Finite Group Databasebeta

← #165017

Subgroups — #165017

Order 512.

981 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36419C211yesmaximal, characteristic
2#22839C211yesmaximal, characteristic, complemented
3#22696C211yesmaximal, characteristic, complemented
4#7730C212yesmaximal, complemented
5#7690C212yesmaximal, complemented
6#7659C211yesmaximal, characteristic, complemented
7#7659C211yesmaximal, characteristic, complemented
8#7653C211yesmaximal, characteristic, complemented
9#7612C212yesmaximal, complemented
10#7578C212yesmaximal, complemented
11#7571C211yesmaximal, characteristic, complemented
12C2×#556C411yescharacteristic
13C2×#550C411yescharacteristic
14C2×C8×D4—21no—
15#2858C2211yescharacteristic
16#2850C2212yes—
17#2568C411yescharacteristic
18#2559—22no—
19#2559—22no—
20#2548C411yescharacteristic
21#2544—21no—
22#2544—21no—
23#2500C412yes—
24#2432C412yes—
25#2367—21no—
26#2367—21no—
27C4×C4⋊Q8C2211yescharacteristic
28#2297C2212yes—
29C42×D4C2211yescharacteristic
30#1954—21no—
31#1939—21no—
32#1924—22no—
33#1850—22no—
34#1831C2211yescharacteristic
35#1831C2211yescharacteristic
36#1817C2212yes—
37#1797C2212yes—
38#1770—21no—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41#1749C2212yes—
42C2×#336C2212yes—
43#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1272C2211yescharacteristic
47#1272C2211yescharacteristic
48#1272C2212yes—
49#1270C2212yes—
50#1270C2212yes—
51#1270C2212yes—
52#1270C2212yes—
53C22×#41C2×C412yes—
54C23×C8—22no—
55#556—44no—
56#550—44no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×#127—21no—
60C2×#127C2×C411yescharacteristic
61C2×#126—22no—
62C2×#124—21no—
63C2×#124—21no—
64C2×#124—21no—
65C2×#124C2×C412yes—
66C2×C22⋊Q8C2×C412yes—
67C2×#121—22no—
68C2×#121C2×C412yes—
69C2×C4×Q8—21no—
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×D4—21no—
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4D411yescharacteristic
75C2×C4×D4D411yescharacteristic
76C2×#117C2×C412yes—
77C22×C42C2312yes—
78C8×D4—44no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×#97—22no—
88C2×C4×C8—21no—
89C2×C4×C8—21no—
90#400—22no—
91#394—22no—
92#394—22no—
93#394—22no—
94#394—22no—
95#394—22no—
96#391—21no—
97#391—21no—
98#391C2×C412yes—
99#391C2311yescharacteristic
100#391C2311yescharacteristic
101#390—21no—
102#390—21no—
103#390—21no—
104#390—21no—
105#390C2×C412yes—
106#390C2×C412yes—
107#390C2312yes—
108#389—21no—
109#389—21no—
110#388—22no—
111#388—22no—
112#388—22no—
113#388—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#386—22no—
118#385—22no—
119#385—22no—
120#384C2×C411yescharacteristic
121#382C2×C412yes—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C4×C22⋊C4—22no—
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
143C2×#41—24no—
144C2×#41—42no—
145C2×#41—44no—
146C2×#41—82no—
147C22×Q8—21no—
148C22×Q8C22×C411yescharacteristic
149C22×Q8C22×C411yescharacteristic
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4C22⋊C411yescharacteristic
155C22×D4C22×C412yes—
156C23×C4—22no—
157C23×C4C22×C412yes—
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—44no—
167C22×C8—44no—
168C22×C8—82no—
169C4⋊Q8—22no—
170C4⋊Q8—41no—
171C4⋊Q8—42no—
172#127—42no—
173#127—44no—
174#126—48no—
175#124—44no—
176#124—42no—
177#124—42no—
178#124—42no—
179#124—42no—
180#124—82no—
181C22⋊Q8—44no—
182#121—48no—
183#121—44no—
184C4×Q8—42no—
185C4×Q8—42no—
186C4×Q8—44no—
187C4×D4—24no—
188C4×D4—22no—
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—22no—
192C4×D4—44no—
193#117—44no—
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⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4C22⋊C411yescharacteristic
203C2×C4⋊C4C22⋊C411yescharacteristic
204C2×C4⋊C4C22⋊C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C412yes—
