Finite Group Databasebeta

← #211847

Subgroups — #211847

Order 512.

584 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22234C211yesmaximal, characteristic
2#20573C211yesmaximal, characteristic, complemented
3#19155C211yesmaximal, characteristic
4#17936C211yesmaximal, characteristic
5#17931C211yesmaximal, characteristic, complemented
6#9751C211yesmaximal, characteristic
7#9627C211yesmaximal, characteristic, complemented
8#9584C211yesmaximal, characteristic, complemented
9#9581C211yesmaximal, characteristic
10#9234C211yesmaximal, characteristic
11#9232C211yesmaximal, characteristic, complemented
12#8663C211yesmaximal, characteristic, complemented
13#8270C211yesmaximal, characteristic, complemented
14#8258C211yesmaximal, characteristic, complemented
15#8251C211yesmaximal, characteristic
16#3131C2211yescharacteristic
17#3104C2211yescharacteristic
18#3103C2211yescharacteristic
19#2960C2211yescharacteristic
20C2×#446C2211yescharacteristic
21C2×#446C2211yescharacteristic
22#2931C2211yescharacteristic
23C42×Q8C2211yescharacteristic
24#1994C2211yescharacteristic
25#1962C2211yescharacteristic
26#1929—21no—
27#1929—21no—
28#1866—21no—
29#1866—21no—
30#1848C2211yescharacteristic
31#1771—21no—
32#1771—21no—
33C8×C4⋊C4C2211yescharacteristic
34C8×C4⋊C4C2211yescharacteristic
35#1767C2211yescharacteristic
36C4×C4⋊C8—21no—
37#1745—21no—
38#1744C2211yescharacteristic
39#1704C412yescomplemented
40#1702C412yescomplemented
41#1700C412yescomplemented
42#1698C412yescomplemented
43#1592—22no—
44#1591—22no—
45#1585—22no—
46#1584—22no—
47#1571C2211yescharacteristic
48#1571C2211yescharacteristic
49#1564C2211yescharacteristic
50#1564C2211yescharacteristic
51#1542C2211yescharacteristic
52#1540C2211yescharacteristic
53#1503C2211yescharacteristic
54#1502C2211yescharacteristic
55#1487C2211yescharacteristic
56#1487C2211yescharacteristic
57#1481C2211yescharacteristic
58#1480C2211yescharacteristic
59#1452C2211yescharacteristic
60#1450C2211yescharacteristic
61#1303C2211yescharacteristic
62#1303C2211yescharacteristic
63#1301C2211yescharacteristic
64#1301C2211yescharacteristic
65#1276C2211yescharacteristic
66#1276C2211yescharacteristic
67C2×#131—21no—
68#533C2311yescharacteristic
69#532C2311yescharacteristic
70C2×C4×Q8—21no—
71C2×C4×Q8C2311yescharacteristic
72#488—21no—
73#488—21no—
74#487C2×C411yescharacteristic
75#487C2×C411yescharacteristic
76#486C2×C411yescharacteristic
77#486C2×C411yescharacteristic
78#484—21no—
79#481—21no—
80#477C2×C412yes—
81#475C2×C412yes—
82#462C2×C412yes—
83#460C2×C412yes—
84#451—21no—
85#448—21no—
86#446—21no—
87#446—21no—
88#446—21no—
89#446—21no—
90#446—21no—
91#446—21no—
92#446C2×C412yes—
93#446C2×C412yes—
94#446D412yes—
95#446D412yes—
96C8×Q8—21no—
97C8×Q8—21no—
98C4×Q16—22no—
99C4×SD16—22no—
100#436—21no—
101#436—21no—
102C8×D4—21no—
103C8×D4—21no—
104#431C2311yescharacteristic
105#431C2311yescharacteristic
106#431C2311yescharacteristic
107#431C2311yescharacteristic
108C2×#108—21no—
109C2×#108—21no—
110#424—21no—
111#423C2311yescharacteristic
112#423C2311yescharacteristic
113C2×C4⋊C8—21no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8C2311yescharacteristic
116C2×C4⋊C8C2311yescharacteristic
117C2×#104—21no—
118C2×#104—21no—
119#416—21no—
120#416—21no—
121#408—21no—
122#405—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#339—21no—
137#339—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336D411yescharacteristic
142#336D411yescharacteristic
143#336D411yescharacteristic
144#336D411yescharacteristic
145#335—22no—
146#335—22no—
147#335—21no—
148#335C2×C412yes—
149#334—21no—
150#334C2×C412yes—
151#333C2×C412yes—
152#332C2×C412yes—
153#332C2×C412yes—
154#331C2×C412yes—
155#328—22no—
156#327—22no—
157Q8⋊C8—22no—
158Q8⋊C8—22no—
159Q8⋊C8—22no—
160D4⋊C8—22no—
161#322—21no—
162#322—21no—
163#322—21no—
164C82—21no—
165C2×#41—21no—
166C22×Q8#4111yescharacteristic
167C2×Q16—22no—
168C2×SD16—22no—
169#131—81no—
170#131—81no—
171C2×M16—21no—
172C2×M16—21no—
173C2×M16—41no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—41no—
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C4⋊Q8C22×C411yescharacteristic
182C4⋊Q8C22×C411yescharacteristic
183C4⋊Q8C22×C411yescharacteristic
184#127C22×C411yescharacteristic
185#125—21no—
186#124—21no—
187C22⋊Q8—21no—
188C22⋊Q8—21no—
189#121—21no—
190#121—21no—
191C4×Q8—21no—
192C4×Q8—21no—
193C4×Q8—21no—
194C4×Q8—21no—
195C4×Q8—44no—
196C4×Q8C22×C412yes—
197C4×Q8C2×D412yes—
