Finite Group Databasebeta

← #500331

Subgroups — #500331

Order 512.

787 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3338C211yesmaximal, characteristic
2C2×#2237C211yesmaximal, characteristic, complemented
3C2×#2159C211yesmaximal, characteristic, complemented
4#27660C211yesmaximal, characteristic
5#13452C211yesmaximal, characteristic, complemented
6#13451C211yesmaximal, characteristic, complemented
7#13383C211yesmaximal, characteristic, complemented
8#13382C211yesmaximal, characteristic, complemented
9#13062C211yesmaximal, characteristic, complemented
10#12977C211yesmaximal, characteristic
11#12938C211yesmaximal, characteristic, complemented
12#12936C211yesmaximal, characteristic, complemented
13#12920C211yesmaximal, characteristic, complemented
14#12919C211yesmaximal, characteristic, complemented
15#12866C211yesmaximal, characteristic, complemented
16C2×#554—21no—
17C22×M32C2211yescharacteristic
18#3338—24no—
19#3338—24no—
20C2×#500C2211yescharacteristic
21C2×#484—21no—
22C2×#484—21no—
23C2×#477—21no—
24C2×#467C2211yescharacteristic
25C2×#467C2211yescharacteristic
26C2×C4×Q16C2211yescharacteristic
27C2×C8×D4C2211yescharacteristic
28C2×#427C2211yescharacteristic
29C22×#107C2211yescharacteristic
30C22×#107C2211yescharacteristic
31#2485—21no—
32#2237—24no—
33#2159—24no—
34C2×#367C2211yescharacteristic
35C2×#367C2211yescharacteristic
36C2×#367C2211yescharacteristic
37C2×#366C2211yescharacteristic
38C2×#366C2211yescharacteristic
39C2×#366C2211yescharacteristic
40C2×#358C2211yescharacteristic
41C2×#358C2211yescharacteristic
42C2×#358C2211yescharacteristic
43C2×#358C2211yescharacteristic
44C2×C22⋊C16C2211yescharacteristic
45C2×C22⋊C16C2211yescharacteristic
46C2×C22⋊C16C2211yescharacteristic
47#2088—21no—
48#2077—21no—
49#1942C2211yescharacteristic
50#1942C2211yescharacteristic
51#1825—21no—
52#1808C2211yescharacteristic
53#1808C2211yescharacteristic
54#1808C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61#1374C2211yescharacteristic
62#1374C2211yescharacteristic
63C22×Q16C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66#554—28no—
67#554—44no—
68C2×C4⋊Q8—21no—
69C2×#125—41no—
70C2×C22⋊Q8—21no—
71C2×C22⋊Q8—21no—
72C2×C22⋊Q8—41no—
73C2×C22⋊Q8—41no—
74C2×C4×Q8—21no—
75C2×C4×D4—41no—
76C2×C4×D4D411yescharacteristic
77C2×#117—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C2×M32—22no—
81C2×M32—22no—
82C2×M32—22no—
83C22×C16C2311yescharacteristic
84C22×C16C2311yescharacteristic
85C22×C16C2311yescharacteristic
86C22×C16C2311yescharacteristic
87#500—22no—
88#500—22no—
89#484—28no—
90#484—28no—
91#477—28no—
92#467—22no—
93#467—22no—
94#467—22no—
95#467D414yes—
96C4×Q16—22no—
97C4×Q16—22no—
98C8×D4—22no—
99C8×D4—22no—
100#427—22no—
101#427—22no—
102C2×#107—21no—
103C2×#107—21no—
104C2×#107—22no—
105C2×#107—21no—
106C2×#107—22no—
107C2×#107—22no—
108C2×#107D414yes—
109C2×#107C2311yescharacteristic
110C2×#107C2311yescharacteristic
111C2×#107C2311yescharacteristic
112C2×#107C2311yescharacteristic
113C2×#107C2311yescharacteristic
114C2×#107C2311yescharacteristic
115C2×#107C2311yescharacteristic
116C2×#106—21no—
117C2×C4⋊C8D411yescharacteristic
118C2×#103—21no—
119C2×#103—21no—
120C2×#103—21no—
121C2×C22⋊C8D411yescharacteristic
122C2×C22⋊C8D411yescharacteristic
123C2×C4×C8C2311yescharacteristic
124#397—41no—
125#394—41no—
126#389—21no—
127#389—21no—
128#388—41no—
129#387—41no—
130#380—21no—
131#380—21no—
132#380—21no—
133#367—22no—
134#367—22no—
135#367—22no—
136#366—22no—
137#366—22no—
138#366—22no—
139#358—22no—
140#358—22no—
141#358—22no—
142#358—22no—
143C22⋊C16—22no—
144C22⋊C16—22no—
145C22⋊C16—22no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151C22×Q8—21no—
152C22×Q8—41no—
153C22×D4D811yescharacteristic
154C23×C4—41no—
155C23×C4—41no—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C2×Q16—21no—
159C2×Q16—21no—
160C2×Q16—21no—
161C2×Q16—21no—
162C2×Q16—21no—
163C2×Q16C2×D412yes—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D412yes—
178C22×C8C2×D412yes—
179C22×C8C2411yescharacteristic
180C4⋊Q8—24no—
181C4⋊Q8—42no—
182#125—48no—
183C22⋊Q8—24no—
184C22⋊Q8—24no—
185C22⋊Q8—48no—
186C22⋊Q8—48no—
187C22⋊Q8—42no—
188C22⋊Q8—42no—
189C4×Q8—24no—
190C4×Q8—42no—
