Finite Group Databasebeta

← #222984

Subgroups — #222984

Order 512.

1042 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34932C211yesmaximal, characteristic
2D4×C4⋊C8C211yesmaximal, characteristic, complemented
3#21264C211yesmaximal, characteristic, complemented
4#9870C211yesmaximal, characteristic, complemented
5#9865C211yesmaximal, characteristic, complemented
6#9076C212yesmaximal, complemented
7#8939C212yesmaximal, complemented
8#8709C212yesmaximal, complemented
9#8563C211yesmaximal, characteristic, complemented
10C4×D4⋊C8C211yesmaximal, characteristic, complemented
11#7893C212yesmaximal, complemented
12C2×#529—21no—
13C2×C8×D4—21no—
14C2×#431C2212yes—
15C22×C4⋊C8C2212yes—
16#2553—22no—
17#2552C411yescharacteristic
18#2547C411yescharacteristic
19#2542C411yescharacteristic
20#2537C411yescharacteristic
21#2496C412yes—
22#2489—21no—
23#2427—21no—
24#2422C412yes—
25#2395—22no—
26C4×C4⋊Q8C2211yescharacteristic
27C4×C22⋊Q8—21no—
28C4×#121C2212yes—
29#2285C2212yes—
30C42×D4C2211yescharacteristic
31C42×D4C2211yescharacteristic
32#1954—21no—
33#1939—21no—
34#1939—21no—
35#1923—22no—
36#1850—22no—
37#1842C2211yescharacteristic
38#1797C2212yes—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
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—
54C2×C4⋊Q8C2×C411yescharacteristic
55#529—44no—
56#529—44no—
57C2×#126—22no—
58C2×#125—21no—
59C2×#124—21no—
60C2×#123—22no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×#121—21no—
64C2×#121C2×C412yes—
65C2×C4×Q8—21no—
66C2×C4×D4—22no—
67C2×C4×D4C2×C411yescharacteristic
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4D411yescharacteristic
71C2×C4×D4D411yescharacteristic
72C2×C4×D4C2311yescharacteristic
73C2×#117—22no—
74C22×C42—21no—
75C22×C42C2312yes—
76C8×D4—44no—
77#431—24no—
78#431—42no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—42no—
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2312yes—
88C2×C4⋊C8C2311yescharacteristic
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C4×C8—21no—
93C2×C4×C8—21no—
94C2×C4×C8—21no—
95C2×C4×C8C2312yes—
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98#400—22no—
99#399—22no—
100#398—22no—
101#397—22no—
102#397—22no—
103#397—22no—
104#394—22no—
105#394—22no—
106#391—21no—
107#391—21no—
108#390C2×C412yes—
109#389—21no—
110#389—21no—
111#389C2×C412yes—
112#389C2×C412yes—
113#388—22no—
114#387—22no—
115#386—22no—
116#385—22no—
117#384C2×C411yescharacteristic
118#384C2×C412yes—
119#384C2×C412yes—
120#382—22no—
121#380—22no—
122#380—21no—
123#380C2×C412yes—
124#380C2×C412yes—
125#380C2×C412yes—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4C2×C411yescharacteristic
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4C2×C411yescharacteristic
131C4×C4⋊C4D411yescharacteristic
132C4×C4⋊C4D411yescharacteristic
133C4×C4⋊C4C2312yes—
134C4×C4⋊C4C2311yescharacteristic
135C4×C22⋊C4—21no—
136C4×C22⋊C4—22no—
137C4×C22⋊C4—21no—
138C4×C22⋊C4—21no—
139C4×C22⋊C4C2×C412yes—
140C4×C22⋊C4D412yes—
141C4×C22⋊C4D412yes—
142C43C2311yescharacteristic
143#336—22no—
144#336—21no—
145#336—21no—
146#336—22no—
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—21no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—44no—
167C22×C8—44no—
168C22×C8—82no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171#126—48no—
172#125—44no—
173#124—44no—
174#123—48no—
175C22⋊Q8—44no—
176C22⋊Q8—44no—
177#121—24no—
178#121—42no—
179#121—44no—
180C4×Q8—42no—
181C4×Q8—42no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—22no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—41no—
192C4×D4—42no—
193C4×D4—44no—
194C4×D4—44no—
195C4×D4D814yes—
196C4×D4SD1614yes—
197#117—44no—
198#117—44no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4C22⋊C411yescharacteristic
209C2×C4⋊C4C22⋊C411yescharacteristic
210C2×C4⋊C4C22⋊C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C4⋊C4C22×C412yes—
215C2×C4⋊C4C22×C412yes—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
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⋊C4—21no—
225C2×C22⋊C4C22⋊C412yes—
226C2×C22⋊C4C22⋊C412yes—
227C2×C22⋊C4C22⋊C412yes—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—22no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42C22×C411yescharacteristic
239C2×C42C22×C412yes—
240C2×C42C22×C411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D412yes—
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D412yes—
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2411yescharacteristic
