Finite Group Databasebeta

← #208449

Subgroups — #208449

Order 512.

1264 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35081C211yesmaximal, characteristic
2#21904C211yesmaximal, characteristic, complemented
3#13749C211yesmaximal, characteristic, complemented
4#9910C212yesmaximal, complemented
5#9712C212yesmaximal, complemented
6#9150C212yesmaximal, complemented
7#9147C212yesmaximal, complemented
8#8218C212yesmaximal, complemented
9C4×#327C212yesmaximal, complemented
10#2856C2211yescharacteristic
11#2851C2211yescharacteristic
12#2847C2212yes—
13#2648—22no—
14#2497C412yescomplemented
15#2490C412yescomplemented
16#2444—22no—
17#2428C412yescomplemented
18#2424—22no—
19#2420C412yescomplemented
20#2333—22no—
21C4×#121C2212yes—
22C2×C4×C22⋊C4—22no—
23C2×C4×C22⋊C4C2211yescharacteristic
24C2×C4×C22⋊C4C2211yescharacteristic
25C22×#95C2211yescharacteristic
26#1924—22no—
27#1923—22no—
28#1797C2212yes—
29#1779C2212yes—
30#1749C2211yescharacteristic
31C4×C22⋊C8C2211yescharacteristic
32C2×#327C2212yes—
33C2×#327C2212yes—
34C2×#327C2212yes—
35C2×#327C2212yes—
36C4×#95—22no—
37C4×#95C2212yes—
38C4×#95C2212yes—
39#1294C2212yes—
40#1294C2212yes—
41#1294C2212yes—
42#1294C2212yes—
43C24×C4—21no—
44C24×C4C2311yescharacteristic
45C2×#127C2×C411yescharacteristic, complemented
46C2×#125C2×C411yescharacteristic, complemented
47C2×#121—22no—
48C2×#121—22no—
49C2×#121C2×C412yes—
50C2×#121C2×C412yes—
51C2×#121C2312yes—
52C2×#120—22no—
53C2×C4×D4—22no—
54C2×C4×D4C2×C412yescomplemented
55C22×C22⋊C4—21no—
56C22×C22⋊C4D412yes—
57C22×C42—21no—
58C22×C42—22no—
59C22×C42—22no—
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C2×C4⋊C8—22no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8C2312yes—
66C2×C22⋊C8C2312yes—
67C2×#97—21no—
68C2×C4×C8—21no—
69#400—22no—
70#397—22no—
71#397—22no—
72#396—22no—
73#394—22no—
74#394—22no—
75#394—22no—
76#392—22no—
77#390—22no—
78#390—22no—
79#390C2×C412yescomplemented
80#390C2×C412yescomplemented
81#389C2×C412yescomplemented
82#389C2×C412yescomplemented
83#388—22no—
84#388—22no—
85#388—22no—
86#387—22no—
87#387—22no—
88#386—22no—
89#386—22no—
90#386—22no—
91#386—22no—
92#386—22no—
93#386—22no—
94#385—22no—
95#384C2×C411yescharacteristic, complemented
96#382C2×C411yescharacteristic, complemented
97#381—22no—
98#380C2×C412yes—
99#380C2×C412yes—
100#380C2312yes—
101C4×C4⋊C4C2×C412yescomplemented
102C4×C22⋊C4—216no—
103C4×C22⋊C4—22no—
104C4×C22⋊C4—24no—
105C4×C22⋊C4C2×C412yes—
106C4×C22⋊C4C2×C412yes—
107C4×C22⋊C4D412yes—
108C4×C22⋊C4D412yes—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—24no—
113C2×#95D412yes—
114C2×#95C2312yes—
115C2×#95C2312yes—
116#336—22no—
117#336—21no—
118#336—21no—
119#327—24no—
120#327—24no—
121#327—24no—
122#327—24no—
123C25C2×D411yescharacteristic
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4C22×C412yes—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—24no—
132C23×C4—28no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—81no—
137C23×C4—81no—
138C23×C4—82no—
139C23×C4—81no—
140C23×C4—81no—
141C23×C4C22×C412yes—
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2411yescharacteristic
147C22×C8—22no—
148C22×C8—22no—
149#127—81no—
150#125—81no—
151#121—22no—
152#121—22no—
153#121—22no—
154#121—22no—
155#121—48no—
156#121—82no—
157#121—82no—
158#121—82no—
159#121—82no—
160#120—82no—
161#120—82no—
162C4×D4—82no—
163C4×D4—82no—
164C4×D4—82no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C22×C412yes—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—28no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4C22⋊C412yes—
187C2×C22⋊C4C22⋊C412yes—
188C2×C22⋊C4C22⋊C412yes—
189C2×C22⋊C4C22⋊C412yes—
190C2×C42—28no—
191C2×C42—28no—
192C2×C42—24no—
193C2×C42—22no—
194C2×C42—81no—
195C2×C42—81no—
196C2×C42—82no—
197C2×C42—82no—
198C2×C42—82no—
199C2×C42—82no—
200C2×C42—81no—
201C2×C42—81no—
202C2×C42C22×C412yes—
203C2×C42C22×C411yescharacteristic
204C2×C42C22×C411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D412yes—
