Finite Group Databasebeta

← #4464094

Subgroups — #4464094

Order 512.

2394 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2632C212yesmaximal, complemented
2#17224C212yesmaximal, complemented
3#17216C212yesmaximal, complemented
4#17104C214yesmaximal, complemented
5#17101C214yesmaximal, complemented
6#17085C211yesmaximal, characteristic, complemented
7#17084C212yesmaximal, complemented
8#17082C211yesmaximal, characteristic
9#15163C212yesmaximal, complemented
10#15120C212yesmaximal, complemented
11#15023C214yesmaximal, complemented
12#14928C214yesmaximal, complemented
13#13850C211yesmaximal, characteristic, complemented
14C22×#124C2212yes—
15C22×C4×Q8C2211yescharacteristic
16#2632—216no—
17C2×#399C2214yes—
18C2×#398C2212yes—
19C2×#396C2214yes—
20C2×#395C2212yes—
21C2×#395C2212yes—
22C2×#395C2214yes—
23C2×#394C2212yes—
24C2×#394C2212yes—
25C2×#394C2212yes—
26C2×#394C2214yes—
27C2×#393C2214yes—
28C2×#393C2214yes—
29C2×#392C2214yes—
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#388C2214yes—
33C2×#388C2214yes—
34C2×#383C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×#376C2211yescharacteristic
37#1446C2211yescharacteristic
38#1446C2211yescharacteristic
39#1445C2211yescharacteristic
40#1445C2211yescharacteristic
41#1445C2212yes—
42#1444C2212yes—
43#1443C2214yes—
44#1442C2212yes—
45#1442C2214yes—
46#1442C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1440C2214yes—
50#1440C2214yes—
51#1440C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1438C2212yes—
59#1438C2212yes—
60#1438C2212yes—
61#1438C2214yes—
62#1438C2214yes—
63#1437C2214yes—
64#1437C2214yes—
65C4⋊#95C2212yes—
66#1434C2212yes—
67C4×#95C2214yes—
68C23×Q8C2312yes—
69C23×Q8C2311yescharacteristic
70C23×D4C2312yes—
71C23×D4C2312yes—
72C23×D4C2312yes—
73C2×#124—28no—
74C2×#124—44no—
75C2×#124—44no—
76C2×C4×Q8—42no—
77C2×C4×Q8—42no—
78C2×C4×Q8D418yescomplemented
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2312yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C42C2312yes—
100C22×C42C2312yes—
101C22×C42C2311yescharacteristic
102C22×C42C2312yes—
103#399—216no—
104#398—44no—
105#396—48no—
106#395—28no—
107#395—28no—
108#395—48no—
109#394—28no—
110#394—28no—
111#394—48no—
112#394—44no—
113#393—216no—
114#393—48no—
115#392—48no—
116#390—44no—
117#390—44no—
118#388—216no—
119#388—216no—
120#383—28no—
121C4×C4⋊C4—42no—
122#376—42no—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2312yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2311yescharacteristic
143C2×#95C2311yescharacteristic
144C2×#95C2312yes—
145C2×#95C2311yescharacteristic
146C2×#95C2311yescharacteristic
147C25C2414yes—
148C22×Q8—24no—
149C22×Q8—41no—
150C22×Q8—41no—
151C22×Q8—44no—
152C22×Q8—42no—
153C22×Q8—42no—
154C22×Q8—44no—
155C22×Q8—44no—
156C22×Q8C2×D414yes—
157C22×D4—24no—
158C22×D4—24no—
159C22×D4—24no—
160C22×D4—42no—
161C22×D4—42no—
162C22×D4—44no—
163C22×D4—44no—
164C22×D4—42no—
165C22×D4—42no—
166C22×D4—42no—
167C22×D4—44no—
168C22×D4—44no—
169C22×D4—44no—
170C22×D4—44no—
171C22×D4—42no—
172C23×C4C2414yes—
173C23×C4C2414yes—
174C23×C4C2414yes—
175C23×C4C2412yes—
176C23×C4C2411yescharacteristic
177C23×C4C2411yescharacteristic
178C23×C4C2412yes—
179C23×C4C2412yes—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2411yescharacteristic
184#124—432no—
185C4×Q8—416no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4C2×D414yes—
208C2×C4⋊C4C2×D418yes—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
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—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×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—42no—
253C2×C42—44no—
254C2×C42C2×D418yes—
255C2×C42C2×D414yes—
256#95—28no—
257#95—28no—
258#95—24no—
259#95—24no—
260#95—28no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—28no—
