Finite Group Databasebeta

← #127139

Subgroups — #127139

Order 512.

826 conjugacy classes of subgroups.

The subgroup lattice has 491 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21740C211yesmaximal, characteristic
2C2×C4×#99C211yesmaximal, characteristic
3#17364C211yesmaximal, characteristic
4#7204C214yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6#2920—21no—
7C22×#99C412yes—
8C2×#407C411yescharacteristic, complemented
9C22×C22⋊C8C411yescharacteristic, complemented
10C2×C4×C22⋊C4—21no—
11#1870—21no—
12#1870—21no—
13#1869C412yescomplemented
14#1869C2212yes—
15#1817—21no—
16#1779—21no—
17C4×#99—21no—
18C4×#99—21no—
19C4×#99C412yes—
20C4×#99C2212yes—
21C2×#342—21no—
22#1466—21no—
23#1466C412yescomplemented
24#1462—21no—
25#1462C412yescomplemented
26#1460—21no—
27#1460C2212yes—
28C23×D4C2×C411yescharacteristic
29C24×C4C2×C411yescharacteristic
30C24×C4D411yescharacteristic
31C22×M16—21no—
32C23×C8C2×C412yescomplemented
33C2×C4×D4—41no—
34C2×C4×D4—41no—
35C2×C4×D4—41no—
36C2×C4×D4—41no—
37C2×C4×D4C2×C411yescharacteristic
38C2×C4×D4C2×C411yescharacteristic
39C2×C4×D4C2×C411yescharacteristic
40C2×C4×D4D411yescharacteristic
41C2×C4×D4D411yescharacteristic
42C2×C4×D4D411yescharacteristic
43C2×C4×D4Q811yescharacteristic
44C2×C4×D4C2311yescharacteristic
45C22×C4⋊C4Q811yescharacteristic
46C22×C22⋊C4—22no—
47C22×C22⋊C4C2×C411yescharacteristic
48C22×C22⋊C4D411yescharacteristic
49C22×C42—41no—
50C22×C42D411yescharacteristic
51#432—22no—
52#432—41no—
53#431—22no—
54#431—41no—
55#423—41no—
56C2×#99—22no—
57C2×#99—28no—
58C2×#99C2×C414yes—
59#407—22no—
60#407—41no—
61#407—41no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—41no—
68C2×C22⋊C8—41no—
69C2×C22⋊C8C2×C412yescomplemented
70C4×M16—41no—
71C4×C22⋊C4—22no—
72C4×C22⋊C4—22no—
73C4×C22⋊C4—41no—
74C4×C22⋊C4—41no—
75C2×#95—41no—
76#342—22no—
77#342—22no—
78#336—22no—
79#336—22no—
80#323—22no—
81#323—22no—
82#323—22no—
83#323C2×C414yescomplemented
84C25C4211yescharacteristic
85C25C22⋊C411yescharacteristic
86C22×D4—24no—
87C22×D4—41no—
88C22×D4—41no—
89C22×D4—42no—
90C22×D4C4211yescharacteristic
91C22×D4C22⋊C411yescharacteristic
92C22×D4C4⋊C411yescharacteristic
93C22×D4C22×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—42no—
103C23×C4—82no—
104C23×C4—82no—
105C23×C4C4211yescharacteristic
106C23×C4C22⋊C411yescharacteristic
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22⋊C411yescharacteristic
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22⋊C411yescharacteristic
111C23×C4C4⋊C411yescharacteristic
112C23×C4C4⋊C411yescharacteristic
113C23×C4C4⋊C411yescharacteristic
114C23×C4C4⋊C411yescharacteristic
115C23×C4C4⋊C411yescharacteristic
116C23×C4C22×C411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C2×M16—42no—
119C2×M16—42no—
120C2×M16—82no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—42no—
124C22×C8—82no—
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—
139C4×D4—41no—
140C4×D4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4C4⋊C411yescharacteristic
144C2×C4⋊C4C4⋊C411yescharacteristic
145C2×C4⋊C4C4⋊C411yescharacteristic
146C2×C4⋊C4C2×Q811yescharacteristic
147C2×C22⋊C4—28no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—41no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4—41no—
153C2×C22⋊C4—44no—
154C2×C22⋊C4—42no—
155C2×C22⋊C4C4211yescharacteristic
156C2×C22⋊C4C22⋊C411yescharacteristic
157C2×C22⋊C4C22⋊C411yescharacteristic
158C2×C22⋊C4C22⋊C411yescharacteristic
159C2×C22⋊C4C4⋊C411yescharacteristic
160C2×C22⋊C4C4⋊C411yescharacteristic
161C2×C22⋊C4C22×C411yescharacteristic
162C2×C22⋊C4C2×D411yescharacteristic
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—42no—
168C2×C42—42no—
169C2×C42—84no—
170C2×C42C22⋊C411yescharacteristic
171C2×C42C22⋊C411yescharacteristic
172C2×C42C4⋊C411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C4⋊C8—42no—
175C4⋊C8—42no—
176#99—28no—
177#99—24no—
178#99M1618yescomplemented
179C22⋊C8—24no—
180C22⋊C8—42no—
