Finite Group Databasebeta

← #504584

Subgroups — #504584

Order 512.

587 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27791C211yesmaximal, characteristic, complemented
2#23027C211yesmaximal, characteristic, complemented
3#13448C211yesmaximal, characteristic, complemented
4#13381C211yesmaximal, characteristic, complemented
5#13363C211yesmaximal, characteristic, complemented
6#13304C211yesmaximal, characteristic
7#13275C211yesmaximal, characteristic, complemented
8#13149C211yesmaximal, characteristic, complemented
9#13131C211yesmaximal, characteristic, complemented
10#13121C211yesmaximal, characteristic, complemented
11#13110C211yesmaximal, characteristic, complemented
12#13012C211yesmaximal, characteristic, complemented
13#12992C211yesmaximal, characteristic, complemented
14#12868C211yesmaximal, characteristic, complemented
15#12637C211yesmaximal, characteristic, complemented
16C22×D16C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#466C2211yescharacteristic
19C2×#466C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C2×#427C2211yescharacteristic
22#2491—21no—
23#2363—21no—
24C2×#367C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×C22⋊C16C2211yescharacteristic
33C2×C22⋊C16C2211yescharacteristic
34C2×#346C2211yescharacteristic
35#2033—21no—
36#2023—21no—
37#1970C2211yescharacteristic
38#1970C2211yescharacteristic
39#1970C2211yescharacteristic
40#1942C2211yescharacteristic
41#1926—21no—
42#1925—21no—
43#1875—21no—
44#1864—21no—
45#1808C2211yescharacteristic
46C4×#107C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C8×C22⋊C4C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C22×D8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C2×C4⋊Q8—21no—
63C2×#124—41no—
64C2×#121—21no—
65C2×#121—21no—
66C2×C4×D4—21no—
67C2×#117—21no—
68C2×D16—22no—
69C2×D16—41no—
70C2×D16—41no—
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75#500—22no—
76#500—41no—
77#466—22no—
78#466—22no—
79#466—22no—
80#466—41no—
81C4×D8—22no—
82C4×D8—41no—
83#427—22no—
84#427—41no—
85C2×#107—21no—
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#106—21no—
93C2×#102—21no—
94C2×#102—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×C22⋊C8D411yescharacteristic
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102#394—41no—
103#393—41no—
104#390—21no—
105#390—21no—
106#390—21no—
107#390—41no—
108#389—21no—
109#388—41no—
110#388—41no—
111#386—41no—
112#380—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C22⋊C4—41no—
116C4×C22⋊C4D411yescharacteristic
117#367—22no—
118C4⋊C16—41no—
119#357—22no—
120#357—22no—
121#357—22no—
122#357—22no—
123#357—41no—
124#357—41no—
125C22⋊C16—41no—
126C22⋊C16—41no—
127#346—41no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#336D411yescharacteristic
133#336D411yescharacteristic
134C22×Q8—41no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—41no—
138C22×D4—41no—
139C23×C4—41no—
140C23×C4C2×D411yescharacteristic
141C2×D8—21no—
142C2×D8—21no—
143C2×D8—21no—
144C2×D8—21no—
145C2×D8—21no—
146C2×D8—21no—
147C2×D8—21no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8#4112yes—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8C2×D411yescharacteristic
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D411yescharacteristic
162C22×C8C2411yescharacteristic
163C4⋊Q8—42no—
164C4⋊Q8—81no—
165#124—84no—
166#121—24no—
167#121—42no—
168#121—42no—
169#121—81no—
170C4×D4—42no—
171C4×D4—81no—
172#117—42no—
173#117—81no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊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⋊C4D811yescharacteristic
194C2×C22⋊C4D811yescharacteristic
195C2×C42—41no—
196C2×C42—41no—
197C2×C42—41no—
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200D16—44no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16—21no—
207C2×C16—21no—
208C2×C16—21no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16—21no—
212C2×C16—21no—
213#107—21no—
214#107—21no—
215#107—21no—
216#107—21no—
217#107—21no—
218#107—21no—
219#107—21no—
220#107—21no—
221#107—21no—
222#107—42no—
223#107#4112yes—
224#107#4112yes—
225#107#4112yes—
226#106—42no—
227#102—24no—
