Finite Group Databasebeta

← #379859

Subgroups — #379859

Order 512.

1791 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#423C211yesmaximal, characteristic, complemented
2#36626C211yesmaximal, characteristic
3C2×#1871C212yesmaximal
4C4×#381C211yesmaximal, characteristic, complemented
5#11209C214yesmaximal, complemented
6#11053C214yesmaximal, complemented
7#10732C212yesmaximal, complemented
8C23×M16C2212yes—
9C2×#525C2211yescharacteristic
10C23×C42C2211yescharacteristic
11C2×#423—22no—
12C2×#423—28no—
13C2×#423C2214yes—
14C22×C4⋊C8—21no—
15C22×C4⋊C8—21no—
16C22×#104C2214yes—
17#2855C411yescharacteristic, complemented
18#2855C412yescomplemented
19#2855C411yescharacteristic, complemented
20#2854C411yescharacteristic, complemented
21#2854C412yescomplemented
22#2854C411yescharacteristic, complemented
23#2853—22no—
24#2849—21no—
25#2849—21no—
26C4×#123—21no—
27C4×#123—21no—
28C4×#123—22no—
29#2297C2212yes—
30#2296C2212yes—
31#2280C2211yescharacteristic
32C2×#381—21no—
33C2×C4×C22⋊C4C2212yes—
34#1871—28no—
35C2×#339C2214yes—
36C2×#339C2218yes—
37#1434—22no—
38C22⋊#95—21no—
39C4×#95C2214yes—
40C22×#41C2311yescharacteristic
41C24×C4D411yescharacteristic
42C24×C4D411yescharacteristic
43C24×C4C2311yescharacteristic
44C22×M16—24no—
45C22×M16—22no—
46C22×M16—48no—
47C22×M16D418yes—
48C22×M16C2314yes—
49C23×C8—22no—
50#525—24no—
51#525—24no—
52C2×#123—21no—
53C2×#123—22no—
54C2×#123—21no—
55C2×#123—21no—
56C2×C22⋊Q8C2×C411yescharacteristic
57C2×C22⋊Q8C2×C411yescharacteristic
58C2×#121C2×C411yescharacteristic
59C2×#121C2×C411yescharacteristic
60C2×#120—21no—
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×#117C2311yescharacteristic
64C2×#117C2312yes—
65C22×C22⋊C4—22no—
66C22×C42—24no—
67C22×C42—24no—
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C411yescharacteristic
70C22×C42C2×C412yes—
71C22×C42D412yes—
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D412yes—
75C22×C42C2314yes—
76C22×C42C2312yes—
77#423—216no—
78#423—216no—
79C2×C4⋊C8—28no—
80C2×C4⋊C8—28no—
81C2×C4⋊C8—24no—
82C2×C4⋊C8—24no—
83C2×#104—28no—
84C2×#104—48no—
85#391C2×C412yes—
86#391C2×C412yes—
87#390C2×C412yes—
88#390C2×C412yes—
89#389C2×C412yes—
90#389C2×C412yes—
91#389C2×C412yes—
92#389C2×C412yes—
93#388—22no—
94#388—22no—
95#388—22no—
96#388—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#386—21no—
102#386—22no—
103#386—21no—
104#386—22no—
105#386—22no—
106#385—21no—
107#385—22no—
108#385—21no—
109#385—22no—
110#385—22no—
111#381—28no—
112#381—21no—
113#381—22no—
114#381—21no—
115#381—21no—
116#380C2×C412yes—
117#380C2×C411yescharacteristic
118#380C2×C411yescharacteristic
119#379—21no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C22⋊C4—21no—
123C4×C22⋊C4—21no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—24no—
127C4×C22⋊C4—24no—
128C2×#95—24no—
129C2×#95—24no—
130C2×#95—22no—
131#339—28no—
132#339—48no—
133C25C2×D411yescharacteristic
134C2×#41—22no—
135C2×#41—22no—
136C2×#41—24no—
137C2×#41—24no—
138C22×Q8C22×C411yescharacteristic
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4C22×C411yescharacteristic
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—21no—
150C23×C4—44no—
151C23×C4—42no—
152C23×C4C22⋊C411yescharacteristic
153C23×C4C22⋊C411yescharacteristic
154C23×C4C22⋊C412yes—
155C23×C4C22⋊C412yes—
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C2×D412yes—
161C23×C4C2×D412yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D414yes—
166C23×C4#4114yes—
167C23×C4C2411yescharacteristic
168C2×M16—28no—
169C2×M16—24no—
170C2×M16—28no—
171C2×M16—28no—
172C2×M16—416no—
173C2×M16—416no—
174C2×M16—48no—
175C2×M16C2×D418yes—
176C22×C8—28no—
177C22×C8—24no—
178C22×C8—48no—
179#123—44no—
180#123—48no—
181#123—44no—
182#123—44no—
183C22⋊Q8—24no—
184C22⋊Q8—24no—
185#121—24no—
186#121—24no—
187#120—44no—
188C4×D4—42no—
189C4×D4—42no—
190C4×D4—42no—
191C4×D4—42no—
192#117—22no—
193#117—22no—
194#117—24no—
195#117—24no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C412yes—
205C2×C4⋊C4C22×C412yes—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C42—24no—
223C2×C42—28no—
224C2×C42—24no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—24no—
228C2×C42—28no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—21no—
235C2×C42—22no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—28no—
239C2×C42—28no—
240C2×C42C22⋊C412yes—
241C2×C42C22⋊C412yes—
242C2×C42C22⋊C412yes—
243C2×C42C22⋊C412yes—
244C2×C42C22×C412yes—
245C2×C42C22×C412yes—
