Finite Group Databasebeta

← #6506764

Subgroups — #6506764

Order 512.

1806 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic
2C2×#2442C211yesmaximal, characteristic
3#29302C212yesmaximal, complemented
4#29301C212yesmaximal, complemented
5#28775C218yesmaximal, complemented
6#27956C212yesmaximal, complemented
7#26068C218yesmaximal, complemented
8#25819C212yesmaximal, complemented
9#21940C212yesmaximal
10C2×#1853C212yesmaximal
11#20526C211yesmaximal, characteristic
12C2×#543—21no—
13C22×#125C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C2×#501C2214yescomplemented
16C22×#106C2214yes—
17C22×C22⋊C8C2212yes—
18#2857C2212yes—
19#2857C2212yes—
20#2659—21no—
21#2659—21no—
22#2549—24no—
23#2442—24no—
24#2442C2218yes—
25#2440C2214yes—
26#2440C2214yes—
27C2×#400—21no—
28C2×#398—21no—
29#2315—21no—
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#385—21no—
33C2×#382C2211yescharacteristic
34C2×#380—21no—
35C2×#380C2212yes—
36C2×#380C2211yescharacteristic
37C2×C4×C22⋊C4C2211yescharacteristic
38#2085C2218yescomplemented
39#2084C2214yescomplemented
40#2083C2214yescomplemented
41#2059C2218yescomplemented
42#2058C2214yescomplemented
43#2057C2214yescomplemented
44#1993C2214yescomplemented
45#1992C2214yescomplemented
46#1944C2218yescomplemented
47#1943C2218yescomplemented
48#1853C2214yes—
49#1853C2218yes—
50#1853C2214yes—
51#1852C2214yes—
52#1852C2214yes—
53#1852C2214yes—
54#1852C2214yes—
55#1847C2214yescomplemented
56#1808C2218yes—
57#1752C2218yes—
58#1749C2212yes—
59#1749C2212yes—
60C2×#340C2214yes—
61C24×C4—21no—
62C24×C4C2311yescharacteristic
63C23×C8C2314yes—
64#543—48no—
65C2×C4⋊Q8C2312yes—
66C2×C4⋊Q8C2312yes—
67C2×#125—22no—
68C2×#125—44no—
69C2×#125C2314yes—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—22no—
72C2×C22⋊Q8—24no—
73C2×C22⋊Q8—24no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—44no—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C22⋊C4—22no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4D411yescharacteristic
91C22×C22⋊C4D411yescharacteristic
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94#501—48no—
95C2×#107C2318yes—
96C2×#107C2318yes—
97C2×#106—48no—
98C2×#106C2318yes—
99C2×#106C2318yes—
100C2×C22⋊C8Q8116yescomplemented
101C2×C22⋊C8C2314yes—
102C2×C22⋊C8C2314yes—
103C2×#97C2314yes—
104C2×#97C2314yes—
105#400—28no—
106#398—28no—
107#398—24no—
108#398—24no—
109#395—24no—
110#395—24no—
111#391—22no—
112#391—22no—
113#389—22no—
114#389—22no—
115#389—22no—
116#389—22no—
117#389C2314yes—
118#389C2314yes—
119#389C2314yes—
120#389C2314yes—
121#385—24no—
122#385—24no—
123#385—28no—
124#384C2312yes—
125#384C2312yes—
126#382—22no—
127#382C2314yes—
128#380—22no—
129#380—22no—
130#380—22no—
131#380—22no—
132#380—22no—
133#380—28no—
134#380C2314yes—
135#380C2314yes—
136#380C2314yes—
137C4×C22⋊C4D414yes—
138C4×C22⋊C4C2314yes—
139C2×#95—21no—
140C2×#95—21no—
141C2×#95—21no—
142C2×#95—21no—
143C2×#95—21no—
144C2×#95—22no—
145C2×#95D411yescharacteristic
146C2×#95D411yescharacteristic
147#340C2318yes—
148#340C2318yes—
149#340C2318yes—
150#340C2318yes—
151#336C2314yes—
152#336C2314yes—
153C25C2×D411yescharacteristic
154C22×Q8—22no—
155C22×Q8—22no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—24no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—44no—
166C23×C4—44no—
167C23×C4—82no—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D412yes—
174C23×C4C2×Q818yes—
175C23×C4C2412yes—
176C23×C4C2411yescharacteristic
177C22×C8—44no—
178C22×C8—44no—
179C22×C8C2×Q8116yes—
180C22×C8C2418yes—
181C4⋊Q8—44no—
182C4⋊Q8—44no—
183#125—28no—
184#125—44no—
185#125—48no—
186C22⋊Q8—416no—
187C22⋊Q8—416no—
188C22⋊Q8—48no—
189C22⋊Q8—48no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—48no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—82no—
222C2×C4⋊C4—82no—
223C2×C4⋊C4C2412yes—
224C2×C4⋊C4C2412yes—
225C2×C4⋊C4C2414yes—
226C2×C4⋊C4C2414yes—
227C2×C4⋊C4C2412yes—
228C2×C4⋊C4C2412yes—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—48no—
236C2×C22⋊C4C2×D414yes—
237C2×C22⋊C4C2×D414yes—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2412yes—
243C2×C42C2412yes—
