Finite Group Databasebeta

← #1495489

Subgroups — #1495489

Order 512.

2598 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2683C211yesmaximal, characteristic
2C2×#2566C211yesmaximal, characteristic
3#15944C212yesmaximal, complemented
4#15849C212yesmaximal, complemented
5#15614C214yesmaximal, complemented
6#15555C214yesmaximal, complemented
7#15197C212yesmaximal, complemented
8#15165C214yesmaximal, complemented
9#14513C212yesmaximal
10#14226C212yesmaximal, complemented
11#14052C214yesmaximal, complemented
12#14016C212yesmaximal
13#13824C211yesmaximal, characteristic
14C22×#125C2211yescharacteristic
15C22×C22⋊Q8C2214yes—
16C22×C4×D4C2211yescharacteristic
17#2683—28no—
18#2566—28no—
19C2×#398C2212yes—
20C2×#396C2214yes—
21C2×#396C2214yes—
22C2×#395C2212yes—
23C2×#395C2212yes—
24C2×#391C2212yes—
25C2×#390C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#389C2212yes—
28C2×#389C2214yes—
29C2×#386C2212yes—
30C2×#386C2212yes—
31C2×#385C2214yes—
32C2×#385C2212yes—
33C2×#384C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4C2214yes—
37C22×#95C2212yes—
38#1446C2214yes—
39#1445C2212yes—
40#1445C2212yes—
41#1444C2214yes—
42#1444C2212yes—
43#1442C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1441C2212yes—
47#1441C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1437C2214yes—
55#1436C2212yes—
56#1436C2212yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61C4⋊#95C2214yes—
62C4⋊#95C2214yes—
63C22⋊#95C2212yes—
64C22⋊#95C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2212yes—
67C22⋊#95C2214yes—
68C4×#95C2212yes—
69C23×Q8C2312yes—
70C23×D4C2311yescharacteristic
71C24×C4C2312yes—
72C24×C4C2312yes—
73C24×C4C2312yes—
74C2×#125—24no—
75C2×#125—42no—
76C2×#125—42no—
77C2×C22⋊Q8—216no—
78C2×C22⋊Q8—216no—
79C2×C22⋊Q8—48no—
80C2×C4×D4—24no—
81C2×C4×D4—42no—
82C2×C4×D4—42no—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C42C2311yescharacteristic
104C22×C42C2314yes—
105C22×C42C2311yescharacteristic
106C22×C42C2312yes—
107C22×C42C2311yescharacteristic
108#398—28no—
109#396—216no—
110#396—216no—
111#395—28no—
112#395—44no—
113#391—44no—
114#390—24no—
115#390D418yes—
116#389—28no—
117#389—48no—
118#386—28no—
119#386Q8116yescomplemented
120#385—216no—
121#385—44no—
122#384—24no—
123#382—24no—
124C4×C4⋊C4—24no—
125C4×C22⋊C4—48no—
126C2×#95—44no—
127C2×#95C2312yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C25C2412yes—
151C22×Q8—24no—
152C22×Q8—28no—
153C22×Q8—44no—
154C22×Q8—42no—
155C22×Q8—42no—
156C22×D4—41no—
157C22×D4—41no—
158C22×D4—42no—
159C22×D4C2×D414yes—
160C22×D4C2×Q818yes—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—28no—
164C23×C4—24no—
165C23×C4—42no—
166C23×C4—44no—
167C23×C4—44no—
168C23×C4—42no—
169C23×C4—44no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—44no—
173C23×C4—44no—
174C23×C4—42no—
175C23×C4C2×Q818yes—
176C23×C4C2412yes—
177C23×C4C2414yes—
178C23×C4C2412yes—
179C23×C4C2412yes—
180C23×C4C2412yes—
181C23×C4C2414yes—
182C23×C4C2414yes—
183C23×C4C2414yes—
184C23×C4C2411yescharacteristic
185C23×C4C2411yescharacteristic
186C23×C4C2411yescharacteristic
187C23×C4C2412yes—
188#125—416no—
189C22⋊Q8—464no—
190C4×D4—416no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4C2×D414yes—
228C2×C4⋊C4#4118yes—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4C2×D4116yes—
248C2×C22⋊C4C2×Q818yes—
249C2×C22⋊C4C2×Q8116yes—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—28no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—44no—
258C2×C42—42no—
259C2×C42—44no—
260C2×C42—44no—
261C2×C42C2×D414yes—
262#95—24no—
263#95—28no—
264#95—28no—
265#95—24no—
266#95—24no—
267#95—28no—
268#95—28no—
269#95—24no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—42no—
283#95—44no—
284#95C2×Q8116yes—
