Finite Group Databasebeta

← #504211

Subgroups — #504211

Order 512.

535 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26060C211yesmaximal, characteristic, complemented
2#23018C211yesmaximal, characteristic
3#13543C211yesmaximal, 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#13001C211yesmaximal, characteristic
11#12950C211yesmaximal, characteristic, complemented
12#12928C211yesmaximal, characteristic, complemented
13#12914C211yesmaximal, characteristic, complemented
14#12823C211yesmaximal, characteristic, complemented
15C2×#1383C211yesmaximal, characteristic, complemented
16C22×M32C2211yescharacteristic
17C2×#498C2211yescharacteristic
18#3050C2211yescharacteristic
19C2×#443C2211yescharacteristic
20#2933—21no—
21C22×#108C2211yescharacteristic
22C22×#106C2211yescharacteristic
23#2440—21no—
24C2×#365C2211yescharacteristic
25C2×#365C2211yescharacteristic
26C2×#364C2211yescharacteristic
27C2×#364C2211yescharacteristic
28C2×#360C2211yescharacteristic
29C2×#360C2211yescharacteristic
30C2×#360C2211yescharacteristic
31C2×#360C2211yescharacteristic
32#2112C2211yescharacteristic
33#2084—21no—
34#2058—21no—
35#1974C2211yescharacteristic
36#1945C2211yescharacteristic
37#1941C2211yescharacteristic
38#1941C2211yescharacteristic
39#1929—21no—
40#1871—21no—
41#1866—21no—
42#1853—21no—
43#1844C2211yescharacteristic
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×C22⋊Q8—41no—
66C22×C4⋊C4—21no—
67C2×M32—21no—
68C2×M32—41no—
69C2×M32—41no—
70C2×M32C2311yescharacteristic
71C2×M32C2311yescharacteristic
72C2×M32C2311yescharacteristic
73C2×M32C2311yescharacteristic
74C22×C16—21no—
75#498—22no—
76#498—41no—
77#467Q811yescharacteristic, complemented
78#467Q811yescharacteristic, complemented
79#466Q811yescharacteristic, complemented
80#466Q811yescharacteristic, complemented
81#465—21no—
82#465—21no—
83#443—22no—
84#443—41no—
85#436—41no—
86#435—41no—
87#435—41no—
88C8×D4—41no—
89C2×#108—22no—
90C2×#108—21no—
91C2×#108—41no—
92C2×#108C2311yescharacteristic
93C2×#108C2311yescharacteristic
94#427C2311yescharacteristic
95C2×#106—22no—
96C2×#106—21no—
97C2×#106—21no—
98C2×#106—21no—
99C2×#106—41no—
100C2×#106C2311yescharacteristic
101C2×#106C2311yescharacteristic
102C2×#106C2311yescharacteristic
103C2×#106C2311yescharacteristic
104#424—21no—
105C2×#104—21no—
106C2×#104—21no—
107#416—21no—
108#408—21no—
109C2×C22⋊C8D411yescharacteristic
110#405—21no—
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#398—41no—
114#395—41no—
115#389—21no—
116#389—21no—
117#385—41no—
118#384—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×C4—41no—
141C23×C4C2×D411yescharacteristic
142#135—21no—
143#135—21no—
144#135—21no—
145#135—21no—
146C2×Q16C2×Q811yescharacteristic
147C2×SD16—21no—
148C2×D8C2×Q811yescharacteristic
149#131—42no—
150#131—81no—
151C2×M16—21no—
152C2×M16—21no—
153C2×M16—41no—
154C2×M16—41no—
155C2×M16—42no—
156C2×M16—81no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—41no—
162C22×C8—41no—
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×Q811yescharacteristic
167C22×C8C2×Q811yescharacteristic
168C22×C8#4111yescharacteristic
169C22×C8#4111yescharacteristic
170C22×C8C2411yescharacteristic
171C4⋊Q8—42no—
172C4⋊Q8—81no—
173#123—41no—
174C22⋊Q8—21no—
175C22⋊Q8—21no—
176C22⋊Q8—84no—
177#121—21no—
178#121—21no—
179#120—41no—
180C4×D4—41no—
181C4×D4—41no—
182#117—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—81no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—41no—
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199M32—21no—
200M32—21no—
201M32—21no—
202M32—21no—
203M32—21no—
204M32—21no—
205M32—21no—
206M32—21no—
207M32—42no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16—21no—
212C2×C16—42no—
213#108—21no—
214#108—21no—
215#108—21no—
216#108—42no—
217#108#4112yes—
218#107C2×Q811yescharacteristic
219#107C2×Q811yescharacteristic
220#106—21no—
221#106—21no—
222#106—21no—
223#106—21no—
224#106—21no—
225#106—21no—
226#106—21no—
227#106—21no—
228#106—42no—
229#106—42no—
230#106—81no—
231#106#4112yes—
232C4⋊C8—41no—
233C4⋊C8—41no—
234#104—42no—
235#104—81no—
236#103—21no—
237#103—21no—
238#102—21no—
