Finite Group Databasebeta

← #213331

Subgroups — #213331

Order 512.

647 conjugacy classes of subgroups.

The subgroup lattice has 463 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21734C211yesmaximal, characteristic, complemented
2#21660C211yesmaximal, characteristic
3#19620C211yesmaximal, characteristic
4#17977C211yesmaximal, characteristic, complemented
5#17976C211yesmaximal, characteristic
6#17958C211yesmaximal, characteristic, complemented
7#17957C211yesmaximal, characteristic
8#9859C211yesmaximal, characteristic, complemented
9#9858C211yesmaximal, characteristic
10#9851C211yesmaximal, characteristic
11#9851C211yesmaximal, characteristic
12#9850C211yesmaximal, characteristic, complemented
13#9850C211yesmaximal, characteristic, complemented
14#8828C211yesmaximal, characteristic, complemented
15#7846C211yesmaximal, characteristic, complemented
16#3170C2211yescharacteristic
17#3170C2211yescharacteristic
18#3167C2211yescharacteristic
19#3167C2211yescharacteristic
20#3160C2211yescharacteristic
21#3131C2211yescharacteristic
22#2960C2211yescharacteristic
23#2960C2211yescharacteristic
24#2960C2211yescharacteristic
25#2931C2211yescharacteristic
26#1994C2211yescharacteristic
27#1994C2211yescharacteristic
28#1994C2211yescharacteristic
29#1962C2211yescharacteristic
30#1946—22no—
31#1938—22no—
32#1848C2211yescharacteristic
33#1837C2211yescharacteristic
34#1773—22no—
35#1765—22no—
36C4×#97C2211yescharacteristic
37C4×#97C2211yescharacteristic
38C4×#97C2211yescharacteristic
39#1723C411yescharacteristic, complemented
40#1722C411yescharacteristic, complemented
41#1720C411yescharacteristic, complemented
42#1719C411yescharacteristic, complemented
43#1718C411yescharacteristic, complemented
44#1718C411yescharacteristic, complemented
45#1716C411yescharacteristic, complemented
46#1714C411yescharacteristic, complemented
47#1591—22no—
48#1590—22no—
49#1584—22no—
50#1584—22no—
51#1583—22no—
52#1582—22no—
53C8×Q16—22no—
54C8×SD16—22no—
55#1567C2211yescharacteristic
56#1567C2211yescharacteristic
57#1566C2211yescharacteristic
58#1566C2211yescharacteristic
59#1493C2211yescharacteristic
60#1493C2211yescharacteristic
61#1493C2211yescharacteristic
62#1493C2211yescharacteristic
63#1493C2211yescharacteristic
64#1493C2211yescharacteristic
65#1492C2211yescharacteristic
66#1492C2211yescharacteristic
67C2×#322C2211yescharacteristic
68C2×#322C2211yescharacteristic
69C2×#322C2211yescharacteristic
70C2×C82C2211yescharacteristic
71C2×#131—21no—
72#533C2311yescharacteristic
73#533C2311yescharacteristic
74#533C2311yescharacteristic
75#532C2311yescharacteristic
76#500C2×C411yescharacteristic
77#500C2×C411yescharacteristic
78#500C2×C411yescharacteristic
79#500C2×C411yescharacteristic
80#499—22no—
81#498C2×C411yescharacteristic
82#498C2×C411yescharacteristic
83#498C2×C411yescharacteristic
84#498C2×C411yescharacteristic
85#494C2×C411yescharacteristic
86#494C2×C411yescharacteristic
87#492C2×C411yescharacteristic
88#492C2×C411yescharacteristic
89#488—22no—
90#487C2×C411yescharacteristic
91#487C2×C411yescharacteristic
92#487C2×C411yescharacteristic
93#487C2×C411yescharacteristic
94#487C2×C411yescharacteristic
95#487C2×C411yescharacteristic
96#487C2×C411yescharacteristic
97#487C2×C411yescharacteristic
98#486C2×C411yescharacteristic
