Finite Group Databasebeta

← #504212

Subgroups — #504212

Order 512.

510 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26061C211yesmaximal, characteristic, complemented
2#23018C211yesmaximal, characteristic
3#13542C211yesmaximal, characteristic, complemented
4#13521C211yesmaximal, characteristic, complemented
5#13467C211yesmaximal, characteristic, complemented
6#13371C211yesmaximal, characteristic, complemented
7#13272C211yesmaximal, characteristic, complemented
8#13116C211yesmaximal, characteristic, complemented
9#13065C211yesmaximal, characteristic
10#13002C211yesmaximal, characteristic
11#12950C211yesmaximal, characteristic, complemented
12#12928C211yesmaximal, characteristic, complemented
13#12914C211yesmaximal, characteristic, complemented
14#12824C211yesmaximal, characteristic, complemented
15#12563C211yesmaximal, characteristic, complemented
16C22×M32C2211yescharacteristic
17C2×#498C2211yescharacteristic
18#3050C2211yescharacteristic
19C2×#443C2211yescharacteristic
20#2933—21no—
21C22×#108C2211yescharacteristic
22C2×#427C2211yescharacteristic
23#2439—21no—
24C2×#365C2211yescharacteristic
25C2×#365C2211yescharacteristic
26C2×#364C2211yescharacteristic
27C2×#364C2211yescharacteristic
28C2×#360C2211yescharacteristic
29C2×#360C2211yescharacteristic
30C2×#360C2211yescharacteristic
31C2×#360C2211yescharacteristic
32#2112C2211yescharacteristic
33#2085—21no—
34#2057—21no—
35#1974C2211yescharacteristic
36#1945C2211yescharacteristic
37#1941C2211yescharacteristic
38#1941C2211yescharacteristic
39#1929—21no—
40#1871—21no—
41#1866—21no—
42#1854—21no—
43#1846C2211yescharacteristic
44#1807C2211yescharacteristic
45#1772C2211yescharacteristic
46C8×C22⋊C4C2211yescharacteristic
47#1386C2211yescharacteristic
48#1386C2211yescharacteristic
49#1386C2211yescharacteristic
50#1386C2211yescharacteristic
51#1383C2211yescharacteristic
52#1383C2211yescharacteristic
53#1383C2211yescharacteristic
54#1383C2211yescharacteristic
55#1383C2211yescharacteristic
56#1383C2211yescharacteristic
57#1376C2211yescharacteristic
58#1376C2211yescharacteristic
59C2×#135C2311yescharacteristic
60C2×#131—21no—
61C22×M16—21no—
62C23×C8C2311yescharacteristic
63C2×C4⋊Q8—21no—
64#525—21no—
65C2×#117—21no—
66C2×M32—21no—
67C2×M32—41no—
68C2×M32—41no—
69C2×M32C2311yescharacteristic
70C2×M32C2311yescharacteristic
71C2×M32C2311yescharacteristic
72C2×M32C2311yescharacteristic
73C22×C16—21no—
74#498—22no—
75#498—41no—
76#467Q811yescharacteristic, complemented
77#467Q811yescharacteristic, complemented
78#466Q811yescharacteristic, complemented
79#466Q811yescharacteristic, complemented
80#465—21no—
81#465—21no—
82#443—22no—
83#443—41no—
84#436—41no—
85#435—41no—
86#435—41no—
87C8×D4—41no—
88C2×#108—22no—
89C2×#108—21no—
90C2×#108—41no—
91C2×#108C2311yescharacteristic
92C2×#108C2311yescharacteristic
93#427—22no—
94#427—41no—
95#427C2311yescharacteristic
96C2×#107—21no—
97C2×#107—21no—
98C2×#106—21no—
99C2×#106C2311yescharacteristic
100C2×#106C2311yescharacteristic
101C2×#106C2311yescharacteristic
102C2×#106C2311yescharacteristic
103#424—21no—
104C2×#104—21no—
105C2×#104—21no—
106#416—21no—
107#408—21no—
108C2×C22⋊C8D411yescharacteristic
109#405—21no—
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112#396—41no—
113#393—41no—
114#391—41no—
115#389—21no—
116#389—21no—
117#387—41no—
118#382—21no—
119#380—21no—
120C4×C22⋊C4D411yescharacteristic
121#365—22no—
122#365—41no—
123#364—41no—
124#364—41no—
125#360—22no—
126#360—41no—
127#360—41no—
128#360—41no—
129C22⋊C16—21no—
130C22⋊C16—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134#339—21no—
135#339—21no—
136#336D411yescharacteristic
137#336D411yescharacteristic
138C2×#41—21no—
139C22×Q8—41no—
140C23×C4C2×D411yescharacteristic
141#135—21no—
142#135—21no—
143#135—21no—
144#135—21no—
145C2×Q16C2×Q811yescharacteristic
146C2×SD16—21no—
147C2×D8C2×Q811yescharacteristic
148#131—42no—
149#131—81no—
150C2×M16—21no—
151C2×M16—21no—
152C2×M16—41no—
153C2×M16—41no—
154C2×M16—42no—
155C2×M16—81no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—41no—
161C22×C8—41no—
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×Q811yescharacteristic
166C22×C8C2×Q811yescharacteristic
167C22×C8#4111yescharacteristic
168C22×C8#4111yescharacteristic
169C22×C8C2411yescharacteristic
170C4⋊Q8—42no—
171C4⋊Q8—81no—
172#123—41no—
173C22⋊Q8—21no—
174C22⋊Q8—21no—
175#121—21no—
176#121—21no—
177#120—41no—
178C4×D4—41no—
179C4×D4—41no—
180#117—21no—
181#117—42no—
182#117—81no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C42—41no—
