Finite Group Databasebeta

← #160907

Subgroups — #160907

Order 512.

982 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34321C211yesmaximal, characteristic
2#22700C211yesmaximal, characteristic, complemented
3#20568C211yesmaximal, characteristic, complemented
4#7633C212yesmaximal, complemented
5#7632C212yesmaximal, complemented
6#7616C212yesmaximal, complemented
7#7613C212yesmaximal, complemented
8#7591C211yesmaximal, characteristic, complemented
9#7574C211yesmaximal, characteristic, complemented
10#7570C211yesmaximal, characteristic, complemented
11#7560C211yesmaximal, characteristic, complemented
12C2×#555C411yescharacteristic
13C2×#436—21no—
14C2×#436—21no—
15#2369C411yescharacteristic
16#2367—22no—
17#2363C412yes—
18#2361—21no—
19#2354C411yescharacteristic
20D4×C4⋊C4C411yescharacteristic
21#2347—21no—
22#2347C412yes—
23#2337—22no—
24#2324—21no—
25C4×C4⋊Q8C2211yescharacteristic
26C4×#121C2212yes—
27#2296C2212yes—
28#2293C2211yescharacteristic
29#2274—21no—
30C42×D4C2211yescharacteristic
31#1939—21no—
32#1851—22no—
33#1831C2211yescharacteristic
34#1797C2212yes—
35#1770—21no—
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38#1753—22no—
39#1749C2212yes—
40#1744C2211yescharacteristic
41#1728C2212yes—
42#1728C2212yes—
43#1271C2211yescharacteristic
44#1271C2211yescharacteristic
45#1271C2212yes—
46#1271C2212yes—
47#1271C2212yes—
48#1271C2212yes—
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2211yescharacteristic
52#1270C2212yes—
53C22×M16—22no—
54C22×M16—22no—
55#555—24no—
56#555—24no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×#124—22no—
59C2×#121—21no—
60C2×#121C2×C412yes—
61C2×C4×Q8—21no—
62C2×C4×D4—22no—
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4C2×C412yes—
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×#117—21no—
69C2×#117—21no—
70C2×#117C2×C412yes—
71C2×#117C2×C412yes—
72C22×C4⋊C4—21no—
73C22×C42C2312yes—
74#436—28no—
75#436—44no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×#97—22no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85C2×C4×C8—21no—
86#391—21no—
87#391—22no—
88#390—21no—
89#390—22no—
90#390C2×C411yescharacteristic
91#390C2311yescharacteristic
92#389—21no—
93#389—21no—
94#389C2×C411yescharacteristic
95#389C2×C412yes—
96#389C2×C412yes—
97#389C2312yes—
98#389C2311yescharacteristic
99#388—22no—
100#388—22no—
101#388—22no—
102#388—21no—
103#388—22no—
104#388—21no—
105#388—22no—
106#386—22no—
107#386—22no—
108#385—22no—
109#385—22no—
110#385—22no—
111#384C2×C411yescharacteristic
112#383—22no—
113#380—21no—
114#380C2×C412yes—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C4×C22⋊C4—22no—
128C4×C22⋊C4—21no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4C2×C412yes—
131C4×C22⋊C4D412yes—
132C4×C22⋊C4D412yes—
133C43C2311yescharacteristic
134#336—21no—
135#336—22no—
136#336—22no—
137#336—21no—
138#336C2311yescharacteristic
139#336C2312yes—
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142#336C2312yes—
143#336C2311yescharacteristic
144C22×Q8—21no—
145C22×D4—22no—
146C22×D4C22⋊C411yescharacteristic
147C22×D4C22×C411yescharacteristic
148C23×C4—21no—
149C23×C4—22no—
150C23×C4C22×C412yes—
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C2×M16—28no—
154C2×M16—44no—
155C2×M16—44no—
156C2×M16—82no—
157C2×M16—82no—
158C2×M16—82no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—21no—
165C4⋊Q8—22no—
166C4⋊Q8M1614yescomplemented
167#124—216no—
168#121—28no—
169#121—24no—
170#121—24no—
171C4×Q8—81no—
172C4×Q8—81no—
173C4×D4—24no—
174C4×D4—22no—
175C4×D4—22no—
176C4×D4—22no—
177C4×D4—22no—
178C4×D4—22no—
179C4×D4—82no—
180C4×D4—82no—
181C4×D4—82no—
182C4×D4M1614yescomplemented
183#117—28no—
184#117—24no—
185#117—24no—
186#117—42no—
187#117—42no—
188#117—82no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4C22⋊C411yescharacteristic
199C2×C4⋊C4C22⋊C411yescharacteristic
200C2×C4⋊C4C22⋊C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C412yes—
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4C22⋊C412yes—
