Finite Group Databasebeta

← #7614145

Subgroups — #7614145

Order 512.

1738 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3387C212yesmaximal
2#59832C211yesmaximal, characteristic
3#59828C211yesmaximal, characteristic
4#31242C214yesmaximal
5#28816C212yesmaximal
6#28284C211yesmaximal, characteristic
7#28283C211yesmaximal, characteristic
8#28274C212yesmaximal
9#26497C214yesmaximal
10#26465C212yesmaximal
11#26462C212yesmaximal
12#24333C212yesmaximal
13#24332C212yesmaximal
14#23187C212yesmaximal
15C4×#498C211yesmaximal, characteristic
16#21712C211yesmaximal, characteristic
17C4×#494C211yesmaximal, characteristic
18C2×Q82C2212yes—
19C2×#557C2212yes—
20#3387—216no—
21C2×#501C2214yes—
22C2×#498C2211yescharacteristic
23C2×#494C2211yescharacteristic
24C2×#479C2214yes—
25C2×#477C2212yes—
26C2×#475C2214yes—
27C2×#475C2212yes—
28C2×#446C2212yes—
29C2×#441C2214yes—
30C2×C4×Q16C2212yes—
31C2×C4×Q16C2212yes—
32#2734—22no—
33#2726—22no—
34#2582C2212yes—
35#2582C2212yes—
36#2552C2212yes—
37#2551—21no—
38#2551C2212yes—
39#2546—22no—
40#2546—21no—
41#2506—22no—
42#2453C2214yes—
43#2451—22no—
44#2451C2214yes—
45#2446—22no—
46#2365—21no—
47#2365C2212yes—
48#2365C2212yes—
49Q8×C4⋊C4—21no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2211yescharacteristic
52C4×C4⋊Q8C2211yescharacteristic
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55#2068C2214yes—
56#2066C2214yes—
57#2066C2214yes—
58#2066C2214yes—
59#2059C2214yes—
60#2057C2214yes—
61#2026C2214yes—
62#2024C2214yes—
63#1990C2211yescharacteristic
64#1990C2211yescharacteristic
65#1984C2212yes—
66#1981C2212yes—
67#1981C2212yes—
68#1971C2214yes—
69#1969C2212yes—
70#1969C2212yes—
71#1956C2211yescharacteristic
72#1956C2211yescharacteristic
73#1948C2212yes—
74#1947C2212yes—
75#1944C2214yes—
76#1941C2212yes—
77#1941C2212yes—
78#1939C2212yes—
79#1939C2212yes—
80#1878C2214yes—
81#1878C2214yes—
82#1865C2214yes—
83#1863C2214yes—
84#1845C2211yescharacteristic
85#1844C2211yescharacteristic
86#1842C2211yescharacteristic
87#1803C2212yes—
88#1803C2212yes—
89C4×#106C2212yes—
90C4×#106C2212yes—
91#1770C2212yes—
92#1770C2212yes—
93C42×C8C2211yescharacteristic
94C22×Q16C2312yes—
95C22×Q16C2312yes—
96Q82—28no—
97Q82—44no—
98#557—28no—
99#557—44no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—24no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8D411yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2314yes—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2311yescharacteristic
110C2×#125—24no—
111C2×#125C2314yes—
112C2×C4×Q8—22no—
113C2×C4×Q8—22no—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2314yes—
119C2×C4×Q8C2314yes—
120C2×C4×Q8C2312yes—
121#501—216no—
122#498D418yes—
123#494Q818yescomplemented
124#479—216no—
125#477—28no—
126#475—28no—
127#475—216no—
128#446—28no—
129#441—216no—
130C4×Q16—28no—
131C4×Q16—44no—
132C2×#107C2314yes—
133C2×#107C2312yes—
134C2×#107C2312yes—
135C2×#106C2314yes—
136C2×#106C2312yes—
137C2×#106C2312yes—
138C2×#106C2312yes—
139C2×#106C2312yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2314yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×#103C2312yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2312yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2312yes—
151C2×#103C2312yes—
152C2×#97C2314yes—
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159#400—24no—
160#398—24no—
161#398—24no—
162#398—24no—
163#398—24no—
164#398—24no—
165#395—24no—
166#395—24no—
167#391—22no—
168#391—22no—
169#391—21no—
170#391—21no—
171#391—24no—
172#391D412yes—
173#391C2312yes—
174#391C2312yes—
175#391C2314yes—
176#391C2314yes—
177#389—22no—
178#389—22no—
179#389—24no—
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#389D412yes—
191#389D412yes—
192#389C2312yes—
193#389C2312yes—
194#389C2314yes—
195#389C2314yes—
196#387—24no—
197#387—24no—
198#387—24no—
199#387—24no—
200#384—22no—
201#384D411yescharacteristic
202#384C2311yescharacteristic
203#384C2311yescharacteristic
204#384C2311yescharacteristic
205#384C2311yescharacteristic
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C43C2311yescharacteristic
219#340C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2312yes—
225#336C2314yes—
226#336C2314yes—
227C22×Q8—21no—
228C22×Q8—21no—
229C22×Q8—24no—
230C22×Q8C2×D412yes—
231C22×Q8C2412yes—
232C22×Q8C2412yes—
233C2×Q16—24no—
234C2×Q16—24no—
235C2×Q16—42no—
236C2×Q16—42no—
237C2×Q16C2×Q818yes—
238C2×Q16C2×Q818yes—
239C22×C8C2411yescharacteristic
240C22×C8C2414yes—
241C22×C8C2411yescharacteristic
242C22×C8C2411yescharacteristic
243C22×C8C2411yescharacteristic
