Finite Group Databasebeta

← #188479

Subgroups — #188479

Order 512.

1488 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#525C211yesmaximal, characteristic
2#20626C211yesmaximal, characteristic, complemented
3C2×#1513C212yesmaximal
4#14682C211yesmaximal, characteristic, complemented
5#8576C214yesmaximal, complemented
6#8575C214yesmaximal, complemented
7#8016C212yesmaximal, complemented
8C2×#525C2211yescharacteristic
9C2×C4×#41C411yescharacteristic, complemented
10C23×C42C2211yescharacteristic
11C2×#423—21no—
12C2×#407C2212yes—
13C4×C22⋊Q8C412yescomplemented
14C4×C22⋊Q8C2212yes—
15C4×#123—22no—
16C4×#121C412yescomplemented
17C4×#121C2212yes—
18C4×#120—22no—
19#2297C411yescharacteristic, complemented
20#2296C411yescharacteristic, complemented
21#2290—21no—
22#2290—21no—
23C2×#382—21no—
24#2280C2211yescharacteristic
25C2×#380C2212yes—
26C42×D4—21no—
27C42×D4—21no—
28C4×#117C411yescharacteristic, complemented
29#1797—22no—
30#1797C2214yes—
31#1728—21no—
32#1513—28no—
33C2×#328C2214yes—
34C2×#327C2214yes—
35C4⋊#95—21no—
36#1434C2214yes—
37C22⋊#95—22no—
38#1294C2218yes—
39C22×#41C2×C411yescharacteristic
40C24×C4D412yes—
41C24×C4C2311yescharacteristic
42C22×M16—22no—
43#525—22no—
44#525—22no—
45#525D414yes—
46#525Q814yes—
47C2×#123—21no—
48C2×C22⋊Q8C2×C411yescharacteristic
49C2×C22⋊Q8C2311yescharacteristic
50C2×#121C2×C411yescharacteristic
51C2×#121C2311yescharacteristic
52C2×#120—21no—
53C4×#41—44no—
54C2×C4×Q8C2×C412yes—
55C2×C4×D4—21no—
56C2×C4×D4—21no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4C2×C412yes—
61C2×#117—21no—
62C2×#117C2×C411yescharacteristic
63C2×#117C2×C411yescharacteristic
64C2×#117C2×C411yescharacteristic
65C22×C4⋊C4—22no—
66C22×C22⋊C4—22no—
67C22×C42—21no—
68C22×C42—44no—
69C22×C42C2×C411yescharacteristic
70C22×C42C2×C411yescharacteristic
71C22×C42C2×C412yes—
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D412yes—
76C22×C42D411yescharacteristic
77C22×C42C2312yes—
78#423—44no—
79C2×C4⋊C8—24no—
80#407—24no—
81#407—24no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8C2314yes—
84#391C2×C412yes—
85#390C2×C412yes—
86#389C2×C412yes—
87#389C2×C412yes—
88#388—22no—
89#388—22no—
90#387—22no—
91#387—22no—
92#386—22no—
93#386—22no—
94#386—22no—
95#385—22no—
96#385—22no—
97#385—22no—
98#382—44no—
99#381—21no—
100#380—24no—
101#380—24no—
102#380C2×C412yes—
103#380C2312yes—
104#379—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4C2×C412yes—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—21no—
114C4×C22⋊C4—21no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—21no—
117C4×C22⋊C4—21no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—22no—
120#376—21no—
121C2×#95—24no—
122C2×#95—24no—
123C2×#95C2314yes—
124C43—21no—
125#336—24no—
126#328—28no—
127#327—28no—
128C25C2×D411yescharacteristic
129C2×#41—22no—
130C2×#41—42no—
131C2×#41—42no—
132C2×#41C4⋊C414yes—
133C22×Q8C22×C411yescharacteristic
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4C22×C411yescharacteristic
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—24no—
141C23×C4—44no—
142C23×C4—44no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—44no—
146C23×C4C22⋊C412yes—
147C23×C4C22⋊C412yes—
148C23×C4C22⋊C412yes—
149C23×C4C22⋊C412yes—
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C2×D412yes—
153C23×C4C2×D412yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4C2×Q812yes—
159C23×C4#4112yes—
160C23×C4#4112yes—
161C23×C4C2411yescharacteristic
162C2×M16—44no—
163C2×M16—44no—
164C2×M16—44no—
165C22×C8—24no—
166#123—24no—
167#123—24no—
168C22⋊Q8—22no—
169C22⋊Q8—21no—
170C22⋊Q8—21no—
171C22⋊Q8C4⋊C414yes—
172C22⋊Q8C2×D412yes—
173C22⋊Q8C2×Q812yes—
174#121—22no—
175#121—21no—
176#121—21no—
177#121C4⋊C414yes—
178#121C2×D412yes—
179#121C2×Q812yes—
180#120—24no—
181#120—24no—
182C4×Q8—44no—
183C4×D4—24no—
184C4×D4—24no—
185C4×D4—24no—
186C4×D4—24no—
187C4×D4—44no—
188C4×D4—48no—
189C4×D4—48no—
190C4×D4—48no—
191#117—22no—
192#117—44no—
193#117—42no—
194#117—42no—
195#117C4⋊C414yes—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C412yes—
207C2×C4⋊C4C22×C412yes—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—48no—
221C2×C22⋊C4—44no—
222C2×C42—21no—
223C2×C42—24no—
224C2×C42—24no—
225C2×C42—24no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—42no—
232C2×C42—44no—
233C2×C42—44no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42—44no—
237C2×C42—44no—
238C2×C42C22⋊C411yescharacteristic
239C2×C42C22⋊C411yescharacteristic
240C2×C42C22⋊C412yes—
