Finite Group Databasebeta

← #500916

Subgroups — #500916

Order 512.

777 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3383C211yesmaximal, characteristic, complemented
2C2×#2228C211yesmaximal, characteristic
3C2×#2175C211yesmaximal, characteristic
4#28263C211yesmaximal, characteristic
5#13453C211yesmaximal, characteristic, complemented
6#13451C211yesmaximal, characteristic, complemented
7#13365C211yesmaximal, characteristic
8#13363C211yesmaximal, characteristic
9#13284C211yesmaximal, characteristic, complemented
10#13264C211yesmaximal, characteristic, complemented
11#13013C211yesmaximal, characteristic
12#13013C211yesmaximal, characteristic
13#12938C211yesmaximal, characteristic, complemented
14#12936C211yesmaximal, characteristic, complemented
15#12812C211yesmaximal, characteristic, complemented
16C2×#557—21no—
17C22×SD32C2211yescharacteristic
18#3383—24no—
19#3383—24no—
20C2×#500C2211yescharacteristic
21C2×#499C2211yescharacteristic
22C2×#499C2211yescharacteristic
23C2×#479—21no—
24C2×#479—21no—
25C2×#475—21no—
26C2×C8×Q8C2211yescharacteristic
27C2×C4×Q16C2211yescharacteristic
28C2×C4×Q16C2211yescharacteristic
29C2×C4×Q16C2211yescharacteristic
30C2×C4×D8C2211yescharacteristic
31#2538—21no—
32#2228—24no—
33#2175—24no—
34C2×#367C2211yescharacteristic
35C2×#366C2211yescharacteristic
36C2×#366C2211yescharacteristic
37C2×C4⋊C16C2211yescharacteristic
38C2×#358C2211yescharacteristic
39C2×#358C2211yescharacteristic
40C2×#358C2211yescharacteristic
41C2×#357C2211yescharacteristic
42C2×#357C2211yescharacteristic
43C2×#357C2211yescharacteristic
44C2×#346C2211yescharacteristic
45#2067—21no—
46#2067—21no—
47#1970C2211yescharacteristic
48#1970C2211yescharacteristic
49#1942C2211yescharacteristic
50#1942C2211yescharacteristic
51#1845C2211yescharacteristic
52#1802—21no—
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61#1374C2211yescharacteristic
62#1374C2211yescharacteristic
63C22×Q16C2311yescharacteristic
64C22×D8C2311yescharacteristic
65#557—28no—
66#557—44no—
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8—41no—
69C2×#125—21no—
70C2×#125—21no—
71C2×#125—41no—
72C2×#125—41no—
73C2×C22⋊Q8—41no—
74C2×C4×Q8—21no—
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×C4×Q8—41no—
78C2×C4×Q8D411yescharacteristic
79C2×C4×D4—21no—
80C2×SD32—22no—
81C2×SD32—22no—
82C2×SD32Q814yescomplemented
83C22×C16C2311yescharacteristic
84C22×C16C2311yescharacteristic
85C22×C16C2311yescharacteristic
86C22×C16C2311yescharacteristic
87#500—22no—
88#500—22no—
89#499—22no—
90#499—22no—
91#499—22no—
92#499D414yes—
93#479—28no—
94#479—28no—
95#475—28no—
96C8×Q8—22no—
97C8×Q8—22no—
98C4×Q16—22no—
99C4×Q16—22no—
100C4×Q16—22no—
101C4×Q16—22no—
102C4×Q16—22no—
103C4×Q16D414yes—
104C4×D8—22no—
105C4×D8—22no—
106C2×#107—21no—
107C2×#107—21no—
108C2×#107C2311yescharacteristic
109C2×#107C2311yescharacteristic
110C2×#107C2311yescharacteristic
111C2×#107C2311yescharacteristic
112C2×#107C2311yescharacteristic
113C2×#107C2311yescharacteristic
114C2×#106—21no—
115C2×#106—21no—
116C2×C4⋊C8D411yescharacteristic
117C2×C4⋊C8D411yescharacteristic
118C2×C4⋊C8D411yescharacteristic
119C2×#103—21no—
120C2×#103—21no—
121C2×#103—21no—
122C2×#102—21no—
123C2×#102—21no—
124C2×#102—21no—
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128#396—41no—
129#396—41no—
130#390—21no—
131#390—21no—
132#389—21no—
133#389—21no—
134#384—21no—
135#380—41no—
136#367—22no—
137#366—22no—
138#366—22no—
139C4⋊C16—22no—
140#358—22no—
141#358—22no—
142#358—22no—
143#357—22no—
144#357—22no—
145#357—22no—
146#346—22no—
147#340—21no—
148#340—21no—
149C22×Q8—21no—
150C22×Q8—41no—
151C22×Q8D811yescharacteristic
152C22×D4—21no—
153C23×C4—41no—
154C2×Q16—21no—
155C2×Q16—21no—
156C2×Q16—21no—
157C2×Q16C2×D412yes—
158C2×Q16C2×Q812yes—
159C2×Q16#4112yes—
160C2×D8—21no—
161C2×D8—21no—
162C2×D8—21no—
163C2×D8—21no—
164C2×D8—21no—
165C2×D8C2×Q812yes—
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2411yescharacteristic
170C4⋊Q8—24no—
171C4⋊Q8—42no—
172C4⋊Q8—48no—
173#125—24no—
174#125—24no—
175#125—42no—
176#125—48no—
177#125—48no—
178#125—42no—
179C22⋊Q8—84no—
180C4×Q8—22no—
181C4×Q8—22no—
182C4×Q8—24no—
183C4×Q8—24no—
184C4×Q8—24no—
185C4×Q8—42no—
186C4×Q8—42no—
187C4×Q8—48no—
188C4×Q8—42no—
189C4×D4—42no—
