Finite Group Databasebeta

← #4468244

Subgroups — #4468244

Order 512.

2138 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2724C211yesmaximal, characteristic
2C2×#2640C211yesmaximal, characteristic
3#17201C214yesmaximal, complemented
4#16993C212yesmaximal, complemented
5#16858C214yesmaximal, complemented
6#16307C214yesmaximal, complemented
7#15983C212yesmaximal
8#15856C212yesmaximal, complemented
9#15783C212yesmaximal, complemented
10#15378C214yesmaximal, complemented
11#15262C212yesmaximal
12#15122C212yesmaximal, complemented
13#14503C211yesmaximal, characteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×#124C2211yescharacteristic
16C22×#123C2212yes—
17#2724—28no—
18#2640—28no—
19C2×#400C2212yes—
20C2×#398C2212yes—
21C2×#397C2214yes—
22C2×#397C2214yes—
23C2×#395C2214yes—
24C2×#393C2214yes—
25C2×#391C2211yescharacteristic
26C2×#391C2211yescharacteristic
27C2×#391C2212yes—
28C2×#390C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#389C2212yes—
31C2×#387C2214yes—
32C2×#385C2214yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#376C2211yescharacteristic
36#1446C2214yes—
37#1446C2212yes—
38#1445C2212yes—
39#1445C2212yes—
40#1445C2214yes—
41#1445C2214yes—
42#1444C2212yes—
43#1444C2214yes—
44#1444C2214yes—
45#1443C2214yes—
46#1442C2214yes—
47#1442C2212yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2214yes—
53#1440C2212yes—
54#1440C2214yes—
55#1440C2214yes—
56#1440C2212yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1438C2212yes—
62#1437C2214yes—
63#1436C2214yes—
64#1436C2212yes—
65#1436C2212yes—
66C4⋊#95C2214yes—
67#1434C2212yes—
68C22⋊#95C2214yes—
69C23×Q8C2311yescharacteristic
70C23×Q8C2312yes—
71C23×D4C2311yescharacteristic
72C24×C4C2312yes—
73C2×C4⋊Q8—24no—
74C2×C4⋊Q8—42no—
75C2×C4⋊Q8—42no—
76C2×#124—24no—
77C2×#124—42no—
78C2×#124—42no—
79C2×#123—28no—
80C2×#123—48no—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C22⋊C4C2312yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C42C2311yescharacteristic
101C22×C42C2311yescharacteristic
102C22×C42C2312yes—
103C22×C42C2311yescharacteristic
104#400—44no—
105#398—28no—
106#397—216no—
107#397—216no—
108#395—48no—
109#393—216no—
110#391—24no—
111#391—24no—
112#391—44no—
113#390—24no—
114#390D418yes—
115#389—28no—
116#387—48no—
117#385—216no—
118#385—44no—
119#385—44no—
120#376—24no—
121C2×#95C2312yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2312yes—
126C2×#95C2312yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C25C2412yes—
148C22×Q8—22no—
149C22×Q8—24no—
150C22×Q8—24no—
151C22×Q8—24no—
152C22×Q8—41no—
153C22×Q8—41no—
154C22×Q8—42no—
155C22×Q8—42no—
156C22×Q8—44no—
157C22×Q8—44no—
158C22×D4—24no—
159C22×D4—41no—
160C22×D4—41no—
161C22×D4—42no—
162C22×D4C2×D414yes—
163C23×C4—24no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4C2412yes—
169C23×C4C2414yes—
170C23×C4C2412yes—
171C23×C4C2412yes—
172C23×C4C2414yes—
173C23×C4C2414yes—
174C23×C4C2412yes—
175C23×C4C2412yes—
176C23×C4C2411yescharacteristic
177C23×C4C2411yescharacteristic
178C23×C4C2414yes—
179C23×C4C2411yescharacteristic
180C4⋊Q8—416no—
181#124—416no—
182#123—432no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4C2×D414yes—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
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—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4C2×D4116yes—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—24no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—44no—
