Finite Group Databasebeta

← #183072

Subgroups — #183072

Order 512.

1438 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2344C211yesmaximal, characteristic
2#20379C212yesmaximal
3C2×#1463C211yesmaximal, characteristic
4C22×#324C211yesmaximal, characteristic
5#8154C214yesmaximal, complemented
6#8153C212yesmaximal, complemented
7C4×#324C212yesmaximal, complemented
8#7913C212yesmaximal
9C22×#117C411yescharacteristic
10C22×#117C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C2×#407C2212yes—
13C22×C22⋊C8C2212yes—
14C2×#405—22no—
15#2344—22no—
16#2344—22no—
17#2344—24no—
18C2×#389C412yes—
19C2×#387C412yes—
20C2×#387C2212yes—
21C2×#381C411yescharacteristic
22C2×#381C2211yescharacteristic
23C2×#380C411yescharacteristic
24C2×C4×C4⋊C4C2212yes—
25C2×C4×C22⋊C4C411yescharacteristic
26#1750—24no—
27#1749C2214yes—
28C4×C22⋊C8C2214yes—
29#1729—22no—
30#1463C418yescomplemented
31C2×#324C418yescomplemented
32C2×#324C2211yescharacteristic
33C2×#324C2211yescharacteristic
34C2×#324C2212yes—
35C2×#324C2212yes—
36C2×#324C2212yes—
37#1280C2214yes—
38#1280C2214yes—
39C24×C4C2×C411yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2311yescharacteristic
42C22×M16—22no—
43C23×C8—22no—
44C2×#117—21no—
45C2×#117—22no—
46C2×#117—21no—
47C2×#117—44no—
48C2×#117—44no—
49C2×#117D412yes—
50C2×#117D412yes—
51C2×#117D414yes—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—24no—
55C22×C4⋊C4C2×C412yes—
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C414yes—
59C22×C4⋊C4C2×C414yes—
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4C2×C412yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2×C411yescharacteristic
68C22×C42C2×C412yes—
69C22×C42C2×C412yes—
70C22×C42C2312yes—
71#407—24no—
72#407C2×C418yes—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8C2×C418yes—
77C2×C22⋊C8C2312yes—
78C2×C22⋊C8C2312yes—
79C2×C22⋊C8C2314yes—
80#405—48no—
81C2×#97—24no—
82C2×C4×C8—24no—
83#389—24no—
84#389—24no—
85#387—24no—
86#387—24no—
87#387—24no—
88#387—24no—
89#381—22no—
90#381—22no—
91#381—22no—
92#381—22no—
93#380—22no—
94#380—22no—
95C4×C4⋊C4—24no—
96C4×C4⋊C4—24no—
97C4×C22⋊C4—21no—
98C4×C22⋊C4—22no—
99C4×C22⋊C4—21no—
100C2×#95C2×C412yes—
101C2×#95C2×C412yes—
102C2×#95C2×C412yes—
103C2×#95C2×C412yes—
104C2×#95C2×C412yes—
105C2×#95C2312yes—
106#336—24no—
107#336—24no—
108#324C2×C414yes—
109#324C2×C414yes—
110#324C2×C418yes—
111#324C2×C418yes—
112#324C2×C418yes—
113C25C22×C411yescharacteristic
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—24no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4—42no—
124C23×C4—42no—
125C23×C4—42no—
126C23×C4C4214yes—
127C23×C4C4214yes—
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C412yes—
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C412yes—
132C23×C4C22×C412yes—
133C23×C4C22×C411yescharacteristic
134C23×C4C22×C411yescharacteristic
135C23×C4C22×C412yes—
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C412yes—
138C23×C4C22×C412yes—
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4#4114yes—
142C23×C4C2411yescharacteristic
143C2×M16—28no—
144C2×M16—44no—
145C2×M16—44no—
146C22×C8—28no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—24no—
150C22×C8—44no—
151C22×C8—44no—
152#117—28no—
153#117—48no—
154#117—44no—
155#117—44no—
156#117—48no—
157#117—44no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—28no—
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—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4C4214yes—
178C2×C4⋊C4C4214yes—
179C2×C4⋊C4C22⋊C414yes—
180C2×C4⋊C4C22⋊C414yes—
181C2×C4⋊C4C22×C412yes—
182C2×C4⋊C4C22×C412yes—
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C412yes—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4C22⋊C411yescharacteristic
194C2×C22⋊C4C22⋊C412yes—
195C2×C22⋊C4C22⋊C412yes—
196C2×C22⋊C4C22⋊C412yes—
197C2×C22⋊C4C22⋊C411yescharacteristic
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—24no—
203C2×C42—44no—
204C2×C42—42no—
205C2×C42—44no—
206C2×C42C2×D414yes—
207C2×C42C2×D414yes—
