Finite Group Databasebeta

← #7562703

Subgroups — #7562703

Order 512.

1791 conjugacy classes of subgroups.

The subgroup lattice has 615 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3363C212yesmaximal
2#59828C211yesmaximal, characteristic
3Q8×C4⋊Q8C211yesmaximal, characteristic
4#30816C214yesmaximal
5#28717C211yesmaximal, characteristic
6#28511C211yesmaximal, characteristic
7#27769C212yesmaximal
8#26171C214yesmaximal
9#26046C212yesmaximal
10#25651C212yesmaximal
11#25471C212yesmaximal
12#24732C212yesmaximal
13#24333C212yesmaximal
14#23149C211yesmaximal, characteristic
15C4⋊C4×Q16C212yesmaximal
16C4×#462C211yesmaximal, characteristic
17C4×#462C211yesmaximal, characteristic
18C2×Q82C2212yes—
19C2×#557C2212yes—
20#3363—216no—
21C2×#487C2212yes—
22C2×#487C2214yes—
23C2×#477C2212yes—
24C2×#477C2214yes—
25C2×#477C2212yes—
26C2×#462C2214yes—
27C2×#462C2211yescharacteristic
28C2×#462C2211yescharacteristic
29C2×C8×Q8C2212yes—
30C2×C4×Q16C2212yes—
31C2×C4×Q16C2214yes—
32C2×C4×Q16C2212yes—
33#2868—21no—
34#2868C2211yescharacteristic
35#2726—22no—
36#2582C2212yes—
37#2567—22no—
38#2567C2212yes—
39#2552—21no—
40#2551—22no—
41#2551—22no—
42#2551C2212yes—
43#2551C2212yes—
44#2546C2212yes—
45#2506—22no—
46#2453—22no—
47#2446C2214yes—
48#2365—21no—
49Q8×C4⋊C4C2212yes—
50Q8×C4⋊C4C2212yes—
51Q8×C4⋊C4C2212yes—
52C4×C4⋊Q8—21no—
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56C42×Q8C2211yescharacteristic
57#2095C2214yes—
58#2087C2214yes—
59#2072C2214yes—
60#2068C2214yes—
61#2057C2214yes—
62#2043C2214yes—
63#2043C2214yes—
64#2024C2214yes—
65#1990C2212yes—
66#1984C2212yes—
67#1981C2211yescharacteristic
68#1981C2211yescharacteristic
69#1956C2212yes—
70#1947C2211yescharacteristic
71#1947C2211yescharacteristic
72#1942C2214yes—
73#1939C2212yes—
74#1933C2214yes—
75#1928C2212yes—
76#1928C2212yes—
77#1916C2214yes—
78#1878C2214yes—
79#1878C2212yes—
80#1878C2212yes—
81#1863C2214yes—
82#1863C2214yes—
83#1863C2214yes—
84#1845C2212yes—
85#1803C2211yescharacteristic
86#1803C2212yes—
87#1803C2212yes—
88#1803C2211yescharacteristic
89C8×C4⋊C4C2212yes—
90C4×C4⋊C8C2211yescharacteristic
91C4×C4⋊C8C2211yescharacteristic
92C4×#103C2212yes—
93C4×#103C2212yes—
94C22×Q16C2312yes—
95C22×Q16C2312yes—
96Q82—28no—
97Q82—44no—
98#557—28no—
99#557—44no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—24no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8D411yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2314yes—
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2312yes—
111C2×C4⋊Q8C2314yes—
112C2×#125—22no—
113C2×#125—24no—
114C2×C4×Q8—22no—
115C2×C4×Q8—22no—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2314yes—
120C2×C4×Q8C2314yes—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2312yes—
124#487—28no—
125#487—216no—
126#477—28no—
127#477—216no—
128#477—44no—
129#462—216no—
130#462—24no—
131#462D418yes—
132C8×Q8—28no—
133C4×Q16—216no—
134C4×Q16—28no—
135C4×Q16—28no—
136C2×#107C2312yes—
137C2×#107C2314yes—
138C2×#107C2312yes—
139C2×#107C2314yes—
