Finite Group Databasebeta

← #186697

Subgroups — #186697

Order 512.

2298 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic
2#21324C211yesmaximal, characteristic
3#20740C211yesmaximal, characteristic
4#20740C211yesmaximal, characteristic
5C2×#1556C211yesmaximal, characteristic
6C2×#1491C211yesmaximal, characteristic
7C2×#1491C211yesmaximal, characteristic
8#8448C218yesmaximal, complemented
9C2×#557—24no—
10C2×#557—22no—
11C2×#557—24no—
12C2×#557—22no—
13C2×#557—24no—
14C2×#557—22no—
15C2×#557C418yescomplemented
16C2×#557C2214yes—
17C22×C4⋊Q8—21no—
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125—21no—
20C22×#125—22no—
21C22×C4×Q8—21no—
22C22×C4×Q8—21no—
23C22×C4×Q8C2211yescharacteristic
24C22×C4×Q8C2211yescharacteristic
25C2×#432C2211yescharacteristic
26C2×#432C2211yescharacteristic
27C2×C4×M16C2211yescharacteristic
28#2285C2211yescharacteristic
29#1993—22no—
30#1990C2214yes—
31#1823—21no—
32#1823—21no—
33#1803C2214yes—
34#1803C2214yes—
35#1787—21no—
36#1787—21no—
37#1763—22no—
38#1763—22no—
39#1736—22no—
40#1556—24no—
41C2×#332C2214yes—
42#1491—24no—
43#1491—24no—
44C2×Q8⋊C8C2214yes—
45C2×Q8⋊C8C2214yes—
46C23×Q8—21no—
47C23×Q8D411yescharacteristic
48C23×Q8D411yescharacteristic
49C22×M16—21no—
50#557—216no—
51#557—216no—
52#557—216no—
53#557—216no—
54#557—28no—
55#557—216no—
56#557—216no—
57#557—28no—
58#557—28no—
59#557C2×C4116yes—
60C2×C4⋊Q8—28no—
61C2×C4⋊Q8—24no—
62C2×C4⋊Q8—28no—
63C2×C4⋊Q8—24no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8C2×C414yes—
69C2×C4⋊Q8D414yes—
70C2×C4⋊Q8D412yes—
71C2×C4⋊Q8C2312yes—
72C2×#125—28no—
73C2×#125—216no—
74C2×#125—28no—
75C2×#125—24no—
76C2×#125—216no—
77C2×#125—28no—
78C2×#125—28no—
79C2×#125—24no—
80C2×C4×Q8—24no—
81C2×C4×Q8—28no—
82C2×C4×Q8—24no—
83C2×C4×Q8—28no—
84C2×C4×Q8—28no—
85C2×C4×Q8—24no—
86C2×C4×Q8—28no—
87C2×C4×Q8—24no—
88C2×C4×Q8—22no—
89C2×C4×Q8—22no—
90C2×C4×Q8—21no—
91C2×C4×Q8—22no—
92C2×C4×Q8—21no—
93C2×C4×Q8—22no—
94C2×C4×Q8—21no—
95C2×C4×Q8—21no—
96C2×C4×Q8C2×C414yes—
97C2×C4×Q8C2×C414yes—
98C2×C4×Q8D414yes—
99C2×C4×Q8D414yes—
100C2×C4×Q8D412yes—
101C2×C4×Q8D412yes—
102C2×C4×Q8C2312yes—
103C2×C4×Q8C2312yes—
104C2×#117—21no—
105C22×C4⋊C4—22no—
106C22×C4⋊C4—22no—
107C22×C4⋊C4—22no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4D411yescharacteristic
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4D412yes—
114C22×C42—22no—
115C22×C42—21no—
116C22×C42—21no—
117C22×C42C2311yescharacteristic
118#432—22no—
119#432—22no—
120#432—22no—
121#432—22no—
122C2×C4⋊C8C2312yes—
123C2×C4⋊C8C2312yes—
124C2×#103—24no—
125C2×#103—24no—
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128C4×M16—22no—
129C4×M16—22no—
130C2×#97—21no—
131C2×#97—21no—
132C2×#97—21no—
133C2×C4×C8C2312yes—
134#384C2312yes—
135#382—21no—
136#380—21no—
137#380—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140#340—24no—
141#340—24no—
142#340—28no—
143#332—28no—
144Q8⋊C8—28no—
145Q8⋊C8—28no—
146C22×Q8—28no—
147C22×Q8—24no—
148C22×Q8—21no—
149C22×Q8—22no—
150C22×Q8—21no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—24no—
154C22×Q8—22no—
155C22×Q8—44no—
156C22×Q8—48no—
157C22×Q8C22⋊C414yes—
158C22×Q8C22⋊C414yes—
159C22×Q8C2×D412yes—
160C22×Q8C2×D412yes—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—22no—
165C23×C4—21no—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C2×M16—24no—
170C2×M16—42no—
171C2×M16—42no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C4⋊Q8—216no—
176C4⋊Q8—232no—
177C4⋊Q8—216no—
178C4⋊Q8—24no—
179C4⋊Q8—24no—
180C4⋊Q8—24no—
181C4⋊Q8—24no—
182C4⋊Q8—24no—
183C4⋊Q8—22no—
184C4⋊Q8—22no—
185C4⋊Q8C22⋊C418yes—
186C4⋊Q8C22×C414yes—
187C4⋊Q8C2×D414yes—
188#125—232no—
189#125—264no—
190#125—216no—
191#125—232no—
192#125—232no—
193#125—216no—
194C4×Q8—216no—
195C4×Q8—216no—
196C4×Q8—232no—
197C4×Q8—232no—
198C4×Q8—216no—
199C4×Q8—216no—
200C4×Q8—24no—
201C4×Q8—24no—
202C4×Q8—24no—
203C4×Q8—24no—
204C4×Q8—24no—
205C4×Q8—24no—
206C4×Q8—24no—
207C4×Q8—24no—
208C4×Q8—24no—
209C4×Q8—24no—
210C4×Q8—22no—
211C4×Q8—22no—
212C4×Q8—22no—
213C4×Q8—22no—
214C4×Q8C22⋊C418yes—
215C4×Q8C22⋊C418yes—
216C4×Q8C22×C414yes—
