Finite Group Databasebeta

← #510926

Subgroups — #510926

Order 512.

498 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27927C211yesmaximal, characteristic
2#27789C211yesmaximal, characteristic
3#13530C211yesmaximal, characteristic, complemented
4#13483C211yesmaximal, characteristic, complemented
5#13450C211yesmaximal, characteristic, complemented
6#13383C211yesmaximal, characteristic
7#13366C211yesmaximal, characteristic, complemented
8#13306C211yesmaximal, characteristic
9#13284C211yesmaximal, characteristic, complemented
10#13240C211yesmaximal, characteristic, complemented
11#13178C211yesmaximal, characteristic, complemented
12#13133C211yesmaximal, characteristic, complemented
13#13121C211yesmaximal, characteristic
14#12994C211yesmaximal, characteristic
15#12921C211yesmaximal, characteristic
16C22×Q32C2211yescharacteristic
17C2×#499C2211yescharacteristic
18C2×#485—21no—
19C2×#467C2211yescharacteristic
20C2×#467C2211yescharacteristic
21C2×#427C2211yescharacteristic
22#2499—21no—
23#2489—21no—
24C2×#367C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×C22⋊C16C2211yescharacteristic
35C2×C22⋊C16C2211yescharacteristic
36C2×#346C2211yescharacteristic
37#2097—21no—
38#2096—21no—
39#2035—21no—
40#2026—21no—
41#1969C2211yescharacteristic
42#1969C2211yescharacteristic
43#1969C2211yescharacteristic
44#1969C2211yescharacteristic
45#1927—21no—
46#1846C2211yescharacteristic
47#1845C2211yescharacteristic
48#1808C2211yescharacteristic
49C4×#107C2211yescharacteristic
50C8×C22⋊C4C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×Q16C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C2×#125—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×#117—21no—
66C2×Q32—22no—
67C2×Q32—41no—
68C2×Q32—41no—
69C22×C16C2311yescharacteristic
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73#499—22no—
74#499—41no—
75#485—44no—
76#467—22no—
77#467—41no—
78#467—41no—
79#467—41no—
80#427—41no—
81#427—41no—
82C2×#107—21no—
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#106—21no—
90C2×#106—21no—
91C2×#106—21no—
92C2×#103—21no—
93C2×#103—21no—
94C2×#103—21no—
95C2×#103—21no—
96C2×#103—21no—
97C2×#103—21no—
98C2×C22⋊C8D411yescharacteristic
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101#400—41no—
102#397—41no—
103#397—41no—
104#397—41no—
105#391—21no—
106#391—21no—
107#391—21no—
108#391—21no—
109#388—41no—
110#387—41no—
111#387—41no—
112#384—21no—
113#382—21no—
114#382—41no—
115#380—21no—
116C4×C4⋊C4—21no—
117C4×C22⋊C4D411yescharacteristic
118#367—41no—
119#366—41no—
120#366—41no—
121C4⋊C16—22no—
122#358—22no—
123#358—41no—
124#358—41no—
125#358—41no—
126#358—41no—
127#358—41no—
128C22⋊C16—22no—
129C22⋊C16—41no—
130#346—41no—
131#340—21no—
132#340—21no—
133#336D411yescharacteristic
134#336D411yescharacteristic
135C22×Q8—21no—
136C22×Q8—21no—
137C23×C4C2×D411yescharacteristic
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16—21no—
144C2×Q16—21no—
145C2×Q16—21no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8C2×D411yescharacteristic
157C22×C8C2×D411yescharacteristic
158C22×C8C2×D411yescharacteristic
159C22×C8C2411yescharacteristic
160#125—42no—
161#125—81no—
162C22⋊Q8—42no—
163C22⋊Q8—42no—
164C22⋊Q8—42no—
165C22⋊Q8—81no—
166#117—42no—
167#117—42no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4D811yescharacteristic
185C2×C22⋊C4D811yescharacteristic
186C2×C42—41no—
187C2×C42—41no—
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190Q32—44no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16—21no—
