Finite Group Databasebeta

← #165123

Subgroups — #165123

Order 512.

933 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36469C211yesmaximal, characteristic
2#22839C211yesmaximal, characteristic, complemented
3#22700C211yesmaximal, characteristic, complemented
4#7698C212yesmaximal, complemented
5#7694C212yesmaximal, complemented
6#7659C211yesmaximal, characteristic, complemented
7#7659C211yesmaximal, characteristic, complemented
8#7652C211yesmaximal, characteristic, complemented
9#7633C212yesmaximal, complemented
10#7599C212yesmaximal, complemented
11#7570C211yesmaximal, characteristic, complemented
12C2×#559—21no—
13C2×#436—21no—
14#2858C2211yescharacteristic
15#2854C2212yes—
16#2602—22no—
17#2592—21no—
18#2557—22no—
19#2547C411yescharacteristic
20#2544—21no—
21#2541C411yescharacteristic
22#2491C412yes—
23#2422C412yes—
24#2367—21no—
25#2366C411yescharacteristic
26#2365C411yescharacteristic
27C4×C4⋊Q8C2211yescharacteristic
28C4×#121C2212yes—
29C42×D4C2211yescharacteristic
30#1954—21no—
31#1939—21no—
32#1924—22no—
33#1851—22no—
34#1831C2211yescharacteristic
35#1831C2211yescharacteristic
36#1817C2212yes—
37#1797C2212yes—
38#1770—21no—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41#1749C2212yes—
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—
54#559—44no—
55#559—44no—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×#127—21no—
59C2×#127C2×C411yescharacteristic
60C2×#124—21no—
61C2×#124—22no—
62C2×#124—21no—
63C2×#124—21no—
64C2×#123—22no—
65C2×#123—22no—
66C2×#121C2×C412yes—
67C2×#121C2×C412yes—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×D4—22no—
71C2×C4×D4—21no—
72C2×C4×D4C2×C412yes—
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4D411yescharacteristic
75C2×C4×D4D411yescharacteristic
76C22×C42C2312yes—
77#436—44no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×#97—22no—
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89#397—22no—
90#397—22no—
91#394—22no—
92#394—22no—
93#393—22no—
94#392—22no—
95#391—21no—
96#391—21no—
97#391—21no—
98#390—21no—
99#390—21no—
100#390—21no—
101#390—21no—
102#390C2×C412yes—
103#390C2×C412yes—
104#390C2312yes—
105#389—21no—
106#389C2×C412yes—
107#389C2311yescharacteristic
108#389C2311yescharacteristic
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#387—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#385—22no—
118#384C2×C411yescharacteristic
119#380C2×C412yes—
120#380C2×C412yes—
121C4×C4⋊C4—21no—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4D411yescharacteristic
128C4×C22⋊C4—22no—
129C4×C22⋊C4C2×C412yes—
130C4×C22⋊C4D412yes—
131C4×C22⋊C4D412yes—
132C43C2311yescharacteristic
133#336—22no—
134#336—21no—
135#336—21no—
136#336—22no—
137#336C2311yescharacteristic
138#336C2312yes—
139#336C2311yescharacteristic
140#336C2312yes—
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143C22×Q8—21no—
144C22×Q8—21no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4C22⋊C411yescharacteristic
150C22×D4C22×C412yes—
151C23×C4—22no—
152C23×C4C22×C412yes—
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C2×M16—44no—
156C2×M16—44no—
157C2×M16—82no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—21no—
164C4⋊Q8—22no—
165C4⋊Q8—41no—
166C4⋊Q8—42no—
167#127—42no—
168#127—44no—
169#124—48no—
170#124—44no—
171#124—42no—
172#124—42no—
173#124—42no—
174#124—42no—
175#123—48no—
176#123—44no—
177#123—44no—
178#121—24no—
179#121—42no—
180#121—82no—
181C4×Q8—44no—
182C4×Q8—42no—
183C4×Q8—42no—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—22no—
187C4×D4—22no—
188C4×D4—42no—
189C4×D4—44no—
190C4×D4—44no—
191C4×D4—44no—
192C4×D4—82no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4C22⋊C411yescharacteristic
201C2×C4⋊C4C22⋊C411yescharacteristic
202C2×C4⋊C4C22⋊C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C4⋊C4C22×C412yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4C22⋊C412yes—
220C2×C22⋊C4C22⋊C412yes—
221C2×C22⋊C4C22⋊C412yes—
