Finite Group Databasebeta

← #415331

Subgroups — #415331

Order 512.

1534 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3165C211yesmaximal, characteristic
2C2×#2862C211yesmaximal, characteristic
3#35281C211yesmaximal, characteristic
4C2×#1997C211yesmaximal, characteristic
5C2×#1994C211yesmaximal, characteristic
6#21313C211yesmaximal, characteristic
7#11339C218yesmaximal
8C4×#339C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3165—24no—
13#3165—24no—
14C2×#501—22no—
15C2×#500—21no—
16C2×#498—21no—
17C2×#428C2214yes—
18C2×#423C2211yescharacteristic
19C2×C4×M16C2211yescharacteristic
20#2868—21no—
21#2862—24no—
22#2862—24no—
23#2862—21no—
24#2660—22no—
25#2567—22no—
26#2546—22no—
27#2502—22no—
28#2434—22no—
29#2317C2214yes—
30C2×#391—22no—
31C2×#391—22no—
32C2×#382C2214yes—
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—24no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—21no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—22no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8D411yescharacteristic
68C2×C4⋊Q8D414yes—
69C2×C4⋊Q8D411yescharacteristic
70C2×#125D412yes—
71C2×C22⋊Q8—21no—
72C2×C22⋊Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×#117C2312yes—
76C2×#117C2312yes—
77C22×C4⋊C4—21no—
78C22×C4⋊C4D412yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C42—21no—
82C22×C42—21no—
83C22×C42C2312yes—
84C22×C42C2311yescharacteristic
85#501—216no—
86#500—28no—
87#498—28no—
88#428—28no—
89#428—28no—
90C2×#107—24no—
91C2×#106—24no—
92#423—22no—
93#423—22no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C4×M16—22no—
97C4×M16D414yes—
98C2×#97—21no—
99C2×C4×C8—21no—
100#398—28no—
101#395—28no—
102#391—24no—
103#391—216no—
104#391—216no—
105#391—24no—
106#389—28no—
107#385—28no—
108#384—24no—
109#384—24no—
110#382—24no—
111#382—28no—
112#382—28no—
113C4×C4⋊C4D412yes—
114C4×C4⋊C4D412yes—
115C4×C22⋊C4—21no—
116C4×C22⋊C4—21no—
117#376—21no—
118C2×#95—28no—
119C43—22no—
120C43—21no—
121C43D414yes—
122#343—28no—
123#340—28no—
124#339—28no—
125#339Q8116yescomplemented
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—22no—
129C22×Q8—24no—
130C22×Q8—24no—
131C22×Q8—22no—
132C22×Q8—24no—
133C22×Q8—48no—
134C22×Q8—48no—
135C22×Q8—41no—
136C22×Q8—41no—
137C22×Q8—42no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D414yes—
143C23×C4C2411yescharacteristic
144C2×M16—22no—
145C2×M16—24no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—28no—
149C2×M16C2×D414yes—
150C2×M16C2×Q818yes—
151C22×C8—22no—
152C22×C8—22no—
153C4⋊Q8—24no—
154C4⋊Q8—24no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—24no—
158C4⋊Q8—24no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—42no—
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—42no—
171C4⋊Q8—42no—
172C4⋊Q8—42no—
173C4⋊Q8—88no—
174#125—24no—
175#125—24no—
176C22⋊Q8—42no—
177C22⋊Q8—42no—
178C22⋊Q8—42no—
179C22⋊Q8—42no—
180C4×Q8—42no—
181C4×Q8—42no—
182C4×Q8—42no—
183C4×Q8—42no—
184#117—24no—
185#117—22no—
186#117—22no—
187#117—22no—
188#117—22no—
189#117C2×Q818yes—
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—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
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—216no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4C2×D414yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D414yes—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—24no—
223C2×C42—22no—
224C2×C42—24no—
225C2×C42—24no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—44no—
229C2×C42—44no—
