Finite Group Databasebeta

← #719288

Subgroups — #719288

Order 512.

2438 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic
2C2×#2453C212yesmaximal
3C2×Q8×C4⋊C4C211yesmaximal, characteristic
4#16523C214yesmaximal
5#15438C214yesmaximal
6#15349C214yesmaximal
7#15218C212yesmaximal
8#15194C212yesmaximal
9#14804C214yesmaximal
10#14276C211yesmaximal, characteristic
11#14150C211yesmaximal, characteristic
12#14017C212yesmaximal
13#13942C211yesmaximal, characteristic
14#13856C211yesmaximal, characteristic
15#13786C211yesmaximal, characteristic
16C22×#125C2212yes—
17C22×C4×Q8C2212yes—
18C22×C4×Q8C2211yescharacteristic
19#2552—28no—
20#2453—216no—
21C2×#400C2214yes—
22C2×#398C2214yes—
23C2×#398C2214yes—
24Q8×C4⋊C4—28no—
25C2×#391C2212yes—
26C2×#391C2211yescharacteristic
27C2×#391C2211yescharacteristic
28C2×#389C2214yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2214yes—
35C2×#387C2214yes—
36C2×#384C2212yes—
37C2×#384C2212yes—
38C2×#384C2211yescharacteristic
39C2×#382C2211yescharacteristic
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×#376C2211yescharacteristic
47#1445C2214yes—
48#1444C2212yes—
49#1444C2212yes—
50#1444C2212yes—
51#1444C2212yes—
52#1444C2212yes—
53#1444C2214yes—
54#1444C2214yes—
55#1444C2214yes—
56#1444C2214yes—
57#1442C2212yes—
58#1442C2214yes—
59#1441C2212yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2212yes—
66#1441C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71C4⋊#95C2212yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2212yes—
74C4⋊#95C2211yescharacteristic
75C4⋊#95C2212yes—
76#1434C2214yes—
77#1434C2214yes—
78C23×Q8C2311yescharacteristic
79C2×#125—28no—
80C2×#125—28no—
81C2×#125—44no—
82C2×C4×Q8—28no—
83C2×C4×Q8—28no—
84C2×C4×Q8—24no—
85C2×C4×Q8—24no—
86C2×C4×Q8—44no—
87C2×C4×Q8Q818yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2314yes—
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119#400—216no—
120#398—216no—
121#398—216no—
122#391—28no—
123#391D418yes—
124#391Q818yes—
125#389—216no—
126#389—28no—
127#389—28no—
128#389—28no—
129#389—28no—
130#389—28no—
131#389—216no—
132#387—216no—
133#384—28no—
134#384—28no—
135#384D418yes—
136#382—24no—
137C4×C4⋊C4—28no—
138C4×C4⋊C4—24no—
139C4×C4⋊C4—28no—
140C4×C4⋊C4—24no—
141C4×C4⋊C4—24no—
142C4×C4⋊C4—44no—
143#376—24no—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C22×Q8—22no—
172C22×Q8—22no—
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8C2×D414yes—
176C22×Q8C2×Q814yes—
177C22×Q8C2×Q814yes—
178C23×C4C2412yes—
179C23×C4C2412yes—
180C23×C4C2412yes—
181C23×C4C2414yes—
182C23×C4C2414yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2411yescharacteristic
191C23×C4C2411yescharacteristic
192C23×C4C2411yescharacteristic
193#125—432no—
194C4×Q8—232no—
195C4×Q8—432no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4C2×D418yes—
253C2×C4⋊C4C2×D414yes—
254C2×C4⋊C4C2×D418yes—
255C2×C4⋊C4C2×D418yes—
256C2×C4⋊C4C2×Q814yes—
257C2×C4⋊C4C2×Q818yes—
258C2×C4⋊C4C2×Q814yes—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—28no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—28no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—44no—
275C2×C42—44no—
276C2×C42C2×D414yes—
277C2×C42C2×D414yes—
278C2×C42C2×Q818yes—
279C2×C42C2×Q814yes—
280C2×C42C2×Q814yes—
281C2×C42#4114yes—
282#95—24no—
283#95—28no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—42no—
