Finite Group Databasebeta

← #189852

Subgroups — #189852

Order 512.

1412 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2285C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic
3#14682C211yesmaximal, characteristic
4#8717C214yesmaximal, complemented
5#8713C214yesmaximal
6#8026C214yesmaximal, complemented
7C22×#117—21no—
8C23×C42C2211yescharacteristic
9C2×#431C2212yes—
10#2285—21no—
11#2285—21no—
12#2285—21no—
13#2285—21no—
14#2285C412yescomplemented
15#2285C412yescomplemented
16#2285C2212yes—
17#2285C2212yes—
18C2×#384C411yescharacteristic, complemented
19C2×#384C411yescharacteristic, complemented
20C2×#382—21no—
21C2×#382C2212yes—
22C2×#380—21no—
23C2×#380—21no—
24C2×#380C2212yes—
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C412yescomplemented
27#1817C2212yes—
28#1797—21no—
29#1797—21no—
30#1779C2212yes—
31C4×C22⋊C8—21no—
32C4×C22⋊C8—21no—
33C4⋊#95—21no—
34#1434—21no—
35#1434C2212yes—
36C22⋊#95—21no—
37C22⋊#95C2212yes—
38#1294C22116yes—
39C24×C4D411yescharacteristic
40C24×C4D411yescharacteristic
41C24×C4C2311yescharacteristic
42C2×#117—22no—
43C2×#117—22no—
44C2×#117—22no—
45C2×#117—22no—
46C2×#117—48no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4C2×C412yes—
53C22×C4⋊C4C2×C412yes—
54C22×C22⋊C4—22no—
55C22×C22⋊C4—22no—
56C22×C42—28no—
57C22×C42—22no—
58C22×C42C2×C411yescharacteristic
59C22×C42C2×C411yescharacteristic
60C22×C42C2×C411yescharacteristic
61C22×C42C2×C411yescharacteristic
62C22×C42D411yescharacteristic
63C22×C42D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70#431—28no—
71C2×C4⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8C2314yes—
75C2×C4×C8—22no—
76#384—21no—
77#384—21no—
78#384—21no—
79#384—21no—
80#384C2×C412yes—
81#384C2×C412yes—
82#384C2×C412yes—
83#384C2×C412yes—
84#382—22no—
85#382—22no—
86#382—22no—
87#382—22no—
88#382—28no—
89#380—22no—
90#380—22no—
91#380—22no—
92#380—22no—
93#380—22no—
94#380—22no—
95#380—44no—
96#380C2×C414yes—
97#380C2314yes—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4C2×C414yes—
105C4×C4⋊C4C2×C414yes—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95C2314yes—
113#336—22no—
114#336—22no—
115C25C2×D411yescharacteristic
116C23×C4—24no—
117C23×C4—24no—
118C23×C4—24no—
119C23×C4—24no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—24no—
123C23×C4—42no—
124C23×C4—42no—
125C23×C4—42no—
126C23×C4—42no—
127C23×C4C22⋊C411yescharacteristic
128C23×C4C22⋊C411yescharacteristic
129C23×C4C22⋊C411yescharacteristic
130C23×C4C22⋊C411yescharacteristic
131C23×C4C22⋊C411yescharacteristic
132C23×C4C22⋊C411yescharacteristic
133C23×C4C22⋊C411yescharacteristic
134C23×C4C22⋊C411yescharacteristic
135C23×C4SD1614yes—
136C23×C4Q1614yes—
137C23×C4C22×C411yescharacteristic
138C23×C4C22×C411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2411yescharacteristic
147C22×C8—24no—
148#117—48no—
149#117—48no—
150#117—48no—
151#117—48no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4C22×C414yes—
172C2×C4⋊C4C22×C414yes—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—48no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—44no—
179C2×C42—24no—
180C2×C42—24no—
181C2×C42—24no—
182C2×C42—24no—
183C2×C42—24no—
184C2×C42—24no—
185C2×C42—24no—
186C2×C42—24no—
187C2×C42—24no—
188C2×C42—24no—
189C2×C42—24no—
190C2×C42—24no—
191C2×C42—24no—
192C2×C42—48no—
193C2×C42C22⋊C411yescharacteristic
194C2×C42C22⋊C411yescharacteristic
195C2×C42C22⋊C411yescharacteristic
196C2×C42C22⋊C411yescharacteristic
197C2×C42C22⋊C411yescharacteristic
198C2×C42C22⋊C411yescharacteristic
199C2×C42C22⋊C411yescharacteristic
200C2×C42C22⋊C411yescharacteristic
201C2×C42C22×C411yescharacteristic
202C2×C42C22×C411yescharacteristic
203C2×C42C22×C411yescharacteristic