99#486C2×C411yescharacteristic
100#486C2×C411yescharacteristic
101#486C2×C411yescharacteristic
102#467—22no—
103#467—22no—
104#465—22no—
105#465—22no—
106#446—21no—
107#446—21no—
108#446—21no—
109#446—21no—
110#446—21no—
111#446—21no—
112C8×Q8—21no—
113C8×Q8—21no—
114C8×Q8—21no—
115C8×Q8—21no—
116C8×Q8—21no—
117C8×Q8—21no—
118C4×Q16—22no—
119C4×Q16—22no—
120C4×SD16—22no—
121C4×SD16—22no—
122#436—21no—
123#436—21no—
124C8×D4—21no—
125C8×D4—21no—
126C2×#108—22no—
127#427—22no—
128#424—21no—
129#424—21no—
130#424—21no—
131#423C2311yescharacteristic
132#423C2311yescharacteristic
133#423C2311yescharacteristic
134#423C2311yescharacteristic
135C2×#104—22no—
136#416—22no—
137#408—21no—
138#405—21no—
139#405—21no—
140#405—21no—
141C2×#97—22no—
142C2×#97—22no—
143C2×#97D412yes—
144C2×#97D412yes—
145C2×C4×C8C2311yescharacteristic
146C2×C4×C8C2311yescharacteristic
147C2×C4×C8C2311yescharacteristic
148C2×C4×C8C2311yescharacteristic
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C43C2311yescharacteristic
152#341—22no—
153#339—22no—
154#339—22no—
155#339—22no—
156#335—22no—
157#335—22no—
158#334—22no—
159#334—22no—
160Q8⋊C8—22no—
161Q8⋊C8—22no—
162Q8⋊C8—22no—
163Q8⋊C8—22no—
164Q8⋊C8—22no—
165Q8⋊C8—22no—
166D4⋊C8—22no—
167D4⋊C8—22no—
168#322—22no—
169#322C2×C411yescharacteristic
170#322C2×C411yescharacteristic
171#322C2×C411yescharacteristic
172#322C2×C411yescharacteristic
173#322D412yes—
174#322D412yes—
175C82D412yes—
176C82D412yes—
177C2×#41—21no—
178C2×Q16—22no—
179C2×Q16—22no—
180C2×SD16—22no—
181C2×SD16—22no—
182#131—44no—
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—41no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—41no—
191C22×C8C2×D412yes—
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C22×C8#4112yes—
195C22×C8#4112yes—
196C4⋊Q8C22×C411yescharacteristic
197C4⋊Q8C22×C411yescharacteristic
198C4⋊Q8C22×C411yescharacteristic
199C4⋊Q8C22×C411yescharacteristic
200C4⋊Q8C22×C411yescharacteristic
201C4⋊Q8C22×C411yescharacteristic
202C4⋊Q8C22×C411yescharacteristic
203#127C22×C411yescharacteristic
204#125—21no—
205#125—21no—
206#125—21no—
207#124—21no—
208C22⋊Q8—21no—
209C22⋊Q8—21no—
210C22⋊Q8—21no—
211C22⋊Q8—21no—
212C22⋊Q8—21no—
213C22⋊Q8—21no—
214#121—21no—
215#121—21no—
216C4×Q8—21no—
217C4×Q8—21no—
218C4×Q8—21no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8—21no—
222C4×D4—21no—
223C4×D4—21no—
224#117—21no—
225#117—21no—
226#117—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42C2×D412yes—
230C2×C42C2411yescharacteristic
231#108—44no—
232#107—22no—
233#107—22no—
234#107—22no—
235#107—22no—
236#106—22no—
237#106—22no—
238#106—22no—
239#106—22no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—41no—
249C4⋊C8—41no—
250C4⋊C8—41no—
251C4⋊C8—41no—
252C4⋊C8—41no—
253C4⋊C8—41no—
254#104—44no—
255#103—22no—
256#103—22no—
257#103—22no—
258#103—22no—
259#103—22no—
260#103—22no—
261#103—22no—
262#103—22no—
263#102—22no—
264#102—22no—
265#102—22no—
266#102—22no—
267C22⋊C8—41no—
268C22⋊C8—41no—
269#97—24no—
270#97—42no—
271#97—44no—
272#97—44no—
273#97—41no—
274#97—41no—
275#97—41no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—41no—
279C4×C8—41no—
280C4×C8—41no—
281C4×C8C22×C411yescharacteristic
282C4×C8C22×C411yescharacteristic
283C4×C8C22×C411yescharacteristic
