Finite Group Databasebeta

← #4310588

Subgroups — #4310588

Order 512.

2502 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2627C212yesmaximal
2#17076C211yesmaximal, characteristic, complemented
3#17073C212yesmaximal, complemented
4#17072C211yesmaximal, characteristic, complemented
5#17036C212yesmaximal, complemented
6#17032C212yesmaximal, complemented
7#16907C214yesmaximal, complemented
8#16827C214yesmaximal, complemented
9#14766C214yesmaximal
10#14758C214yesmaximal
11#14685C212yesmaximal
12#14339C212yesmaximal
13C2×C4⋊#95C211yesmaximal, characteristic
14C22×C22⋊Q8C2212yes—
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C42C2211yescharacteristic
18#2627—216no—
19C2×#400C2212yes—
20C2×#400C2212yes—
21C2×#398C2214yes—
22C2×#398C2214yes—
23C2×#398C2214yes—
24C2×#395C2212yes—
25C2×#395C2212yes—
26C2×#395C2214yes—
27C2×#385C2214yes—
28C2×#381C2214yes—
29C2×#380C2212yes—
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#380C2212yes—
34C22×#95C2214yes—
35#1446C2214yes—
36#1445C2214yes—
37#1444C2214yes—
38#1444C2214yes—
39#1444C2214yes—
40#1444C2214yes—
41#1442C2214yes—
42#1442C2214yes—
43#1441C2212yes—
44#1441C2214yes—
45#1441C2214yes—
46#1441C2212yes—
47#1441C2212yes—
48#1441C2212yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55C4⋊#95C2212yes—
56C4⋊#95C2214yes—
57C4⋊#95C2212yes—
58C4⋊#95C2212yes—
59C4⋊#95C2212yes—
60C4⋊#95C2211yescharacteristic
61C4⋊#95C2211yescharacteristic
62C4⋊#95C2211yescharacteristic
63C4⋊#95C2211yescharacteristic
64#1434C2214yes—
65#1434C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C23×Q8C2312yes—
69C24×C4C2312yes—
70C24×C4C2311yescharacteristic
71C24×C4C2312yes—
72C24×C4C2312yes—
73C2×C22⋊Q8—28no—
74C2×C22⋊Q8—44no—
75C2×C22⋊Q8—44no—
76C22×C4⋊C4—42no—
77C22×C4⋊C4—42no—
78C22×C4⋊C4—42no—
79C22×C4⋊C4—42no—
80C22×C4⋊C4D418yes—
81C22×C4⋊C4Q818yescomplemented
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C42—42no—
103C22×C42—42no—
104C22×C42Q818yescomplemented
105C22×C42C2311yescharacteristic
106C22×C42C2312yes—
107C22×C42C2311yescharacteristic
108#400—28no—
109#400—28no—
110#398—216no—
111#398—216no—
112#398—48no—
113#395—28no—
114#395—28no—
115#395—48no—
116#385—216no—
117#381—216no—
118#380—28no—
119#380—44no—
120#380—44no—
121#380—44no—
122#380—44no—
123C2×#95—48no—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2312yes—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C25C2411yescharacteristic
150C22×Q8—24no—
151C22×Q8—42no—
152C22×Q8—42no—
153C22×Q8—44no—
154C22×Q8—44no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—44no—
170C23×C4—44no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4C2×D414yes—
174C23×C4C2×D418yes—
175C23×C4C2×Q818yes—
176C23×C4C2×Q814yes—
177C23×C4C2×Q814yes—
178C23×C4C2×Q818yes—
179C23×C4C2414yes—
180C23×C4C2414yes—
181C23×C4C2414yes—
182C23×C4C2414yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2411yescharacteristic
189C23×C4C2411yescharacteristic
190C23×C4C2412yes—
191C22⋊Q8—432no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—416no—
202C2×C4⋊C4—416no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4C2×D414yes—
241C2×C4⋊C4C2×D418yes—
242C2×C4⋊C4C2×D414yes—
243C2×C4⋊C4C2×Q818yes—
244C2×C4⋊C4C2×Q814yes—
245C2×C4⋊C4C2×Q814yes—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C42—416no—
253C2×C42—41no—
254C2×C42—41no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—41no—
258C2×C42—41no—
259C2×C42C2×Q814yes—
260C2×C42C2×Q818yes—
261C2×C42C2×Q814yes—
262#95—24no—
263#95—24no—
264#95—28no—
265#95—28no—
266#95—28no—
267#95—28no—
268#95—28no—
269#95—28no—
270#95—28no—
271#95—28no—
272#95—28no—
273#95—28no—
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—42no—
284#95—44no—
285#95—44no—
286#95—44no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24C22×D412yes—
292C24C22×Q812yes—
293C24C22×Q812yes—
294C24H(5,𝔽2)12yes—
