Finite Group Databasebeta

← #97539

Subgroups — #97539

Order 512.

853 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2726C211yesmaximal, characteristic
2#11207C211yesmaximal, characteristic
3#11207C211yesmaximal, characteristic
4#11207C211yesmaximal, characteristic
5C2×#1271C211yesmaximal, characteristic
6C2×#1271C211yesmaximal, characteristic
7C2×#1271C211yesmaximal, characteristic
8C22×#125—21no—
9#2726—22no—
10#2726—22no—
11#2726—22no—
12#2726C414yescomplemented
13C2×#398—21no—
14C2×#398—21no—
15C2×#398—21no—
16C2×#389—21no—
17C2×#389C2211yescharacteristic
18C2×#389C2211yescharacteristic
19C2×#389C2211yescharacteristic
20C2×#387—21no—
21C2×#384C2211yescharacteristic
22C2×#340—21no—
23C2×#340—21no—
24C2×#340—21no—
25C2×#340—21no—
26C2×#340—21no—
27C2×#340—21no—
28C2×#336C2211yescharacteristic
29C2×#336C2211yescharacteristic
30C2×#336C2211yescharacteristic
31#1271—22no—
32#1271—22no—
33#1271—22no—
34C23×C8—21no—
35C23×C8—21no—
36C23×C8—21no—
37C2×#125—24no—
38C2×#125—24no—
39C2×#125—42no—
40C2×#125—42no—
41C2×#125—42no—
42C2×#125—42no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4D411yescharacteristic
59C22×C42—21no—
60C22×C42C2311yescharacteristic
61#398—24no—
62#398—24no—
63#398—24no—
64#398—24no—
65#398—24no—
66#398—24no—
67#389—24no—
68#389—24no—
69#389—21no—
70#389—21no—
71#389—21no—
72#389—21no—
73#389—21no—
74#389—21no—
75#389C2×C412yes—
76#389C2×C412yes—
77#389C2×C412yes—
78#389D412yes—
79#389D412yes—
80#389D412yes—
81#387—24no—
82#387—24no—
83#384—42no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—21no—
89C2×#95—21no—
90#340—42no—
91#340—42no—
92#340—42no—
93#340—42no—
94#340—42no—
95#340—42no—
96#336—21no—
97#336—21no—
98#336—21no—
99#336—21no—
100#336—21no—
101#336—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—41no—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—41no—
116C22×C8—41no—
117C22×C8—41no—
118C22×C8—41no—
119C22×C8—41no—
120C22×C8—41no—
121C22×C8—41no—
122C22×C8—41no—
123C22×C8—41no—
124C22×C8—41no—
125C22×C8—41no—
126C22×C8—41no—
127C22×C8—41no—
128C22×C8—41no—
129C22×C8—41no—
130C22×C8—41no—
131C22×C8—41no—
132C22×C8—41no—
133#125—48no—
134#125—48no—
135#125—48no—
136#125—48no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—81no—
204C2×C4⋊C4—81no—
205C2×C4⋊C4C22⋊C412yes—
206C2×C4⋊C4C22⋊C412yes—
207C2×C4⋊C4C22⋊C412yes—
208C2×C4⋊C4C22⋊C412yes—
209C2×C4⋊C4C22⋊C412yes—
210C2×C4⋊C4C22⋊C412yes—
211C2×C4⋊C4D812yes—
212C2×C4⋊C4SD1612yes—
213C2×C4⋊C4SD1612yes—
214C2×C4⋊C4Q1612yes—
215C2×C42—24no—
216C2×C42—41no—
217C2×C42—41no—
218C2×C42—42no—
219C2×C42—42no—
220C2×C42C22×C411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224#95—24no—
225#95—24no—
226#95—24no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233C24#12011yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—81no—
290C22×C4—81no—
291C22×C4—81no—
292C22×C4—81no—
293C22×C4—81no—
294C22×C4—81no—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4#12011yescharacteristic
299C22×C4#12011yescharacteristic
300C22×C4#12011yescharacteristic
301C22×C4C2×D811yescharacteristic
302C22×C4C2×SD1611yescharacteristic
303C22×C4C2×SD1611yescharacteristic
304C22×C4C2×Q1611yescharacteristic
305C22×C4#13511yescharacteristic
306C22×C4#13511yescharacteristic
307C22×C4AGL(1,ℤ/8)11yescharacteristic
308C22×C4AGL(1,ℤ/8)11yescharacteristic
309C22×C4AGL(1,ℤ/8)11yescharacteristic
310C22×C4#13711yescharacteristic
311C22×C4#13711yescharacteristic
312C22×C4#13711yescharacteristic
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—81no—
323C2×C8—81no—
324C2×C8—81no—
