Finite Group Databasebeta

← #6434767

Subgroups — #6434767

Order 512.

1743 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35264C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#30072C214yesmaximal, complemented
4#30071C214yesmaximal, complemented
5#30064C214yesmaximal, complemented
6#30059C214yesmaximal, complemented
7#23734C211yesmaximal, characteristic, complemented
8C2×#2093C211yesmaximal, characteristic, complemented
9#23555C212yesmaximal, complemented
10#23509C212yesmaximal
11#23454C211yesmaximal, characteristic, complemented
12#23451C211yesmaximal, characteristic, complemented
13#21067C212yesmaximal, complemented
14#21063C212yesmaximal, complemented
15#20774C211yesmaximal, characteristic, complemented
16C2×#544C2211yescharacteristic
17C2×#535C2211yescharacteristic
18C2×#533C2211yescharacteristic
19C2×#428C2212yes—
20C22×#107C2212yes—
21C2×#417C2212yes—
22C2×#418C2212yes—
23C2×#416C2212yes—
24C22×#102C2212yes—
25#2861C2212yes—
26#2858C2212yes—
27#2641C2212yes—
28#2640C2212yes—
29#2491—22no—
30#2490—22no—
31#2472—24no—
32#2360—24no—
33#2355—22no—
34Q8×C4⋊C4—22no—
35#2330C2214yescomplemented
36#2317C2212yes—
37#2311C2211yescharacteristic
38#2309C2211yescharacteristic
39C4×C4⋊Q8C2214yescomplemented
40#2297C2211yescharacteristic
41C2×#390C2211yescharacteristic
42C2×#384C2211yescharacteristic
43#2274C2212yes—
44C4×#117C2211yescharacteristic
45#2096C2212yescomplemented
46#2095C2212yescomplemented
47#2094C2212yescomplemented
48#2093C2214yes—
49#2093C2214yes—
50#2093C2212yescomplemented
51#2059C2218yescomplemented
52#2057C2218yescomplemented
53#2042C2218yes—
54#2041C2218yes—
55#2026C2212yescomplemented
56#2025C2214yes—
57#2025C2212yescomplemented
58#2024C2212yescomplemented
59#2023C2214yes—
60#2023C2212yescomplemented
61#1928C2218yescomplemented
62#1926C2218yes—
63#1925C2218yes—
64#1925C2218yescomplemented
65#1840C2214yescomplemented
66#1839C2214yescomplemented
67#1736C2212yes—
68#1735C2212yes—
69#1729C2212yes—
70#1729C2212yes—
71#1728C2211yescharacteristic
72C2×#336C2211yescharacteristic
73C22×#41C2311yescharacteristic
74C23×D4C2311yescharacteristic
75C22×M16C2312yes—
76C23×C8C2312yes—
77#544—44no—
78#535—44no—
79#533—28no—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8C2312yes—
82C2×C4⋊Q8C2312yes—
83C2×#127C2312yes—
84C2×#125—21no—
85C2×#125C2312yes—
86C2×#125C2312yes—
87C2×#124—24no—
88C2×#124C2312yes—
89C2×C22⋊Q8—22no—
90C2×C22⋊Q8—22no—
91C2×#121—21no—
92C2×#121—21no—
93C2×#120—22no—
94C2×C4×Q8—22no—
95C2×C4×Q8—24no—
96C2×#117C2311yescharacteristic
97C2×#117C2312yes—
98C2×#117C2311yescharacteristic
99C2×#117C2311yescharacteristic
100C22×C4⋊C4—22no—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C22⋊C4—22no—
104C22×C42C2311yescharacteristic
105#428—44no—
106C2×#107—44no—
107C2×#107C2314yes—
108C2×#107C2318yes—
109C2×#106C2314yes—
110#418—28no—
111#417—44no—
112#416—28no—
113C2×#103C2314yes—
114C2×#103C2314yes—
115C2×#102—44no—
116C2×#102C2314yes—
117C2×#102C2318yes—
118C2×#102C2318yes—
119C2×#102C2314yes—
120#397—24no—
121#394—24no—
122#394—24no—
123#393—24no—
124#391—24no—
125#391C2312yes—
126#391C2311yescharacteristic
127#391C2311yescharacteristic
128#390—22no—
129#390—24no—
130#390—22no—
131#390C2312yes—
132#390C2312yes—
133#390C2312yes—
134#390C2311yescharacteristic
