Finite Group Databasebeta

← #6433749

Subgroups — #6433749

Order 512.

2000 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34633C211yesmaximal, characteristic
2C2×#2628C211yesmaximal, characteristic, complemented
3#30195C218yesmaximal, complemented
4#30193C214yesmaximal, complemented
5#30192C214yesmaximal, complemented
6#23747C212yesmaximal, complemented
7#23631C212yesmaximal, complemented
8#23508C212yesmaximal, complemented
9#23452C212yesmaximal, complemented
10#20971C214yesmaximal, complemented
11#20335C211yesmaximal, characteristic, complemented
12C23×M16C2212yes—
13C2×#532C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic
15C2×#525C2212yes—
16C22×#117C2211yescharacteristic
17C2×#417C2214yes—
18C2×#417C2214yes—
19#2628—22no—
20#2628—22no—
21#2628C2214yes—
22#2628C2214yes—
23#2621C2212yes—
24#2620C2214yes—
25#2619C2212yes—
26#2618—21no—
27#2618—21no—
28#2617—21no—
29#2617—21no—
30C2×#400—21no—
31C2×#399—21no—
32#2392—22no—
33#2392—22no—
34C2×#395—21no—
35C2×#393—21no—
36#2297C2212yes—
37C2×#389C2212yes—
38#2285C2211yescharacteristic
39C2×#382C2211yescharacteristic
40C2×#380C2212yes—
41#2101C2218yescomplemented
42#2100C2214yescomplemented
43#2099C2214yescomplemented
44#2074C2218yescomplemented
45#2073C2218yescomplemented
46#2043C2214yescomplemented
47#2042C2218yescomplemented
48#2041C2214yescomplemented
49#2024C2218yescomplemented
50#2023C2218yescomplemented
51#1895C2218yes—
52#1894C2218yes—
53#1893C2218yes—
54#1892C2218yes—
55#1736C2214yes—
56#1736C2214yes—
57#1729C2214yes—
58#1729C2214yes—
59#1728C2212yes—
60#1728C2212yes—
61C22×#41C2312yes—
62C23×Q8—21no—
63C24×C4C2311yescharacteristic
64C22×M16—28no—
65C22×M16—44no—
66C22×M16—44no—
67C22×M16C2314yes—
68C22×M16C2314yes—
69C23×C8C2314yes—
70#532—44no—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8—44no—
73C2×C4⋊Q8C2311yescharacteristic
74C2×C4⋊Q8C2314yes—
75C2×#127C2311yescharacteristic
76C2×#124C2312yes—
77#525—216no—
78C2×#123—22no—
79C2×C22⋊Q8C2314yes—
80C2×#121—21no—
81C2×#121—21no—
82C2×#121C2314yes—
83C2×#120—22no—
84C2×C4×D4—24no—
85C2×C4×D4—21no—
86C2×C4×D4—21no—
87C2×#117—44no—
88C2×#117D412yes—
89C2×#117D412yes—
90C2×#117C2312yes—
91C2×#117C2312yes—
92C2×#117C2312yes—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—22no—
95C22×C4⋊C4D411yescharacteristic
96C22×C4⋊C4D411yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C22⋊C4—22no—
100C22×C22⋊C4D411yescharacteristic
101C22×C22⋊C4D411yescharacteristic
102C22×C42C2311yescharacteristic
103C22×C42C2311yescharacteristic
104#417—48no—
105#417—48no—
106C2×#103C2318yes—
107C2×#103C2318yes—
108C2×#102C2318yes—
109C2×#102C2318yes—
110#400—24no—
111#400—24no—
112#399—24no—
113#399—24no—
114#398—22no—
115#398—22no—
116#397—22no—
117#397—22no—
118#396—24no—
119#396—24no—
120#395—24no—
121#395—24no—
122#394—22no—
123#394—22no—
124#394—24no—
125#394—24no—
126#393—24no—
127#393—24no—
128#392—22no—
129#392—22no—
130#391C2314yes—
131#390C2314yes—
132#389—22no—
133#389—22no—
134#389C2314yes—
135#389C2314yes—
136#386—22no—
137#386—22no—
138#384C2312yes—
139#382—22no—
140#382C2314yes—
141#382C2312yes—
142#380—21no—
143#380—21no—
144#380—24no—
145#380C2318yes—
146C4×C4⋊C4—21no—
147C4×C4⋊C4—21no—
148C4×C22⋊C4—24no—
149C2×#95—21no—
150C2×#95—22no—
151C2×#95—22no—
152C2×#95—21no—
153#340C2318yes—
154#340C2318yes—
155#340C2318yes—
156#340C2318yes—
157#336C2318yes—
158#336C2314yes—
159#336C2314yes—
160C25C2×D411yescharacteristic
161C2×#41—28no—
162C2×#41—28no—
163C2×#41—24no—
164C2×#41—24no—
165C22×Q8—24no—
166C22×Q8—44no—
167C22×Q8—84no—
168C22×Q8C2412yes—
169C22×D4—22no—
170C22×D4—24no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4C2412yes—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D412yes—
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4#4118yes—
187C23×C4C2412yes—
188C23×C4C2411yescharacteristic
189C2×M16—28no—
190C2×M16—28no—
191C2×M16—28no—
192C2×M16—28no—
193C2×M16—44no—
194C2×M16—44no—
195C2×M16—432no—
196C22×C8—28no—
197C22×C8—44no—
198C22×C8—44no—
199C22×C8C2418yes—
200C4⋊Q8—44no—
201C4⋊Q8—44no—
202C4⋊Q8—42no—
203C4⋊Q8—48no—
204#127—42no—
205#124—44no—
206#123—48no—
207C22⋊Q8—28no—
208C22⋊Q8—28no—
209#121—28no—
210#121—28no—
211#121—44no—
212#121—44no—
213#120—48no—
214C4×D4—44no—
215C4×D4—44no—
216C4×D4—416no—
217#117—28no—
218#117—44no—
219#117—44no—
220#117—42no—
221#117—42no—
222#117—42no—
223#117—42no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—84no—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4C2414yes—
246C2×C4⋊C4C2414yes—