244#107—48no—
245#107—48no—
246#106—216no—
247#106—48no—
248C22⋊C8C2×Q8116yes—
249C22⋊C8C2×Q8116yes—
250#97—44no—
251#97—44no—
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—
261#95C2×D414yes—
262#95C2×D414yes—
263C24—41no—
264C24—41no—
265C24—41no—
266C24—41no—
267C24C4⋊Q814yes—
268C24AGL(1,ℤ/8)14yes—
269C24#13714yes—
270C24C22×D412yes—
271C24C22×D411yescharacteristic
272C2×Q8—44no—
273C2×Q8—44no—
274C2×Q8—82no—
275C2×Q8—82no—
276C2×Q8—82no—
277C2×Q8—82no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—416no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—48no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—48no—
318C22×C4—42no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—84no—
322C22×C4—82no—
323C22×C4—82no—
324C22×C4—82no—
325C22×C4—82no—
326C22×C4—816no—
327C22×C4C4⋊Q814yes—
328C22×C4C4⋊Q818yes—
329C22×C4C22×D412yes—
330C22×C4C22×D412yes—
331C22×C4C22×D412yes—
332C22×C4C22×D412yes—
333C22×C4C22×D412yes—
334C22×C4C22×D411yescharacteristic
335C22×C4C22×Q814yes—
336C22×C4C22×Q814yes—
337C22×C4C22×Q814yes—
338C22×C4C2511yescharacteristic
339C2×C8—28no—
340C2×C8—28no—
341C2×C8—416no—
342C2×C8C22×Q8116yes—
343C4⋊C4—24no—
344C4⋊C4—22no—
345C4⋊C4—28no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—28no—
350C4⋊C4—48no—
351C4⋊C4—48no—
352C4⋊C4—48no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—48no—
359C4⋊C4—48no—
360C4⋊C4—82no—
361C4⋊C4—82no—
362C4⋊C4—82no—
363C4⋊C4—82no—
364C4⋊C4—84no—
365C4⋊C4—84no—
366C4⋊C4—82no—
367C4⋊C4—82no—
368C4⋊C4H(5,𝔽2)14yes—
369C22⋊C4—28no—
370C22⋊C4—28no—
371C22⋊C4—48no—
372C22⋊C4—48no—
373C22⋊C4—48no—
374C42—24no—
375C42—24no—
376C42—42no—
377C42—42no—
378C23—21no—
379C23—22no—
380C23—21no—
381C23—22no—
382C23—21no—
383C23—22no—
384C23—21no—
385C23—22no—
386C23—44no—
387C23—44no—
388C23—44no—
389C23—44no—
390C23—44no—
391C23—44no—
392C23#501116yescomplemented
393C23C2×C4⋊Q812yes—
394C23C2×C4⋊Q814yes—
395C23C2×AGL(1,ℤ/8)14yes—
396C23C2×AGL(1,ℤ/8)12yes—
397C23C2×#13714yes—
398C23C2×#13712yes—
399C23C23×D411yescharacteristic
400Q8—164no—
401Q8—164no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—28no—
407C2×C4—28no—
408C2×C4—28no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—48no—
423C2×C4—44no—
424C2×C4—48no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—816no—
447C2×C4—84no—
448C2×C4—84no—
449C2×C4—84no—
450C2×C4—164no—
451C2×C4—164no—
452C2×C4C2×C4⋊Q814yes—
453C2×C4C2×C4⋊Q812yes—
454C2×C4C23×Q812yes—
455C2×C4C2×H(5,𝔽2)12yes—
456C2×C4#58512yes—
457C8—216no—
458C22—22no—
459C22—24no—
460C22—22no—
461C22—24no—
462C22—24no—
463C22—22no—
464C22—22no—
465C22—24no—
466C22—22no—
467C22—24no—
468C22—22no—
469C22—24no—
470C22—416no—
471C22C2×#50118yes—
472C22C2×#50114yes—
473C22C22×C4⋊Q811yescharacteristic
474C22#348411yescharacteristic
475C22#348511yescharacteristic
476C22C22×AGL(1,ℤ/8)11yescharacteristic
477C22C22×#13711yescharacteristic
478C22#358511yescharacteristic
479C22#358611yescharacteristic
480C4—22no—
481C4—24no—
482C4—42no—
483C4—42no—
484C4—42no—
485C4—44no—
486C4—42no—
487C4—82no—
488C4—84no—
489C4—82no—
490C4—82no—
491C4—84no—
492C4#350114yes—
493C2—24no—
494C2—28no—
495C2#3539111yescharacteristic
496C2C22×#50111yescharacteristic
497C2#6066411yescharacteristic
498C2#6157011yescharacteristic
499C2#6157211yescharacteristic
500C2#6170511yescharacteristic
501C2#6170711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
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
3. #29302C2semidirect
4. #29301C2semidirect
5. #28775C2semidirect
6. #27956C2semidirect
7. #26068C2semidirect
8. #25819C2semidirect
15. C2×#501C22semidirect
38. #2085C22semidirect
39. #2084C22semidirect
40. #2083C22semidirect
41. #2059C22semidirect
42. #2058C22semidirect
43. #2057C22semidirect
44. #1993C22semidirect
45. #1992C22semidirect
46. #1944C22semidirect
47. #1943C22semidirect
55. #1847C22semidirect
100. C2×C22⋊C8Q8semidirect
392. C23#501semidirect

Maximal subgroups

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

subgroupindex
#350022
C2×#24422
#293022
#293012
#287752
#279562
#260682
#258192
#219402
C2×#18532
#205262