Finite Group Databasebeta

← #6468693

Subgroups — #6468693

Order 512.

2188 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C212yesmaximal
2#29416C214yesmaximal, complemented
3#29244C214yesmaximal, complemented
4#28124C214yesmaximal, complemented
5#27600C211yesmaximal, characteristic
6#24003C212yesmaximal
7#23921C211yesmaximal, characteristic
8#23699C214yesmaximal, complemented
9C4×#483C214yesmaximal, complemented
10C4×#451C214yesmaximal, complemented
11#20714C211yesmaximal, characteristic
12C22×C22⋊Q8C2212yes—
13C22×C4×Q8C2212yes—
14C23×C42C2211yescharacteristic
15C2×#483C2214yes—
16C2×#451C2214yes—
17C2×C4×Q16C2214yescomplemented
18C2×#431C2212yes—
19C4×C22⋊Q8—22no—
20C4×C22⋊Q8—22no—
21C4×C22⋊Q8—22no—
22C4×C22⋊Q8—22no—
23C4×C22⋊Q8C2214yes—
24C4×C22⋊Q8C2214yes—
25C4×C22⋊Q8C2214yes—
26C4×C22⋊Q8C2214yes—
27C2×#391C2212yes—
28C2×#389C2212yes—
29C2×#387—22no—
30C2×#385—22no—
31C2×#380C2212yes—
32C2×C4×C4⋊C4—22no—
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C22⋊C4—22no—
35#2086C2214yescomplemented
36#2075C2214yes—
37#2066C2218yescomplemented
38#2056C2214yescomplemented
39#2035C2218yes—
40#2001C2214yes—
41#1927C2218yescomplemented
42#1920C2218yescomplemented
43#1878C2218yescomplemented
44#1872C2214yes—
45#1872C2214yes—
46#1863C2218yescomplemented
47#1853C2214yes—
48#1852C2214yes—
49#1823C2214yes—
50#1817C2212yes—
51#1797C2214yes—
52#1787C2214yes—
53#1779C2211yescharacteristic
54#1779C2211yescharacteristic
55C4×#106C2214yescomplemented
56C4×#103C2218yescomplemented
57C4×C22⋊C8C2214yes—
58C23×Q8C2312yes—
59C24×C4D411yescharacteristic
60C24×C4D411yescharacteristic
61C24×C4C2311yescharacteristic
62C22×Q16C2314yes—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—216no—
66C2×C22⋊Q8C2314yes—
67C2×C22⋊Q8C2314yes—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—48no—
71C2×C4×Q8C2314yes—
72C2×C4×Q8C2314yes—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C42—22no—
82C22×C42—22no—
83C22×C42—44no—
84C22×C42D412yes—
85C22×C42D412yes—
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42C2312yes—
89C22×C42C2311yescharacteristic
90C22×C42C2312yes—
91C22×C42C2311yescharacteristic
92#483—216no—
93#451—216no—
94C4×Q16—48no—
95#431—44no—
96C2×#107C2314yes—
97C2×#106C2318yes—
98C2×C4⋊C8C2314yes—
99C2×#103C2318yes—
100C2×#103C2318yes—
101C2×#103C2318yes—
102C2×C22⋊C8C2314yes—
103C2×C22⋊C8C2314yes—
104C2×C22⋊C8C2312yes—
105C2×C22⋊C8C2312yes—
106C2×C4×C8C2314yes—
107#391—22no—
108#391—22no—
109#391C2314yes—
110#391C2314yes—
111#389—22no—
112#389—22no—
113#389C2314yes—
114#389C2314yes—
115#387—24no—
116#387—24no—
117#387—24no—
118#387—24no—
119#385—24no—
120#385—24no—
121#385—24no—
122#385—24no—
123#380—22no—
124#380—22no—
125#380C2314yes—
126#380C2314yes—
127C4×C4⋊C4—24no—
128C4×C4⋊C4—24no—
129C4×C4⋊C4—24no—
130C4×C4⋊C4—24no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4C2314yes—
134C4×C4⋊C4C2314yes—
135C4×C22⋊C4—24no—
136C4×C22⋊C4—24no—
137C4×C22⋊C4—24no—
138C4×C22⋊C4—24no—
139C2×#95—22no—
140C2×#95—22no—
141C2×#95—22no—
142C2×#95—22no—
143C2×#95—22no—
144#340C2314yes—
145#340C2318yes—
146#340C2318yes—
147#340C2314yes—
148#336C2318yes—
149C25C2×D411yescharacteristic
150C22×Q8—28no—
151C22×Q8—22no—
152C22×Q8—28no—
153C22×Q8—44no—
154C22×Q8C2414yes—
155C23×C4—24no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—28no—
159C23×C4—28no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D412yes—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D412yes—
175C23×C4C2×D411yescharacteristic
176C23×C4C2411yescharacteristic
177C23×C4C2412yes—
178C2×Q16—216no—
179C2×Q16—44no—
180C22×C8C2414yes—
181C22×C8C2414yes—
182C22⋊Q8—216no—
183C22⋊Q8—216no—
184C22⋊Q8—28no—
185C22⋊Q8—28no—
186C22⋊Q8—28no—
187C22⋊Q8#41116yes—
188C4×Q8—216no—
189C4×Q8—48no—
190C4×Q8—48no—
191C4×Q8—48no—
192C2×C4⋊C4—216no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—48no—
206C2×C4⋊C4—48no—
207C2×C4⋊C4C2414yes—
208C2×C4⋊C4C2414yes—
209C2×C4⋊C4C2414yes—
210C2×C22⋊C4—216no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—48no—
