Finite Group Databasebeta

← #6434117

Subgroups — #6434117

Order 512.

1852 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35254C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic, complemented
3#29916C218yesmaximal, complemented
4#29914C214yesmaximal, complemented
5#29907C214yesmaximal, complemented
6#23745C211yesmaximal, characteristic, complemented
7#23744C211yesmaximal, characteristic, complemented
8#23731C211yesmaximal, characteristic, complemented
9C2×#2094C211yesmaximal, characteristic, complemented
10#23661C212yesmaximal, complemented
11#23632C212yesmaximal, complemented
12#20914C212yesmaximal
13#20848C212yesmaximal, complemented
14#20684C211yesmaximal, characteristic, complemented
15C2×#535C2211yescharacteristic
16C2×#520C2212yes—
17C2×C4×#41C2211yescharacteristic
18C2×#428C2212yes—
19C22×#106C2212yes—
20C2×#418C2212yes—
21C22×#102C2212yes—
22C2×#408C2212yes—
23#2854C2214yes—
24#2624C2214yes—
25#2448—22no—
26#2445—22no—
27#2418—24no—
28Q8×C4⋊C4—22no—
29D4×C4⋊C4—24no—
30D4×C4⋊C4—22no—
31#2317C2212yes—
32#2317C2214yescomplemented
33#2316C2211yescharacteristic
34#2311C2211yescharacteristic
35#2309C2211yescharacteristic
36#2296C2211yescharacteristic
37C2×#390C2211yescharacteristic
38#2285C2211yescharacteristic
39#2285C2212yescomplemented
40C2×#384C2211yescharacteristic
41C2×#384C2212yescomplemented
42#2101C2214yescomplemented
43#2100C2212yescomplemented
44#2100C2214yescomplemented
45#2099C2212yescomplemented
46#2099C2214yescomplemented
47#2094C2214yes—
48#2094C2214yes—
49#2094C2212yes—
50#2093C2212yes—
51#2092C2214yes—
52#2085C2218yescomplemented
53#2084C2218yescomplemented
54#2073C2218yes—
55#2071C2218yes—
56#1875C2218yes—
57#1874C2218yes—
58#1854C2218yescomplemented
59#1853C2218yescomplemented
60#1818C2214yescomplemented
61#1817C2214yescomplemented
62#1736C2212yes—
63#1736C2212yes—
64#1735C2212yes—
65#1735C2212yes—
66#1729C2212yes—
67#1728C2211yescharacteristic
68C2×#336C2211yescharacteristic
69C22×#41C2311yescharacteristic
70C23×D4C2311yescharacteristic
71C22×M16C2312yes—
72C23×C8C2312yes—
73#535—28no—
74C2×#127—21no—
75C2×#124—21no—
76C2×#123—24no—
77C2×#121C2314yes—
78C2×#120—22no—
79#520—44no—
80#520—44no—
81C4×#41—28no—
82C2×C4×Q8—22no—
83C2×C4×Q8D412yes—
84C2×C4×D4—24no—
85C2×C4×D4—22no—
86C2×C4×D4D414yes—
87C2×C4×D4D412yes—
88C2×#117C2311yescharacteristic
89C2×#117C2311yescharacteristic
90C2×#117C2312yes—
91C2×#117C2314yes—
92C2×#117C2311yescharacteristic
93C2×#117C2312yes—
94C22×C4⋊C4—22no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C22⋊C4—22no—
100C22×C42C2312yes—
101C22×C42C2311yescharacteristic
102#428—44no—
103C2×#107C2314yes—
104C2×#106—44no—
105C2×#106C2314yes—
106C2×#106C2318yes—
107#418—44no—
108C2×#102—44no—
109C2×#102C2314yes—
110C2×#102C2318yes—
111C2×#102C2314yes—
112#408—28no—
113C2×C22⋊C8C2318yes—
114#399—24no—
115#397—24no—
116#394—24no—
117#394—24no—
118#391—21no—
119#390—21no—
120#390—22no—
121#390—22no—
122#390C2314yes—
123#390C2312yes—
124#390C2312yes—
125#389—22no—
126#389—24no—
127#389—28no—
128#389C2314yes—
129#389C2312yes—
130#389C2312yes—
131#388—24no—
132#388—24no—
133#386—22no—
134#386—24no—
135#385—22no—
136#385—24no—