222C2×C42—22no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—82no—
226C2×C42C22×C411yescharacteristic
227C2×C42C22×C411yescharacteristic
228C2×C42C22×C412yes—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2411yescharacteristic
236C4⋊C8—42no—
237C4⋊C8—42no—
238C4⋊C8—42no—
239C4⋊C8—42no—
240C4⋊C8—82no—
241C22⋊C8—44no—
242C22⋊C8—44no—
243C22⋊C8—82no—
244#97—82no—
245C4×C8—42no—
246C4×C8—42no—
247#95—22no—
248#95—21no—
249#95—22no—
250#95—21no—
251#95—22no—
252#95—22no—
253#95—21no—
254#95—21no—
255#95C22⋊C412yes—
256#95C22⋊C412yes—
257#95C22⋊C412yes—
258C24—22no—
259C24C2×C22⋊C412yes—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—81no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—44no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—44no—
276C2×D4—44no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—81no—
281C2×D4—81no—
282C2×D4—82no—
283C2×D4—82no—
284C2×D4—82no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
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×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C23×C411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315M16—88no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—42no—
333C2×C8—42no—
334C4⋊C4—21no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—42no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—42no—
344C4⋊C4—44no—
345C4⋊C4—42no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—42no—
351C4⋊C4—81no—
352C4⋊C4#13112yes—
353C22⋊C4—24no—
354C22⋊C4—24no—
355C22⋊C4—42no—
356C22⋊C4—44no—
357C22⋊C4—44no—
358C22⋊C4—44no—
359C22⋊C4—44no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—44no—
365C22⋊C4—44no—
366C22⋊C4—82no—
367C22⋊C4—82no—
368C22⋊C4—82no—
369C42—21no—
370C42—21no—
371C42—22no—
372C42—42no—
373C42—41no—
374C42—41no—
375C42—42no—
376C42—41no—
377C42—42no—
378C42—44no—
379C42—42no—
380C42—42no—
381C42AGL(1,ℤ/8)12yes—
382C42AGL(1,ℤ/8)12yes—
383C42AGL(1,ℤ/8)12yes—
384C42#13712yes—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23C22×C22⋊C411yescharacteristic
400Q8—84no—
401Q8—84no—
402D4—44no—
403D4—82no—
404D4—82no—
405D4—88no—
406D4—84no—
407D4—162no—
408D4—164no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—164no—
453C2×C4#41312yes—
454C2×C4#41712yes—
455C2×C4#41712yes—
456C2×C4#41812yes—
457C2×C4#41812yes—
458C2×C4C2×#13111yescharacteristic
459C2×C4#56811yescharacteristic
460C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
461C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
462C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
463C2×C4C2×#13711yescharacteristic
464C8—42no—
465C8—42no—
466C8—81no—
467C8—82no—
468C8—81no—
469C8—82no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22—82no—
478C22—82no—
479C22—82no—
480C22—82no—
481C22—82no—
482C22—82no—
483C22—82no—
484C22#288011yescharacteristic
485C22#288311yescharacteristic
486C22C2×#41311yescharacteristic
487C22C2×#41811yescharacteristic
488C22C2×#41711yescharacteristic
489C22#290111yescharacteristic
490C22#290111yescharacteristic
491C4—21no—
492C4—21no—
493C4—22no—
494C4—22no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—42no—
500C4—41no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—82no—
505C4—81no—
506C4—81no—
507C4—81no—
508C4#155714yes—
509C2—82no—
510C2—162no—
511C2#1742911yescharacteristic
512C2#1751111yescharacteristic
513C2#1751511yescharacteristic
514C2#1762611yescharacteristic
515C2#1763011yescharacteristic
516C2#1769111yescharacteristic
517C2C2×#155711yescharacteristic
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. #22839C2semidirect
3. #22700C2semidirect
4. #7698C2semidirect
5. #7694C2semidirect
6. #7659C2semidirect
7. #7659C2semidirect
8. #7652C2semidirect
9. #7633C2semidirect
10. #7599C2semidirect
11. #7570C2semidirect

Maximal subgroups

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

subgroupindex
#364692
#228392
#227002
#76982
#76942
#76592
#76522
#76332
#75992
#75702