Finite Group Databasebeta

← #414893

Subgroups — #414893

Order 512.

1218 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35281C211yesmaximal, characteristic
2C2×#1997C211yesmaximal, characteristic
3#21312C211yesmaximal, characteristic
4C2×#1986C211yesmaximal, characteristic
5C4×#428C211yesmaximal, characteristic
6#14155C211yesmaximal, characteristic
7#11276C214yesmaximal
8#11217C214yesmaximal
9#10852C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C2×#428C2212yes—
13C2×#423C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15#2868—21no—
16#2862—21no—
17#2660—22no—
18#2567—22no—
19#2546—21no—
20#2546—21no—
21#2502—22no—
22#2434—22no—
23#2317C2214yes—
24C2×#391—22no—
25C4×#117C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1997—24no—
29#1989—22no—
30#1988—21no—
31#1987—21no—
32#1986—24no—
33#1776—22no—
34C4×#107—21no—
35C4×#106—21no—
36C2×#343C2212yes—
37C2×#343C2212yes—
38#1736C2212yes—
39#1736C2214yes—
40C2×#339C2212yes—
41C2×#339C2214yes—
42C2×#339C2212yes—
43#1445—22no—
44C4⋊#95C2214yes—
45C23×Q8D412yes—
46C22×M16C2312yes—
47C22×M16C2311yescharacteristic
48C22×M16C2311yescharacteristic
49#533—22no—
50#533—22no—
51#533—22no—
52#533—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8D411yescharacteristic
64C2×C4⋊Q8D411yescharacteristic
65C2×#125D412yes—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×#117C2312yes—
71C2×#117C2312yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4D412yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C42—21no—
77C22×C42C2312yes—
78C22×C42C2311yescharacteristic
79#428—24no—
80#428Q818yescomplemented
81C2×#107—22no—
82C2×#106—22no—
83#423—22no—
84#423—22no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C4×M16—22no—
88C4×M16D414yes—
89C2×#97—21no—
90C2×C4×C8—21no—
91#398—24no—
92#398—24no—
93#395—28no—
94#391—24no—
95#391—22no—
96#391—22no—
97#391—48no—
98#389—24no—
99#389—24no—
100#385—28no—
101#384—24no—
102#384—24no—
103#382—24no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4D412yes—
111C4×C4⋊C4D411yescharacteristic
112C4×C4⋊C4D411yescharacteristic
113C4×C22⋊C4—21no—
114C4×C22⋊C4—21no—
115#376—21no—
116C2×#95—28no—
117C2×#95—21no—
118C2×#95—21no—
119C43—21no—
120#343—24no—
121#343—24no—
122#340—24no—
123#340—24no—
124#340—22no—
125#340—22no—
126#339—24no—
127#339—24no—
128#339—28no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—24no—
134C22×Q8—24no—
135C22×Q8—44no—
136C22×Q8—44no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D414yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2411yescharacteristic
143C2×M16—21no—
144C2×M16—22no—
145C2×M16—21no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—24no—
150C2×M16—22no—
151C2×M16—24no—
152C2×M16C2×D412yes—
153C2×M16C2×D412yes—
154C2×M16C2×Q814yes—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—42no—
163C4⋊Q8—42no—
164C4⋊Q8—42no—
165C4⋊Q8—42no—
166C4⋊Q8—42no—
167C4⋊Q8—42no—
168C4⋊Q8—42no—
169C4⋊Q8—42no—
170C4⋊Q8—84no—
171C4⋊Q8—84no—
172#125—24no—
173#125—24no—
174C22⋊Q8—42no—
175C22⋊Q8—42no—
176C22⋊Q8—42no—
177C22⋊Q8—42no—
178C4×Q8—42no—
179C4×Q8—42no—
180C4×Q8—42no—
181C4×Q8—42no—
182#117—22no—
183#117—22no—
184#117—22no—
185#117—22no—
186#117—22no—
187#117—22no—
188#117C2×Q814yes—
189#117#4114yes—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×Q814yes—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C42—22no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—24no—
228C2×C42—22no—
229C2×C42—24no—
230C2×C42—44no—
231C2×C42—41no—
232C2×C42—41no—
233C2×C42C2×D412yes—
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42#4112yes—
239C2×C42#4112yes—
240C2×C42#4112yes—
241#107—28no—
242#106—28no—
243C4⋊C8—42no—
244C4⋊C8—42no—
245#97—24no—
246C4×C8—24no—
247#95—28no—
248#95—28no—
249#95—22no—
250#95—21no—
