Finite Group Databasebeta

← #461437

Subgroups — #461437

Order 512.

1134 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1878C211yesmaximal, characteristic
2C2×#1803C211yesmaximal, characteristic
3#15199C211yesmaximal, characteristic
4#13931C211yesmaximal, characteristic
5#11357C211yesmaximal, characteristic
6#11267C212yesmaximal
7#11255C211yesmaximal, characteristic
8#11190C212yesmaximal
9#11077C211yesmaximal, characteristic
10#11050C212yesmaximal
11#11024C212yesmaximal
12C23×Q16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2211yescharacteristic
18C22×#103C2211yescharacteristic
19C2×#391C2211yescharacteristic
20C2×#391C2212yes—
21C2×#391C2212yes—
22C2×#384C2211yescharacteristic
23C2×#384C2212yes—
24#1878—24no—
25#1803—24no—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34#1445—22no—
35#1444—21no—
36#1444—22no—
37#1444—21no—
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40#1434—21no—
41#1434—21no—
42C4×#95C2211yescharacteristic
43C23×Q8C2311yescharacteristic
44C23×Q8C2311yescharacteristic
45C22×Q16—22no—
46C22×Q16—22no—
47C22×Q16—24no—
48C22×Q16—22no—
49C22×Q16—42no—
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×C4×Q8—22no—
54C2×C4×Q8—24no—
55C2×C4×Q8—42no—
56C2×C4×Q8D414yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C42—21no—
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69C2×C4⋊C8—22no—
70C2×C4⋊C8—42no—
71C2×#103—24no—
72C2×#103—22no—
73C2×#103—24no—
74C2×#103—22no—
75C2×#103—42no—
76C2×#103—42no—
77C2×#103—42no—
78C2×#103—42no—
79C2×#103—42no—
80C2×#103—42no—
81#391—24no—
82#391—22no—
83#391—22no—
84#391—42no—
85#391—42no—
86#391—42no—
87#384—22no—
88#384—22no—
89#384—42no—
90#384—42no—
91C2×#95—21no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95D411yescharacteristic
102C2×#95D412yes—
103C2×#95D411yescharacteristic
104#340—24no—
105#340—22no—
106#340—42no—
107#340—42no—
108#340—42no—
109#336—24no—
110#336—22no—
111#336—42no—
112C22×Q8—21no—
113C22×Q8—22no—
114C22×Q8—21no—
115C22×Q8—21no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—22no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8C2×D412yes—
130C22×Q8#4112yes—
131C22×Q8#4114yes—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2411yescharacteristic
143C2×Q16—28no—
144C2×Q16—48no—
145C2×Q16—48no—
146C2×Q16—44no—
147C2×Q16—44no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8#4112yes—
165C4×Q8—28no—
166C4×Q8—44no—
167C4×Q8—48no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4—81no—
195C2×C4⋊C4—81no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4C2×D412yes—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—24no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—44no—
211C2×C42C2×D412yes—
212C2×C42#4112yes—
213C4⋊C8—44no—
214#103—44no—
215#103—48no—
216#103—44no—
217#103—48no—
218#95—24no—
219#95—22no—
220#95—44no—
221#95—44no—
222#95—44no—
223#95—44no—
224#95—42no—
225#95—42no—
226#95—44no—
227#95—42no—
228#95—82no—
229#95—81no—
230#95Q1614yescomplemented
231C24C22×D411yescharacteristic
232C2×Q8—22no—
233C2×Q8—24no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—24no—
240C2×Q8—24no—
241C2×Q8—24no—
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—24no—
245C2×Q8—22no—
246C2×Q8—41no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8#12314yes—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4#12112yes—
321C22×C4#12112yes—