241C2×C42C22⋊C411yescharacteristic
242C2×C42C22⋊C412yes—
243C2×C42C22⋊C411yescharacteristic
244C2×C42C22×C412yes—
245C2×C42C22×C412yes—
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248C4⋊C8—48no—
249C22⋊C8—24no—
250C22⋊C8—24no—
251C22⋊C8—44no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—24no—
261#95—24no—
262#95—48no—
263#95—48no—
264C24—24no—
265C24—21no—
266C24—41no—
267C24—41no—
268C24—42no—
269C24C2×C22⋊C411yescharacteristic
270C24C2×C22⋊C411yescharacteristic
271C24#12112yes—
272C24C22⋊Q812yes—
273C24C22×D411yescharacteristic
274#41—48no—
275#41—48no—
276C2×Q8—21no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8C2×C4⋊C412yes—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—21no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—42no—
288C2×D4—44no—
289C2×D4—42no—
290C2×D4—44no—
291C2×D4C2×C4⋊C412yes—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—432no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—48no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—48no—
343C22×C4C2×C22⋊C412yes—
344C22×C4C2×C22⋊C412yes—
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×C22⋊C412yes—
347C22×C4C2×C22⋊C412yes—
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C4⋊C412yes—
350C22×C4C2×C4⋊C412yes—
351C22×C4#11712yes—
352C22×C4#11712yes—
353C22×C4#12112yes—
354C22×C4#12112yes—
355C22×C4#12112yes—
356C22×C4C22⋊Q812yes—
357C22×C4C22⋊Q812yes—
358C22×C4C22⋊Q812yes—
359C22×C4C23×C411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×Q811yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4111yescharacteristic
365M16—88no—
366C2×C8—44no—
367C2×C8—44no—
368C2×C8—44no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—88no—
380C4⋊C4C2×C4⋊C412yes—
381C4⋊C4C2×C4⋊C412yes—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—24no—
385C22⋊C4—24no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—24no—
389C22⋊C4—44no—
390C22⋊C4—44no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—88no—
397C42—24no—
398C42—24no—
399C42—22no—
400C42—24no—
401C42—22no—
402C42—22no—
403C42—24no—
404C42—42no—
405C42—42no—
406C42—42no—
407C42—44no—
408C42—44no—
409C42—42no—
410C42—44no—
411C42—42no—
412C42#9914yes—
413C23—21no—
414C23—21no—
415C23—22no—
416C23—24no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—24no—
426C23—22no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—48no—
432C23—48no—
433C23—41no—
434C23—41no—
435C23—42no—
436C23—42no—
437C23#38012yes—
438C23#38012yes—
439C23C22×C22⋊C411yescharacteristic
440C23C2×#12111yescharacteristic
441C23C2×C22⋊Q811yescharacteristic
442C23#57514yes—
443Q8—42no—
444Q8—42no—
445D4—42no—
446D4—42no—
447D4—48no—
448D4—48no—
449D4—44no—
450D4—44no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—48no—
485C2×C4—48no—
486C2×C4—48no—
487C2×C4—48no—
488C2×C4—48no—
489C2×C4—48no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—48no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—88no—
502C2×C4—84no—
503C2×C4—88no—
504C2×C4—84no—
505C2×C4—88no—
506C2×C4—84no—
507C2×C4#38012yes—
508C2×C4#38012yes—
509C2×C4C2×#9914yes—
510C2×C4C2×#9912yes—
511C2×C4#41012yes—
512C2×C4#42114yes—
513C2×C4C22×C4⋊C411yescharacteristic
514C2×C4C2×#11711yescharacteristic
515C2×C4C2×#12111yescharacteristic
516C2×C4C2×C22⋊Q811yescharacteristic
517C8—84no—
518C22—24no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—24no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—48no—
531C22—48no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—82no—
540C22—82no—
541C22C2×#38011yescharacteristic
542C22C22×#9911yescharacteristic
543C22C2×#41011yescharacteristic
544C22#289612yes—
545C22C2×#42112yes—
546C22#304012yes—
547C22#308012yes—
548C4—24no—
549C4—24no—
550C4—21no—
551C4—21no—
552C4—21no—
553C4—21no—
554C4—22no—
555C4—42no—
556C4—42no—
557C4—42no—
558C4—42no—
559C4—42no—
560C4—82no—
561C4—82no—
562C4—164no—
563C4#179914yes—
564C2—22no—
565C2—22no—
566C2—42no—
567C2—42no—
568C2—82no—
569C2#1755912yes—
570C2#1759412yes—
571C2C2×#179911yescharacteristic
572C2#2064911yescharacteristic
573C2#2065411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20626C2semidirect
4. #14682C2semidirect
5. #8576C2semidirect
6. #8575C2semidirect
7. #8016C2semidirect
9. C2×C4×#41C4semidirect
13. C4×C22⋊Q8C4semidirect
16. C4×#121C4semidirect
19. #2297C4semidirect
20. #2296C4semidirect
28. C4×#117C4semidirect

Maximal subgroups

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

subgroupindex
C4×#5252
#206262
C2×#15132
#146822
#85762
#85752
#80162