244C2×C42C2×D414yes—
245#95—24no—
246#95—24no—
247#95—28no—
248#95—28no—
249#95—28no—
250#95—28no—
251#95—24no—
252#95—28no—
253#95—28no—
254#95—28no—
255#95—28no—
256#95—28no—
257#95—44no—
258#95—42no—
259#95—44no—
260#95—42no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—42no—
270#95—44no—
271C24—22no—
272C24—22no—
273C24—24no—
274C24—24no—
275C24—24no—
276C24—24no—
277C24—24no—
278C24—24no—
279C24C22×D414yes—
280C24C2511yescharacteristic
281C2×Q8—48no—
282C2×Q8—44no—
283C2×Q8—44no—
284C2×Q8—48no—
285C2×Q8—44no—
286C2×Q8—84no—
287C2×Q8—84no—
288C2×Q8—88no—
289C2×Q8—84no—
290C2×Q8—88no—
291C2×D4—28no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—84no—
295C2×D4—84no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
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—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—48no—
356C22×C4—84no—
357C22×C4—84no—
358C22×C4—88no—
359C22×C4—88no—
360C22×C4C22×D414yes—
361C22×C4C22×D414yes—
362C22×C4C22×D412yes—
363C22×C4H(5,𝔽2)12yes—
364C22×C4H(5,𝔽2)12yes—
365C22×C4H(5,𝔽2)14yes—
366C22×C42−1+418yes—
367C4⋊C4—44no—
368C4⋊C4—48no—
369C4⋊C4—48no—
370C4⋊C4—416no—
371C4⋊C4—44no—
372C4⋊C4—416no—
373C4⋊C4—84no—
374C4⋊C4—84no—
375C4⋊C4—88no—
376C4⋊C4—88no—
377C22⋊C4—416no—
378C22⋊C4—48no—
379C22⋊C4—48no—
380C22⋊C4—48no—
381C22⋊C4—48no—
382C22⋊C4—416no—
383C22⋊C4—48no—
384C22⋊C4—88no—
385C22⋊C4—88no—
386C42—28no—
387C42—44no—
388C42—44no—
389C42—84no—
390C23—24no—
391C23—24no—
392C23—28no—
393C23—24no—
394C23—28no—
395C23—44no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—44no—
401C23—44no—
402C23—44no—
403C23—44no—
404C23—44no—
405C23—44no—
406C23—44no—
407C23—44no—
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+414yes—
416C23#58514yes—
417C23#58512yes—
418Q8—88no—
419Q8—816no—
420D4—88no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—28no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—28no—
434C2×C4—28no—
435C2×C4—24no—
436C2×C4—28no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
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—
488C22—44no—
489C22—44no—
490C22—44no—
491C22—48no—
492C22—48no—
493C22—84no—
494C22—84no—
495C22—84no—
496C22—84no—
497C22C2×#53511yescharacteristic
498C22#348411yescharacteristic
499C22#348512yes—
500C22#348511yescharacteristic
501C22#349912yes—
502C22#353312yes—
503C22#353812yes—
504C22#354012yes—
505C22#354114yes—
506C22#354412yes—
507C22#355414yes—
508C22#355714yes—
509C22#355812yes—
510C22#355812yes—
511C22#356414yes—
512C4—24no—
513C4—42no—
514C4—44no—
515C4—42no—
516C4—48no—
517C4—44no—
518C4—82no—
519C4—84no—
520C4—82no—
521C4—82no—
522C4—84no—
523C4—84no—
524C2—84no—
525C2#3696711yescharacteristic
526C2#4264812yes—
527C2#4681812yes—
528C2#5154211yescharacteristic
529C2#5205411yescharacteristic
530C2#5645814yes—
531C2#5675914yes—
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
3. #17201C2semidirect
4. #16993C2semidirect
5. #16858C2semidirect
6. #16307C2semidirect
8. #15856C2semidirect
9. #15783C2semidirect
10. #15378C2semidirect
12. #15122C2semidirect

Maximal subgroups

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

subgroupindex
C2×#27242
C2×#26402
#172012
#169932
#168582
#163072
#159832
#158562
#157832
#153782
#152622
#151222
#145032