Finite Group Databasebeta

← #164503

Subgroups — #164503

Order 512.

871 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36463C211yesmaximal, characteristic
2#22012C211yesmaximal, characteristic, complemented
3#20567C211yesmaximal, characteristic, complemented
4#7694C212yesmaximal, complemented
5#7679C211yesmaximal, characteristic, complemented
6#7679C211yesmaximal, characteristic, complemented
7#7658C212yesmaximal, complemented
8#7636C211yesmaximal, characteristic, complemented
9#7616C212yesmaximal, complemented
10#7582C212yesmaximal, complemented
11#7568C211yesmaximal, characteristic, complemented
12C2×#436—21no—
13#2858C2211yescharacteristic
14#2854C2212yes—
15#2563C411yescharacteristic
16#2557—22no—
17#2541C411yescharacteristic
18#2513—21no—
19#2490C412yes—
20#2459—22no—
21#2445—21no—
22#2420C412yes—
23#2369C411yescharacteristic
24#2366—21no—
25#2366C411yescharacteristic
26#2357—21no—
27C4×#125C2211yescharacteristic
28C4×#121C2212yes—
29C42×D4C2211yescharacteristic
30#1954—21no—
31#1939—21no—
32#1924—22no—
33#1817C2212yes—
34#1797C2212yes—
35C8×C4⋊C4—21no—
36#1767C2211yescharacteristic
37#1767C2211yescharacteristic
38#1753—22no—
39C4×C22⋊C8C2212yes—
40#1744C2211yescharacteristic
41#1744C2211yescharacteristic
42#1728C2212yes—
43#1271C2211yescharacteristic
44#1271C2212yes—
45#1271C2211yescharacteristic
46#1271C2211yescharacteristic
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1270C2212yes—
50#1270C2212yes—
51#1270C2212yes—
52#1270C2212yes—
53C22×M16—22no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×#127—21no—
56C2×#127C2×C411yescharacteristic
57C2×#125C2×C411yescharacteristic
58C2×#124—21no—
59C2×#124—21no—
60C2×#123—22no—
61C2×#121C2×C412yes—
62C2×#121C2×C412yes—
63C2×C4×D4—22no—
64C2×C4×D4—21no—
65C2×C4×D4C2×C412yes—
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C22×C42C2312yes—
70#436—28no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×#97—21no—
78C2×#97—21no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81C2×C4×C8—22no—
82#397—22no—
83#397—22no—
84#394—22no—
85#394—22no—
86#393—22no—
87#392—22no—
88#391—22no—
89#390—21no—
90#390—21no—
91#390—21no—
92#390—22no—
93#390—21no—
94#390—21no—
95#390C2×C412yes—
96#390C2×C412yes—
97#390C2312yes—
98#389—21no—
99#389C2×C412yes—
100#389C2311yescharacteristic
101#389C2311yescharacteristic
102#388—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#387—21no—
107#387—21no—
108#386—22no—
109#386—22no—
110#386—22no—
111#386—22no—
112#382—21no—
113#382C2×C411yescharacteristic
114#381—22no—
115#380C2×C412yes—
116#380C2×C412yes—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C22⋊C4—22no—
127C4×C22⋊C4C2×C412yes—
128C4×C22⋊C4D412yes—
129C4×C22⋊C4D412yes—
130C43C2311yescharacteristic
131#336—22no—
132#336—21no—
133#336—22no—
134#336—21no—
135#336C2311yescharacteristic
136#336C2312yes—
137#336C2311yescharacteristic
138#336C2311yescharacteristic
139#336C2312yes—
140#336C2311yescharacteristic
141C22×Q8—21no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4C22⋊C411yescharacteristic
147C22×D4C22×C412yes—
148C23×C4—22no—
149C23×C4C22×C412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C2×M16—28no—
153C2×M16—82no—
154C2×M16—82no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—22no—
161C4⋊Q8—42no—
162#127—42no—
163#127—44no—
164#125—22no—
165#125—41no—
166#124—81no—
167#124—81no—
168#124—81no—
169#124—81no—
170#123—82no—
171#123—82no—
172#121—24no—
173#121—42no—
174#121—82no—
175C4×D4—24no—
176C4×D4—42no—
177C4×D4—42no—
178C4×D4—44no—
179C4×D4—82no—
180C4×D4—82no—
181C4×D4—82no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4C22⋊C411yescharacteristic
189C2×C4⋊C4C22⋊C411yescharacteristic
190C2×C4⋊C4C22⋊C411yescharacteristic
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C412yes—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4C22⋊C412yes—
208C2×C22⋊C4C22⋊C412yes—