216C2×C42—28no—
217C2×C42—24no—
218C2×C42—24no—
219C2×C42—24no—
220C2×C42—24no—
221C2×C42—24no—
222C2×C42—44no—
223C2×C42—44no—
224C2×C42—42no—
225C2×C42—42no—
226C2×C42—44no—
227C2×C42—42no—
228C2×C42—48no—
229C2×C42—48no—
230C2×C42C2×D412yes—
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D412yes—
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42C2412yes—
238C2×C42C2411yescharacteristic
239C2×C42C2411yescharacteristic
240#107—44no—
241#106—216no—
242C4⋊C8—44no—
243#103—216no—
244#103—216no—
245#103—48no—
246C22⋊C8—28no—
247C22⋊C8—28no—
248C22⋊C8—42no—
249C22⋊C8—42no—
250C4×C8—28no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—24no—
261C24—24no—
262C24—24no—
263C24—41no—
264C24—41no—
265C24#13514yes—
266C24C22×D411yescharacteristic
267C24C22×D412yes—
268C2×Q8—28no—
269C2×Q8—24no—
270C2×Q8—28no—
271C2×Q8—28no—
272C2×Q8—432no—
273C2×Q8—44no—
274C2×Q8—44no—
275C2×Q8C2×#4118yes—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—28no—
284C22×C4—28no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—416no—
313C22×C4—416no—
314C22×C4—416no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4#13718yes—
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D412yes—
329C22×C4C22×D412yes—
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D412yes—
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C2×#4118yes—
335C22×C4C2511yescharacteristic
336Q16—416no—
337C2×C8—28no—
338C2×C8—28no—
339C2×C8—24no—
340C2×C8—24no—
341C4⋊C4—28no—
342C4⋊C4—216no—
343C4⋊C4—216no—
344C4⋊C4—24no—
345C4⋊C4—28no—
346C4⋊C4—28no—
347C4⋊C4—48no—
348C4⋊C4—44no—
349C4⋊C4—48no—
350C4⋊C4—44no—
351C4⋊C4—48no—
352C4⋊C4—48no—
353C4⋊C4C2×#4118yes—
354C22⋊C4—216no—
355C22⋊C4—216no—
356C22⋊C4—48no—
357C22⋊C4—48no—
358C42—22no—
359C42—22no—
360C42—24no—
361C42—24no—
362C42—22no—
363C42—24no—
364C42—42no—
365C42—42no—
366C42—42no—
367C42—48no—
368C42—48no—
369C42—48no—
370C42—48no—
371C42H(5,𝔽2)14yes—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—216no—
377C23—22no—
378C23—21no—
379C23—22no—
380C23—21no—
381C23—44no—
382C23—44no—
383C23—44no—
384C23—48no—
385C23#52514yes—
386C23C2×#13512yes—
387C23C2×#13514yes—
388C23C2×#13714yes—
389C23C23×D411yescharacteristic
390Q8—216no—
391Q8—416no—
392Q8—48no—
393C2×C4—28no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—216no—
403C2×C4—216no—
404C2×C4—28no—
405C2×C4—28no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—48no—
416C2×C4—48no—
417C2×C4—48no—
418C2×C4—48no—
419C2×C4—416no—
420C2×C4—44no—
421C2×C4—416no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—88no—
435C2×C4—88no—
436C2×C4#52514yes—
437C2×C4#53412yes—
438C2×C4#53412yes—
439C2×C4C2×#13714yes—
440C2×C4C2×#13714yes—
441C2×C4#57512yes—
442C2×C4C22×#4114yes—
443C2×C4C2×H(5,𝔽2)12yes—
444C8—44no—
445C8—44no—
446C22—24no—
447C22—24no—
448C22—24no—
449C22—24no—
450C22—24no—
451C22—24no—
452C22—22no—
453C22—24no—
454C22—416no—
455C22—41no—
456C22—41no—
457C22—42no—
458C22#310818yes—
459C22C2×#52512yes—
460C22C2×#53411yescharacteristic
461C22C22×#13511yescharacteristic
462C22C22×#13712yes—
463C22C2×#57511yescharacteristic
464C4—22no—
465C4—22no—
466C4—24no—
467C4—24no—
468C4—42no—
469C4—41no—
470C4—41no—
471C4—48no—
472C4—48no—
473C4—84no—
474C4—84no—
475C4#319514yes—
476C4#348114yes—
477C2—28no—
478C2—42no—
479C2#3495611yescharacteristic
480C2C2×#310812yes—
481C2#6056212yes—
482C2#6073411yescharacteristic
483C2C2×#319511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #29416C2semidirect
3. #29244C2semidirect
4. #28124C2semidirect
8. #23699C2semidirect
9. C4×#483C2semidirect
10. C4×#451C2semidirect
17. C2×C4×Q16C22semidirect
35. #2086C22semidirect
37. #2066C22semidirect
38. #2056C22semidirect
41. #1927C22semidirect
42. #1920C22semidirect
43. #1878C22semidirect
46. #1863C22semidirect
55. C4×#106C22semidirect
56. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
#294162
#292442
#281242
#276002
#240032
#239212
#236992
C4×#4832
C4×#4512
#207142