Finite Group Databasebeta

← #496963

Subgroups — #496963

Order 512.

656 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2210C211yesmaximal, characteristic
2C4×#495C211yesmaximal, characteristic, complemented
3#21275C211yesmaximal, characteristic, complemented
4#13254C211yesmaximal, characteristic, complemented
5#13239C211yesmaximal, characteristic
6#13238C211yesmaximal, characteristic, complemented
7#13178C211yesmaximal, characteristic, complemented
8#13150C211yesmaximal, characteristic, complemented
9#13149C211yesmaximal, characteristic, complemented
10#13013C211yesmaximal, characteristic, complemented
11#13012C211yesmaximal, characteristic
12#12938C211yesmaximal, characteristic, complemented
13#12935C211yesmaximal, characteristic, complemented
14#12813C211yesmaximal, characteristic, complemented
15#12653C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C22×D16C2211yescharacteristic
18C2×#495C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×C4×SD16—21no—
21C2×C4×D8C2211yescharacteristic
22C2×C4×D8C2211yescharacteristic
23C4×#124—21no—
24#2293—21no—
25#2210C418yescomplemented
26C2×#367C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×C4⋊C16C2211yescharacteristic
29C2×C4⋊C16C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#346C2211yescharacteristic
39C2×#346C2211yescharacteristic
40#1970C2211yescharacteristic
41#1970C2211yescharacteristic
42#1970C2211yescharacteristic
43#1969C2211yescharacteristic
44#1968—21no—
45#1958—21no—
46#1942C2211yescharacteristic
47#1942C2211yescharacteristic
48#1934—21no—
49#1934—21no—
50#1846C2211yescharacteristic
51C4×#107C2211yescharacteristic
52C4×#102—21no—
53C42×C8C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60C22×Q16C2311yescharacteristic
61C22×SD16—21no—
62C22×D8C2311yescharacteristic
63C22×D8C2311yescharacteristic
64C2×#124—21no—
65C2×C4×Q8—21no—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×#117—41no—
69C2×SD32—41no—
70C2×SD32—41no—
71C2×SD32C2×C414yescomplemented
72C2×D16—41no—
73C2×D16—41no—
74C2×D16C2×C414yes—
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78C22×C16C2311yescharacteristic
79#495—22no—
80#495C2×C414yes—
81C4×Q16—41no—
82C4×Q16—41no—
83C4×SD16—44no—
84C4×D8—22no—
85C4×D8—41no—
86C4×D8—41no—
87C4×D8C2×C414yescomplemented
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×#106—21no—
94C2×#103—21no—
95C2×#103—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×C4×C8D411yescharacteristic
102C2×C4×C8D411yescharacteristic
103C2×C4×C8D411yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#391—21no—
108#390—21no—
109#390—21no—
110#390—21no—
111#389—21no—
112#389—21no—
113#388—41no—
114#388—41no—
115#388—41no—
116#388—41no—
117#382—21no—
118C4×C4⋊C4—21no—
119C4×C22⋊C4—41no—
120C4×C22⋊C4—41no—
121C4×C22⋊C4—41no—
122C43D411yescharacteristic
123#367—41no—
124#366—41no—
125C4⋊C16—22no—
126C4⋊C16C2×C414yescomplemented
127#358—41no—
128#358—41no—
129#358C2×C414yescomplemented
130#357—41no—
131#357—41no—
132#357—41no—
133#357—41no—
134#357C2×C414yes—
135#346—41no—
136#346—41no—
137#340—21no—
138#340—21no—
139#340—21no—
140C22×Q8—21no—
141C22×D4—21no—
142C22×D4—21no—
143C23×C4—41no—
144C23×C4—41no—
145C2×Q16—21no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16C22×C412yes—
151C2×SD16—24no—
152C2×SD16—42no—
153C2×SD16—42no—
154C2×D8—21no—
155C2×D8—21no—
156C2×D8—21no—
157C2×D8—21no—
158C2×D8—21no—
159C2×D8—21no—
160C2×D8—21no—
161C2×D8—21no—
162C2×D8—21no—
163C2×D8—21no—
164C2×D8C22×C412yes—
165C2×D8C22×C412yes—
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2411yescharacteristic
170#124—24no—
171#124—42no—
172C4×Q8—42no—
173C4×Q8—42no—
174C4×D4—24no—
175C4×D4—42no—
176C4×D4—42no—
177C4×D4—42no—
178#117—84no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C42—41no—
198C2×C42—41no—
199C2×C42D811yescharacteristic
200C2×C42D811yescharacteristic
201C2×C42D811yescharacteristic
202C2×C42D811yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206SD32—44no—
207D16—44no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16—21no—
212C2×C16—21no—
213C2×C16—21no—
214C2×C16—21no—
215C2×C16—21no—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16C22×C412yes—
219C2×C16C22×C412yes—
220#107—21no—
221#107—21no—
222#107—21no—
