Finite Group Databasebeta

← #208461

Subgroups — #208461

Order 512.

1104 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35081C211yesmaximal, characteristic
2#21904C211yesmaximal, characteristic, complemented
3#13796C211yesmaximal, characteristic, complemented
4#9913C212yesmaximal, complemented
5#9715C212yesmaximal, complemented
6#9150C212yesmaximal, complemented
7#9147C212yesmaximal, complemented
8#8220C212yesmaximal, complemented
9#8216C212yesmaximal, 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⋊C4C2211yescharacteristic
23C2×C4×C22⋊C4C2211yescharacteristic
24C22×#95C2211yescharacteristic
25#1924—22no—
26#1923—22no—
27#1797C2212yes—
28#1779C2212yes—
29#1749C2211yescharacteristic
30C4×C22⋊C8C2211yescharacteristic
31C2×#327C2212yes—
32C2×#327C2212yes—
33C2×#327C2212yes—
34C2×#327C2212yes—
35#1434—22no—
36#1434C2212yes—
37#1434C2212yes—
38C22⋊#95—22no—
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×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C2×C4⋊C8—22no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8C2312yes—
64C2×C22⋊C8C2312yes—
65C2×#97—21no—
66C2×C4×C8—21no—
67#400—22no—
68#397—22no—
69#397—22no—
70#396—22no—
71#394—22no—
72#394—22no—
73#394—22no—
74#392—22no—
75#390—22no—
76#390—22no—
77#390C2×C412yescomplemented
78#390C2×C412yescomplemented
79#389C2×C412yescomplemented
80#389C2×C412yescomplemented
81#388—22no—
82#388—22no—
83#388—22no—
84#387—22no—
85#387—22no—
86#386—22no—
87#386—22no—
88#386—22no—
89#386—22no—
90#386—22no—
91#386—22no—
92#385—22no—
93#384C2×C411yescharacteristic, complemented
94#382C2×C411yescharacteristic, complemented
95#381—22no—
96#380C2×C412yes—
97#380C2×C412yes—
98#380C2312yes—
99C4×C4⋊C4C2×C412yescomplemented
100C4×C22⋊C4—22no—
101C4×C22⋊C4—24no—
102C4×C22⋊C4C2×C412yes—
103C4×C22⋊C4C2×C412yes—
104C4×C22⋊C4D412yes—
105C4×C22⋊C4D412yes—
106C2×#95—21no—
107C2×#95—24no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95D412yes—
113C2×#95C2312yes—
114C2×#95C2312yes—
115#336—22no—
116#336—21no—
117#336—21no—
118#327—24no—
119#327—24no—
120#327—24no—
121#327—24no—
122C25C2×D411yescharacteristic
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4C22×C412yes—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—24no—
133C23×C4—22no—
134C23×C4—44no—
135C23×C4—81no—
136C23×C4—81no—
137C23×C4—82no—
138C23×C4—81no—
139C23×C4—81no—
140C23×C4C22×C412yes—
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D412yes—
145C23×C4C2411yescharacteristic
146C22×C8—22no—
147C22×C8—22no—
148#127—81no—
149#125—81no—
150#121—22no—
151#121—22no—
152#121—22no—
153#121—22no—
154#121—48no—
155#121—82no—
156#121—82no—
157#121—82no—
158#121—82no—
159#120—82no—
160#120—82no—
161C4×D4—82no—
162C4×D4—82no—
163C4×D4—82no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C22×C412yes—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
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—44no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4C22⋊C412yes—
186C2×C22⋊C4C22⋊C412yes—
187C2×C22⋊C4C22⋊C412yes—
188C2×C22⋊C4C22⋊C412yes—
189C2×C42—24no—
190C2×C42—22no—
191C2×C42—81no—
192C2×C42—81no—
193C2×C42—81no—
194C2×C42—81no—
195C2×C42C22×C412yes—
196C2×C42C22×C411yescharacteristic
197C2×C42C22×C411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D412yes—
200C2×C42C2×D411yescharacteristic
201C4⋊C8—82no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8—22no—
205C22⋊C8—22no—
206C22⋊C8—82no—
207C22⋊C8—82no—
208#97—81no—
209C4×C8—81no—
