Finite Group Databasebeta

← #231849

Subgroups — #231849

Order 512.

567 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23234C211yesmaximal, characteristic
2#23171C211yesmaximal, characteristic, complemented
3#19620C211yesmaximal, characteristic
4#18072C211yesmaximal, characteristic, complemented
5#18067C211yesmaximal, characteristic
6#9754C211yesmaximal, characteristic
7#9749C211yesmaximal, characteristic, complemented
8#9722C211yesmaximal, characteristic
9#9718C211yesmaximal, characteristic, complemented
10#9056C211yesmaximal, characteristic, complemented
11#9056C211yesmaximal, characteristic, complemented
12#9055C211yesmaximal, characteristic
13#9054C211yesmaximal, characteristic
14#8296C211yesmaximal, characteristic, complemented
15#8288C211yesmaximal, characteristic, complemented
16#3170C2211yescharacteristic
17#3170C2211yescharacteristic
18#3167C2211yescharacteristic
19#3167C2211yescharacteristic
20#3160C2211yescharacteristic
21#3131C2211yescharacteristic
22#2962C2211yescharacteristic
23#2962C2211yescharacteristic
24#2962C2211yescharacteristic
25#2935C2211yescharacteristic
26#1986C2211yescharacteristic
27#1986C2211yescharacteristic
28#1986C2211yescharacteristic
29#1980C2211yescharacteristic
30#1946—21no—
31#1937—21no—
32#1929—21no—
33#1922—21no—
34#1922—21no—
35#1866—21no—
36C4×#108C2211yescharacteristic
37C4×#107C2211yescharacteristic
38#1773—21no—
39#1770C2211yescharacteristic
40C8×C4⋊C4C2211yescharacteristic
41#1764—21no—
42#1723C411yescharacteristic, complemented
43#1722C411yescharacteristic, complemented
44#1720C411yescharacteristic, complemented
45#1719C411yescharacteristic, complemented
46#1718C411yescharacteristic, complemented
47#1718C411yescharacteristic, complemented
48#1716C411yescharacteristic, complemented
49#1714C411yescharacteristic, complemented
50#1593—21no—
51#1590—21no—
52#1587—21no—
53#1587—21no—
54#1583—21no—
55#1582—21no—
56#1579—21no—
57C8×SD16—21no—
58#1564C2211yescharacteristic
59C2×#334C2211yescharacteristic
60#1503C2211yescharacteristic
61#1503C2211yescharacteristic
62#1503C2211yescharacteristic
63#1502C2211yescharacteristic
64#1481C2211yescharacteristic
65#1481C2211yescharacteristic
66#1481C2211yescharacteristic
67#1480C2211yescharacteristic
68#1451C2211yescharacteristic
69C8×M16C2211yescharacteristic
70C2×#322C2211yescharacteristic
71#1279C2211yescharacteristic
72#1279C2211yescharacteristic
73#1278C2211yescharacteristic
74#1278C2211yescharacteristic
75C2×#131—21no—
76#533C2311yescharacteristic
77#533C2311yescharacteristic
78#533C2311yescharacteristic
79#532C2311yescharacteristic
80#500C2×C411yescharacteristic
81#500C2×C411yescharacteristic
82#500C2×C411yescharacteristic
83#500C2×C411yescharacteristic
84#499—21no—
85#499—21no—
86#498C2×C411yescharacteristic
87#498C2×C411yescharacteristic
88#498C2×C411yescharacteristic
89#498C2×C411yescharacteristic
90#494C2×C411yescharacteristic
91#494C2×C411yescharacteristic
92#492C2×C411yescharacteristic
93#492C2×C411yescharacteristic
94#488—21no—
95#488—21no—
96#487C2×C411yescharacteristic
97#487C2×C411yescharacteristic
98#487C2×C411yescharacteristic
99#487C2×C411yescharacteristic
100#487C2×C411yescharacteristic
101#487C2×C411yescharacteristic
102#487C2×C411yescharacteristic
103#487C2×C411yescharacteristic
104#486C2×C411yescharacteristic
105#486C2×C411yescharacteristic
106#486C2×C411yescharacteristic
107#486C2×C411yescharacteristic
108#467—21no—
109#467—21no—
110#467—41no—
111#465—21no—
112#465—21no—
113#465—41no—
114#446—21no—
115#446—21no—
116#446—21no—
117#446—41no—
118#446—41no—
119#446—41no—
120C8×Q8—21no—
121C8×Q8—21no—
122C8×Q8—21no—
123C4×Q16—21no—
124C4×Q16—21no—
