Finite Group Databasebeta

← #6470262

Subgroups — #6470262

Order 512.

2930 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3007C212yesmaximal
2#35011C211yesmaximal, characteristic, complemented
3C2×#2341C211yesmaximal, characteristic
4#32906C214yesmaximal, complemented
5#29714C214yesmaximal, complemented
6#24546C218yesmaximal, complemented
7#24374C218yesmaximal, complemented
8#21934C211yesmaximal, characteristic, complemented
9#21765C212yesmaximal, complemented
10C23×#41C2211yescharacteristic
11C22×#137C2214yes—
12C22×C22⋊Q8C2212yes—
13C22×C4×Q8C2212yes—
14C2×#462C2218yescomplemented
15#3007—216no—
16C2×#452C2218yes—
17C2×#451C2214yes—
18C2×#450C2214yes—
19C2×#417C2214yes—
20C2×#407C2212yes—
21#2857C2214yes—
22#2659—21no—
23#2659—21no—
24#2583—24no—
25#2502C2214yescomplemented
26#2434C2214yescomplemented
27#2341—24no—
28#2341C2218yes—
29#2315—22no—
30C2×#390—22no—
31C2×#390C2212yes—
32C2×#380—21no—
33C2×#380C2211yescharacteristic
34C2×C4×C22⋊C4—21no—
35C2×C4×C22⋊C4C2211yescharacteristic
36#2071C2218yescomplemented
37#2055C22116yescomplemented
38#2013C2218yescomplemented
39#1894C2218yescomplemented
40#1878C22116yescomplemented
41#1872C2218yes—
42#1851C2218yescomplemented
43#1823C2214yescomplemented
44#1803C2218yescomplemented
45#1797C2214yes—
46#1787C2214yescomplemented
47C22×#41—22no—
48C22×#41—24no—
49C22×#41—48no—
50C22×#41D418yes—
51C22×#41C2314yes—
52C23×Q8C2311yescharacteristic
53C23×Q8C2312yes—
54C23×D4—21no—
55C23×D4C2311yescharacteristic
56C24×C4—21no—
57C24×C4C2311yescharacteristic
58C2×#137—216no—
59C2×#137—416no—
60C22×Q16C2318yes—
61C22×SD16C2318yes—
62C22×M16C2314yes—
63C2×C4⋊Q8C2314yes—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—28no—
67C2×C22⋊Q8—28no—
68C2×C22⋊Q8C2314yes—
69C2×C22⋊Q8C2314yes—
70C2×C4×Q8—24no—
71C2×C4×Q8—48no—
72C2×C4×Q8C2318yes—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4D412yes—
82C22×C42—22no—
83C22×C42C2312yes—
84#462—416no—
85#452—232no—
86#451—216no—
87#450—216no—
88C2×C4⋊C8C2318yes—
89#417—216no—
90C2×#103C2318yes—
91C2×#103C23116yes—
92C2×#102C2318yes—
93#407—28no—
94C2×C22⋊C8C2314yes—
95C2×C22⋊C8C2314yes—
96#398—24no—
97#398—24no—
98#395—24no—
99#395—24no—
100#391C2318yes—
101#390—216no—
102#390—24no—
103#390C2318yes—
104#389—24no—
105#389—24no—
106#385—28no—
107#385—28no—
108#384C2314yes—
109#380—24no—
110#380—24no—
111#380—22no—
112#380—28no—
113#380C2314yes—
114C4×C22⋊C4—28no—
115C4×C22⋊C4D414yes—
116C4×C22⋊C4C2314yes—
117C2×#95—21no—
118C2×#95D411yescharacteristic
119C2×#95D411yescharacteristic
120#340C23116yes—
121#336C2318yes—
122C25—21no—
123C25C2×D411yescharacteristic
124C2×#41—216no—
125C2×#41—28no—
126C2×#41—216no—
127C2×#41—48no—
128C2×#41—432no—
129C2×#41—432no—
130C2×#41—416no—
131C2×#41C2×D4116yes—
132C22×Q8—28no—
133C22×Q8—21no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—24no—
137C22×Q8—24no—
138C22×Q8—48no—
139C22×Q8C2×D414yes—
140C22×Q8C2414yes—
141C22×Q8C2412yes—
142C22×D4—28no—
143C22×D4—24no—
144C22×D4—21no—
145C22×D4—22no—
146C22×D4—44no—
147C22×D4—48no—
148C22×D4C2×D414yes—
149C22×D4C2412yes—
150C23×C4—21no—
151C23×C4—28no—
152C23×C4—24no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—44no—
157C23×C4—48no—
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D414yes—
162C23×C4C2×D412yes—
163C23×C4#4114yes—
164C23×C4C2411yescharacteristic
165C23×C4C2412yes—
166#137—464no—
167C2×Q16—216no—
168C2×Q16—232no—
169C2×SD16—216no—
170C2×SD16—416no—
171C2×M16—28no—
172C2×M16—48no—
173C22×C8C2418yes—
174C4⋊Q8—48no—
175C22⋊Q8—28no—
176C22⋊Q8—28no—
177C22⋊Q8—28no—
178C22⋊Q8—28no—
179C22⋊Q8—48no—
180C22⋊Q8—48no—
181C4×Q8—216no—
182C4×Q8—48no—
183C4×Q8—416no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—48no—
197C2×C4⋊C4—48no—
198C2×C4⋊C4C2414yes—
199C2×C4⋊C4C2414yes—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—48no—
207C2×C22⋊C4—48no—
208C2×C22⋊C4—84no—
209C2×C22⋊C4—82no—
210C2×C22⋊C4C2×D418yes—
211C2×C42—28no—
212C2×C42—44no—
213C2×C42—48no—
214C2×C42C2×D414yes—
215C2×C42C2414yes—
216C4⋊C8—48no—
217#103—216no—
218#103—416no—
219#102—216no—
220C22⋊C8—28no—
221C22⋊C8—28no—
222#95—24no—
223#95—28no—
224#95—28no—
225#95C2×D414yes—
226#95C2×D414yes—
227C24—24no—
228C24—24no—
229C24—22no—