137#385—28no—
138#384—22no—
139#384—24no—
140#384C2314yes—
141#384C2312yes—
142#384C2311yescharacteristic
143#384C2312yes—
144#384C2311yescharacteristic
145#382C2314yes—
146#382C2312yes—
147#380—22no—
148#380—24no—
149#380—22no—
150#380C2314yes—
151#379—22no—
152C4×C4⋊C4—22no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4—22no—
155#340C2314yes—
156#340C2318yes—
157#340C2314yes—
158#340C2314yes—
159#340C2318yes—
160#340C2314yes—
161#336C2318yes—
162#336C2312yes—
163#336C2312yes—
164#336C2312yes—
165#336C2312yes—
166C25—21no—
167C2×#41—24no—
168C2×#41—22no—
169C2×#41—22no—
170C2×#41#4118yes—
171C22×Q8C2×D411yescharacteristic
172C22×D4—24no—
173C22×D4—21no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—22no—
177C22×D4—22no—
178C22×D4C2×D412yes—
179C22×D4C2×D411yescharacteristic
180C22×D4C2412yes—
181C23×C4—21no—
182C23×C4—24no—
183C23×C4—22no—
184C23×C4C2×D412yes—
185C23×C4C2×D414yes—
186C23×C4C2412yes—
187C23×C4C2411yescharacteristic
188C2×M16—24no—
189C2×M16—24no—
190C2×M16—24no—
191C22×C8—24no—
192C22×C8—24no—
193C22×C8—24no—
194C22×C8C2414yes—
195C22×C8C2414yes—
196#127—44no—
197#124—44no—
198#123—88no—
199#121—216no—
200#120—48no—
201C4×Q8—28no—
202C4×Q8—48no—
203C4×D4—216no—
204C4×D4—28no—
205C4×D4—416no—
206C4×D4—48no—
207#117—28no—
208#117—24no—
209#117—24no—
210#117—24no—
211#117—24no—
212#117—22no—
213#117—22no—
214#117—44no—
215#117—44no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—84no—
233C2×C4⋊C4—88no—
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4C2×D414yes—
236C2×C4⋊C4C2414yes—
237C2×C4⋊C4C2412yes—
238C2×C4⋊C4C2412yes—
239C2×C4⋊C4C2412yes—
240C2×C4⋊C4C2412yes—
241C2×C4⋊C4C2412yes—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—84no—
253C2×C22⋊C4C2×D412yes—
254C2×C22⋊C4C2×D414yes—
255C2×C42—28no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—22no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42C2×D412yes—
262C2×C42C2×D414yes—
263C2×C42C2412yes—
264C2×C42C2411yescharacteristic
265C2×C42C2411yescharacteristic
266#107—44no—
267#106—48no—
268#106—44no—
269#102—48no—
270#102—44no—
271#102—44no—
272C22⋊C8—216no—
273#95—24no—
274#95—28no—
275#95—22no—
276#95—22no—
277#95—24no—
278#95—24no—
279C24—22no—
280C24—42no—
281C24—42no—
282C24—42no—
283C24—82no—
284C24—82no—
285C24C22×D412yes—
286C24C22×D411yescharacteristic
287#41—216no—
288#41—216no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8#12314yes—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—48no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—48no—
305C2×D4—48no—
306C2×D4—84no—
307C2×D4—84no—
308C2×D4#12318yes—
309C2×D4#12314yes—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—28no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—28no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—44no—
324C22×C4—42no—
325C22×C4—48no—
326C22×C4—44no—
327C22×C4—42no—
328C22×C4—82no—
329C22×C4—84no—
330C22×C4—88no—
331C22×C4C22×D414yes—
332C22×C4C22×D412yes—
333C22×C4C22×D412yes—
334C22×C4C22×D412yes—
335C22×C4C22×D411yescharacteristic
336C22×C4C2×#4118yes—
337C22×C4C2×#4114yes—
338C22×C4H(5,𝔽2)14yes—
339C22×C42−1+412yes—
340C22×C42−1+412yes—