325C2×C8—81no—
326C2×C8—81no—
327C2×C8—81no—
328C2×C8—81no—
329C2×C8—81no—
330C2×C8—81no—
331C2×C8—81no—
332C2×C8—81no—
333C2×C8—81no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—48no—
346C4⋊C4—48no—
347C4⋊C4—48no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—41no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—41no—
367C4⋊C4—42no—
368C4⋊C4—81no—
369C4⋊C4—81no—
370C4⋊C4—81no—
371C4⋊C4—81no—
372C4⋊C4—81no—
373C4⋊C4—81no—
374C4⋊C4—81no—
375C4⋊C4—81no—
376C4⋊C4—81no—
377C4⋊C4—84no—
378C4⋊C4#10214yes—
379C4⋊C4#10314yes—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42—48no—
385C23#37911yescharacteristic
386C23#44711yescharacteristic
387C23#44811yescharacteristic
388C23#44911yescharacteristic
389C23#45011yescharacteristic
390C23#45111yescharacteristic
391C23#45211yescharacteristic
392C23#45411yescharacteristic
393C23#45411yescharacteristic
394C23#45411yescharacteristic
395C23#45611yescharacteristic
396C23#45611yescharacteristic
397C23#45611yescharacteristic
398C23B(4,𝔽2)11yescharacteristic
399C23#45811yescharacteristic
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—161no—
482C2×C4—161no—
483C2×C4—161no—
484C2×C4—161no—
485C2×C4C2×#10211yescharacteristic
486C2×C4C2×#10311yescharacteristic
487C2×C4#41611yescharacteristic
488C2×C4#41711yescharacteristic
489C2×C4#41811yescharacteristic
490C2×C4#41911yescharacteristic
491C2×C4#44711yescharacteristic
492C2×C4#44811yescharacteristic
493C2×C4#44911yescharacteristic
494C2×C4#45011yescharacteristic
495C2×C4#45111yescharacteristic
496C2×C4#45211yescharacteristic
497C8—82no—
498C8—82no—
499C8—82no—
500C22#159711yescharacteristic
501C22#159811yescharacteristic
502C22#159911yescharacteristic
503C22#160311yescharacteristic
504C22#160411yescharacteristic
505C22#160511yescharacteristic
506C22#160911yescharacteristic
507C22#161011yescharacteristic
508C22#161111yescharacteristic
509C22#161511yescharacteristic
510C22#161611yescharacteristic
511C22#161711yescharacteristic
512C22#162511yescharacteristic
513C22#162611yescharacteristic
514C22#162711yescharacteristic
515C22#163511yescharacteristic
516C22#163611yescharacteristic
517C22#163711yescharacteristic
518C22#164711yescharacteristic
519C22#164811yescharacteristic
520C22#164911yescharacteristic
521C22#165111yescharacteristic
522C22#165311yescharacteristic
523C22#165411yescharacteristic
524C22#165811yescharacteristic
525C22#166311yescharacteristic
526C22#166311yescharacteristic
527C22#166311yescharacteristic
528C22#178611yescharacteristic
529C22#178711yescharacteristic
530C22#178811yescharacteristic
531C22#179011yescharacteristic
532C22#179311yescharacteristic
533C22#179311yescharacteristic
534C22#179311yescharacteristic
535C4—21no—
536C4—21no—
537C4—21no—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—44no—
548C4—44no—
549C4—82no—
550C4—82no—
551C4—82no—
552C4—82no—
553C4—162no—
554C2#773911yescharacteristic
555C2#776311yescharacteristic
556C2#776811yescharacteristic
557C2#777211yescharacteristic
558C2#779211yescharacteristic
559C2#779811yescharacteristic
560C2#781511yescharacteristic
561C2#782711yescharacteristic
562C2#840411yescharacteristic
563C2#840611yescharacteristic
564C2#840711yescharacteristic
565C2#844611yescharacteristic
566C2#844711yescharacteristic
567C2#844811yescharacteristic
568C2#850911yescharacteristic
subgroupG/H
center, socle, prime radicalC24#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #2726C4semidirect

Maximal subgroups

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

subgroupindex
C2×#27262
#112072
C2×#12712