Finite Group Databasebeta

← #222964

Subgroups — #222964

Order 512.

1136 conjugacy classes of subgroups.

The subgroup lattice has 571 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#555C211yesmaximal, characteristic
2#22795C211yesmaximal, characteristic, complemented
3C4×C8×D4C211yesmaximal, characteristic, complemented
4#9870C211yesmaximal, characteristic, complemented
5#9864C211yesmaximal, characteristic, complemented
6#9072C212yesmaximal, complemented
7#8940C212yesmaximal, complemented
8#8708C212yesmaximal, complemented
9#8563C211yesmaximal, characteristic, complemented
10C4×D4⋊C8C211yesmaximal, characteristic, complemented
11#7890C212yesmaximal, complemented
12C2×#555C411yescharacteristic
13C2×C8×D4—21no—
14C2×#431C2212yes—
15C22×C4×C8C2212yes—
16#2372—22no—
17#2369C411yescharacteristic
18#2363—21no—
19#2363C412yes—
20#2354C411yescharacteristic
21#2354C411yescharacteristic
22#2346—21no—
23#2346C412yes—
24C4×C4⋊Q8C2211yescharacteristic
25C4×#124—22no—
26C4×#121—21no—
27C4×#121C2212yes—
28C42×D4C2211yescharacteristic
29C42×D4C2211yescharacteristic
30C4×#117—21no—
31C4×#117C2212yes—
32#1954—21no—
33#1923—22no—
34#1842C2211yescharacteristic
35#1797C2212yes—
36C8×C4⋊C4—21no—
37C8×C4⋊C4—21no—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C8×C22⋊C4—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
53C23×C8—22no—
54#555—24no—
55#555C818yescomplemented
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×#124—22no—
58C2×#121—21no—
59C2×#121C2×C412yes—
60C2×C4×Q8—21no—
61C2×C4×D4—22no—
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4C2×C411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4C2311yescharacteristic
68C2×#117—21no—
69C2×#117—21no—
70C2×#117C2×C412yes—
71C2×#117C2×C412yes—
72C22×C42—21no—
73C22×C42C2312yes—
74C8×D4—28no—
75#431—24no—
76#431—42no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8C2311yescharacteristic
81C2×C4⋊C8C2312yes—
82C2×C4⋊C8C2311yescharacteristic
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C4×C8—21no—
87C2×C4×C8—24no—
88C2×C4×C8—22no—
89C2×C4×C8—22no—
90C2×C4×C8—21no—
91C2×C4×C8—21no—
92C2×C4×C8—42no—
93C2×C4×C8C2312yes—
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96#391—21no—
97#391—22no—
98#390—21no—
99#390—22no—
100#390C2×C412yes—
101#389—21no—
102#389—21no—
103#389C2×C412yes—
104#389C2×C412yes—
105#388—22no—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#386—22no—
112#386—22no—
113#384C2×C411yescharacteristic
114#382—22no—
115#380—21no—
116#380C2×C412yes—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4C2312yes—
128C4×C4⋊C4C2311yescharacteristic
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—21no—
132C4×C22⋊C4—21no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4C2×C412yes—
137C4×C22⋊C4D412yes—
138C4×C22⋊C4D412yes—
139#376—22no—
140#376—22no—
141C43—22no—
142C43C2311yescharacteristic
143#336—22no—
144#336—21no—
145#336—22no—
146#336—21no—
147D4⋊C8—22no—
148D4⋊C8—22no—
149D4⋊C8—42no—
150C22×Q8—21no—
151C22×D4—22no—
152C22×D4C22⋊C411yescharacteristic
153C22×D4C22×C411yescharacteristic
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4C22×C412yes—
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C22×C8—21no—
161C22×C8—28no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—44no—
168C22×C8—44no—
169C4⋊Q8—22no—
170C4⋊Q8C2×C814yescomplemented
171#124—216no—
172#121—28no—
173#121—24no—
174#121C2×C818yescomplemented
175C4×Q8—81no—
176C4×Q8—81no—
177C4×D4—24no—
178C4×D4—22no—
179C4×D4—22no—
180C4×D4—22no—
181C4×D4—22no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—41no—
187C4×D4—82no—
188C4×D4—82no—
189C4×D4C2×C814yes—
190C4×D4C2×C814yescomplemented
191C4×D4M1614yes—
192#117—24no—
193#117—28no—
194#117—24no—
195#117—42no—
196#117—44no—
197#117C2×C818yescomplemented
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4C22⋊C411yescharacteristic
204C2×C4⋊C4C22⋊C411yescharacteristic
205C2×C4⋊C4C22⋊C411yescharacteristic
206C2×C4⋊C4C22×C412yes—
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C412yes—
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4C22⋊C412yes—
221C2×C22⋊C4C22⋊C412yes—
222C2×C22⋊C4C22⋊C412yes—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—21no—
234C2×C42—22no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42C22×C411yescharacteristic
238C2×C42C22×C412yes—
239C2×C42C22×C411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D412yes—
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D412yes—
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2411yescharacteristic
247C4⋊C8—24no—
248C4⋊C8—21no—
249C4⋊C8—22no—
250C4⋊C8—21no—
251C4⋊C8—22no—
252C4⋊C8—21no—
253C4⋊C8—42no—
254C4⋊C8—81no—
255C4⋊C8#4112yes—
256C22⋊C8—28no—
257C22⋊C8—44no—
258C22⋊C8—82no—
259C4×C8—24no—
260C4×C8—21no—
