Finite Group Databasebeta

← #510125

Subgroups — #510125

Order 512.

532 conjugacy classes of subgroups.

The subgroup lattice has 473 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28705C211yesmaximal, characteristic
2#24781C211yesmaximal, characteristic
3#13493C211yesmaximal, characteristic, complemented
4#13451C211yesmaximal, characteristic, complemented
5#13383C211yesmaximal, characteristic, complemented
6#13383C211yesmaximal, characteristic, complemented
7#13383C211yesmaximal, characteristic, complemented
8#13364C211yesmaximal, characteristic, complemented
9#13133C211yesmaximal, characteristic, complemented
10#13112C211yesmaximal, characteristic, complemented
11#13060C211yesmaximal, characteristic, complemented
12#12976C211yesmaximal, characteristic
13#12919C211yesmaximal, characteristic, complemented
14#12867C211yesmaximal, characteristic, complemented
15#12834C211yesmaximal, characteristic, complemented
16C23×C16C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#467C2211yescharacteristic
19C2×#467C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C22×#107C2211yescharacteristic
22#2563—21no—
23#2417—21no—
24C2×#366C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×C22⋊C16C2211yescharacteristic
33C2×C22⋊C16C2211yescharacteristic
34C2×C22⋊C16C2211yescharacteristic
35#2076—21no—
36#2076—21no—
37#2068—21no—
38#2036—21no—
39#1987—21no—
40#1969C2211yescharacteristic
41#1942C2211yescharacteristic
42#1872—21no—
43#1808C2211yescharacteristic
44#1808C2211yescharacteristic
45#1808C2211yescharacteristic
46#1808C2211yescharacteristic
47#1808C2211yescharacteristic
48C4×#107C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C2×C4⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C4×D4D411yescharacteristic
66C22×C4⋊C4—21no—
67C22×C16—22no—
68C22×C16—41no—
69C22×C16—41no—
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74#500—41no—
75#500—41no—
76#467—22no—
77#467—41no—
78#467—41no—
79#467—41no—
80C8×D4—41no—
81C8×D4—41no—
82C2×#107—22no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—41no—
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#107C2311yescharacteristic
92C2×#107C2311yescharacteristic
93C2×C4⋊C8D411yescharacteristic
94C2×#103—21no—
95C2×#103—21no—
96C2×#103—21no—
97C2×C22⋊C8D411yescharacteristic
98C2×C22⋊C8D411yescharacteristic
99C2×C4×C8C2311yescharacteristic
100#398—41no—
101#396—41no—
102#393—41no—
103#393—41no—
104#391—21no—
105#390—41no—
106#389—21no—
107#386—41no—
108#382—41no—
109#380—21no—
110#380—21no—
111#380—21no—
112#380—21no—
113#380—21no—
114#380—41no—
115C4×C4⋊C4—21no—
116#366—22no—
117#366—41no—
118#366—41no—
119#366—41no—
120#358—22no—
121#358—41no—
122#358—41no—
123#358—41no—
124C22⋊C16—22no—
125C22⋊C16—41no—
126C22⋊C16—41no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134C22×Q8—21no—
135C22×Q8—41no—
136C22×D4D811yescharacteristic
137C23×C4—41no—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16—21no—
144C2×Q16—21no—
145C2×Q16—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8C2×D411yescharacteristic
159C22×C8C2×D411yescharacteristic
160C22×C8C2×D411yescharacteristic
161C22×C8C2411yescharacteristic
162C4⋊Q8—42no—
163C4⋊Q8—42no—
164C22⋊Q8—42no—
165C22⋊Q8—42no—
166C22⋊Q8—42no—
167C22⋊Q8—81no—
168C4×D4—41no—
169C4×D4—41no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—81no—
184C2×C4⋊C4D811yescharacteristic
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⋊C4D811yescharacteristic
193C2×C22⋊C4D811yescharacteristic
194C2×C42—41no—
195C2×C42C2×D411yescharacteristic
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16—21no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16—21no—