248C4⋊C8—21no—
249C4⋊C8—22no—
250C4⋊C8—21no—
251C4⋊C8—22no—
252C4⋊C8—21no—
253C4⋊C8—44no—
254C4⋊C8—44no—
255C4⋊C8—42no—
256C4⋊C8—42no—
257C4⋊C8—81no—
258C4⋊C8#4112yes—
259C22⋊C8—44no—
260C22⋊C8—44no—
261C22⋊C8—44no—
262C4×C8—21no—
263C4×C8—22no—
264C4×C8—21no—
265C4×C8—22no—
266C4×C8—21no—
267C4×C8—42no—
268C4×C8—42no—
269C4×C8—42no—
270C4×C8#4112yes—
271#95—21no—
272#95—21no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—21no—
279#95—22no—
280#95C22⋊C412yes—
281#95C22⋊C412yes—
282#95C22⋊C412yes—
283C24—21no—
284C24C2×C22⋊C412yes—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—44no—
296C2×D4—44no—
297C2×D4—82no—
298C2×D4#10214yes—
299C2×D4#13112yes—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
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—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—81no—
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C412yes—
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C412yes—
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×D814yes—
339C22×C4C2×SD1614yes—
340C22×C4C23×C411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—41no—
361C2×C8—88no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—21no—
373C4⋊C4—22no—
374C4⋊C4—44no—
375C4⋊C4—42no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—42no—
380C4⋊C4—44no—
381C4⋊C4—81no—
382C4⋊C4—82no—
383C4⋊C4#10214yes—
384C22⋊C4—24no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—44no—
388C22⋊C4—44no—
389C22⋊C4—44no—
390C22⋊C4—44no—
391C22⋊C4—44no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—42no—
395C22⋊C4—44no—
396C22⋊C4#10218yes—
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—41no—
410C42—42no—
411C42—41no—
412C42—42no—
413C42—42no—
414C42—42no—
415C42#13112yes—
416C42#13112yes—
417C42C2×D812yes—
418C42C2×SD1612yes—
419C42C2×#4111yescharacteristic
420C42C2×#4111yescharacteristic
421C42H(5,𝔽2)11yescharacteristic
422C422−1+411yescharacteristic
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23C2×#10214yes—
437C23C22×C22⋊C411yescharacteristic
438Q8—84no—
439D4—22no—
440D4—22no—
441D4—28no—
442D4—41no—
443D4—41no—
444D4—88no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—84no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—84no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4#40812yes—
494C2×C4#40812yes—
495C2×C4C2×#10212yes—
496C2×C4C2×#10214yes—
497C2×C4C2×#10212yes—
498C2×C4#42112yes—
499C2×C4C4×#4111yescharacteristic
500C2×C4#52011yescharacteristic
501C2×C4#54611yescharacteristic
502C2×C4#54611yescharacteristic
503C2×C4#54711yescharacteristic
504C2×C4#54711yescharacteristic
505C2×C4C2×#13111yescharacteristic
506C2×C4C2×#13111yescharacteristic
507C2×C4C2×#13111yescharacteristic
508C2×C4#56811yescharacteristic
509C2×C4C22×D811yescharacteristic
510C2×C4C22×SD1611yescharacteristic
511C2×C4#57711yescharacteristic
512C2×C4#57811yescharacteristic
513C8—42no—
514C8—42no—
515C8—42no—
516C8—82no—
517C8—81no—
518C8—82no—
519C22—24no—
520C22—24no—
521C22—24no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22#233911yescharacteristic
534C22C2×#40811yescharacteristic
535C22#288011yescharacteristic
536C22C22×#10211yescharacteristic
537C22#289811yescharacteristic
538C22C2×#42111yescharacteristic
539C22#290111yescharacteristic
540C4—22no—
541C4—22no—
542C4—24no—
543C4—21no—
544C4—21no—
545C4—41no—
546C4—42no—
547C4—42no—
548C4—81no—
549C4—82no—
550C4—81no—
551C4—82no—
552C4—81no—
553C4#148014yes—
554C4#296611yescharacteristic
555C4#296911yescharacteristic
556C4#299311yescharacteristic
557C4#299411yescharacteristic
558C4#329411yescharacteristic
559C4#329811yescharacteristic
560C4#333411yescharacteristic
561C4#333511yescharacteristic
562C2—44no—
563C2—82no—
564C2C2×#148011yescharacteristic
565C2#1746911yescharacteristic
566C2#1750711yescharacteristic
567C2#1752311yescharacteristic
568C2#2247411yescharacteristic
569C2#2248511yescharacteristic
570C2#2250711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radical#2537C4
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. D4×C4⋊C8C2semidirect
3. #21264C2semidirect
4. #9870C2semidirect
5. #9865C2semidirect
6. #9076C2semidirect
7. #8939C2semidirect
8. #8709C2semidirect
9. #8563C2semidirect
10. C4×D4⋊C8C2semidirect
11. #7893C2semidirect

Maximal subgroups

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

subgroupindex
#349322
D4×C4⋊C82
#212642
#98702
#98652
#90762
#89392
#87092
#85632
C4×D4⋊C82
#78932