125C4×Q16—41no—
126C4×SD16—21no—
127C4×SD16—21no—
128C4×SD16—41no—
129#436—21no—
130#435—41no—
131C8×D4—21no—
132#431C2311yescharacteristic
133#431C2311yescharacteristic
134C2×#108D411yescharacteristic
135C2×#108D411yescharacteristic
136#427—21no—
137#427—21no—
138C2×#107—21no—
139#424—21no—
140#424—21no—
141#424—21no—
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2311yescharacteristic
144C2×#104—21no—
145C2×#104—21no—
146#416—21no—
147#416—21no—
148#408—21no—
149#405—21no—
150#405—21no—
151#405—21no—
152C4×M16C2311yescharacteristic
153C4×M16C2311yescharacteristic
154C2×#97—21no—
155C2×C4×C8—21no—
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159C4×C4⋊C4C2311yescharacteristic
160C4×C4⋊C4C2311yescharacteristic
161#341D411yescharacteristic
162#341D411yescharacteristic
163#340—21no—
164#339—21no—
165#339—21no—
166#339—21no—
167#339—21no—
168#339—21no—
169#339—21no—
170#336—21no—
171#336—21no—
172#335—21no—
173#334—21no—
174#334D411yescharacteristic
175#334D411yescharacteristic
176#334D411yescharacteristic
177#334D411yescharacteristic
178#328—41no—
179#328—41no—
180#328—41no—
181#327—41no—
182Q8⋊C8—21no—
183Q8⋊C8—21no—
184Q8⋊C8—21no—
185Q8⋊C8—21no—
186Q8⋊C8—21no—
187Q8⋊C8—21no—
188D4⋊C8—21no—
189D4⋊C8—21no—
190#322—21no—
191#322—21no—
192#322—21no—
193#322C2×C411yescharacteristic
194#322C2×C411yescharacteristic
195#322C2×C411yescharacteristic
196#322C2×C411yescharacteristic
197C82—21no—
198C2×#41—21no—
199C2×Q16—21no—
200C2×Q16—21no—
201C2×Q16—41no—
202C2×SD16—21no—
203C2×SD16—21no—
204C2×SD16—41no—
205#131—42no—
206#131—81no—
207C2×M16—41no—
208C2×M16C2×D411yescharacteristic
209C2×M16C2×D411yescharacteristic
210C2×M16C2×D411yescharacteristic
211C2×M16C2×D411yescharacteristic
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—41no—
216C22×C8C2×D411yescharacteristic
217C22×C8C2×D411yescharacteristic
218C22×C8#4111yescharacteristic
219C22×C8#4111yescharacteristic
220C4⋊Q8C22×C411yescharacteristic
221C4⋊Q8C22×C411yescharacteristic
222C4⋊Q8C22×C411yescharacteristic
223C4⋊Q8C22×C411yescharacteristic
224C4⋊Q8C22×C411yescharacteristic
225C4⋊Q8C22×C411yescharacteristic
226C4⋊Q8C22×C411yescharacteristic
227#127C22×C411yescharacteristic
228#125—21no—
229#125—21no—
230#125—21no—
231#124—21no—
232C22⋊Q8—21no—
233C22⋊Q8—21no—
234C22⋊Q8—21no—
235C22⋊Q8—21no—
236C22⋊Q8—21no—
237C22⋊Q8—21no—
238#121—21no—
239#121—21no—
240C4×Q8—21no—
241C4×Q8—21no—
242C4×Q8—21no—
243C4×Q8—21no—
244C4×Q8—21no—
245C4×Q8—21no—
246C4×D4—21no—
247C4×D4—21no—
248#117—21no—
249#117—21no—
250#117—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42C2411yescharacteristic
256#108—22no—
257#108—41no—
258#107—21no—
259#107—21no—
260#107—21no—
261#107—21no—
262#107—21no—
263#107—21no—
264#107—41no—
265#107—42no—
266#106—21no—
267#106—21no—
268#106—21no—
269#106—21no—
270#106—21no—
271#106—21no—
272#106—41no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—41no—
278C4⋊C8—41no—
279C4⋊C8—41no—
280C4⋊C8—41no—
281C4⋊C8—41no—
282C4⋊C8—41no—
283C4⋊C8C2×D411yescharacteristic
284C4⋊C8C2×D411yescharacteristic
285C4⋊C8C2×D411yescharacteristic
286C4⋊C8C2×D411yescharacteristic
287#104—42no—
288#104—81no—
289#103—21no—
290#103—21no—
291#103—21no—
292#103—21no—
293#103—21no—
294#103—21no—
295#103—21no—
296#103—21no—
297#103—21no—
298#103—21no—
299#103—41no—
300#103—41no—
301#103—41no—
302#102—21no—
303#102—21no—
304#102—21no—
305#102—21no—
306#102—21no—
307#102—21no—
308#102—41no—
309C22⋊C8—21no—
310C22⋊C8—21no—
311C22⋊C8—41no—
312C22⋊C8—41no—
313#97—21no—
314#97—21no—
315#97—41no—
316#97—41no—
317#97—41no—
318#97—42no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—21no—
323C4×C8—21no—
