Finite Group Databasebeta

← #497974

Subgroups — #497974

Order 512.

563 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33423C211yesmaximal, characteristic, complemented
2#33398C211yesmaximal, characteristic, complemented
3#20510C211yesmaximal, characteristic
4#13125C211yesmaximal, characteristic, complemented
5#13124C211yesmaximal, characteristic, complemented
6#13106C211yesmaximal, characteristic, complemented
7#13106C211yesmaximal, characteristic, complemented
8#12980C211yesmaximal, characteristic
9#12980C211yesmaximal, characteristic
10#12884C211yesmaximal, characteristic, complemented
11#12787C211yesmaximal, characteristic, complemented
12#12787C211yesmaximal, characteristic, complemented
13#12654C211yesmaximal, characteristic, complemented
14#12615C211yesmaximal, characteristic, complemented
15#12614C211yesmaximal, characteristic, complemented
16C2×#504C2211yescharacteristic
17C2×#504C2211yescharacteristic
18C2×#444C411yescharacteristic
19C2×#443C2211yescharacteristic
20#2943C411yescharacteristic
21C4×#131C2211yescharacteristic
22C4×#131C2211yescharacteristic
23#2164—21no—
24#2164—21no—
25#2163—21no—
26#2162—21no—
27#2151—21no—
28#2151—21no—
29#2151C2211yescharacteristic
30#2151C2211yescharacteristic
31#2150—21no—
32#2150C2211yescharacteristic
33#2150C2211yescharacteristic
34C2×C4⋊C16—21no—
35C2×C4⋊C16C2211yescharacteristic
36C2×C4⋊C16C2211yescharacteristic
37C2×#350C2211yescharacteristic
38C2×#350C2211yescharacteristic
39C2×#350C2211yescharacteristic
40C2×#350C2211yescharacteristic
41#2113C2211yescharacteristic
42#2113C2211yescharacteristic
43#2108C2211yescharacteristic
44#2108C2211yescharacteristic
45C2×#346C2211yescharacteristic
46C2×C4×C16C2211yescharacteristic
47#1778C411yescharacteristic
48#1773C411yescharacteristic
49#1772C2211yescharacteristic
50#1765C411yescharacteristic
51#1765C411yescharacteristic
52#1764C2211yescharacteristic
53#1764C2211yescharacteristic
54#1759C411yescharacteristic
55#1759C411yescharacteristic
56#1726C2211yescharacteristic
57#1379C2211yescharacteristic
58#1379C2211yescharacteristic
59#1379C2211yescharacteristic
60#1379C2211yescharacteristic
61#1376C2211yescharacteristic
62#1376C2211yescharacteristic
63#1376C2211yescharacteristic
64#1376C2211yescharacteristic
65#1374C2211yescharacteristic
66#1374C2211yescharacteristic
67C2×#135C2×C411yescharacteristic
68C2×#131C2×C411yescharacteristic
69C2×#131C2×C411yescharacteristic
70C2×#131C2311yescharacteristic
71C2×#131C2311yescharacteristic
72C4×#41C2×C411yescharacteristic
73C4×#41C2×C411yescharacteristic
74#504—22no—
75#504—22no—
76#504—41no—
77#504—41no—
78C2×M32—21no—
79C2×M32—21no—
80C2×M32C2311yescharacteristic
81C2×M32C2311yescharacteristic
82C2×M32C2311yescharacteristic
83C2×M32C2311yescharacteristic
84C22×C16—21no—
85C22×C16—21no—
86C22×C16C2311yescharacteristic
87C22×C16C2311yescharacteristic
88C22×C16C2311yescharacteristic
89C22×C16C2311yescharacteristic
90#444—24no—
91#443—22no—
92#443—22no—
93#442C811yescharacteristic
94#442C811yescharacteristic
95#441C811yescharacteristic
96#441C811yescharacteristic
97#440C811yescharacteristic
98#440C811yescharacteristic
99#440C811yescharacteristic
100#440C811yescharacteristic
101C2×#108C2×C411yescharacteristic
102C2×#108C2×C411yescharacteristic
103#427C2×C411yescharacteristic
104C2×#104C2×C411yescharacteristic
105C2×#104C2×C411yescharacteristic
106C2×#104C2×C411yescharacteristic
107C2×#104C2×C411yescharacteristic
108#416C2×C411yescharacteristic
109#416C2×C411yescharacteristic
110#405—21no—
111#405—21no—
112#405—21no—
113#405—21no—
114#405C2×C411yescharacteristic
115#405C2×C411yescharacteristic
116#405C2311yescharacteristic
117#405C2311yescharacteristic
118C4×M16—21no—
119C4×M16—21no—
120C4×M16C2×C411yescharacteristic
121C4×M16C2×C411yescharacteristic