251#95—21no—
252#95—416no—
253#95—41no—
254#95—41no—
255#95—42no—
256C24C22×D411yescharacteristic
257C2×Q8—44no—
258C2×Q8—416no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—44no—
262C2×Q8—44no—
263C2×Q8—42no—
264C2×Q8—44no—
265C2×Q8—42no—
266C2×Q8—44no—
267C2×Q8—82no—
268C2×Q8—82no—
269C2×Q8—88no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—42no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—44no—
291C22×C4—42no—
292C22×C4#12011yescharacteristic
293C22×C4#12011yescharacteristic
294C22×C4#12011yescharacteristic
295C22×C4#12011yescharacteristic
296C22×C4#12112yes—
297C22×C4#12112yes—
298C22×C4#12312yes—
299C22×C4#12312yes—
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4C22×Q812yes—
304C22×C4C2×#4111yescharacteristic
305C22×C4C2×#4111yescharacteristic
306C22×C4C2×#4112yes—
307C22×C4C2×#4111yescharacteristic
308C22×C42−1+412yes—
309C22×C42−1+412yes—
310M16—22no—
311M16—24no—
312M16—22no—
313M16—22no—
314M16—42no—
315M16—42no—
316M16—42no—
317M16—44no—
318M16C4⋊Q814yescomplemented
319C2×C8—22no—
320C2×C8—24no—
321C2×C8—22no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—44no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—24no—
333C4⋊C4—48no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—44no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—88no—
347C4⋊C4—88no—
348C4⋊C4C22⋊Q814yes—
349C4⋊C4C22⋊Q814yes—
350C22⋊C4—24no—
351C22⋊C4—24no—
352C22⋊C4—42no—
353C22⋊C4—42no—
354C42—24no—
355C42—24no—
356C42—44no—
357C42—42no—
358C42—42no—
359C42—42no—
360C42—42no—
361C42—42no—
362C42—42no—
363C42—44no—
364C42—82no—
365C42—82no—
366C42#12112yes—
367C42#12112yes—
368C42#12112yes—
369C42#12112yes—
370C42#12312yes—
371C42#12312yes—
372C42#12412yes—
373C42#12412yes—
374C23—21no—
375C23—21no—
376C23—22no—
377C23C2×#12011yescharacteristic
378C23C2×#12111yescharacteristic
379C23C2×#12311yescharacteristic
380C23#54812yes—
381C23D4×Q812yes—
382Q8—84no—
383Q8—88no—
384Q8—84no—
385Q8—88no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—48no—
402C2×C4—48no—
403C2×C4—48no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—84no—
414C2×C4—82no—
415C2×C4—84no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—84no—
420C2×C4#39412yes—
421C2×C4#39412yes—
422C2×C4#39412yes—
423C2×C4#39412yes—
424C2×C4#45312yes—
425C2×C4#45512yes—
426C2×C4C2×#12111yescharacteristic
427C2×C4C2×#12111yescharacteristic
428C2×C4C2×C22⋊Q812yes—
429C2×C4C2×#12311yescharacteristic
430C2×C4C2×#12411yescharacteristic
431C2×C4C2×C4⋊Q811yescharacteristic
432C2×C4#53211yescharacteristic
433C2×C4#53312yes—
434C2×C4#53612yes—
435C2×C4#53611yescharacteristic
436C2×C4#53611yescharacteristic
437C2×C4#54112yes—
438C2×C4#54812yes—
439C2×C4D4×Q812yes—
440C2×C4#55612yes—
441C2×C4#55612yes—
442C2×C4#55712yes—
443C2×C4#55712yes—
444C8—24no—
445C8—42no—
446C8—82no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22C2×#39411yescharacteristic
456C22#240912yes—
457C22#244612yes—
458C22#254511yescharacteristic
459C22#254611yescharacteristic
460C22C2×#45311yescharacteristic
461C22C2×#45511yescharacteristic
462C22#302012yes—
463C4—22no—
464C4—22no—
465C4—22no—
466C4—44no—
467C4—44no—
468C4—84no—
469C4—84no—
470C4—84no—
471C4—82no—
472C4—82no—
473C4#203714yes—
474C4#244612yes—
475C4#254511yescharacteristic
476C4#254611yescharacteristic
477C4#256812yes—
478C4#262111yescharacteristic
479C4#285711yescharacteristic
480C2—21no—
481C2—22no—
482C2—21no—
483C2#1518011yescharacteristic
484C2C2×#203711yescharacteristic
485C2#2349411yescharacteristic
486C2#2432012yes—
487C2#2432212yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
80. #428Q8semidirect
318. M16C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#352812
C2×#19972
#213122
C2×#19862
C4×#4282
#141552
#112762
#112172
#108522