322C22×C4C2×Q1614yes—
323C22×C4C2×Q1612yes—
324C22×C4#13714yes—
325C22×C4#13712yes—
326C22×C4C22×D411yescharacteristic
327C22×C4C2×#4112yes—
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C22×C4H(5,𝔽2)12yes—
332C22×C4H(5,𝔽2)12yes—
333C22×C4H(5,𝔽2)12yes—
334C22×C42−1+411yescharacteristic
335C22×C42−1+411yescharacteristic
336Q16—416no—
337C2×C8—22no—
338C2×C8—24no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—41no—
344C2×C8—42no—
345C2×C8—41no—
346C2×C8—42no—
347C2×C8—41no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—48no—
357C4⋊C4—42no—
358C4⋊C4—41no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—41no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—84no—
370C4⋊C4—84no—
371C4⋊C4—88no—
372C4⋊C4—88no—
373C42—22no—
374C42—22no—
375C42—24no—
376C42—48no—
377C42—42no—
378C42—42no—
379C42—42no—
380C42—42no—
381C23C2×#12111yescharacteristic
382C23#53512yes—
383C23#54612yes—
384C23#54711yescharacteristic
385C23#54811yescharacteristic
386C23C22×Q1611yescharacteristic
387C23C2×#13712yes—
388C23C2×#13711yescharacteristic
389C23#57611yescharacteristic
390C23#57712yes—
391C23#57711yescharacteristic
392Q8—24no—
393Q8—24no—
394Q8—42no—
395Q8—42no—
396Q8—44no—
397Q8—44no—
398Q8—42no—
399Q8—42no—
400Q8—44no—
401Q8—81no—
402Q8—81no—
403Q8—82no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—82no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—81no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—81no—
458C2×C4—82no—
459C2×C4—81no—
460C2×C4—82no—
461C2×C4#46214yes—
462C2×C4C2×#12311yescharacteristic
463C2×C4#52511yescharacteristic
464C2×C4#53411yescharacteristic
465C2×C4#53511yescharacteristic
466C2×C4#53511yescharacteristic
467C2×C4#53611yescharacteristic
468C2×C4#53812yes—
469C2×C4#53912yes—
470C2×C4#53912yes—
471C2×C4#54012yes—
472C2×C4#54012yes—
473C2×C4#55112yes—
474C2×C4#55112yes—
475C2×C4#55112yes—
476C2×C4#55211yescharacteristic
477C2×C4#55611yescharacteristic
478C2×C4#56212yes—
479C2×C4#56212yes—
480C8—42no—
481C8—81no—
482C8—82no—
483C22#241211yescharacteristic
484C22#241511yescharacteristic
485C22#261911yescharacteristic
486C22#269612yes—
487C22#270612yes—
488C22C2×#46211yescharacteristic
489C22#303911yescharacteristic
490C22#304311yescharacteristic
491C22#304511yescharacteristic
492C22#320912yes—
493C22#321812yes—
494C22#322212yes—
495C22#322412yes—
496C22#329912yes—
497C22#331612yes—
498C22#331612yes—
499C22#333312yes—
500C22#333811yescharacteristic
501C22#334412yes—
502C22#335611yescharacteristic
503C22#336311yescharacteristic
504C22#336912yes—
505C22#337511yescharacteristic
506C4—24no—
507C4—44no—
508C4—44no—
509C4—44no—
510C4—42no—
511C4—82no—
512C4—82no—
513C4—161no—
514C4—162no—
515C4—162no—
516C4#248211yescharacteristic
517C4#262311yescharacteristic
518C4#263511yescharacteristic
519C4#274012yes—
520C4#274312yes—
521C4#281611yescharacteristic
522C2#1639211yescharacteristic
523C2#2453211yescharacteristic
524C2#2454611yescharacteristic
525C2#2468211yescharacteristic
526C2#2469811yescharacteristic
527C2#2979911yescharacteristic
528C2#2980311yescharacteristic
529C2#3164012yes—
530C2#3165212yes—
531C2#3193612yes—
532C2#3195612yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
230. #95Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#18782
C2×#18032
#151992
#139312
#113572
#112672
#112552
#111902
#110772
#110502
#110242