Finite Group Databasebeta

← #215918

Subgroups — #215918

Order 512.

810 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35104C211yesmaximal, characteristic
2#21912C211yesmaximal, characteristic
3#13949C211yesmaximal, characteristic
4#9913C212yesmaximal
5#9910C212yesmaximal
6#9885C211yesmaximal, characteristic
7#9885C211yesmaximal, characteristic
8#9884C211yesmaximal, characteristic
9#9884C211yesmaximal, characteristic
10#8714C212yesmaximal
11#8714C212yesmaximal
12#2857C2211yescharacteristic
13#2857C2211yescharacteristic
14#2851C2212yes—
15#2627—21no—
16#2627—21no—
17#2606—22no—
18#2605—22no—
19#2570C412yescomplemented
20#2570C412yescomplemented
21#2550C412yescomplemented
22#2550C412yescomplemented
23#2329—21no—
24#2329—21no—
25C2×#387—21no—
26#2283C2212yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28#1916—22no—
29#1916—22no—
30#1817C2212yes—
31#1797C2212yes—
32C4×C22⋊C8C2211yescharacteristic
33C4×C22⋊C8C2211yescharacteristic
34#1442—21no—
35#1441—21no—
36#1436—21no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39#1434C2211yescharacteristic
40#1434C2211yescharacteristic
41#1434C2211yescharacteristic
42C4×#95C2211yescharacteristic
43#1294C2212yes—
44#1294C2212yes—
45#1294C2212yes—
46#1294C2212yes—
47#1294C2212yes—
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×#125C2×C412yes—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C42—21no—
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C2×C4⋊C8—22no—
65C2×C22⋊C8C2312yes—
66C2×C22⋊C8C2312yes—
67C2×C4×C8—21no—
68C2×C4×C8—21no—
69#400—22no—
70#400—22no—
71#398—22no—
72#398—22no—
73#398—22no—
74#398—22no—
75#395—22no—
76#395—22no—
77#391—21no—
78#391—21no—
79#391—22no—
80#391—22no—
81#389—21no—
82#389—21no—
83#389—22no—
84#389—22no—
85#389C2×C412yes—
86#389C2×C412yes—
87#389C2×C412yes—
88#389C2×C412yes—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#387—44no—
96#385—22no—
97#385—22no—
98#384C2×C411yescharacteristic
99#384C2×C411yescharacteristic
100#383—22no—
101#382C2×C412yes—
102#382C2×C412yes—
103#382C2×C412yes—
104#381—22no—
105#380—21no—
106#380—21no—
107#380C2312yes—
108#380C2312yes—
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4C2×C412yes—
111C4×C4⋊C4C2×C412yes—
112C4×C4⋊C4C2×C412yes—
113C4×C4⋊C4D412yes—
114C4×C4⋊C4D412yes—
115C4×C22⋊C4—22no—
116#376—22no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95D411yescharacteristic
127C2×#95D411yescharacteristic
128C2×#95C2311yescharacteristic
129C2×#95C2311yescharacteristic
130C2×#95C2311yescharacteristic
131C2×#95C2311yescharacteristic
132#336—22no—
133#336—24no—
134#336—21no—
135#336—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2411yescharacteristic
149C22×C8—22no—
150C22×C8—22no—
151C4⋊Q8—81no—
152C4⋊Q8—81no—
153#125—82no—
154C22⋊Q8—42no—
155C22⋊Q8—42no—
156C22⋊Q8—42no—
157C22⋊Q8—42no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—81no—
179C2×C4⋊C4—81no—
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C22×C412yes—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—42no—
195C2×C42—42no—
196C2×C42—81no—
197C2×C42—81no—
198C2×C42—81no—
199C2×C42C22×C412yes—
200C2×C42C22×C412yes—
201C2×C42C22×C411yescharacteristic
202C2×C42C22×C411yescharacteristic
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C4⋊C8—82no—
208C22⋊C8—22no—
209C22⋊C8—22no—
210C22⋊C8—22no—
211C22⋊C8—22no—
212C4×C8—81no—
213C4×C8—81no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—42no—
233#95—42no—
234#95—42no—
235#95—42no—
236#95—42no—
237#95—42no—
238#95—42no—
239#95—42no—
240#95—81no—
241#95C22⋊C412yes—
242#95C22⋊C412yes—
243#95C22⋊C412yes—
244#95C22⋊C412yes—
245C24C22×D411yescharacteristic
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—81no—
249C2×Q8—81no—
250C2×Q8—81no—
251C2×Q8—81no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
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—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4#13512yes—
317C22×C4#13512yes—
318C22×C4AGL(1,ℤ/8)12yes—
319C22×C4#13712yes—
320C22×C4C23×C411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C42−1+412yes—
323C22×C42−1+412yes—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—44no—
340C4⋊C4—42no—
341C4⋊C4—44no—
342C4⋊C4—81no—
343C4⋊C4—81no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C4⋊C4—82no—
347C4⋊C4—82no—
348C4⋊C4—81no—
349C4⋊C4—81no—
350C4⋊C4—84no—
351C4⋊C4—84no—
352C22⋊C4—42no—
353C22⋊C4—42no—
354C22⋊C4—42no—
355C22⋊C4—42no—
356C42—24no—
357C42—24no—
358C42—41no—
359C42—41no—
360C42—42no—
361C42—42no—
362C42—41no—
363C42—41no—
364C42—82no—
365C42—82no—
366C42—84no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23C22×C22⋊C411yescharacteristic
372C23#53611yescharacteristic
373C23#53611yescharacteristic
374C23C2×#13511yescharacteristic
375C23C2×#13511yescharacteristic
376C23C2×AGL(1,ℤ/8)11yescharacteristic
377C23C2×#13711yescharacteristic
378Q8—162no—
379Q8—162no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4#41612yes—
461C2×C4#41612yes—
462C2×C4#41712yes—
463C2×C4#41712yes—
464C2×C4#51912yes—
465C2×C4#53612yes—
466C8—82no—
467C8—82no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22#231011yescharacteristic
481C22#288311yescharacteristic
482C22#288411yescharacteristic
483C22C2×#41611yescharacteristic
484C22C2×#41711yescharacteristic
485C22#290111yescharacteristic
486C22#290111yescharacteristic
487C22#322811yescharacteristic
488C22#322811yescharacteristic
489C22#323111yescharacteristic
490C22#323111yescharacteristic
491C4—22no—
492C4—22no—
493C4—22no—
494C4—22no—
495C4—22no—
496C4—22no—
497C4—42no—
498C4—42no—
499C4—42no—
500C4—42no—
501C4—82no—
502C4—82no—
503C4—82no—
504C4—82no—
505C4—82no—
506C4—82no—
507C4—161no—
508C4—161no—
509C2—21no—
510C2—21no—
511C2—21no—
512C2—21no—
513C2#1761511yescharacteristic
514C2#1761611yescharacteristic
515C2#1762211yescharacteristic
516C2#1762811yescharacteristic
517C2#2182811yescharacteristic
518C2#2183511yescharacteristic
519C2#2184611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
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
19. #2570C4semidirect
20. #2570C4semidirect
21. #2550C4semidirect
22. #2550C4semidirect

Maximal subgroups

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

subgroupindex
#351042
#219122
#139492
#99132
#99102
#98852
#98842
#87142