Finite Group Databasebeta

← #212976

Subgroups — #212976

Order 512.

762 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21714C211yesmaximal, characteristic, complemented
2C4×#497C211yesmaximal, characteristic
3#21633C211yesmaximal, characteristic
4C2×#1591C211yesmaximal, characteristic, complemented
5C2×#1584C211yesmaximal, characteristic, complemented
6C2×#1582C211yesmaximal, characteristic
7C2×C8×SD16C211yesmaximal, characteristic
8#9666C211yesmaximal, characteristic
9#9664C211yesmaximal, characteristic, complemented
10#9660C211yesmaximal, characteristic
11#9658C211yesmaximal, characteristic, complemented
12#9647C211yesmaximal, characteristic, complemented
13#9646C211yesmaximal, characteristic
14#8808C211yesmaximal, characteristic, complemented
15#7846C211yesmaximal, characteristic, complemented
16C2×#497C411yescharacteristic
17C2×#446C2211yescharacteristic
18C2×#446C2211yescharacteristic
19C2×C8×Q8C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C2×C4×Q16C2211yescharacteristic
22C2×C4×SD16C2211yescharacteristic
23C2×C4×SD16C2211yescharacteristic
24C2×C8×D4C2211yescharacteristic
25C4×C4⋊Q8C2211yescharacteristic
26C4×#124C2211yescharacteristic
27#1984C2211yescharacteristic
28#1984C2211yescharacteristic
29#1984C2211yescharacteristic
30#1973C411yescharacteristic
31#1973C411yescharacteristic
32#1971C411yescharacteristic
33#1971C411yescharacteristic
34#1960C411yescharacteristic
35#1959C411yescharacteristic
36#1954C2211yescharacteristic
37#1940—22no—
38#1924—22no—
39#1847C411yescharacteristic
40#1842C2211yescharacteristic
41#1831C2211yescharacteristic
42#1771—22no—
43#1752—22no—
44C4×#97C2211yescharacteristic
45C4×#97C2211yescharacteristic
46C4×#97C2211yescharacteristic
47#1591—24no—
48#1584—24no—
49#1582—24no—
50C8×SD16—24no—
51C2×#335C2211yescharacteristic
52C2×#335C2211yescharacteristic
53C2×#334C2211yescharacteristic
54C2×#334C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×D4⋊C8C2211yescharacteristic
62C2×D4⋊C8C2211yescharacteristic
63C2×#322C2211yescharacteristic
64C2×#322C2211yescharacteristic
65C2×#322C2211yescharacteristic
66C2×C82C2211yescharacteristic
67C22×Q16C2×C411yescharacteristic
68C22×Q16C2×C411yescharacteristic
69C22×SD16C2×C411yescharacteristic
70C22×SD16C2×C411yescharacteristic
71C23×C8—21no—
72C2×C4⋊Q8C2×C411yescharacteristic
73C2×#124C2×C411yescharacteristic
74C2×C4×Q8C2311yescharacteristic
75C2×C4×Q8C2311yescharacteristic
76C2×C4×Q8C2311yescharacteristic
77C2×C4×D4C2311yescharacteristic
78#497—24no—
79#446—22no—
80#446—22no—
81#446—22no—
82#446—22no—
83C8×Q8—22no—
84C8×Q8—22no—
85C4×Q16—22no—
86C4×Q16—22no—
87C4×Q16—22no—
88C4×Q16—22no—
89C4×SD16—22no—
90C4×SD16—22no—
91C4×SD16—22no—
92C4×SD16—22no—
93C8×D4—22no—
94C8×D4—22no—
95C2×#107C2×C411yescharacteristic
96C2×#107C2×C411yescharacteristic
97C2×#106C2×C411yescharacteristic
98C2×#106C2×C411yescharacteristic
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×#103C2×C411yescharacteristic
108C2×#103C2×C411yescharacteristic
109C2×#103C2×C411yescharacteristic
110C2×#103C2×C411yescharacteristic
111C2×#103C2×C411yescharacteristic
