Finite Group Databasebeta

← #4100359

Subgroups — #4100359

Order 512.

2646 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2687C214yesmaximal, complemented
2#17137C211yesmaximal, characteristic, complemented
3#17120C212yesmaximal, complemented
4#15966C211yesmaximal, characteristic, complemented
5#15921C214yesmaximal, complemented
6#15511C212yesmaximal, complemented
7#15306C214yesmaximal, complemented
8#15260C212yesmaximal
9#15004C214yesmaximal, complemented
10#14934C212yesmaximal, complemented
11#14410C212yesmaximal, complemented
12#14229C212yesmaximal, complemented
13#14152C211yesmaximal, characteristic
14C22×#123C2212yes—
15C22×C22⋊Q8C2212yes—
16C22×#121C2211yescharacteristic
17#2687—232no—
18C2×#400C2214yes—
19C2×#398C2214yes—
20C2×#398C2214yes—
21C2×#397C2212yes—
22C2×#394C2214yes—
23C2×#393C2214yes—
24C2×#391C2214yes—
25C2×#389C2212yes—
26C2×#388C2214yes—
27C2×#388C2214yes—
28C2×#386C2214yes—
29C2×#386C2212yes—
30C2×#385C2214yes—
31C2×#385C2212yes—
32C2×#381C2214yes—
33C2×#380C2214yes—
34C2×C4×C22⋊C4C2211yescharacteristic
35C22×#95C2211yescharacteristic
36#1446C2212yes—
37#1446C2212yes—
38#1445C2211yescharacteristic
39#1445C2211yescharacteristic
40#1445C2212yes—
41#1444C2214yes—
42#1443C2214yes—
43#1442C2214yes—
44#1442C2212yes—
45#1442C2214yes—
46#1442C2212yes—
47#1442C2212yes—
48#1441C2214yes—
49#1441C2214yes—
50#1440C2214yes—
51#1440C2212yes—
52#1439C2212yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1438C2212yes—
57#1438C2211yescharacteristic
58#1438C2211yescharacteristic
59#1438C2212yes—
60#1437C2214yes—
61#1437C2214yes—
62#1436C2214yes—
63#1434C2214yes—
64#1434C2214yes—
65C22⋊#95C2212yes—
66C22⋊#95C2214yes—
67C4×#95C2212yes—
68C23×Q8C2312yes—
69C23×D4C2312yes—
70C23×D4C2311yescharacteristic
71C24×C4C2312yes—
72C24×C4C2311yescharacteristic
73C24×C4C2312yes—
74C2×#123—216no—
75C2×#123—44no—
76C2×C22⋊Q8—216no—
77C2×C22⋊Q8—44no—
78C2×#121—24no—
79C2×#121—42no—
80C2×#121—42no—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C22⋊C4C2312yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C42C2312yes—
101C22×C42C2314yes—
102#400—216no—
103#398—216no—
104#398—216no—
105#397D4116yescomplemented
106#394—216no—
107#393—216no—
108#391—216no—
109#389—44no—
110#388—216no—
111#388—216no—
112#386—28no—
113#386—216no—
114#385—216no—
115#385—44no—
116#381—216no—
117#380—216no—
118C4×C22⋊C4—42no—
119C2×#95—42no—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2311yescharacteristic
140C2×#95C2312yes—
141C2×#95C2311yescharacteristic
142C2×#95C2311yescharacteristic
143C2×#95C2311yescharacteristic
144C2×#95C2312yes—
145C2×#95C2314yes—
146C25C2412yes—
147C25C2411yescharacteristic
148C22×Q8—28no—
149C22×Q8—24no—
150C22×Q8—42no—
151C22×Q8—44no—
152C22×Q8—42no—
153C22×D4—24no—
154C22×D4—28no—
155C22×D4—24no—
156C22×D4—41no—
157C22×D4—41no—
158C22×D4—42no—
159C22×D4—44no—
160C22×D4—44no—
161C22×D4#4114yes—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—28no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—28no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—44no—
173C23×C4—44no—
174C23×C4—44no—
175C23×C4—42no—
176C23×C4C2414yes—
177C23×C4C2414yes—
178C23×C4C2412yes—
179C23×C4C2412yes—
180C23×C4C2414yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2414yes—
184C23×C4C2411yescharacteristic
185C23×C4C2411yescharacteristic
186C23×C4C2412yes—
187#123—432no—
188C22⋊Q8—432no—
189#121—416no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4C2×D418yes—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4C2×D4116yes—
241C2×C22⋊C4C2×D418yes—
242C2×C42—28no—
243C2×C42—24no—
244C2×C42—44no—
245C2×C42—44no—
246C2×C42—44no—
247#95—28no—
248#95—28no—
249#95—28no—
250#95—28no—
251#95—28no—
252#95—28no—
253#95—28no—
254#95—28no—
255#95—28no—
256#95—28no—
257#95—28no—
258#95—24no—
259#95—42no—
260#95—41no—
