Finite Group Databasebeta

← #498234

Subgroups — #498234

Order 512.

557 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33646C211yesmaximal, characteristic
2#23019C211yesmaximal, characteristic
3#22617C211yesmaximal, characteristic, complemented
4#13174C211yesmaximal, characteristic, complemented
5#13153C211yesmaximal, characteristic, complemented
6#13143C211yesmaximal, characteristic, complemented
7#13119C211yesmaximal, characteristic, complemented
8#13118C211yesmaximal, characteristic, complemented
9#13065C211yesmaximal, characteristic
10#12955C211yesmaximal, characteristic, complemented
11#12928C211yesmaximal, characteristic, complemented
12#12677C211yesmaximal, characteristic, complemented
13#12650C211yesmaximal, characteristic, complemented
14#12616C211yesmaximal, characteristic, complemented
15C2×#1387C211yesmaximal, characteristic, complemented
16C22×M32C2211yescharacteristic
17#3050C2211yescharacteristic
18C2×#444C2211yescharacteristic
19#2933—21no—
20C22×#108C2211yescharacteristic
21C22×#107C2211yescharacteristic
22#2344—21no—
23#2236C412yescomplemented
24#2235C412yescomplemented
25#2233C412yescomplemented
26#2232C412yescomplemented
27#2159C2211yescharacteristic
28#2159C2211yescharacteristic
29C2×#365C2211yescharacteristic
30C2×#365C2211yescharacteristic
31C2×#360C2211yescharacteristic
32C2×#360C2211yescharacteristic
33#2139C2211yescharacteristic
34#2139C2211yescharacteristic
35#2112C2211yescharacteristic
36C2×#347C2211yescharacteristic
37C2×#347C2211yescharacteristic
38#1975C2211yescharacteristic
39#1946C2211yescharacteristic
40#1943C2211yescharacteristic
41#1943C2211yescharacteristic
42#1929—21no—
43#1921—21no—
44#1921—21no—
45#1871—21no—
46#1866—21no—
47#1823—21no—
48#1807C2211yescharacteristic
49#1776C2211yescharacteristic
50#1776C2211yescharacteristic
51#1773C2211yescharacteristic
52#1752C2211yescharacteristic
53#1387C2211yescharacteristic
54#1387C2211yescharacteristic
55#1387C2211yescharacteristic
56#1387C2211yescharacteristic
57#1386C2211yescharacteristic
58#1386C2211yescharacteristic
59#1383C2211yescharacteristic
60#1383C2211yescharacteristic
61#1375C2211yescharacteristic
62#1375C2211yescharacteristic
63C2×#135C2311yescharacteristic
64C2×#131—21no—
65C22×M16—21no—
66C23×C8C2311yescharacteristic
67#525—21no—
68C2×#117—41no—
69C22×C4⋊C4—21no—
70C2×M32—21no—
71C2×M32—22no—
72C2×M32—41no—
73C2×M32C2311yescharacteristic
74C2×M32C2311yescharacteristic
75C2×M32C2311yescharacteristic
76C2×M32C2311yescharacteristic
77C22×C16—21no—
78#467C2×C411yescharacteristic
79#467C2×C411yescharacteristic
80#466C2×C411yescharacteristic
81#466C2×C411yescharacteristic
82#465—21no—
83#465—21no—
84#444—22no—
85#444—41no—
86#436—41no—
87#435—41no—
88#435—41no—
89C8×D4—41no—
90C2×#108—22no—
91C2×#108—21no—
92C2×#108—41no—
93C2×#108C2311yescharacteristic
94C2×#108C2311yescharacteristic
95#427C2311yescharacteristic
96C2×#107—21no—
97C2×#107—21no—
98C2×#107—21no—
99C2×#107C2×C412yes—
100C2×#107C2×C412yes—
101C2×#107C2311yescharacteristic
102C2×#107C2311yescharacteristic
103C2×#106C2311yescharacteristic
104C2×#106C2311yescharacteristic
105#424—21no—
106C2×#104—21no—
107C2×#104—21no—
108#416—21no—
109#408—21no—
110C2×C22⋊C8D411yescharacteristic
111#405—21no—
112C2×#97C2311yescharacteristic
113C2×#97C2311yescharacteristic
114#389—21no—
115#389—21no—
116#387—41no—
117#387—41no—
118#381—41no—
119#380—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C22⋊C4D411yescharacteristic
123#367C2×C412yes—
124#367C2×C412yes—
125#366C2×C412yes—
126#366C2×C412yes—
127#365—22no—
128#365—41no—
129#360—22no—
130#360—41no—
131#358C2×C412yes—
132#358C2×C412yes—
133#357C2×C412yes—
134#357C2×C412yes—
135C22⋊C16C2×C412yes—
136C22⋊C16C2×C412yes—
137#347—22no—
138#347—41no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#339—21no—
145#339—21no—
146#336D411yescharacteristic
147#336D411yescharacteristic
148C2×#41—21no—
149C23×C4—41no—
150C23×C4C2×D411yescharacteristic
151#135—21no—
152#135—21no—
153#135—21no—
154#135—21no—
155C2×Q16C22×C411yescharacteristic
156C2×SD16—21no—
157C2×D8C22×C411yescharacteristic
158#131—42no—
159#131—81no—
160C2×M16—21no—
161C2×M16—21no—
162C2×M16—41no—
163C2×M16—41no—
164C2×M16—42no—
165C2×M16—81no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—41no—
171C22×C8—41no—
172C22×C8C22×C411yescharacteristic
173C22×C8C22×C411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8#4111yescharacteristic
178C22×C8#4111yescharacteristic
179C22×C8C2411yescharacteristic
180#123—41no—
181C22⋊Q8—21no—
182C22⋊Q8—21no—
183#121—21no—
184#121—21no—
185#120—41no—
186C4×D4—41no—
187C4×D4—41no—
188#117—21no—
