Finite Group Databasebeta

← #222982

Subgroups — #222982

Order 512.

1102 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#555C211yesmaximal, characteristic
2#22650C211yesmaximal, characteristic, complemented
3C4×#436C211yesmaximal, characteristic, complemented
4#9870C211yesmaximal, characteristic, complemented
5#9864C211yesmaximal, characteristic, complemented
6#9078C212yesmaximal, complemented
7#8944C212yesmaximal, complemented
8#8708C212yesmaximal, complemented
9#8563C211yesmaximal, characteristic, complemented
10C4×D4⋊C8C211yesmaximal, characteristic, complemented
11#7903C212yesmaximal, complemented
12C2×#555C411yescharacteristic
13C2×#436—21no—
14C2×#436—21no—
15C2×#423C2212yes—
16C2×C4×M16C2212yes—
17#2372—22no—
18#2369C411yescharacteristic
19#2363—21no—
20#2363C412yes—
21#2354C411yescharacteristic
22#2354C411yescharacteristic
23#2346—21no—
24#2346C412yes—
25C4×C4⋊Q8C2211yescharacteristic
26C4×#124—22no—
27C4×#121—21no—
28C4×#121C2212yes—
29C42×D4C2211yescharacteristic
30C42×D4C2211yescharacteristic
31C4×#117—21no—
32C4×#117C2212yes—
33#1939—21no—
34#1851—22no—
35#1842C2211yescharacteristic
36#1797C2212yes—
37#1770—21no—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40#1753—22no—
41C4×C22⋊C8C2212yes—
42C42×C8C2211yescharacteristic
43C2×D4⋊C8C2211yescharacteristic
44C2×D4⋊C8C2211yescharacteristic
45C2×D4⋊C8C2212yes—
46#1279C2211yescharacteristic
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52#1279C2211yescharacteristic
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×C411yescharacteristic
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×#117—21no—
70C2×#117—21no—
71C2×#117C2×C412yes—
72C2×#117C2×C412yes—
73C22×C42—21no—
74C22×C42C2312yes—
75#436—28no—
76#436—44no—
77#423—24no—
78#423—42no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2312yes—
85C2×C4⋊C8C2311yescharacteristic
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C4×M16—24no—
89C4×M16—42no—
90C2×#97—22no—
91C2×C4×C8—21no—
92C2×C4×C8—21no—
93C2×C4×C8—21no—
94C2×C4×C8C2312yes—
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97#391—21no—
98#391—22no—
99#390—21no—
100#390—22no—
101#390C2×C412yes—
102#389—21no—
103#389—21no—
104#389C2×C412yes—
105#389C2×C412yes—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#386—22no—
113#386—22no—
114#384C2×C411yescharacteristic
115#382—22no—
116#380—21no—
117#380C2×C412yes—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4D411yescharacteristic
128C4×C4⋊C4C2312yes—
129C4×C4⋊C4C2311yescharacteristic
130C4×C22⋊C4—22no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—21no—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—22no—
137C4×C22⋊C4C2×C412yes—
138C4×C22⋊C4D412yes—
139C4×C22⋊C4D412yes—
140#376—22no—
141#376—22no—
142C43—22no—
143C43C2311yescharacteristic
144#336—22no—
145#336—21no—
146#336—22no—
147#336—21no—
148D4⋊C8—22no—
149D4⋊C8—22no—
150D4⋊C8—42no—
151C22×Q8—21no—
152C22×D4—22no—
153C22×D4C22⋊C411yescharacteristic
154C22×D4C22×C411yescharacteristic
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4C22×C412yes—
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C2×M16—28no—
162C2×M16—44no—
163C2×M16—44no—
164C2×M16—44no—
165C2×M16—44no—
166C2×M16—82no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—21no—
173C4⋊Q8—22no—
174C4⋊Q8M1614yescomplemented
175#124—216no—
176#121—28no—
177#121—24no—
178#121—24no—
179C4×Q8—81no—
180C4×Q8—81no—
181C4×D4—24no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—22no—
187C4×D4—22no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—41no—
193C4×D4—82no—
194C4×D4—82no—
195C4×D4M1614yescomplemented
196#117—28no—
197#117—24no—
198#117—24no—
199#117—24no—
200#117—42no—
201#117—44no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4C22⋊C411yescharacteristic
208C2×C4⋊C4C22⋊C411yescharacteristic
209C2×C4⋊C4C22⋊C411yescharacteristic
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C412yes—
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4C22⋊C412yes—
225C2×C22⋊C4C22⋊C412yes—
226C2×C22⋊C4C22⋊C412yes—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—22no—
237C2×C42—21no—
238C2×C42—22no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42C22×C411yescharacteristic
242C2×C42C22×C412yes—
243C2×C42C22×C411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D412yes—
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D412yes—
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2411yescharacteristic
251C4⋊C8—24no—
252C4⋊C8—21no—
253C4⋊C8—22no—
254C4⋊C8—21no—
255C4⋊C8—22no—
256C4⋊C8—21no—
257C4⋊C8—44no—