122C2×#97C2×C411yescharacteristic
123C2×#97C2×C411yescharacteristic
124C2×#97C2×C411yescharacteristic
125C2×C4×C8—21no—
126C2×C4×C8—21no—
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130#376C2×C411yescharacteristic
131C4⋊C16—21no—
132C4⋊C16—21no—
133C4⋊C16—21no—
134C4⋊C16—21no—
135C4⋊C16—21no—
136C4⋊C16—21no—
137C4⋊C16—21no—
138C4⋊C16—21no—
139C4⋊C16—21no—
140C4⋊C16—21no—
141C4⋊C16D411yescharacteristic
142C4⋊C16D411yescharacteristic
143C4⋊C16D411yescharacteristic
144C4⋊C16D411yescharacteristic
145#350—22no—
146#350—22no—
147#350—41no—
148#350—41no—
149C22⋊C16—21no—
150C22⋊C16—21no—
151C22⋊C16—21no—
152C22⋊C16—21no—
153#346—21no—
154#346—21no—
155#346—41no—
156C4×C16—21no—
157C4×C16—21no—
158C4×C16—41no—
159#341C2×C411yescharacteristic
160#341C2×C411yescharacteristic
161#339C2×C411yescharacteristic
162#339C2×C411yescharacteristic
163#339C2×C411yescharacteristic
164#339C2×C411yescharacteristic
165C2×#41C22×C411yescharacteristic
166C2×#41C22×C411yescharacteristic
167#135—22no—
168#135—22no—
169C2×Q16C2×C811yescharacteristic
170C2×SD16C2×C811yescharacteristic
171C2×SD16C2×C811yescharacteristic
172C2×D8C2×C811yescharacteristic
173#131—21no—
174#131—21no—
175#131—22no—
176#131—22no—
177#131—21no—
178#131—21no—
179#131—21no—
180#131—21no—
181#131—41no—
182#131—41no—
183#131#4112yes—
184#131#4112yes—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16C22×C411yescharacteristic
190C2×M16C22×C411yescharacteristic
191C2×M16C22×C411yescharacteristic
192C2×M16C22×C411yescharacteristic
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8C22×C411yescharacteristic
198C22×C8C22×C411yescharacteristic
199C22×C8C22×C411yescharacteristic
200C22×C8C2411yescharacteristic
201C4×Q8C2×C811yescharacteristic
202C4×Q8C2×C811yescharacteristic
203C4×Q8C2×C811yescharacteristic
204C4×Q8C2×C811yescharacteristic
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4C2×C811yescharacteristic
210C4×D4C2×C811yescharacteristic
211C4×D4C2×C811yescharacteristic
212C4×D4C2×C811yescharacteristic
213#117—21no—
214#117—21no—
215#117C22×C411yescharacteristic
216#117C22×C411yescharacteristic
217C2×C42—21no—
218C2×C42—21no—
219C2×C42C22×C411yescharacteristic
220C2×C42C22×C411yescharacteristic
221C2×C42C22×C411yescharacteristic
222M32—21no—
223M32—21no—
224M32—21no—
225M32—21no—
226M32—21no—
227M32—21no—
228M32—21no—
229M32—21no—
230M32—42no—
231M32—42no—
232C2×C16—21no—
233C2×C16—21no—
234C2×C16—21no—
235C2×C16—21no—
236C2×C16—21no—
237C2×C16—21no—
238C2×C16—21no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—21no—
246C2×C16—42no—
247C2×C16—42no—
248C2×C16C2×D411yescharacteristic
249C2×C16C2×D411yescharacteristic
250C2×C16C2×D411yescharacteristic
251C2×C16C2×D411yescharacteristic
252#108—22no—
253#108—22no—
254#107C2×C811yescharacteristic
255#107C2×C811yescharacteristic
256#106C2×C811yescharacteristic
257#106C2×C811yescharacteristic
258#104—22no—
259#104—22no—
260#104—22no—
261#104M1614yes—
262#103C2×C811yescharacteristic
263#103C2×C811yescharacteristic
264#103C2×C811yescharacteristic
265#103C2×C811yescharacteristic
266#102C2×C811yescharacteristic
267#102C2×C811yescharacteristic
268#102C2×C811yescharacteristic
269#102C2×C811yescharacteristic
270#97—21no—
271#97—21no—
272#97—21no—
273#97—21no—
274#97—21no—
275#97—21no—
276#97—22no—
277#97—41no—
278#97C2×C811yescharacteristic
279#97C2×C811yescharacteristic
280#97C2×C811yescharacteristic
281#97C2×C811yescharacteristic
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8—21no—
287C4×C8—21no—
288C4×C8—21no—
289C4×C8—21no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8C2×D411yescharacteristic
293C4×C8C2×D411yescharacteristic
294C4×C8#4111yescharacteristic
295C4×C8#4111yescharacteristic
296#95—21no—
297#95—21no—
298#41—21no—
299#41—21no—
300#41—21no—
301#41—21no—
302#41—21no—
