Finite Group Databasebeta

← #509009

Subgroups — #509009

Order 512.

542 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33266C211yesmaximal, characteristic, complemented
2#33240C211yesmaximal, characteristic, complemented
3#13476C211yesmaximal, characteristic, complemented
4#13475C211yesmaximal, characteristic, complemented
5#13453C211yesmaximal, characteristic, complemented
6#13449C211yesmaximal, characteristic, complemented
7#13448C211yesmaximal, characteristic
8#13408C211yesmaximal, characteristic, complemented
9#13365C211yesmaximal, characteristic, complemented
10#13326C211yesmaximal, characteristic
11#13237C211yesmaximal, characteristic, complemented
12#13235C211yesmaximal, characteristic
13#13148C211yesmaximal, characteristic, complemented
14#13147C211yesmaximal, characteristic, complemented
15C4×C4⋊C16C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C22×D16C2211yescharacteristic
18C2×#499C2211yescharacteristic
19C2×#495C2211yescharacteristic
20C2×#488—21no—
21#2861—21no—
22#2860—21no—
23C2×#367C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×C4⋊C16C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×C4×C16C2211yescharacteristic
36C2×C4×C16C2211yescharacteristic
37#1970C2211yescharacteristic
38#1970C2211yescharacteristic
39#1970C2211yescharacteristic
40#1970C2211yescharacteristic
41#1970C2211yescharacteristic
42#1969C2211yescharacteristic
43#1969C2211yescharacteristic
44#1968—21no—
45#1968—21no—
46#1942C2211yescharacteristic
47#1942C2211yescharacteristic
48#1942C2211yescharacteristic
49#1934—21no—
50#1934—21no—
51#1934—21no—
52C42×C8C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×Q16C2311yescharacteristic
60C22×SD16—21no—
61C22×D8C2311yescharacteristic
62C22×D8C2311yescharacteristic
63C2×#126—41no—
64C2×#125—21no—
65C2×#124—21no—
66C2×SD32—22no—
67C2×SD32—41no—
68C2×SD32—41no—
69C2×D16—22no—
70C2×D16—41no—
71C2×D16—41no—
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76#499—41no—
77#499—41no—
78#495—41no—
79#495—41no—
80#488—44no—
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#106—21no—
87C2×#103—21no—
88C2×#103—21no—
89C2×#102—21no—
90C2×#102—21no—
91C2×#102—21no—
92C2×#102—21no—
93C2×#102—21no—
94C2×C4×C8D411yescharacteristic
95C2×C4×C8D411yescharacteristic
96C2×C4×C8D411yescharacteristic
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100#391—21no—
101#391—21no—
102#390—21no—
103#390—21no—
104#390—21no—
105#390—21no—
106#390—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#388—41no—
111#388—41no—
112#388—41no—
113#388—41no—
114#388—41no—
115#388—41no—
116#388—41no—
117C43D411yescharacteristic
118#367—41no—
119#366—41no—
120C4⋊C16—22no—
121C4⋊C16—22no—
122#358—22no—
123#358—41no—
124#358—41no—
125#357—22no—
126#357—41no—
127#357—41no—
128#357—41no—
129#357—41no—
130C4×C16—41no—
131C4×C16—41no—
132#340—21no—
133#340—21no—
134#340—21no—
135C22×Q8—21no—
136C22×D4—21no—
137C22×D4—21no—
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16—21no—
144C2×SD16—42no—
145C2×SD16—42no—
146C2×SD16—42no—
147C2×D8—21no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8—21no—
153C2×D8—21no—
154C2×D8—21no—
155C2×D8—21no—
156C2×D8—21no—
157C2×D8—21no—
158C2×D8—21no—
159C22×C8C2×D411yescharacteristic
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D411yescharacteristic
162C22×C8C2411yescharacteristic
163#126—84no—
164#125—42no—
165#125—42no—
166#124—42no—
167#124—42no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊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×C42D811yescharacteristic
191C2×C42D811yescharacteristic
192C2×C42D811yescharacteristic
193C2×C42D811yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197SD32—44no—
198D16—44no—
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—21no—