198C4×D4—21no—
199C4×D4—21no—
200#117—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4#4111yescharacteristic
205C2×C42—21no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2411yescharacteristic
214#108—81no—
215#108—81no—
216#107—22no—
217#106—22no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—41no—
231C4⋊C8—41no—
232C4⋊C8C22×C412yes—
233C4⋊C8C22×C412yes—
234C4⋊C8C22×C412yes—
235C4⋊C8C2×D412yes—
236#104—81no—
237#104—81no—
238#103—22no—
239#103—22no—
240#103—21no—
241#103—21no—
242#103—21no—
243#103—21no—
244#103—22no—
245#102—21no—
246#102—21no—
247#102—21no—
248#102—21no—
249#102—22no—
250C22⋊C8—21no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253C22⋊C8—21no—
254C22⋊C8—41no—
255C22⋊C8—41no—
256#97—22no—
257#97—22no—
258#97—22no—
259#97—22no—
260#97—41no—
261#97—81no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—41no—
267C4×C8—81no—
268C4×C8—81no—
269C4×C8C22×C411yescharacteristic
270C4×C8C22×C411yescharacteristic
271C4×C8C2×D411yescharacteristic
272C4×C8C2×D411yescharacteristic
273C4×C8#4111yescharacteristic
274C4×C8#4111yescharacteristic
275C4×C8#4111yescharacteristic
276C4×C8#4111yescharacteristic
277#95—22no—
278#95#4112yes—
279#41—41no—
280#41—41no—
281#41—41no—
282#41—41no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—42no—
288C2×Q8C4×D412yes—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—41no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—41no—
299C22×C4C22×D411yescharacteristic
300C22×C4C2×#4111yescharacteristic
301C22×C4C2×#4111yescharacteristic
302Q16—82no—
303SD16—82no—
304M16—41no—
305M16—41no—
306M16—41no—
307M16—41no—
308M16—82no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—82no—
330C2×C8C4×D411yescharacteristic
331C2×C8C4×D411yescharacteristic
332C2×C8C4×D411yescharacteristic
333C2×C8C4×D411yescharacteristic
334C2×C8#12111yescharacteristic
335C2×C8#12111yescharacteristic
336C2×C8#12111yescharacteristic
337C2×C8#12111yescharacteristic
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—21no—
341C4⋊C4—22no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—44no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4C4×D412yes—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C42—22no—
356C42—22no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—21no—
361C42—42no—
362C42—44no—
363C42—42no—
364C42—44no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42C23×C411yescharacteristic
369C42C22×D411yescharacteristic
370C42C2×#4111yescharacteristic
371C42C2×#4111yescharacteristic
372C23—41no—
373C23#52511yescharacteristic
374Q8—22no—
375Q8—22no—
376Q8—42no—
377Q8—41no—
378Q8—41no—
379Q8—42no—
380Q8—81no—
381D4—41no—
382D4—41no—
383D4—42no—
384D4—81no—
385D4—81no—
386D4—81no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4C2×C4×D411yescharacteristic
408C2×C4C2×C4×D411yescharacteristic
409C2×C4C4×#4111yescharacteristic
410C2×C4C2×#12111yescharacteristic
411C2×C4#52911yescharacteristic
412C2×C4#53211yescharacteristic
413C2×C4#57611yescharacteristic
414C2×C4#57711yescharacteristic
415C2×C4#57811yescharacteristic
416C2×C4#57811yescharacteristic
417C8—21no—
418C8—21no—
419C8—21no—
420C8—21no—
421C8—21no—
422C8—21no—
423C8—21no—
424C8—21no—
425C8—42no—
426C8—42no—
427C8—42no—
428C8—42no—
429C8—81no—
430C8—81no—
431C8—81no—
432C8—81no—
433C22—21no—
434C22—21no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22C4×#12111yescharacteristic
439C22#312111yescharacteristic
440C22#312411yescharacteristic
441C4—21no—
442C4—21no—
443C4—21no—
444C4—44no—
445C4—42no—
446C4—42no—
447C4—42no—
448C4—81no—
449C4—81no—
450C4—81no—
451C4—81no—
452C4#294711yescharacteristic
453C4#295111yescharacteristic
454C4#295612yes—
455C4#304411yescharacteristic
456C4#304511yescharacteristic
457C2—21no—
458C2—21no—
459C2—161no—
460C2#2151211yescharacteristic
461C2#2155111yescharacteristic
462C2#2155811yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C4×#121
prime radical#532C23
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
5. #17931C2semidirect
7. #9627C2semidirect
8. #9584C2semidirect
11. #9232C2semidirect
12. #8663C2semidirect
13. #8270C2semidirect
14. #8258C2semidirect
39. #1704C4semidirect
40. #1702C4semidirect
41. #1700C4semidirect
42. #1698C4semidirect

Maximal subgroups

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

subgroupindex
#222342
#205732
#191552
#179362
#179312
#97512
#96272
#95842
#95812
#92342
#92322
#86632
#82702
#82582
#82512