247C2×C4⋊C4C2414yes—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—84no—
259C2×C22⋊C4C2×D412yes—
260C2×C22⋊C4C2×D412yes—
261C2×C22⋊C4C2×D412yes—
262C2×C22⋊C4C2×D412yes—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—22no—
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D412yes—
269C2×C42C2411yescharacteristic
270C2×C42C2411yescharacteristic
271C2×C42C2412yes—
272#103—48no—
273#103—48no—
274#102—48no—
275#102—48no—
276#95—28no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—24no—
283#95—22no—
284#95—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—42no—
289C24—42no—
290C24#12414yes—
291C24C22×D412yes—
292C24C22×D411yescharacteristic
293#41—416no—
294#41—416no—
295C2×Q8—24no—
296C2×Q8—22no—
297C2×Q8—24no—
298C2×Q8—22no—
299C2×Q8—48no—
300C2×Q8—88no—
301C2×Q8—816no—
302C2×D4—24no—
303C2×D4—22no—
304C2×D4—24no—
305C2×D4—22no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—48no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4—44no—
312C2×D4—82no—
313C2×D4—82no—
314C2×D4—84no—
315C2×D4—84no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—216no—
321C22×C4—216no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4#12418yes—
348C22×C4#12414yes—
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D412yes—
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D412yes—
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D412yes—
356C22×C4C22×D411yescharacteristic
357C22×C4C2×#4114yes—
358C22×C4C2×#4114yes—
359C22×C4C2×#4114yes—
360C22×C4H(5,𝔽2)14yes—
361C22×C4H(5,𝔽2)12yes—
362C22×C42−1+412yes—
363C22×C4C2511yescharacteristic
364M16—416no—
365M16—416no—
366C2×C8—28no—
367C2×C8—28no—
368C2×C8—28no—
369C2×C8—416no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—28no—
374C4⋊C4—24no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—48no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—48no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—82no—
386C4⋊C4—82no—
387C4⋊C4—84no—
388C4⋊C4—84no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—48no—
397C22⋊C4—48no—
398C22⋊C4—84no—
399C22⋊C4—84no—
400C22⋊C4—88no—
401C42—22no—
402C42—22no—
403C42—24no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—44no—
409C23—22no—
410C23—22no—
411C23—24no—
412C23—24no—
413C23—22no—
414C23—22no—
415C23—28no—
416C23—48no—
417C23—44no—
418C23—44no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—82no—
423C23—82no—
424C23—84no—
425C23C2×#12414yes—
426C23C2×#12412yes—
427C23#53412yes—
428C23#53712yes—
429C23C23×D411yescharacteristic
430Q8—44no—
431Q8—44no—
432Q8—816no—
433D4—44no—
434D4—44no—
435D4—84no—
436D4—84no—
437D4—816no—
438D4—88no—
439C2×C4—28no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—216no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—88no—
476C2×C4—88no—
477C2×C4—88no—
478C2×C4C2×#12414yes—
479C2×C4C2×#12412yes—
480C2×C4#53412yes—
481C2×C4#53712yes—
482C2×C4#53818yes—
483C2×C4#53918yes—
484C2×C4C22×#4112yes—
485C2×C4C2×H(5,𝔽2)11yescharacteristic
486C2×C4C2×H(5,𝔽2)12yes—
487C2×C4C2×2−1+411yescharacteristic
488C8—48no—
489C22—22no—
490C22—22no—
491C22—24no—
492C22—24no—
493C22—24no—
494C22—28no—
495C22—42no—
496C22—42no—
497C22—44no—
498C22—88no—
499C22—82no—
500C22—82no—
501C22—82no—
502C22—84no—
503C22—84no—
504C22#263914yes—
505C22C22×#12411yescharacteristic
506C22C2×#53411yescharacteristic
507C22C2×#53711yescharacteristic
508C22#358512yes—
509C22#358612yes—
510C4—24no—
511C4—42no—
512C4—41no—
513C4—41no—
514C4—42no—
515C4—41no—
516C4—41no—
517C4—82no—
518C4—84no—
519C4—84no—
520C4—84no—
521C4#263914yes—
522C4C2×#53812yes—
523C4C2×#53912yes—
524C2—24no—
525C2—44no—
526C2—162no—
527C2C2×#263911yescharacteristic
528C2#6062112yes—
529C2#6075311yescharacteristic
530C2#6075411yescharacteristic
531C2#6091912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#34633C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2628C2semidirect
3. #30195C2semidirect
4. #30193C2semidirect
5. #30192C2semidirect
6. #23747C2semidirect
7. #23631C2semidirect
8. #23508C2semidirect
9. #23452C2semidirect
10. #20971C2semidirect
11. #20335C2semidirect
41. #2101C22semidirect
42. #2100C22semidirect
43. #2099C22semidirect
44. #2074C22semidirect
45. #2073C22semidirect
46. #2043C22semidirect
47. #2042C22semidirect
48. #2041C22semidirect
49. #2024C22semidirect
50. #2023C22semidirect

Maximal subgroups

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

subgroupindex
#346332
C2×#26282
#301952
#301932
#301922
#237472
#236312
#235082
#234522
#209712
#203352