324C4×C8—41no—
325C4×C8—41no—
326C4×C8—41no—
327C4×C8—81no—
328C4×C8C22×C411yescharacteristic
329C4×C8C22×C411yescharacteristic
330C4×C8C22×C411yescharacteristic
331C4×C8C22×C411yescharacteristic
332C4×C8C22×C411yescharacteristic
333C4×C8C22×C411yescharacteristic
334C4×C8C2×D411yescharacteristic
335C4×C8C2×D411yescharacteristic
336C4×C8#4111yescharacteristic
337C4×C8#4111yescharacteristic
338#95—21no—
339#95—21no—
340#41—41no—
341#41—41no—
342#41—41no—
343#41—41no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4C22×D411yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4H(5,𝔽2)11yescharacteristic
361Q16—42no—
362Q16—81no—
363Q16—82no—
364SD16—42no—
365SD16—81no—
366SD16—82no—
367M16—21no—
368M16—21no—
369M16—21no—
370M16—21no—
371M16—21no—
372M16—21no—
373M16—21no—
374M16—82no—
375M16#13712yes—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—42no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—82no—
400C2×C8C4×D411yescharacteristic
401C2×C8C4×D411yescharacteristic
402C2×C8C4×D411yescharacteristic
403C2×C8C4×D411yescharacteristic
404C2×C8C4×D411yescharacteristic
405C2×C8C4×D411yescharacteristic
406C2×C8C4×D411yescharacteristic
407C2×C8C4×D411yescharacteristic
408C4⋊C4—22no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C42—41no—
438C42—41no—
439C42—41no—
440C42—41no—
441C42—41no—
442C42—41no—
443C42—41no—
444C42C23×C411yescharacteristic
445C42C22×D411yescharacteristic
446C42C2×#4111yescharacteristic
447C42H(5,𝔽2)11yescharacteristic
448C23—41no—
449C23#55011yescharacteristic
450Q8—41no—
451Q8—41no—
452Q8—41no—
453Q8—41no—
454Q8—41no—
455Q8—41no—
456Q8—41no—
457Q8—41no—
458Q8—81no—
459Q8—81no—
460Q8—81no—
461D4—41no—
462D4—41no—
463D4—41no—
464D4—41no—
465D4—81no—
466D4—81no—
467D4—81no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4C2×C4×D411yescharacteristic
487C2×C4C2×C4×D411yescharacteristic
488C2×C4#51811yescharacteristic
489C2×C4D4211yescharacteristic
490C2×C4#55011yescharacteristic
491C2×C4#55911yescharacteristic
492C2×C4C2×#13711yescharacteristic
493C2×C4#57511yescharacteristic
494C2×C4#57711yescharacteristic
495C2×C4#57811yescharacteristic
496C8—22no—
497C8—21no—
498C8—21no—
499C8—21no—
500C8—21no—
501C8—21no—
502C8—21no—
503C8—21no—
504C8—21no—
505C8—22no—
506C8—41no—
507C8—41no—
508C8—41no—
509C8—41no—
510C8—41no—
511C8—41no—
512C8—41no—
513C8—41no—
514C8—81no—
515C8—81no—
516C22—21no—
517C22—21no—
518C22—81no—
519C22—81no—
520C22—81no—
521C22#236611yescharacteristic
522C22#339811yescharacteristic
523C22#340111yescharacteristic
524C4—21no—
525C4—21no—
526C4—21no—
527C4—82no—
528C4—81no—
529C4—81no—
530C4—81no—
531C4—81no—
532C4—81no—
533C4—81no—
534C4—81no—
535C4—161no—
536C4C4×#13711yescharacteristic
537C4#294711yescharacteristic
538C4#295511yescharacteristic
539C4#295611yescharacteristic
540C4#326611yescharacteristic
541C4#326911yescharacteristic
542C2—21no—
543C2—21no—
544C2—161no—
545C2#2318111yescharacteristic
546C2#2319211yescharacteristic
547C2#2319611yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2366
prime radical#532C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #23171C2semidirect
4. #18072C2semidirect
7. #9749C2semidirect
9. #9718C2semidirect
10. #9056C2semidirect
11. #9056C2semidirect
14. #8296C2semidirect
15. #8288C2semidirect
42. #1723C4semidirect
43. #1722C4semidirect
44. #1720C4semidirect
45. #1719C4semidirect
46. #1718C4semidirect
47. #1718C4semidirect
48. #1716C4semidirect
49. #1714C4semidirect

Maximal subgroups

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

subgroupindex
#232342
#231712
#196202
#180722
#180672
#97542
#97492
#97222
#97182
#90562
#90552
#90542
#82962
#82882