Finite Group Databasebeta

← #211770

Subgroups — #211770

Order 512.

599 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22280C211yesmaximal, characteristic
2#20571C211yesmaximal, characteristic, complemented
3#19114C211yesmaximal, characteristic
4#17936C211yesmaximal, characteristic, complemented
5#17929C211yesmaximal, characteristic, complemented
6#9753C211yesmaximal, characteristic
7#9632C211yesmaximal, characteristic, complemented
8#9578C211yesmaximal, characteristic, complemented
9#9577C211yesmaximal, characteristic, complemented
10#9238C211yesmaximal, characteristic, complemented
11#9238C211yesmaximal, characteristic, complemented
12#8661C211yesmaximal, characteristic, complemented
13#8270C211yesmaximal, characteristic, complemented
14#8257C211yesmaximal, characteristic, complemented
15#8251C211yesmaximal, characteristic, complemented
16#3147C2211yescharacteristic
17#3103C2211yescharacteristic
18#3103C2211yescharacteristic
19C2×#446C2211yescharacteristic
20C2×C8×Q8C2211yescharacteristic
21#2931C2211yescharacteristic
22#2931C2211yescharacteristic
23#2295C2211yescharacteristic
24#1962C2211yescharacteristic
25#1962C2211yescharacteristic
26#1876—21no—
27#1876—21no—
28#1876—21no—
29#1876—21no—
30#1837C2211yescharacteristic
31#1771—21no—
32#1770C2211yescharacteristic
33C8×C4⋊C4—21no—
34C8×C4⋊C4C2211yescharacteristic
35#1768—21no—
36C4×C4⋊C8C2211yescharacteristic
37#1746—21no—
38#1744C2211yescharacteristic
39#1670C412yescomplemented
40#1668C412yescomplemented
41#1666C412yescomplemented
42#1666C412yescomplemented
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×Q8C2311yescharacteristic
74#495—21no—
75#495—21no—
76#494C2×C411yescharacteristic
77#493C2×C411yescharacteristic
78#492C2×C411yescharacteristic
79#492C2×C411yescharacteristic
80#481—21no—
81#481—21no—
82#475C2×C412yes—
83#475C2×C412yes—
84#460C2×C412yes—
85#460C2×C412yes—
86#448—21no—
87#448—21no—
88#446—21no—
89#446—21no—
90#446D412yes—
91#446D412yes—
92C8×Q8—21no—
93C8×Q8—21no—
94C8×Q8C2×C412yes—
95C8×Q8C2×C412yes—
96C4×SD16—22no—
97C4×SD16—22no—
98#436—21no—
99#436—21no—
100#436—21no—
101#436—21no—
102C8×D4—21no—
103C8×D4—21no—
104C8×D4—21no—
105C8×D4—21no—
106#431C2311yescharacteristic
107#431C2311yescharacteristic
108#431C2311yescharacteristic
109#431C2311yescharacteristic
110#423C2311yescharacteristic
111#423C2311yescharacteristic
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8C2311yescharacteristic
115C2×C4⋊C8C2311yescharacteristic
116C2×#104—21no—
117C2×#104—21no—
118C2×#104—21no—
119C2×#104—21no—
120#408—21no—
121#408—21no—
122C2×#97—21no—
123C2×#97—21no—
124C2×C4×C8—21no—
125C2×C4×C8—21no—
126C2×C4×C8—21no—
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129#391—21no—
130#389—21no—
131#387—22no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4C2311yescharacteristic
134#376—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×Q8C22×C412yes—
211C4×Q8C2×D412yes—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—21no—
215C4×D4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4#4111yescharacteristic
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2411yescharacteristic
227#106—22no—
228#106—22no—
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8—22no—
232C4⋊C8—22no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8C22×C412yes—
242C4⋊C8C22×C412yes—
243C4⋊C8C22×C412yes—
244C4⋊C8C2×D412yes—
245#104—81no—
246#104—81no—
247#104—81no—
248#104—81no—
249#103—22no—
250#103—22no—
251#102—22no—
252#102—22no—
253C22⋊C8—21no—
254C22⋊C8—21no—
255C22⋊C8—21no—
256C22⋊C8—21no—
257C22⋊C8—41no—
258C22⋊C8—41no—
259C22⋊C8—41no—
260C22⋊C8—41no—
261#97—22no—
262#97—22no—
263#97—81no—
264C4×C8—22no—
265C4×C8—22no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—81no—