207C2×C42C2×D411yescharacteristic
208C4⋊C8—82no—
209C22⋊C8—22no—
210C22⋊C8—22no—
211C22⋊C8—22no—
212C22⋊C8—22no—
213C22⋊C8—82no—
214C22⋊C8—82no—
215#97—81no—
216C4×C8—81no—
217#95—24no—
218#95—28no—
219#95—28no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95C22⋊C412yes—
230#95C22⋊C412yes—
231#95C22⋊C412yes—
232#95C22⋊C412yes—
233#95#4114yes—
234#95#4114yes—
235C24—21no—
236C24—21no—
237C24—24no—
238C24—22no—
239C24—22no—
240C24—22no—
241C24—41no—
242C24—41no—
243C24C2×C22⋊C412yes—
244C24C22×D411yescharacteristic
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4—44no—
258C2×D4—82no—
259C2×D4—82no—
260C2×D4—82no—
261C2×D4—82no—
262C2×D4—82no—
263C2×D4—82no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—21no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—216no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—44no—
303C22×C4—82no—
304C22×C4—82no—
305C22×C4—84no—
306C22×C4—84no—
307C22×C4—88no—
308C22×C4—88no—
309C22×C4—84no—
310C22×C4—84no—
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C412yes—
316C22×C4C23×C411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C2×#4112yes—
319C22×C4C2×#4112yes—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—44no—
331C4⋊C4—82no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C22⋊C4—216no—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—44no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C22⋊C4—82no—
348C22⋊C4—82no—
349C22⋊C4—82no—
350C22⋊C4—82no—
351C22⋊C4#10418yes—
352C42—41no—
353C42—41no—
354C42—42no—
355C42—42no—
356C42—41no—
357C42—41no—
358C42—82no—
359C42—88no—
360C42—88no—
361C42—84no—
362C23—21no—
363C23—21no—
364C23—22no—
365C23—24no—
366C23—24no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—44no—
374C23—44no—
375C23—44no—
376C23—44no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—41no—
381C23—41no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23C2×#10414yes—
390C23C22×C22⋊C411yescharacteristic
391C23#52511yescharacteristic
392C23#52511yescharacteristic
393C23#57612yes—
394C23#57712yes—
395D4—82no—
396D4—82no—
397D4—82no—
398D4—82no—
399D4—82no—
400D4—82no—
401D4—164no—
402D4—164no—
403D4—164no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—28no—
413C2×C4—28no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—84no—
453C2×C4—84no—
454C2×C4—816no—
455C2×C4—164no—
456C2×C4C2×#10414yes—
457C2×C4C2×#10414yes—
458C2×C4C4×#4112yes—
459C2×C4#53212yes—
460C8—82no—
461C8—82no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—24no—
471C22—24no—
472C22—24no—
473C22—44no—
474C22—48no—
475C22—44no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—82no—
481C22—82no—
482C22—82no—
483C22—82no—
484C22—82no—
485C22—82no—
486C22—82no—
487C22#229711yescharacteristic
488C22#288311yescharacteristic
489C22#288311yescharacteristic
490C22#289712yes—
491C22C22×#10412yes—
492C22#311912yes—
493C22#312211yescharacteristic
494C22#312611yescharacteristic
495C4—24no—
496C4—24no—
497C4—22no—
498C4—22no—
499C4—48no—
500C4—48no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—82no—
505C4—82no—
506C4—82no—
507C2—21no—
508C2—22no—
509C2—21no—
510C2—44no—
511C2—162no—
512C2#1754712yes—
513C2#1754712yes—
514C2#2139911yescharacteristic
515C2#2141511yescharacteristic
516C2#2142211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#35081C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21904C2semidirect
3. #13749C2semidirect
4. #9910C2semidirect
5. #9712C2semidirect
6. #9150C2semidirect
7. #9147C2semidirect
8. #8218C2semidirect
9. C4×#327C2semidirect
14. #2497C4semidirect
15. #2490C4semidirect
17. #2428C4semidirect
19. #2420C4semidirect
45. C2×#127C2×C4semidirect
46. C2×#125C2×C4semidirect
54. C2×C4×D4C2×C4semidirect
79. #390C2×C4semidirect
80. #390C2×C4semidirect
81. #389C2×C4semidirect
82. #389C2×C4semidirect
95. #384C2×C4semidirect
96. #382C2×C4semidirect
101. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#350812
#219042
#137492
#99102
#97122
#91502
#91472
#82182
C4×#3272