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248C4⋊C8—216no—
249C4⋊C8—216no—
250#104—416no—
251#95—216no—
252#95—21no—
253#95—22no—
254#95—21no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—48no—
264#95—44no—
265C24—24no—
266C24—22no—
267C24—22no—
268C24—21no—
269C24—42no—
270C24C2×C22⋊C411yescharacteristic
271C24C2×C22⋊C411yescharacteristic
272C24#12114yes—
273C24C22×D411yescharacteristic
274#41—44no—
275#41—44no—
276#41—48no—
277C2×Q8—22no—
278C2×Q8—24no—
279C2×D4—22no—
280C2×D4—24no—
281C2×D4—44no—
282C2×D4—42no—
283C2×D4—44no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—28no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—28no—
300C22×C4—216no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—28no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—416no—
326C22×C4—416no—
327C22×C4—48no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—48no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4#10414yescomplemented
336C22×C4C2×C22⋊C412yes—
337C22×C4C2×C22⋊C412yes—
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C412yes—
341C22×C4C2×C22⋊C412yes—
342C22×C4C4×D414yes—
343C22×C4C4×D414yes—
344C22×C4#12012yes—
345C22×C4#12012yes—
346C22×C4#12114yes—
347C22×C4#12312yes—
348C22×C4#12312yes—
349C22×C4C23×C411yescharacteristic
350C22×C4C22×D412yes—
351C22×C4C22×D411yescharacteristic
352C22×C4C2×#4112yes—
353C22×C4H(5,𝔽2)14yes—
354C22×C42−1+414yes—
355M16—28no—
356M16—28no—
357M16—416no—
358M16—48no—
359C2×C8—28no—
360C2×C8—416no—
361C2×C8—48no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C22⋊C4—44no—
373C22⋊C4—44no—
374C22⋊C4—416no—
375C22⋊C4—44no—
376C22⋊C4—42no—
377C22⋊C4—44no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—44no—
385C22⋊C4—44no—
386C42—24no—
387C42—24no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42—24no—
393C42—24no—
394C42—24no—
395C42—24no—
396C42—41no—
397C42—41no—
398C42—42no—
399C42—42no—
400C42—44no—
401C42—44no—
402C42—48no—
403C23—22no—
404C23—21no—
405C23—22no—
406C23—22no—
407C23—21no—
408C23—22no—
409C23—28no—
410C23—28no—
411C23—21no—
412C23—22no—
413C23—21no—
414C23—44no—
415C23—48no—
416C23—48no—
417C23—42no—
418C23—41no—
419C23—42no—
420C23—41no—
421C23—41no—
422C23#38612yes—
423C23#38612yes—
424C23C2×#10412yes—
425C23#42112yes—
426C23C22×C22⋊C411yescharacteristic
427C23C2×#12112yes—
428C23#53714yes—
429Q8—44no—
430D4—44no—
431D4—84no—
432D4—84no—
433D4—84no—
434C2×C4—28no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—28no—
438C2×C4—28no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—28no—
444C2×C4—28no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—48no—
458C2×C4—48no—
459C2×C4—48no—
460C2×C4—48no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—48no—
464C2×C4—416no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—88no—
479C2×C4—84no—
480C2×C4—84no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—82no—
484C2×C4#38612yes—
485C2×C4#38612yes—
486C2×C4C2×#10414yes—
487C2×C4#42114yes—
488C2×C4C2×C4×D412yes—
489C2×C4#52014yes—
490C2×C4C2×#12011yescharacteristic
491C2×C4C2×#12311yescharacteristic
492C2×C4#53414yes—
493C2×C4#53512yes—
494C2×C4#53614yes—
495C2×C4#53612yes—
496C2×C4#53918yes—
497C8—48no—
498C22—24no—
499C22—24no—
500C22—24no—
501C22—24no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—416no—
510C22—42no—
511C22—42no—
512C22—82no—
513C22—82no—
514C22—81no—
515C22—81no—
516C22—81no—
517C22#187114yes—
518C22C2×#38611yescharacteristic
519C22#231112yes—
520C22#261814yes—
521C22C22×#10411yescharacteristic
522C22C2×#42112yes—
523C22C2×#42111yescharacteristic
524C4—24no—
525C4—24no—
526C4—22no—
527C4—22no—
528C4—42no—
529C4—42no—
530C4—88no—
531C4—82no—
532C4—82no—
533C4—81no—
534C4—81no—
535C4—81no—
536C4#232214yes—
537C4#261311yescharacteristic
538C4#261411yescharacteristic
539C4#262312yes—
540C2—24no—
541C2—24no—
542C2—42no—
543C2—161no—
544C2#1425611yescharacteristic
545C2C2×#187111yescharacteristic
546C2#2090211yescharacteristic
547C2#2185612yes—
548C2#2185712yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#423C2semidirect
4. C4×#381C2semidirect
5. #11209C2semidirect
6. #11053C2semidirect
7. #10732C2semidirect
17. #2855C4semidirect
18. #2855C4semidirect
19. #2855C4semidirect
20. #2854C4semidirect
21. #2854C4semidirect
22. #2854C4semidirect
335. C22×C4#104semidirect

Maximal subgroups

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

subgroupindex
C22×#4232
#366262
C2×#18712
C4×#3812
#112092
#110532
#107322