2×#381C411yescharacteristic
23C2×#380C411yescharacteristic
24C2×#380C411yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×#376C2211yescharacteristic
27#1917—24no—
28#1916—24no—
29#1817C2212yes—
30#1817C2212yes—
31#1797C2212yes—
32#1797C2212yes—
33#1749C2212yes—
34C4×C22⋊C8C2212yes—
35C2×#323C2214yes—
36C2×#323C2214yes—
37#1280C2214yes—
38#1280C2214yes—
39C23×D4C2×C411yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2×C411yescharacteristic
42C2×C4×D4—24no—
43C2×C4×D4—24no—
44C2×C4×D4—22no—
45C2×C4×D4—22no—
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C411yescharacteristic
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⋊C4C2×C411yescharacteristic
61C22×C22⋊C4C2311yescharacteristic
62C22×C22⋊C4C2311yescharacteristic
63C22×C42C2×C411yescharacteristic
64C22×C42C2×C411yescharacteristic
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C22⋊C8C2314yes—
71C2×C22⋊C8C2314yes—
72C2×#97—22no—
73C2×C4×C8—22no—
74#390—24no—
75#390—24no—
76#389—24no—
77#389—24no—
78#388—22no—
79#388—22no—
80#388—22no—
81#388—22no—
82#388—22no—
83#388—22no—
84#388—22no—
85#388—22no—
86#381—24no—
87#381—24no—
88#380—24no—
89#380—24no—
90C4×C4⋊C4D414yes—
91C4×C4⋊C4D414yes—
92#376—22no—
93#376—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×#95C2311yescharacteristic
103C2×#95C2311yescharacteristic
104#336—22no—
105#336—22no—
106#336—24no—
107#336—24no—
108#336—22no—
109#336—22no—
110#323—28no—
111#323—28no—
112C25C22×C411yescharacteristic
113C25C22×C411yescharacteristic
114C22×D4—22no—
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—24no—
118C22×D4—42no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—81no—
126C23×C4—81no—
127C23×C4—81no—
128C23×C4—81no—
129C23×C4C22×C411yescharacteristic
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C411yescharacteristic
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×C4C2411yescharacteristic
142C22×C8—24no—
143C22×C8—24no—
144C4×D4—216no—
145C4×D4—216no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—81no—
156C2×C4⋊C4—81no—
157C2×C4⋊C4—81no—
158C2×C4⋊C4—81no—
159C2×C4⋊C4C22⋊C414yes—
160C2×C4⋊C4C22⋊C414yes—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—81no—
176C2×C22⋊C4—81no—
177C2×C22⋊C4—81no—
178C2×C22⋊C4—81no—
179C2×C22⋊C4—81no—
180C2×C22⋊C4—81no—
181C2×C22⋊C4—81no—
182C2×C22⋊C4—81no—
183C2×C22⋊C4#4114yes—
184C2×C22⋊C4#4114yes—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—24no—
190C2×C42—42no—
191C2×C42—81no—
192C2×C42—81no—
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C4⋊C8—82no—
200C4⋊C8—82no—
201C22⋊C8—24no—
202C22⋊C8—24no—
203C22⋊C8—24no—
204C22⋊C8—24no—
205#97—82no—
206C4×C8—82no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95C22⋊C414yes—
218#95C22⋊C414yes—
219C24—21no—
220C24—21no—
221C24—21no—
222C24—21no—
223C24—22no—
224C24—22no—
225C24—22no—
226C24—22no—
227C24—41no—
228C24—41no—
229C24—41no—
230C24—41no—
231C24—41no—
232C24—41no—
233C24—41no—
234C24—41no—
235C24#11712yes—
236C24#11712yes—
237C24C23×C411yescharacteristic
238C2×D4—28no—
239C2×D4—48no—
240C2×D4—48no—
241C2×D4—44no—
242C2×D4—44no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—28no—
253C22×C4—28no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
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—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4—84no—
295C22×C4—84no—
296C22×C4—84no—
297C22×C4—84no—
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C412yes—
303C22×C4C2×C22⋊C412yes—
304C22×C4#11712yes—
305C22×C4#11712yes—
306C22×C4#13114yes—
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C2×#4112yes—
310C22×C4C2×#4112yes—
311C2×C8—44no—
312C2×C8—44no—
313C2×C8—44no—
314C2×C8—44no—
315C2×C8—44no—
316C2×C8—44no—
317C4⋊C4—28no—
318C4⋊C4—28no—
319C4⋊C4—44no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C22⋊C4—24no—
325C22⋊C4—24no—
326C22⋊C4—24no—
327C22⋊C4—24no—
328C22⋊C4—82no—
329C22⋊C4—82no—
330C22⋊C4—82no—
331C22⋊C4—82no—
332C22⋊C4—82no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C42—24no—
337C42—24no—
338C42—42no—
339C42—42no—
340C42—42no—
341C42—42no—
342C42—44no—
343C42—44no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—22no—
349C23—22no—
350C23—24no—
351C23—24no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—42no—
373C23—42no—
374C23—82no—
375C23—82no—
376C23—82no—
377C23—82no—
378C23—82no—
379C23—82no—
380C23—82no—
381C23—82no—
382C23C22×C22⋊C411yescharacteristic
383C23C2×#11711yescharacteristic
384C23C2×#11711yescharacteristic
385C23C2×#13112yes—
386C23#56812yes—
387D4—416no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
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—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
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—82no—
438C2×C4—82no—
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#40814yes—
448C2×C4#52512yes—
449C2×C4#52512yes—
450C8—84no—
451C8—84no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22—84no—
467C22—84no—
468C22—84no—
469C22—84no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22—81no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22#228011yescharacteristic
479C22C2×#40811yescharacteristic
480C22#287911yescharacteristic
481C22#288311yescharacteristic
482C22#288411yescharacteristic
483C22#288811yescharacteristic
484C22#288811yescharacteristic
485C22#292412yes—
486C22#292412yes—
487C4—24no—
488C4—22no—
489C4—22no—
490C4—42no—
491C4—42no—
492C4—42no—
493C4—42no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—81no—
498C4—81no—
499C4—81no—
500C4—81no—
501C2—22no—
502C2—22no—
503C2—82no—
504C2—82no—
505C2#1738911yescharacteristic
506C2#1739011yescharacteristic
507C2#1740811yescharacteristic
508C2#1740911yescharacteristic
509C2#2071711yescharacteristic
510C2#2072511yescharacteristic
511C2#2072911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21916C2semidirect
3. #21912C2semidirect
4. #8699C2semidirect
5. #8561C2semidirect
6. #8561C2semidirect
8. #8155C2semidirect
9. #8155C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23192
#219162
#219122
#86992
#85612
#83022
#81552