261C4×C8—22no—
262C4×C8—21no—
263C4×C8—22no—
264C4×C8—21no—
265C4×C8—44no—
266C4×C8—44no—
267C4×C8—42no—
268C4×C8—42no—
269C4×C8#4112yes—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—21no—
276#95—21no—
277#95—21no—
278#95—22no—
279#95C22⋊C412yes—
280#95C22⋊C412yes—
281#95C22⋊C412yes—
282C24—21no—
283C24C2×C22⋊C412yes—
284C2×Q8—24no—
285C2×Q8—81no—
286C2×Q8—81no—
287C2×D4—28no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—82no—
296C2×D4—82no—
297C2×D4C22⋊C814yes—
298C2×D4C22×C812yes—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
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—24no—
315C22×C4—24no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—81no—
330C22×C4C2×C22⋊C412yes—
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C412yes—
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C22×C814yes—
338C22×C4C2×M1614yes—
339C22×C4C23×C411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—24no—
347C2×C8—22no—
348C2×C8—416no—
349C2×C8—41no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—42no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—42no—
359C2×C8—41no—
360C2×C8—42no—
361C4⋊C4—21no—
362C4⋊C4—22no—
363C4⋊C4—21no—
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—42no—
371C4⋊C4—82no—
372C4⋊C4—81no—
373C4⋊C4—82no—
374C4⋊C4C22⋊C814yes—
375C4⋊C4#10414yes—
376C4⋊C4C22×C812yes—
377C4⋊C4C22×C814yes—
378C4⋊C4#13112yes—
379C22⋊C4—28no—
380C22⋊C4—28no—
381C22⋊C4—24no—
382C22⋊C4—24no—
383C22⋊C4—24no—
384C22⋊C4—42no—
385C22⋊C4—44no—
386C22⋊C4—41no—
387C22⋊C4—41no—
388C22⋊C4—44no—
389C22⋊C4—82no—
390C22⋊C4—82no—
391C22⋊C4C22⋊C818yes—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—22no—
398C42—21no—
399C42—21no—
400C42—21no—
401C42—21no—
402C42—28no—
403C42—22no—
404C42—41no—
405C42—44no—
406C42—41no—
407C42—41no—
408C42—42no—
409C42—42no—
410C42—42no—
411C42—42no—
412C42—44no—
413C42—82no—
414C42C22×C812yes—
415C42C2×M1612yes—
416C42#13512yes—
417C42#13512yes—
418C42C2×#4111yescharacteristic
419C42C2×#4111yescharacteristic
420C42H(5,𝔽2)11yescharacteristic
421C422−1+411yescharacteristic
422C23—22no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23C2×C22⋊C814yes—
436C23C22×C22⋊C411yescharacteristic
437Q8—84no—
438D4—22no—
439D4—22no—
440D4—28no—
441D4—41no—
442D4—41no—
443D4—88no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—48no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—84no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4C2×C22⋊C814yes—
492C2×C4C2×C22⋊C812yes—
493C2×C4C2×C22⋊C812yes—
494C2×C4#41612yes—
495C2×C4#41612yes—
496C2×C4C2×#10414yes—
497C2×C4C2×#10412yes—
498C2×C4C4×#4111yescharacteristic
499C2×C4#52011yescharacteristic
500C2×C4#54611yescharacteristic
501C2×C4#54611yescharacteristic
502C2×C4#54711yescharacteristic
503C2×C4#54711yescharacteristic
504C2×C4C23×C811yescharacteristic
505C2×C4C22×M1611yescharacteristic
506C2×C4C2×#13111yescharacteristic
507C2×C4#56811yescharacteristic
508C2×C4C2×#13511yescharacteristic
509C2×C4C2×#13511yescharacteristic
510C2×C4#57611yescharacteristic
511C2×C4#57711yescharacteristic
512C8—24no—
513C8—42no—
514C8—44no—
515C8—42no—
516C8—81no—
517C8—82no—
518C22—24no—
519C22—24no—
520C22—24no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22#233911yescharacteristic
533C22C22×C22⋊C811yescharacteristic
534C22#288011yescharacteristic
535C22C2×#41611yescharacteristic
536C22#289711yescharacteristic
537C22C22×#10411yescharacteristic
538C22#290111yescharacteristic
539C4—22no—
540C4—22no—
541C4—24no—
542C4—21no—
543C4—21no—
544C4—24no—
545C4—42no—
546C4—44no—
547C4—42no—
548C4—41no—
549C4—42no—
550C4—82no—
551C4—81no—
552C4#147614yes—
553C4C8×#4111yescharacteristic
554C4#297011yescharacteristic
555C4#298811yescharacteristic
556C4#299011yescharacteristic
557C4#330311yescharacteristic
558C4#330511yescharacteristic
559C4#334011yescharacteristic
560C4#334111yescharacteristic
561C2—44no—
562C2—82no—
563C2C2×#147611yescharacteristic
564C2#1743811yescharacteristic
565C2#1750811yescharacteristic
566C2#1752011yescharacteristic
567C2#2246911yescharacteristic
568C2#2248911yescharacteristic
569C2#2250511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#555C4
LCSG ▷ C2×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. #22795C2semidirect
3. C4×C8×D4C2semidirect
4. #9870C2semidirect
5. #9864C2semidirect
6. #9072C2semidirect
7. #8940C2semidirect
8. #8708C2semidirect
9. #8563C2semidirect
10. C4×D4⋊C8C2semidirect
11. #7890C2semidirect
55. #555C8semidirect
170. C4⋊Q8C2×C8semidirect
174. #121C2×C8semidirect
190. C4×D4C2×C8semidirect
197. #117C2×C8semidirect

Maximal subgroups

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

subgroupindex
C4×#5552
#227952
C4×C8×D42
#98702
#98642
#90722
#89402
#87082
#85632
C4×D4⋊C82
#78902