Finite Group Databasebeta

← #6475939

Subgroups — #6475939

Order 512.

2046 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C212yesmaximal
2#32931C214yesmaximal
3#30291C211yesmaximal, characteristic, complemented
4#30242C211yesmaximal, characteristic, complemented
5#30187C212yesmaximal, complemented
6#29160C214yesmaximal, complemented
7#28575C214yesmaximal, complemented
8#27858C214yesmaximal, complemented
9#25498C214yesmaximal, complemented
10#22182C214yesmaximal, complemented
11C22×#336C211yesmaximal, characteristic
12C24×C8C2212yes—
13C2×#533C2212yes—
14C23×C42C2211yescharacteristic
15C2×#467C2214yes—
16C2×#467C2214yes—
17C2×#467C2214yes—
18#2857C2212yes—
19#2857C2212yes—
20#2857C2212yes—
21#2857C2212yes—
22#2855—22no—
23#2855C2214yes—
24#2851—22no—
25#2851—22no—
26#2848—22no—
27#2848C2214yes—
28C4×C22⋊Q8—22no—
29C4×C22⋊Q8—22no—
30C4×C22⋊Q8C2214yes—
31C4×C22⋊Q8C2214yes—
32#2296—22no—
33#2285C2212yes—
34#2280—22no—
35#2100C2214yescomplemented
36#2095C2214yescomplemented
37#2083C2214yescomplemented
38#2072C2214yescomplemented
39#2057C2214yescomplemented
40#2043C2214yescomplemented
41#2024C2214yescomplemented
42#2016C2214yescomplemented
43#1928C2218yescomplemented
44#1921C2218yescomplemented
45#1892C2218yes—
46#1892C2218yes—
47#1892C2218yes—
48#1878C2218yescomplemented
49#1863C2218yescomplemented
50#1808C2214yes—
51#1808C2214yes—
52#1808C2214yes—
53C2×#336C2211yescharacteristic
54C2×#336C2214yes—
55C2×#336C2214yes—
56C2×#336C2212yes—
57C2×#336C2211yescharacteristic
58C24×C4D411yescharacteristic
59C24×C4D411yescharacteristic
60C24×C4C2311yescharacteristic
61C22×Q16C2314yes—
62C23×C8—28no—
63C23×C8—44no—
64C23×C8—44no—
65C23×C8C2314yes—
66C23×C8C2314yes—
67C23×C8C2312yes—
68C23×C8C2312yes—
69#533—48no—
70C2×C4⋊Q8C2312yes—
71C2×C4⋊Q8C2312yes—
72C2×#125—22no—
73C2×C22⋊Q8—22no—
74C2×C22⋊Q8—22no—
75C2×C22⋊Q8—22no—
76C2×C22⋊Q8C2314yes—
77C2×C22⋊Q8C2314yes—
78C2×C22⋊Q8C2314yes—
79C2×C4×Q8—22no—
80C2×C4×Q8C2314yes—
81C2×#117—22no—
82C2×#117—22no—
83C22×C42—44no—
84C22×C42D411yescharacteristic
85C22×C42D412yes—
86C22×C42D412yes—
87C22×C42D411yescharacteristic
88C22×C42C2312yes—
89C22×C42C2311yescharacteristic
90C22×C42C2312yes—
91C22×C42C2311yescharacteristic
92#467—216no—
93#467—48no—
94#467—48no—
95C2×#107C2314yes—
96C2×#107C2314yes—
97C2×#107C2314yes—
98C2×#103C2318yes—
99C2×#103C2318yes—
100C2×#103C2318yes—
101#391—22no—
102#391—22no—
103#391C2314yes—
104#391C2314yes—
105#389—22no—
106#389—22no—
107#389—22no—
108#389—22no—
109#389—22no—
110#389—22no—
111#389C2314yes—
112#389C2314yes—
113#387—24no—
114#387—24no—
115#387—24no—
116#387—24no—
117#385—24no—
118#385—24no—
119#385—24no—
120#385—24no—
121#385—24no—
122#385—24no—
123#384C2312yes—
124#384C2312yes—
125#382—22no—
126#381—24no—
127#380—22no—
128#380—22no—
129#380—22no—
130#380—22no—
131#380—22no—
132#380—22no—
133#380—22no—
134#380C2314yes—
135#380C2314yes—
136#380C2314yes—
137#379—24no—
138C4×C4⋊C4—22no—
139C4×C4⋊C4—24no—
140C4×C4⋊C4—24no—
141C4×C4⋊C4C2314yes—
142C4×C22⋊C4—24no—
143C4×C22⋊C4—24no—
144#340C2318yes—
145#340C2318yes—
146#340C2318yes—
147#336C2314yes—
148#336C2314yes—
149#336C2318yes—
150C25C2×D411yescharacteristic
151C22×Q8—22no—
152C22×Q8C2414yes—
153C23×C4—24no—
154C23×C4—22no—
155C23×C4—24no—
156C23×C4—22no—
157C23×C4—24no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4Q1618yescomplemented
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D412yes—
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2411yescharacteristic
175C23×C4C2411yescharacteristic
176C23×C4C2411yescharacteristic
177C2×Q16—28no—
178C2×Q16—44no—
179C2×Q16—44no—
180C22×C8—24no—
181C22×C8—28no—
182C22×C8—28no—
183C22×C8—28no—
184C22×C8—28no—
185C22×C8—42no—
186C22×C8—42no—
187C22×C8—44no—
188C22×C8—44no—
189C22×C8—42no—
190C22×C8—42no—
191C22×C8—432no—
192C22×C8#4118yes—
193C22×C8C2414yes—
194C22×C8C2414yes—
195C4⋊Q8—44no—
196C4⋊Q8—44no—
197#125—48no—
198C22⋊Q8—216no—
199C22⋊Q8—216no—
200C22⋊Q8—28no—
201C22⋊Q8—28no—
202C22⋊Q8—48no—
203C22⋊Q8—48no—
204C22⋊Q8—48no—
205C4×Q8—48no—
206C4×Q8—48no—
207#117—48no—
208#117—48no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4C2414yes—
223C2×C4⋊C4C2414yes—
224C2×C4⋊C4C2414yes—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C42—28no—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—42no—