208C22⋊C8—28no—
209C22⋊C8—28no—
210C22⋊C8—22no—
211C22⋊C8—22no—
212C22⋊C8—24no—
213C22⋊C8C22×C414yes—
214C22⋊C8C22×C414yes—
215C22⋊C8C22×C418yes—
216#97—48no—
217C4×C8—48no—
218#95—22no—
219#95—22no—
220#95—24no—
221#95—24no—
222#95—24no—
223#95—24no—
224#95—24no—
225C24—22no—
226C24—22no—
227C24—22no—
228C24—22no—
229C24—42no—
230C24C2×C4212yes—
231C24C2×C22⋊C411yescharacteristic
232C24C2×C22⋊C411yescharacteristic
233C24#11712yes—
234C24C23×C411yescharacteristic
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—28no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—44no—
268C22×C4—41no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—42no—
272C22×C4—48no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—48no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—48no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—48no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4C2×C4212yes—
295C22×C4C2×C4212yes—
296C22×C4C2×C4212yes—
297C22×C4C2×C4212yes—
298C22×C4C2×C4212yes—
299C22×C4C2×C22⋊C414yes—
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C412yes—
303C22×C4#11712yes—
304C22×C4C4×D414yes—
305C22×C4C4×D414yes—
306C22×C4C4×D414yes—
307C22×C4C4×D414yes—
308C22×C4#13114yes—
309C22×C4C23×C411yescharacteristic
310C22×C4C23×C411yescharacteristic
311C22×C4C22×D412yes—
312C22×C4C2×#4112yes—
313M16—416no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—28no—
317C2×C8—42no—
318C2×C8—44no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—416no—
323C2×C8—44no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—28no—
329C4⋊C4—28no—
330C4⋊C4—28no—
331C4⋊C4—44no—
332C4⋊C4—48no—
333C4⋊C4—44no—
334C4⋊C4—44no—
335C22⋊C4—22no—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4#9914yes—
344C42—28no—
345C42—44no—
346C42—44no—
347C42—44no—
348C42—44no—
349C42—44no—
350C23—22no—
351C23—21no—
352C23—21no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—21no—
359C23—21no—
360C23—22no—
361C23—24no—
362C23—24no—
363C23—24no—
364C23—42no—
365C23—42no—
366C23—44no—
367C23—44no—
368C23—44no—
369C23—44no—
370C23C4×C22⋊C412yes—
371C23C4×C22⋊C412yes—
372C23C2×#9912yes—
373C23#41012yes—
374C23C22×C4211yescharacteristic
375C23C22×C22⋊C411yescharacteristic
376C23C2×#11711yescharacteristic
377C23C2×#13112yes—
378C23#56812yes—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—28no—
389C2×C4—28no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—48no—
412C2×C4—48no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—48no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—48no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4C4×C22⋊C414yes—
438C2×C4#40514yescomplemented
439C2×C4C2×#9914yes—
440C2×C4C2×C4×D412yes—
441C2×C4C2×C4×D412yes—
442C8—44no—
443C8—44no—
444C8—48no—
445C22—22no—
446C22—22no—
447C22—24no—
448C22—22no—
449C22—22no—
450C22—24no—
451C22—22no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—22no—
456C22—21no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—44no—
461C22—44no—
462C22—48no—
463C22—42no—
464C22—42no—
465C22C4×#9914yes—
466C22C2×C4×C22⋊C411yescharacteristic
467C22C2×#40511yescharacteristic
468C22#287311yescharacteristic
469C22#288012yes—
470C22C22×#9911yescharacteristic
471C22C2×#41011yescharacteristic
472C22#288912yes—
473C22#292412yes—
474C4—24no—
475C4—24no—
476C4—22no—
477C4—22no—
478C4—24no—
479C4—42no—
480C4—44no—
481C4—42no—
482C4—42no—
483C4—42no—
484C4—82no—
485C4—82no—
486C2—21no—
487C2—21no—
488C2—22no—
489C2—24no—
490C2—42no—
491C2#1741612yes—
492C2#1741712yes—
493C2#2037911yescharacteristic
494C2C2×C4×#9911yescharacteristic
495C2#2039911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C22×C42
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #8154C2semidirect
6. #8153C2semidirect
7. C4×#324C2semidirect
30. #1463C4semidirect
31. C2×#324C4semidirect
438. C2×C4#405semidirect

Maximal subgroups

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

subgroupindex
C2×#23442
#203792
C2×#14632
C22×#3242
#81542
#81532
C4×#3242
#79132