244C4⋊Q8—28no—
245C4⋊Q8—24no—
246C4⋊Q8—22no—
247C4⋊Q8—22no—
248C4⋊Q8—22no—
249C4⋊Q8—22no—
250C4⋊Q8—44no—
251C4⋊Q8—48no—
252C4⋊Q8—416no—
253C4⋊Q8—48no—
254C4⋊Q8—84no—
255C4⋊Q8C2×D414yes—
256C4⋊Q8C2×D414yes—
257C4⋊Q8C2×Q814yes—
258C4⋊Q8C2×Q814yes—
259#125—28no—
260#125—44no—
261#125—416no—
262C4×Q8—24no—
263C4×Q8—24no—
264C4×Q8—28no—
265C4×Q8—28no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—44no—
271C4×Q8—44no—
272C4×Q8—48no—
273C4×Q8—48no—
274C4×Q8—44no—
275C4×Q8—82no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D414yes—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—24no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2414yes—
324C2×C42C2411yescharacteristic
325C2×C42C2411yescharacteristic
326C2×C42C2411yescharacteristic
327#107—24no—
328#107—28no—
329#107—42no—
330#106—24no—
331#106—28no—
332#106—42no—
333#106C2×D418yes—
334#106C2×D418yes—
335C4⋊C8—28no—
336C4⋊C8—24no—
337C4⋊C8—24no—
338C4⋊C8—42no—
339#103—24no—
340#103—24no—
341#103—28no—
342#103—28no—
343#103—44no—
344#103—44no—
345#103—42no—
346#103—42no—
347#97—28no—
348C4×C8—22no—
349C4×C8—22no—
350C4×C8—22no—
351C4×C8—41no—
352C4×C8C2×D414yes—
353C4×C8C2×Q814yes—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—24no—
359#95—24no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95C2×D414yes—
369#95C2×D414yes—
370C2×Q8—24no—
371C2×Q8—24no—
372C2×Q8—24no—
373C2×Q8—24no—
374C2×Q8—24no—
375C2×Q8—22no—
376C2×Q8—22no—
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—48no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—81no—
389C2×Q8—81no—
390C2×Q8—84no—
391C2×Q8—84no—
392C2×Q8C22×Q814yes—
393C2×Q8C22×Q814yes—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D414yes—
404C22×C4C22×D414yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C2511yescharacteristic
411Q16—216no—
412Q16—48no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—24no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8C22×D412yes—
424C2×C8C22×D412yes—
425C2×C8C22×Q812yes—
426C2×C8C22×Q812yes—
427C2×C82−1+412yes—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—28no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—42no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—81no—
463C4⋊C4—81no—
464C4⋊C4—81no—
465C4⋊C4—84no—
466C4⋊C4—81no—
467C4⋊C4—82no—
468C4⋊C4—84no—
469C4⋊C4C22×D414yes—
470C4⋊C4C22×D414yes—
471C42—22no—
472C42—22no—
473C42—24no—
474C42—24no—
475C42—21no—
476C42—21no—
477C42—41no—
478C42—48no—
479C42—48no—
480C42—44no—
481C42—44no—
482C42—44no—
483C42—44no—
484C42#13714yes—
485C42C22×D412yes—
486C42C22×Q812yes—
487C42H(5,𝔽2)12yes—
488C422−1+412yes—
489C23C23×D411yescharacteristic
490Q8—24no—
491Q8—24no—
492Q8—24no—
493Q8—24no—
494Q8—42no—
495Q8—42no—
496Q8—48no—
497Q8—48no—
498Q8—42no—
499Q8—42no—
500Q8—84no—
501Q8—168no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—21no—
516C2×C4—24no—
517C2×C4—21no—
518C2×C4—24no—
519C2×C4—21no—
520C2×C4—24no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
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×C4D4212yes—
542C2×C4D4212yes—
543C2×C4D4×Q812yes—
544C2×C4D4×Q812yes—
545C2×C4C2×#13712yes—
546C2×C4C2×#13712yes—
547C2×C4C2×#13712yes—
548C2×C4#57712yes—
549C2×C4#57812yes—
550C2×C4C23×D411yescharacteristic
551C2×C4C23×Q811yescharacteristic
552C2×C4C2×H(5,𝔽2)11yescharacteristic
553C2×C4C2×2−1+411yescharacteristic
554C2×C4C2×2−1+411yescharacteristic
555C2×C4#58511yescharacteristic
556C8—22no—
557C8—22no—
558C8—44no—
559C8—41no—
560C8D4×Q814yes—
561C22C2×D4211yescharacteristic
562C22C2×D4×Q811yescharacteristic
563C22#349711yescharacteristic
564C22C22×#13711yescharacteristic
565C22C2×#57711yescharacteristic
566C22C2×#57811yescharacteristic
567C22#358411yescharacteristic
568C4—22no—
569C4—22no—
570C4—22no—
571C4—44no—
572C4—44no—
573C4—44no—
574C4—44no—
575C4—44no—
576C4—44no—
577C4—41no—
578C4—82no—
579C4—82no—
580C4—82no—
581C4—82no—
582C4#327614yes—
583C4#338714yescomplemented
584C4C2×D4×Q811yescharacteristic
585C4#349711yescharacteristic
586C4#350111yescharacteristic
587C4#355011yescharacteristic
588C2#5981711yescharacteristic
589C2C2×#327611yescharacteristic
590C2#6108511yescharacteristic
591C2C2×#338711yescharacteristic
592C2#6137211yescharacteristic
593C2#6236611yescharacteristic
594C2#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
123. #494Q8semidirect
583. C4#3387semidirect

Maximal subgroups

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

subgroupindex
C2×#33872
#598322
#598282
#312422
#288162
#282842
#282832
#282742
#264972
#264652
#264622
#243332
#243322
#231872
C4×#4982
#217122
C4×#4942