Finite Group Databasebeta

← #455019

Subgroups — #455019

Order 512.

1388 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#462C211yesmaximal, characteristic
2C2×#2056C212yesmaximal
3C2×#1863C211yesmaximal, characteristic
4#15188C211yesmaximal, characteristic
5#13931C211yesmaximal, characteristic
6#11322C211yesmaximal, characteristic
7#11228C211yesmaximal, characteristic
8#11124C212yesmaximal
9#11077C212yesmaximal
10#11024C212yesmaximal
11#11010C211yesmaximal, characteristic
12C23×Q16C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C2×#462—28no—
16C2×#462—24no—
17C22×#107C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×C4⋊C8C2212yes—
20C22×#103C2211yescharacteristic
21C22×#103C2212yes—
22C22×#103C2211yescharacteristic
23C22×#103C2212yes—
24C22×#103C2212yes—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#389C2211yescharacteristic
29#2056—28no—
30#1863—24no—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36#1445—22no—
37#1442—21no—
38#1442—21no—
39#1441—22no—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2212yes—
42#1434—21no—
43#1434—21no—
44C4×#95C2211yescharacteristic
45C23×Q8C2312yes—
46C23×Q8C2311yescharacteristic
47C22×Q16—24no—
48C22×Q16—24no—
49C22×Q16—42no—
50C22×Q16—42no—
51C22×Q16—42no—
52C22×Q16—42no—
53C22×Q16D418yes—
54C23×C8C2311yescharacteristic
55C23×C8C2312yes—
56C23×C8C2311yescharacteristic
57C2×C4⋊Q8—24no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—42no—
61C2×C4×Q8—22no—
62C2×C4×Q8—24no—
63C2×C4×Q8—42no—
64C2×C4×Q8D414yes—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C42—21no—
74C22×C42C2312yes—
75C22×C42C2311yescharacteristic
76#462—232no—
77C2×#107—22no—
78C2×#107—42no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—24no—
82C2×C4⋊C8—42no—
83C2×C4⋊C8—42no—
84C2×#103—22no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—22no—
88C2×#103—24no—
89C2×#103—22no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—22no—
93C2×#103—24no—
94C2×#103—42no—
95C2×#103—42no—
96C2×#103—42no—
97C2×#103—42no—
98#391—24no—
99#391—24no—
100#391—24no—
101#391—42no—
102#391—42no—
103#391—42no—
104#389—22no—
105#389—22no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—21no—
117C2×#95D411yescharacteristic
118C2×#95D412yes—
119C2×#95D411yescharacteristic
120#340—24no—
121#340—22no—
122#340—42no—
123#336—22no—
124#336—42no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8C2×D414yes—
147C22×Q8C2×D414yes—
148C22×Q8C2×D412yes—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2411yescharacteristic
160C2×Q16—216no—
161C2×Q16—48no—
162C2×Q16—48no—
163C2×Q16—48no—
164C2×Q16—48no—
165C2×Q16—48no—
166C2×Q16—48no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—21no—
178C22×C8—22no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—21no—
182C22×C8C2×D414yes—
183C22×C8#4112yes—
184C4⋊Q8—28no—
185C4⋊Q8—48no—
186C4⋊Q8—44no—
187C4×Q8—28no—
188C4×Q8—44no—
189C4×Q8—48no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4#4112yes—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—24no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—44no—
228C2×C42C2×D412yes—
229C2×C42#4112yes—
230#107—44no—
231C4⋊C8—28no—
232C4⋊C8—48no—
233#103—28no—
234#103—216no—
235#103—48no—
236#103—48no—
237#103—44no—
238#95—22no—
239#95—42no—
240#95—44no—
241#95—44no—
242#95—44no—
243#95—44no—
244#95—44no—
245#95—44no—
246#95—44no—
247#95—42no—
248#95—42no—
249#95—82no—
250#95—81no—
251#95Q1614yescomplemented
252C24C22×D411yescharacteristic
253C2×Q8—22no—
254C2×Q8—24no—
255C2×Q8—24no—
