Finite Group Databasebeta

← #211850

Subgroups — #211850

Order 512.

617 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22280C211yesmaximal, characteristic
2#20573C211yesmaximal, characteristic, complemented
3#19146C211yesmaximal, characteristic
4#17936C211yesmaximal, characteristic, complemented
5#17929C211yesmaximal, characteristic, complemented
6#9750C211yesmaximal, characteristic
7#9629C211yesmaximal, characteristic, complemented
8#9582C211yesmaximal, characteristic, complemented
9#9581C211yesmaximal, characteristic, complemented
10#9234C211yesmaximal, characteristic, complemented
11#9234C211yesmaximal, characteristic, complemented
12#8665C211yesmaximal, characteristic, complemented
13#8270C211yesmaximal, characteristic, complemented
14#8257C211yesmaximal, characteristic, complemented
15#8251C211yesmaximal, characteristic, complemented
16#3147C2211yescharacteristic
17#3103C2211yescharacteristic
18#3103C2211yescharacteristic
19C2×#446C2211yescharacteristic
20C2×#446C2211yescharacteristic
21#2931C2211yescharacteristic
22#2931C2211yescharacteristic
23C42×Q8C2211yescharacteristic
24#1962C2211yescharacteristic
25#1962C2211yescharacteristic
26#1876—21no—
27#1876—21no—
28#1876—21no—
29#1876—21no—
30#1837C2211yescharacteristic
31#1771—21no—
32#1771—21no—
33C8×C4⋊C4C2211yescharacteristic
34C8×C4⋊C4C2211yescharacteristic
35#1767C2211yescharacteristic
36C4×C4⋊C8—21no—
37#1745—21no—
38#1744C2211yescharacteristic
39#1690C412yescomplemented
40#1688C412yescomplemented
41#1686C412yescomplemented
42#1686C412yescomplemented
43#1592—22no—
44#1590—22no—
45#1585—22no—
46#1582—22no—
47#1571C2211yescharacteristic
48#1570C2211yescharacteristic
49#1564C2211yescharacteristic
50#1564C2211yescharacteristic
51#1540C2211yescharacteristic
52#1540C2211yescharacteristic
53#1502C2211yescharacteristic
54#1502C2211yescharacteristic
55#1487C2211yescharacteristic
56#1487C2211yescharacteristic
57#1480C2211yescharacteristic
58#1480C2211yescharacteristic
59#1452C2211yescharacteristic
60#1450C2211yescharacteristic
61#1303C2211yescharacteristic
62#1303C2211yescharacteristic
63#1301C2211yescharacteristic
64#1301C2211yescharacteristic
65#1276C2211yescharacteristic
66#1276C2211yescharacteristic
67C2×#135—21no—
68C2×#135—21no—
69C2×#131—21no—
70C2×#131—21no—
71#532C2311yescharacteristic
72#532C2311yescharacteristic
73C2×C4×Q8—21no—
74C2×C4×Q8C2311yescharacteristic
75#495—21no—
76#495—21no—
77#494C2×C411yescharacteristic
78#493C2×C411yescharacteristic
79#492C2×C411yescharacteristic
80#492C2×C411yescharacteristic
81#481—21no—
82#481—21no—
83#475C2×C412yes—
84#475C2×C412yes—
85#460C2×C412yes—
86#460C2×C412yes—
87#448—21no—
88#448—21no—
89#446—21no—
90#446—21no—
91#446—21no—
92#446—21no—
93#446C2×C412yes—
94#446C2×C412yes—
95#446D412yes—
96#446D412yes—
97C4×SD16—22no—
98C4×SD16—22no—
99#436—21no—
100#436—21no—
101#436—21no—
102#436—21no—
103C8×D4—21no—
104C8×D4—21no—
105C8×D4—21no—
106C8×D4—21no—
107#431C2311yescharacteristic
108#431C2311yescharacteristic
109#431C2311yescharacteristic
110#431C2311yescharacteristic
111#423C2311yescharacteristic
112#423C2311yescharacteristic
113C2×C4⋊C8—21no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8C2311yescharacteristic
