Finite Group Databasebeta

← #475790

Subgroups — #475790

Order 512.

1284 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2077C212yesmaximal, complemented
2C2×#1955C211yesmaximal, characteristic
3C2×#1944C211yesmaximal, characteristic, complemented
4#15129C212yesmaximal
5#11228C212yesmaximal, complemented
6#11096C212yesmaximal, complemented
7#11096C212yesmaximal, complemented
8#11095C212yesmaximal, complemented
9#10863C211yesmaximal, characteristic, complemented
10C22×C4×D4C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C22×#107C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C22×#97C2211yescharacteristic
19C2×#394—22no—
20C2×#389C2212yes—
21C2×#380C2212yes—
22C2×#380C2212yes—
23C2×#380C2212yes—
24C2×#380C2212yes—
25#2077—28no—
26#1955—24no—
27#1944—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34#1441—22no—
35#1440—22no—
36#1440—22no—
37C4⋊#95C2212yes—
38C23×D4D411yescharacteristic
39C24×C4C2312yes—
40C23×C8C2312yes—
41C23×C8C2311yescharacteristic
42C23×C8C2311yescharacteristic
43C2×C4×D4—22no—
44C2×C4×D4—22no—
45C2×C4×D4—41no—
46C2×C4×D4—41no—
47C2×C4×D4—41no—
48C2×C4×D4—41no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4D412yes—
60C22×C42C2311yescharacteristic
61C2×#107—22no—
62C2×#107—22no—
63C2×#107—41no—
64C2×#107—41no—
65C2×#107—41no—
66C2×#107—41no—
67C2×#106—22no—
68C2×#106—22no—
69C2×#106—22no—
70C2×#106—41no—
71C2×#106—41no—
72C2×#106D414yescomplemented
73C2×C4⋊C8—22no—
74C2×C4⋊C8—41no—
75C2×C4⋊C8—41no—
76C2×C22⋊C8—24no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8—24no—
79C2×C22⋊C8—24no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—24no—
82C2×#97—41no—
83C2×#97—41no—
84C2×#97D414yes—
85#394—48no—
86#389—24no—
87#389—24no—
88#380—24no—
89#380—24no—
90#380—42no—
91#380—42no—
92#380—42no—
93#380—42no—
94#380—42no—
95#380—42no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103#340—24no—
104#340—24no—
105#340—24no—
106#340—24no—
107#340—42no—
108#340—42no—
109C25C2×D412yes—
110C22×D4—24no—
111C22×D4—22no—
112C22×D4—41no—
113C22×D4—41no—
114C22×D4—42no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2411yescharacteristic
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8C2×D412yes—
153C22×C8C2×D412yes—
154C22×C8C2×D412yes—
155C22×C8#4112yes—
156C22×C8#4112yes—
157C22×C8#4112yes—
158C4×D4—44no—
159C4×D4—44no—
160C4×D4—44no—
161C4×D4—44no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
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—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4C2×D414yes—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—82no—
206C2×C22⋊C4—82no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42C2×D412yes—
212C2×C42#4112yes—
213#107—44no—
214#107—44no—
215#106—28no—
216#106—44no—
217C4⋊C8—44no—
218C22⋊C8—48no—
219C22⋊C8—48no—
220#97—44no—
221#95—44no—
222#95—44no—
223#95—44no—
224#95—44no—
225#95—44no—
226#95—44no—
227#95—82no—
228C24—22no—
229C24—22no—
230C24—22no—
231C24—22no—
232C24—22no—
233C24—22no—
234C24—22no—
235C24—22no—
236C24—22no—
237C24—22no—
238C24—22no—
239C24—22no—
240C24—22no—
241C24—22no—
242C24AGL(1,ℤ/8)14yes—
243C24C22×D411yescharacteristic
244C2×D4—48no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—48no—
248C2×D4—44no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
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—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4#12112yes—
321C22×C4#12112yes—
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C2×#4111yescharacteristic
325C22×C4C2×#4111yescharacteristic
326C22×C4C2×#4111yescharacteristic
327C22×C4C2×#4111yescharacteristic
328C22×C4H(5,𝔽2)11yescharacteristic
329C22×C4H(5,𝔽2)11yescharacteristic