228#102—42no—
229#102—42no—
230#102—42no—
231#102—81no—
232#102—81no—
233C22⋊C8—41no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95D811yescharacteristic
243#95D811yescharacteristic
244C24—41no—
245C24—41no—
246C24C2×D811yescharacteristic
247C2×Q8—82no—
248C2×Q8—82no—
249C2×Q8—161no—
250C2×D4—22no—
251C2×D4—44no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—82no—
264C2×D4—82no—
265C2×D4—161no—
266C2×D4—161no—
267C2×D4—161no—
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×C4—41no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4—161no—
283C22×C4C2×D811yescharacteristic
284C22×C4C2×D811yescharacteristic
285C22×C4C2×D811yescharacteristic
286C22×C4C2×D811yescharacteristic
287C22×C4C2×D811yescharacteristic
288C22×C4C22×D411yescharacteristic
289D8—22no—
290D8—22no—
291D8—41no—
292D8—41no—
293D8—41no—
294D8—41no—
295D8—41no—
296D8—41no—
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—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8C2×#4111yescharacteristic
311C2×C8C2×#4111yescharacteristic
312C2×C8C2×#4111yescharacteristic
313C2×C8C2×#4111yescharacteristic
314C2×C8H(5,𝔽2)11yescharacteristic
315C2×C8H(5,𝔽2)11yescharacteristic
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—82no—
329C4⋊C4—161no—
330C22⋊C4—22no—
331C22⋊C4—41no—
332C22⋊C4—44no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C22⋊C4—82no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—161no—
342C22⋊C4—161no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—21no—
347C42—82no—
348C42—82no—
349C42—161no—
350C16—41no—
351C16—41no—
352C16—41no—
353C16—41no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
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—
374C23—81no—
375C23C22×D811yescharacteristic
376Q8—164no—
377D4—42no—
378D4—42no—
379D4—81no—
380D4—81no—
381D4—81no—
382D4—81no—
383D4—81no—
384D4—81no—
385D4—164no—
386D4—164no—
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—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
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—81no—
422C2×C4—81no—
423C2×C4—164no—
424C2×C4#50912yes—
425C2×C4#52511yescharacteristic
426C2×C4#54611yescharacteristic
427C2×C4#54611yescharacteristic
428C2×C4#57511yescharacteristic
429C2×C4#57711yescharacteristic
430C2×C4#57711yescharacteristic
431C8—41no—
432C8—41no—
433C8—41no—
434C8#52911yescharacteristic
435C8#53811yescharacteristic
436C8#55511yescharacteristic
437C8#55911yescharacteristic
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—82no—
446C22—82no—
447C22—82no—
448C22—161no—
449C22—161no—
450C22—161no—
451C22—161no—
452C22—161no—
453C22—161no—
454C22—161no—
455C22—161no—
456C22—161no—
457C22—161no—
458C22—161no—
459C22#310011yescharacteristic
460C22#329411yescharacteristic
461C22#329411yescharacteristic
462C22C2×#50911yescharacteristic
463C22#341411yescharacteristic
464C22#341511yescharacteristic
465C22#341611yescharacteristic
466C4—22no—
467C4—41no—
468C4—41no—
469C4—41no—
470C4—41no—
471C4—82no—
472C4—82no—
473C4—82no—
474C4—161no—
475C4—161no—
476C4—161no—
477C4#245911yescharacteristic
478C4#312511yescharacteristic
479C4#332111yescharacteristic
480C4#332611yescharacteristic
481C2—81no—
482C2—162no—
483C2—321no—
484C2#2672011yescharacteristic
485C2#3367511yescharacteristic
486C2#3367811yescharacteristic
487C2#3388511yescharacteristic
488C2#3388611yescharacteristic
489C2#3388811yescharacteristic
490C2#3389911yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#13304C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊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. #27791C2semidirect
2. #23027C2semidirect
3. #13448C2semidirect
4. #13381C2semidirect
5. #13363C2semidirect
7. #13275C2semidirect
8. #13149C2semidirect
9. #13131C2semidirect
10. #13121C2semidirect
11. #13110C2semidirect
12. #13012C2semidirect
13. #12992C2semidirect
14. #12868C2semidirect
15. #12637C2semidirect

Maximal subgroups

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

subgroupindex
#277912
#230272
#134482
#133812
#133632
#133042
#132752
#131492
#131312
#131212
#131102
#130122
#129922
#128682
#126372