Finite Group Databasebeta

← #195025

Subgroups — #195025

Order 512.

1609 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#60099C211yesmaximal, characteristic, complemented
2#34935C211yesmaximal, characteristic
3C2×#1486C212yesmaximal, complemented
4#17349C211yesmaximal, characteristic, complemented
5#9035C218yesmaximal, complemented
6#8095C212yesmaximal, complemented
7C2×#568—21no—
8C2×#525—21no—
9C2×C4×#41C2211yescharacteristic
10C22×C4×D4C2211yescharacteristic
11C2×#431C2212yes—
12#2906—24no—
13#2906—24no—
14#2905—21no—
15#2905C2214yescomplemented
16C2×#423—22no—
17C2×#423C2212yes—
18#2538—21no—
19#2537—21no—
20#2536C412yes—
21#2535C412yes—
22#2534—22no—
23#2486C412yes—
24#2408C412yes—
25#2392C2214yes—
26C4×#121—22no—
27#2285C2211yescharacteristic
28#2280—21no—
29#1486—24no—
30#1486C2218yes—
31C2×D4⋊C8—24no—
32#1461—21no—
33#1460—21no—
34#1459—22no—
35#1458C2214yescomplemented
36#1303C2218yescomplemented
37C22×#41D411yescharacteristic
38C23×D4C2×C411yescharacteristic
39C24×C4—21no—
40#568—44no—
41#568—44no—
42C2×#131—28no—
43C22×M16—24no—
44C22×M16—22no—
45#525—44no—
46#525—44no—
47C2×#123—22no—
48C2×C22⋊Q8—22no—
49C2×#121—22no—
50C2×#121—22no—
51C2×#120—22no—
52C4×#41—22no—
53C4×#41—22no—
54C4×#41—22no—
55C4×#41—22no—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8D411yescharacteristic
58C2×C4×D4—24no—
59C2×C4×D4—21no—
60C2×C4×D4—21no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4C2×C412yes—
66C2×C4×D4C2×C412yes—
67C2×C4×D4C2×C411yescharacteristic
68C2×C4×D4D412yes—
69C2×C4×D4D411yescharacteristic
70C2×C4×D4C2312yes—
71C2×#117—21no—
72C2×#117D411yescharacteristic
73C2×#117C2312yes—
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C22⋊C4—21no—
76C22×C42D412yes—
77C22×C42C2311yescharacteristic
78#431—22no—
79#431—22no—
80#431—22no—
81#431C2314yes—
82#424—28no—
83#424—216no—
84#423—24no—
85#423—24no—
86#423—22no—
87#423—24no—
88#423—22no—
89#423D414yescomplemented
90#423C2314yes—
91C2×C4⋊C8—24no—
92C2×C4⋊C8—24no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8D414yescomplemented
96C2×C22⋊C8—22no—
97C2×C4×C8—22no—
98#398C2×C414yes—
99#397C2×C414yes—
100#396—24no—
101#394—24no—
102#394C2×C414yes—
103#392C2×C414yes—
104#390—22no—
105#389—22no—
106#388—24no—
107#386—24no—
108#384—21no—
109#382C2×C412yes—
110#381—22no—
111#380—22no—
112#380—21no—
113#380—22no—
114#380—22no—
115#380C2×C412yes—
116#379—22no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4C2312yes—
119C4×C22⋊C4—24no—
120D4⋊C8—216no—
121#324—24no—
122#323—24no—
123C25—21no—
124C2×#41—22no—
125C2×#41—22no—
126C2×#41—22no—
127C2×#41—22no—
128C2×#41D814yes—
129C2×#41SD1614yes—
130C22×Q8C22⋊C411yescharacteristic
131C22×D4—24no—
132C22×D4—21no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—44no—
136C22×D4C22⋊C412yes—
137C22×D4C22⋊C411yescharacteristic
138C22×D4C22×C412yes—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4C22⋊C412yes—
149C23×C4C22⋊C412yes—
150C23×C4C22×C411yescharacteristic
151C23×C4C2×D412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154#131—432no—
155C2×M16—28no—
156C2×M16—24no—
157C2×M16—24no—
158C2×M16—48no—
159C2×M16—48no—
160C2×M16—44no—
161C2×M16—44no—
162C22×C8—28no—
163C22×C8—24no—
164C22×C8—42no—
165#123—48no—
166C22⋊Q8—48no—
167#121—48no—
168#121—48no—
169#120—48no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×D4—24no—
175C4×D4—24no—
176C4×D4—24no—
177C4×D4—24no—
178C4×D4—22no—
179C4×D4—22no—
180C4×D4—24no—
181C4×D4—24no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—24no—
185C4×D4—24no—
186C4×D4—48no—
187C4×D4—48no—
188C4×D4—41no—
189C4×D4—41no—
190C4×D4—42no—
191C4×D4—42no—
192C4×D4—41no—
193C4×D4—41no—
194#117—22no—
195#117—22no—
196#117—24no—
197#117—24no—
198#117—42no—
199#117—42no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4C22⋊C411yescharacteristic
208C2×C4⋊C4C22⋊C411yescharacteristic
209C2×C4⋊C4C22×C412yes—
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4C22⋊C412yes—
222C2×C22⋊C4C22×C412yes—
223C2×C22⋊C4C2×D412yes—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42C22⋊C412yes—
230C2×C42C22⋊C412yes—
231C2×C42C22×C411yescharacteristic
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42#4112yes—
238C2×C42#4112yes—
239C2×C42C2411yescharacteristic
240C4⋊C8—28no—
241C4⋊C8—28no—
242C4⋊C8—24no—
243C4⋊C8—24no—
244C4⋊C8—24no—
245C4⋊C8—24no—
246C4⋊C8—42no—
247C4⋊C8C2×D418yes—
248C22⋊C8—24no—
249C22⋊C8—42no—
250C4×C8—24no—
251C4×C8—42no—
252#95—24no—
253#95—24no—
254#95—22no—
255#95C22×C414yes—
256C24—22no—