341C22×C4C2511yescharacteristic
342M16—48no—
343C2×C8—24no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—48no—
350C4⋊C4—28no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—48no—
363C4⋊C4—48no—
364C4⋊C4—44no—
365C4⋊C4—42no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—82no—
370C4⋊C4—88no—
371C4⋊C4—84no—
372C4⋊C4—84no—
373C22⋊C4—28no—
374C22⋊C4—24no—
375C22⋊C4—48no—
376C22⋊C4—42no—
377C22⋊C4—44no—
378C22⋊C4—42no—
379C22⋊C4—48no—
380C22⋊C4—48no—
381C22⋊C4—84no—
382C22⋊C4—84no—
383C22⋊C4—84no—
384C22⋊C4—84no—
385C42—28no—
386C42—24no—
387C42—22no—
388C42—24no—
389C42—22no—
390C42—48no—
391C42—42no—
392C42—44no—
393C42—42no—
394C23—22no—
395C23—22no—
396C23—24no—
397C23—22no—
398C23—22no—
399C23—24no—
400C23—44no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—44no—
405C23—88no—
406C23—84no—
407C23—164no—
408C23—162no—
409C23—162no—
410C23C2×#12314yes—
411C23C2×#12312yes—
412C23#53512yes—
413C23#53612yes—
414C23C23×D411yescharacteristic
415Q8—28no—
416D4—216no—
417D4—28no—
418D4—44no—
419D4—44no—
420D4—84no—
421D4—168no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—28no—
427C2×C4—28no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—48no—
435C2×C4—416no—
436C2×C4—42no—
437C2×C4—48no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—162no—
451C2×C4—164no—
452C2×C4—168no—
453C2×C4C2×#12314yes—
454C2×C4C2×#12312yes—
455C2×C4#53512yes—
456C2×C4#53612yes—
457C2×C4#53918yes—
458C2×C4#53918yes—
459C2×C4#57714yes—
460C2×C4C22×#4112yes—
461C2×C4C2×H(5,𝔽2)12yes—
462C2×C4C2×2−1+411yescharacteristic
463C2×C4C2×2−1+411yescharacteristic
464C8—44no—
465C8—44no—
466C22—24no—
467C22—22no—
468C22—22no—
469C22—24no—
470C22—24no—
471C22—28no—
472C22—42no—
473C22—42no—
474C22—44no—
475C22—82no—
476C22—84no—
477C22—82no—
478C22—82no—
479C22—84no—
480C22—168no—
481C22#262314yes—
482C22C22×#12311yescharacteristic
483C22C2×#53511yescharacteristic
484C22C2×#53611yescharacteristic
485C22C2×#57712yes—
486C22#358512yes—
487C4—24no—
488C4—24no—
489C4—44no—
490C4—42no—
491C4—82no—
492C4—82no—
493C4—82no—
494C4—84no—
495C4—82no—
496C4—82no—
497C4#262314yes—
498C4C2×#53912yes—
499C4C2×#53912yes—
500C2—24no—
501C2—44no—
502C2—162no—
503C2C2×#262311yescharacteristic
504C2#6052512yes—
505C2#6082011yescharacteristic
506C2#6082711yescharacteristic
507C2#6089212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34183C2semidirect
3. #29916C2semidirect
4. #29914C2semidirect
5. #29907C2semidirect
6. #23745C2semidirect
7. #23744C2semidirect
8. #23731C2semidirect
9. C2×#2094C2semidirect
10. #23661C2semidirect
11. #23632C2semidirect
13. #20848C2semidirect
14. #20684C2semidirect
32. #2317C22semidirect
39. #2285C22semidirect
41. C2×#384C22semidirect
42. #2101C22semidirect
43. #2100C22semidirect
44. #2100C22semidirect
45. #2099C22semidirect
46. #2099C22semidirect
52. #2085C22semidirect
53. #2084C22semidirect
58. #1854C22semidirect
59. #1853C22semidirect
60. #1818C22semidirect
61. #1817C22semidirect

Maximal subgroups

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

subgroupindex
#352542
#341832
#299162
#299142
#299072
#237452
#237442
#237312
C2×#20942
#236612
#236322
#209142
#208482
#206842