Finite Group Databasebeta

← #7604163

Subgroups — #7604163

Order 512.

1780 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3363C212yesmaximal
2#59832C211yesmaximal, characteristic
3#59828C211yesmaximal, characteristic
4#31151C214yesmaximal
5#28284C211yesmaximal, characteristic
6#28255C211yesmaximal, characteristic
7#28187C212yesmaximal
8#27769C212yesmaximal
9#26822C214yesmaximal
10#26046C212yesmaximal
11#25370C212yesmaximal
12#24333C212yesmaximal
13#24191C212yesmaximal
14#22998C212yesmaximal
15C8×C4⋊Q8C211yesmaximal, characteristic
16C4×#494C211yesmaximal, characteristic
17C4×#487C211yesmaximal, characteristic
18C2×Q82C2212yes—
19C2×#557C2212yes—
20#3363—216no—
21C2×#494C2211yescharacteristic
22C2×#487C2214yes—
23C2×#487C2211yescharacteristic
24C2×#477C2212yes—
25C2×#477C2214yes—
26C2×#462C2212yes—
27C2×#462C2214yes—
28C2×#462C2212yes—
29C2×C8×Q8C2212yes—
30C2×C4×Q16C2212yes—
31C2×C4×Q16C2214yes—
32C2×C4×Q16C2212yes—
33#2734—22no—
34#2726—22no—
35#2582C2212yes—
36#2582C2212yes—
37#2552—21no—
38#2551—21no—
39#2551C2212yes—
40#2546—22no—
41#2546C2212yes—
42#2506—22no—
43#2453—22no—
44#2451—22no—
45#2451C2214yes—
46#2446C2214yes—
47#2365—21no—
48#2365—21no—
49#2365C2212yes—
50Q8×C4⋊C4C2212yes—
51C4×C4⋊Q8C2211yescharacteristic
52C4×C4⋊Q8C2211yescharacteristic
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56#2087C2214yes—
57#2068C2214yes—
58#2057C2214yes—
59#2056C2214yes—
60#2056C2214yes—
61#2043C2214yes—
62#2024C2214yes—
63#2016C2214yes—
64#1990C2211yescharacteristic
65#1984C2212yes—
66#1981C2212yes—
67#1969C2212yes—
68#1969C2212yes—
69#1956C2211yescharacteristic
70#1956C2211yescharacteristic
71#1956C2211yescharacteristic
72#1947C2212yes—
73#1939C2212yes—
74#1939C2212yes—
75#1933C2214yes—
76#1933C2214yes—
77#1933C2212yes—
78#1933C2212yes—
79#1878C2214yes—
80#1878C2214yes—
81#1863C2214yes—
82#1863C2214yes—
83#1845C2211yescharacteristic
84#1842C2211yescharacteristic
85#1834C2211yescharacteristic
86#1803C2212yes—
87#1803C2212yes—
88#1803C2212yes—
89C8×C4⋊C4C2212yes—
90C8×C4⋊C4C2212yes—
91C4×#103C2212yes—
92C4×#103C2212yes—
93C42×C8C2211yescharacteristic
94C22×Q16C2312yes—
95C22×Q16C2314yes—
96C22×Q16C2312yes—
97Q82—28no—
98Q82—44no—
99#557—28no—
100#557—44no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—24no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8D411yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2314yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2314yes—
110C2×C4⋊Q8C2311yescharacteristic
111C2×#125—22no—
112C2×#125—24no—
113C2×C4×Q8—22no—
114C2×C4×Q8—22no—
115C2×C4×Q8—21no—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2314yes—
120C2×C4×Q8C2314yes—
121C2×C4×Q8C2312yes—
122#494—24no—
123#487—216no—
124#487D418yes—
125#477—28no—
126#477—216no—
127#462—28no—
128#462—28no—
129#462—216no—
130C8×Q8—28no—
131C4×Q16—216no—
132C4×Q16—28no—
133C4×Q16—44no—
134C2×#107C2314yes—
135C2×#107C2312yes—
136C2×#107C2312yes—
137C2×C4⋊C8C2314yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2312yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×#103C2312yes—
146C2×#103C2312yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2314yes—
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2311yescharacteristic
160#400—24no—
161#398—24no—
162#398—24no—
163#398—24no—
164#398—24no—
165#398—24no—
166#395—24no—
167#395—24no—
168#391—22no—
169#391—22no—
170#391—21no—
171#391—24no—
172#391D412yes—
173#391C2312yes—
174#391C2312yes—
175#391C2312yes—
176#391C2314yes—
177#391C2314yes—
178#389—22no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—22no—
183#389—22no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—24no—
192#389D412yes—
193#389D412yes—
194#389C2312yes—
195#389C2314yes—
196#389C2314yes—
197#387—24no—
198#387—24no—
199#387—24no—
200#387—24no—
201#384—22no—
202#384D411yescharacteristic
203#384C2311yescharacteristic
204#384C2311yescharacteristic
205#384C2311yescharacteristic
206#384C2311yescharacteristic
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2312yes—
219C43C2311yescharacteristic
220#340C2312yes—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#336C2314yes—
227#336C2314yes—
228C22×Q8—21no—
229C22×Q8—24no—
230C22×Q8C2×D412yes—
231C22×Q8C2412yes—
232C22×Q8C2412yes—
233C22×Q8C2412yes—
234C2×Q16—28no—
235C2×Q16—28no—
236C2×Q16—24no—
237C2×Q16—24no—
238C2×Q16—24no—
239C2×Q16—24no—
240C2×Q16—42no—
241C2×Q16—44no—
242C2×Q16—42no—
243C22×C8C2414yes—
244C22×C8C2411yescharacteristic
245C22×C8C2411yescharacteristic
246C22×C8C2411yescharacteristic
247C22×C8C2411yescharacteristic
248C4⋊Q8—28no—
249C4⋊Q8—28no—
250C4⋊Q8—22no—