217C4×Q8C22×C414yes—
218C4×Q8C2×D414yes—
219C4×Q8C2×D414yes—
220#117—44no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—216no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—216no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—216no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4C22⋊C414yes—
249C2×C4⋊C4C22⋊C414yes—
250C2×C4⋊C4C22⋊C418yes—
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4C2×D414yes—
254C2×C22⋊C4—41no—
255C2×C42—28no—
256C2×C42—216no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—24no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—41no—
266C2×C42C22×C412yes—
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42C2×D412yes—
273C2×C42C2411yescharacteristic
274C4⋊C8—22no—
275C4⋊C8—22no—
276C4⋊C8—22no—
277C4⋊C8—22no—
278#103—48no—
279#103—48no—
280C22⋊C8—42no—
281C22⋊C8—42no—
282#97—42no—
283#97—42no—
284#97—42no—
285C4×C8—22no—
286C4×C8—22no—
287#95—41no—
288#95—41no—
289C24#12011yescharacteristic
290C2×Q8—216no—
291C2×Q8—28no—
292C2×Q8—216no—
293C2×Q8—24no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—28no—
301C2×Q8—24no—
302C2×Q8—24no—
303C2×Q8—48no—
304C2×Q8—44no—
305C2×Q8—48no—
306C2×Q8—416no—
307C2×Q8—48no—
308C2×Q8—416no—
309C2×Q8—48no—
310C2×Q8C2×C22⋊C414yes—
311C2×Q8C2×C22⋊C414yes—
312C2×Q8#13718yes—
313C22×C4—28no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—28no—
328C22×C4—44no—
329C22×C4—48no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4C2×C22⋊C412yes—
336C22×C4C2×C22⋊C412yes—
337C22×C4C2×C22⋊C412yes—
338C22×C4#12012yes—
339C22×C4#12012yes—
340C22×C4#12012yes—
341C22×C4#12011yescharacteristic
342C22×C4#12011yescharacteristic
343C22×C4#12011yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347M16—48no—
348C2×C8—24no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—44no—
354C2×C8—44no—
355C4⋊C4—232no—
356C4⋊C4—232no—
357C4⋊C4—232no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—24no—
367C4⋊C4—28no—
368C4⋊C4—28no—
369C4⋊C4—28no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—28no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4C2×C22⋊C414yes—
377C4⋊C4C2×C22⋊C414yes—
378C4⋊C4C2×C22⋊C418yes—
379C22⋊C4—82no—
380C42—232no—
381C42—24no—
382C42—24no—
383C42—24no—
384C42—24no—
385C42—28no—
386C42—28no—
387C42—82no—
388C42C2×C22⋊C412yes—
389C42C2×C22⋊C412yes—
390C42C2×C22⋊C412yes—
391C42#12012yes—
392C42#12012yes—
393C42#12012yes—
394C42C23×C411yescharacteristic
395C42C22×D411yescharacteristic
396C42C22×D411yescharacteristic
397C42C22×D411yescharacteristic
398C23—22no—
399C23—22no—
400C23#37912yes—
401C23#45412yes—
402C23#45612yes—
403C23C2×#12011yescharacteristic
404Q8—216no—
405Q8—24no—
406Q8—28no—
407Q8—24no—
408Q8—24no—
409Q8—48no—
410Q8—416no—
411Q8—48no—
412Q8—44no—
413Q8—48no—
414Q8#41918yes—
415C2×C4—216no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—28no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—28no—
434C2×C4—48no—
435C2×C4—416no—
436C2×C4—48no—
437C2×C4—48no—
438C2×C4—44no—
439C2×C4—48no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4#37912yes—
448C2×C4#37912yes—
449C2×C4#37912yes—
450C2×C4C22×C22⋊C411yescharacteristic
451C2×C4C22×C22⋊C411yescharacteristic
452C2×C4C22×C22⋊C411yescharacteristic
453C2×C4C2×#12011yescharacteristic
454C2×C4C2×#12011yescharacteristic
455C2×C4C2×#12011yescharacteristic
456C2×C4C2×#13714yes—
457C2×C4#57514yes—
458C8—44no—
459C8—82no—
460C8—82no—
461C22—24no—
462C22—24no—
463C22—22no—
464C22—22no—
465C22#179314yes—
466C22C2×#37911yescharacteristic
467C22#300412yes—
468C22#300512yes—
469C22C2×#45411yescharacteristic
470C22C2×#45611yescharacteristic
471C4—28no—
472C4—22no—
473C4—22no—
474C4—21no—
475C4—21no—
476C4—22no—
477C4—44no—
478C4—48no—
479C4—44no—
480C4—44no—
481C4—162no—
482C4C2×#41912yes—
483C4#289612yes—
484C4#300412yes—
485C4#300512yes—
486C2—24no—
487C2#1853512yes—
488C2#1854112yes—
489C2#2059911yescharacteristic
490C2#2060011yescharacteristic
491C2C2×#179311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #8448C2semidirect
15. C2×#557C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5572
#213242
#207402
C2×#15562
C2×#14912
#84482