230C2×C42C2×D414yes—
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D412yes—
238C2×C42C2×Q818yes—
239C2×C42#4112yes—
240#107—216no—
241#106—216no—
242C4⋊C8—42no—
243C4⋊C8—42no—
244#97—24no—
245C4×C8—24no—
246#95—232no—
247#95—216no—
248#95—22no—
249#95—22no—
250C24C22×D411yescharacteristic
251C2×Q8—416no—
252C2×Q8—416no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—44no—
256C2×Q8—44no—
257C2×Q8—44no—
258C2×Q8—44no—
259C2×Q8—44no—
260C2×Q8—44no—
261C2×Q8—48no—
262C2×Q8—84no—
263C2×Q8—88no—
264C2×Q8—84no—
265C2×Q8—88no—
266C2×Q8—84no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—28no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—48no—
281C22×C4—48no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4#12012yes—
285C22×C4#12012yes—
286C22×C4#12011yescharacteristic
287C22×C4#12011yescharacteristic
288C22×C4#12011yescharacteristic
289C22×C4#12011yescharacteristic
290C22×C4#12112yes—
291C22×C4#12112yes—
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C22×Q814yes—
298C22×C4C2×#4111yescharacteristic
299C22×C42−1+414yes—
300M16—24no—
301M16—28no—
302M16—44no—
303M16—44no—
304M16C4⋊Q818yescomplemented
305C2×C8—24no—
306C2×C8—28no—
307C2×C8—42no—
308C2×C8—44no—
309C4⋊C4—24no—
310C4⋊C4—22no—
311C4⋊C4—24no—
312C4⋊C4—22no—
313C4⋊C4—24no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—816no—
331C4⋊C4C4⋊Q818yes—
332C22⋊C4—28no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C42—24no—
336C42—28no—
337C42—28no—
338C42—48no—
339C42—48no—
340C42—42no—
341C42—42no—
342C42—44no—
343C42—44no—
344C42#12012yes—
345C42#12012yes—
346C42#12112yes—
347C42#12112yes—
348C42#12112yes—
349C42#12112yes—
350C42#12712yes—
351C42#12712yes—
352C42C4⋊Q818yescomplemented
353C23—22no—
354C23—22no—
355C23C2×#12011yescharacteristic
356C23C2×#12011yescharacteristic
357C23C2×#12111yescharacteristic
358C23D4×Q814yes—
359Q8—88no—
360Q8—88no—
361Q8—84no—
362Q8—88no—
363Q8—84no—
364Q8—88no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—28no—
372C2×C4—28no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—416no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—48no—
379C2×C4—48no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—48no—
386C2×C4—44no—
387C2×C4—84no—
388C2×C4—84no—
389C2×C4—84no—
390C2×C4#39212yes—
391C2×C4#39212yes—
392C2×C4#39212yes—
393C2×C4#39212yes—
394C2×C4#45414yes—
395C2×C4#45514yes—
396C2×C4C2×#12011yescharacteristic
397C2×C4C2×#12111yescharacteristic
398C2×C4C2×#12111yescharacteristic
399C2×C4C2×#12711yescharacteristic
400C2×C4C2×C4⋊Q812yes—
401C2×C4C2×C4⋊Q812yes—
402C2×C4C2×C4⋊Q812yes—
403C2×C4#53612yes—
404C2×C4#53612yes—
405C2×C4#53612yes—
406C2×C4D4×Q814yes—
407C2×C4D4×Q814yes—
408C2×C4#55714yes—
409C8—28no—
410C8—82no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—28no—
416C22C2×#39211yescharacteristic
417C22#243412yes—
418C22#243412yes—
419C22#254612yes—
420C22C2×#45512yes—
421C22C2×#45412yes—
422C4—22no—
423C4—22no—
424C4—48no—
425C4—48no—
426C4—84no—
427C4—82no—
428C4—84no—
429C4—82no—
430C4#200314yes—
431C4#200414yes—
432C4#243412yes—
433C4#254612yes—
434C4#254612yes—
435C4#265811yescharacteristic
436C4#286811yescharacteristic
437C2—22no—
438C2—22no—
439C2#1519011yescharacteristic
440C2C2×#200411yescharacteristic
441C2C2×#200311yescharacteristic
442C2#2582914yes—
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
125. #339Q8semidirect
304. M16C4⋊Q8semidirect
352. 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×#28622
#352812
C2×#19972
C2×#19942
#213132
#113392
C4×#3392