Finite Group Databasebeta

← #186622

Subgroups — #186622

Order 512.

2314 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic
2#21834C211yesmaximal, characteristic
3#20734C211yesmaximal, characteristic
4#20647C211yesmaximal, characteristic
5C2×#1549C211yesmaximal, characteristic
6C2×#1487C211yesmaximal, characteristic
7C2×#1479C211yesmaximal, 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×#431C2211yescharacteristic
26C2×#431C2211yescharacteristic
27C2×#423C2211yescharacteristic
28C2×#419—21no—
29#2285C2211yescharacteristic
30#1990C2214yes—
31#1987—22no—
32#1824—22no—
33#1824—21no—
34#1805—22no—
35#1803C2214yes—
36#1803C2214yes—
37#1787—21no—
38C4×#103—22no—
39#1736—21no—
40#1549—24no—
41C2×#332C2214yes—
42#1487—24no—
43#1479—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⋊C4D412yes—
113C22×C4⋊C4D411yescharacteristic
114C22×C42—22no—
115C22×C42—21no—
116C22×C42—21no—
117C22×C42C2311yescharacteristic
118#431—22no—
119#431—22no—
120#431—22no—
121#431—22no—
122#423—22no—
123#423—22no—
124C2×C4⋊C8—21no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2312yes—
128#419—44no—
129C2×#103—24no—
130C2×#103—24no—
131C2×C22⋊C8—21no—
132C2×C22⋊C8—21no—
133C2×C4×C8—21no—
134C2×C4×C8C2312yes—
135#384C2312yes—
136#382—21no—
137#380—21no—
138#380—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141#340—24no—
142#340—24no—
143#340—28no—
144#332—28no—
145Q8⋊C8—28no—
146Q8⋊C8—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—22no—
156C22×Q8—22no—
157C22×Q8—44no—
158C22×Q8—48no—
159C22×Q8C22⋊C414yes—
160C22×Q8C22⋊C414yes—
161C22×Q8SD1614yes—
162C22×Q8Q1614yes—
163C22×Q8C2×D412yes—
164C22×Q8C2×D412yes—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4—21no—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C2×M16—42no—
174C2×M16—42no—
175C2×M16—42no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C4⋊Q8—216no—
180C4⋊Q8—232no—
181C4⋊Q8—216no—
182C4⋊Q8—24no—
183C4⋊Q8—24no—
184C4⋊Q8—24no—
185C4⋊Q8—24no—
186C4⋊Q8—24no—
187C4⋊Q8—22no—
188C4⋊Q8—22no—
189C4⋊Q8C22⋊C418yes—
190C4⋊Q8C22×C414yes—
191C4⋊Q8C2×D414yes—
192#125—232no—
193#125—264no—
194#125—216no—
195#125—232no—
196#125—232no—
197#125—216no—
198C4×Q8—216no—
199C4×Q8—216no—
200C4×Q8—232no—
201C4×Q8—232no—
202C4×Q8—216no—
203C4×Q8—216no—
204C4×Q8—24no—
205C4×Q8—24no—
206C4×Q8—24no—
207C4×Q8—24no—
208C4×Q8—24no—
209C4×Q8—24no—
210C4×Q8—24no—
211C4×Q8—24no—
212C4×Q8—24no—
213C4×Q8—24no—
214C4×Q8—22no—
215C4×Q8—22no—
216C4×Q8—22no—
217C4×Q8—22no—
218C4×Q8C22⋊C418yes—
219C4×Q8C22⋊C418yes—
220C4×Q8C22×C414yes—
221C4×Q8C22×C414yes—
222C4×Q8C2×D414yes—
223C4×Q8C2×D414yes—
224#117—42no—
225#117—42no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—216no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—216no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—216no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4C22⋊C414yes—
254C2×C4⋊C4C22⋊C414yes—
255C2×C4⋊C4C22⋊C418yes—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2×D414yes—
259C2×C22⋊C4—41no—
260C2×C42—28no—
261C2×C42—216no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—22no—
266C2×C42—24no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—41no—
271C2×C42C22×C412yes—
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D412yes—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42C2411yescharacteristic
279C4⋊C8—22no—
280C4⋊C8—22no—
281C4⋊C8—22no—
282C4⋊C8—22no—
283C4⋊C8—42no—
284C4⋊C8—42no—
285#103—48no—
286#103—48no—
287C22⋊C8—42no—
288C22⋊C8—42no—
289C4×C8—22no—
290C4×C8—22no—
291C4×C8—42no—
292#95—41no—
293#95—41no—
294C24#12011yescharacteristic
295C2×Q8—24no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—24no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—24no—
302C2×Q8—24no—
303C2×Q8—28no—