181C22⋊C8—42no—
182C22⋊C8—41no—
183C22⋊C8—41no—
184C22⋊C8—41no—
185C22⋊C8—41no—
186C22⋊C8—42no—
187C22⋊C8—82no—
188C22⋊C8—82no—
189#97—42no—
190C4×C8—42no—
191#95—42no—
192#95—42no—
193C24—21no—
194C24—21no—
195C24—22no—
196C24—22no—
197C24—21no—
198C24—21no—
199C24—22no—
200C24—41no—
201C24—42no—
202C24—42no—
203C24—81no—
204C24—81no—
205C24#9511yescharacteristic
206C24#9511yescharacteristic
207C24#9511yescharacteristic
208C24C2×C4211yescharacteristic
209C24C2×C22⋊C411yescharacteristic
210C2×D4—22no—
211C2×D4—22no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—48no—
222C2×D4—48no—
223C2×D4C2×M1614yes—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
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—42no—
250C22×C4—41no—
251C22×C4—44no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4—48no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—81no—
270C22×C4—81no—
271C22×C4—81no—
272C22×C4—84no—
273C22×C4—84no—
274C22×C4#9511yescharacteristic
275C22×C4#9511yescharacteristic
276C22×C4#9511yescharacteristic
277C22×C4#9511yescharacteristic
278C22×C4#9511yescharacteristic
279C22×C4#9911yescharacteristic
280C22×C4#9911yescharacteristic
281C22×C4#9911yescharacteristic
282C22×C4#9911yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C4⋊C411yescharacteristic
286C22×C4C2×C4⋊C411yescharacteristic
287C22×C4C2×C4⋊C411yescharacteristic
288M16—84no—
289M16—84no—
290C2×C8—44no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—84no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C22⋊C4—28no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—48no—
312C22⋊C4—44no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—21no—
321C42—84no—
322C42—84no—
323C23—24no—
324C23—24no—
325C23—22no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—41no—
338C23—42no—
339C23—42no—
340C23—41no—
341C23—41no—
342C23—42no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—42no—
348C23—41no—
349C23—41no—
350C23—42no—
351C23—42no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23—84no—
357C23—84no—
358C23—84no—
359C23—84no—
360C23#34211yescharacteristic
361C23#34211yescharacteristic
362C23#34412yes—
363C23C2×#9511yescharacteristic
364C23C4×M1612yes—
365C23#40712yes—
366D4—44no—
367D4—44no—
368D4—44no—
369D4—44no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—48no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—84no—
419C2×C4—816no—
420C2×C4—84no—
421C2×C4#34211yescharacteristic
422C2×C4#34211yescharacteristic
423C2×C4C2×#9911yescharacteristic
424C2×C4C2×#9911yescharacteristic
425C2×C4#42312yes—
426C8—82no—
427C8—84no—
428C8—82no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—22no—
436C22—21no—
437C22—22no—
438C22—22no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—48no—
445C22—42no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—81no—
456C22—81no—
457C22—84no—
458C22—81no—
459C22—81no—
460C22—84no—
461C22—84no—
462C22#139112yes—
463C22#172811yescharacteristic
464C22C2×#34211yescharacteristic
465C22C2×#34411yescharacteristic
466C4—21no—
467C4—21no—
468C4—21no—
469C4—22no—
470C4—41no—
471C4—41no—
472C4—41no—
473C4—41no—
474C4—41no—
475C4—41no—
476C4—41no—
477C4—88no—
478C4#212112yes—
479C2—22no—
480C2—21no—
481C2—21no—
482C2—42no—
483C2—41no—
484C2—42no—
485C2—42no—
486C2—81no—
487C2#792911yescharacteristic
488C2#1257411yescharacteristic
489C2C2×#139111yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC24C2×C42
Frattini, cosocleC2×C4×D4C23
socleC22C2×#342
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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. #7204C2semidirect
8. C2×#407C4semidirect
9. C22×C22⋊C8C4semidirect
13. #1869C4semidirect
23. #1466C4semidirect
25. #1462C4semidirect
32. C23×C8C2×C4semidirect
69. C2×C22⋊C8C2×C4semidirect
83. #323C2×C4semidirect
178. #99M16semidirect

Maximal subgroups

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

subgroupindex
#217402
C2×C4×#992
#173642
#72042