116C2×C4⋊C8C2311yescharacteristic
117C2×#104—21no—
118C2×#104—21no—
119C2×#104—21no—
120C2×#104—21no—
121#408—21no—
122#408—21no—
123C2×#97—21no—
124C2×#97—21no—
125C2×#97—21no—
126C2×C4×C8—21no—
127C2×C4×C8—21no—
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4C2311yescharacteristic
134C43—21no—
135C43C2311yescharacteristic
136#341—21no—
137#341—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336D411yescharacteristic
142#336D411yescharacteristic
143#336D411yescharacteristic
144#336D411yescharacteristic
145#335—22no—
146#335—21no—
147#335C2×C412yes—
148#334—22no—
149#334—21no—
150#334C2×C412yes—
151#332C2×C412yes—
152#332C2×C412yes—
153#331C2×C412yes—
154#331C2×C412yes—
155#327—22no—
156#327—22no—
157Q8⋊C8—22no—
158Q8⋊C8—22no—
159D4⋊C8—22no—
160D4⋊C8—22no—
161#322—21no—
162#322—21no—
163#322—21no—
164C82—21no—
165C2×#41—21no—
166C2×#41—21no—
167C22×Q8#4111yescharacteristic
168#135—81no—
169#135—81no—
170#135—81no—
171#135—81no—
172C2×Q16—21no—
173C2×Q16—21no—
174C2×SD16—22no—
175C2×SD16—22no—
176C2×SD16—21no—
177C2×SD16—21no—
178C2×SD16—21no—
179C2×SD16—21no—
180C2×D8—21no—
181C2×D8—21no—
182#131—81no—
183#131—81no—
184#131—81no—
185#131—81no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—41no—
189C2×M16—41no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—41no—
193C22×C8—41no—
194C22×C8C2×D411yescharacteristic
195C22×C8C2×D411yescharacteristic
196C22×C8C2×D411yescharacteristic
197C22×C8C2×D411yescharacteristic
198C4⋊Q8C22×C411yescharacteristic
199C4⋊Q8C22×C411yescharacteristic
200#127C22×C411yescharacteristic
201#127C22×C411yescharacteristic
202#124—21no—
203#124—21no—
204#121—21no—
205#121—21no—
206#121—21no—
207#121—21no—
208C4×Q8—21no—
209C4×Q8—21no—
210C4×Q8—44no—
211C4×Q8C22×C412yes—
212C4×Q8C2×D412yes—
213C4×D4—21no—
214C4×D4—21no—
215C4×D4—21no—
216C4×D4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4#4111yescharacteristic
221C2×C42—21no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2411yescharacteristic
230#106—22no—
231#106—22no—
232C4⋊C8—22no—
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8—22no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8C22×C412yes—
245C4⋊C8C22×C412yes—
246C4⋊C8C22×C412yes—
247C4⋊C8C2×D412yes—
248#104—81no—
249#104—81no—
250#104—81no—
251#104—81no—
252#103—22no—
253#103—22no—
254#102—22no—
255#102—22no—
256C22⋊C8—21no—
257C22⋊C8—21no—
258C22⋊C8—21no—
259C22⋊C8—21no—
260C22⋊C8—41no—
261C22⋊C8—41no—
262C22⋊C8—41no—
263C22⋊C8—41no—
264#97—22no—
265#97—22no—
266#97—22no—
267#97—22no—
268#97—81no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—81no—
274C4×C8—81no—
275C4×C8C22×C411yescharacteristic
276C4×C8C22×C411yescharacteristic
277C4×C8C2×D411yescharacteristic
278C4×C8C2×D411yescharacteristic
279C4×C8#4111yescharacteristic
280C4×C8#4111yescharacteristic
281C4×C8#4111yescharacteristic
282C4×C8#4111yescharacteristic
283#95—22no—
284#95#4112yes—
285#41—41no—
286#41—41no—
287#41—41no—
288#41—41no—
289#41—41no—
290#41—41no—
291#41—41no—
292#41—41no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—42no—
297C2×Q8C4×D412yes—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—41no—
305C2×D4—41no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4C22×D411yescharacteristic