230C24—41no—
231C24—42no—
232C24—41no—
233C24—42no—
234C24—44no—
235C24—42no—
236C24—44no—
237C24#12114yes—
238C24C22×D412yes—
239C24C22×D411yescharacteristic
240#41—232no—
241#41—416no—
242#41—432no—
243#41—464no—
244C2×Q8—28no—
245C2×Q8—216no—
246C2×Q8—216no—
247C2×Q8—24no—
248C2×Q8—22no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—216no—
253C2×Q8—416no—
254C2×Q8—44no—
255C2×Q8—84no—
256C2×Q8—88no—
257C2×Q8C22×D414yes—
258C2×Q8H(5,𝔽2)18yes—
259C2×D4—216no—
260C2×D4—24no—
261C2×D4—22no—
262C2×D4—28no—
263C2×D4—48no—
264C2×D4—416no—
265C2×D4—432no—
266C2×D4—44no—
267C2×D4—48no—
268C2×D4C22×D414yes—
269C22×C4—28no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—28no—
274C22×C4—24no—
275C22×C4—216no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—48no—
280C22×C4—48no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—44no—
284C22×C4—42no—
285C22×C4—416no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—432no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4—84no—
297C22×C4#12118yes—
298C22×C4#12114yes—
299C22×C4#13718yescomplemented
300C22×C4C22×D414yes—
301C22×C4C22×D412yes—
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D412yes—
304C22×C4C22×D414yes—
305C22×C4C22×D414yes—
306C22×C4C2×#4114yes—
307C22×C4C2×#4112yes—
308C22×C4C2511yescharacteristic
309Q16—432no—
310SD16—432no—
311M16—416no—
312C2×C8—28no—
313C2×C8—216no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—28no—
317C4⋊C4—416no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—48no—
321C4⋊C4—48no—
322C4⋊C4—84no—
323C4⋊C4—84no—
324C4⋊C4—84no—
325C4⋊C4—84no—
326C22⋊C4—216no—
327C22⋊C4—48no—
328C22⋊C4—48no—
329C22⋊C4—416no—
330C22⋊C4—88no—
331C22⋊C4—84no—
332C42—28no—
333C42—44no—
334C42—416no—
335C23—21no—
336C23—21no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—24no—
341C23—24no—
342C23—28no—
343C23—24no—
344C23—44no—
345C23—44no—
346C23—48no—
347C23—48no—
348C23—42no—
349C23—42no—
350C23—44no—
351C23—48no—
352C23—44no—
353C23—84no—
354C23—84no—
355C23—88no—
356C23—88no—
357C23C2×#12114yes—
358C23C2×#12112yes—
359C23C2×#13714yes—
360C23C23×D411yescharacteristic
361Q8—216no—
362Q8—24no—
363Q8—28no—
364Q8—48no—
365Q8—48no—
366Q8—88no—
367Q8#53418yes—
368D4—28no—
369D4—432no—
370D4—48no—
371D4—44no—
372C2×C4—216no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—28no—
377C2×C4—22no—
378C2×C4—24no—
379C2×C4—28no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—28no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—44no—
387C2×C4—48no—
388C2×C4—48no—
389C2×C4—432no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—48no—
397C2×C4—44no—
398C2×C4—84no—
399C2×C4—88no—
400C2×C4—88no—
401C2×C4—88no—
402C2×C4C2×#12112yes—
403C2×C4C2×#12114yes—
404C2×C4C2×#13718yes—
405C2×C4C23×D411yescharacteristic
406C2×C4C22×#4111yescharacteristic
407C2×C4C2×H(5,𝔽2)14yes—
408C8—48no—
409C22—22no—
410C22—24no—
411C22—24no—
412C22—22no—
413C22—24no—
414C22—24no—
415C22—24no—
416C22—44no—
417C22—44no—
418C22—44no—
419C22—44no—
420C22—44no—
421C22—88no—
422C22—816no—
423C22—84no—
424C22—84no—
425C22#303914yes—
426C22C22×#12111yescharacteristic
427C22#348412yes—
428C22C22×#13712yes—
429C22#358511yescharacteristic
430C22#358611yescharacteristic
431C4—28no—
432C4—28no—
433C4—24no—
434C4—22no—
435C4—44no—
436C4—84no—
437C4—84no—
438C4—84no—
439C4C2×#53412yes—
440C4#349012yes—
441C2—24no—
442C2—44no—
443C2—84no—
444C2#3498111yescharacteristic
445C2C2×#303911yescharacteristic
446C2#6042511yescharacteristic
447C2#6157312yes—
448C2#6158812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35011C2semidirect
4. #32906C2semidirect
5. #29714C2semidirect
6. #24546C2semidirect
7. #24374C2semidirect
8. #21934C2semidirect
9. #21765C2semidirect
14. C2×#462C22semidirect
25. #2502C22semidirect
26. #2434C22semidirect
36. #2071C22semidirect
37. #2055C22semidirect
38. #2013C22semidirect
39. #1894C22semidirect
40. #1878C22semidirect
42. #1851C22semidirect
43. #1823C22semidirect
44. #1803C22semidirect
46. #1787C22semidirect
299. C22×C4#137semidirect

Maximal subgroups

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

subgroupindex
C2×#30072
#350112
C2×#23412
#329062
#297142
#245462
#243742
#219342
#217652