330C22×C4H(5,𝔽2)11yescharacteristic
331C22×C4H(5,𝔽2)11yescharacteristic
332C22×C4H(5,𝔽2)11yescharacteristic
333C22×C42−1+411yescharacteristic
334C22×C42−1+411yescharacteristic
335C22×C42−1+411yescharacteristic
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8#12114yes—
356C2×C8#12414yes—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—44no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—88no—
375C22⋊C4—48no—
376C22⋊C4—88no—
377C22⋊C4—88no—
378C22⋊C4—88no—
379C22⋊C4—88no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—22no—
384C23—24no—
385C23—24no—
386C23—24no—
387C23—24no—
388C23—24no—
389C23—24no—
390C23—24no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23C2×#12111yescharacteristic
420C23#53411yescharacteristic
421C23#53411yescharacteristic
422C23D4211yescharacteristic
423C23#54711yescharacteristic
424C23#54711yescharacteristic
425C23#54711yescharacteristic
426C23C2×AGL(1,ℤ/8)12yes—
427C23#57512yes—
428C23#57712yes—
429C23#57812yes—
430D4—416no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4C2×#12111yescharacteristic
485C2×C4C2×#12411yescharacteristic
486C2×C4#53411yescharacteristic
487C2×C4#53511yescharacteristic
488C2×C4#53611yescharacteristic
489C2×C4#53711yescharacteristic
490C2×C4#53711yescharacteristic
491C2×C4#53711yescharacteristic
492C2×C4#53811yescharacteristic
493C2×C4#53811yescharacteristic
494C2×C4#53811yescharacteristic
495C2×C4#53811yescharacteristic
496C2×C4#53911yescharacteristic
497C2×C4#53911yescharacteristic
498C2×C4#53911yescharacteristic
499C2×C4#53911yescharacteristic
500C2×C4#54711yescharacteristic
501C2×C4#55011yescharacteristic
502C2×C4#55011yescharacteristic
503C2×C4#55511yescharacteristic
504C2×C4#55611yescharacteristic
505C2×C4#55611yescharacteristic
506C2×C4#55911yescharacteristic
507C2×C4#55911yescharacteristic
508C2×C4#56211yescharacteristic
509C2×C4#56211yescharacteristic
510C2×C4#56311yescharacteristic
511C2×C4#56311yescharacteristic
512C8—24no—
513C8—24no—
514C8—82no—
515C22—44no—
516C22—44no—
517C22—44no—
518C22—44no—
519C22—44no—
520C22—44no—
521C22—44no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22#253711yescharacteristic
531C22#254711yescharacteristic
532C22#261711yescharacteristic
533C22#268211yescharacteristic
534C22#268211yescharacteristic
535C22#272411yescharacteristic
536C22#272411yescharacteristic
537C22#305512yes—
538C22#306312yes—
539C22#319211yescharacteristic
540C22#319412yes—
541C22#319711yescharacteristic
542C22#320112yes—
543C22#320211yescharacteristic
544C22#320411yescharacteristic
545C22#326512yes—
546C22#326912yes—
547C22#334312yes—
548C22#334312yes—
549C22#334512yes—
550C22#334812yes—
551C22#334912yes—
552C22#335012yes—
553C4—42no—
554C4—42no—
555C4—84no—
556C4—82no—
557C4—162no—
558C4—162no—
559C4—162no—
560C4#254811yescharacteristic
561C4#261811yescharacteristic
562C4#263911yescharacteristic
563C4#272711yescharacteristic
564C4#272711yescharacteristic
565C4#280811yescharacteristic
566C4#280811yescharacteristic
567C4#282011yescharacteristic
568C2—84no—
569C2#1706911yescharacteristic
570C2#2823612yes—
571C2#2871412yes—
572C2#2974211yescharacteristic
573C2#2974611yescharacteristic
574C2#3119912yes—
575C2#3120411yescharacteristic
576C2#3120511yescharacteristic
577C2#3246612yes—
578C2#3246911yescharacteristic
579C2#3247111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2077C2semidirect
3. C2×#1944C2semidirect
5. #11228C2semidirect
6. #11096C2semidirect
7. #11096C2semidirect
8. #11095C2semidirect
9. #10863C2semidirect
72. C2×#106D4semidirect

Maximal subgroups

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

subgroupindex
C2×#20772
C2×#19552
C2×#19442
#151292
#112282
#110962
#110952
#108632