190C4×D4—42no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4D811yescharacteristic
210C2×C4⋊C4D811yescharacteristic
211C2×C4⋊C4D811yescharacteristic
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C42—41no—
218C2×C42—41no—
219C2×C42—41no—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223SD32—28no—
224C2×C16—21no—
225C2×C16—21no—
226C2×C16—21no—
227C2×C16—21no—
228C2×C16—21no—
229C2×C16—21no—
230C2×C16—21no—
231C2×C16—21no—
232C2×C16—21no—
233C2×C16—21no—
234C2×C16—21no—
235C2×C16C2×Q812yes—
236#107—24no—
237#107—21no—
238#107—21no—
239#107—21no—
240#107—21no—
241#107—21no—
242#107—21no—
243#107—21no—
244#107—21no—
245#107—21no—
246#107—81no—
247#107C2×D412yes—
248#107C2×D412yes—
249#107C2×D412yes—
250#106—24no—
251#106—24no—
252C4⋊C8—22no—
253C4⋊C8—22no—
254C4⋊C8—22no—
255#103—24no—
256#103—24no—
257#103—24no—
258#102—42no—
259#102—81no—
260#102—81no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8C2×D412yes—
266C4×C8C2×D412yes—
267#95—41no—
268#95—41no—
269C24—41no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—41no—
276C2×Q8—44no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—82no—
281C2×Q8—82no—
282C2×D4—22no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4C2×D811yescharacteristic
299C22×C4C2×D811yescharacteristic
300C22×C4C2×D811yescharacteristic
301C22×C4C2×D811yescharacteristic
302C22×C4C2×D811yescharacteristic
303C22×C4C2×D811yescharacteristic
304C22×C4C22×D411yescharacteristic
305Q16—22no—
306Q16—22no—
307Q16—22no—
308Q16C22⋊Q814yescomplemented
309D8—22no—
310D8—22no—
311D8—41no—
312D8—41no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8#12012yes—
322C2×C8C22×D411yescharacteristic
323C2×C8C22×D411yescharacteristic
324C2×C8C22×Q811yescharacteristic
325C2×C8C2×#4111yescharacteristic
326C2×C82−1+411yescharacteristic
327C2×C82−1+411yescharacteristic
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—44no—
338C4⋊C4—44no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—82no—
352C4⋊C4—82no—
353C4⋊C4—82no—
354C4⋊C4—161no—
355C22⋊C4—82no—
356C22⋊C4—82no—
357C22⋊C4—82no—
358C22⋊C4—161no—
359C22⋊C4—161no—
360C42—21no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42—44no—
365C42—44no—
366C42—44no—
367C42#12012yes—
368C42AGL(1,ℤ/8)12yes—
369C16—22no—
370C16—41no—
371C16—41no—
372C16—41no—
373C23—42no—
374C23—81no—
375C23—81no—
376C23—81no—
377C23—81no—
378C23—81no—
379C23—81no—
380C23C22×D811yescharacteristic
381Q8—24no—
382Q8—44no—
383Q8—42no—
384Q8—42no—
385Q8—42no—
386Q8—88no—
387D4—42no—
388D4—42no—
389D4—81no—
390D4—81no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—164no—
428C2×C4#44712yes—
429C2×C4#44712yes—
430C2×C4#50912yes—
431C2×C4C2×#12011yescharacteristic
432C2×C4#54811yescharacteristic
433C2×C4D4×Q811yescharacteristic
434C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
435C2×C4#57711yescharacteristic
436C2×C4#57811yescharacteristic
437C8—22no—
438C8—41no—
439C8—41no—
440C8C2×C22⋊Q811yescharacteristic
441C8#53611yescharacteristic
442C8#54811yescharacteristic
443C8D4×Q811yescharacteristic
444C22—82no—
445C22—82no—
446C22—82no—
447C22—161no—
448C22—161no—
449C22—161no—
450C22—161no—
451C22C2×#44711yescharacteristic
452C22#335811yescharacteristic
453C22Q8×D811yescharacteristic
454C22C2×#50911yescharacteristic
455C22#341411yescharacteristic
456C22#341611yescharacteristic
457C22#341711yescharacteristic
458C4—22no—
459C4—22no—
460C4—22no—
461C4—44no—
462C4—41no—
463C4—41no—
464C4—41no—
465C4—82no—
466C4—82no—
467C4—82no—
468C4—161no—
469C4—161no—
470C4—161no—
471C4#240911yescharacteristic
472C4#301311yescharacteristic
473C4#336711yescharacteristic
474C4#338411yescharacteristic
475C2—162no—
476C2#2432611yescharacteristic
477C2#3352011yescharacteristic
478C2#3352711yescharacteristic
479C2#3393511yescharacteristic
480C2#3394411yescharacteristic
481C2#3395311yescharacteristic
482C2#3395511yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3383C2semidirect
5. #13453C2semidirect
6. #13451C2semidirect
9. #13284C2semidirect
10. #13264C2semidirect
13. #12938C2semidirect
14. #12936C2semidirect
15. #12812C2semidirect
82. C2×SD32Q8semidirect
308. Q16C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#33832
C2×#22282
C2×#21752
#282632
#134532
#134512
#133652
#133632
#132842
#132642
#130132
#129382
#129362
#128122