209C2×C22⋊C4C22⋊C412yes—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—22no—
213C2×C42—82no—
214C2×C42C22×C411yescharacteristic
215C2×C42C22×C411yescharacteristic
216C2×C42C22×C412yes—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42C2411yescharacteristic
224C4⋊C8—24no—
225C4⋊C8—42no—
226C4⋊C8—82no—
227C22⋊C8—28no—
228C22⋊C8—82no—
229C22⋊C8—82no—
230#97—81no—
231#97—81no—
232C4×C8—24no—
233C4×C8—42no—
234C4×C8—82no—
235#95—22no—
236#95—21no—
237#95—22no—
238#95—21no—
239#95—21no—
240#95—22no—
241#95—21no—
242#95—22no—
243#95—21no—
244#95—21no—
245#95C22⋊C412yes—
246#95C22⋊C412yes—
247#95C22⋊C412yes—
248C24—22no—
249C24C2×C22⋊C412yes—
250C2×Q8—42no—
251C2×Q8—81no—
252C2×Q8—81no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—44no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—44no—
263C2×D4—44no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—81no—
268C2×D4—81no—
269C2×D4—82no—
270C2×D4—82no—
271C2×D4—82no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—44no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C23×C411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D411yescharacteristic
302M16—88no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—24no—
306C2×C8—41no—
307C2×C8—42no—
308C2×C8—41no—
309C2×C8—42no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—41no—
318C2×C8—42no—
319C2×C8—41no—
320C2×C8—41no—
321C4⋊C4—21no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—42no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—42no—
336C4⋊C4—82no—
337C4⋊C4#10414yes—
338C4⋊C4#13112yes—
339C22⋊C4—24no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—44no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—44no—
348C22⋊C4—82no—
349C22⋊C4—82no—
350C22⋊C4—82no—
351C22⋊C4—82no—
352C22⋊C4—82no—
353C22⋊C4—82no—
354C22⋊C4—82no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—21no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—42no—
365C42—42no—
366C42—41no—
367C42—42no—
368C42—42no—
369C42—42no—
370C42—82no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23C22×C22⋊C411yescharacteristic
386Q8—162no—
387D4—44no—
388D4—82no—
389D4—82no—
390D4—88no—
391D4—84no—
392D4—162no—
393D4—164no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—164no—
439C2×C4#41312yes—
440C2×C4C2×#10414yes—
441C2×C4C2×#10412yes—
442C2×C4#42112yes—
443C2×C4C2×#13111yescharacteristic
444C2×C4#56811yescharacteristic
445C2×C4#57611yescharacteristic
446C2×C4#57611yescharacteristic
447C2×C4#57711yescharacteristic
448C2×C4#57711yescharacteristic
449C8—42no—
450C8—44no—
451C8—42no—
452C8—81no—
453C8—82no—
454C8—81no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—82no—
463C22—82no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22—82no—
468C22—82no—
469C22#288011yescharacteristic
470C22#288311yescharacteristic
471C22C2×#41311yescharacteristic
472C22#289711yescharacteristic
473C22#289711yescharacteristic
474C22C22×#10411yescharacteristic
475C22C2×#42111yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—24no—
481C4—21no—
482C4—21no—
483C4—42no—
484C4—41no—
485C4—41no—
486C4—41no—
487C4—42no—
488C4—42no—
489C4—81no—
490C4—82no—
491C4—81no—
492C4—81no—
493C4#152914yes—
494C2—82no—
495C2—162no—
496C2#1742911yescharacteristic
497C2#1752011yescharacteristic
498C2#1752211yescharacteristic
499C2#1754711yescharacteristic
500C2#1759911yescharacteristic
501C2C2×#152911yescharacteristic
502C2#1777611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2541C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22012C2semidirect
3. #20567C2semidirect
4. #7694C2semidirect
5. #7679C2semidirect
6. #7679C2semidirect
7. #7658C2semidirect
8. #7636C2semidirect
9. #7616C2semidirect
10. #7582C2semidirect
11. #7568C2semidirect

Maximal subgroups

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

subgroupindex
#364632
#220122
#205672
#76942
#76792
#76582
#76362
#76162
#75822
#75682