Finite Group Databasebeta

← #364052

Subgroups — #364052

Order 512.

1286 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34109C211yesmaximal, characteristic
2#20694C211yesmaximal, characteristic
3C2×#1740C211yesmaximal, characteristic
4#11348C212yesmaximal, complemented
5#11347C212yesmaximal, complemented
6#10824C212yesmaximal, complemented
7#10816C212yesmaximal, complemented
8#10733C212yesmaximal
9#10716C212yesmaximal
10C22×#117C2212yes—
11C22×#117C2211yescharacteristic
12C2×#407C2212yes—
13#2347—21no—
14#2346—21no—
15#2345—21no—
16#2344—21no—
17#2343—22no—
18#2342—22no—
19#2298C412yescomplemented
20#2298C412yescomplemented
21#2296C2211yescharacteristic
22#2296C2211yescharacteristic
23#2283C412yescomplemented
24#2280C2211yescharacteristic
25C4×#117C412yescomplemented
26C4×#117C2211yescharacteristic
27C4×#117C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29#1822—21no—
30#1822—21no—
31#1821—21no—
32#1821—21no—
33#1742C2212yes—
34#1742C2212yes—
35#1740—22no—
36#1740C2212yes—
37#1740C2212yes—
38C2×#342—22no—
39C2×#339C2214yes—
40C2×#339C2214yes—
41C2×H(3,ℤ/4)C2214yes—
42C2×H(3,ℤ/4)C2214yes—
43C24×C4C2311yescharacteristic
44C22×M16C2314yes—
45C2×#117—22no—
46C2×#117—22no—
47C2×#117—22no—
48C2×#117—22no—
49C2×#117—22no—
50C2×#117—22no—
51C2×#117—22no—
52C2×#117—28no—
53C2×#117—22no—
54C2×#117—44no—
55C2×#117C2×C412yes—
56C2×#117D411yescharacteristic
57C2×#117D411yescharacteristic
58C2×#117D412yes—
59C2×#117C2314yes—
60C2×#117C2311yescharacteristic
61C2×#117C2311yescharacteristic
62C2×#117C2311yescharacteristic
63C2×#117C2311yescharacteristic
64C22×C4⋊C4—22no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C22⋊C4—22no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4D412yes—
70C22×C42—22no—
71C22×C42C2×C412yes—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74#407—22no—
75#407—22no—
76#407—22no—
77#407—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80#389—21no—
81#389—21no—
82#389—21no—
83#389—21no—
84#388—22no—
85#388—22no—
86#388—22no—
87#388—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#385—22no—
93#385—22no—
94#383—22no—
95#382—22no—
96#381—22no—
97#381—21no—
98#380—22no—
99#380—21no—
100#380—21no—
101#379—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4C2×C412yescomplemented
108C4×C4⋊C4C2×C412yescomplemented
109C4×C4⋊C4C2×C412yescomplemented
110C4×C4⋊C4C2×C412yescomplemented
111C4×C22⋊C4—22no—
112C4×C22⋊C4—21no—
113C4×C22⋊C4—21no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4C2×C411yescharacteristic
117C4×C22⋊C4C2×C411yescharacteristic
118C4×C22⋊C4C2×C411yescharacteristic
119C4×C22⋊C4C2×C411yescharacteristic
120C4×C22⋊C4C2×C412yes—
121C4×C22⋊C4C2×C412yes—
122#376—22no—
123#376C2×C412yescomplemented
124#376C2×C412yescomplemented
125C2×#95—21no—
126C43C2×C412yescomplemented
127C43C2×C412yescomplemented
128#343—22no—
129#343—22no—
130#343—22no—
131#343—22no—
132#342—24no—
133#342—24no—
134#339—28no—
135#339—44no—
136H(3,ℤ/4)—28no—
137H(3,ℤ/4)—44no—
138C25C2×D411yescharacteristic
139C23×C4—24no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—42no—
145C23×C4C22⋊C412yes—
146C23×C4C22⋊C412yes—
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4#4112yes—
153C23×C4#4112yes—
154C23×C4#4112yes—
155C23×C4#4112yes—
156C23×C4C2411yescharacteristic
157C2×M16—24no—
158C2×M16—24no—
159C2×M16—24no—
160C2×M16—24no—
161C2×M16—24no—
162C2×M16—24no—
163C22×C8—24no—
164#117—24no—
165#117—24no—
166#117—24no—
167#117—24no—
168#117—22no—
169#117—22no—
170#117—22no—
171#117—22no—
172#117—22no—
173#117—22no—
174#117—22no—
175#117—22no—
176#117—24no—
177#117—24no—
178#117—22no—
179#117—22no—
180#117—28no—
181#117—24no—
182#117—24no—
183#117—48no—
184#117—48no—
185#117—48no—
186#117—44no—
187#117#4118yes—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—48no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4C22⋊C412yes—
202C2×C4⋊C4C22⋊C412yes—
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C22⋊C4—21no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4C22⋊C412yes—
223C2×C22⋊C4C22⋊C412yes—
224C2×C22⋊C4C2×D412yes—
225C2×C22⋊C4C2×D412yes—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—48no—
241C2×C42—42no—
242C2×C42C22⋊C412yes—
243C2×C42C22⋊C412yes—
244C2×C42C22×C412yes—
245C2×C42C22×C412yes—
246C2×C42C22×C411yescharacteristic
247C2×C42C22×C411yescharacteristic
248C2×C42C22×C411yescharacteristic
249C2×C42C22×C411yescharacteristic
250C2×C42C22×C412yes—
251C2×C42C22×C412yes—
252C22⋊C8—42no—
253C22⋊C8—42no—
254C22⋊C8—42no—
