Finite Group Databasebeta

← #6436697

Subgroups — #6436697

Order 512.

1902 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3110C211yesmaximal, characteristic
2C2×#3101C211yesmaximal, characteristic
3#35322C211yesmaximal, characteristic, complemented
4#34058C211yesmaximal, characteristic
5#29865C212yesmaximal, complemented
6#29829C212yesmaximal, complemented
7#29828C212yesmaximal, complemented
8#29798C212yesmaximal, complemented
9#29764C214yesmaximal, complemented
10#29723C214yesmaximal
11#23689C218yesmaximal, complemented
12#21392C211yesmaximal, characteristic, complemented
13#21389C211yesmaximal, characteristic, complemented
14#20815C211yesmaximal, characteristic, complemented
15C2×#533C2211yescharacteristic
16C2×#533C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18#3110—28no—
19#3101—28no—
20C2×#484C2212yes—
21C2×#483C2212yes—
22C2×#476C2212yes—
23C2×#474C2212yes—
24C2×#461C2212yes—
25C2×#461C2212yes—
26C2×#450C2212yes—
27C2×#450C2212yes—
28C2×#440C2214yes—
29C2×C4×SD16C2214yes—
30C2×#432C2211yescharacteristic
31C2×#431C2211yescharacteristic
32#2868C2211yescharacteristic, complemented
33#2862C2211yescharacteristic, complemented
34#2862C2211yescharacteristic, complemented
35#2862C2211yescharacteristic, complemented
36#2621C2214yes—
37#2565—22no—
38#2565—22no—
39#2499—22no—
40#2499—22no—
41#2327C2214yescomplemented
42#2319—22no—
43#2319C2214yes—
44#2316C2212yes—
45#2312—22no—
46#2309C2212yes—
47#2293—22no—
48#2293C2212yes—
49#2293C2212yes—
50#2280—22no—
51C4×#117C2211yescharacteristic
52C4×#117C2211yescharacteristic
53#2085C2218yescomplemented
54#2073C2218yes—
55#2068C2218yescomplemented
56#2066C2218yescomplemented
57#2055C2218yes—
58#2055C2218yes—
59#2043C2218yescomplemented
60#2013C2218yes—
61#1988C2212yescomplemented
62#1987C2212yescomplemented
63#1981C2212yescomplemented
64#1981C2212yescomplemented
65#1873C2214yes—
66#1873C2214yes—
67#1842C2212yescomplemented
68#1831C2212yescomplemented
69#1798C2214yes—
70#1768C2214yescomplemented
71#1763C2214yescomplemented
72C4×#103C2214yescomplemented
73C23×D4C2311yescharacteristic
74C24×C4—21no—
75C22×SD16C2314yes—
76C22×SD16C2314yes—
77#533—24no—
78#533—24no—
79#533—24no—
80#533—24no—
81C2×C4⋊Q8C2311yescharacteristic
82C2×C4⋊Q8C2311yescharacteristic
83C2×C4⋊Q8C2314yes—
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2311yescharacteristic
86C2×#125—21no—
87C2×#125—21no—
88C2×C22⋊Q8—21no—
89C2×C22⋊Q8—21no—
90C2×C22⋊Q8C2312yes—
91C2×C22⋊Q8C2312yes—
92C2×C4×Q8—21no—
93C2×C4×Q8—21no—
94C2×C4×Q8C2312yes—
95C2×C4×Q8C2312yes—
96C2×C4×D4—21no—
97C2×C4×D4—21no—
98C2×C4×D4—24no—
99C2×C4×D4D418yes—
100C2×C4×D4C2312yes—
101C2×C4×D4C2312yes—
102C2×#117—21no—
103C2×#117—21no—
104C2×#117—22no—
105C2×#117—22no—
106C2×#117C2312yes—
107C2×#117C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4—21no—
110C22×C42C2311yescharacteristic
111#484—28no—
112#483—28no—
113#476—28no—
114#474—28no—
115#461—28no—
116#461—28no—
117#450—28no—
118#450—28no—
119#440—216no—
120C4×SD16—216no—
121#432—24no—
122#431—24no—
123C2×#107C2314yes—
124C2×#106C2314yes—
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2318yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2318yes—