251C4⋊Q8—22no—
252C4⋊Q8—22no—
253C4⋊Q8—22no—
254C4⋊Q8—22no—
255C4⋊Q8—44no—
256C4⋊Q8—44no—
257C4⋊Q8—48no—
258C4⋊Q8—416no—
259C4⋊Q8—84no—
260C4⋊Q8Q1618yescomplemented
261C4⋊Q8C2×D414yes—
262C4⋊Q8C2×D414yes—
263C4⋊Q8#4114yes—
264#125—416no—
265#125—48no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8—28no—
269C4×Q8—28no—
270C4×Q8—24no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—44no—
277C4×Q8—44no—
278C4×Q8—48no—
279C4×Q8—48no—
280C4×Q8—82no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D414yes—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C42—22no—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2414yes—
329C2×C42C2411yescharacteristic
330C2×C42C2411yescharacteristic
331C2×C42C2411yescharacteristic
332#107—24no—
333#107—28no—
334#107—42no—
335C4⋊C8—28no—
336C4⋊C8—24no—
337C4⋊C8—24no—
338C4⋊C8—42no—
339#103—28no—
340#103—28no—
341#103—24no—
342#103—24no—
343#103—24no—
344#103—28no—
345#103—42no—
346#103—44no—
347#103—42no—
348#103—42no—
349#103C2×D418yes—
350#103C2×D418yes—
351C4×C8—28no—
352C4×C8—22no—
353C4×C8—22no—
354C4×C8—22no—
355C4×C8—22no—
356C4×C8—41no—
357C4×C8C2×D414yes—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—24no—
365#95—24no—
366#95—22no—
367#95—22no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95C2×D414yes—
373#95C2×D414yes—
374C2×Q8—24no—
375C2×Q8—24no—
376C2×Q8—24no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
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—81no—
395C2×Q8—81no—
396C2×Q8—84no—
397C2×Q8—84no—
398C2×Q8C2×Q1618yes—
399C2×Q8C22×D414yes—
400C2×Q8C2×#4114yes—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D414yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D414yes—
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C2511yescharacteristic
418Q16—48no—
419Q16—48no—
420Q16—416no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—24no—
433C2×C8C22×D412yes—
434C2×C8C22×D412yes—
435C2×C82−1+412yes—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—28no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
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—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—81no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C4⋊C4—84no—
472C4⋊C4—81no—
473C4⋊C4—82no—
474C4⋊C4—84no—
475C4⋊C4C2×Q1618yes—
476C4⋊C4C2×Q1618yes—
477C4⋊C4C22×D414yes—
478C42—22no—
479C42—22no—
480C42—24no—
481C42—24no—
482C42—21no—
483C42—21no—
484C42—41no—
485C42—48no—
486C42—48no—
487C42—44no—
488C42—44no—
489C42—44no—
490C42—44no—
491C42C2×Q1614yes—
492C42C22×D412yes—
493C42C2×#4112yes—
494C42H(5,𝔽2)12yes—
495C422−1+412yes—
496C23C23×D411yescharacteristic
497Q8—216no—
498Q8—24no—
499Q8—24no—
500Q8—24no—
501Q8—24no—
502Q8—42no—
503Q8—42no—
504Q8—42no—
505Q8—42no—
506Q8—42no—
507Q8—42no—
508Q8—42no—
509Q8—42no—
510Q8—84no—
511Q8—168no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—24no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—24no—
528C2×C4—21no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4D4212yes—
551C2×C4D4212yes—
552C2×C4#54812yes—
553C2×C4#54812yes—
554C2×C4C22×Q1612yes—
555C2×C4C22×Q1618yes—
556C2×C4C22×Q1612yes—
557C2×C4C22×Q1612yes—
558C2×C4#57712yes—
559C2×C4#57812yes—
560C2×C4C23×D411yescharacteristic
561C2×C4C22×#4111yescharacteristic
562C2×C4C2×H(5,𝔽2)11yescharacteristic
563C2×C4C2×2−1+411yescharacteristic
564C2×C4C2×2−1+411yescharacteristic
565C2×C4#58511yescharacteristic
566C8—22no—
567C8—22no—
568C8—22no—
569C8—44no—
570C8—41no—
571C22C2×D4211yescharacteristic
572C22C2×#54811yescharacteristic
573C22#349711yescharacteristic
574C22C23×Q1611yescharacteristic
575C22C2×#57711yescharacteristic
576C22C2×#57811yescharacteristic
577C22#358411yescharacteristic
578C4—28no—
579C4—22no—
580C4—22no—
581C4—22no—
582C4—44no—
583C4—44no—
584C4—44no—
585C4—44no—
586C4—41no—
587C4—82no—
588C4—82no—
589C4—82no—
590C4—82no—
591C4—82no—
592C4D4×Q1614yes—
593C4#336314yescomplemented
594C4D4×#4111yescharacteristic
595C4#349911yescharacteristic
596C4#352011yescharacteristic
597C4#354711yescharacteristic
598C2#5965411yescharacteristic
599C2C2×D4×Q1611yescharacteristic
600C2#6108511yescharacteristic
601C2C2×#336311yescharacteristic
602C2#6132811yescharacteristic
603C2#6236011yescharacteristic
604C2#6238211yescharacteristic
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
260. C4⋊Q8Q16semidirect
593. C4#3363semidirect

Maximal subgroups

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

subgroupindex
C2×#33632
#598322
#598282
#311512
#282842
#282552
#281872
#277692
#268222
#260462
#253702
#243332
#241912
#229982
C8×C4⋊Q82
C4×#4942
C4×#4872