Finite Group Databasebeta

← #500875

Subgroups — #500875

Order 512.

775 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3164C211yesmaximal, characteristic
2C2×#2229C211yesmaximal, characteristic, complemented
3C2×#2225C211yesmaximal, characteristic, complemented
4#21525C211yesmaximal, characteristic
5#13453C211yesmaximal, characteristic, complemented
6#13452C211yesmaximal, characteristic, complemented
7#13366C211yesmaximal, characteristic, complemented
8#13364C211yesmaximal, characteristic, complemented
9#13277C211yesmaximal, characteristic, complemented
10#13276C211yesmaximal, characteristic, complemented
11#12994C211yesmaximal, characteristic
12#12993C211yesmaximal, characteristic
13#12921C211yesmaximal, characteristic
14#12920C211yesmaximal, characteristic
15#12784C211yesmaximal, characteristic
16C2×#544—21no—
17#3164—24no—
18#3164—24no—
19C2×#501—21no—
20C2×#501—21no—
21C2×#500C2211yescharacteristic
22C2×#500C2211yescharacteristic
23C2×#499C2211yescharacteristic
24C2×#499C2211yescharacteristic
25C2×#467C2211yescharacteristic
26C2×C4×Q16C2211yescharacteristic
27C2×#428—21no—
28C2×#427C2211yescharacteristic
29C22×#107C2211yescharacteristic
30C2×#405C2211yescharacteristic
31#2323—21no—
32#2229—24no—
33#2225—24no—
34C2×#367C2211yescharacteristic
35C2×#367C2211yescharacteristic
36C2×#367C2211yescharacteristic
37C2×#366C2211yescharacteristic
38C2×C4⋊C16C2211yescharacteristic
39C2×C4⋊C16C2211yescharacteristic
40C2×#358C2211yescharacteristic
41C2×#358C2211yescharacteristic
42C2×#358C2211yescharacteristic
43C2×#358C2211yescharacteristic
44C2×C22⋊C16C2211yescharacteristic
45C2×C22⋊C16C2211yescharacteristic
46#1969C2211yescharacteristic
47#1942C2211yescharacteristic
48#1935—21no—
49#1894—21no—
50#1808C2211yescharacteristic
51C4×#107C2211yescharacteristic
52#1763—21no—
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61#1374C2211yescharacteristic
62#1374C2211yescharacteristic
63C22×Q16C2311yescharacteristic
64C22×M16D411yescharacteristic
65C23×C8C2311yescharacteristic
66#544—28no—
67#544—44no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8—41no—
71C2×#125—21no—
72C2×#125—21no—
73C2×#125—41no—
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8—41no—
76C2×C22⋊Q8—41no—
77C2×C4×Q8—21no—
78C2×#117—21no—
79C2×#117D411yescharacteristic
80C22×C4⋊C4—21no—
81C22×C16C2311yescharacteristic
82C22×C16C2311yescharacteristic
83C22×C16C2311yescharacteristic
84C22×C16C2311yescharacteristic
85#501—28no—
86#501—28no—
87#500—22no—
88#500—22no—
89#500—22no—
90#500D414yes—
91#499—22no—
92#499—22no—
93#499—22no—
94#499D414yes—
95#467—22no—
96#467Q814yescomplemented
97C4×Q16—22no—
98C4×Q16—22no—
99#428—28no—
100#427—22no—
101#427—22no—
102C2×#107—21no—
103C2×#107—21no—
104C2×#107—21no—
105C2×#107—22no—
106C2×#107—22no—
107C2×#107C2311yescharacteristic
108C2×#107C2311yescharacteristic
109C2×#107C2311yescharacteristic
110C2×#107C2311yescharacteristic
111C2×#107C2311yescharacteristic
112C2×#107C2311yescharacteristic
113C2×#107C2311yescharacteristic
114C2×#106—21no—
115C2×#106—21no—
116C2×#106—21no—
117C2×#103—21no—
118C2×#103—21no—
119C2×#103—21no—
120#405—22no—
121#405—22no—
122C2×#97D411yescharacteristic
123C2×#97D411yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126#391—21no—
127#389—21no—
128#389—41no—
129#388—41no—
130#388—41no—
131#387—41no—
132#380—21no—
133C4×C4⋊C4—21no—
134#367—22no—
135#367—22no—
136#367—22no—
137#366—22no—
138C4⋊C16—22no—
139C4⋊C16—22no—
140#358—22no—
141#358—22no—
142#358—22no—
143#358—22no—
144C22⋊C16—22no—
145C22⋊C16—22no—
146#340—21no—
147#340—21no—
148#340—21no—
149C22×Q8—21no—
150C22×Q8—41no—
151C22×Q8—41no—
152C23×C4—41no—
153C23×C4C2×D411yescharacteristic
154C2×Q16—21no—
155C2×Q16—21no—
156C2×Q16—21no—
157C2×Q16—21no—
158C2×Q16—21no—
159C2×Q16C2×Q812yes—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×Q812yes—
171C22×C8#4112yes—
172C22×C8C2411yescharacteristic
173C4⋊Q8—24no—
174C4⋊Q8—24no—
175C4⋊Q8—42no—
176C4⋊Q8—48no—
177C4⋊Q8—42no—
178#125—24no—
179#125—24no—
180#125—42no—
181#125—48no—
182#125—42no—
183C22⋊Q8—24no—
184C22⋊Q8—48no—
185C22⋊Q8—48no—
186C22⋊Q8—42no—
187C4×Q8—42no—
188C4×Q8—42no—
189#117—22no—
190#117—22no—
191#117—24no—
192#117—42no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—42no—
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⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4D811yescharacteristic