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2314yes—
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2311yescharacteristic
144C2×C4⋊C8C2311yescharacteristic
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#103C2314yes—
156C2×#103C2312yes—
157C2×C4×C8C2312yes—
158C2×C4×C8C2314yes—
159C2×C4×C8C2312yes—
160#400—24no—
161#398—24no—
162#398—24no—
163#398—24no—
164#395—24no—
165#395—24no—
166#395—24no—
167#391—22no—
168#391—22no—
169#391D412yes—
170#391C2312yes—
171#391C2312yes—
172#391C2314yes—
173#389—22no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—24no—
183#389—24no—
184#389—24no—
185#389D412yes—
186#389D412yes—
187#389C2314yes—
188#389C2314yes—
189#389C2314yes—
190#387—24no—
191#387—24no—
192#384—22no—
193#384—22no—
194#384—22no—
195#384—22no—
196#384—21no—
197#384—21no—
198#384D411yescharacteristic
199#384C2311yescharacteristic
200#384C2311yescharacteristic
201#384C2312yes—
202#384C2312yes—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C43—21no—
219C43C2311yescharacteristic
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#336C2314yes—
227#336C2312yes—
228#336C2312yes—
229#336C2314yes—
230C22×Q8—24no—
231C22×Q8—22no—
232C22×Q8C2×D412yes—
233C22×Q8C2412yes—
234C22×Q8C2412yes—
235C22×Q8C2411yescharacteristic
236C2×Q16—24no—
237C2×Q16—24no—
238C2×Q16—24no—
239C2×Q16—24no—
240C2×Q16—24no—
241C2×Q16C2×D418yes—
242C22×C8C2412yes—
243C22×C8C2412yes—
244C22×C8C2412yes—
245C22×C8C2412yes—
246C4⋊Q8—28no—
247C4⋊Q8—28no—
248C4⋊Q8—24no—
249C4⋊Q8—22no—
250C4⋊Q8—22no—
251C4⋊Q8—24no—
252C4⋊Q8—22no—
253C4⋊Q8—44no—
254C4⋊Q8—48no—
255C4⋊Q8—44no—
256C4⋊Q8—416no—
257C4⋊Q8—82no—
258C4⋊Q8—84no—
259C4⋊Q8—82no—
260C4⋊Q8—84no—
261C4⋊Q8Q1618yescomplemented
262C4⋊Q8C2×D414yes—
263#125—416no—
264#125—48no—
265C4×Q8—22no—
266C4×Q8—22no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8—28no—
272C4×Q8—28no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—44no—
278C4×Q8—44no—
279C4×Q8—48no—
280C4×Q8—48no—
281C4×Q8C2×D414yes—
282C4×Q8#4114yes—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D414yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2414yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—22no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—22no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2414yes—
336C2×C42C2411yescharacteristic
337C2×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339#107—28no—
340#107—24no—
341#107—24no—
342#107—44no—
343C4⋊C8—28no—
344C4⋊C8—22no—
345C4⋊C8—22no—
346C4⋊C8—22no—
347C4⋊C8C2×D414yes—
348#103—24no—
349#103—28no—
350#103—28no—
351#103—24no—
352#103—28no—
353#103—24no—
354#103—24no—
355#103—24no—
356#103—24no—
357#103—24no—
358#103—44no—
359#103C2×D418yes—
360C4×C8—28no—
361C4×C8—24no—
362C4×C8—24no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95C2×D414yes—
376#95C2×D414yes—
377C2×Q8—24no—
378C2×Q8—24no—
379C2×Q8—21no—
380C2×Q8—22no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—48no—
394C2×Q8—44no—
395C2×Q8—82no—
396C2×Q8—82no—
397C2×Q8—84no—
