Finite Group Databasebeta

← #188480

Subgroups — #188480

Order 512.

1560 conjugacy classes of subgroups.

The subgroup lattice has 599 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#14686C211yesmaximal, characteristic, complemented
5#8578C214yesmaximal, complemented
6#8577C214yesmaximal, 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×#384—21no—
24#2280C2211yescharacteristic
25C2×#380—22no—
26C2×#380C2212yes—
27C42×D4—21no—
28C42×D4—21no—
29C4×#117C411yescharacteristic, complemented
30#1797—22no—
31#1797C2214yes—
32#1728—21no—
33#1513—28no—
34C2×#328C2214yes—
35C2×#327C2214yes—
36C4⋊#95—21no—
37C4⋊#95C2214yes—
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×C4⋊C4—24no—
67C22×C22⋊C4—22no—
68C22×C42—21no—
69C22×C42—44no—
70C22×C42C2×C411yescharacteristic
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C412yes—
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D412yes—
77C22×C42D411yescharacteristic
78C22×C42C2312yes—
79#423—44no—
80C2×C4⋊C8—24no—
81#407—24no—
82#407—24no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8C2314yes—
85#391C2×C412yes—
86#390C2×C412yes—
87#389C2×C412yes—
88#389C2×C412yes—
89#388—22no—
90#388—22no—
91#387—22no—
92#387—22no—
93#386—22no—
94#386—22no—
95#386—22no—
96#385—22no—
97#385—22no—
98#385—22no—
99#384—44no—
100#381—21no—
101#380—24no—
102#380—24no—
103#380—48no—
104#380C2×C412yes—
105#380C2312yes—
106#379—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4C2×C412yes—
112C4×C4⋊C4C2×C412yes—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—21no—
116C4×C22⋊C4—21no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—21no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—22no—
122#376—21no—
123C2×#95—24no—
124C2×#95C2314yes—
125C43—21no—
126#336—24no—
127#328—28no—
128#327—28no—
129C25C2×D411yescharacteristic
130C2×#41—22no—
131C2×#41—42no—
132C2×#41—42no—
133C2×#41C4⋊C414yescomplemented
134C22×Q8C22×C411yescharacteristic
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4C22×C411yescharacteristic
139C23×C4—21no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—22no—
143C23×C4—24no—
144C23×C4—44no—
145C23×C4—44no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4C22⋊C412yes—
149C23×C4C22⋊C412yes—
150C23×C4C22⋊C412yes—
151C23×C4C22⋊C412yes—
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×Q812yes—
161C23×C4#4112yes—
162C23×C4#4112yes—
163C23×C4C2411yescharacteristic
164C2×M16—44no—
165C2×M16—44no—
166C2×M16—44no—
167C22×C8—24no—
168#123—24no—
169#123—24no—
170C22⋊Q8—22no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173C22⋊Q8C4⋊C414yescomplemented
174C22⋊Q8C2×D412yes—
175C22⋊Q8C2×Q812yes—
176#121—22no—
177#121—21no—
178#121—21no—
179#121C4⋊C414yescomplemented
180#121C2×D412yes—
181#121C2×Q812yes—
182#120—24no—
183#120—24no—
184C4×Q8—44no—
185C4×D4—24no—
186C4×D4—24no—
187C4×D4—24no—
188C4×D4—24no—
189C4×D4—44no—
190C4×D4—48no—
191C4×D4—48no—
192C4×D4—44no—
193C4×D4—44no—
194#117—22no—
195#117—44no—
196#117—42no—
197#117—42no—
198#117C4⋊C414yescomplemented
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—48no—
206C2×C4⋊C4—48no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—48no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C412yes—
213C2×C4⋊C4C22×C412yes—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—48no—
227C2×C22⋊C4—44no—
228C2×C42—21no—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—42no—
238C2×C42—44no—
239C2×C42—44no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—44no—
243C2×C42—44no—
244C2×C42C22⋊C411yescharacteristic
245C2×C42C22⋊C411yescharacteristic
246C2×C42C22⋊C411yescharacteristic
247C2×C42C22⋊C412yes—
248C2×C42C22⋊C412yes—
249C2×C42C22⋊C411yescharacteristic
250C2×C42C22×C412yes—
251C2×C42C22×C412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C4⋊C8—48no—
255C22⋊C8—24no—
256C22⋊C8—24no—
257C22⋊C8—44no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—24no—
267#95—24no—
268#95—48no—
269C24—22no—
270C24—22no—
271C24—21no—
272C24—41no—
273C24—41no—
274C24—42no—
275C24C2×C22⋊C411yescharacteristic
276C24C2×C22⋊C411yescharacteristic
277C24#12112yes—
278C24C22⋊Q812yes—
279C24C22×D411yescharacteristic
