Finite Group Databasebeta

← #6506548

Subgroups — #6506548

Order 512.

1806 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35011C211yesmaximal, characteristic
2C2×#2439C211yesmaximal, characteristic
3#29335C212yesmaximal, complemented
4#29329C212yesmaximal, complemented
5#27983C212yesmaximal, complemented
6#27721C218yesmaximal, complemented
7#27277C218yesmaximal, complemented
8#25902C212yesmaximal, complemented
9#21765C212yesmaximal
10#20911C212yesmaximal
11#20524C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C22×#117C2211yescharacteristic
15C2×#487C2214yescomplemented
16C2×#419C2214yes—
17C2×#407C2212yes—
18#2857C2212yes—
19#2857C2212yes—
20#2659—21no—
21#2659—21no—
22#2583—24no—
23#2502C2214yes—
24#2439—24no—
25#2439C2218yes—
26#2434C2214yes—
27C2×#396—21no—
28C2×#393—21no—
29#2315—21no—
30#2315—21no—
31C2×#391—21no—
32C2×#389C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#387—21no—
35C2×#382C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C2×C4×C22⋊C4C2211yescharacteristic
38#2100C2214yescomplemented
39#2099C2214yescomplemented
40#2074C2218yescomplemented
41#2043C2214yescomplemented
42#2043C2214yescomplemented
43#2024C2218yescomplemented
44#1990C2214yescomplemented
45#1960C2214yescomplemented
46#1935C2218yescomplemented
47#1933C2218yescomplemented
48#1894C2218yes—
49#1873C2218yes—
50#1872C2214yes—
51#1872C2214yes—
52#1872C2214yes—
53#1872C2214yes—
54#1836C2214yescomplemented
55#1823C2214yes—
56#1787C2214yes—
57#1753C2218yes—
58#1749C2212yes—
59C4×C22⋊C8C2212yes—
60#1736C2214yes—
61C23×Q8—21no—
62C23×Q8C2312yes—
63C24×C4C2311yescharacteristic
64C22×M16C2314yes—
65C2×C4⋊Q8—22no—
66C2×C4⋊Q8—44no—
67C2×C4⋊Q8C2312yes—
68C2×C4⋊Q8C2312yes—
69C2×C4⋊Q8C2314yes—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—22no—
72C2×C22⋊Q8—28no—
73C2×C22⋊Q8—28no—
74C2×C22⋊Q8C2314yes—
75C2×C22⋊Q8C2314yes—
76C2×#117—22no—
77C2×#117—44no—
78C2×#117C2314yes—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C22⋊C4—22no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4—21no—
91C22×C22⋊C4D411yescharacteristic
92C22×C22⋊C4D411yescharacteristic
93C22×C42—21no—
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96#487—48no—
97#419—48no—
98C2×#103C2318yes—
99C2×#103C2318yes—
100C2×#103C2318yes—
101C2×#103C2318yes—
102#407—28no—
103C2×C22⋊C8C2314yes—
104C2×C22⋊C8C2314yes—
105C2×#97C2314yes—
106C2×C4×C8C2314yes—
107#398—24no—
108#398—24no—
109#396—28no—
110#395—24no—
111#395—24no—
112#393—28no—
113#391—28no—
114#391C2314yes—
115#391C2314yes—
116#389—22no—
117#389—22no—
118#389—22no—
119#389—22no—
120#389—22no—
121#389—22no—
122#389C2314yes—
123#389C2314yes—
124#387—28no—
125#385—24no—
126#385—24no—
127#385—24no—
128#385—24no—
129#384C2312yes—
130#384C2312yes—
131#382—22no—
132#382C2314yes—
133#380—22no—
134#380—22no—
135#380—22no—
136#380—22no—
137#380—22no—
138#380C2314yes—
139C4×C22⋊C4D414yes—
140C4×C22⋊C4C2314yes—
141C2×#95—21no—
142C2×#95—21no—
143C2×#95—21no—
144C2×#95—21no—
145C2×#95—21no—
146C2×#95D411yescharacteristic
147C2×#95D411yescharacteristic
148#340C2318yes—
149#340C2318yes—
150#340C2318yes—
151#340C2318yes—
152#336C2314yes—
153#336C2314yes—
154C25C2×D411yescharacteristic
155C22×Q8—24no—
156C22×Q8—22no—
157C22×Q8—24no—
158C22×Q8—24no—
159C22×Q8—44no—
160C22×Q8—44no—
161C22×Q8—44no—
162C22×Q8—44no—
163C22×Q8—82no—
164C22×Q8C2414yes—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D412yes—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4#4118yes—
180C23×C4C2411yescharacteristic
181C23×C4C2412yes—
182C2×M16—28no—
183C2×M16—44no—
184C2×M16—44no—
185C22×C8C2418yes—
186C4⋊Q8—28no—
187C4⋊Q8—44no—
188C4⋊Q8—44no—
189C4⋊Q8—44no—
190C4⋊Q8—48no—
191C22⋊Q8—28no—
192C22⋊Q8—28no—
193C22⋊Q8—28no—
194C22⋊Q8—28no—
195C22⋊Q8—48no—
196C22⋊Q8—48no—
197#117—28no—
198#117—44no—
199#117—48no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—48no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—82no—
228C2×C4⋊C4C2412yes—
229C2×C4⋊C4C2414yes—
230C2×C4⋊C4C2412yes—
231C2×C4⋊C4C2412yes—
232C2×C4⋊C4C2412yes—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—48no—