285C24—24no—
286C24—24no—
287C24—22no—
288C24—22no—
289C24—24no—
290C24—22no—
291C24—24no—
292C24—22no—
293C24C22×D414yes—
294C24C22×Q814yes—
295C24C22×Q814yes—
296C24C2511yescharacteristic
297C2×Q8—416no—
298C2×Q8—48no—
299C2×Q8—84no—
300C2×Q8—88no—
301C2×Q8—84no—
302C2×D4—44no—
303C2×D4—44no—
304C2×D4—48no—
305C2×D4—48no—
306C2×D4C4⋊Q8116yes—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—21no—
313C22×C4—24no—
314C22×C4—21no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
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—24no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
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—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—416no—
364C22×C4—48no—
365C22×C4—48no—
366C22×C4—48no—
367C22×C4—416no—
368C22×C4—48no—
369C22×C4—88no—
370C22×C4—84no—
371C22×C4—84no—
372C22×C4—88no—
373C22×C4—84no—
374C22×C4—84no—
375C22×C4—84no—
376C22×C4—88no—
377C22×C4—88no—
378C22×C4C22×D414yes—
379C22×C4C22×D412yes—
380C22×C4C22×D414yes—
381C22×C4C22×Q818yes—
382C22×C4C22×Q818yes—
383C22×C4C22×Q814yes—
384C22×C4C2×#4118yes—
385C22×C4C2×#4114yes—
386C22×C4H(5,𝔽2)12yes—
387C22×C4H(5,𝔽2)12yes—
388C22×C4H(5,𝔽2)14yes—
389C4⋊C4—216no—
390C4⋊C4—44no—
391C4⋊C4—416no—
392C4⋊C4—416no—
393C4⋊C4—48no—
394C4⋊C4—48no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—48no—
398C4⋊C4—416no—
399C4⋊C4—84no—
400C4⋊C4—88no—
401C4⋊C4—88no—
402C22⋊C4—232no—
403C22⋊C4—416no—
404C22⋊C4—48no—
405C22⋊C4—48no—
406C22⋊C4—416no—
407C22⋊C4—84no—
408C22⋊C4—88no—
409C42—28no—
410C42—44no—
411C42—44no—
412C42—88no—
413C42—84no—
414C23—28no—
415C23—28no—
416C23—28no—
417C23—24no—
418C23—24no—
419C23—28no—
420C23—24no—
421C23—28no—
422C23—24no—
423C23—28no—
424C23—24no—
425C23—24no—
426C23—42no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23C2×C4⋊Q818yes—
437C23C23×D411yescharacteristic
438C23C23×Q812yes—
439C23C22×#4112yes—
440C23C2×H(5,𝔽2)12yes—
441C23C2×H(5,𝔽2)11yescharacteristic
442C23C2×H(5,𝔽2)11yescharacteristic
443C23#58514yes—
444C23#58512yes—
445Q8—816no—
446D4—416no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—28no—
454C2×C4—28no—
455C2×C4—28no—
456C2×C4—28no—
457C2×C4—28no—
458C2×C4—28no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—28no—
463C2×C4—28no—
464C2×C4—28no—
465C2×C4—28no—
466C2×C4—28no—
467C2×C4—24no—
468C2×C4—28no—
469C2×C4—24no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—816no—
513C2×C4—816no—
514C2×C4—816no—
515C2×C4C2×C4⋊Q814yes—
516C2×C4#54618yes—
517C22—28no—
518C22—216no—
519C22—44no—
520C22—44no—
521C22—44no—
522C22—48no—
523C22—48no—
524C22—48no—
525C22—44no—
526C22—44no—
527C22—44no—
528C22C22×C4⋊Q811yescharacteristic
529C22C2×#54612yes—
530C22Q8×#4114yes—
531C22#348111yescharacteristic
532C22#348411yescharacteristic
533C22#348511yescharacteristic
534C22#350012yes—
535C22#350112yes—
536C22#350112yes—
537C22#350114yes—
538C22#351214yes—
539C22#351912yes—
540C22#353312yes—
541C22#354911yescharacteristic
542C22#355112yes—
543C22#356014yes—
544C4—24no—
545C4—28no—
546C4—42no—
547C4—48no—
548C4—42no—
549C4—48no—
550C4—44no—
551C4—48no—
552C4—82no—
553C4—82no—
554C4—84no—
555C4—82no—
556C2—48no—
557C2#3539111yescharacteristic
558C2#3627612yes—
559C2#4313914yes—
560C2#4444612yes—
561C2#4883812yes—
562C2#5197811yescharacteristic
563C2#5904111yescharacteristic
564C2#5993912yes—
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. #15944C2semidirect
4. #15849C2semidirect
5. #15614C2semidirect
6. #15555C2semidirect
7. #15197C2semidirect
8. #15165C2semidirect
10. #14226C2semidirect
11. #14052C2semidirect
119. #386Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26832
C2×#25662
#159442
#158492
#156142
#155552
#151972
#151652
#145132
#142262
#140522
#140162
#138242