271C4×C8—81no—
272C4×C8C22×C411yescharacteristic
273C4×C8C22×C411yescharacteristic
274C4×C8C2×D411yescharacteristic
275C4×C8C2×D411yescharacteristic
276C4×C8#4111yescharacteristic
277C4×C8#4111yescharacteristic
278C4×C8#4111yescharacteristic
279C4×C8#4111yescharacteristic
280#95—22no—
281#95—22no—
282#95—22no—
283#95#4112yes—
284#41—41no—
285#41—41no—
286#41—41no—
287#41—41no—
288#41—41no—
289#41—41no—
290#41—41no—
291#41—41no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—42no—
296C2×Q8C4×D412yes—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—41no—
304C2×D4—41no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4C22×D411yescharacteristic
313C22×C4C2×#4111yescharacteristic
314C22×C4C2×#4111yescharacteristic
315Q16—81no—
316Q16—81no—
317SD16—81no—
318SD16—81no—
319SD16—81no—
320SD16—81no—
321SD16—82no—
322SD16—82no—
323D8—81no—
324D8—81no—
325M16—41no—
326M16—41no—
327M16—41no—
328M16—41no—
329M16—82no—
330M16—82no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—82no—
352C2×C8—82no—
353C2×C8C4×D411yescharacteristic
354C2×C8C4×D411yescharacteristic
355C2×C8C4×D411yescharacteristic
356C2×C8C4×D411yescharacteristic
357C2×C8#12111yescharacteristic
358C2×C8#12111yescharacteristic
359C2×C8#12111yescharacteristic
360C2×C8#12111yescharacteristic
361C4⋊C4—22no—
362C4⋊C4—21no—
363C4⋊C4—22no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—42no—
369C4⋊C4—44no—
370C4⋊C4C4×D412yes—
371C22⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C42—22no—
376C42—22no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—42no—
382C42—42no—
383C42—41no—
384C42—41no—
385C42C23×C411yescharacteristic
386C42C22×D411yescharacteristic
387C42C2×#4111yescharacteristic
388C42C2×#4111yescharacteristic
389C23—41no—
390C23—41no—
391C23#52511yescharacteristic
392Q8—22no—
393Q8—22no—
394Q8—42no—
395Q8—41no—
396Q8—41no—
397Q8—41no—
398Q8—41no—
399D4—41no—
400D4—41no—
401D4—41no—
402D4—41no—
403D4—42no—
404D4—42no—
405D4—81no—
406D4—81no—
407D4—81no—
408D4—81no—
409D4—81no—
410D4—81no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4C2×C4×D411yescharacteristic
433C2×C4C2×C4×D411yescharacteristic
434C2×C4C4×#4111yescharacteristic
435C2×C4C2×#12111yescharacteristic
436C2×C4#52911yescharacteristic
437C2×C4#53211yescharacteristic
438C2×C4#57611yescharacteristic
439C2×C4#57711yescharacteristic
440C2×C4#57711yescharacteristic
441C2×C4#57711yescharacteristic
442C8—21no—
443C8—21no—
444C8—21no—
445C8—21no—
446C8—21no—
447C8—21no—
448C8—21no—
449C8—21no—
450C8—42no—
451C8—42no—
452C8—42no—
453C8—42no—
454C8—81no—
455C8—81no—
456C8—81no—
457C8—81no—
458C22—21no—
459C22—21no—
460C22—81no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22C4×#12111yescharacteristic
467C22#311911yescharacteristic
468C22#312611yescharacteristic
469C4—21no—
470C4—21no—
471C4—21no—
472C4—42no—
473C4—42no—
474C4—42no—
475C4—82no—
476C4—81no—
477C4—81no—
478C4—81no—
479C4#294611yescharacteristic
480C4#294911yescharacteristic
481C4#295712yes—
482C4#304311yescharacteristic
483C4#304511yescharacteristic
484C2—21no—
485C2—21no—
486C2—161no—
487C2—161no—
488C2#2151511yescharacteristic
489C2#2154711yescharacteristic
490C2#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. #20571C2semidirect
4. #17936C2semidirect
5. #17929C2semidirect
7. #9632C2semidirect
8. #9578C2semidirect
9. #9577C2semidirect
10. #9238C2semidirect
11. #9238C2semidirect
12. #8661C2semidirect
13. #8270C2semidirect
14. #8257C2semidirect
15. #8251C2semidirect
39. #1670C4semidirect
40. #1668C4semidirect
41. #1666C4semidirect
42. #1666C4semidirect

Maximal subgroups

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

subgroupindex
#222802
#205712
#191142
#179362
#179292
#97532
#96322
#95782
#95772
#92382
#86612
#82702
#82572
#82512