295C24H(5,𝔽2)12yes—
296C24H(5,𝔽2)12yes—
297C242−1+412yes—
298C24C2511yescharacteristic
299C2×Q8—48no—
300C2×Q8—88no—
301C2×Q8—88no—
302C2×Q8—84no—
303C2×Q8—84no—
304C22×C4—28no—
305C22×C4—28no—
306C22×C4—28no—
307C22×C4—28no—
308C22×C4—28no—
309C22×C4—28no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—216no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—216no—
346C22×C4—216no—
347C22×C4—216no—
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—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—48no—
377C22×C4—416no—
378C22×C4—416no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—48no—
382C22×C4—48no—
383C22×C4—88no—
384C22×C4—88no—
385C22×C4—84no—
386C22×C4—84no—
387C22×C4—84no—
388C22×C4—84no—
389C22×C4C4⋊Q8116yes—
390C22×C4C22×D414yes—
391C22×C4C22×D414yes—
392C22×C4C22×D412yes—
393C22×C4C22×D412yes—
394C22×C4C22×Q814yes—
395C22×C4C22×Q814yes—
396C22×C4C22×Q814yes—
397C22×C4C22×Q812yes—
398C22×C4C22×Q812yes—
399C22×C4C22×Q812yes—
400C22×C4C22×Q812yes—
401C22×C4C22×Q814yes—
402C22×C4H(5,𝔽2)14yes—
403C22×C42−1+414yes—
404C4⋊C4—44no—
405C4⋊C4—48no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—48no—
409C4⋊C4—44no—
410C4⋊C4—416no—
411C4⋊C4—48no—
412C4⋊C4—84no—
413C4⋊C4—84no—
414C4⋊C4—84no—
415C4⋊C4—84no—
416C4⋊C4—88no—
417C4⋊C4—88no—
418C4⋊C4—88no—
419C4⋊C4—88no—
420C4⋊C4—84no—
421C4⋊C4—84no—
422C22⋊C4—416no—
423C22⋊C4—416no—
424C42—44no—
425C42—48no—
426C42—44no—
427C23—24no—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—24no—
442C23—24no—
443C23C2×C4⋊Q814yes—
444C23#53414yes—
445C23#53714yes—
446C23#54314yes—
447C23#54314yes—
448C23#54414yes—
449C23#54414yes—
450C23C23×D411yescharacteristic
451C23C23×Q811yescharacteristic
452C23C23×Q811yescharacteristic
453C23C2×H(5,𝔽2)11yescharacteristic
454C23C2×H(5,𝔽2)11yescharacteristic
455C23C2×H(5,𝔽2)11yescharacteristic
456C23C2×H(5,𝔽2)12yes—
457C23C2×2−1+411yescharacteristic
458C23C2×2−1+412yes—
459C23#58512yes—
460C23#58512yes—
461Q8—816no—
462C2×C4—232no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—28no—
489C2×C4—28no—
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—432no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—816no—
538C2×C4—816no—
539C2×C4C2×C4⋊Q814yes—
540C2×C4C2×C4⋊Q814yes—
541C2×C4#54418yes—
542C22—24no—
543C22—28no—
544C22—24no—
545C22—28no—
546C22—24no—
547C22—24no—
548C22—28no—
549C22—28no—
550C22—24no—
551C22—24no—
552C22#265918yes—
553C22C22×C4⋊Q811yescharacteristic
554C22C2×#53411yescharacteristic
555C22C2×#53711yescharacteristic
556C22C2×#54311yescharacteristic
557C22C2×#54311yescharacteristic
558C22C2×#54411yescharacteristic
559C22C2×#54411yescharacteristic
560C22C2×#54412yes—
561C22#345812yes—
562C22#349612yes—
563C22#349712yes—
564C22#350112yes—
565C22#350212yes—
566C22#353412yes—
567C22#353412yes—
568C22#353512yes—
569C22#353512yes—
570C22#354311yescharacteristic
571C22#354411yescharacteristic
572C22#354511yescharacteristic
573C22#354511yescharacteristic
574C22#355212yes—
575C22#355312yes—
576C4—24no—
577C4—24no—
578C4—28no—
579C4—44no—
580C4—44no—
581C4—44no—
582C4—44no—
583C4—44no—
584C4—84no—
585C4—84no—
586C4—84no—
587C4—84no—
588C2—28no—
589C2C2×#265911yescharacteristic
590C2#3516712yes—
591C2#3712911yescharacteristic
592C2#3713012yes—
593C2#3713411yescharacteristic
594C2#3713512yes—
595C2#3719011yescharacteristic
596C2#3719411yescharacteristic
597C2#3747812yes—
598C2#3760911yescharacteristic
599C2#3762311yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #17076C2semidirect
3. #17073C2semidirect
4. #17072C2semidirect
5. #17036C2semidirect
6. #17032C2semidirect
7. #16907C2semidirect
8. #16827C2semidirect
81. C22×C4⋊C4Q8semidirect
104. C22×C42Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26272
#170762
#170732
#170722
#170362
#170322
#169072
#168272
#147662
#147582
#146852
#143392
C2×C4⋊#952