Finite Group Databasebeta

← #128123

Subgroups — #128123

Order 512.

875 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20898C211yesmaximal, characteristic
2C2×#1799C211yesmaximal, characteristic
3#17364C211yesmaximal, characteristic
4#7320C214yesmaximal, complemented
5C23×M16C411yescharacteristic, complemented
6C22×C4×D4C2211yescharacteristic
7#2920—21no—
8C22×#99C412yes—
9C2×#407—21no—
10C2×#407C411yescharacteristic, complemented
11C2×#380—21no—
12#1870C412yescomplemented
13#1870C2212yes—
14#1869—21no—
15#1869—21no—
16#1799—21no—
17#1799—21no—
18#1799C412yes—
19#1799C2212yes—
20C2×#342—21no—
21#1728—21no—
22#1466—21no—
23#1466C2212yes—
24#1462—21no—
25#1462C412yescomplemented
26#1460—21no—
27#1460C412yescomplemented
28C23×D4C2×C411yescharacteristic
29C24×C4C2×C411yescharacteristic
30C24×C4D411yescharacteristic
31C22×M16—21no—
32C22×M16—21no—
33C22×M16—44no—
34C22×M16—41no—
35C22×M16—41no—
36C22×M16C2×C412yescomplemented
37C23×C8—21no—
38C2×C4×D4—41no—
39C2×C4×D4—41no—
40C2×C4×D4—41no—
41C2×C4×D4—41no—
42C2×C4×D4C2×C411yescharacteristic
43C2×C4×D4C2×C411yescharacteristic
44C2×C4×D4C2×C411yescharacteristic
45C2×C4×D4D411yescharacteristic
46C2×C4×D4D411yescharacteristic
47C2×C4×D4D411yescharacteristic
48C2×C4×D4Q811yescharacteristic
49C2×C4×D4C2311yescharacteristic
50C22×C4⋊C4—41no—
51C22×C4⋊C4Q811yescharacteristic
52C22×C22⋊C4—22no—
53C22×C22⋊C4C2×C411yescharacteristic
54C22×C22⋊C4D411yescharacteristic
55C22×C42D411yescharacteristic
56#432—22no—
57#432—41no—
58#431—22no—
59#431—41no—
60#423—41no—
61C2×#99—22no—
62C2×#99—44no—
63C2×#99C2×C414yes—
64#407—22no—
65#407—44no—
66#407—41no—
67#407—41no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8C2×C412yescomplemented
72C4×M16—41no—
73#380—22no—
74#380—22no—
75#380—41no—
76#380—41no—
77C2×#95—41no—
78#342—22no—
79#342—22no—
80#336—22no—
81#336—22no—
82#323—22no—
83#323—22no—
84#323—22no—
85#323C2×C414yescomplemented
86C25C4211yescharacteristic
87C25C22⋊C411yescharacteristic
88C22×D4—22no—
89C22×D4—22no—
90C22×D4—41no—
91C22×D4—41no—
92C22×D4—42no—
93C22×D4C4211yescharacteristic
94C22×D4C22⋊C411yescharacteristic
95C22×D4C4⋊C411yescharacteristic
96C22×D4C22×C411yescharacteristic
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—42no—
106C23×C4—42no—
107C23×C4—42no—
108C23×C4—42no—
109C23×C4—42no—
110C23×C4C4211yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4C22⋊C411yescharacteristic
113C23×C4C22⋊C411yescharacteristic
114C23×C4C22⋊C411yescharacteristic
115C23×C4C22⋊C411yescharacteristic
116C23×C4C4⋊C411yescharacteristic
117C23×C4C4⋊C411yescharacteristic
118C23×C4C4⋊C411yescharacteristic
119C23×C4C4⋊C411yescharacteristic
120C23×C4C4⋊C411yescharacteristic
121C23×C4C22×C411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C2×M16—24no—
124C2×M16—44no—
125C2×M16—42no—
126C2×M16—48no—
127C2×M16—44no—
128C2×M16—41no—
129C2×M16—41no—
130C2×M16—44no—
131C2×M16—42no—
132C2×M16—42no—
133C2×M16—42no—
134C2×M16—41no—
135C2×M16—41no—
136C2×M16—42no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—44no—
140C22×C8—42no—
141C4×D4—41no—
142C4×D4—41no—
143C4×D4—41no—
144C4×D4—41no—
145C4×D4—41no—
146C4×D4—41no—
147C4×D4—41no—
148C4×D4—41no—
149C4×D4—41no—
150C4×D4—41no—
151C4×D4—41no—
152C4×D4—41no—
153C4×D4—41no—
154C4×D4—41no—
155C4×D4—41no—
156C4×D4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4—82no—
163C2×C4⋊C4C4⋊C411yescharacteristic
164C2×C4⋊C4C4⋊C411yescharacteristic
165C2×C4⋊C4C4⋊C411yescharacteristic
166C2×C4⋊C4C2×Q811yescharacteristic
167C2×C22⋊C4—24no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—41no—
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—44no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4C4211yescharacteristic
176C2×C22⋊C4C22⋊C411yescharacteristic
177C2×C22⋊C4C22⋊C411yescharacteristic
178C2×C22⋊C4C22⋊C411yescharacteristic
179C2×C22⋊C4C4⋊C411yescharacteristic
180C2×C22⋊C4C4⋊C411yescharacteristic
181C2×C22⋊C4C22×C411yescharacteristic
182C2×C22⋊C4C2×D411yescharacteristic
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42C22⋊C411yescharacteristic
188C2×C42C22⋊C411yescharacteristic
189C2×C42C4⋊C411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C4⋊C8—42no—
192C4⋊C8—42no—
193#99—44no—
194#99—44no—
195C22⋊C8—24no—
196C22⋊C8—44no—
197C22⋊C8—44no—
198C22⋊C8—42no—
