Finite Group Databasebeta

← #500874

Subgroups — #500874

Order 512.

803 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3164C211yesmaximal, characteristic, complemented
2C2×#2228C211yesmaximal, characteristic, complemented
3C2×#2226C211yesmaximal, characteristic, complemented
4#21524C211yesmaximal, characteristic
5#13453C211yesmaximal, characteristic, complemented
6#13452C211yesmaximal, characteristic, complemented
7#13365C211yesmaximal, characteristic, complemented
8#13363C211yesmaximal, characteristic, complemented
9#13277C211yesmaximal, characteristic, complemented
10#13275C211yesmaximal, characteristic, complemented
11#12992C211yesmaximal, characteristic
12#12991C211yesmaximal, characteristic
13#12921C211yesmaximal, characteristic, complemented
14#12920C211yesmaximal, characteristic, complemented
15#12784C211yesmaximal, characteristic, complemented
16C2×#544—21no—
17#3164—24no—
18#3164—24no—
19C2×#501—21no—
20C2×#501—21no—
21C2×#500C2211yescharacteristic
22C2×#500C2211yescharacteristic
23C2×#499C2211yescharacteristic
24C2×#499C2211yescharacteristic
25C2×#466C2211yescharacteristic
26C2×C4×D8C2211yescharacteristic
27C2×#428—21no—
28C2×#427C2211yescharacteristic
29C22×#107C2211yescharacteristic
30C2×#405C2211yescharacteristic
31#2324—21no—
32#2228—24no—
33#2226—24no—
34C2×#367C2211yescharacteristic
35C2×#367C2211yescharacteristic
36C2×#367C2211yescharacteristic
37C2×#366C2211yescharacteristic
38C2×C4⋊C16C2211yescharacteristic
39C2×C4⋊C16C2211yescharacteristic
40C2×#357C2211yescharacteristic
41C2×#357C2211yescharacteristic
42C2×#357C2211yescharacteristic
43C2×#357C2211yescharacteristic
44C2×C22⋊C16C2211yescharacteristic
45C2×C22⋊C16C2211yescharacteristic
46#1970C2211yescharacteristic
47#1942C2211yescharacteristic
48#1936—21no—
49#1895—21no—
50#1808C2211yescharacteristic
51C4×#107C2211yescharacteristic
52#1762—21no—
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61#1374C2211yescharacteristic
62#1374C2211yescharacteristic
63C22×D8C2311yescharacteristic
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—41no—
75C2×C22⋊Q8—41no—
76C2×#121—21no—
77C2×C4×D4—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#500D414yescomplemented
91#499—22no—
92#499—22no—
93#499—22no—
94#499D414yescomplemented
95#466—22no—
96#466Q814yescomplemented
97C4×D8—22no—
98C4×D8—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×#102—21no—
118C2×#102—21no—
119C2×#102—21no—
120#405—22no—
121#405—22no—
122C2×#97D411yescharacteristic
123C2×#97D411yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126#390—21no—
127#390—41no—
128#389—21no—
129#388—41no—
130#386—41no—
131#385—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#357—22no—
141#357—22no—
142#357—22no—
143#357—22no—
144C22⋊C16—22no—
145C22⋊C16—22no—
146#340—21no—
147#340—21no—
148#340—21no—
149C22×Q8—41no—
150C22×Q8—41no—
151C22×D4—21no—
152C22×D4—41no—
153C23×C4—41no—
154C23×C4—41no—
155C23×C4C2×D411yescharacteristic
156C2×D8—21no—
157C2×D8—21no—
158C2×D8—21no—
159C2×D8—21no—
160C2×D8—21no—
161C2×D8C2×Q812yes—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×Q812yes—
173C22×C8#4112yes—
174C22×C8C2411yescharacteristic
175C4⋊Q8—24no—
176C4⋊Q8—24no—
177C4⋊Q8—42no—
178C4⋊Q8—48no—
179C4⋊Q8—42no—
180#125—24no—
181#125—24no—
182#125—42no—
183#125—48no—
184#125—42no—
185C22⋊Q8—48no—
186C22⋊Q8—48no—
187#121—24no—
188#121—42no—
189C4×D4—42no—
190C4×D4—42no—
191#117—22no—
192#117—22no—
193#117—24no—
194#117—42no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—42no—
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⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4D811yescharacteristic
221C2×C22⋊C4D811yescharacteristic
222C2×C42—41no—
223C2×C42—41no—
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
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—
236C2×C16—21no—
237C2×C16—21no—
238#107—24no—
239#107—24no—
240#107—24no—
241#107—21no—
242#107—21no—
243#107—21no—
244#107—21no—
245#107—21no—
246#107—21no—
247#107—21no—
248#107—21no—
249#107—21no—
250#107C2×D412yes—
251#107C2×D412yes—
252#107C2×D412yes—
253#107C2×D412yes—