131C2×#103C2318yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2312yes—
136C2×#97C2312yes—
137C2×C4×C8C2312yes—
138#398—28no—
139#397—28no—
140#391—22no—
141#391—22no—
142#391—22no—
143#391—22no—
144#391C2314yes—
145#391C2314yes—
146#390—24no—
147#390C2314yes—
148#390C2314yes—
149#389—24no—
150#389C2314yes—
151#389C2314yes—
152#388—24no—
153#388—24no—
154#388—24no—
155#388—24no—
156#387—28no—
157#387—28no—
158#384—22no—
159#384—22no—
160#382—22no—
161#382—22no—
162#382C2314yes—
163#381—24no—
164#380—24no—
165#380—24no—
166#379—24no—
167C4×C4⋊C4D412yes—
168C4×C4⋊C4D412yes—
169C4×C4⋊C4C2312yes—
170C4×C4⋊C4C2312yes—
171C4×C22⋊C4—21no—
172C4×C22⋊C4—21no—
173C4×C22⋊C4—22no—
174C4×C22⋊C4—22no—
175C4×C22⋊C4D412yes—
176C4×C22⋊C4D412yes—
177#376—21no—
178#376C2312yes—
179C43—21no—
180C43C2311yescharacteristic
181C43C2311yescharacteristic
182#340C2318yes—
183#340C2318yes—
184#336C2318yes—
185C25—21no—
186C22×Q8—21no—
187C22×Q8—21no—
188C22×Q8C2412yes—
189C22×Q8C2412yes—
190C22×D4—21no—
191C22×D4—22no—
192C22×D4—44no—
193C22×D4C2×D414yes—
194C22×D4C2412yes—
195C23×C4—21no—
196C23×C4—22no—
197C23×C4—28no—
198C23×C4—22no—
199C23×C4—84no—
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2411yescharacteristic
203C2×SD16—28no—
204C2×SD16—28no—
205C2×SD16—48no—
206C2×SD16—48no—
207C22×C8C2414yes—
208C22×C8C2414yes—
209C4⋊Q8—22no—
210C4⋊Q8—22no—
211C4⋊Q8—22no—
212C4⋊Q8—22no—
213C4⋊Q8—22no—
214C4⋊Q8—22no—
215C4⋊Q8—22no—
216C4⋊Q8—22no—
217C4⋊Q8—48no—
218#125—44no—
219#125—44no—
220C22⋊Q8—24no—
221C22⋊Q8—24no—
222C22⋊Q8—24no—
223C22⋊Q8—24no—
224C22⋊Q8—44no—
225C22⋊Q8—44no—
226C4×Q8—24no—
227C4×Q8—24no—
228C4×Q8—24no—
229C4×Q8—24no—
230C4×Q8—44no—
231C4×Q8—44no—
232C4×D4—28no—
233C4×D4—28no—
234C4×D4—24no—
235C4×D4—24no—
236C4×D4C2×D418yes—
237C4×D4C2×D418yes—
238#117—44no—
239#117—44no—
240#117—44no—
241#117—44no—
242#117—84no—
243#117—84no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2414yes—
262C2×C4⋊C4C2412yes—
263C2×C4⋊C4C2412yes—
264C2×C4⋊C4C2412yes—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4C2×D414yes—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D414yes—
288C2×C42#4114yes—
289C2×C42C2412yes—
290C2×C42C2412yes—
291C2×C42C2411yescharacteristic
292C2×C42C2411yescharacteristic
293#107—28no—
294#106—28no—
295C4⋊C8—24no—
296C4⋊C8—24no—
297C4⋊C8—48no—
298#103—28no—
299#103—28no—
300#103—48no—
301#103—48no—
302#102—28no—
303#102—28no—
304C22⋊C8—24no—
305C22⋊C8—24no—
306#97—24no—
307C4×C8—24no—
308#95—22no—
309#95—22no—
310#95—24no—
311#95—24no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95C2×D414yes—
316#95C2×D414yes—
317C24—22no—
318C24—24no—
319C24—44no—
320C24—82no—
321C24—82no—
322C24C22×D411yescharacteristic
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—28no—
328C2×Q8—28no—
329C2×Q8—44no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—44no—
333C2×D4—24no—
334C2×D4—22no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—48no—
338C2×D4—44no—
339C2×D4—48no—
340C2×D4C22×D414yes—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—28no—
346C22×C4—28no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—42no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—816no—