199C22⋊C8—42no—
200C22⋊C8—42no—
201#97—42no—
202C4×C8—42no—
203#95—42no—
204#95—42no—
205C24—21no—
206C24—21no—
207C24—22no—
208C24—21no—
209C24—21no—
210C24—21no—
211C24—21no—
212C24—22no—
213C24—41no—
214C24—42no—
215C24—42no—
216C24—81no—
217C24—81no—
218C24#9511yescharacteristic
219C24#9511yescharacteristic
220C24#9511yescharacteristic
221C24C2×C4211yescharacteristic
222C24C2×C22⋊C411yescharacteristic
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—22no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—44no—
236C2×D4—44no—
237C2×D4—44no—
238C2×D4—44no—
239C2×D4#10012yes—
240C2×D4#10112yes—
241C22×C4—22no—
242C22×C4—22no—
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—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—44no—
271C22×C4—42no—
272C22×C4—44no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—44no—
292C22×C4—81no—
293C22×C4—81no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4—81no—
297C22×C4—84no—
298C22×C4—84no—
299C22×C4—84no—
300C22×C4#9511yescharacteristic
301C22×C4#9511yescharacteristic
302C22×C4#9511yescharacteristic
303C22×C4#9511yescharacteristic
304C22×C4#9511yescharacteristic
305C22×C4#9911yescharacteristic
306C22×C4#9911yescharacteristic
307C22×C4#9911yescharacteristic
308C22×C4#9911yescharacteristic
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C4⋊C411yescharacteristic
312C22×C4C2×C4⋊C411yescharacteristic
313C22×C4C2×C4⋊C411yescharacteristic
314M16—44no—
315M16—44no—
316M16—44no—
317M16—44no—
318M16—82no—
319M16—82no—
320M16—84no—
321C2×C8—24no—
322C2×C8—42no—
323C2×C8—48no—
324C2×C8—42no—
325C2×C8—44no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C4⋊C4—84no—
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—44no—
345C22⋊C4—44no—
346C22⋊C4—84no—
347C42—21no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—21no—
352C42—21no—
353C42—21no—
354C42—21no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—41no—
372C23—42no—
373C23—42no—
374C23—41no—
375C23—41no—
376C23—42no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—42no—
382C23—41no—
383C23—41no—
384C23—42no—
385C23—42no—
386C23—81no—
387C23—81no—
388C23—81no—
389C23—81no—
390C23—84no—
391C23—82no—
392C23—82no—
393C23—84no—
394C23—84no—
395C23#34211yescharacteristic
396C23#34211yescharacteristic
397C23#34312yes—
398C23#34412yes—
399C23C2×#9511yescharacteristic
400C23C2×#10011yescharacteristic
401C23C2×#10111yescharacteristic
402D4—42no—
403D4—42no—
404D4—44no—
405D4—44no—
406D4—42no—
407D4—42no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—22no—
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—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—84no—
461C2×C4—84no—
462C2×C4—84no—
463C2×C4—168no—
464C2×C4#34211yescharacteristic
465C2×C4#34211yescharacteristic
466C2×C4#34312yes—
467C2×C4AΣL(1,𝔽16)12yes—
468C2×C4#35212yes—
469C2×C4C2×#9911yescharacteristic
470C2×C4C2×#9911yescharacteristic
471C8—44no—
472C8—82no—
473C8—84no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—22no—
483C22—21no—
484C22—22no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—41no—
490C22—41no—
491C22—44no—
492C22—44no—
493C22—42no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—81no—
504C22—81no—
505C22—84no—
506C22—81no—
507C22—81no—
508C22—82no—
509C22—82no—
510C22—84no—
511C22#139012yes—
512C22C2×#34211yescharacteristic
513C22C2×#34311yescharacteristic
514C22C2×#34411yescharacteristic
515C4—21no—
516C4—21no—
517C4—22no—
518C4—41no—
519C4—41no—
520C4—41no—
521C4—41no—
522C4—41no—
523C4—41no—
524C4—41no—
525C4—164no—
526C4#139012yes—
527C4C2×AΣL(1,𝔽16)11yescharacteristic
528C4C2×#35211yescharacteristic
529C2—22no—
530C2—21no—
531C2—21no—
532C2—41no—
533C2—41no—
534C2—42no—
535C2—41no—
536C2—42no—
537C2—81no—
538C2#810711yescharacteristic
539C2C2×#139011yescharacteristic
540C2#1257711yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC24C2×C42
Frattini, cosocleC2×C4×D4C23
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7320C2semidirect
5. C23×M16C4semidirect
10. C2×#407C4semidirect
12. #1870C4semidirect
25. #1462C4semidirect
27. #1460C4semidirect
36. C22×M16C2×C4semidirect
71. C2×C22⋊C8C2×C4semidirect
85. #323C2×C4semidirect

Maximal subgroups

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

subgroupindex
#208982
C2×#17992
#173642
#73202