235C2×C42—44no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42—44no—
239C2×C42—42no—
240C2×C42—44no—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247C2×C42C2412yes—
248C2×C42C2412yes—
249#107—28no—
250#107—44no—
251#107—44no—
252#103—216no—
253#103—48no—
254#103—48no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24C2×Q1614yes—
272C24C22×D411yescharacteristic
273C24C22×D411yescharacteristic
274C24C22×D411yescharacteristic
275C2×Q8—28no—
276C2×Q8—24no—
277C2×Q8—28no—
278C2×Q8—24no—
279C2×Q8—44no—
280C2×Q8—44no—
281C2×Q8—44no—
282C22×C4—216no—
283C22×C4—216no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—416no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—416no—
326C22×C4C2×Q1618yes—
327C22×C4C2×Q1614yes—
328C22×C4C2×Q1614yes—
329C22×C4C2×Q1618yes—
330C22×C4#13514yes—
331C22×C4C22×D412yes—
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D412yes—
336C22×C4C22×D412yes—
337C22×C4C22×D412yes—
338C22×C4C2×#4114yes—
339C22×C4H(5,𝔽2)14yes—
340C22×C4H(5,𝔽2)12yes—
341C22×C42−1+412yes—
342C22×C4C2511yescharacteristic
343Q16—88no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—48no—
350C2×C8—416no—
351C2×C8—416no—
352C2×C8—48no—
353C2×C8C2×#4118yes—
354C4⋊C4—28no—
355C4⋊C4—24no—
356C4⋊C4—28no—
357C4⋊C4—24no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—48no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—48no—
366C4⋊C4—48no—
367C4⋊C4—48no—
368C4⋊C4—48no—
369C4⋊C4—44no—
370C4⋊C4—48no—
371C22⋊C4—48no—
372C22⋊C4—48no—
373C22⋊C4—48no—
374C22⋊C4—48no—
375C22⋊C4—48no—
376C22⋊C4—48no—
377C42—24no—
378C42—24no—
379C42—24no—
380C42—24no—
381C42—42no—
382C42—42no—
383C42—42no—
384C42—42no—
385C42—48no—
386C42—48no—
387C23—28no—
388C23—28no—
389C23—28no—
390C23—28no—
391C23—28no—
392C23—21no—
393C23—21no—
394C23—28no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—28no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23#53212yes—
407C23#53412yes—
408C23#53712yes—
409C23C22×Q1612yes—
410C23C22×Q1612yes—
411C23C22×Q1612yes—
412C23C2×#13512yes—
413C23#57512yes—
414C23#57512yes—
415C23C23×D411yescharacteristic
416Q8—48no—
417Q8—48no—
418Q8—88no—
419C2×C4—28no—
420C2×C4—28no—
421C2×C4—28no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—216no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—44no—
445C2×C4—48no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—48no—
449C2×C4—48no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4#53212yes—
464C2×C4#53412yes—
465C2×C4#53712yes—
466C2×C4C22×Q1614yes—
467C2×C4C22×Q1614yes—
468C2×C4C2×#13514yes—
469C2×C4C22×#4112yes—
470C2×C4C2×H(5,𝔽2)11yescharacteristic
471C2×C4C2×H(5,𝔽2)12yes—
472C2×C4C2×2−1+411yescharacteristic
473C8—28no—
474C8—44no—
475C22—232no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22#317014yes—
495C22#318914yes—
496C22#324914yes—
497C22C2×#53211yescharacteristic
498C22C2×#53411yescharacteristic
499C22C2×#53711yescharacteristic
500C22C23×Q1611yescharacteristic
501C22C22×#13511yescharacteristic
502C22C2×#57511yescharacteristic
503C22C2×#57511yescharacteristic
504C4—24no—
505C4—24no—
506C4—24no—
507C4—22no—
508C4—22no—
509C4—22no—
510C4—42no—
511C4—84no—
512C4—84no—
513C4—84no—
514C4—84no—
515C4#349012yes—
516C4#349212yes—
517C2—24no—
518C2—24no—
519C2—24no—
520C2#3505111yescharacteristic
521C2C2×#317011yescharacteristic
522C2#6068811yescharacteristic
523C2C2×#318911yescharacteristic
524C2#6073411yescharacteristic
525C2C2×#324911yescharacteristic
526C2#6090011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #30291C2semidirect
4. #30242C2semidirect
5. #30187C2semidirect
6. #29160C2semidirect
7. #28575C2semidirect
8. #27858C2semidirect
9. #25498C2semidirect
10. #22182C2semidirect
35. #2100C22semidirect
36. #2095C22semidirect
37. #2083C22semidirect
38. #2072C22semidirect
39. #2057C22semidirect
40. #2043C22semidirect
41. #2024C22semidirect
42. #2016C22semidirect
43. #1928C22semidirect
44. #1921C22semidirect
48. #1878C22semidirect
49. #1863C22semidirect
162. C23×C4Q16semidirect

Maximal subgroups

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

subgroupindex
#366252
#329312
#302912
#302422
#301872
#291602
#285752
#278582
#254982
#221822
C22×#3362