112C2×#103C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115C2×C22⋊C8—21no—
116C2×C22⋊C8—22no—
117C2×#97—21no—
118C2×#97—21no—
119C2×#97—22no—
120C2×#97C2×C411yescharacteristic
121C2×#97C2×C411yescharacteristic
122C2×#97C2×C411yescharacteristic
123C2×#97C2×C411yescharacteristic
124C2×#97D412yes—
125C2×#97D412yes—
126C2×C4×C8—21no—
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133#391C2×C411yescharacteristic
134#391C2×C411yescharacteristic
135#391C2×C411yescharacteristic
136#390C2×C411yescharacteristic
137#389—22no—
138#388—22no—
139#384C2×C411yescharacteristic
140#382C2×C411yescharacteristic
141C4×C4⋊C4—22no—
142C4×C22⋊C4—22no—
143C43C2311yescharacteristic
144#336—22no—
145#336—22no—
146#336—22no—
147#336—22no—
148#335—22no—
149#335—22no—
150#334—22no—
151#334—22no—
152Q8⋊C8—22no—
153Q8⋊C8—22no—
154Q8⋊C8—22no—
155Q8⋊C8—22no—
156Q8⋊C8—22no—
157Q8⋊C8—22no—
158D4⋊C8—22no—
159D4⋊C8—22no—
160#322—22no—
161#322—22no—
162#322D414yes—
163C82D414yes—
164C22×Q8C22×C411yescharacteristic
165C22×Q8C22×C411yescharacteristic
166C22×Q8C22×C411yescharacteristic
167C22×D4C22×C411yescharacteristic
168C23×C4—21no—
169C2×Q16—22no—
170C2×Q16—22no—
171C2×Q16—22no—
172C2×Q16—22no—
173C2×Q16—22no—
174C2×Q16—41no—
175C2×SD16—22no—
176C2×SD16—22no—
177C2×SD16—22no—
178C2×SD16—22no—
179C2×SD16—41no—
180C2×SD16M1614yes—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—42no—
188C22×C8—44no—
189C22×C8C22×C411yescharacteristic
190C22×C8C22×C411yescharacteristic
191C22×C8C22×C411yescharacteristic
192C22×C8C22×C411yescharacteristic
193C22×C8C2×D412yes—
194C22×C8C2×D411yescharacteristic
195C22×C8C2×D411yescharacteristic
196C4⋊Q8—22no—
197C4⋊Q8—41no—
198#124—22no—
199#124—41no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×Q8—21no—
203C4×Q8—21no—
204C4×Q8—21no—
205C4×Q8—21no—
206C4×Q8—21no—
207C4×Q8—21no—
208C4×Q8—21no—
209C4×Q8—21no—
210C4×Q8—21no—
211C4×Q8—21no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—21no—
215C4×D4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C22⋊C4—22no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—22no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42C22×C411yescharacteristic
232C2×C42C22×C411yescharacteristic
233C2×C42C2×D412yes—
234C2×C42#4112yes—
235C2×C42C2411yescharacteristic
236#107—22no—
237#107—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—44no—
249C4⋊C8—42no—
250C4⋊C8—42no—
251C4⋊C8—42no—
252#103—22no—
253#103—22no—
254#103—22no—
255#103—22no—
256#103—22no—
257#103—22no—
258#102—22no—
259#102—22no—
260C22⋊C8—44no—
261C22⋊C8—42no—
262#97—22no—
263#97—22no—
264#97—24no—
265#97—41no—
266#97—41no—
267#97—42no—
268#97—44no—
269#97—42no—
270#97—42no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—42no—
276C4×C8C2×D412yes—
277C4×C8C2×D412yes—
278C4×C8C2×D412yes—
279C4×C8C2×D412yes—
280C4×C8C2×D412yes—
281C4×C8C2×D412yes—
282C4×C8#4112yes—
283C4×C8#4112yes—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288C24—21no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8C2×M1612yes—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4C2×M1612yes—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4C4×D412yes—