303#41—21no—
304#41#11712yes—
305#41C2×M1612yes—
306C2×Q8C22×C811yescharacteristic
307C2×Q8C22×C811yescharacteristic
308C2×D4—21no—
309C2×D4—21no—
310C2×D4C22×C811yescharacteristic
311C2×D4C22×C811yescharacteristic
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4#13111yescharacteristic
321C22×C4#13111yescharacteristic
322C22×C4C23×C411yescharacteristic
323Q16—22no—
324SD16—22no—
325SD16—22no—
326D8—22no—
327M16—21no—
328M16—21no—
329M16—22no—
330M16—22no—
331M16—21no—
332M16—21no—
333M16—21no—
334M16—21no—
335M16—42no—
336M16—42no—
337M16—41no—
338M16—41no—
339M16#11712yes—
340M16C2×M1612yes—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8C4×D411yescharacteristic
358C2×C8C4×D411yescharacteristic
359C2×C8C22×C811yescharacteristic
360C2×C8C22×C811yescharacteristic
361C2×C8C22×C811yescharacteristic
362C2×C8C22×C811yescharacteristic
363C2×C8C22×D411yescharacteristic
364C2×C8C2×#4111yescharacteristic
365C2×C8C2×#4111yescharacteristic
366C2×C8C2×#4111yescharacteristic
367C2×C8H(5,𝔽2)11yescharacteristic
368C2×C82−1+411yescharacteristic
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4C22×C811yescharacteristic
374C4⋊C4C22×C811yescharacteristic
375C4⋊C4C22×C811yescharacteristic
376C4⋊C4C22×C811yescharacteristic
377C22⋊C4—21no—
378C22⋊C4—21no—
379C22⋊C4—21no—
380C22⋊C4—21no—
381C22⋊C4—21no—
382C22⋊C4—21no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42C4×D411yescharacteristic
393C42C4×D411yescharacteristic
394C42C22×C811yescharacteristic
395C42C22×C811yescharacteristic
396C42C2×M1612yes—
397C16—21no—
398C16—21no—
399C16—21no—
400C16—21no—
401C16—41no—
402C16—41no—
403C16—41no—
404C16—41no—
405C16—41no—
406C16—41no—
407C23—21no—
408C23—21no—
409C23C2×#13111yescharacteristic
410Q8—21no—
411Q8—21no—
412Q8—21no—
413Q8#43112yes—
414D4—22no—
415D4—22no—
416D4—21no—
417D4—21no—
418D4—21no—
419D4—41no—
420D4—41no—
421D4#43112yes—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4C8×D411yescharacteristic
445C2×C4C8×D411yescharacteristic
446C2×C4C8×D411yescharacteristic
447C2×C4C8×D411yescharacteristic
448C2×C4C2×#11711yescharacteristic
449C2×C4C2×C4×D411yescharacteristic
450C2×C4#52011yescharacteristic
451C2×C4C23×C811yescharacteristic
452C2×C4C22×M1611yescharacteristic
453C2×C4#56811yescharacteristic
454C8—22no—
455C8—21no—
456C8—21no—
457C8—21no—
458C8—21no—
459C8—22no—
460C8—21no—
461C8—21no—
462C8—41no—
463C8—41no—
464C8#54611yescharacteristic
465C8#54811yescharacteristic
466C8#55211yescharacteristic
467C8#55311yescharacteristic
468C22—21no—
469C22—21no—
470C22—22no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22#292511yescharacteristic
477C22C2×C8×D411yescharacteristic
478C22#299011yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—22no—
484C4—42no—
485C4—41no—
486C4—41no—
487C4—41no—
488C4—41no—
489C4—41no—
490C4—81no—
491C4#235211yescharacteristic
492C4C2×#43111yescharacteristic
493C4D4×M1611yescharacteristic
494C4#298811yescharacteristic
495C2—21no—
496C2—21no—
497C2—42no—
498C2—81no—
499C2#2274811yescharacteristic
500C2#3344111yescharacteristic
501C2#3344211yescharacteristic
subgroupG/H
centerC2×C8C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC22×C8C24
socleC22C2×C8×D4
prime radicalC2×#135C2×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C2×C4⋊C16 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33423C2semidirect
2. #33398C2semidirect
4. #13125C2semidirect
5. #13124C2semidirect
6. #13106C2semidirect
7. #13106C2semidirect
10. #12884C2semidirect
11. #12787C2semidirect
12. #12787C2semidirect
13. #12654C2semidirect
14. #12615C2semidirect
15. #12614C2semidirect

Maximal subgroups

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

subgroupindex
#334232
#333982
#205102
#131252
#131242
#131062
#129802
#128842
#127872
#126542
#126152
#126142