Finite Group Databasebeta

← #200392

Subgroups — #200392

Order 512.

1479 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60276C211yesmaximal, characteristic, complemented
2#34937C211yesmaximal, characteristic
3#20632C211yesmaximal, characteristic, complemented
4#17574C212yesmaximal, complemented
5#9384C218yesmaximal, complemented
6#8014C212yesmaximal, complemented
7C2×#532—21no—
8C2×#532C2211yescharacteristic
9C2×C4×#41C2211yescharacteristic
10#2988—24no—
11#2982—21no—
12#2972—21no—
13#2972C2214yes—
14C8×#41—24no—
15C2×#431—22no—
16C2×#431C2212yes—
17C2×#410—21no—
18#2548C411yescharacteristic, complemented
19#2547C411yescharacteristic, complemented
20#2546C411yescharacteristic, complemented
21#2545—21no—
22#2544—21no—
23#2543C2214yes—
24#2541C411yescharacteristic, complemented
25#2416—22no—
26#2392C414yescomplemented
27C4×#121—22no—
28#2297—21no—
29#2285C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31#1799—22no—
32#1799C2214yes—
33#1740—21no—
34#1519C2218yes—
35#1517—24no—
36#1502—24no—
37#1294C2218yes—
38C22×#41—21no—
39C22×#41D411yescharacteristic
40#532—22no—
41#532—22no—
42#532—22no—
43#532—22no—
44#532—44no—
45#532—44no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8C2×C411yescharacteristic
48C2×#127—21no—
49C2×#127C2×C411yescharacteristic
50C2×#124—21no—
51C2×#124—21no—
52C2×#124—21no—
53C2×#121—24no—
54C2×#121—22no—
55C2×#121—21no—
56C2×#121C2312yes—
57C4×#41—22no—
58C4×#41—22no—
59C4×#41—22no—
60C4×#41—22no—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×D4—24no—
64C2×C4×D4—21no—
65C2×C4×D4C2×C412yes—
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4D412yes—
68C2×C4×D4D411yescharacteristic
69C2×C4×D4C2312yes—
70C2×#117—22no—
71C2×#117C2×C412yes—
72C2×#117D411yescharacteristic
73C22×C4⋊C4—21no—
74C22×C42—21no—
75C22×C42D412yes—
76C22×C42C2311yescharacteristic
77C8×Q8—28no—
78C8×D4—28no—
79C8×D4—216no—
80#431—24no—
81#431—24no—
82#431—22no—
83#431—24no—
84#431—22no—
85#431D414yescomplemented
86#431D414yescomplemented
87C2×C4⋊C8—24no—
88C2×C4⋊C8—22no—
89#410—44no—
90C2×#99—24no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8C2314yes—
93C2×C4×C8—24no—
94C2×C4×C8—22no—
95#398C2×C414yes—
96#397C2×C414yes—
97#396—24no—
98#394—24no—
99#394C2×C414yes—
100#392C2×C414yes—
101#391—21no—
102#391—21no—
103#390—21no—
104#390—21no—
105#390—22no—
106#389—22no—
107#388—24no—
108#386—24no—
109#386—24no—
110#384—22no—
111#382—22no—
112#380—22no—
113#380—22no—
114#380C2314yes—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C22⋊C4—24no—
123C2×#95C2312yes—
124#342—24no—
125#327—216no—
126C2×#41—22no—
127C2×#41—22no—
128C2×#41—24no—
129C2×#41—42no—
130C2×#41—42no—
131C2×#41—42no—
132C2×#41—42no—
133C2×#41—44no—
134C2×#41D814yes—
135C2×#41SD1614yes—
136C22×Q8—21no—
137C22×Q8C22⋊C411yescharacteristic
138C22×D4—22no—
139C22×D4—24no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4C22⋊C411yescharacteristic
143C22×D4C22×C412yes—
144C22×D4C2×D412yes—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4C22⋊C412yes—
148C23×C4C22⋊C412yes—
149C23×C4C2×D412yes—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2411yescharacteristic
153C22×C8—28no—
154C22×C8—24no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—44no—
158#127—22no—
159#127—22no—
160#127—44no—
161#124—44no—
162#124—44no—
163#124—42no—
164#124—42no—
165#121—24no—
166#121—24no—
167#121—48no—
168#121—48no—
169#121—48no—
170#121—42no—
171#121—42no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×Q8—22no—
175C4×Q8—22no—
176C4×D4—24no—
177C4×D4—24no—
178C4×D4—24no—
179C4×D4—24no—
180C4×D4—22no—
181C4×D4—22no—
182C4×D4—24no—
183C4×D4—24no—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—416no—
187C4×D4—41no—
188C4×D4—41no—
189C4×D4—41no—
190C4×D4—41no—
191#117—22no—
192#117—22no—
193#117—24no—
194#117—24no—
195#117—44no—
196#117—44no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4C22⋊C412yes—
207C2×C4⋊C4C22⋊C411yescharacteristic
208C2×C4⋊C4C22⋊C411yescharacteristic
209C2×C4⋊C4C22×C412yes—
210C2×C4⋊C4C22×C412yes—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4C22⋊C412yes—
219C2×C22⋊C4C22×C412yes—
220C2×C22⋊C4C2×D412yes—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42C22⋊C412yes—
232C2×C42C22⋊C412yes—
233C2×C42C22×C411yescharacteristic
234C2×C42C22×C411yescharacteristic
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42#4112yes—
241C2×C42#4112yes—
242C4⋊C8—28no—
243C4⋊C8—24no—
244C4⋊C8—24no—
245#99—48no—
246C22⋊C8—28no—
247C22⋊C8—24no—
