Finite Group Databasebeta

← #359949

Subgroups — #359949

Order 512.

1456 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2924C211yesmaximal, characteristic
2C22×#428C211yesmaximal, characteristic
3#20702C211yesmaximal, characteristic
4C2×#1824C211yesmaximal, characteristic
5C2×#1821C211yesmaximal, characteristic
6#11216C214yesmaximal, complemented
7#10896C214yesmaximal, complemented
8#10739C211yesmaximal, characteristic
9#10734C211yesmaximal, characteristic
10C23×M16C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13#2924—24no—
14#2924—24no—
15C2×#433—22no—
16C2×#432—21no—
17C2×#431—21no—
18C2×#428—24no—
19C2×#428—21no—
20C2×#428—21no—
21C2×#428C418yescomplemented
22C2×#428C2212yes—
23C2×#428C2212yes—
24C22×#107—21no—
25C22×#106—21no—
26C2×#424C2212yes—
27C2×#407C2211yescharacteristic
28C2×#407C2211yescharacteristic
29C22×C22⋊C8C2211yescharacteristic
30C2×#405C2212yes—
31C2×#380C2211yescharacteristic
32#1825—21no—
33#1825—21no—
34#1824—21no—
35#1824—21no—
36#1824—22no—
37#1824—21no—
38#1823—21no—
39#1821—24no—
40C2×#343C2214yes—
41C2×#343C2214yes—
42C2×#341C2214yes—
43#1736C2212yes—
44#1736C2212yes—
45C2×#340C2214yes—
46C24×C4—21no—
47C24×C4C2311yescharacteristic
48C22×M16—22no—
49C22×M16—21no—
50C22×M16—28no—
51C22×M16C2×C414yes—
52C22×M16C2312yes—
53C22×M16C2312yes—
54C22×M16C2312yes—
55C23×C8—21no—
56C23×C8C2312yes—
57C2×#117—21no—
58C2×#117—22no—
59C2×#117—22no—
60C2×#117—21no—
61C2×#117C2×C414yes—
62C2×#117D414yes—
63C2×#117D411yescharacteristic
64C2×#117D411yescharacteristic
65C2×#117C2311yescharacteristic
66C2×#117C2311yescharacteristic
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4C2×C414yes—
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C42—21no—
83#433—216no—
84#432—28no—
85#431—28no—
86#428—28no—
87#428—28no—
88#428—24no—
89#428—24no—
90#428C2×C418yes—
91#428C2×C418yes—
92C2×#107—28no—
93C2×#107—22no—
94C2×#107—22no—
95C2×#106—28no—
96C2×#106—22no—
97C2×#106—22no—
98#424—24no—
99#424—24no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—22no—
102#407—22no—
103#407—22no—
104#407—22no—
105#407D414yes—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8D411yescharacteristic
113C2×C22⋊C8D411yescharacteristic
114C2×C22⋊C8D412yes—
115#405—24no—
116#405—24no—
117C2×#97—22no—
118C2×C4×C8—22no—
119#380—21no—
120#380—21no—
121#380—21no—
122#380—21no—
123C2×#95—21no—
124#343—28no—
125#343—28no—
126#341—28no—
127#340—24no—
128#340—24no—
129#340—22no—
130#340—22no—
131#340—22no—
132#340—22no—
133C25#4111yescharacteristic
134C23×C4—24no—
135C23×C4—24no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—24no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—41no—
142C23×C4—41no—
143C23×C4—42no—
144C23×C4—84no—
145C23×C4C22×C412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D412yes—
152C23×C4#4111yescharacteristic
153C23×C4#4112yes—
154C23×C4C2411yescharacteristic
155C2×M16—216no—
156C2×M16—28no—
157C2×M16—24no—
158C2×M16—22no—
159C2×M16—24no—
160C2×M16—216no—
161C2×M16—22no—
162C2×M16—24no—
163C2×M16—24no—
164C2×M16—22no—
165C2×M16—24no—
166C2×M16—22no—
167C2×M16—24no—
168C2×M16C22×C414yes—
169C2×M16C2×D414yes—
170C2×M16#4118yes—
171C22×C8—28no—
172C22×C8—24no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—24no—
178C22×C8—24no—
179C22×C8—22no—
180C22×C8C2×D412yes—
181C22×C8C2×D412yes—
182#117—24no—
183#117—24no—
184#117—24no—
185#117—22no—
186#117—22no—
187#117—24no—
188#117—21no—
189#117—21no—
190#117—21no—
191#117—21no—
192#117C22⋊C414yes—
193#117C22⋊C414yes—
194#117C22×C412yes—
195#117C22×C412yes—
196#117C2×D412yes—
197#117C2×D412yes—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4C22⋊C414yes—
224C2×C4⋊C4C22⋊C414yes—
225C2×C4⋊C4C22×C414yes—
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C4⋊C4C2×D411yescharacteristic
229C2×C4⋊C4C2×D411yescharacteristic
230C2×C22⋊C4—24no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—82no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245#107—28no—