207C2×C4⋊C4C22×C411yescharacteristic
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—22no—
249#95—22no—
250#95—22no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95C22⋊C412yes—
256#95C22⋊C412yes—
257#95C22⋊C412yes—
258C24—22no—
259C24C2×C22⋊C412yes—
260#41—88no—
261#41—88no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8#13112yes—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—44no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—44no—
282C2×D4—44no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—81no—
287C2×D4—81no—
288C2×D4—82no—
289C2×D4—82no—
290C2×D4—82no—
291C2×D4#10014yes—
292C22×C4—21no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—82no—
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C23×C411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C2×C8—22no—
323C2×C8—22no—
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—41no—
333C2×C8—41no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—41no—
337C2×C8—42no—
338C2×C8—41no—
339C2×C8—42no—
340C2×C8—88no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—41no—
347C4⋊C4—42no—
348C4⋊C4—41no—
349C4⋊C4—42no—
350C4⋊C4—44no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—81no—
354C4⋊C4—81no—
355C22⋊C4—24no—
356C22⋊C4—24no—
357C22⋊C4—24no—
358C22⋊C4—44no—
359C22⋊C4—44no—
360C22⋊C4—44no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—44no—
366C22⋊C4—44no—
367C22⋊C4—82no—
368C22⋊C4—82no—
369C22⋊C4—82no—
370C22⋊C4—82no—
371C42—21no—
372C42—21no—
373C42—22no—
374C42—42no—
375C42—41no—
376C42—41no—
377C42—42no—
378C42—41no—
379C42—42no—
380C42—44no—
381C42—42no—
382C42—42no—
383C42#13512yes—
384C42#13512yes—
385C42AGL(1,ℤ/8)12yes—
386C42#13712yes—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23C2×#10014yes—
401C23C22×C22⋊C411yescharacteristic
402Q8—42no—
403Q8—42no—
404Q8—81no—
405Q8—81no—
406Q8—84no—
407D4—28no—
408D4—82no—
409D4—82no—
410D4—88no—
411D4—84no—
412D4—162no—
413D4—164no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—88no—
458C2×C4C2×#10012yes—
459C2×C4#41612yes—
460C2×C4#41612yes—
461C2×C4#41712yes—
462C2×C4#41712yes—
463C2×C4C2×#13111yescharacteristic
464C2×C4#56811yescharacteristic
465C2×C4C2×#13511yescharacteristic
466C2×C4C2×#13511yescharacteristic
467C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
468C2×C4C2×#13711yescharacteristic
469C8—42no—
470C8—42no—
471C8—81no—
472C8—82no—
473C8—81no—
474C8—82no—
475C22—24no—
476C22—24no—
477C22—24no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—82no—
483C22—82no—
484C22—82no—
485C22—82no—
486C22—82no—
487C22—82no—
488C22—82no—
489C22#288011yescharacteristic
490C22#288311yescharacteristic
491C22C22×#10011yescharacteristic
492C22C2×#41611yescharacteristic
493C22C2×#41711yescharacteristic
494C22#290111yescharacteristic
495C22#290111yescharacteristic
496C4—21no—
497C4—21no—
498C4—22no—
499C4—22no—
500C4—21no—
501C4—21no—
502C4—21no—
503C4—21no—
504C4—42no—
505C4—42no—
506C4—41no—
507C4—42no—
508C4—42no—
509C4—81no—
510C4—81no—
511C4—81no—
512C4—82no—
513C4#155114yes—
514C2—44no—
515C2—162no—
516C2#1742111yescharacteristic
517C2#1751011yescharacteristic
518C2#1751511yescharacteristic
519C2#1761711yescharacteristic
520C2#1763011yescharacteristic
521C2#1768511yescharacteristic
522C2C2×#155111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#550C4
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. #22696C2semidirect
4. #7730C2semidirect
5. #7690C2semidirect
6. #7659C2semidirect
7. #7659C2semidirect
8. #7653C2semidirect
9. #7612C2semidirect
10. #7578C2semidirect
11. #7571C2semidirect

Maximal subgroups

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

subgroupindex
#364192
#228392
#226962
#77302
#76902
#76592
#76532
#76122
#75782
#75712