284C4×C8C22×C411yescharacteristic
285C4×C8C22×C411yescharacteristic
286C4×C8C22×C411yescharacteristic
287C4×C8C2×D412yes—
288C4×C8C2×D412yes—
289C4×C8C2×D412yes—
290C4×C8C2×D412yes—
291C4×C8C2×D411yescharacteristic
292C4×C8C2×D411yescharacteristic
293C4×C8C2×D411yescharacteristic
294C4×C8C2×D411yescharacteristic
295C4×C8#4112yes—
296#41—42no—
297#41—42no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—41no—
309C22×C4—22no—
310C22×C4—41no—
311C22×C4C22×D411yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C22×C4C2×#4111yescharacteristic
314Q16—44no—
315Q16—82no—
316SD16—44no—
317SD16—82no—
318M16—42no—
319M16—42no—
320M16—42no—
321M16—42no—
322M16—82no—
323C2×C8—24no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—24no—
329C2×C8—22no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—82no—
337C2×C8C4×D412yes—
338C2×C8C4×D412yes—
339C2×C8C4×D412yes—
340C2×C8C4×D412yes—
341C2×C8C4×D412yes—
342C2×C8C4×D412yes—
343C2×C8#12711yescharacteristic
344C2×C8#12711yescharacteristic
345C2×C8#12711yescharacteristic
346C2×C8#12711yescharacteristic
347C2×C8#13714yes—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—41no—
371C22⋊C4—41no—
372C22⋊C4—41no—
373C42—22no—
374C42—22no—
375C42—42no—
376C42—42no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42C23×C411yescharacteristic
381C42C22×D411yescharacteristic
382C42C22×D411yescharacteristic
383C42C2×#4111yescharacteristic
384C23—41no—
385C23#53211yescharacteristic
386Q8—42no—
387Q8—42no—
388Q8—42no—
389Q8—42no—
390Q8—42no—
391Q8—42no—
392Q8—42no—
393D4—42no—
394D4—42no—
395D4—42no—
396D4—82no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4C2×C4×D411yescharacteristic
412C2×C4C2×C4×D411yescharacteristic
413C2×C4C2×C4×D411yescharacteristic
414C2×C4C2×#12711yescharacteristic
415C2×C4#53211yescharacteristic
416C2×C4#53211yescharacteristic
417C2×C4C2×#13712yes—
418C2×C4#57512yes—
419C8—24no—
420C8—21no—
421C8—21no—
422C8—22no—
423C8—22no—
424C8—22no—
425C8—22no—
426C8—22no—
427C8—42no—
428C8—81no—
429C8—81no—
430C8—81no—
431C8—81no—
432C8#44414yes—
433C8#49712yes—
434C8#49712yes—
435C22—21no—
436C22—21no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22C4×#12711yescharacteristic
441C22#317212yes—
442C4—21no—
443C4—21no—
444C4—82no—
445C4—81no—
446C4—81no—
447C4—81no—
448C4—81no—
449C4—81no—
450C4—81no—
451C4—81no—
452C4C4×#13712yes—
453C4C2×#44412yes—
454C4#295612yes—
455C4C2×#49711yescharacteristic
456C4#315011yescharacteristic
457C2—22no—
458C2—161no—
459C2C4×#49711yescharacteristic
460C2#2170611yescharacteristic
461C2#2171011yescharacteristic
subgroupG/H
centerC2×C4C2×#127
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C4×#127
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
1. #21734C2semidirect
4. #17977C2semidirect
6. #17958C2semidirect
8. #9859C2semidirect
12. #9850C2semidirect
13. #9850C2semidirect
14. #8828C2semidirect
15. #7846C2semidirect
39. #1723C4semidirect
40. #1722C4semidirect
41. #1720C4semidirect
42. #1719C4semidirect
43. #1718C4semidirect
44. #1718C4semidirect
45. #1716C4semidirect
46. #1714C4semidirect

Maximal subgroups

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

subgroupindex
#217342
#216602
#196202
#179772
#179762
#179582
#179572
#98592
#98582
#98512
#98502
#88282
#78462