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196M32—21no—
197M32—21no—
198M32—21no—
199M32—21no—
200M32—21no—
201M32—21no—
202M32—21no—
203M32—21no—
204M32—42no—
205C2×C16—21no—
206C2×C16—21no—
207C2×C16—21no—
208C2×C16—21no—
209C2×C16—42no—
210#108—21no—
211#108—21no—
212#108—21no—
213#108—42no—
214#108#4112yes—
215#107—42no—
216#107—81no—
217#107C2×Q811yescharacteristic
218#107C2×Q811yescharacteristic
219#106—21no—
220#106—21no—
221#106—21no—
222#106—21no—
223#106—21no—
224#106—21no—
225#106—21no—
226#106—21no—
227#106—42no—
228#106#4112yes—
229C4⋊C8—41no—
230C4⋊C8—41no—
231#104—42no—
232#104—81no—
233#103—21no—
234#103—21no—
235#102—21no—
236#102—21no—
237C22⋊C8—41no—
238C22⋊C8—41no—
239C22⋊C8SD1611yescharacteristic
240C22⋊C8SD1611yescharacteristic
241C22⋊C8SD1611yescharacteristic
242C22⋊C8SD1611yescharacteristic
243#97—41no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—41no—
249#95—21no—
250#95—41no—
251#95—41no—
252#95—41no—
253#95—41no—
254#95—41no—
255C24AGL(1,ℤ/8)11yescharacteristic
256#41—41no—
257#41—41no—
258#41—41no—
259#41—41no—
260C2×Q8—21no—
261C2×Q8—82no—
262C2×Q8—82no—
263C2×Q8—161no—
264C2×D4—21no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4C22⋊Q811yescharacteristic
280C22×C4C22⋊Q811yescharacteristic
281C22×C4C2×SD1611yescharacteristic
282C22×C4C2×SD1611yescharacteristic
283C22×C4AGL(1,ℤ/8)11yescharacteristic
284C22×C4C22×D411yescharacteristic
285Q16—21no—
286Q16—21no—
287SD16—41no—
288SD16—41no—
289D8—21no—
290D8—21no—
291M16—22no—
292M16—41no—
293M16—41no—
294M16—41no—
295M16—82no—
296M16—82no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—82no—
314C2×C8—82no—
315C2×C8C22⋊Q811yescharacteristic
316C2×C8C22⋊Q811yescharacteristic
317C2×C8C2×SD1611yescharacteristic
318C2×C8C2×SD1611yescharacteristic
319C2×C8C2×SD1611yescharacteristic
320C2×C8C2×SD1611yescharacteristic
321C2×C8C22×Q811yescharacteristic
322C2×C8C2×#4111yescharacteristic
323C2×C8C2×#4111yescharacteristic
324C2×C8C2×#4111yescharacteristic
325C2×C8H(5,𝔽2)11yescharacteristic
326C2×C8H(5,𝔽2)11yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—22no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—82no—
340C4⋊C4—82no—
341C4⋊C4—161no—
342C22⋊C4—42no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—82no—
349C22⋊C4—161no—
350C42—21no—
351C42—21no—
352C42—21no—
353C42—21no—
354C42—41no—
355C42—82no—
356C16—41no—
357C16—41no—
358C16—41no—
359C16—41no—
360C23—21no—
361C23—21no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23#47611yescharacteristic
366C23#47611yescharacteristic
367C23C2×AGL(1,ℤ/8)11yescharacteristic
368Q8—41no—
369Q8—41no—
370Q8—164no—
371D4—41no—
372D4—41no—
373D4—82no—
374D4—81no—
375D4—81no—
376D4—161no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4#47611yescharacteristic
411C2×C4#47611yescharacteristic
412C2×C4C2×C22⋊Q811yescharacteristic
413C2×C4#54611yescharacteristic
414C2×C4#54611yescharacteristic
415C2×C4C22×SD1611yescharacteristic
416C2×C4#57611yescharacteristic
417C2×C4#57811yescharacteristic
418C8—21no—
419C8—21no—
420C8—21no—
421C8—21no—
422C8—21no—
423C8—21no—
424C8—42no—
425C8—81no—
426C8#52911yescharacteristic
427C8#53811yescharacteristic
428C8#55411yescharacteristic
429C8#55411yescharacteristic
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22C2×#47611yescharacteristic
444C22#330811yescharacteristic
445C22#331811yescharacteristic
446C4—21no—
447C4—21no—
448C4—42no—
449C4—81no—
450C4—82no—
451C4—81no—
452C4—81no—
453C4—81no—
454C4—161no—
455C4—161no—
456C4—161no—
457C4#245011yescharacteristic
458C4#308311yescharacteristic
459C4#329511yescharacteristic
460C4#329811yescharacteristic
461C2—21no—
462C2—21no—
463C2—41no—
464C2—41no—
465C2—161no—
466C2#2641011yescharacteristic
467C2#3363711yescharacteristic
468C2#3363811yescharacteristic
subgroupG/H
center, socleC22C2×#476
commutator, Frattini, cosocleC22×C8C24
prime radical#3050C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #26061C2semidirect
3. #13542C2semidirect
4. #13521C2semidirect
5. #13467C2semidirect
6. #13371C2semidirect
7. #13272C2semidirect
8. #13116C2semidirect
11. #12950C2semidirect
12. #12928C2semidirect
13. #12914C2semidirect
14. #12824C2semidirect
15. #12563C2semidirect
76. #467Q8semidirect
77. #467Q8semidirect
78. #466Q8semidirect
79. #466Q8semidirect

Maximal subgroups

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

subgroupindex
#260612
#230182
#135422
#135212
#134672
#133712
#132722
#131162
#130652
#130022
#129502
#129282
#129142
#128242
#125632