255C22⋊C8—42no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270C24—22no—
271C24—21no—
272C24—21no—
273C24—42no—
274C24—81no—
275C24C2×C22⋊C411yescharacteristic
276C24C2×C22⋊C411yescharacteristic
277C24#12312yes—
278C24#12312yes—
279C24C22×D411yescharacteristic
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—44no—
312C22×C4—48no—
313C22×C4—44no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—81no—
318C22×C4—81no—
319C22×C4—82no—
320C22×C4—84no—
321C22×C4—84no—
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C412yes—
328C22×C4#11711yescharacteristic
329C22×C4#11711yescharacteristic
330C22×C4#11712yes—
331C22×C4#11711yescharacteristic
332C22×C4#11711yescharacteristic
333C22×C4#11712yes—
334C22×C4#12312yes—
335C22×C4#12312yes—
336C22×C4#12312yes—
337C22×C4#12312yes—
338C22×C4#12312yes—
339C22×C4#12312yes—
340C22×C4C23×C411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C2×#4112yes—
343C22×C4C2×#4112yes—
344C22×C4C2×#4114yes—
345C22×C4H(5,𝔽2)12yes—
346C22×C42−1+412yes—
347M16—44no—
348M16—44no—
349M16—44no—
350M16—44no—
351C2×C8—44no—
352C2×C8—44no—
353C2×C8—44no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—28no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—48no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—44no—
367C4⋊C4—82no—
368C4⋊C4—82no—
369C22⋊C4—24no—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C22⋊C4—24no—
373C22⋊C4—24no—
374C22⋊C4—28no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—48no—
379C22⋊C4—44no—
380C22⋊C4—42no—
381C22⋊C4—44no—
382C22⋊C4—41no—
383C22⋊C4—41no—
384C22⋊C4—82no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—24no—
396C42—22no—
397C42—22no—
398C42—24no—
399C42—28no—
400C42—44no—
401C42—44no—
402C42—48no—
403C42—42no—
404C42—42no—
405C42—44no—
406C42—82no—
407C42—82no—
408C23—21no—
409C23—22no—
410C23—24no—
411C23—22no—
412C23—22no—
413C23—21no—
414C23—21no—
415C23—22no—
416C23—22no—
417C23—21no—
418C23—41no—
419C23—41no—
420C23—42no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—42no—
425C23—42no—
426C23—84no—
427C23—82no—
428C23—82no—
429C23#38112yes—
430C23#38111yescharacteristic
431C23#38111yescharacteristic
432C23C22×C22⋊C411yescharacteristic
433C23C2×#12312yes—
434C23#52512yes—
435C23#53511yescharacteristic
436C23#53611yescharacteristic
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—44no—
464C2×C4—41no—
465C2×C4—44no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—48no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—84no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—84no—
490C2×C4—84no—
491C2×C4—816no—
492C2×C4#38111yescharacteristic
493C2×C4#38111yescharacteristic
494C2×C4#38112yes—
495C2×C4C2×#11711yescharacteristic
496C2×C4C2×#11711yescharacteristic
497C2×C4C4×#4112yes—
498C2×C4#52012yes—
499C2×C4C2×#12312yes—
500C2×C4#53212yes—
501C2×C4#53712yes—
502C2×C4#55114yes—
503C2×C4#55114yes—
504C2×C4#55214yes—
505C2×C4#55214yes—
506C8—84no—
507C22—21no—
508C22—21no—
509C22—22no—
510C22—21no—
511C22—22no—
512C22—21no—
513C22—21no—
514C22—22no—
515C22—21no—
516C22—22no—
517C22—21no—
518C22—22no—
519C22—42no—
520C22—41no—
521C22—41no—
522C22—44no—
523C22—42no—
524C22—42no—
525C22—44no—
526C22—81no—
527C22—81no—
528C22—82no—
529C22—82no—
530C22#182712yes—
531C22C2×#38111yescharacteristic
532C22#229711yescharacteristic
533C22#231111yescharacteristic
534C22#248212yes—
535C22#248412yes—
536C4—21no—
537C4—22no—
538C4—21no—
539C4—42no—
540C4—42no—
541C4—42no—
542C4—42no—
543C4—42no—
544C4—42no—
545C4—82no—
546C4—82no—
547C4—81no—
548C4—81no—
549C4—88no—
550C4#234312yes—
551C4#234312yes—
552C4#249814yes—
553C2—21no—
554C2—21no—
555C2—22no—
556C2—22no—
557C2—22no—
558C2—41no—
559C2—42no—
560C2#1395811yescharacteristic
561C2C2×#182711yescharacteristic
562C2#2070811yescharacteristic
subgroupG/H
centerC2×C4C2×#117
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#381
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ #2280 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11348C2semidirect
5. #11347C2semidirect
6. #10824C2semidirect
7. #10816C2semidirect
19. #2298C4semidirect
20. #2298C4semidirect
23. #2283C4semidirect
25. C4×#117C4semidirect
107. C4×C4⋊C4C2×C4semidirect
108. C4×C4⋊C4C2×C4semidirect
109. C4×C4⋊C4C2×C4semidirect
110. C4×C4⋊C4C2×C4semidirect
123. #376C2×C4semidirect
124. #376C2×C4semidirect
126. C43C2×C4semidirect
127. C43C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341092
#206942
C2×#17402
#113482
#113472
#108242
#108162
#107332
#107162