Finite Group Databasebeta

← #4598534

Subgroups — #4598534

Order 512.

2134 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2822C211yesmaximal, characteristic, complemented
2C2×#2563C211yesmaximal, characteristic
3#17153C214yesmaximal, complemented
4#16688C212yesmaximal, complemented
5#16633C212yesmaximal
6#16378C214yesmaximal, complemented
7#16304C214yesmaximal, complemented
8#15814C212yesmaximal
9#15783C212yesmaximal, complemented
10#15490C214yesmaximal, complemented
11#15071C212yesmaximal, complemented
12#14824C212yesmaximal
13#14476C211yesmaximal, characteristic, complemented
14C22×C4⋊Q8C2211yescharacteristic
15C22×#125C2212yes—
16C22×C4×D4C2211yescharacteristic
17#2822—28no—
18#2563—28no—
19C2×#400C2214yes—
20C2×#400C2212yes—
21C2×#398C2212yes—
22C2×#398C2212yes—
23C2×#398C2214yes—
24C2×#397C2212yes—
25C2×#394C2214yes—
26C2×#393C2214yes—
27C2×#391C2212yes—
28C2×#390C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#387C2214yes—
31C2×#382C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#380C2214yes—
34C2×#380C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36#1446C2212yes—
37#1446C2214yes—
38#1444C2212yes—
39#1444C2214yes—
40#1443C2214yes—
41#1442C2214yes—
42#1442C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1442C2212yes—
47#1441C2214yes—
48#1441C2212yes—
49#1441C2214yes—
50#1441C2212yes—
51#1441C2212yes—
52#1440C2214yes—
53#1439C2212yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1437C2214yes—
62#1436C2212yes—
63#1436C2212yes—
64#1436C2214yes—
65#1434C2214yes—
66#1434C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2214yes—
69C23×Q8C2312yes—
70C23×D4C2311yescharacteristic
71C24×C4C2312yes—
72C2×C4⋊Q8—24no—
73C2×C4⋊Q8—42no—
74C2×C4⋊Q8—42no—
75C2×#125—28no—
76C2×#125—48no—
77C2×C4×D4—24no—
78C2×C4×D4—42no—
79C2×C4×D4—42no—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C42C2311yescharacteristic
101C22×C42C2311yescharacteristic
102C22×C42C2311yescharacteristic
103C22×C42C2312yes—
104#400—216no—
105#400—44no—
106#398—28no—
107#398—28no—
108#398—216no—
109#397—44no—
110#394—48no—
111#393—216no—
112#391—44no—
113#390—24no—
114#390—24no—
115#387—48no—
116#382—24no—
117#382D418yes—
118#380—216no—
119#380—44no—
120C4×C4⋊C4—24no—
121C2×#95C2312yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2312yes—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C25C2412yes—
148C22×Q8—24no—
149C22×Q8—24no—
150C22×Q8—42no—
151C22×Q8—44no—
152C22×Q8—44no—
153C22×D4—24no—
154C22×D4—22no—
155C22×D4—24no—
156C22×D4—41no—
157C22×D4—41no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—28no—
161C23×C4—42no—
162C23×C4—44no—
163C23×C4C2414yes—
164C23×C4C2414yes—
165C23×C4C2412yes—
166C23×C4C2412yes—
167C23×C4C2414yes—
168C23×C4C2414yes—
169C23×C4C2412yes—
170C23×C4C2412yes—
171C23×C4C2412yes—
172C23×C4C2411yescharacteristic
173C23×C4C2411yescharacteristic
174C23×C4C2411yescharacteristic
175C4⋊Q8—416no—
176#125—432no—
177C4×D4—416no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4C2×D414yes—
213C2×C4⋊C4C2×D414yes—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—24no—
239C2×C42—22no—
240C2×C42—24no—
241C2×C42—42no—
242C2×C42—44no—
243C2×C42C2×D414yes—
244#95—24no—
245#95—28no—
246#95—28no—
247#95—28no—