254#107C2×Q812yes—
255#106—24no—
256#106—24no—
257#106—24no—
258#102—24no—
259#102—42no—
260#102—42no—
261#97—22no—
262#97—22no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8C2×D412yes—
266C4×C8C2×D412yes—
267#95—41no—
268#95—41no—
269#95—41no—
270C24—41no—
271C24C2×D811yescharacteristic
272C2×Q8—44no—
273C2×Q8—44no—
274C2×Q8—161no—
275C2×Q8—161no—
276C2×Q8—161no—
277C2×Q8—161no—
278C2×D4—22no—
279C2×D4—44no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—161no—
286C2×D4—161no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—44no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—82no—
300C22×C4—161no—
301C22×C4—161no—
302C22×C4—161no—
303C22×C4—161no—
304C22×C4C22⋊Q812yes—
305C22×C4C2×D811yescharacteristic
306C22×C4C2×D811yescharacteristic
307C22×C4C2×D811yescharacteristic
308C22×C4C2×D811yescharacteristic
309C22×C4C2×D811yescharacteristic
310C22×C4#13712yes—
311C22×C4C22×D411yescharacteristic
312D8—22no—
313D8—22no—
314D8—41no—
315D8—41no—
316M16—28no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8#12012yes—
328C2×C8C22⋊Q812yes—
329C2×C8#13712yes—
330C2×C8C22×D411yescharacteristic
331C2×C8C22×D411yescharacteristic
332C2×C8C22×Q811yescharacteristic
333C2×C8C2×#4111yescharacteristic
334C2×C82−1+411yescharacteristic
335C2×C82−1+411yescharacteristic
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—44no—
346C4⋊C4—44no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—161no—
360C22⋊C4—22no—
361C22⋊C4—22no—
362C22⋊C4—44no—
363C22⋊C4—44no—
364C22⋊C4—82no—
365C22⋊C4—161no—
366C22⋊C4—161no—
367C22⋊C4—161no—
368C42—21no—
369C42—21no—
370C42—44no—
371C42—161no—
372C42#12012yes—
373C42AGL(1,ℤ/8)12yes—
374C16—41no—
375C16—41no—
376C16—41no—
377C16—41no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—42no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23—81no—
389C23—81no—
390C23—81no—
391C23#47412yes—
392C23C22×D811yescharacteristic
393Q8—164no—
394Q8—164no—
395D4—42no—
396D4—42no—
397D4—81no—
398D4—81no—
399D4—164no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
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—164no—
438C2×C4—164no—
439C2×C4#44712yes—
440C2×C4#44712yes—
441C2×C4#47412yes—
442C2×C4C2×#12011yescharacteristic
443C2×C4C2×C22⋊Q811yescharacteristic
444C2×C4#53611yescharacteristic
445C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
446C2×C4C2×#13711yescharacteristic
447C2×C4C2×#13711yescharacteristic
448C8—22no—
449C8—22no—
450C8—22no—
451C8#54811yescharacteristic
452C8#54811yescharacteristic
453C8D4×Q811yescharacteristic
454C8D4×Q811yescharacteristic
455C22—22no—
456C22—22no—
457C22—22no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—82no—
463C22—82no—
464C22—82no—
465C22—161no—
466C22—161no—
467C22—161no—
468C22—161no—
469C22C2×#44711yescharacteristic
470C22C2×#47411yescharacteristic
471C22#322611yescharacteristic
472C22#341511yescharacteristic
473C22#341611yescharacteristic
474C22#341611yescharacteristic
475C22#341611yescharacteristic
476C4—22no—
477C4—22no—
478C4—22no—
479C4—41no—
480C4—41no—
481C4—82no—
482C4—82no—
483C4—82no—
484C4—82no—
485C4—82no—
486C4—161no—
487C4—161no—
488C4#240911yescharacteristic
489C4#300411yescharacteristic
490C4#307911yescharacteristic
491C4#323911yescharacteristic
492C2—42no—
493C2—162no—
494C2#2429811yescharacteristic
495C2#3352411yescharacteristic
496C2#3352611yescharacteristic
497C2#3363111yescharacteristic
498C2#3363411yescharacteristic
499C2#3380711yescharacteristic
500C2#3381011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
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
1. C2×#3164C2semidirect
2. C2×#2228C2semidirect
3. C2×#2226C2semidirect
5. #13453C2semidirect
6. #13452C2semidirect
7. #13365C2semidirect
8. #13363C2semidirect
9. #13277C2semidirect
10. #13275C2semidirect
13. #12921C2semidirect
14. #12920C2semidirect
15. #12784C2semidirect
90. #500D4semidirect
94. #499D4semidirect
96. #466Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#31642
C2×#22282
C2×#22262
#215242
#134532
#134522
#133652
#133632
#132772
#132752
#129922
#129912
#129212
#129202
#127842