241C2×C22⋊C4—82no—
242C2×C22⋊C4C2×D414yes—
243C2×C22⋊C4C2×D414yes—
244C2×C42—24no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—44no—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C2×C42C2412yes—
251C2×C42C2412yes—
252#103—216no—
253#103—48no—
254#103—48no—
255#103—48no—
256C22⋊C8—28no—
257C22⋊C8—28no—
258#97—44no—
259C4×C8—44no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95C2×D414yes—
270#95C2×D414yes—
271C24—22no—
272C24—22no—
273C24—41no—
274C24—41no—
275C24—41no—
276C24—41no—
277C24#12414yes—
278C24C22×D412yes—
279C24C22×D411yescharacteristic
280C2×Q8—24no—
281C2×Q8—24no—
282C2×Q8—28no—
283C2×Q8—28no—
284C2×Q8—48no—
285C2×Q8—416no—
286C2×Q8—44no—
287C2×Q8—48no—
288C2×Q8—82no—
289C2×Q8—82no—
290C2×Q8—84no—
291C2×Q8—88no—
292C2×Q8—816no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—416no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—82no—
333C22×C4—82no—
334C22×C4—82no—
335C22×C4—82no—
336C22×C4#12418yes—
337C22×C4#12414yes—
338C22×C4#13714yes—
339C22×C4#13714yes—
340C22×C4C22×D412yes—
341C22×C4C22×D412yes—
342C22×C4C22×D412yes—
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D412yes—
345C22×C4C22×D412yes—
346C22×C4C2×#4114yes—
347C22×C4C2×#4114yes—
348C22×C4C2×#4114yes—
349C22×C4C2511yescharacteristic
350M16—88no—
351C2×C8—28no—
352C2×C8—28no—
353C2×C8—28no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—48no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—44no—
366C4⋊C4—48no—
367C4⋊C4—48no—
368C4⋊C4—48no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4—82no—
372C4⋊C4—82no—
373C4⋊C4—84no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C4⋊C4—82no—
377C4⋊C4—82no—
378C4⋊C4H(5,𝔽2)14yes—
379C22⋊C4—28no—
380C22⋊C4—28no—
381C22⋊C4—48no—
382C22⋊C4—48no—
383C22⋊C4—48no—
384C22⋊C4—84no—
385C42—24no—
386C42—24no—
387C42—42no—
388C42—42no—
389C42—48no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—28no—
401C23—44no—
402C23—44no—
403C23—82no—
404C23—82no—
405C23—82no—
406C23—82no—
407C23C2×#12414yes—
408C23C2×#12412yes—
409C23C2×#13712yes—
410C23C2×#13712yes—
411C23C23×D411yescharacteristic
412Q8—48no—
413Q8—48no—
414Q8—816no—
415Q8—88no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—28no—
425C2×C4—28no—
426C2×C4—28no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—48no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—48no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—84no—
459C2×C4—84no—
460C2×C4—84no—
461C2×C4—88no—
462C2×C4—88no—
463C2×C4C2×#12412yes—
464C2×C4C2×#12414yes—
465C2×C4C2×#13714yes—
466C2×C4C2×#13714yes—
467C2×C4C22×#4112yes—
468C2×C4C2×H(5,𝔽2)12yes—
469C2×C4#58512yes—
470C8—48no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—24no—
478C22—24no—
479C22—24no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—88no—
485C22#313914yes—
486C22C22×#12411yescharacteristic
487C22#348411yescharacteristic
488C22#348511yescharacteristic
489C22C22×#13711yescharacteristic
490C22C22×#13711yescharacteristic
491C22#358511yescharacteristic
492C22#358611yescharacteristic
493C4—22no—
494C4—24no—
495C4—24no—
496C4—24no—
497C4—42no—
498C4—42no—
499C4—44no—
500C4—82no—
501C4—84no—
502C4—82no—
503C4—82no—
504C4—84no—
505C4#350514yes—
506C2—24no—
507C2—44no—
508C2#3537711yescharacteristic
509C2C2×#313911yescharacteristic
510C2#6062111yescharacteristic
511C2#6157011yescharacteristic
512C2#6157311yescharacteristic
513C2#6172111yescharacteristic
514C2#6172311yescharacteristic
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. #29335C2semidirect
4. #29329C2semidirect
5. #27983C2semidirect
6. #27721C2semidirect
7. #27277C2semidirect
8. #25902C2semidirect
15. C2×#487C22semidirect
38. #2100C22semidirect
39. #2099C22semidirect
40. #2074C22semidirect
41. #2043C22semidirect
42. #2043C22semidirect
43. #2024C22semidirect
44. #1990C22semidirect
45. #1960C22semidirect
46. #1935C22semidirect
47. #1933C22semidirect
54. #1836C22semidirect

Maximal subgroups

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

subgroupindex
#350112
C2×#24392
#293352
#293292
#279832
#277212
#272772
#259022
#217652
#209112
#205242