204C2×C42C22×C411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42#4118yes—
210C4⋊C8—44no—
211C22⋊C8—28no—
212C22⋊C8—44no—
213C22⋊C8—44no—
214C4×C8—44no—
215#95—24no—
216#95—24no—
217#95—24no—
218#95—24no—
219#95—28no—
220#95—44no—
221#95—44no—
222#95—44no—
223#95—44no—
224C24—22no—
225C24—22no—
226C24—22no—
227C24—22no—
228C24—41no—
229C24—41no—
230C24C2×C22⋊C411yescharacteristic
231C24C2×C22⋊C411yescharacteristic
232C24C2×SD1612yes—
233C24C2×Q1612yes—
234C24C22×D411yescharacteristic
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—216no—
252C22×C4—216no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—24no—
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—416no—
284C22×C4—416no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4#10314yes—
299C22×C4#10314yes—
300C22×C4#10314yes—
301C22×C4#10314yes—
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4#12314yes—
313C22×C4#12314yes—
314C22×C4C2×SD1612yes—
315C22×C4C2×SD1612yes—
316C22×C4C2×Q1612yes—
317C22×C4C2×Q1612yes—
318C22×C4AGL(1,ℤ/8)14yes—
319C22×C4C23×C411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C2×#4114yes—
322C2×C8—44no—
323C2×C8—44no—
324C2×C8—44no—
325C4⋊C4—44no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—44no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—88no—
338C22⋊C4—48no—
339C22⋊C4—48no—
340C22⋊C4—88no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—48no—
354C42—48no—
355C42#11714yes—
356C42#11714yes—
357C42#12314yes—
358C42#12314yes—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—24no—
368C23—24no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—24no—
376C23—24no—
377C23—42no—
378C23—42no—
379C23—44no—
380C23—44no—
381C23—44no—
382C23—44no—
383C23#41012yes—
384C23C2×#10312yes—
385C23C2×#10312yes—
386C23#42112yes—
387C23C22×C22⋊C411yescharacteristic
388C23C2×#12312yes—
389C23C22×SD1611yescharacteristic
390C23C22×Q1611yescharacteristic
391C23C2×AGL(1,ℤ/8)12yes—
392C2×C4—28no—
393C2×C4—28no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—28no—
413C2×C4—28no—
414C2×C4—28no—
415C2×C4—28no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—88no—
439C2×C4—88no—
440C2×C4—84no—
441C2×C4—84no—
442C2×C4—84no—
443C2×C4—84no—
444C2×C4—84no—
445C2×C4—84no—
446C2×C4—84no—
447C2×C4#38112yes—
448C2×C4#38112yes—
449C2×C4#38112yes—
450C2×C4#38112yes—
451C2×C4#41012yes—
452C2×C4C2×#10312yes—
453C2×C4C2×#10312yes—
454C2×C4C2×#10312yes—
455C2×C4C2×#10312yes—
456C2×C4#41812yes—
457C2×C4#41812yes—
458C2×C4#42112yes—
459C2×C4#48118yes—
460C2×C4#48418yes—
461C2×C4C2×#11712yes—
462C2×C4C2×#12312yes—
463C8—84no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—48no—
482C22—48no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22#151814yes—
488C22C2×#38111yescharacteristic
489C22C2×#41011yescharacteristic
490C22C2×#41011yescharacteristic
491C22C22×#10311yescharacteristic
492C22C2×#41811yescharacteristic
493C22C2×#42111yescharacteristic
494C22C2×#42111yescharacteristic
495C22C2×#48112yes—
496C22C2×#48412yes—
497C4—22no—
498C4—22no—
499C4—22no—
500C4—22no—
501C4—22no—
502C4—22no—
503C4—22no—
504C4—22no—
505C4—41no—
506C4—41no—
507C4—84no—
508C4—84no—
509C4—164no—
510C4#182414yes—
511C4#182414yes—
512C2—22no—
513C2—22no—
514C2—24no—
515C2—42no—
516C2C2×#151811yescharacteristic
517C2#1757511yescharacteristic
518C2#1758311yescharacteristic
519C2#1758911yescharacteristic
520C2#2068811yescharacteristic
521C2C2×#182411yescharacteristic
522C2#2070511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8717C2semidirect
6. #8026C2semidirect
14. #2285C4semidirect
15. #2285C4semidirect
18. C2×#384C4semidirect
19. C2×#384C4semidirect
26. C2×C4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22852
#207142
#146822
#87172
#87132
#80262