257C24—44no—
258C24—42no—
259C24—42no—
260C24—42no—
261C24—82no—
262C24C2×C22⋊C412yes—
263C24C2×C22⋊C411yescharacteristic
264#41—28no—
265#41—28no—
266#41—28no—
267#41—28no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8#10214yes—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—28no—
274C2×D4—24no—
275C2×D4—24no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—42no—
279C2×D4—44no—
280C2×D4—44no—
281C2×D4—48no—
282C2×D4—48no—
283C2×D4—48no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—84no—
287C2×D4#10218yes—
288C2×D4#10214yes—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—42no—
300C22×C4—44no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—81no—
314C22×C4—82no—
315C22×C4—84no—
316C22×C4—84no—
317C22×C4—84no—
318C22×C4—84no—
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C412yes—
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C22⋊C412yes—
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C4×D412yes—
326C22×C4C4×D412yes—
327C22×C4#12012yes—
328C22×C4#12112yes—
329C22×C4#12112yes—
330C22×C4#12312yes—
331C22×C4C2×D814yes—
332C22×C4C2×D812yes—
333C22×C4C2×SD1614yes—
334C22×C4C2×SD1612yes—
335C22×C4AGL(1,ℤ/8)12yes—
336C22×C4#13712yes—
337C22×C4C23×C411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340M16—416no—
341M16—44no—
342M16—44no—
343M16—48no—
344C2×C8—28no—
345C2×C8—416no—
346C2×C8—48no—
347C2×C8—44no—
348C2×C8—84no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—24no—
354C4⋊C4—44no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—44no—
358C4⋊C4—48no—
359C4⋊C4—82no—
360C22⋊C4—24no—
361C22⋊C4—24no—
362C22⋊C4—24no—
363C22⋊C4—48no—
364C22⋊C4—44no—
365C22⋊C4—48no—
366C22⋊C4—44no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—48no—
370C22⋊C4—84no—
371C42—22no—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—42no—
378C42—42no—
379C42C4×D412yes—
380C42C4×D412yes—
381C42#12011yescharacteristic
382C42#12011yescharacteristic
383C42#12112yes—
384C42#12112yes—
385C42#12311yescharacteristic
386C42#12311yescharacteristic
387C42C22×D412yes—
388C42C2×#4112yes—
389C23—22no—
390C23—22no—
391C23—24no—
392C23—41no—
393C23—41no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—48no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—84no—
404C23—88no—
405C23—88no—
406C23—84no—
407C23#38612yes—
408C23C2×#10214yes—
409C23C2×#10212yes—
410C23#41712yes—
411C23C22×C22⋊C411yescharacteristic
412Q8—28no—
413D4—216no—
414D4—28no—
415D4—28no—
416D4—44no—
417D4—44no—
418D4—44no—
419D4—816no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—84no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—88no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4#38611yescharacteristic
455C2×C4#38611yescharacteristic
456C2×C4#38611yescharacteristic
457C2×C4#38612yes—
458C2×C4#38611yescharacteristic
459C2×C4C2×#10214yes—
460C2×C4C2×#10212yes—
461C2×C4#41712yes—
462C2×C4#44714yes—
463C2×C4#44814yes—
464C2×C4#48014yes—
465C2×C4#48114yes—
466C2×C4C2×C4×D412yes—
467C2×C4C2×#12011yescharacteristic
468C2×C4C2×#12112yes—
469C2×C4C2×#12311yescharacteristic
470C2×C4C22×D811yescharacteristic
471C2×C4C22×SD1611yescharacteristic
472C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
473C2×C4C2×#13711yescharacteristic
474C8—48no—
475C8—84no—
476C22—21no—
477C22—22no—
478C22—21no—
479C22—22no—
480C22—24no—
481C22—42no—
482C22—42no—
483C22—44no—
484C22—44no—
485C22—44no—
486C22—42no—
487C22—42no—
488C22—44no—
489C22—82no—
490C22—82no—
491C22—82no—
492C22—816no—
493C22—82no—
494C22—82no—
495C22#187414yes—
496C22C2×#38611yescharacteristic
497C22C22×#10211yescharacteristic
498C22C2×#41711yescharacteristic
499C4—24no—
500C4—22no—
501C4—21no—
502C4—21no—
503C4—21no—
504C4—22no—
505C4—42no—
506C4—42no—
507C4—84no—
508C4—82no—
509C4#187414yes—
510C4#189112yes—
511C4C2×#44711yescharacteristic
512C4C2×#44811yescharacteristic
513C4C2×#48011yescharacteristic
514C4C2×#48111yescharacteristic
515C2—21no—
516C2—21no—
517C2—24no—
518C2—44no—
519C2—84no—
520C2C2×#187411yescharacteristic
521C2#2094711yescharacteristic
522C2C2×#189111yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radical#34935C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60099C2semidirect
3. C2×#1486C2semidirect
4. #17349C2semidirect
5. #9035C2semidirect
6. #8095C2semidirect
15. #2905C22semidirect
35. #1458C22semidirect
36. #1303C22semidirect
89. #423D4semidirect
95. C2×C4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
#600992
#349352
C2×#14862
#173492
#90352
#80952