356C22×C4—82no—
357C22×C4—82no—
358C22×C4—82no—
359C22×C4#12114yes—
360C22×C4#12114yes—
361C22×C4C22×D414yes—
362C22×C4C22×D414yes—
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D412yes—
368C22×C4C22×D411yescharacteristic
369C22×C4C2×#4112yes—
370C22×C4C2511yescharacteristic
371SD16—416no—
372SD16—416no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—28no—
376C2×C8—28no—
377C4⋊C4—24no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—44no—
386C4⋊C4—416no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—84no—
398C4⋊C4—84no—
399C4⋊C4C22×D414yes—
400C22⋊C4—28no—
401C22⋊C4—28no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—84no—
407C22⋊C4—84no—
408C22⋊C4—84no—
409C22⋊C4—84no—
410C42—24no—
411C42—24no—
412C42—24no—
413C42—24no—
414C42—42no—
415C42—42no—
416C42—44no—
417C42—44no—
418C42—44no—
419C42—44no—
420C42—82no—
421C42#12114yes—
422C42#12114yes—
423C42C22×D412yes—
424C42C22×D412yes—
425C42C2×#4112yes—
426C42C2×#4112yes—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—24no—
431C23—48no—
432C23—44no—
433C23—44no—
434C23—42no—
435C23—42no—
436C23—84no—
437C23—84no—
438C23—816no—
439C23C2×#12112yes—
440C23C23×D411yescharacteristic
441Q8—44no—
442Q8—44no—
443Q8—44no—
444Q8—44no—
445Q8—84no—
446Q8—84no—
447D4—24no—
448D4—24no—
449D4—48no—
450D4—44no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—28no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—416no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—88no—
481C2×C4—84no—
482C2×C4C2×#12112yes—
483C2×C4C2×#12112yes—
484C2×C4C2×#12112yes—
485C2×C4C2×#12112yes—
486C2×C4C2×#12112yes—
487C2×C4#57714yes—
488C2×C4C23×D411yescharacteristic
489C2×C4C22×#4111yescharacteristic
490C2×C4#58514yes—
491C8—44no—
492C8—44no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—44no—
497C22—44no—
498C22—44no—
499C22—44no—
500C22—816no—
501C22—84no—
502C22—84no—
503C22C22×#12111yescharacteristic
504C22#344912yes—
505C22C2×#57712yes—
506C22#358612yes—
507C4—22no—
508C4—22no—
509C4—22no—
510C4—22no—
511C4—44no—
512C4—44no—
513C4—44no—
514C4—82no—
515C4—82no—
516C4—84no—
517C4—82no—
518C4—82no—
519C4#304214yes—
520C4#344912yes—
521C4#345512yes—
522C2—42no—
523C2—84no—
524C2#3463311yescharacteristic
525C2C2×#304211yescharacteristic
526C2#6042711yescharacteristic
527C2#6099114yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #35322C2semidirect
5. #29865C2semidirect
6. #29829C2semidirect
7. #29828C2semidirect
8. #29798C2semidirect
9. #29764C2semidirect
11. #23689C2semidirect
12. #21392C2semidirect
13. #21389C2semidirect
14. #20815C2semidirect
32. #2868C22semidirect
33. #2862C22semidirect
34. #2862C22semidirect
35. #2862C22semidirect
41. #2327C22semidirect
53. #2085C22semidirect
55. #2068C22semidirect
56. #2066C22semidirect
59. #2043C22semidirect
61. #1988C22semidirect
62. #1987C22semidirect
63. #1981C22semidirect
64. #1981C22semidirect
67. #1842C22semidirect
68. #1831C22semidirect
70. #1768C22semidirect
71. #1763C22semidirect
72. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31102
C2×#31012
#353222
#340582
#298652
#298292
#298282
#297982
#297642
#297232
#236892
#213922
#213892
#208152