Finite Group Databasebeta

← #4467971

Subgroups — #4467971

Order 512.

2386 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2683C211yesmaximal, characteristic
2C2×#2640C211yesmaximal, characteristic
3#17122C214yesmaximal, complemented
4#16989C212yesmaximal, complemented
5#16859C214yesmaximal, complemented
6#16262C214yesmaximal, complemented
7#16032C212yesmaximal
8#15854C212yesmaximal, complemented
9#15762C212yesmaximal, complemented
10#15342C212yesmaximal
11#15265C214yesmaximal, complemented
12#15121C212yesmaximal, complemented
13#14503C211yesmaximal, characteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×C4⋊Q8C2212yes—
16C22×#124C2211yescharacteristic
17C22×C22⋊Q8C2214yes—
18#2683—28no—
19#2640—28no—
20C2×#397C2214yes—
21C2×#397C2214yes—
22C2×#396C2214yes—
23C2×#395C2212yes—
24C2×#395C2212yes—
25C2×#395C2214yes—
26C2×#393C2214yes—
27C2×#391C2211yescharacteristic
28C2×#391C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#390C2211yescharacteristic
31C2×#389C2212yes—
32C2×#386C2212yes—
33C2×#385C2214yes—
34C2×#385C2212yes—
35C2×#381C2212yes—
36C2×#376C2211yescharacteristic
37#1446C2214yes—
38#1446C2214yes—
39#1446C2212yes—
40#1445C2212yes—
41#1445C2212yes—
42#1444C2212yes—
43#1444C2214yes—
44#1444C2214yes—
45#1444C2214yes—
46#1443C2214yes—
47#1442C2212yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1440C2214yes—
55#1440C2214yes—
56#1440C2212yes—
57#1440C2212yes—
58#1440C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1438C2212yes—
62#1437C2214yes—
63#1436C2214yes—
64#1436C2212yes—
65#1436C2212yes—
66#1436C2214yes—
67#1434C2212yes—
68#1434C2214yes—
69C23×Q8C2312yes—
70C23×Q8C2311yescharacteristic
71C23×Q8C2312yes—
72C23×D4C2311yescharacteristic
73C24×C4C2312yes—
74C2×C4⋊Q8—24no—
75C2×C4⋊Q8—42no—
76C2×C4⋊Q8—42no—
77C2×C4⋊Q8—44no—
78C2×C4⋊Q8—48no—
79C2×#124—24no—
80C2×#124—42no—
81C2×#124—42no—
82C2×C22⋊Q8—216no—
83C2×C22⋊Q8—48no—
84C2×C22⋊Q8—48no—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2312yes—
107C22×C42C2311yescharacteristic
108#397—216no—
109#397—48no—
110#396—216no—
111#395—28no—
112#395—44no—
113#395—48no—
114#393—216no—
115#391—24no—
116#391—24no—
117#390—24no—
118#390D418yes—
119#389—28no—
120#386—28no—
121#385—48no—
122#385—44no—
123#381—44no—
124#376—24no—
125C2×#95C2312yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C25C2412yes—
151C22×Q8—22no—
152C22×Q8—24no—
153C22×Q8—24no—
154C22×Q8—28no—
155C22×Q8—24no—
156C22×Q8—24no—
157C22×Q8—44no—
158C22×Q8—41no—
159C22×Q8—41no—
160C22×Q8—42no—
161C22×Q8—42no—
162C22×Q8—44no—
163C22×Q8—42no—
164C22×Q8—42no—
165C22×Q8—44no—
166C22×D4—24no—
167C22×D4—41no—
168C22×D4—41no—
169C22×D4—42no—
170C22×D4C2×D414yes—
171C23×C4—24no—
172C23×C4—28no—
173C23×C4—44no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4C2412yes—
177C23×C4C2414yes—
178C23×C4C2414yes—
179C23×C4C2412yes—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2411yescharacteristic
184C23×C4C2411yescharacteristic
185C23×C4C2414yes—
186C23×C4C2414yes—
187C23×C4C2411yescharacteristic
188C4⋊Q8—416no—
189C4⋊Q8—432no—
190#124—416no—
191C22⋊Q8—464no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4C2×D414yes—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4C2×D4116yes—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—24no—
250C2×C42—44no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42C2×D414yes—
254#95—24no—
255#95—24no—
256#95—28no—
257#95—28no—
258#95—28no—
259#95—24no—
