Finite Group Databasebeta

← #127202

Subgroups — #127202

Order 512.

908 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20722C211yesmaximal, characteristic
2#17362C211yesmaximal, characteristic
3#13785C211yesmaximal, characteristic
4#7213C214yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6C2×#407—21no—
7C2×#390—21no—
8#2280—21no—
9C2×C4×C4⋊C4C411yescharacteristic, complemented
10C2×#376C411yescharacteristic, complemented
11#1819C2212yes—
12#1799—21no—
13#1799C412yescomplemented
14#1781—21no—
15C4×#99—21no—
16C4×#99C412yescomplemented
17#1465—21no—
18#1465C412yescomplemented
19#1463—21no—
20#1462C2212yes—
21#1459—21no—
22#1459C412yescomplemented
23#1437C412yescomplemented
24#1437C2212yes—
25C4⋊#95—21no—
26C22⋊#95—21no—
27C22⋊#95—21no—
28C23×D4D411yescharacteristic
29C24×C4D411yescharacteristic
30C24×C4D411yescharacteristic
31C22×M16—41no—
32C2×C4×D4—41no—
33C2×C4×D4—41no—
34C2×C4×D4—41no—
35C2×C4×D4—41no—
36C2×C4×D4C2×C411yescharacteristic
37C2×C4×D4C2×C411yescharacteristic
38C2×C4×D4C2×C411yescharacteristic
39C2×C4×D4C2×C411yescharacteristic
40C2×C4×D4D411yescharacteristic
41C2×C4×D4D411yescharacteristic
42C2×C4×D4Q811yescharacteristic
43C2×C4×D4Q811yescharacteristic
44C2×#117—41no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4D411yescharacteristic
47C22×C22⋊C4—22no—
48C22×C22⋊C4—22no—
49C22×C22⋊C4C2×C411yescharacteristic
50C22×C22⋊C4C2311yescharacteristic
51C22×C42—21no—
52C22×C42C2×C411yescharacteristic
53C22×C42C2×C412yes—
54C2×#99—22no—
55C2×#99—22no—
56#407—22no—
57#407—22no—
58#407—41no—
59#407—41no—
60C2×C22⋊C8—22no—
61C2×C22⋊C8—41no—
62#390—28no—
63#381—41no—
64#380—22no—
65#380—41no—
66#379—22no—
67C4×C4⋊C4—41no—
68C4×C4⋊C4—41no—
69C4×C22⋊C4—22no—
70C4×C22⋊C4—41no—
71#376—41no—
72#376—41no—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95C2×C412yes—
78#342—22no—
79#342C2×C414yes—
80#324—22no—
81#324—22no—
82#323—22no—
83#323C2×C414yes—
84C25C22⋊C411yescharacteristic
85C25C2×D411yescharacteristic
86C22×D4—24no—
87C22×D4—42no—
88C22×D4—41no—
89C22×D4—41no—
90C22×D4C22⋊C411yescharacteristic
91C22×D4C22⋊C411yescharacteristic
92C22×D4C4⋊C411yescharacteristic
93C22×D4C4⋊C411yescharacteristic
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—82no—
105C23×C4—82no—
106C23×C4C22⋊C411yescharacteristic
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22⋊C411yescharacteristic
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4C22⋊C411yescharacteristic
113C23×C4C4⋊C411yescharacteristic
114C23×C4C4⋊C411yescharacteristic
115C23×C4C4⋊C411yescharacteristic
116C23×C4C4⋊C411yescharacteristic
117C23×C4C22×C411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C2×M16—42no—
120C2×M16—42no—
121C2×M16—84no—
122C22×C8—42no—
123C4×D4—41no—
124C4×D4—41no—
125C4×D4—41no—
126C4×D4—41no—
127C4×D4—41no—
128C4×D4—41no—
129C4×D4—41no—
130C4×D4—41no—
131C4×D4—41no—
132C4×D4—41no—
133C4×D4—41no—
134C4×D4—41no—
135C4×D4—41no—
136C4×D4—41no—
137C4×D4—41no—
138C4×D4—41no—
139#117—84no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4C22⋊C411yescharacteristic
147C2×C4⋊C4C22⋊C411yescharacteristic
148C2×C4⋊C4C4⋊C411yescharacteristic
149C2×C4⋊C4C4⋊C411yescharacteristic
150C2×C22⋊C4—24no—
151C2×C22⋊C4—28no—
152C2×C22⋊C4—28no—
153C2×C22⋊C4—41no—
154C2×C22⋊C4—41no—
155C2×C22⋊C4—41no—
156C2×C22⋊C4—41no—
157C2×C22⋊C4—42no—
158C2×C22⋊C4—42no—
159C2×C22⋊C4—82no—
160C2×C22⋊C4—82no—
161C2×C22⋊C4—82no—
162C2×C22⋊C4—82no—
163C2×C22⋊C4C4211yescharacteristic
164C2×C22⋊C4C4211yescharacteristic
165C2×C22⋊C4C4⋊C411yescharacteristic
166C2×C22⋊C4C4⋊C411yescharacteristic
167C2×C22⋊C4C22×C411yescharacteristic
168C2×C22⋊C4C22×C411yescharacteristic
169C2×C22⋊C4C2×D411yescharacteristic
170C2×C22⋊C4C2×Q811yescharacteristic
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—24no—
176C2×C42—42no—
177C2×C42—42no—
178C2×C42—42no—
179C2×C42—42no—
180C2×C42—82no—
181C2×C42C4211yescharacteristic
182C2×C42C4211yescharacteristic
183C2×C42C4⋊C411yescharacteristic
184C2×C42C4⋊C411yescharacteristic
185#99—44no—
186#99—44no—
187C22⋊C8—24no—
188C22⋊C8—42no—
189C22⋊C8—42no—
190C22⋊C8—42no—
191#95—42no—
192#95—42no—
193#95—42no—
194#95—41no—
195#95—41no—
196#95—41no—
197#95—41no—
198#95—82no—
199#95—82no—