314C22×C4C2×#4111yescharacteristic
315C22×C4C2×#4111yescharacteristic
316Q16—81no—
317Q16—81no—
318SD16—81no—
319SD16—81no—
320SD16—81no—
321SD16—81no—
322SD16—82no—
323SD16—82no—
324D8—81no—
325D8—81no—
326M16—41no—
327M16—41no—
328M16—41no—
329M16—41no—
330M16—82no—
331M16—82no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—82no—
353C2×C8—82no—
354C2×C8C4×D411yescharacteristic
355C2×C8C4×D411yescharacteristic
356C2×C8C4×D411yescharacteristic
357C2×C8C4×D411yescharacteristic
358C2×C8#12111yescharacteristic
359C2×C8#12111yescharacteristic
360C2×C8#12111yescharacteristic
361C2×C8#12111yescharacteristic
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—21no—
365C4⋊C4—22no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—44no—
371C4⋊C4C4×D412yes—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C42—22no—
377C42—22no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—42no—
383C42—44no—
384C42—42no—
385C42—44no—
386C42—41no—
387C42—41no—
388C42C23×C411yescharacteristic
389C42C22×D411yescharacteristic
390C42C2×#4111yescharacteristic
391C42C2×#4111yescharacteristic
392C23—41no—
393C23—41no—
394C23#52511yescharacteristic
395Q8—22no—
396Q8—22no—
397Q8—42no—
398Q8—41no—
399Q8—41no—
400Q8—41no—
401Q8—41no—
402D4—41no—
403D4—41no—
404D4—41no—
405D4—41no—
406D4—42no—
407D4—42no—
408D4—81no—
409D4—81no—
410D4—81no—
411D4—81no—
412D4—81no—
413D4—81no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4C2×C4×D411yescharacteristic
436C2×C4C2×C4×D411yescharacteristic
437C2×C4C4×#4111yescharacteristic
438C2×C4C2×#12111yescharacteristic
439C2×C4#52911yescharacteristic
440C2×C4#53211yescharacteristic
441C2×C4#57611yescharacteristic
442C2×C4#57711yescharacteristic
443C2×C4#57711yescharacteristic
444C2×C4#57711yescharacteristic
445C8—21no—
446C8—21no—
447C8—21no—
448C8—21no—
449C8—21no—
450C8—21no—
451C8—21no—
452C8—21no—
453C8—42no—
454C8—42no—
455C8—42no—
456C8—42no—
457C8—81no—
458C8—81no—
459C8—81no—
460C8—81no—
461C22—21no—
462C22—21no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22C4×#12111yescharacteristic
470C22#311911yescharacteristic
471C22#312611yescharacteristic
472C4—21no—
473C4—21no—
474C4—21no—
475C4—44no—
476C4—42no—
477C4—42no—
478C4—42no—
479C4—81no—
480C4—81no—
481C4—81no—
482C4#294611yescharacteristic
483C4#294911yescharacteristic
484C4#295612yes—
485C4#304311yescharacteristic
486C4#304511yescharacteristic
487C2—21no—
488C2—21no—
489C2—161no—
490C2—161no—
491C2#2151511yescharacteristic
492C2#2155111yescharacteristic
493C2#2155811yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C4×#121
prime radical#3147C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20573C2semidirect
4. #17936C2semidirect
5. #17929C2semidirect
7. #9629C2semidirect
8. #9582C2semidirect
9. #9581C2semidirect
10. #9234C2semidirect
11. #9234C2semidirect
12. #8665C2semidirect
13. #8270C2semidirect
14. #8257C2semidirect
15. #8251C2semidirect
39. #1690C4semidirect
40. #1688C4semidirect
41. #1686C4semidirect
42. #1686C4semidirect

Maximal subgroups

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

subgroupindex
#222802
#205732
#191462
#179362
#179292
#97502
#96292
#95822
#95812
#92342
#86652
#82702
#82572
#82512