223#107—21no—
224#107—21no—
225#107—21no—
226#107—21no—
227#107—21no—
228#107—21no—
229#107C22×C412yes—
230#106—42no—
231#103—42no—
232#103—42no—
233#102—24no—
234#102—42no—
235#102—42no—
236#102—42no—
237#102—42no—
238C4×C8—22no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—22no—
244C4×C8—41no—
245C4×C8C22×C412yes—
246C4×C8#4112yes—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251#95—41no—
252#95—41no—
253#95—41no—
254C24—41no—
255C24—41no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—44no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4C2×D811yescharacteristic
287C22×C4C2×D811yescharacteristic
288C22×C4C2×D811yescharacteristic
289C22×C4C2×D811yescharacteristic
290C22×C4C2×D811yescharacteristic
291C22×C4C2×D811yescharacteristic
292C22×C4C22×D411yescharacteristic
293Q16—22no—
294Q16—22no—
295Q16—41no—
296Q16—41no—
297SD16—48no—
298D8—22no—
299D8—22no—
300D8—22no—
301D8—22no—
302D8—41no—
303D8—41no—
304D8—41no—
305D8—41no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8C4×D412yes—
314C2×C8AGL(1,ℤ/8)12yes—
315C2×C8C23×C411yescharacteristic
316C2×C8C2×#4111yescharacteristic
317C2×C8H(5,𝔽2)11yescharacteristic
318C2×C8H(5,𝔽2)11yescharacteristic
319C2×C8H(5,𝔽2)11yescharacteristic
320C2×C82−1+411yescharacteristic
321C4⋊C4—22no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—82no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C22⋊C4—82no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C42—21no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42—22no—
349C42—41no—
350C42—41no—
351C42—41no—
352C42—82no—
353C42—82no—
354C42C4×D412yes—
355C42#13512yes—
356C16—22no—
357C16—22no—
358C16—41no—
359C16—41no—
360C23—42no—
361C23—42no—
362C23—81no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23C22×D811yescharacteristic
375Q8—42no—
376Q8—42no—
377Q8—81no—
378Q8—81no—
379D4—42no—
380D4—42no—
381D4—42no—
382D4—42no—
383D4—81no—
384D4—81no—
385D4—81no—
386D4—81no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
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—42no—
403C2×C4—42no—
404C2×C4—88no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
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—164no—
422C2×C4C4×D812yes—
423C2×C4C4×D812yes—
424C2×C4#50912yes—
425C2×C4C2×C4×D411yescharacteristic
426C2×C4#53511yescharacteristic
427C2×C4#53711yescharacteristic
428C2×C4C2×#13511yescharacteristic
429C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
430C2×C4#57511yescharacteristic
431C8—22no—
432C8—22no—
433C8—41no—
434C8#51811yescharacteristic
435C8#52011yescharacteristic
436C8#54011yescharacteristic
437C8#54211yescharacteristic
438C22—82no—
439C22—82no—
440C22—82no—
441C22—82no—
442C22—82no—
443C22—82no—
444C22—161no—
445C22—161no—
446C22—161no—
447C22—161no—
448C22—161no—
449C22—161no—
450C22—161no—
451C22—161no—
452C22C2×C4×D811yescharacteristic
453C22#320511yescharacteristic
454C22#324511yescharacteristic
455C22C2×#50911yescharacteristic
456C22#341411yescharacteristic
457C22#341411yescharacteristic
458C22#341411yescharacteristic
459C4—22no—
460C4—22no—
461C4—22no—
462C4—41no—
463C4—41no—
464C4—41no—
465C4—82no—
466C4—82no—
467C4—161no—
468C4—161no—
469C4—161no—
470C4—161no—
471C4#232411yescharacteristic
472C4C4×AGL(1,ℤ/8)11yescharacteristic
473C4#321211yescharacteristic
474C4#324711yescharacteristic
475C2—162no—
476C2—162no—
477C2#2214511yescharacteristic
478C2#3345111yescharacteristic
479C2C4×#50911yescharacteristic
480C2#3377911yescharacteristic
481C2#3378511yescharacteristic
482C2#3381711yescharacteristic
483C2#3382211yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×D16C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#495C2semidirect
3. #21275C2semidirect
4. #13254C2semidirect
6. #13238C2semidirect
7. #13178C2semidirect
8. #13150C2semidirect
9. #13149C2semidirect
10. #13013C2semidirect
12. #12938C2semidirect
13. #12935C2semidirect
14. #12813C2semidirect
15. #12653C2semidirect
25. #2210C4semidirect
71. C2×SD32C2×C4semidirect
87. C4×D8C2×C4semidirect
126. C4⋊C16C2×C4semidirect
129. #358C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22102
C4×#4952
#212752
#132542
#132392
#132382
#131782
#131502
#131492
#130132
#130122
#129382
#129352
#128132
#126532