200C24—22no—
201C24—22no—
202C24—21no—
203C24—21no—
204C24—21no—
205C24—21no—
206C24—41no—
207C24—41no—
208C24—41no—
209C24—41no—
210C24—42no—
211C24—42no—
212C24—41no—
213C24—41no—
214C24#9511yescharacteristic
215C24#9511yescharacteristic
216C24C2×C22⋊C411yescharacteristic
217C24C2×C22⋊C411yescharacteristic
218C24C2×C4⋊C411yescharacteristic
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—48no—
231C2×D4—48no—
232C2×D4#10414yes—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
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—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—42no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—44no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4—81no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4—81no—
287C22×C4—81no—
288C22×C4—81no—
289C22×C4—81no—
290C22×C4—81no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4#9511yescharacteristic
296C22×C4#9511yescharacteristic
297C22×C4#9511yescharacteristic
298C22×C4#9511yescharacteristic
299C22×C4#9511yescharacteristic
300C22×C4#9511yescharacteristic
301C22×C4#9911yescharacteristic
302C22×C4#9911yescharacteristic
303C22×C4#9911yescharacteristic
304C22×C4#9911yescharacteristic
305C22×C4C2×C4211yescharacteristic
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C4⋊C411yescharacteristic
308C22×C4C2×C4⋊C411yescharacteristic
309M16—84no—
310M16—84no—
311C2×C8—44no—
312C2×C8—84no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—48no—
318C4⋊C4—84no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—44no—
328C22⋊C4—44no—
329C22⋊C4—88no—
330C22⋊C4—88no—
331C22⋊C4—82no—
332C22⋊C4—82no—
333C22⋊C4—84no—
334C42—21no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42—21no—
341C42—21no—
342C42—48no—
343C42—84no—
344C42—84no—
345C23—21no—
346C23—21no—
347C23—22no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—44no—
373C23—44no—
374C23—44no—
375C23—44no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—42no—
381C23—42no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—82no—
387C23—82no—
388C23—82no—
389C23—82no—
390C23#33912yes—
391C23C2×#9511yescharacteristic
392C23C2×#9911yescharacteristic
393C23C2×#9911yescharacteristic
394C23C2×#10412yes—
395D4—44no—
396D4—44no—
397D4—44no—
398D4—44no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—22no—
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—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
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—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
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—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—84no—
461C2×C4—84no—
462C2×C4—84no—
463C2×C4H(3,ℤ/4)12yes—
464C2×C4#33912yes—
465C2×C4#34211yescharacteristic
466C2×C4#34211yescharacteristic
467C2×C4#34211yescharacteristic
468C2×C4#34211yescharacteristic
469C8—84no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—22no—
481C22—22no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—41no—
487C22—41no—
488C22—41no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—81no—
497C22—81no—
498C22—82no—
499C22—82no—
500C22—84no—
501C22—84no—
502C22—82no—
503C22—82no—
504C22—81no—
505C22—81no—
506C22C2×H(3,ℤ/4)11yescharacteristic
507C22C2×#33911yescharacteristic
508C22C2×#34211yescharacteristic
509C22#212211yescharacteristic
510C22#212311yescharacteristic
511C22#212712yes—
512C4—22no—
513C4—22no—
514C4—21no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—41no—
519C4—41no—
520C4—41no—
521C4—41no—
522C4—41no—
523C4—41no—
524C4—84no—
525C4—84no—
526C4—164no—
527C2—21no—
528C2—22no—
529C2—21no—
530C2—21no—
531C2—21no—
532C2—42no—
533C2—42no—
534C2—81no—
535C2—81no—
536C2#794211yescharacteristic
537C2#1257511yescharacteristic
538C2#1258211yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC22×C22⋊C4C23
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7213C2semidirect
9. C2×C4×C4⋊C4C4semidirect
10. C2×#376C4semidirect
13. #1799C4semidirect
16. C4×#99C4semidirect
18. #1465C4semidirect
22. #1459C4semidirect
23. #1437C4semidirect

Maximal subgroups

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

subgroupindex
#207222
#173622
#137852
#72132