208C2×C16—21no—
209C2×C16#4112yes—
210C2×C16#4112yes—
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#106—42no—
222#103—42no—
223#103—42no—
224#102—42no—
225#102—42no—
226#102—42no—
227#102—42no—
228#102—42no—
229C4×C8—22no—
230C4×C8—22no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95—41no—
243#95—41no—
244#95—41no—
245C24—41no—
246C24—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—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—41no—
264C2×D4—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×C4C2×D811yescharacteristic
277C22×C4C22×D411yescharacteristic
278Q16—41no—
279Q16—41no—
280Q16—41no—
281Q16—41no—
282SD16—84no—
283D8—41no—
284D8—41no—
285D8—41no—
286D8—41no—
287D8—41no—
288D8—41no—
289D8—41no—
290D8—41no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8#13512yes—
299C2×C8AGL(1,ℤ/8)12yes—
300C2×C8C2×#4111yescharacteristic
301C2×C8C2×#4111yescharacteristic
302C2×C8H(5,𝔽2)11yescharacteristic
303C2×C8H(5,𝔽2)11yescharacteristic
304C2×C8H(5,𝔽2)11yescharacteristic
305C2×C82−1+411yescharacteristic
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—
321C22⋊C4—82no—
322C22⋊C4—82no—
323C22⋊C4—82no—
324C22⋊C4—82no—
325C22⋊C4—82no—
326C22⋊C4—82no—
327C22⋊C4—82no—
328C22⋊C4—82no—
329C22⋊C4—82no—
330C22⋊C4—82no—
331C22⋊C4—82no—
332C22⋊C4—82no—
333C42—21no—
334C42—21no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—41no—
343C16—22no—
344C16—22no—
345C16—41no—
346C16—41no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23—81no—
358C23—81no—
359C23—81no—
360C23—81no—
361C23C22×D811yescharacteristic
362Q8—81no—
363Q8—81no—
364Q8—81no—
365Q8—81no—
366D4—81no—
367D4—81no—
368D4—81no—
369D4—81no—
370D4—81no—
371D4—81no—
372D4—81no—
373D4—81no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
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#50812yes—
408C2×C4#50812yes—
409C2×C4#50912yes—
410C2×C4#53211yescharacteristic
411C2×C4#53511yescharacteristic
412C2×C4#53711yescharacteristic
413C2×C4C2×#13511yescharacteristic
414C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
415C2×C4#57511yescharacteristic
416C8—22no—
417C8—22no—
418C8—41no—
419C8#53811yescharacteristic
420C8#53811yescharacteristic
421C8#54211yescharacteristic
422C8#54211yescharacteristic
423C22—161no—
424C22—161no—
425C22—161no—
426C22—161no—
427C22—161no—
428C22—161no—
429C22—161no—
430C22—161no—
431C22—161no—
432C22—161no—
433C22—161no—
434C22—161no—
435C22—161no—
436C22—161no—
437C22#316911yescharacteristic
438C22#320511yescharacteristic
439C22#324511yescharacteristic
440C22C2×#50811yescharacteristic
441C22C2×#50811yescharacteristic
442C22C2×#50911yescharacteristic
443C22#341411yescharacteristic
444C4—22no—
445C4—22no—
446C4—22no—
447C4—41no—
448C4—41no—
449C4—41no—
450C4—161no—
451C4—161no—
452C4—161no—
453C4—161no—
454C4—161no—
455C4—161no—
456C4#266211yescharacteristic
457C4#317111yescharacteristic
458C4#321111yescharacteristic
459C4#324711yescharacteristic
460C2—321no—
461C2—321no—
462C2#3033611yescharacteristic
463C2#3373511yescharacteristic
464C2#3373711yescharacteristic
465C2#3377311yescharacteristic
466C2#3377811yescharacteristic
467C2#3381311yescharacteristic
468C2#3381611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D16C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33266C2semidirect
2. #33240C2semidirect
3. #13476C2semidirect
4. #13475C2semidirect
5. #13453C2semidirect
6. #13449C2semidirect
8. #13408C2semidirect
9. #13365C2semidirect
11. #13237C2semidirect
13. #13148C2semidirect
14. #13147C2semidirect
15. C4×C4⋊C16C2semidirect

Maximal subgroups

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

subgroupindex
#332662
#332402
#134762
#134752
#134532
#134492
#134482
#134082
#133652
#133262
#132372
#132352
#131482
#131472
C4×C4⋊C162