248#95—24no—
249#95—24no—
250#95—28no—
251#95—28no—
252#95—28no—
253#95—28no—
254#95—42no—
255#95—42no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—44no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—42no—
267#95—42no—
268#95—44no—
269#95C2×D4116yes—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—24no—
276C24—22no—
277C24—24no—
278C24—24no—
279C24—24no—
280C242−1+414yes—
281C24C2511yescharacteristic
282C2×Q8—48no—
283C2×Q8—48no—
284C2×Q8—88no—
285C2×Q8—84no—
286C2×Q8—88no—
287C2×D4—28no—
288C2×D4—48no—
289C2×D4—48no—
290C2×D4—44no—
291C2×D4—44no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—48no—
352C22×C4—416no—
353C22×C4—48no—
354C22×C4—84no—
355C22×C4—88no—
356C22×C4C22×D414yes—
357C22×C4C22×D414yes—
358C22×C4C22×D414yes—
359C22×C4C22×D412yes—
360C22×C4H(5,𝔽2)12yes—
361C22×C4H(5,𝔽2)14yes—
362C22×C4H(5,𝔽2)12yes—
363C22×C42−1+414yes—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—48no—
367C4⋊C4—48no—
368C4⋊C4—48no—
369C4⋊C4—416no—
370C4⋊C4—48no—
371C4⋊C4—44no—
372C4⋊C4—416no—
373C4⋊C4—416no—
374C4⋊C4—84no—
375C4⋊C4—84no—
376C4⋊C4—84no—
377C22⋊C4—48no—
378C22⋊C4—88no—
379C22⋊C4—88no—
380C22⋊C4—88no—
381C22⋊C4—88no—
382C22⋊C4—88no—
383C22⋊C4—88no—
384C42—28no—
385C42—28no—
386C42—44no—
387C42—84no—
388C23—24no—
389C23—24no—
390C23—24no—
391C23—28no—
392C23—28no—
393C23—42no—
394C23—42no—
395C23—44no—
396C23—44no—
397C23—42no—
398C23—44no—
399C23—42no—
400C23—42no—
401C23—44no—
402C23—44no—
403C23—42no—
404C23—44no—
405C23—44no—
406C23—42no—
407C23—42no—
408C23—44no—
409C23—44no—
410C23—44no—
411C23C23×D411yescharacteristic
412C23C2×H(5,𝔽2)12yes—
413C23C2×H(5,𝔽2)11yescharacteristic
414C23C2×H(5,𝔽2)11yescharacteristic
415C23C2×2−1+412yes—
416C23C2×2−1+412yes—
417C23#58512yes—
418C23#58514yes—
419Q8—816no—
420D4—416no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—28no—
435C2×C4—24no—
436C2×C4—28no—
437C2×C4—24no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—816no—
487C2×C4#53514yes—
488C2×C4#56314yes—
489C22—44no—
490C22—44no—
491C22—48no—
492C22—48no—
493C22—44no—
494C22—82no—
495C22—82no—
496C22—82no—
497C22—84no—
498C22—82no—
499C22—84no—
500C22C2×#53511yescharacteristic
501C22#348411yescharacteristic
502C22#348611yescharacteristic
503C22#348912yes—
504C22#350012yes—
505C22C2×#56311yescharacteristic
506C22#353312yes—
507C22#353612yes—
508C22#353612yes—
509C22#354014yes—
510C22#354511yescharacteristic
511C22#354714yes—
512C22#355112yes—
513C22#355314yes—
514C22#356112yes—
515C22#356314yes—
516C4—24no—
517C4—24no—
518C4—44no—
519C4—42no—
520C4—44no—
521C4—44no—
522C4—82no—
523C4—84no—
524C4—84no—
525C4—84no—
526C4—82no—
527C4—84no—
528C2—84no—
529C2#3670711yescharacteristic
530C2#4299512yes—
531C2#4613611yescharacteristic
532C2#5089812yes—
533C2#5197811yescharacteristic
534C2#5301712yes—
535C2#5397912yes—
536C2#5498014yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2822C2semidirect
3. #17153C2semidirect
4. #16688C2semidirect
6. #16378C2semidirect
7. #16304C2semidirect
9. #15783C2semidirect
10. #15490C2semidirect
11. #15071C2semidirect
13. #14476C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28222
C2×#25632
#171532
#166882
#166332
#163782
#163042
#158142
#157832
#154902
#150712
#148242
#144762