217C2×C22⋊C4C22⋊C412yes—
218C2×C22⋊C4C22⋊C412yes—
219C2×C42—22no—
220C2×C42—21no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—82no—
225C2×C42C22×C412yes—
226C2×C42C22×C411yescharacteristic
227C2×C42C22×C411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D411yescharacteristic
234C2×C42C2411yescharacteristic
235C4⋊C8—24no—
236C4⋊C8—42no—
237C4⋊C8—42no—
238C4⋊C8—82no—
239C22⋊C8—28no—
240C22⋊C8—44no—
241#97—82no—
242C4×C8—24no—
243C4×C8—42no—
244C4×C8—81no—
245#95—22no—
246#95—22no—
247#95—21no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—21no—
252#95—21no—
253#95—22no—
254#95C22⋊C412yes—
255#95C22⋊C412yes—
256#95C22⋊C412yes—
257C24—21no—
258C24C2×C22⋊C412yes—
259C2×Q8—24no—
260C2×Q8—81no—
261C2×Q8—81no—
262C2×D4—28no—
263C2×D4—21no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—82no—
270C2×D4—82no—
271C2×D4C2×M1612yes—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—44no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C23×C411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305M16—88no—
306M16—88no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—24no—
310C2×C8—22no—
311C2×C8—41no—
312C2×C8—42no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—42no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—41no—
324C2×C8—42no—
325C4⋊C4—21no—
326C4⋊C4—22no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—42no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—44no—
339C4⋊C4—42no—
340C4⋊C4—82no—
341C4⋊C4—81no—
342C4⋊C4#10414yes—
343C4⋊C4C2×M1612yes—
344C4⋊C4C2×M1614yes—
345C22⋊C4—28no—
346C22⋊C4—28no—
347C22⋊C4—24no—
348C22⋊C4—24no—
349C22⋊C4—24no—
350C22⋊C4—24no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—42no—
356C22⋊C4—81no—
357C22⋊C4—82no—
358C22⋊C4—82no—
359C22⋊C4—82no—
360C42—28no—
361C42—21no—
362C42—21no—
363C42—22no—
364C42—42no—
365C42—41no—
366C42—42no—
367C42—41no—
368C42—42no—
369C42—41no—
370C42—42no—
371C42—82no—
372C42—82no—
373C42—82no—
374C42C2×M1612yes—
375C42C2×M1612yes—
376C42AGL(1,ℤ/8)12yes—
377C42#13712yes—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23C22×C22⋊C411yescharacteristic
393Q8—84no—
394D4—42no—
395D4—42no—
396D4—44no—
397D4—88no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—48no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—164no—
442C2×C4#40712yes—
443C2×C4#40712yes—
444C2×C4#41712yes—
445C2×C4#41712yes—
446C2×C4C2×#10414yes—
447C2×C4C2×#10412yes—
448C2×C4C22×M1611yescharacteristic
449C2×C4C22×M1611yescharacteristic
450C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
451C2×C4C2×#13711yescharacteristic
452C2×C4#57611yescharacteristic
453C2×C4#57711yescharacteristic
454C8—42no—
455C8—44no—
456C8—42no—
457C8—42no—
458C8—81no—
459C8—82no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22C2×#40711yescharacteristic
475C22#288311yescharacteristic
476C22#288911yescharacteristic
477C22C2×#41711yescharacteristic
478C22#289711yescharacteristic
479C22C22×#10411yescharacteristic
480C22#290111yescharacteristic
481C4—22no—
482C4—22no—
483C4—21no—
484C4—21no—
485C4—24no—
486C4—21no—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—42no—
491C4—44no—
492C4—42no—
493C4—42no—
494C4—41no—
495C4—81no—
496C4—82no—
497C4—81no—
498C4#148914yes—
499C2—82no—
500C2—82no—
501C2#1737011yescharacteristic
502C2C2×#148911yescharacteristic
503C2#1745311yescharacteristic
504C2#1754711yescharacteristic
505C2#1763011yescharacteristic
506C2#1765411yescharacteristic
507C2#1776511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#555C4
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. #22700C2semidirect
3. #20568C2semidirect
4. #7633C2semidirect
5. #7632C2semidirect
6. #7616C2semidirect
7. #7613C2semidirect
8. #7591C2semidirect
9. #7574C2semidirect
10. #7570C2semidirect
11. #7560C2semidirect
166. C4⋊Q8M16semidirect
182. C4×D4M16semidirect

Maximal subgroups

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

subgroupindex
#343212
#227002
#205682
#76332
#76322
#76162
#76132
#75912
#75742
#75702
#75602