189#117—84no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—41no—
203C2×C42—41no—
204C2×C42—41no—
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207M32—21no—
208M32—21no—
209M32—21no—
210M32—21no—
211M32—21no—
212M32—21no—
213M32—21no—
214M32—42no—
215M32#4112yes—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16—42no—
219C2×C16C22×C412yes—
220C2×C16C22×C412yes—
221#108—21no—
222#108—21no—
223#108—21no—
224#108—42no—
225#108#4112yes—
226#107—22no—
227#107—22no—
228#107—21no—
229#107—21no—
230#107C22×C412yes—
231#107C22×C412yes—
232#107C22×C411yescharacteristic
233#107C22×C411yescharacteristic
234#106—21no—
235#106—21no—
236#106—21no—
237#106—21no—
238#106—21no—
239C4⋊C8—41no—
240C4⋊C8—41no—
241#104—42no—
242#104—81no—
243#103—21no—
244#103—21no—
245#102—21no—
246#102—21no—
247C22⋊C8—21no—
248C22⋊C8—21no—
249C22⋊C8—41no—
250C22⋊C8—41no—
251#97—21no—
252#97—21no—
253#97—21no—
254#97—21no—
255#97—41no—
256C4×C8—41no—
257#95—21no—
258#95—41no—
259#95—41no—
260#95—41no—
261#95—41no—
262#95—41no—
263C24AGL(1,ℤ/8)11yescharacteristic
264#41—41no—
265#41—41no—
266#41—41no—
267#41—41no—
268C2×Q8—21no—
269C2×D4—21no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4C4×D411yescharacteristic
289C22×C4C4×D411yescharacteristic
290C22×C4AGL(1,ℤ/8)11yescharacteristic
291C22×C4AGL(1,ℤ/8)11yescharacteristic
292C22×C4AGL(1,ℤ/8)11yescharacteristic
293C22×C4C22×D411yescharacteristic
294Q16—21no—
295Q16—21no—
296SD16—41no—
297SD16—41no—
298D8—21no—
299D8—21no—
300M16—22no—
301M16—41no—
302M16—41no—
303M16—41no—
304M16—82no—
305M16—82no—
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—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—82no—
325C2×C8—82no—
326C2×C8C4×D411yescharacteristic
327C2×C8C4×D411yescharacteristic
328C2×C8C23×C411yescharacteristic
329C2×C8C2×#4111yescharacteristic
330C2×C8C2×#4111yescharacteristic
331C2×C8C2×#4111yescharacteristic
332C2×C8H(5,𝔽2)11yescharacteristic
333C2×C82−1+411yescharacteristic
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—82no—
349C4⋊C4—82no—
350C22⋊C4—42no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—82no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—21no—
361C42—41no—
362C42—82no—
363C42—82no—
364C16—22no—
365C16—22no—
366C16—22no—
367C16—41no—
368C23—21no—
369C23—22no—
370C23—21no—
371C23—41no—
372C23—41no—
373C23#44211yescharacteristic
374C23#44211yescharacteristic
375C23C2×AGL(1,ℤ/8)11yescharacteristic
376Q8—41no—
377Q8—41no—
378D4—41no—
379D4—41no—
380D4—81no—
381D4—82no—
382D4—81no—
383D4—161no—
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—21no—
393C2×C4—21no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—84no—
410C2×C4—84no—
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#44211yescharacteristic
420C2×C4#44211yescharacteristic
421C2×C4C2×C4×D411yescharacteristic
422C2×C4#54611yescharacteristic
423C2×C4#54711yescharacteristic
424C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
425C2×C4#57711yescharacteristic
426C2×C4#57811yescharacteristic
427C8—21no—
428C8—21no—
429C8—41no—
430C8—41no—
431C8—42no—
432C8—81no—
433C8C4×#4111yescharacteristic
434C8#52011yescharacteristic
435C8#55111yescharacteristic
436C8#55211yescharacteristic
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—42no—
445C22—42no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22AGL(1,ℤ/16)12yes—
450C22C2×#44211yescharacteristic
451C22#330811yescharacteristic
452C22#334511yescharacteristic
453C4—21no—
454C4—21no—
455C4—81no—
456C4—81no—
457C4—81no—
458C4—82no—
459C4—82no—
460C4—81no—
461C4—81no—
462C4—161no—
463C4—161no—
464C4#235911yescharacteristic
465C4#294711yescharacteristic
466C4#331011yescharacteristic
467C4#334311yescharacteristic
468C2—21no—
469C2—21no—
470C2—22no—
471C2—41no—
472C2—161no—
473C2#2293811yescharacteristic
474C2C2×AGL(1,ℤ/16)11yescharacteristic
475C2#3347011yescharacteristic
subgroupG/H
center, socleC22C2×#442
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radical#3050C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1752 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #22617C2semidirect
4. #13174C2semidirect
5. #13153C2semidirect
6. #13143C2semidirect
7. #13119C2semidirect
8. #13118C2semidirect
10. #12955C2semidirect
11. #12928C2semidirect
12. #12677C2semidirect
13. #12650C2semidirect
14. #12616C2semidirect
15. C2×#1387C2semidirect
23. #2236C4semidirect
24. #2235C4semidirect
25. #2233C4semidirect
26. #2232C4semidirect

Maximal subgroups

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

subgroupindex
#336462
#230192
#226172
#131742
#131532
#131432
#131192
#131182
#130652
#129552
#129282
#126772
#126502
#126162
C2×#13872