Finite Group Databasebeta

← #403529

Subgroups — #403529

Order 512.

1430 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C8×Q8C211yesmaximal, characteristic, complemented
2C2×#2067C212yesmaximal
3C2×#1934C211yesmaximal, characteristic
4#15158C211yesmaximal, characteristic, complemented
5#15157C211yesmaximal, characteristic
6#11232C212yesmaximal, complemented
7#11227C211yesmaximal, characteristic, complemented
8#11076C211yesmaximal, characteristic
9#10925C212yesmaximal
10#10880C212yesmaximal, complemented
11#10862C211yesmaximal, characteristic, complemented
12C22×C4×Q8C2211yescharacteristic
13C2×C8×Q8—28no—
14C2×C8×Q8—24no—
15C22×#107C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×C4⋊C8C2212yes—
18C22×#102C2211yescharacteristic
19C22×#102C2212yes—
20C22×#102C2211yescharacteristic
21C22×C4×C8C2212yes—
22C22×C4×C8C2211yescharacteristic
23C2×#395—21no—
24C2×#393—21no—
25C2×#390C2212yes—
26C2×#390C2211yescharacteristic
27C2×#389C2212yes—
28C2×#389C2211yescharacteristic
29#2067—28no—
30#1934—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37#1441—22no—
38#1441—21no—
39#1439—22no—
40#1439—21no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2212yes—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C23×Q8D411yescharacteristic
46C23×D4C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C4×Q8—24no—
51C2×C4×Q8—22no—
52C2×C4×Q8—24no—
53C2×C4×Q8—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4D412yes—
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—21no—
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69C8×Q8—232no—
70C2×#107—24no—
71C2×#107—24no—
72C2×#107—42no—
73C2×C4⋊C8—24no—
74C2×C4⋊C8—24no—
75C2×C4⋊C8—24no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8D418yes—
78C2×#102—22no—
79C2×#102—22no—
80C2×#102—24no—
81C2×#102—24no—
82C2×#102—22no—
83C2×#102—24no—
84C2×#102—42no—
85C2×#102Q814yescomplemented
86C2×C4×C8—24no—
87C2×C4×C8—24no—
88C2×C4×C8—24no—
89C2×C4×C8—24no—
90C2×C4×C8D414yes—
91#395—44no—
92#393—44no—
93#390—24no—
94#390—22no—
95#390—24no—
96#390—22no—
97#389—24no—
98#389—22no—
99#389—24no—
100#389—22no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95—22no—
111#340—24no—
112#340—22no—
113#340—22no—
114#340—24no—
115#340—24no—
116#340—42no—
117C25—21no—
118C22×Q8—24no—
119C22×Q8—22no—
120C22×Q8—24no—
121C22×Q8—22no—
122C22×Q8D814yes—
123C22×D4—21no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—21no—
130C22×D4—22no—
131C22×D4—21no—
132C22×D4C2×Q812yes—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4C2411yescharacteristic
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8C2×D414yes—
155C22×C8C2×D412yes—
156C22×C8C2×D414yes—
157C22×C8C2×D412yes—
158C22×C8C2×Q812yes—
159C22×C8#4112yes—
160C4×Q8—28no—
161C4×Q8—216no—
162C4×Q8—28no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4C2×Q812yes—
199C2×C4⋊C4#4114yes—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—82no—
209C2×C22⋊C4—82no—
210C2×C22⋊C4—82no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42C2×D414yes—
218C2×C42C2×D412yes—
219C2×C42#4114yes—
220C2×C42#4112yes—
221#107—48no—
222C4⋊C8—28no—
223C4⋊C8—216no—
224#102—28no—
225#102—48no—
226#102—44no—
227C4×C8—216no—
228C4×C8—28no—
229#95—44no—
230#95—42no—
231#95—44no—
232#95—42no—
233#95—42no—
234#95—44no—
235#95—81no—
236#95—81no—
237#95—82no—
238#95—82no—
239C24—22no—
240C24—42no—
241C24—41no—
242C24—42no—
243C24—41no—
244C24—41no—
245C24—41no—
246C24—42no—
247C24—41no—
248C24—42no—
249C24—41no—
250C24C22×D411yescharacteristic
251C2×Q8—28no—
252C2×Q8—28no—
253C2×Q8—216no—
254C2×Q8—28no—
255C2×Q8—216no—
256C2×D4—22no—
257C2×D4—24no—
258C2×D4—24no—
259C2×D4—22no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—41no—
274C22×C4—22no—
275C22×C4—22no—
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—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4#12112yes—
322C22×C4#12114yes—
323C22×C4#12114yes—
324C22×C4#12112yes—
325C22×C4C2×D814yes—
326C22×C4C2×D812yes—
327C22×C4#13514yes—