304C2×Q8—28no—
305C2×Q8—24no—
306C2×Q8—28no—
307C2×Q8—24no—
308C2×Q8—28no—
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—48no—
312C2×Q8—44no—
313C2×Q8—48no—
314C2×Q8—416no—
315C2×Q8—48no—
316C2×Q8—416no—
317C2×Q8—48no—
318C2×Q8#103116yescomplemented
319C2×Q8C2×C22⋊C414yes—
320C2×Q8C2×C22⋊C414yes—
321C2×Q8C2×SD1614yes—
322C2×Q8C2×Q1614yes—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—28no—
336C22×C4—42no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—48no—
341C22×C4—42no—
342C22×C4C2×C22⋊C412yes—
343C22×C4C2×C22⋊C412yes—
344C22×C4C2×C22⋊C412yes—
345C22×C4#12012yes—
346C22×C4#12011yescharacteristic
347C22×C4#12011yescharacteristic
348C22×C4#12012yes—
349C22×C4#12012yes—
350C22×C4#12011yescharacteristic
351C22×C4C2×SD1612yes—
352C22×C4C2×Q1612yes—
353C22×C4#13712yes—
354C22×C4#13712yes—
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358M16—84no—
359C2×C8—44no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—44no—
366C4⋊C4—232no—
367C4⋊C4—232no—
368C4⋊C4—232no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—24no—
378C4⋊C4—28no—
379C4⋊C4—28no—
380C4⋊C4—28no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—28no—
384C4⋊C4—44no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—44no—
388C4⋊C4C2×C22⋊C414yes—
389C4⋊C4C2×C22⋊C414yes—
390C4⋊C4C2×C22⋊C418yes—
391C22⋊C4—82no—
392C42—232no—
393C42—24no—
394C42—24no—
395C42—24no—
396C42—24no—
397C42—28no—
398C42—28no—
399C42—82no—
400C42C2×C22⋊C412yes—
401C42C2×C22⋊C412yes—
402C42C2×C22⋊C412yes—
403C42#12012yes—
404C42#12012yes—
405C42#12012yes—
406C42C23×C411yescharacteristic
407C42C22×D411yescharacteristic
408C42C22×D411yescharacteristic
409C42C22×D411yescharacteristic
410C23—22no—
411C23—22no—
412C23#37912yes—
413C23#44812yes—
414C23#45112yes—
415C23C2×#12011yescharacteristic
416Q8—24no—
417Q8—24no—
418Q8—24no—
419Q8—216no—
420Q8—24no—
421Q8—24no—
422Q8—48no—
423Q8—416no—
424Q8—48no—
425Q8—44no—
426Q8—48no—
427Q8C2×#10318yes—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—216no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—28no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—48no—
444C2×C4—48no—
445C2×C4—48no—
446C2×C4—416no—
447C2×C4—44no—
448C2×C4—48no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4#37912yes—
455C2×C4#37912yes—
456C2×C4#37912yes—
457C2×C4C2×#10314yes—
458C2×C4#41914yes—
459C2×C4#44814yes—
460C2×C4#44812yes—
461C2×C4#45114yes—
462C2×C4#45112yes—
463C2×C4C22×C22⋊C411yescharacteristic
464C2×C4C22×C22⋊C411yescharacteristic
465C2×C4C22×C22⋊C411yescharacteristic
466C2×C4C2×#12011yescharacteristic
467C2×C4C2×#12011yescharacteristic
468C2×C4C2×#12011yescharacteristic
469C2×C4C22×SD1611yescharacteristic
470C2×C4C22×Q1611yescharacteristic
471C2×C4C2×#13711yescharacteristic
472C2×C4C2×#13711yescharacteristic
473C2×C4#57514yes—
474C8—82no—
475C8—82no—
476C8—82no—
477C22—24no—
478C22—24no—
479C22—22no—
480C22—22no—
481C22#178714yes—
482C22C2×#37911yescharacteristic
483C22C2×#44811yescharacteristic
484C22C2×#45111yescharacteristic
485C22#300512yes—
486C22#302012yes—
487C4—28no—
488C4—22no—
489C4—21no—
490C4—21no—
491C4—22no—
492C4—44no—
493C4—48no—
494C4—44no—
495C4—44no—
496C4—162no—
497C4#178714yes—
498C4C22×#10311yescharacteristic
499C4C2×#41911yescharacteristic
500C4#289612yes—
501C4C2×#44811yescharacteristic
502C4C2×#45111yescharacteristic
503C4#300512yes—
504C2—24no—
505C2#1877812yes—
506C2#1877912yes—
507C2C2×#178711yescharacteristic
508C2#2060011yescharacteristic
509C2#2061711yescharacteristic
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
318. C2×Q8#103semidirect

Maximal subgroups

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

subgroupindex
C22×#5572
#218342
#207342
#206472
C2×#15492
C2×#14872
C2×#14792
#84482