248C22⋊C8C2×D418yes—
249C4×C8—28no—
250C4×C8—24no—
251C4×C8—24no—
252#95—24no—
253#95—24no—
254#95—22no—
255#95—24no—
256#95—42no—
257#95C22×C412yes—
258#95C22×C412yes—
259C24—22no—
260C24C2×C22⋊C412yes—
261C24C22×D411yescharacteristic
262#41—28no—
263#41—28no—
264#41—28no—
265#41—28no—
266#41—84no—
267#41—84no—
268#41—84no—
269#41—84no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8#10214yes—
276C2×D4—24no—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—44no—
282C2×D4—48no—
283C2×D4—44no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—44no—
287C2×D4—48no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—84no—
294C2×D4—82no—
295C2×D4#10214yes—
296C2×D4C2×D814yes—
297C2×D4C2×SD1614yes—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—42no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—81no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4#10218yes—
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C412yes—
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4#12112yes—
330C22×C4#12112yes—
331C22×C4#12112yes—
332C22×C4#12112yes—
333C22×C4C2×D812yes—
334C22×C4C2×SD1612yes—
335C22×C4#13514yes—
336C22×C4C23×C411yescharacteristic
337C22×C4C22×D412yes—
338C22×C4C22×D411yescharacteristic
339C22×C4C2×#4112yes—
340C2×C8—28no—
341C2×C8—84no—
342C2×C8—88no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—24no—
349C4⋊C4—42no—
350C4⋊C4—44no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—48no—
354C4⋊C4—84no—
355C4⋊C4—84no—
356C4⋊C4—84no—
357C22⋊C4—24no—
358C22⋊C4—24no—
359C22⋊C4—24no—
360C22⋊C4—48no—
361C22⋊C4—48no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—44no—
365C22⋊C4—48no—
366C22⋊C4—44no—
367C22⋊C4—44no—
368C22⋊C4—84no—
369C42—22no—
370C42—22no—
371C42—22no—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—44no—
376C42—42no—
377C42—42no—
378C42—84no—
379C42C4×D412yes—
380C42C4×D412yes—
381C42C4×D412yes—
382C42C4×D412yes—
383C42#12012yes—
384C42#12011yescharacteristic
385C42#12011yescharacteristic
386C42#12311yescharacteristic
387C42#12312yes—
388C42#12311yescharacteristic
389C23—22no—
390C23—22no—
391C23—41no—
392C23—41no—
393C23—42no—
394C23—42no—
395C23—44no—
396C23—44no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23C2×#10214yes—
401C23C22×C22⋊C411yescharacteristic
402C23C2×#12112yes—
403C23C22×D811yescharacteristic
404C23C22×SD1611yescharacteristic
405C23C2×#13512yes—
406Q8—28no—
407Q8—84no—
408D4—28no—
409D4—28no—
410D4—28no—
411D4—84no—
412D4—82no—
413D4—816no—
414D4—84no—
415D4—84no—
416D4—82no—
417D4—84no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—28no—
423C2×C4—28no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—84no—
448C2×C4—84no—
449C2×C4—84no—
450C2×C4—84no—
451C2×C4—84no—
452C2×C4—84no—
453C2×C4—84no—
454C2×C4#38611yescharacteristic
455C2×C4#38612yes—
456C2×C4#38611yescharacteristic
457C2×C4#38611yescharacteristic
458C2×C4#38612yes—
459C2×C4#38611yescharacteristic
460C2×C4C2×#10214yes—
461C2×C4C2×#10212yes—
462C2×C4C2×#10212yes—
463C2×C4#41612yes—
464C2×C4#41612yes—
465C2×C4#46514yes—
466C2×C4#46514yes—
467C2×C4#46614yes—
468C2×C4#46614yes—
469C2×C4C2×C4×D412yes—
470C2×C4C2×#12011yescharacteristic
471C2×C4C2×#12311yescharacteristic
472C8—48no—
473C22—21no—
474C22—22no—
475C22—21no—
476C22—22no—
477C22—24no—
478C22—41no—
479C22—41no—
480C22—42no—
481C22—44no—
482C22—42no—
483C22—42no—
484C22—82no—
485C22—84no—
486C22—84no—
487C22—82no—
488C22—82no—
489C22—82no—
490C22C2×#38611yescharacteristic
491C22C22×#10211yescharacteristic
492C22C2×#41611yescharacteristic
493C22C2×#46512yes—
494C22C2×#46612yes—
495C4—21no—
496C4—21no—
497C4—22no—
498C4—21no—
499C4—22no—
500C4—24no—
501C4—44no—
502C4—42no—
503C4—42no—
504C4—82no—
505C4—82no—
506C4—164no—
507C4#188412yes—
508C4#189312yes—
509C4#189312yes—
510C4#189314yes—
511C2—21no—
512C2—22no—
513C2—21no—
514C2—22no—
515C2—44no—
516C2—162no—
517C2C2×#188411yescharacteristic
518C2C2×#189311yescharacteristic
519C2#2101411yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#386
prime radical#34937C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60276C2semidirect
3. #20632C2semidirect
4. #17574C2semidirect
5. #9384C2semidirect
6. #8014C2semidirect
18. #2548C4semidirect
19. #2547C4semidirect
20. #2546C4semidirect
24. #2541C4semidirect
26. #2392C4semidirect
85. #431D4semidirect
86. #431D4semidirect

Maximal subgroups

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

subgroupindex
#602762
#349372
#206322
#175742
#93842
#80142