135#390C2312yes—
136#390C2312yes—
137#390C2311yescharacteristic
138#389—24no—
139#389—28no—
140#389C2318yes—
141#388—24no—
142#388—24no—
143#387—28no—
144#386—22no—
145#386—22no—
146#385—24no—
147#384—22no—
148#384—22no—
149#384C2312yes—
150#384C2312yes—
151#382—22no—
152#380—22no—
153#379—22no—
154C4×C4⋊C4—24no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4D412yes—
157C4×C4⋊C4D412yes—
158C4×C4⋊C4C2318yes—
159C4×C22⋊C4—24no—
160C4×C22⋊C4—22no—
161C4×C22⋊C4D414yes—
162#376C2312yes—
163C43C2312yes—
164#340C2314yes—
165#340C2318yes—
166#340C2314yes—
167#336C2318yes—
168#336C2312yes—
169#336C2312yes—
170#336C2312yes—
171#336C2312yes—
172#336C2318yes—
173C25—21no—
174C2×#41—24no—
175C2×#41—24no—
176C2×#41—22no—
177C2×#41—22no—
178C22×Q8—22no—
179C22×Q8—22no—
180C22×Q8C2411yescharacteristic
181C22×D4—21no—
182C22×D4—21no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4C2412yes—
190C22×D4C2411yescharacteristic
191C23×C4—21no—
192C23×C4—22no—
193C23×C4—21no—
194C23×C4C2×D412yes—
195C23×C4C2411yescharacteristic
196C2×M16—24no—
197C2×M16—24no—
198C2×M16—24no—
199C22×C8—24no—
200C22×C8—24no—
201C22×C8—24no—
202C22×C8C2414yes—
203C22×C8C2414yes—
204C4⋊Q8—28no—
205C4⋊Q8—44no—
206C4⋊Q8—44no—
207#127—44no—
208#125—28no—
209#125—44no—
210#125—44no—
211#124—44no—
212#124—88no—
213C22⋊Q8—216no—
214C22⋊Q8—48no—
215#121—44no—
216#121—44no—
217#120—48no—
218C4×Q8—216no—
219C4×Q8—88no—
220#117—24no—
221#117—24no—
222#117—22no—
223#117—22no—
224#117—24no—
225#117#4118yes—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—84no—
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2412yes—
244C2×C4⋊C4C2414yes—
245C2×C4⋊C4C2411yescharacteristic
246C2×C4⋊C4C2411yescharacteristic
247C2×C4⋊C4C2414yes—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—84no—
260C2×C22⋊C4C2×D412yes—
261C2×C22⋊C4C2×D412yes—
262C2×C42—22no—
263C2×C42—24no—
264C2×C42—21no—
265C2×C42—28no—
266C2×C42C2×D414yes—
267C2×C42C2×D412yes—
268C2×C42C2×D414yes—
269C2×C42C2412yes—
270C2×C42C2412yes—
271C2×C42C2411yescharacteristic
272C2×C42C2411yescharacteristic
273#107—48no—
274#107—44no—
275#106—44no—
276#103—28no—
277#103—44no—
278#102—28no—
279#102—216no—
280#102—48no—
281#102—44no—
282#95—28no—
283#95—22no—
284#95—22no—
285#95—28no—
286#95—24no—
287#95C2×D414yes—
288#95C2×D414yes—
289C24—22no—
290C24—21no—
291C24—42no—
292C24—42no—
293C24—42no—
294C24—82no—
295C24—82no—
296C24C22×D411yescharacteristic
297#41—48no—
298#41—48no—
299C2×Q8—28no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—44no—
305C2×Q8—44no—
306C2×Q8—44no—
307C2×Q8—84no—
308C2×D4—24no—
309C2×D4—22no—
310C2×D4—24no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—44no—
317C2×D4—44no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—48no—
325C2×D4—48no—
326C2×D4—84no—
327C2×D4—84no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—21no—
332C22×C4—24no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—44no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—82no—
343C22×C4—84no—
344C22×C4C22×D412yes—
345C22×C4C22×D414yes—
346C22×C4C22×D412yes—
347C22×C4C22×D412yes—
348C22×C4C22×D412yes—
349C22×C4C22×D411yescharacteristic
350C22×C4C2×#4114yes—
351C22×C4H(5,𝔽2)14yes—
352C22×C42−1+412yes—
353C22×C42−1+412yes—
354C22×C4C2511yescharacteristic