239#102—21no—
240C22⋊C8—41no—
241C22⋊C8—41no—
242C22⋊C8SD1611yescharacteristic
243C22⋊C8SD1611yescharacteristic
244C22⋊C8SD1611yescharacteristic
245C22⋊C8SD1611yescharacteristic
246#97—41no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—41no—
252#95—21no—
253#95—41no—
254#95—41no—
255#95—41no—
256C24AGL(1,ℤ/8)11yescharacteristic
257#41—41no—
258#41—41no—
259#41—41no—
260#41—41no—
261C2×Q8—21no—
262C2×Q8—82no—
263C2×Q8—82no—
264C2×Q8—161no—
265C2×D4—21no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4—161no—
283C22×C4C22⋊Q811yescharacteristic
284C22×C4C22⋊Q811yescharacteristic
285C22×C4C2×SD1611yescharacteristic
286C22×C4C2×SD1611yescharacteristic
287C22×C4AGL(1,ℤ/8)11yescharacteristic
288C22×C4C22×D411yescharacteristic
289Q16—21no—
290Q16—21no—
291SD16—41no—
292SD16—41no—
293D8—21no—
294D8—21no—
295M16—22no—
296M16—41no—
297M16—41no—
298M16—41no—
299M16—82no—
300M16—82no—
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—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—82no—
318C2×C8—82no—
319C2×C8C22⋊Q811yescharacteristic
320C2×C8C22⋊Q811yescharacteristic
321C2×C8C2×SD1611yescharacteristic
322C2×C8C2×SD1611yescharacteristic
323C2×C8C2×SD1611yescharacteristic
324C2×C8C2×SD1611yescharacteristic
325C2×C8C22×Q811yescharacteristic
326C2×C8C2×#4111yescharacteristic
327C2×C8C2×#4111yescharacteristic
328C2×C8C2×#4111yescharacteristic
329C2×C8H(5,𝔽2)11yescharacteristic
330C2×C8H(5,𝔽2)11yescharacteristic
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—22no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—82no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C4⋊C4—82no—
347C4⋊C4—82no—
348C4⋊C4—161no—
349C22⋊C4—24no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—82no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—41no—
361C16—41no—
362C16—41no—
363C16—41no—
364C16—41no—
365C23—22no—
366C23—21no—
367C23—21no—
368C23—41no—
369C23—41no—
370C23#47611yescharacteristic
371C23#47611yescharacteristic
372C23C2×AGL(1,ℤ/8)11yescharacteristic
373Q8—41no—
374Q8—41no—
375Q8—164no—
376D4—41no—
377D4—41no—
378D4—82no—
379D4—81no—
380D4—81no—
381D4—161no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
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—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
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—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—164no—
416C2×C4#47611yescharacteristic
417C2×C4#47611yescharacteristic
418C2×C4C2×C22⋊Q811yescharacteristic
419C2×C4#54611yescharacteristic
420C2×C4#54611yescharacteristic
421C2×C4C22×SD1611yescharacteristic
422C2×C4#57611yescharacteristic
423C2×C4#57811yescharacteristic
424C8—21no—
425C8—21no—
426C8—21no—
427C8—21no—
428C8—21no—
429C8—21no—
430C8—42no—
431C8—81no—
432C8#52911yescharacteristic
433C8#53811yescharacteristic
434C8#55411yescharacteristic
435C8#55411yescharacteristic
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—42no—
444C22—42no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22#223012yes—
449C22C2×#47611yescharacteristic
450C22#330811yescharacteristic
451C22#331811yescharacteristic
452C4—21no—
453C4—21no—
454C4—42no—
455C4—42no—
456C4—82no—
457C4—81no—
458C4—81no—
459C4—81no—
460C4—161no—
461C4—161no—
462C4—161no—
463C4#245011yescharacteristic
464C4#308311yescharacteristic
465C4#329511yescharacteristic
466C4#329811yescharacteristic
467C2—21no—
468C2—21no—
469C2—22no—
470C2—41no—
471C2—161no—
472C2#2641011yescharacteristic
473C2C2×#223011yescharacteristic
474C2#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. #26060C2semidirect
3. #13543C2semidirect
4. #13521C2semidirect
5. #13467C2semidirect
6. #13371C2semidirect
7. #13272C2semidirect
8. #13116C2semidirect
11. #12950C2semidirect
12. #12928C2semidirect
13. #12914C2semidirect
14. #12823C2semidirect
15. C2×#1383C2semidirect
77. #467Q8semidirect
78. #467Q8semidirect
79. #466Q8semidirect
80. #466Q8semidirect

Maximal subgroups

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

subgroupindex
#260602
#230182
#135432
#135212
#134672
#133712
#132722
#131162
#130652
#130012
#129502
#129282
#129142
#128232
C2×#13832