398C2×Q8—84no—
399C2×Q8C2×Q1618yes—
400C2×Q8C22×D414yes—
401C2×Q8C22×D412yes—
402C2×Q8C2×#4112yes—
403C2×Q82−1+412yes—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D414yes—
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D414yes—
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C2511yescharacteristic
422Q16—216no—
423Q16—48no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8C22×D414yes—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—28no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—42no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—81no—
469C4⋊C4—81no—
470C4⋊C4—84no—
471C4⋊C4—82no—
472C4⋊C4—84no—
473C4⋊C4—82no—
474C4⋊C4—84no—
475C4⋊C4C2×Q1618yes—
476C4⋊C4C2×Q1618yes—
477C4⋊C4C22×D412yes—
478C4⋊C4C2×#4112yes—
479C4⋊C42−1+412yes—
480C42—24no—
481C42—24no—
482C42—24no—
483C42—22no—
484C42—22no—
485C42—24no—
486C42—24no—
487C42—21no—
488C42—21no—
489C42—41no—
490C42—48no—
491C42—48no—
492C42—42no—
493C42—44no—
494C42—44no—
495C42—42no—
496C42—42no—
497C42C2×Q1614yes—
498C42C22×D412yes—
499C42C2×#4112yes—
500C42H(5,𝔽2)12yes—
501C422−1+412yes—
502C23C23×D411yescharacteristic
503Q8—22no—
504Q8—22no—
505Q8—216no—
506Q8—22no—
507Q8—24no—
508Q8—24no—
509Q8—42no—
510Q8—42no—
511Q8—42no—
512Q8—42no—
513Q8—42no—
514Q8—42no—
515Q8—164no—
516Q8—168no—
517Q8#54714yes—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—24no—
538C2×C4—21no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4D4212yes—
560C2×C4D4212yes—
561C2×C4#54812yes—
562C2×C4#54812yes—
563C2×C4C22×Q1612yes—
564C2×C4C22×Q1618yes—
565C2×C4C22×Q1612yes—
566C2×C4C22×Q1612yes—
567C2×C4#57712yes—
568C2×C4#57812yes—
569C2×C4C23×D411yescharacteristic
570C2×C4C22×#4111yescharacteristic
571C2×C4C2×H(5,𝔽2)11yescharacteristic
572C2×C4C2×2−1+411yescharacteristic
573C2×C4C2×2−1+411yescharacteristic
574C2×C4C2×2−1+411yescharacteristic
575C8—24no—
576C8—42no—
577C8—42no—
578C8—42no—
579C22C2×D4211yescharacteristic
580C22C2×#54811yescharacteristic
581C22#348611yescharacteristic
582C22C23×Q1611yescharacteristic
583C22C2×#57711yescharacteristic
584C22C2×#57811yescharacteristic
585C22#358511yescharacteristic
586C4—28no—
587C4—22no—
588C4—22no—
589C4—22no—
590C4—24no—
591C4—42no—
592C4—44no—
593C4—42no—
594C4—41no—
595C4—42no—
596C4—44no—
597C4—82no—
598C4—82no—
599C4—84no—
600C4—82no—
601C4D4×Q1614yes—
602C4#336314yescomplemented
603C4C2×#54711yescharacteristic
604C4#349911yescharacteristic
605C4#351111yescharacteristic
606C4#355211yescharacteristic
607C2#5912511yescharacteristic
608C2C2×D4×Q1611yescharacteristic
609C2#6109511yescharacteristic
610C2C2×#336311yescharacteristic
611C2#6134811yescharacteristic
612C2#6182111yescharacteristic
613C2#6186911yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
261. C4⋊Q8Q16semidirect
602. C4#3363semidirect

Maximal subgroups

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

subgroupindex
C2×#33632
#598282
Q8×C4⋊Q82
#308162
#287172
#285112
#277692
#261712
#260462
#256512
#254712
#247322
#243332
#231492
C4⋊C4×Q162
C4×#4622