205C2×C16—21no—
206C2×C16—21no—
207C2×C16—44no—
208C2×C16#4112yes—
209#107—21no—
210#107—21no—
211#107—21no—
212#107—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—42no—
224#107#4112yes—
225C4⋊C8—41no—
226#103—42no—
227#103—42no—
228#103—81no—
229C22⋊C8—41no—
230C22⋊C8—41no—
231C4×C8—21no—
232C4×C8—21no—
233#95—41no—
234#95—41no—
235#95—41no—
236#95—41no—
237#95—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242C24C2×D811yescharacteristic
243C24C2×D811yescharacteristic
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—82no—
251C2×Q8—161no—
252C2×Q8—161no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4Q3214yes—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4—161no—
279C22×C4C2×D811yescharacteristic
280C22×C4C2×D811yescharacteristic
281C22×C4C2×D811yescharacteristic
282C22×C4C2×D811yescharacteristic
283C22×C4C22×D411yescharacteristic
284Q16—41no—
285Q16—41no—
286Q16—41no—
287Q16—41no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8C2×#4111yescharacteristic
306C2×C8C2×#4111yescharacteristic
307C2×C8H(5,𝔽2)11yescharacteristic
308C2×C8H(5,𝔽2)11yescharacteristic
309C2×C8H(5,𝔽2)11yescharacteristic
310C2×C82−1+411yescharacteristic
311C4⋊C4—22no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—82no—
327C4⋊C4—82no—
328C4⋊C4—82no—
329C4⋊C4—82no—
330C4⋊C4—161no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C22⋊C4—161no—
336C22⋊C4—161no—
337C22⋊C4—161no—
338C22⋊C4—161no—
339C22⋊C4—161no—
340C42—21no—
341C42—21no—
342C42—82no—
343C16—22no—
344C16—41no—
345C16—41no—
346C16—41no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23C2×Q3212yes—
360C23C2×Q3212yes—
361C23C22×D811yescharacteristic
362Q8—81no—
363Q8—81no—
364Q8—81no—
365Q8—81no—
366Q8—164no—
367D4—44no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
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—81no—
411C2×C4—164no—
412C2×C4C2×Q3212yes—
413C2×C4#53411yescharacteristic
414C2×C4#54611yescharacteristic
415C2×C4#54711yescharacteristic
416C2×C4#57511yescharacteristic
417C2×C4#57711yescharacteristic
418C2×C4#57811yescharacteristic
419C8—41no—
420C8—41no—
421C8—41no—
422C8#53811yescharacteristic
423C8#54211yescharacteristic
424C8#55511yescharacteristic
425C8#56211yescharacteristic
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22#318611yescharacteristic
441C22#329411yescharacteristic
442C22#333411yescharacteristic
443C22C22×Q3211yescharacteristic
444C22#341411yescharacteristic
445C22#341511yescharacteristic
446C22#341611yescharacteristic
447C4—22no—
448C4—41no—
449C4—41no—
450C4—41no—
451C4—82no—
452C4—161no—
453C4—161no—
454C4—161no—
455C4—161no—
456C4—161no—
457C4—161no—
458C4—161no—
459C4#268611yescharacteristic
460C4#320111yescharacteristic
461C4#331311yescharacteristic
462C4#335011yescharacteristic
463C2—42no—
464C2—42no—
465C2#3131811yescharacteristic
466C2#3375511yescharacteristic
467C2#3376411yescharacteristic
468C2#3386611yescharacteristic
469C2#3388611yescharacteristic
470C2#3390511yescharacteristic
471C2#3391511yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D4D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13493C2semidirect
4. #13451C2semidirect
5. #13383C2semidirect
6. #13383C2semidirect
7. #13383C2semidirect
8. #13364C2semidirect
9. #13133C2semidirect
10. #13112C2semidirect
11. #13060C2semidirect
13. #12919C2semidirect
14. #12867C2semidirect
15. #12834C2semidirect

Maximal subgroups

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

subgroupindex
#287052
#247812
#134932
#134512
#133832
#133642
#131332
#131122
#130602
#129762
#129192
#128672
#128342