Finite Group Databasebeta

← #4464008

Subgroups — #4464008

Order 512.

2726 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2626C212yesmaximal
2#17198C211yesmaximal, characteristic, complemented
3#17168C211yesmaximal, characteristic, complemented
4#17124C214yesmaximal, complemented
5#17122C214yesmaximal, complemented
6#17110C214yesmaximal, complemented
7#17106C212yesmaximal, complemented
8#14826C214yesmaximal, complemented
9#14820C214yesmaximal
10#14758C214yesmaximal, complemented
11C2×#1440C211yesmaximal, characteristic
12C22×C22⋊Q8C2212yes—
13C23×C22⋊C4C2212yes—
14C23×C22⋊C4C2211yescharacteristic
15#2626—216no—
16C2×#400C2214yescomplemented
17C2×#399C2212yes—
18C2×#399C2212yes—
19C2×#397C2214yes—
20C2×#397C2214yes—
21C2×#396C2214yes—
22C2×#396C2214yes—
23C2×#395C2214yescomplemented
24C2×#393C2212yes—
25C2×#393C2212yes—
26C2×#385C2214yes—
27C2×#380C2212yes—
28C2×#380C2214yes—
29C2×#379C2214yes—
30C22×#95C2214yes—
31#1446C2214yescomplemented
32#1445C2214yescomplemented
33#1444C2214yescomplemented
34#1444C2214yescomplemented
35#1444C2214yescomplemented
36#1443C2214yescomplemented
37#1443C2214yescomplemented
38#1442C2214yescomplemented
39#1442C2214yescomplemented
40#1442C2214yescomplemented
41#1441C2214yescomplemented
42#1441C2214yescomplemented
43#1441C2214yescomplemented
44#1441C2214yescomplemented
45#1440C2211yescharacteristic
46#1440C2211yescharacteristic
47#1440C2214yes—
48#1440C2212yes—
49#1440C2214yes—
50#1440C2214yes—
51#1440C2214yes—
52#1440C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55C22⋊#95C2214yes—
56C22⋊#95C2214yes—
57C22⋊#95C2214yes—
58C26C2311yescharacteristic
59C23×Q8C2312yes—
60C24×C4C2314yes—
61C24×C4C2312yes—
62C2×C22⋊Q8—28no—
63C2×C22⋊Q8—44no—
64C2×C22⋊Q8—44no—
65C22×C4⋊C4C2314yes—
66C22×C4⋊C4C2314yes—
67C22×C4⋊C4C2314yes—
68C22×C4⋊C4C2314yes—
69C22×C4⋊C4C2314yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2314yes—
72C22×C22⋊C4—48no—
73C22×C22⋊C4—44no—
74C22×C22⋊C4—44no—
75C22×C22⋊C4D418yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2314yes—
78C22×C22⋊C4C2312yes—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2312yes—
85C22×C22⋊C4C2311yescharacteristic
86C22×C22⋊C4C2312yes—
87C22×C22⋊C4C2311yescharacteristic
88#400—48no—
89#399—28no—
90#399—28no—
91#397—48no—
92#397—48no—
93#396—216no—
94#396—216no—
95#395—48no—
96#393—28no—
97#393—28no—
98#385—216no—
99#380—28no—
100#380—48no—
101#379—216no—
102C2×#95—48no—
103C2×#95C2314yes—
104C2×#95C2314yes—
105C2×#95C2312yes—
106C2×#95C2312yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C25—24no—
126C25—28no—
127C25—44no—
128C25—44no—
129C25C2×D414yes—
130C25C2411yescharacteristic
131C25C2411yescharacteristic
132C25C2411yescharacteristic
133C22×Q8—24no—
134C22×Q8—42no—
135C22×Q8—42no—
136C22×Q8—44no—
137C22×Q8—44no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—28no—
141C23×C4—28no—
142C23×C4—28no—
143C23×C4—28no—
144C23×C4—44no—
145C23×C4—44no—
146C23×C4—44no—
147C23×C4—44no—
148C23×C4—44no—
149C23×C4C2×D418yes—
150C23×C4C2414yes—
151C23×C4C2414yes—
152C23×C4C2414yes—
153C23×C4C2414yes—
154C23×C4C2414yes—
155C23×C4C2412yes—
156C23×C4C2412yes—
157C23×C4C2414yes—
158C22⋊Q8—432no—
159C2×C4⋊C4—28no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—28no—
182C2×C22⋊C4—28no—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—28no—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—416no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—432no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4C2×D414yes—
212C2×C22⋊C4C2×D418yes—
213C2×C22⋊C4C2×D414yes—
214#95—24no—
215#95—24no—
216#95—28no—
217#95—28no—
218#95—28no—
219#95—28no—
220#95—28no—
221#95—28no—