213C2×C4⋊C4D811yescharacteristic
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4D811yescharacteristic
218C2×C22⋊C4D811yescharacteristic
219C2×C42—41no—
220C2×C42—41no—
221C2×C42—41no—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
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×C16—21no—
236#107—24no—
237#107—24no—
238#107—24no—
239#107—21no—
240#107—21no—
241#107—21no—
242#107—21no—
243#107—21no—
244#107—21no—
245#107—21no—
246#107—21no—
247#107—21no—
248#107C2×D412yes—
249#107C2×D412yes—
250#107C2×D412yes—
251#107C2×D412yes—
252#107C2×Q812yes—
253#106—24no—
254#106—24no—
255#106—24no—
256#103—24no—
257#103—42no—
258#103—42no—
259#97—22no—
260#97—22no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8C2×D412yes—
264C4×C8C2×D412yes—
265#95—41no—
266#95—41no—
267#95—41no—
268#95—41no—
269#95—41no—
270C24C2×D811yescharacteristic
271C2×Q8—22no—
272C2×Q8—44no—
273C2×Q8—41no—
274C2×Q8—44no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—161no—
280C2×Q8—161no—
281C2×Q8—161no—
282C2×Q8—161no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—44no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—161no—
297C22×C4—161no—
298C22×C4C22⋊Q812yes—
299C22×C4C2×D811yescharacteristic
300C22×C4C2×D811yescharacteristic
301C22×C4C2×D811yescharacteristic
302C22×C4C2×D811yescharacteristic
303C22×C4C2×D811yescharacteristic
304C22×C4#13712yes—
305C22×C4C22×D411yescharacteristic
306Q16—22no—
307Q16—22no—
308Q16—41no—
309Q16—41no—
310M16—28no—
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#12012yes—
322C2×C8C22⋊Q812yes—
323C2×C8#13712yes—
324C2×C8C22×D411yescharacteristic
325C2×C8C22×D411yescharacteristic
326C2×C8C22×Q811yescharacteristic
327C2×C8C2×#4111yescharacteristic
328C2×C82−1+411yescharacteristic
329C2×C82−1+411yescharacteristic
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—44no—
338C4⋊C4—44no—
339C4⋊C4—44no—
340C4⋊C4—44no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—44no—
347C4⋊C4—44no—
348C4⋊C4—44no—
349C4⋊C4—44no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—82no—
355C4⋊C4—161no—
356C22⋊C4—22no—
357C22⋊C4—22no—
358C22⋊C4—44no—
359C22⋊C4—44no—
360C22⋊C4—161no—
361C42—21no—
362C42—21no—
363C42—44no—
364C42—82no—
365C42—161no—
366C42#12012yes—
367C42AGL(1,ℤ/8)12yes—
368C16—41no—
369C16—41no—
370C16—41no—
371C16—41no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23#47412yes—
379C23C22×D811yescharacteristic
380Q8—42no—
381Q8—42no—
382Q8—81no—
383Q8—81no—
384Q8—164no—
385Q8—164no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
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—164no—
427C2×C4#44712yes—
428C2×C4#44712yes—
429C2×C4#47412yes—
430C2×C4C2×#12011yescharacteristic
431C2×C4C2×C22⋊Q811yescharacteristic
432C2×C4#53611yescharacteristic
433C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
434C2×C4C2×#13711yescharacteristic
435C2×C4C2×#13711yescharacteristic
436C8—22no—
437C8—22no—
438C8—22no—
439C8#54811yescharacteristic
440C8#54811yescharacteristic
441C8D4×Q811yescharacteristic
442C8D4×Q811yescharacteristic
443C22—22no—
444C22—22no—
445C22—22no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22C2×#44711yescharacteristic
451C22C2×#47411yescharacteristic
452C22#322611yescharacteristic
453C22#341611yescharacteristic
454C22#341611yescharacteristic
455C22#341611yescharacteristic
456C22#341711yescharacteristic
457C4—22no—
458C4—22no—
459C4—22no—
460C4—41no—
461C4—41no—
462C4—82no—
463C4—82no—
464C4—82no—
465C4—82no—
466C4—82no—
467C4—82no—
468C4—161no—
469C4—161no—
470C4#240911yescharacteristic
471C4#300411yescharacteristic
472C4#307911yescharacteristic
473C4#323911yescharacteristic
474C2—42no—
475C2#2429811yescharacteristic
476C2#3352311yescharacteristic
477C2#3352711yescharacteristic
478C2#3363211yescharacteristic
479C2#3363411yescharacteristic
480C2#3380811yescharacteristic
481C2#3381011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2229C2semidirect
3. C2×#2225C2semidirect
5. #13453C2semidirect
6. #13452C2semidirect
7. #13366C2semidirect
8. #13364C2semidirect
9. #13277C2semidirect
10. #13276C2semidirect
96. #467Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#31642
C2×#22292
C2×#22252
#215252
#134532
#134522
#133662
#133642
#132772
#132762
#129942
#129932
#129212
#129202
#127842