325C22×C4C4×D412yes—
326C22×C4C23×C411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C2×#4111yescharacteristic
329Q16—44no—
330Q16—44no—
331SD16—28no—
332SD16—44no—
333C2×C8—24no—
334C2×C8—22no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—42no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—84no—
356C2×C8C4×D412yes—
357C2×C8C4×D412yes—
358C2×C8C4×D412yes—
359C2×C8C4×D412yes—
360C2×C8#12712yes—
361C2×C8#12712yes—
362C2×C8C2×M1612yes—
363C2×C8#13112yes—
364C2×C8#13714yes—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—44no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—82no—
381C22⋊C4—44no—
382C22⋊C4—41no—
383C22⋊C4—41no—
384C22⋊C4—82no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—22no—
390C42—22no—
391C42—42no—
392C42—41no—
393C42—41no—
394C42—41no—
395C42—41no—
396C42—41no—
397C42—41no—
398C42C22×D411yescharacteristic
399C42C22×D411yescharacteristic
400C42C2×#4111yescharacteristic
401C42C2×#4111yescharacteristic
402C23—22no—
403C23—41no—
404C23—41no—
405C23—42no—
406C23—42no—
407C23C2×C4×D411yescharacteristic
408Q8—22no—
409Q8—22no—
410Q8—41no—
411Q8—41no—
412Q8—41no—
413Q8—41no—
414Q8—41no—
415Q8—41no—
416Q8—41no—
417Q8—41no—
418Q8—41no—
419Q8—41no—
420D4—22no—
421D4—22no—
422D4—41no—
423D4—41no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—82no—
451C2×C4C2×C4×D411yescharacteristic
452C2×C4C2×C4×D411yescharacteristic
453C2×C4C2×#12711yescharacteristic
454C2×C4#53211yescharacteristic
455C2×C4#53211yescharacteristic
456C2×C4#53211yescharacteristic
457C2×C4C22×M1611yescharacteristic
458C2×C4C2×#13111yescharacteristic
459C2×C4#56811yescharacteristic
460C2×C4#56811yescharacteristic
461C2×C4C2×#13712yes—
462C2×C4#57512yes—
463C8—24no—
464C8—22no—
465C8—22no—
466C8—22no—
467C8—22no—
468C8—42no—
469C8—42no—
470C8—81no—
471C8—81no—
472C8—81no—
473C8—81no—
474C8#43514yes—
475C8#49714yescomplemented
476C22—44no—
477C22—42no—
478C22—82no—
479C22—81no—
480C22—81no—
481C22C4×#12711yescharacteristic
482C22D4×M1611yescharacteristic
483C22#293511yescharacteristic
484C22C4×#13712yes—
485C22#295612yes—
486C4—21no—
487C4—21no—
488C4—42no—
489C4—42no—
490C4—42no—
491C4—81no—
492C4—81no—
493C4—81no—
494C4—81no—
495C4—81no—
496C4—81no—
497C4C2×#43511yescharacteristic
498C4#293211yescharacteristic
499C4D4×M1611yescharacteristic
500C4#293511yescharacteristic
501C4C2×#49711yescharacteristic
502C4#315011yescharacteristic
503C4#317212yes—
504C2—82no—
505C2#1804712yes—
506C2#1807712yes—
507C2#2168211yescharacteristic
508C2C4×#49711yescharacteristic
509C2#2171011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#97 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21714C2semidirect
4. C2×#1591C2semidirect
5. C2×#1584C2semidirect
9. #9664C2semidirect
11. #9658C2semidirect
12. #9647C2semidirect
14. #8808C2semidirect
15. #7846C2semidirect
475. C8#497semidirect

Maximal subgroups

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

subgroupindex
#217142
C4×#4972
#216332
C2×#15912
C2×#15842
C2×#15822
C2×C8×SD162
#96662
#96642
#96602
#96582
#96472
#96462
#88082
#78462