258C4⋊C8—42no—
259C4⋊C8—81no—
260C4⋊C8#4112yes—
261C22⋊C8—28no—
262C22⋊C8—44no—
263#97—44no—
264C4×C8—24no—
265C4×C8—21no—
266C4×C8—22no—
267C4×C8—21no—
268C4×C8—22no—
269C4×C8—21no—
270C4×C8—42no—
271C4×C8—42no—
272C4×C8#4112yes—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—21no—
279#95—21no—
280#95—21no—
281#95—22no—
282#95C22⋊C412yes—
283#95C22⋊C412yes—
284#95C22⋊C412yes—
285C24—21no—
286C24C2×C22⋊C412yes—
287C2×Q8—24no—
288C2×Q8—81no—
289C2×Q8—81no—
290C2×D4—28no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—82no—
300C2×D4—82no—
301C2×D4C2×M1612yes—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—81no—
335C22×C4C2×C22⋊C412yes—
336C22×C4C2×C22⋊C412yes—
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C412yes—
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C23×C411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345M16—416no—
346M16—88no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—24no—
352C2×C8—22no—
353C2×C8—41no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—42no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—41no—
365C4⋊C4—21no—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—24no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—42no—
376C4⋊C4—82no—
377C4⋊C4—81no—
378C4⋊C4—82no—
379C4⋊C4#10414yes—
380C4⋊C4C2×M1612yes—
381C4⋊C4C2×M1614yes—
382C4⋊C4#13112yes—
383C22⋊C4—28no—
384C22⋊C4—28no—
385C22⋊C4—24no—
386C22⋊C4—24no—
387C22⋊C4—24no—
388C22⋊C4—24no—
389C22⋊C4—42no—
390C22⋊C4—44no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—44no—
394C22⋊C4—82no—
395C22⋊C4—82no—
396C42—28no—
397C42—22no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—21no—
404C42—21no—
405C42—21no—
406C42—21no—
407C42—22no—
408C42—41no—
409C42—44no—
410C42—41no—
411C42—41no—
412C42—42no—
413C42—42no—
414C42—42no—
415C42—42no—
416C42—44no—
417C42—82no—
418C42C2×M1612yes—
419C42C2×M1612yes—
420C42AGL(1,ℤ/8)12yes—
421C42#13712yes—
422C42C2×#4111yescharacteristic
423C42C2×#4111yescharacteristic
424C42H(5,𝔽2)11yescharacteristic
425C422−1+411yescharacteristic
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23C22×C22⋊C411yescharacteristic
441Q8—84no—
442D4—22no—
443D4—22no—
444D4—41no—
445D4—41no—
446D4—44no—
447D4—88no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—48no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—84no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4#40712yes—
497C2×C4#40712yes—
498C2×C4#41712yes—
499C2×C4#41712yes—
500C2×C4C2×#10414yes—
501C2×C4C2×#10412yes—
502C2×C4C4×#4111yescharacteristic
503C2×C4#52011yescharacteristic
504C2×C4#54611yescharacteristic
505C2×C4#54611yescharacteristic
506C2×C4#54711yescharacteristic
507C2×C4#54711yescharacteristic
508C2×C4C22×M1611yescharacteristic
509C2×C4C22×M1611yescharacteristic
510C2×C4C2×#13111yescharacteristic
511C2×C4#56811yescharacteristic
512C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
513C2×C4C2×#13711yescharacteristic
514C2×C4#57611yescharacteristic
515C2×C4#57711yescharacteristic
516C8—24no—
517C8—42no—
518C8—44no—
519C8—42no—
520C8—81no—
521C8—82no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—81no—
536C22#233911yescharacteristic
537C22C2×#40711yescharacteristic
538C22#288011yescharacteristic
539C22C2×#41711yescharacteristic
540C22#289711yescharacteristic
541C22C22×#10411yescharacteristic
542C22#290111yescharacteristic
543C4—22no—
544C4—22no—
545C4—21no—
546C4—21no—
547C4—24no—
548C4—42no—
549C4—44no—
550C4—42no—
551C4—41no—
552C4—42no—
553C4—42no—
554C4—82no—
555C4—81no—
556C4#148914yes—
557C4#296511yescharacteristic
558C4#297111yescharacteristic
559C4#298911yescharacteristic
560C4#299111yescharacteristic
561C4#332511yescharacteristic
562C4#332911yescharacteristic
563C4#335411yescharacteristic
564C4#335711yescharacteristic
565C2—82no—
566C2—82no—
567C2C2×#148911yescharacteristic
568C2#1745311yescharacteristic
569C2#1751211yescharacteristic
570C2#1752011yescharacteristic
571C2#2247311yescharacteristic
572C2#2250311yescharacteristic
573C2#2250511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#555C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22650C2semidirect
3. C4×#436C2semidirect
4. #9870C2semidirect
5. #9864C2semidirect
6. #9078C2semidirect
7. #8944C2semidirect
8. #8708C2semidirect
9. #8563C2semidirect
10. C4×D4⋊C8C2semidirect
11. #7903C2semidirect
174. C4⋊Q8M16semidirect
195. C4×D4M16semidirect

Maximal subgroups

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

subgroupindex
C4×#5552
#226502
C4×#4362
#98702
#98642
#90782
#89442
#87082
#85632
C4×D4⋊C82
#79032