306#95—42no—
307#95C2×D4116yes—
308#95C2×Q8116yes—
309C24C2511yescharacteristic
310C2×Q8—28no—
311C2×Q8—28no—
312C2×Q8—28no—
313C2×Q8—28no—
314C2×Q8—28no—
315C2×Q8—28no—
316C2×Q8C4⋊Q8116yes—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4C22×D414yes—
390C22×C4C22×D414yes—
391C22×C4C22×D418yes—
392C22×C4C22×D412yes—
393C22×C4C22×D412yes—
394C22×C4C22×D414yes—
395C22×C4C22×D412yes—
396C22×C4C22×D412yes—
397C22×C4C22×Q814yes—
398C22×C4C22×Q814yes—
399C22×C4C22×Q814yes—
400C22×C4C22×Q812yes—
401C22×C4C22×Q812yes—
402C22×C4C22×Q814yes—
403C22×C4C22×Q814yes—
404C22×C4C22×Q812yes—
405C22×C4C22×Q812yes—
406C22×C4C2×#4112yes—
407C22×C4C2×#4112yes—
408C22×C4C2×#4112yes—
409C22×C4H(5,𝔽2)12yes—
410C22×C42−1+414yes—
411C22×C42−1+412yes—
412C4⋊C4—28no—
413C4⋊C4—216no—
414C4⋊C4—28no—
415C4⋊C4—216no—
416C4⋊C4—28no—
417C4⋊C4—28no—
418C4⋊C4—48no—
419C4⋊C4—48no—
420C4⋊C4—48no—
421C4⋊C4—416no—
422C4⋊C4—416no—
423C4⋊C4—416no—
424C4⋊C4—416no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—416no—
429C4⋊C4—48no—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C4⋊C4—48no—
433C4⋊C4—88no—
434C4⋊C4—88no—
435C42—216no—
436C42—28no—
437C42—28no—
438C42—416no—
439C42—48no—
440C42—48no—
441C42—48no—
442C42C22⋊Q8116yes—
443C23C23×D411yescharacteristic
444C23C23×D411yescharacteristic
445C23C23×Q811yescharacteristic
446C23C23×Q811yescharacteristic
447C23C22×#4111yescharacteristic
448C23C2×H(5,𝔽2)11yescharacteristic
449C23C2×2−1+411yescharacteristic
450C23C2×2−1+412yes—
451C23#58512yes—
452C23#58512yes—
453C23#58512yes—
454Q8—232no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—28no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—28no—
476C2×C4—28no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—28no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—28no—
485C2×C4—28no—
486C2×C4—24no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4C2×C22⋊Q814yes—
541C2×C4C2×C22⋊Q814yes—
542C2×C4C2×C22⋊Q814yes—
543C2×C4C2×C4⋊Q818yes—
544C2×C4C2×C4⋊Q814yes—
545C2×C4#54414yes—
546C2×C4D4214yes—
547C2×C4#54714yes—
548C2×C4D4×Q814yes—
549C2×C4#55414yes—
550C22C22×C22⋊Q811yescharacteristic
551C22C22×C4⋊Q811yescharacteristic
552C22C2×#54411yescharacteristic
553C22#346112yes—
554C22C2×D4211yescharacteristic
555C22C2×#54711yescharacteristic
556C22C2×D4×Q811yescharacteristic
557C22C2×#55411yescharacteristic
558C22#348511yescharacteristic
559C22#348511yescharacteristic
560C22#348611yescharacteristic
561C22#348711yescharacteristic
562C22#348712yes—
563C22#348912yes—
564C22#350212yes—
565C22#350212yes—
566C22#350412yes—
567C22#350412yes—
568C22#351212yes—
569C22#353912yes—
570C22#354112yes—
571C22#355512yes—
572C22#355712yes—
573C4—24no—
574C4—24no—
575C4—28no—
576C4—44no—
577C4—44no—
578C4—48no—
579C4—44no—
580C4—44no—
581C4—44no—
582C4—44no—
583C4—82no—
584C4—82no—
585C4—84no—
586C4—82no—
587C4#244018yes—
588C2C2×#244011yescharacteristic
589C2#3515411yescharacteristic
590C2#3563911yescharacteristic
591C2#3595712yes—
592C2#3779912yes—
593C2#3969312yes—
594C2#4281912yes—
595C2#4513911yescharacteristic
596C2#4526711yescharacteristic
597C2#4538711yescharacteristic
598C2#4538911yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#24532
C2×Q8×C4⋊C42
#165232
#154382
#153492
#152182
#151942
#148042
#142762
#141502
#140172
#139422
#138562
#137862