256C2×Q8—22no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8#12014yes—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4#12112yes—
353C22×C4#12112yes—
354C22×C4C2×Q1614yes—
355C22×C4C2×Q1612yes—
356C22×C4#13712yes—
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D412yes—
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4H(5,𝔽2)12yes—
363C22×C4H(5,𝔽2)11yescharacteristic
364C22×C4H(5,𝔽2)11yescharacteristic
365C22×C4H(5,𝔽2)12yes—
366C22×C42−1+412yes—
367Q16—432no—
368C2×C8—24no—
369C2×C8—22no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—41no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—41no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—41no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—41no—
384C2×C8—41no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—48no—
392C4⋊C4—41no—
393C4⋊C4—42no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—42no—
397C4⋊C4—41no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—84no—
403C42—22no—
404C42—22no—
405C42—24no—
406C42—48no—
407C42—42no—
408C42—42no—
409C42—42no—
410C42—42no—
411C23C2×#12111yescharacteristic
412C23#53712yes—
413C23D4212yes—
414C23#54611yescharacteristic
415C23#55011yescharacteristic
416C23C22×Q1611yescharacteristic
417C23C2×#13711yescharacteristic
418C23#57512yes—
419C23#57711yescharacteristic
420C23#57811yescharacteristic
421C23#57812yes—
422Q8—24no—
423Q8—24no—
424Q8—42no—
425Q8—42no—
426Q8—44no—
427Q8—44no—
428Q8—44no—
429Q8—44no—
430Q8—44no—
431Q8—44no—
432Q8—82no—
433Q8—82no—
434Q8—82no—
435Q8—82no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—81no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4#46214yes—
494C2×C4C2×#12011yescharacteristic
495C2×C4#52511yescharacteristic
496C2×C4#53411yescharacteristic
497C2×C4#53412yes—
498C2×C4#53412yes—
499C2×C4#53511yescharacteristic
500C2×C4#53511yescharacteristic
501C2×C4#53611yescharacteristic
502C2×C4#53812yes—
503C2×C4#54012yes—
504C2×C4#54212yes—
505C2×C4#54612yes—
506C2×C4#54712yes—
507C2×C4#54712yes—
508C2×C4#55311yescharacteristic
509C2×C4#55511yescharacteristic
510C2×C4#56212yes—
511C2×C4#56212yes—
512C8—44no—
513C8—42no—
514C8—81no—
515C22#241011yescharacteristic
516C22#241411yescharacteristic
517C22#262011yescharacteristic
518C22#267112yes—
519C22#268212yes—
520C22C2×#46211yescharacteristic
521C22#304011yescharacteristic
522C22#304411yescharacteristic
523C22#304611yescharacteristic
524C22#324912yes—
525C22#325212yes—
526C22#325812yes—
527C22#326112yes—
528C22#327012yes—
529C22#327112yes—
530C22#327712yes—
531C22D4×Q1612yes—
532C22#329911yescharacteristic
533C22#331512yes—
534C22#332811yescharacteristic
535C22#339511yescharacteristic
536C22#340011yescharacteristic
537C22#340212yes—
538C4—24no—
539C4—42no—
540C4—44no—
541C4—44no—
542C4—82no—
543C4—82no—
544C4—84no—
545C4—82no—
546C4—161no—
547C4—162no—
548C4#240611yescharacteristic
549C4#261311yescharacteristic
550C4#263511yescharacteristic
551C4#268612yes—
552C4#270112yes—
553C4#281611yescharacteristic
554C2#1625911yescharacteristic
555C2#2436011yescharacteristic
556C2#2437511yescharacteristic
557C2#2463211yescharacteristic
558C2#2464511yescharacteristic
559C2#2983011yescharacteristic
560C2#2983711yescharacteristic
561C2#3060012yes—
562C2#3061612yes—
563C2#3121312yes—
564C2#3122612yes—
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
251. #95Q16semidirect

Maximal subgroups

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

subgroupindex
C22×#4622
C2×#20562
C2×#18632
#151882
#139312
#113222
#112282
#111242
#110772
#110242
#110102