265#95—28no—
266#95—24no—
267#95—44no—
268#95—44no—
269#95—42no—
270#95—44no—
271#95—42no—
272#95—41no—
273#95—41no—
274#95—41no—
275#95—41no—
276#95—44no—
277#95—42no—
278#95—44no—
279#95—42no—
280C24—24no—
281C24—24no—
282C24—24no—
283C24—24no—
284C24—24no—
285C24—24no—
286C24—24no—
287C24—24no—
288C24—24no—
289C24—24no—
290C24—24no—
291C24—24no—
292C24—24no—
293C24—24no—
294C24—24no—
295C24C2511yescharacteristic
296C2×Q8—44no—
297C2×Q8—48no—
298C2×Q8—48no—
299C2×Q8—48no—
300C2×Q8—48no—
301C2×Q8—88no—
302C2×Q8—88no—
303C2×Q8—84no—
304C2×Q8—84no—
305C2×D4—48no—
306C2×D4—48no—
307C2×D4—48no—
308C2×D4—88no—
309C2×D4—84no—
310C2×D4—84no—
311C2×D4—88no—
312C2×D4—88no—
313C2×D4—88no—
314C2×D4—84no—
315C2×D4—88no—
316C2×D4—84no—
317C2×D4—84no—
318C2×D4—84no—
319C2×D4—88no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—22no—
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—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—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4C22×D414yes—
376C22×C4C22×D412yes—
377C22×C4C22×D414yes—
378C22×C4C22×D412yes—
379C22×C4C22×D412yes—
380C22×C4H(5,𝔽2)14yes—
381C22×C4H(5,𝔽2)14yes—
382C22×C4H(5,𝔽2)12yes—
383C22×C4H(5,𝔽2)12yes—
384C22×C42−1+414yes—
385C4⋊C4—48no—
386C4⋊C4—44no—
387C4⋊C4—48no—
388C4⋊C4—88no—
389C4⋊C4—88no—
390C4⋊C4—84no—
391C4⋊C4—88no—
392C4⋊C4—84no—
393C22⋊C4—416no—
394C22⋊C4—416no—
395C22⋊C4—416no—
396C22⋊C4—416no—
397C22⋊C4—88no—
398C22⋊C4—88no—
399C22⋊C4—88no—
400C22⋊C4—88no—
401C22⋊C4—88no—
402C22⋊C4—88no—
403C22⋊C4—88no—
404C22⋊C4—88no—
405C42—28no—
406C42—216no—
407C42—48no—
408C42—84no—
409C23—44no—
410C23—44no—
411C23—44no—
412C23—44no—
413C23—44no—
414C23—44no—
415C23—44no—
416C23—44no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23C23×D411yescharacteristic
445C23C2×H(5,𝔽2)11yescharacteristic
446C23C2×H(5,𝔽2)11yescharacteristic
447C23C2×H(5,𝔽2)12yes—
448C23C2×H(5,𝔽2)12yes—
449C23C2×2−1+412yes—
450C23#58514yes—
451C23#58512yes—
452Q8—416no—
453Q8—816no—
454D4—816no—
455D4—816no—
456D4—816no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—28no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—28no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
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—44no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4#53414yes—
522C22—84no—
523C22—84no—
524C22—84no—
525C22—84no—
526C22—84no—
527C22—84no—
528C22—84no—
529C22—84no—
530C22—84no—
531C22—84no—
532C22—84no—
533C22—84no—
534C22—84no—
535C22—84no—
536C22—84no—
537C22C2×#53411yescharacteristic
538C22#344912yes—
539C22#349712yes—
540C22#349812yes—
541C22#352912yes—
542C22#352912yes—
543C22#353512yes—
544C22#353712yes—
545C22#354411yescharacteristic
546C22#354511yescharacteristic
547C22#354514yes—
548C22#354714yes—
549C22#354911yescharacteristic
550C22#354911yescharacteristic
551C22#356014yes—
552C22#356114yes—
553C4—24no—
554C4—44no—
555C4—42no—
556C4—42no—
557C4—44no—
558C4—44no—
559C4—44no—
560C4—44no—
561C4—84no—
562C4—84no—
563C4—84no—
564C4—82no—
565C2—164no—
566C2#3665211yescharacteristic
567C2#3719011yescharacteristic
568C2#3719214yes—
569C2#3729711yescharacteristic
570C2#5291612yes—
571C2#5293012yes—
572C2#5294312yes—
573C2#5295312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#17082C2
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×#2632C2semidirect
2. #17224C2semidirect
3. #17216C2semidirect
4. #17104C2semidirect
5. #17101C2semidirect
6. #17085C2semidirect
7. #17084C2semidirect
9. #15163C2semidirect
10. #15120C2semidirect
11. #15023C2semidirect
12. #14928C2semidirect
13. #13850C2semidirect
78. C2×C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26322
#172242
#172162
#171042
#171012
#170852
#170842
#170822
#151632
#151202
#150232
#149282
#138502