Finite Group Databasebeta

← #216582

Subgroups — #216582

Order 512.

962 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35006C211yesmaximal, characteristic
2#21955C211yesmaximal, characteristic, complemented
3#13950C211yesmaximal, characteristic, complemented
4#9912C211yesmaximal, characteristic, complemented
5#9909C211yesmaximal, characteristic, complemented
6#9715C211yesmaximal, characteristic
7#9712C211yesmaximal, characteristic
8#9118C212yesmaximal, complemented
9#9118C212yesmaximal, complemented
10#8575C212yesmaximal, complemented
11#8573C212yesmaximal, complemented
12C2×#539—21no—
13C2×#432C2212yes—
14#2857C2211yescharacteristic
15#2856C2211yescharacteristic
16#2854C2212yes—
17#2639—21no—
18#2618—21no—
19#2560—22no—
20#2558—22no—
21#2512C412yescomplemented
22#2506C412yescomplemented
23#2452C412yescomplemented
24#2445C412yescomplemented
25#2325—21no—
26C4×#121C2212yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28#1940—22no—
29#1940—22no—
30#1797C2212yes—
31#1797C2211yescharacteristic
32#1797C2211yescharacteristic
33C2×#327C2212yes—
34C2×#327C2212yes—
35C2×#327C2212yes—
36C2×#327C2212yes—
37#1445—21no—
38#1443—21no—
39#1436—21no—
40#1434—21no—
41#1434C2211yescharacteristic
42#1434C2211yescharacteristic
43#1434C2211yescharacteristic
44#1434C2211yescharacteristic
45#1434C2211yescharacteristic
46C4×#95C2211yescharacteristic
47#1294C2212yes—
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51#539—44no—
52#539—44no—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#127C2×C411yescharacteristic
55C2×#126—22no—
56C2×#125—22no—
57C2×#123—22no—
58C2×#123—22no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—22no—
61C2×#121—21no—
62C2×#121—21no—
63C2×#121C2312yes—
64C2×C4×D4—22no—
65C2×C4×D4C2×C412yes—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C42—21no—
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72#432—42no—
73#432—42no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C22⋊C8C2312yes—
79C2×C22⋊C8C2312yes—
80C2×#97—22no—
81#400—22no—
82#398—22no—
83#397—22no—
84#396—22no—
85#396—22no—
86#394—22no—
87#394—22no—
88#393—22no—
89#391—21no—
90#390—21no—
91#390—21no—
92#390C2×C412yes—
93#390C2×C412yes—
94#390C2×C412yes—
95#390C2×C412yes—
96#389—21no—
97#389C2×C412yes—
98#389C2×C412yes—
99#389C2×C412yes—
100#389C2×C412yes—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#387—22no—
106#386—22no—
107#386—22no—
108#386—22no—
109#385—22no—
110#384C2×C411yescharacteristic
111#384C2×C411yescharacteristic
112#380—21no—
113#380C2312yes—
114C4×C4⋊C4C2×C412yes—
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4D412yes—
118C4×C4⋊C4D412yes—
119C4×C22⋊C4—22no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95D411yescharacteristic
132C2×#95D411yescharacteristic
133C2×#95C2311yescharacteristic
134C2×#95C2311yescharacteristic
135C2×#95C2311yescharacteristic
136C2×#95C2311yescharacteristic
137#336—22no—
138#336—21no—
139#336—21no—
140#327—42no—
141#327—42no—
142#327—42no—
143#327—42no—
144C22×Q8—21no—
145C22×D4—22no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4C22×C412yes—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2411yescharacteristic
161C22×C8—22no—
162C22×C8—22no—
163C4⋊Q8—81no—
164#127—81no—
165#126—48no—
166#125—48no—
167#123—48no—
168#123—44no—
169#123—44no—
170C22⋊Q8—48no—
171C22⋊Q8—42no—
172C22⋊Q8—42no—
173#121—22no—
174#121—22no—
175#121—22no—
176#121—22no—
177#121—42no—
178#121—42no—
179#121—42no—
180#121—42no—
181C4×D4—44no—
182C4×D4—44no—
183C4×D4—82no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C22⋊C412yes—
199C2×C4⋊C4C22⋊C412yes—
200C2×C4⋊C4C22⋊C412yes—
201C2×C4⋊C4C22×C412yes—
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C22×C412yes—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—22no—
220C2×C42—42no—
221C2×C42—42no—
222C2×C42—81no—
223C2×C42—81no—
224C2×C42—81no—
225C2×C42C22×C412yes—
226C2×C42C22×C412yes—