222#95—28no—
223#95—44no—
224#95—44no—
225#95—44no—
226#95—44no—
227#95—44no—
228#95—44no—
229#95—44no—
230#95—44no—
231#95—44no—
232#95—44no—
233#95—44no—
234#95—44no—
235#95—44no—
236C24—24no—
237C24—24no—
238C24—28no—
239C24—24no—
240C24—24no—
241C24—22no—
242C24—22no—
243C24—24no—
244C24—24no—
245C24—24no—
246C24—24no—
247C24—216no—
248C24—416no—
249C24—48no—
250C24—48no—
251C24—416no—
252C24—416no—
253C24—88no—
254C24—88no—
255C24—88no—
256C24—88no—
257C24C22×D412yes—
258C24C22×D412yes—
259C24C22×D412yes—
260C24H(5,𝔽2)14yes—
261C242−1+418yes—
262C24C2511yescharacteristic
263C2×Q8—48no—
264C2×Q8—88no—
265C2×Q8—84no—
266C2×Q8—84no—
267C2×Q8—88no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—216no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—416no—
319C22×C4—416no—
320C22×C4—416no—
321C22×C4—416no—
322C22×C4—416no—
323C22×C4—84no—
324C22×C4—84no—
325C22×C4—88no—
326C22×C4—88no—
327C22×C4—88no—
328C22×C4—88no—
329C22×C4C22×D414yes—
330C22×C4C22×D414yes—
331C22×C4H(5,𝔽2)18yes—
332C4⋊C4—416no—
333C4⋊C4—48no—
334C4⋊C4—88no—
335C4⋊C4—88no—
336C4⋊C4—88no—
337C4⋊C4—88no—
338C4⋊C4—88no—
339C22⋊C4—44no—
340C22⋊C4—48no—
341C22⋊C4—44no—
342C22⋊C4—48no—
343C22⋊C4—416no—
344C22⋊C4—48no—
345C22⋊C4—416no—
346C22⋊C4—416no—
347C22⋊C4—88no—
348C22⋊C4—88no—
349C22⋊C4—88no—
350C22⋊C4—88no—
351C23—28no—
352C23—24no—
353C23—24no—
354C23—28no—
355C23—28no—
356C23—28no—
357C23—28no—
358C23—28no—
359C23—22no—
360C23—22no—
361C23—24no—
362C23—24no—
363C23—28no—
364C23—28no—
365C23—432no—
366C23—416no—
367C23—44no—
368C23—44no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—44no—
374C23—44no—
375C23—44no—
376C23—44no—
377C23—44no—
378C23—44no—
379C23—44no—
380C23—44no—
381C23—44no—
382C23—44no—
383C23—832no—
384C23—832no—
385C23—832no—
386C23#53414yes—
387C23#53618yes—
388C23#563116yes—
389C23C23×D411yescharacteristic
390C23C2×H(5,𝔽2)12yes—
391C23C2×H(5,𝔽2)14yes—
392C23C2×2−1+414yes—
393C23#58514yes—
394Q8—816no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—28no—
398C2×C4—28no—
399C2×C4—28no—
400C2×C4—28no—
401C2×C4—28no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
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—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—816no—
448C2×C4—832no—
449C22—216no—
450C22—28no—
451C22—216no—
452C22—216no—
453C22—42no—
454C22—42no—
455C22—44no—
456C22—44no—
457C22—48no—
458C22—48no—
459C22—832no—
460C22—84no—
461C22—84no—
462C22—84no—
463C22—84no—
464C22#281718yes—
465C22C2×#53411yescharacteristic
466C22C2×#53612yes—
467C22#349614yes—
468C22C2×#56314yes—
469C22#353212yes—
470C22#353312yes—
471C22#353514yes—
472C22#353614yes—
473C22#354214yes—
474C22#356114yes—
475C22#356414yes—
476C4—48no—
477C4—44no—
478C4—44no—
479C4—84no—
480C4—84no—
481C4—84no—
482C4—84no—
483C4—84no—
484C2—28no—
485C2—82no—
486C2—82no—
487C2C2×#281711yescharacteristic
488C2#3673712yes—
489C2#3674014yes—
490C2#5231414yes—
491C2#5236814yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17198C2semidirect
3. #17168C2semidirect
4. #17124C2semidirect
5. #17122C2semidirect
6. #17110C2semidirect
7. #17106C2semidirect
8. #14826C2semidirect
10. #14758C2semidirect
16. C2×#400C22semidirect
23. C2×#395C22semidirect
31. #1446C22semidirect
32. #1445C22semidirect
33. #1444C22semidirect
34. #1444C22semidirect
35. #1444C22semidirect
36. #1443C22semidirect
37. #1443C22semidirect
38. #1442C22semidirect
39. #1442C22semidirect
40. #1442C22semidirect
41. #1441C22semidirect
42. #1441C22semidirect
43. #1441C22semidirect
44. #1441C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26262
#171982
#171682
#171242
#171222
#171102
#171062
#148262
#148202
#147582
C2×#14402