246#107—28no—
247#106—28no—
248#106—28no—
249C4⋊C8—28no—
250C4⋊C8—28no—
251C22⋊C8—22no—
252C22⋊C8—22no—
253C22⋊C8—24no—
254C22⋊C8—24no—
255C22⋊C8—24no—
256C22⋊C8—42no—
257C22⋊C8—42no—
258#97—28no—
259C4×C8—28no—
260#95—41no—
261#95—41no—
262C24—24no—
263C24—22no—
264C24—22no—
265C24—42no—
266C24C4×D412yes—
267C24#12111yescharacteristic
268C24#12111yescharacteristic
269C24#12312yes—
270C24C2×#4111yescharacteristic
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—28no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—28no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—42no—
298C22×C4—44no—
299C22×C4—82no—
300C22×C4—84no—
301C22×C4—84no—
302C22×C4—88no—
303C22×C4—84no—
304C22×C4—88no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C412yes—
309C22×C4C4×D412yes—
310C22×C4C4×D412yes—
311C22×C4C4×D412yes—
312C22×C4#12012yes—
313C22×C4#12012yes—
314C22×C4#12112yes—
315C22×C4#12112yes—
316C22×C4#12111yescharacteristic
317C22×C4#12111yescharacteristic
318C22×C4#12312yes—
319C22×C4C23×C411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C2×#4114yes—
325C22×C4C2×#4111yescharacteristic
326C22×C42−1+414yes—
327M16—216no—
328M16—24no—
329M16—28no—
330M16—28no—
331M16—42no—
332M16—42no—
333M16—44no—
334M16—44no—
335M16#11718yes—
336C2×C8—216no—
337C2×C8—24no—
338C2×C8—28no—
339C2×C8—22no—
340C2×C8—24no—
341C2×C8—22no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—44no—
347C2×C8—44no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—44no—
366C4⋊C4C2×C22⋊C412yes—
367C4⋊C4C2×C22⋊C412yes—
368C4⋊C4C2×C22⋊C412yes—
369C4⋊C4C2×C22⋊C412yes—
370C22⋊C4—22no—
371C22⋊C4—22no—
372C22⋊C4—22no—
373C22⋊C4—22no—
374C22⋊C4—24no—
375C22⋊C4—24no—
376C22⋊C4—82no—
377C22⋊C4—82no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—24no—
383C42—24no—
384C23—22no—
385C23—21no—
386C23—21no—
387C23—22no—
388C23—22no—
389C23—24no—
390C23—22no—
391C23—24no—
392C23—24no—
393C23—24no—
394C23—21no—
395C23—21no—
396C23—22no—
397C23—42no—
398C23—42no—
399C23—48no—
400C23—48no—
401C23#38612yes—
402C23#38612yes—
403C23#47112yes—
404C23#47312yes—
405C23C2×C4×D411yescharacteristic
406C23C2×#12111yescharacteristic
407C23C2×#12311yescharacteristic
408C23#55214yes—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—48no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—84no—
436C2×C4—88no—
437C2×C4—88no—
438C2×C4—84no—
439C2×C4—84no—
440C2×C4—84no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—164no—
445C2×C4#38612yes—
446C2×C4#38612yes—
447C2×C4C22×C22⋊C411yescharacteristic
448C2×C4C2×#11712yes—
449C2×C4C2×C4×D411yescharacteristic
450C2×C4#51912yes—
451C2×C4C2×#12011yescharacteristic
452C2×C4C2×#12111yescharacteristic
453C2×C4#52512yes—
454C2×C4#52512yes—
455C2×C4#53612yes—
456C2×C4#53612yes—
457C2×C4#54714yes—
458C2×C4#54714yes—
459C2×C4#55214yes—
460C8—28no—
461C8—42no—
462C8—42no—
463C8—44no—
464C8—82no—
465C22—21no—
466C22—21no—
467C22—22no—
468C22—21no—
469C22—21no—
470C22—22no—
471C22—24no—
472C22—24no—
473C22—24no—
474C22—24no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—48no—
479C22—48no—
480C22—42no—
481C22—42no—
482C22#190814yes—
483C22C2×#38611yescharacteristic
484C22#234412yes—
485C22#241312yes—
486C22#248412yes—
487C22C2×#47111yescharacteristic
488C22C2×#47311yescharacteristic
489C22#306812yes—
490C4—22no—
491C4—22no—
492C4—42no—
493C4—42no—
494C4—42no—
495C4—42no—
496C4—84no—
497C4—162no—
498C4#228011yescharacteristic
499C4#231011yescharacteristic
500C4#234412yes—
501C4#240612yes—
502C4#241312yes—
503C2—22no—
504C2—24no—
505C2—22no—
506C2—42no—
507C2#1387011yescharacteristic
508C2C2×#190811yescharacteristic
509C2#2100511yescharacteristic
510C2#2462212yes—
511C2#2462312yes—
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11216C2semidirect
7. #10896C2semidirect
21. C2×#428C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29242
C22×#4282
#207022
C2×#18242
C2×#18212
#112162
#108962
#107392
#107342