227C2×C42C22×C411yescharacteristic
228C2×C42C22×C411yescharacteristic
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C4⋊C8—44no—
234C4⋊C8—81no—
235C4⋊C8—81no—
236C4⋊C8—82no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239C22⋊C8—22no—
240C22⋊C8—22no—
241#97—44no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—42no—
257#95—42no—
258#95—42no—
259#95—42no—
260#95—42no—
261#95—42no—
262#95—42no—
263#95—42no—
264#95—81no—
265#95—81no—
266#95—81no—
267#95C22⋊C412yes—
268#95C22⋊C412yes—
269#95C22⋊C412yes—
270#95C22⋊C412yes—
271C24—22no—
272C24C22×D411yescharacteristic
273C2×Q8—42no—
274C2×Q8—81no—
275C2×Q8—81no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—81no—
289C2×D4—81no—
290C2×D4—81no—
291C2×D4—81no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—44no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4C2×C22⋊C412yes—
353C22×C4C2×C22⋊C412yes—
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C412yes—
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C412yes—
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4AGL(1,ℤ/8)12yes—
361C22×C4AGL(1,ℤ/8)12yes—
362C22×C4C23×C411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4H(5,𝔽2)11yescharacteristic
365C22×C4H(5,𝔽2)11yescharacteristic
366C22×C42−1+411yescharacteristic
367C22×C42−1+411yescharacteristic
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—44no—
383C4⋊C4—42no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—42no—
387C4⋊C4—44no—
388C4⋊C4—42no—
389C4⋊C4—81no—
390C4⋊C4—81no—
391C22⋊C4—44no—
392C22⋊C4—44no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—42no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C22⋊C4—42no—
399C22⋊C4—44no—
400C22⋊C4—82no—
401C22⋊C4—82no—
402C22⋊C4—82no—
403C22⋊C4—82no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—41no—
409C42—41no—
410C42—42no—
411C42—42no—
412C42—41no—
413C42—41no—
414C42—44no—
415C42—84no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23C22×C22⋊C411yescharacteristic
428C23#53711yescharacteristic
429C23#53711yescharacteristic
430C23C2×AGL(1,ℤ/8)11yescharacteristic
431C23C2×AGL(1,ℤ/8)11yescharacteristic
432C23#57511yescharacteristic
433C23#57511yescharacteristic
434Q8—162no—
435D4—82no—
436D4—82no—
437D4—82no—
438D4—82no—
439D4—82no—
440D4—82no—
441D4—162no—
442D4—162no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—164no—
525C2×C4#41812yes—
526C2×C4#41812yes—
527C2×C4#52011yescharacteristic
528C2×C4#52011yescharacteristic
529C2×C4#53511yescharacteristic
530C2×C4#53611yescharacteristic
531C8—82no—
532C8—82no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22#231111yescharacteristic
553C22#288311yescharacteristic
554C22#288411yescharacteristic
555C22C2×#41811yescharacteristic
556C22#289611yescharacteristic
557C22#290111yescharacteristic
558C22#290111yescharacteristic
559C22#325111yescharacteristic
560C22#325111yescharacteristic
561C22#325111yescharacteristic
562C22#325111yescharacteristic
563C4—21no—
564C4—21no—
565C4—21no—
566C4—21no—
567C4—22no—
568C4—22no—
569C4—42no—
570C4—42no—
571C4—42no—
572C4—42no—
573C4—41no—
574C4—41no—
575C4—82no—
576C4—82no—
577C4—82no—
578C4—82no—
579C4—82no—
580C4—161no—
581C4—161no—
582C2—21no—
583C2—21no—
584C2—21no—
585C2—21no—
586C2—162no—
587C2#1762011yescharacteristic
588C2#1762411yescharacteristic
589C2#1762511yescharacteristic
590C2#1763111yescharacteristic
591C2#2185811yescharacteristic
592C2#2187711yescharacteristic
593C2#2189111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21955C2semidirect
3. #13950C2semidirect
4. #9912C2semidirect
5. #9909C2semidirect
8. #9118C2semidirect
9. #9118C2semidirect
10. #8575C2semidirect
11. #8573C2semidirect
21. #2512C4semidirect
22. #2506C4semidirect
23. #2452C4semidirect
24. #2445C4semidirect

Maximal subgroups

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

subgroupindex
#350062
#219552
#139502
#99122
#99092
#97152
#97122
#91182
#85752
#85732