200C2×C16—21no—
201C2×C16—21no—
202C2×C16#4112yes—
203#107—21no—
204#107—21no—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—21no—
209#107—21no—
210#107—21no—
211#107—21no—
212#107—21no—
213#107—21no—
214#107—42no—
215#107#4112yes—
216#106—42no—
217#106—42no—
218#106—42no—
219#103—42no—
220#103—42no—
221#103—42no—
222#103—42no—
223#103—42no—
224#103—81no—
225C22⋊C8—22no—
226C4×C8—21no—
227C4×C8—21no—
228C4×C8—21no—
229C4×C8—21no—
230#95—41no—
231#95—41no—
232#95—41no—
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—
242#95D811yescharacteristic
243#95D811yescharacteristic
244C24C2×D811yescharacteristic
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4C2×D811yescharacteristic
272C22×C4C2×D811yescharacteristic
273C22×C4C2×D811yescharacteristic
274C22×C4C2×D811yescharacteristic
275C22×C4C2×D811yescharacteristic
276C22×C4C22×D411yescharacteristic
277Q16—41no—
278Q16—41no—
279Q16—41no—
280Q16—41no—
281Q16—41no—
282Q16—41no—
283Q16—41no—
284Q16—41no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8AGL(1,ℤ/8)12yes—
298C2×C8C2×#4111yescharacteristic
299C2×C8C2×#4111yescharacteristic
300C2×C8H(5,𝔽2)11yescharacteristic
301C2×C8H(5,𝔽2)11yescharacteristic
302C2×C8H(5,𝔽2)11yescharacteristic
303C2×C82−1+411yescharacteristic
304C4⋊C4—22no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—82no—
325C22⋊C4—82no—
326C22⋊C4—82no—
327C22⋊C4—82no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—82no—
333C42—82no—
334C16—22no—
335C16—41no—
336C16—41no—
337C16—41no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23C22×D811yescharacteristic
346Q8—81no—
347Q8—81no—
348Q8—81no—
349Q8—81no—
350Q8—81no—
351Q8—81no—
352Q8—81no—
353Q8—81no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
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#53511yescharacteristic
398C2×C4#54611yescharacteristic
399C2×C4#54711yescharacteristic
400C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
401C2×C4#57711yescharacteristic
402C2×C4#57811yescharacteristic
403C8—22no—
404C8—41no—
405C8—41no—
406C8#53811yescharacteristic
407C8#54211yescharacteristic
408C8#55911yescharacteristic
409C8#56211yescharacteristic
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22#320511yescharacteristic
418C22#329411yescharacteristic
419C22#333411yescharacteristic
420C22#341411yescharacteristic
421C22#341411yescharacteristic
422C22#341711yescharacteristic
423C22#341711yescharacteristic
424C4—41no—
425C4—41no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—82no—
430C4—161no—
431C4—161no—
432C4—161no—
433C4—161no—
434C4—161no—
435C4—161no—
436C4—161no—
437C4#271011yescharacteristic
438C4#322511yescharacteristic
439C4#330811yescharacteristic
440C4#335411yescharacteristic
441C2—81no—
442C2#3204611yescharacteristic
443C2#3378511yescharacteristic
444C2#3378911yescharacteristic
445C2#3388211yescharacteristic
446C2#3389411yescharacteristic
447C2#3391611yescharacteristic
448C2#3391611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
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
3. #13530C2semidirect
4. #13483C2semidirect
5. #13450C2semidirect
7. #13366C2semidirect
9. #13284C2semidirect
10. #13240C2semidirect
11. #13178C2semidirect
12. #13133C2semidirect

Maximal subgroups

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

subgroupindex
#279272
#277892
#135302
#134832
#134502
#133832
#133662
#133062
#132842
#132402
#131782
#131332
#131212
#129942
#129212