Finite Group Databasebeta

← #415332

Subgroups — #415332

Order 512.

1774 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3165C211yesmaximal, characteristic
2C2×#2868C211yesmaximal, characteristic
3#35281C211yesmaximal, characteristic
4C2×#1997C211yesmaximal, characteristic
5C2×#1994C211yesmaximal, characteristic
6#21313C211yesmaximal, characteristic
7#11340C218yesmaximal
8C4×#339C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8—22no—
11C22×C4⋊Q8C2211yescharacteristic
12C22×C4⋊Q8C2211yescharacteristic
13#3165—24no—
14#3165—24no—
15C2×#501—22no—
16C2×#500—21no—
17C2×#498—21no—
18C2×#428C2214yes—
19C2×#423C2211yescharacteristic
20C2×C4×M16C2211yescharacteristic
21#2868—24no—
22#2868—24no—
23#2868—21no—
24#2862—21no—
25#2660—22no—
26#2567—22no—
27#2546—22no—
28#2502—22no—
29#2434—22no—
30#2317C2214yes—
31C2×#384—22no—
32C2×#384C2214yes—
33C4×#117C2211yescharacteristic
34C2×C43C2211yescharacteristic
35#1997—24no—
36#1994—24no—
37#1988—22no—
38#1987—22no—
39C2×#343C2214yes—
40#1736C2214yes—
41C2×#339C2214yes—
42C2×#339C2214yes—
43C23×Q8—21no—
44C23×Q8D412yes—
45C22×M16C2312yes—
46C22×M16C2312yes—
47#533—22no—
48#533—22no—
49#533—22no—
50#533—22no—
51C2×C4⋊Q8—216no—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—416no—
69C2×C4⋊Q8D411yescharacteristic
70C2×C4⋊Q8D411yescharacteristic
71C2×C4⋊Q8D414yes—
72C2×#125D412yes—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8—21no—
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×#117C2312yes—
78C2×#117C2312yes—
79C22×C4⋊C4—28no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4D412yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C42—21no—
85C22×C42—21no—
86C22×C42C2312yes—
87C22×C42C2311yescharacteristic
88#501—216no—
89#500—28no—
90#498—28no—
91#428—28no—
92#428—28no—
93C2×#107—24no—
94C2×#106—24no—
95#423—22no—
96#423—22no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C4×M16—22no—
100C4×M16D414yes—
101C2×#97—21no—
102C2×C4×C8—21no—
103#398—28no—
104#395—28no—
105#391—24no—
106#391—24no—
107#389—28no—
108#385—28no—
109#384—24no—
110#384—216no—
111#384—24no—
112#384—28no—
113#384—28no—
114#382—24no—
115C4×C4⋊C4D412yes—
116C4×C4⋊C4D412yes—
117C4×C22⋊C4—21no—
118C4×C22⋊C4—21no—
119#376—21no—
120C43—21no—
121C43—22no—
122C43D414yes—
123#343—28no—
124#340—28no—
125#339—28no—
126#339Q8116yescomplemented
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—24no—
130C22×Q8—24no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—24no—
134C22×Q8—48no—
135C22×Q8—48no—
136C22×Q8—41no—
137C22×Q8—41no—
138C22×Q8—42no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D414yes—
144C23×C4C2411yescharacteristic
145C2×M16—22no—
146C2×M16—24no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—28no—
150C2×M16C2×D414yes—
151C2×M16C2×Q818yes—
152C22×C8—22no—
153C22×C8—22no—
154C4⋊Q8—24no—
155C4⋊Q8—24no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—24no—
159C4⋊Q8—24no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—464no—
163C4⋊Q8—42no—
164C4⋊Q8—42no—
165C4⋊Q8—42no—
166C4⋊Q8—42no—
167C4⋊Q8—42no—
168C4⋊Q8—42no—
169C4⋊Q8—42no—
170C4⋊Q8—42no—
171C4⋊Q8—42no—
172C4⋊Q8—42no—
173C4⋊Q8—42no—
174C4⋊Q8—42no—
175C4⋊Q8—88no—
176#125—24no—
177#125—24no—
178C22⋊Q8—42no—
179C22⋊Q8—42no—
180C22⋊Q8—42no—
181C22⋊Q8—42no—
182C4×Q8—42no—
183C4×Q8—42no—
184C4×Q8—42no—
185C4×Q8—42no—
186#117—24no—
187#117—22no—
188#117—22no—
189#117—22no—
190#117—22no—
191#117C2×Q818yes—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—232no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—216no—
214C2×C4⋊C4—432no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4C2×D414yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D414yes—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C42—28no—
225C2×C42—28no—