328C22×C4#13512yes—
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D412yes—
331C22×C4C22×Q811yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4112yes—
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4112yes—
336C22×C4H(5,𝔽2)11yescharacteristic
337C22×C42−1+412yes—
338C22×C42−1+411yescharacteristic
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—24no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8#12014yes—
355C2×C8C22⋊Q814yes—
356C4⋊C4—216no—
357C4⋊C4—28no—
358C4⋊C4—22no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—22no—
362C4⋊C4—41no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—84no—
377C4⋊C4—88no—
378C22⋊C4—84no—
379C22⋊C4—88no—
380C22⋊C4—88no—
381C22⋊C4—84no—
382C42—24no—
383C42—22no—
384C42—22no—
385C42—24no—
386C42—24no—
387C42—24no—
388C42—24no—
389C23—42no—
390C23—42no—
391C23—44no—
392C23—44no—
393C23—42no—
394C23—82no—
395C23—81no—
396C23—82no—
397C23—82no—
398C23—82no—
399C23—81no—
400C23—81no—
401C23—82no—
402C23—82no—
403C23—81no—
404C23—82no—
405C23—81no—
406C23—82no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23—82no—
411C23—82no—
412C23C2×#12112yes—
413C23C2×#12111yescharacteristic
414C23#54812yes—
415C23D4×Q811yescharacteristic
416C23#55011yescharacteristic
417C23C22×D811yescharacteristic
418C23C2×#13512yes—
419C23C2×#13511yescharacteristic
420C23#57611yescharacteristic
421C23#57712yes—
422C23#57811yescharacteristic
423Q8—232no—
424D4—42no—
425D4—42no—
426D4—44no—
427D4—81no—
428D4—81no—
429D4—82no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—81no—
459C2×C4—82no—
460C2×C4—81no—
461C2×C4—82no—
462C2×C4—81no—
463C2×C4—82no—
464C2×C4—81no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—81no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—82no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—82no—
483C2×C4#46614yes—
484C2×C4#46618yes—
485C2×C4C2×#12011yescharacteristic
486C2×C4C2×C22⋊Q811yescharacteristic
487C2×C4#52511yescharacteristic
488C2×C4#52512yes—
489C2×C4#52912yes—
490C2×C4#53211yescharacteristic
491C2×C4#53312yes—
492C2×C4#53611yescharacteristic
493C2×C4#53711yescharacteristic
494C2×C4#54112yes—
495C2×C4#54212yes—
496C2×C4#54612yes—
497C2×C4#54712yes—
498C2×C4#54812yes—
499C2×C4D4×Q812yes—
500C2×C4#55411yescharacteristic
501C2×C4#55612yes—
502C2×C4#55611yescharacteristic
503C8—24no—
504C8—24no—
505C8—44no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22—84no—
510C22—84no—
511C22—161no—
512C22—161no—
513C22—162no—
514C22—162no—
515C22—161no—
516C22—161no—
517C22#242612yes—
518C22#242912yes—
519C22#253811yescharacteristic
520C22#254511yescharacteristic
521C22#284711yescharacteristic
522C22C2×#46611yescharacteristic
523C22C2×#46612yes—
524C22#305011yescharacteristic
525C22#305012yes—
526C22#305711yescharacteristic
527C22#306111yescharacteristic
528C22#306212yes—
529C22#306312yes—
530C22#335812yes—
531C22#336112yes—
532C22#336412yes—
533C22#336512yes—
534C22Q8×D811yescharacteristic
535C22#338011yescharacteristic
536C22#338112yes—
537C22#339111yescharacteristic
538C22#339311yescharacteristic
539C22#339412yes—
540C4—28no—
541C4—24no—
542C4—42no—
543C4—44no—
544C4—82no—
545C4—84no—
546C4—161no—
547C4—162no—
548C4—161no—
549C4#239611yescharacteristic
550C4#240911yescharacteristic
551C4#243011yescharacteristic
552C4#246512yes—
553C4#246912yes—
554C4#263711yescharacteristic
555C2—162no—
556C2#1487711yescharacteristic
557C2#2529212yes—
558C2#2529712yes—
559C2#2546912yes—
560C2#2547312yes—
561C2#2826311yescharacteristic
562C2#2826611yescharacteristic
563C2#2863711yescharacteristic
564C2#2864111yescharacteristic
565C2#3293011yescharacteristic
566C2#3293811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×C8×Q8C2semidirect
4. #15158C2semidirect
6. #11232C2semidirect
7. #11227C2semidirect
10. #10880C2semidirect
11. #10862C2semidirect
85. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C22×C8×Q82
C2×#20672
C2×#19342
#151582
#151572
#112322
#112272
#110762
#109252
#108802
#108622