260#95—28no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—28no—
265#95—28no—
266#95—42no—
267#95—44no—
268#95—42no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—42no—
276#95—44no—
277#95—44no—
278#95—44no—
279C24—24no—
280C24—22no—
281C24—22no—
282C24—24no—
283C24—24no—
284C24—24no—
285C24—24no—
286C24—24no—
287C24C22×D414yes—
288C24C2511yescharacteristic
289C2×Q8—416no—
290C2×Q8—48no—
291C2×Q8—48no—
292C2×Q8—44no—
293C2×Q8—48no—
294C2×Q8—44no—
295C2×Q8—44no—
296C2×Q8—84no—
297C2×Q8—84no—
298C2×Q8—88no—
299C2×Q8—84no—
300C2×Q8—84no—
301C2×Q8—88no—
302C2×Q8—84no—
303C2×Q8—88no—
304C2×D4—28no—
305C2×D4—44no—
306C2×D4—44no—
307C2×D4—84no—
308C2×D4—84no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—24no—
327C22×C4—22no—
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—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—48no—
368C22×C4—416no—
369C22×C4—88no—
370C22×C4—84no—
371C22×C4—84no—
372C22×C4C22×D414yes—
373C22×C4C22×D414yes—
374C22×C4C22×D412yes—
375C22×C4H(5,𝔽2)12yes—
376C22×C4H(5,𝔽2)18yes—
377C22×C4H(5,𝔽2)12yes—
378C22×C42−1+418yes—
379C22×C42−1+414yes—
380C4⋊C4—216no—
381C4⋊C4—44no—
382C4⋊C4—416no—
383C4⋊C4—48no—
384C4⋊C4—416no—
385C4⋊C4—416no—
386C4⋊C4—44no—
387C4⋊C4—416no—
388C4⋊C4—84no—
389C4⋊C4—88no—
390C22⋊C4—416no—
391C22⋊C4—48no—
392C22⋊C4—416no—
393C22⋊C4—48no—
394C22⋊C4—48no—
395C22⋊C4—416no—
396C22⋊C4—88no—
397C22⋊C4—84no—
398C22⋊C4—84no—
399C42—28no—
400C42—44no—
401C42—48no—
402C42—44no—
403C23—24no—
404C23—28no—
405C23—24no—
406C23—24no—
407C23—28no—
408C23—44no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—44no—
414C23—44no—
415C23—44no—
416C23—44no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23C23×D411yescharacteristic
425C23C2×H(5,𝔽2)11yescharacteristic
426C23C2×H(5,𝔽2)14yes—
427C23C2×H(5,𝔽2)11yescharacteristic
428C23C2×2−1+412yes—
429C23C2×2−1+414yes—
430C23#58512yes—
431Q8—88no—
432Q8—816no—
433Q8—816no—
434D4—88no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—28no—
443C2×C4—28no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—28no—
448C2×C4—24no—
449C2×C4—28no—
450C2×C4—28no—
451C2×C4—28no—
452C2×C4—24no—
453C2×C4—28no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—44no—
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—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—816no—
501C2×C4#53614yes—
502C22—44no—
503C22—44no—
504C22—44no—
505C22—48no—
506C22—48no—
507C22—84no—
508C22—84no—
509C22—84no—
510C22—84no—
511C22C2×#53611yescharacteristic
512C22#348411yescharacteristic
513C22#348412yes—
514C22#348511yescharacteristic
515C22#349912yes—
516C22#353212yes—
517C22#353712yes—
518C22#353812yes—
519C22#354114yes—
520C22#354412yes—
521C22#354714yes—
522C22#355014yes—
523C22#355214yes—
524C22#356112yes—
525C22#356112yes—
526C4—24no—
527C4—48no—
528C4—42no—
529C4—44no—
530C4—42no—
531C4—44no—
532C4—48no—
533C4—82no—
534C4—84no—
535C4—82no—
536C4—84no—
537C4—82no—
538C2—84no—
539C2#3696611yescharacteristic
540C2#4242712yes—
541C2#4721412yes—
542C2#5154211yescharacteristic
543C2#5200411yescharacteristic
544C2#5631414yes—
545C2#5895714yes—
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. #17122C2semidirect
4. #16989C2semidirect
5. #16859C2semidirect
6. #16262C2semidirect
8. #15854C2semidirect
9. #15762C2semidirect
11. #15265C2semidirect
12. #15121C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26832
C2×#26402
#171222
#169892
#168592
#162622
#160322
#158542
#157622
#153422
#152652
#151212
#145032