226C2×C42—24no—
227C2×C42—24no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—24no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42C2×D414yes—
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D412yes—
242C2×C42C2×Q818yes—
243C2×C42#4112yes—
244#107—216no—
245#106—216no—
246C4⋊C8—42no—
247C4⋊C8—42no—
248#97—24no—
249C4×C8—24no—
250#95—216no—
251#95—22no—
252#95—22no—
253C24C22×D411yescharacteristic
254C2×Q8—416no—
255C2×Q8—416no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—44no—
259C2×Q8—44no—
260C2×Q8—44no—
261C2×Q8—44no—
262C2×Q8—44no—
263C2×Q8—48no—
264C2×Q8—44no—
265C2×Q8—84no—
266C2×Q8—88no—
267C2×Q8—84no—
268C2×Q8—88no—
269C2×Q8—84no—
270C22×C4—28no—
271C22×C4—28no—
272C22×C4—24no—
273C22×C4—28no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—48no—
284C22×C4—48no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4#12012yes—
288C22×C4#12012yes—
289C22×C4#12011yescharacteristic
290C22×C4#12011yescharacteristic
291C22×C4#12011yescharacteristic
292C22×C4#12011yescharacteristic
293C22×C4#12112yes—
294C22×C4#12112yes—
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×Q814yes—
301C22×C4C2×#4111yescharacteristic
302C22×C42−1+414yes—
303M16—24no—
304M16—28no—
305M16—44no—
306M16—44no—
307M16C4⋊Q818yescomplemented
308C2×C8—24no—
309C2×C8—28no—
310C2×C8—42no—
311C2×C8—44no—
312C4⋊C4—24no—
313C4⋊C4—22no—
314C4⋊C4—24no—
315C4⋊C4—22no—
316C4⋊C4—24no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—464no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—816no—
335C4⋊C4C4⋊Q818yescomplemented
336C22⋊C4—28no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C42—216no—
340C42—216no—
341C42—24no—
342C42—28no—
343C42—28no—
344C42—42no—
345C42—42no—
346C42—44no—
347C42—44no—
348C42#12012yes—
349C42#12012yes—
350C42#12112yes—
351C42#12112yes—
352C42#12112yes—
353C42#12112yes—
354C42#12712yes—
355C42#12712yes—
356C42C4⋊Q818yescomplemented
357C23—22no—
358C23—22no—
359C23C2×#12011yescharacteristic
360C23C2×#12011yescharacteristic
361C23C2×#12111yescharacteristic
362C23D4×Q814yes—
363Q8—88no—
364Q8—88no—
365Q8—84no—
366Q8—88no—
367Q8—88no—
368Q8—84no—
369C2×C4—28no—
370C2×C4—28no—
371C2×C4—232no—
372C2×C4—28no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—28no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—48no—
383C2×C4—48no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—48no—
390C2×C4—44no—
391C2×C4—84no—
392C2×C4—84no—
393C2×C4—84no—
394C2×C4#39212yes—
395C2×C4#39212yes—
396C2×C4#39212yes—
397C2×C4#39212yes—
398C2×C4#45314yes—
399C2×C4#45614yes—
400C2×C4C2×#12011yescharacteristic
401C2×C4C2×#12111yescharacteristic
402C2×C4C2×#12111yescharacteristic
403C2×C4C2×#12711yescharacteristic
404C2×C4C2×C4⋊Q812yes—
405C2×C4C2×C4⋊Q812yes—
406C2×C4C2×C4⋊Q812yes—
407C2×C4#53612yes—
408C2×C4#53612yes—
409C2×C4#53612yes—
410C2×C4D4×Q814yes—
411C2×C4D4×Q814yes—
412C2×C4#55714yes—
413C8—28no—
414C8—82no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—28no—
420C22C2×#39211yescharacteristic
421C22#243412yes—
422C22#243412yes—
423C22#254612yes—
424C22C2×#45312yes—
425C22C2×#45612yes—
426C4—216no—
427C4—22no—
428C4—22no—
429C4—48no—
430C4—84no—
431C4—84no—
432C4—82no—
433C4—82no—
434C4#200214yes—
435C4#200514yes—
436C4#243412yes—
437C4#254612yes—
438C4#254612yes—
439C4#265811yescharacteristic
440C4#286811yescharacteristic
441C2—22no—
442C2—22no—
443C2#1519011yescharacteristic
444C2C2×#200211yescharacteristic
445C2C2×#200511yescharacteristic
446C2#2583014yes—
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
126. #339Q8semidirect
307. M16C4⋊Q8semidirect
335. C4⋊C4C4⋊Q8semidirect
356. C42C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#31652
C2×#28682
#352812
C2×#19972
C2×#19942
#213132
#113402
C4×#3392