261#95—41no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—42no—
266#95—42no—
267#95—44no—
268#95—44no—
269#95—41no—
270#95—41no—
271#95C2×D4116yes—
272#95C2×D418yes—
273C24—21no—
274C24—21no—
275C24—22no—
276C24—22no—
277C24—24no—
278C24—22no—
279C24—24no—
280C24—22no—
281C24—22no—
282C24—24no—
283C24—24no—
284C24—24no—
285C24—24no—
286C24—24no—
287C24C22×D414yes—
288C24C2×#4114yes—
289C24H(5,𝔽2)12yes—
290C24C2511yescharacteristic
291C2×Q8—48no—
292C2×Q8—416no—
293C2×Q8—88no—
294C2×Q8—84no—
295C2×Q8—84no—
296C2×D4—44no—
297C2×D4—48no—
298C2×D4—48no—
299C2×D4—416no—
300C2×D4—48no—
301C2×D4—88no—
302C2×D4—84no—
303C2×D4—84no—
304C2×D4—84no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—21no—
310C22×C4—24no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—28no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—48no—
361C22×C4—416no—
362C22×C4—416no—
363C22×C4—48no—
364C22×C4—48no—
365C22×C4—84no—
366C22×C4—88no—
367C22×C4—84no—
368C22×C4—84no—
369C22×C4—88no—
370C22×C4—84no—
371C22×C4—84no—
372C22×C4—84no—
373C22×C4C22×D414yes—
374C22×C4C22×D418yes—
375C22×C4C22×D414yes—
376C22×C4C22×D418yes—
377C22×C4C2×#4112yes—
378C22×C4H(5,𝔽2)12yes—
379C22×C42−1+414yes—
380C4⋊C4—48no—
381C4⋊C4—416no—
382C4⋊C4—48no—
383C4⋊C4—416no—
384C4⋊C4—44no—
385C4⋊C4—88no—
386C4⋊C4—88no—
387C4⋊C4—84no—
388C22⋊C4—232no—
389C22⋊C4—48no—
390C22⋊C4—416no—
391C22⋊C4—48no—
392C22⋊C4—416no—
393C22⋊C4—48no—
394C22⋊C4—48no—
395C22⋊C4—88no—
396C22⋊C4—88no—
397C22⋊C4—84no—
398C22⋊C4—88no—
399C42—88no—
400C42—84no—
401C23—24no—
402C23—28no—
403C23—22no—
404C23—24no—
405C23—28no—
406C23—24no—
407C23—24no—
408C23—22no—
409C23—28no—
410C23—24no—
411C23—24no—
412C23—24no—
413C23—28no—
414C23—42no—
415C23—42no—
416C23—41no—
417C23—42no—
418C23—41no—
419C23—44no—
420C23—44no—
421C23—42no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—42no—
428C23—44no—
429C23—42no—
430C23—44no—
431C23—44no—
432C23—42no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23D4214yes—
438C23C23×D412yes—
439C23C22×#4111yescharacteristic
440C23C2×H(5,𝔽2)11yescharacteristic
441C23C2×H(5,𝔽2)11yescharacteristic
442C23C2×2−1+412yes—
443C23#58512yes—
444C23#58514yes—
445C23#58512yes—
446Q8—816no—
447D4—88no—
448D4—816no—
449C2×C4—28no—
450C2×C4—28no—
451C2×C4—28no—
452C2×C4—28no—
453C2×C4—28no—
454C2×C4—28no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—28no—
461C2×C4—28no—
462C2×C4—28no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—88no—
511C2×C4—816no—
512C2×C4—816no—
513C22—24no—
514C22—28no—
515C22—48no—
516C22—44no—
517C22—44no—
518C22—44no—
519C22—42no—
520C22—42no—
521C22—48no—
522C22—48no—
523C22—44no—
524C22—44no—
525C22—44no—
526C22—84no—
527C22—84no—
528C22—84no—
529C22—84no—
530C22C2×D4211yescharacteristic
531C22D4×#4112yes—
532C22#348914yes—
533C22#349011yescharacteristic
534C22#349112yes—
535C22#349612yes—
536C22#350014yes—
537C22#351512yes—
538C22#352911yescharacteristic
539C22#353512yes—
540C22#354511yescharacteristic
541C22#354512yes—
542C22#354512yes—
543C22#355011yescharacteristic
544C22#356514yes—
545C22#356714yes—
546C4—44no—
547C4—44no—
548C4—48no—
549C4—44no—
550C4—48no—
551C4—42no—
552C4—42no—
553C4—84no—
554C4—84no—
555C4—82no—
556C4—82no—
557C2—44no—
558C2—84no—
559C2#4457914yes—
560C2#4476012yes—
561C2#4623612yes—
562C2#4629311yescharacteristic
563C2#4683711yescharacteristic
564C2#4745912yes—
565C2#5561112yes—
566C2#5981411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2687C2semidirect
2. #17137C2semidirect
3. #17120C2semidirect
4. #15966C2semidirect
5. #15921C2semidirect
6. #15511C2semidirect
7. #15306C2semidirect
9. #15004C2semidirect
10. #14934C2semidirect
11. #14410C2semidirect
12. #14229C2semidirect
105. #397D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26872
#171372
#171202
#159662
#159212
#155112
#153062
#152602
#150042
#149342
#144102
#142292
#141522