355M16—48no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—48no—
363C4⋊C4—216no—
364C4⋊C4—28no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—22no—
372C4⋊C4—24no—
373C4⋊C4—22no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—42no—
379C4⋊C4—48no—
380C4⋊C4—84no—
381C4⋊C4—88no—
382C4⋊C4C2×#4118yes—
383C22⋊C4—28no—
384C22⋊C4—24no—
385C22⋊C4—24no—
386C22⋊C4—44no—
387C22⋊C4—44no—
388C22⋊C4—42no—
389C22⋊C4—48no—
390C22⋊C4—44no—
391C22⋊C4—84no—
392C22⋊C4—82no—
393C22⋊C4—84no—
394C22⋊C4—84no—
395C22⋊C4—84no—
396C42—24no—
397C42—28no—
398C42—28no—
399C42—22no—
400C42—22no—
401C42—24no—
402C42—22no—
403C42—22no—
404C42—44no—
405C42—42no—
406C42—88no—
407C23—22no—
408C23—22no—
409C23—42no—
410C23—44no—
411C23—41no—
412C23—42no—
413C23—42no—
414C23—41no—
415C23—41no—
416C23—42no—
417C23—42no—
418C23—44no—
419C23—88no—
420C23—84no—
421C23—162no—
422C23—162no—
423C23—164no—
424C23#52512yes—
425C23#53512yes—
426C23#53612yes—
427C23C23×D411yescharacteristic
428Q8—416no—
429Q8—42no—
430Q8—42no—
431Q8—164no—
432D4—44no—
433D4—44no—
434D4—42no—
435D4—42no—
436D4—84no—
437D4—84no—
438D4—164no—
439D4—164no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—28no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—48no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—41no—
464C2×C4—48no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—84no—
468C2×C4—164no—
469C2×C4—162no—
470C2×C4#52512yes—
471C2×C4#53512yes—
472C2×C4#53612yes—
473C2×C4#54218yes—
474C2×C4#54218yes—
475C2×C4#57614yes—
476C2×C4C22×#4112yes—
477C2×C4C2×H(5,𝔽2)12yes—
478C2×C4C2×2−1+411yescharacteristic
479C2×C4C2×2−1+411yescharacteristic
480C8—44no—
481C8—44no—
482C22—22no—
483C22—22no—
484C22—24no—
485C22—24no—
486C22—81no—
487C22—82no—
488C22—81no—
489C22—82no—
490C22—81no—
491C22—82no—
492C22—82no—
493C22—84no—
494C22—82no—
495C22—84no—
496C22—168no—
497C22#263214yes—
498C22C2×#52511yescharacteristic
499C22C2×#53511yescharacteristic
500C22C2×#53611yescharacteristic
501C22C2×#57612yes—
502C22#358612yes—
503C4—24no—
504C4—42no—
505C4—42no—
506C4—42no—
507C4—48no—
508C4—44no—
509C4—81no—
510C4—84no—
511C4—82no—
512C4—81no—
513C4—81no—
514C4#263214yes—
515C4C2×#54212yes—
516C4C2×#54212yes—
517C2—24no—
518C2—161no—
519C2—162no—
520C2C2×#263211yescharacteristic
521C2#6058412yes—
522C2#6081711yescharacteristic
523C2#6082611yescharacteristic
524C2#6088812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34214C2semidirect
3. #30072C2semidirect
4. #30071C2semidirect
5. #30064C2semidirect
6. #30059C2semidirect
7. #23734C2semidirect
8. C2×#2093C2semidirect
9. #23555C2semidirect
11. #23454C2semidirect
12. #23451C2semidirect
13. #21067C2semidirect
14. #21063C2semidirect
15. #20774C2semidirect
35. #2330C22semidirect
39. C4×C4⋊Q8C22semidirect
45. #2096C22semidirect
46. #2095C22semidirect
47. #2094C22semidirect
50. #2093C22semidirect
51. #2059C22semidirect
52. #2057C22semidirect
55. #2026C22semidirect
57. #2025C22semidirect
58. #2024C22semidirect
60. #2023C22semidirect
61. #1928C22semidirect
64. #1925C22semidirect
65. #1840C22semidirect
66. #1839C22semidirect

Maximal subgroups

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

subgroupindex
#352642
#342142
#300722
#300712
#300642
#300592
#237342
C2×#20932
#235552
#235092
#234542
#234512
#210672
#210632
#207742