191C4×D4—22no—
192C4×D4—22no—
193C4×D4—48no—
194#117—42no—
195#117—42no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4D811yescharacteristic
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4D811yescharacteristic
223C2×C22⋊C4D811yescharacteristic
224C2×C42—41no—
225C2×C42—41no—
226C2×C42C2×D411yescharacteristic
227M32—28no—
228C2×C16—21no—
229C2×C16—21no—
230C2×C16—21no—
231C2×C16—21no—
232C2×C16—21no—
233C2×C16—21no—
234C2×C16—21no—
235C2×C16—21no—
236C2×C16—21no—
237C2×C16—21no—
238C2×C16#4112yes—
239C2×C16#4112yes—
240#107—24no—
241#107—24no—
242#107—24no—
243#107—21no—
244#107—21no—
245#107—21no—
246#107—21no—
247#107—21no—
248#107—21no—
249#107—21no—
250#107—21no—
251#107—21no—
252#107—21no—
253#107—21no—
254#107C2×D412yes—
255#107C2×D412yes—
256#107C2×D412yes—
257#106—42no—
258C4⋊C8—22no—
259#103—24no—
260#103—24no—
261#103—24no—
262C22⋊C8—22no—
263C22⋊C8—22no—
264C4×C8—21no—
265C4×C8—21no—
266#95—41no—
267#95—41no—
268#95—41no—
269#95—41no—
270#95—41no—
271C24C2×D811yescharacteristic
272C24C2×D811yescharacteristic
273C2×Q8—22no—
274C2×Q8—44no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—161no—
281C2×Q8—161no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—161no—
306C22×C4—161no—
307C22×C4—161no—
308C22×C4—161no—
309C22×C4#12012yes—
310C22×C4C2×D811yescharacteristic
311C22×C4C2×D811yescharacteristic
312C22×C4C2×D811yescharacteristic
313C22×C4C2×D811yescharacteristic
314C22×C4AGL(1,ℤ/8)12yes—
315C22×C4C22×D411yescharacteristic
316Q16—22no—
317Q16—22no—
318Q16—41no—
319Q16—41no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8#12012yes—
337C2×C8C22×D411yescharacteristic
338C2×C8C22×D411yescharacteristic
339C2×C8C2×#4111yescharacteristic
340C2×C8C2×#4111yescharacteristic
341C2×C82−1+411yescharacteristic
342C2×C82−1+411yescharacteristic
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—44no—
348C4⋊C4—44no—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—82no—
365C4⋊C4—161no—
366C22⋊C4—22no—
367C22⋊C4—22no—
368C22⋊C4—44no—
369C22⋊C4—44no—
370C22⋊C4—82no—
371C22⋊C4—82no—
372C22⋊C4—161no—
373C22⋊C4—161no—
374C42—21no—
375C42—21no—
376C42—44no—
377C42—82no—
378C16—22no—
379C16—22no—
380C16—41no—
381C16—41no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23#44712yes—
395C23#51012yes—
396C23C22×D811yescharacteristic
397Q8—42no—
398Q8—42no—
399Q8—81no—
400Q8—81no—
401Q8—164no—
402D4—28no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—164no—
446C2×C4—164no—
447C2×C4#44712yes—
448C2×C4#51012yes—
449C2×C4C2×#12011yescharacteristic
450C2×C4#54711yescharacteristic
451C2×C4#54711yescharacteristic
452C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
453C2×C4#57811yescharacteristic
454C2×C4#57811yescharacteristic
455C8—22no—
456C8—41no—
457C8—41no—
458C8#52511yescharacteristic
459C8#53611yescharacteristic
460C8#54711yescharacteristic
461C8#54711yescharacteristic
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22C2×#44711yescharacteristic
477C22#333411yescharacteristic
478C22#333411yescharacteristic
479C22C2×#51011yescharacteristic
480C22C2×#51011yescharacteristic
481C22#341511yescharacteristic
482C22#341611yescharacteristic
483C4—22no—
484C4—22no—
485C4—41no—
486C4—41no—
487C4—82no—
488C4—82no—
489C4—82no—
490C4—82no—
491C4—161no—
492C4—161no—
493C4—161no—
494C4—161no—
495C4#240611yescharacteristic
496C4#301111yescharacteristic
497C4#334311yescharacteristic
498C4#334311yescharacteristic
499C2—42no—
500C2—42no—
501C2#2408811yescharacteristic
502C2#3351811yescharacteristic
503C2#3352611yescharacteristic
504C2#3391111yescharacteristic
505C2#3391311yescharacteristic
506C2#3392311yescharacteristic
507C2#3392411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D4D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2237C2semidirect
3. C2×#2159C2semidirect
5. #13452C2semidirect
6. #13451C2semidirect
7. #13383C2semidirect
8. #13382C2semidirect
9. #13062C2semidirect
11. #12938C2semidirect
12. #12936C2semidirect
13. #12920C2semidirect
14. #12919C2semidirect
15. #12866C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33382
C2×#22372
C2×#21592
#276602
#134522
#134512
#133832
#133822
#130622
#129772
#129382
#129362
#129202
#129192
#128662