210#95—24no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—82no—
221#95—82no—
222#95—82no—
223#95—82no—
224#95C22⋊C412yes—
225#95C22⋊C412yes—
226#95C22⋊C412yes—
227#95C22⋊C412yes—
228#95#4114yes—
229#95#4114yes—
230C24—21no—
231C24—21no—
232C24—24no—
233C24—22no—
234C24—22no—
235C24—22no—
236C24—41no—
237C24—41no—
238C24C2×C22⋊C412yes—
239C24C22×D411yescharacteristic
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—44no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—44no—
252C2×D4—44no—
253C2×D4—82no—
254C2×D4—82no—
255C2×D4—82no—
256C2×D4—82no—
257C2×D4—82no—
258C2×D4—82no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—48no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—84no—
301C22×C4—84no—
302C22×C4—84no—
303C22×C4—84no—
304C22×C4—88no—
305C22×C4—88no—
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C412yes—
311C22×C4C23×C411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C2×#4112yes—
314C22×C4C2×#4112yes—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—44no—
326C4⋊C4—82no—
327C4⋊C4—82no—
328C4⋊C4—82no—
329C22⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—44no—
334C22⋊C4—48no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C42—41no—
348C42—41no—
349C42—42no—
350C42—42no—
351C42—41no—
352C42—41no—
353C42—82no—
354C42—84no—
355C23—21no—
356C23—21no—
357C23—22no—
358C23—24no—
359C23—24no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—22no—
366C23—44no—
367C23—44no—
368C23—44no—
369C23—44no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—41no—
374C23—41no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23#42114yes—
383C23C22×C22⋊C411yescharacteristic
384C23#52511yescharacteristic
385C23#52511yescharacteristic
386C23#57612yes—
387C23#57712yes—
388D4—82no—
389D4—82no—
390D4—82no—
391D4—82no—
392D4—82no—
393D4—82no—
394D4—164no—
395D4—164no—
396D4—164no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—84no—
436C2×C4—84no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—816no—
450C2×C4—164no—
451C2×C4C4×#4112yes—
452C2×C4#53212yes—
453C8—82no—
454C8—82no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—24no—
464C22—24no—
465C22—24no—
466C22—44no—
467C22—48no—
468C22—44no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—82no—
474C22—82no—
475C22—82no—
476C22—82no—
477C22—82no—
478C22—82no—
479C22—82no—
480C22#229711yescharacteristic
481C22#288311yescharacteristic
482C22#288311yescharacteristic
483C22#289712yes—
484C22C2×#42112yes—
485C22#311912yes—
486C22#312211yescharacteristic
487C22#312611yescharacteristic
488C4—22no—
489C4—22no—
490C4—42no—
491C4—42no—
492C4—42no—
493C4—42no—
494C4—42no—
495C4—84no—
496C4—84no—
497C4—82no—
498C4—82no—
499C4—82no—
500C2—21no—
501C2—22no—
502C2—21no—
503C2—44no—
504C2—162no—
505C2#1759712yes—
506C2#1759712yes—
507C2#2139911yescharacteristic
508C2#2141711yescharacteristic
509C2#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. #13796C2semidirect
4. #9913C2semidirect
5. #9715C2semidirect
6. #9150C2semidirect
7. #9147C2semidirect
8. #8220C2semidirect
9. #8216C2semidirect
14. #2497C4semidirect
15. #2490C4semidirect
17. #2428C4semidirect
19. #2420C4semidirect
45. C2×#127C2×C4semidirect
46. C2×#125C2×C4semidirect
54. C2×C4×D4C2×C4semidirect
77. #390C2×C4semidirect
78. #390C2×C4semidirect
79. #389C2×C4semidirect
80. #389C2×C4semidirect
93. #384C2×C4semidirect
94. #382C2×C4semidirect
99. 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
#137962
#99132
#97152
#91502
#91472
#82202
#82162