280#41—48no—
281#41—48no—
282C2×Q8—21no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8C2×C4⋊C412yes—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—21no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—42no—
294C2×D4—44no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4C2×C4⋊C412yes—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—416no—
340C22×C4—416no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—48no—
349C22×C4—44no—
350C22×C4—48no—
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C412yes—
353C22×C4C2×C22⋊C412yes—
354C22×C4C2×C22⋊C412yes—
355C22×C4C2×C22⋊C412yes—
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C4⋊C412yes—
358C22×C4C2×C4⋊C412yes—
359C22×C4#11712yes—
360C22×C4#11712yes—
361C22×C4#12112yes—
362C22×C4#12112yes—
363C22×C4#12112yes—
364C22×C4C22⋊Q812yes—
365C22×C4C22⋊Q812yes—
366C22×C4C22⋊Q812yes—
367C22×C4AGL(1,ℤ/8)14yes—
368C22×C4#13714yes—
369C22×C4C23×C411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×Q811yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375M16—88no—
376C2×C8—44no—
377C2×C8—44no—
378C2×C8—44no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—24no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—816no—
390C4⋊C4—88no—
391C4⋊C4C2×C4⋊C412yes—
392C4⋊C4C2×C4⋊C412yes—
393C22⋊C4—22no—
394C22⋊C4—22no—
395C22⋊C4—24no—
396C22⋊C4—24no—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—24no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—44no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—88no—
409C42—24no—
410C42—24no—
411C42—22no—
412C42—24no—
413C42—22no—
414C42—22no—
415C42—24no—
416C42—42no—
417C42—42no—
418C42—42no—
419C42—44no—
420C42—44no—
421C42—42no—
422C42—42no—
423C42—44no—
424C42#9914yes—
425C23—21no—
426C23—21no—
427C23—22no—
428C23—24no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—41no—
449C23—41no—
450C23—42no—
451C23—42no—
452C23#38012yes—
453C23#38012yes—
454C23C22×C22⋊C411yescharacteristic
455C23C2×#12111yescharacteristic
456C23C2×C22⋊Q811yescharacteristic
457C23C2×AGL(1,ℤ/8)12yes—
458C23C2×#13712yes—
459Q8—42no—
460Q8—42no—
461D4—42no—
462D4—42no—
463D4—48no—
464D4—48no—
465D4—44no—
466D4—42no—
467D4—42no—
468C2×C4—28no—
469C2×C4—28no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—48no—
504C2×C4—48no—
505C2×C4—48no—
506C2×C4—48no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—48no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—88no—
519C2×C4—84no—
520C2×C4—88no—
521C2×C4—88no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4#38012yes—
525C2×C4#38012yes—
526C2×C4C2×#9914yes—
527C2×C4C2×#9912yes—
528C2×C4#41012yes—
529C2×C4#41714yes—
530C2×C4#41714yes—
531C2×C4#42114yes—
532C2×C4C22×C4⋊C411yescharacteristic
533C2×C4C2×#11711yescharacteristic
534C2×C4C2×#12111yescharacteristic
535C2×C4C2×C22⋊Q811yescharacteristic
536C8—84no—
537C22—21no—
538C22—21no—
539C22—22no—
540C22—22no—
541C22—21no—
542C22—21no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—44no—
552C22—48no—
553C22—44no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—82no—
562C22—82no—
563C22C2×#38011yescharacteristic
564C22C22×#9911yescharacteristic
565C22C2×#41011yescharacteristic
566C22C2×#41712yes—
567C22C2×#42112yes—
568C22#303811yescharacteristic
569C22#303911yescharacteristic
570C22#307912yes—
571C4—21no—
572C4—21no—
573C4—21no—
574C4—21no—
575C4—22no—
576C4—24no—
577C4—24no—
578C4—24no—
579C4—24no—
580C4—42no—
581C4—42no—
582C4—42no—
583C4—82no—
584C4—82no—
585C4—164no—
586C4#179914yes—
587C2—21no—
588C2—22no—
589C2—21no—
590C2—42no—
591C2—42no—
592C2—82no—
593C2#1756312yes—
594C2#1759212yes—
595C2C2×#179911yescharacteristic
596C2#2064811yescharacteristic
597C2#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. #14686C2semidirect
5. #8578C2semidirect
6. #8577C2semidirect
7. #8016C2semidirect
9. C2×C4×#41C4semidirect
13. C4×C22⋊Q8C4semidirect
16. C4×#121C4semidirect
19. #2297C4semidirect
20. #2296C4semidirect
29. C4×#117C4semidirect
133. C2×#41C4⋊C4semidirect
173. C22⋊Q8C4⋊C4semidirect
179. #121C4⋊C4semidirect
198. #117C4⋊C4semidirect

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
#146862
#85782
#85772
#80162