Finite Group Databasebeta

← #507243

Subgroups — #507243

Order 512.

518 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27792C211yesmaximal, characteristic
2#25033C211yesmaximal, characteristic
3#13494C211yesmaximal, characteristic
4#13483C211yesmaximal, characteristic, complemented
5#13364C211yesmaximal, characteristic, complemented
6#13306C211yesmaximal, characteristic
7#13283C211yesmaximal, characteristic, complemented
8#13240C211yesmaximal, characteristic, complemented
9#13150C211yesmaximal, characteristic, complemented
10#13111C211yesmaximal, characteristic, complemented
11#13060C211yesmaximal, characteristic
12#13060C211yesmaximal, characteristic
13#13015C211yesmaximal, characteristic, complemented
14#12638C211yesmaximal, characteristic, complemented
15#12611C211yesmaximal, characteristic
16C22×Q32C2211yescharacteristic
17C23×C16C2211yescharacteristic
18C2×#500C2211yescharacteristic
19C2×#467C2211yescharacteristic
20C2×#467C2211yescharacteristic
21C2×C4×Q16C2211yescharacteristic
22#2492—21no—
23#2421—21no—
24C2×#367C2211yescharacteristic
25C2×#358C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×C22⋊C16C2211yescharacteristic
32C2×#346C2211yescharacteristic
33C2×#346C2211yescharacteristic
34#2074—21no—
35#2036—21no—
36#2035—21no—
37#2024—21no—
38#1969C2211yescharacteristic
39#1969C2211yescharacteristic
40#1969C2211yescharacteristic
41#1942C2211yescharacteristic
42#1928—21no—
43#1865—21no—
44#1846C2211yescharacteristic
45#1808C2211yescharacteristic
46#1808C2211yescharacteristic
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×Q16C2311yescharacteristic
60C22×Q16C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C2×C4⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C4×Q8—21no—
66C2×Q32—22no—
67C2×Q32—22no—
68C2×Q32—41no—
69C22×C16—22no—
70C22×C16—41no—
71C22×C16—41no—
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76#500—22no—
77#500—41no—
78#467—22no—
79#467—22no—
80#467—41no—
81#467—41no—
82C4×Q16—22no—
83C4×Q16—41no—
84C2×#107—21no—
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
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#400—41no—
103#397—41no—
104#393—41no—
105#391—21no—
106#391—21no—
107#391—21no—
108#391—41no—
109#389—21no—
110#388—41no—
111#388—41no—
112#387—41no—
113#382—21no—
114#380—21no—
115#380—21no—
116C4×C4⋊C4—21no—
117C4×C22⋊C4D411yescharacteristic
118#367—22no—
119#358—22no—
120#358—22no—
121#358—22no—
122#358—41no—
123#358—41no—
124#358—41no—
125C22⋊C16—22no—
126#346—41no—
127#346—41no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#336D411yescharacteristic
133#336D411yescharacteristic
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—41no—
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#4112yes—
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
160C4⋊Q8—42no—
161C4⋊Q8—81no—
162C22⋊Q8—42no—
163C22⋊Q8—42no—
164C22⋊Q8—81no—
165C22⋊Q8—81no—
166C4×Q8—42no—
167C4×Q8—81no—
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×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4D811yescharacteristic
184C2×C22⋊C4D811yescharacteristic
185C2×C42—41no—
186C2×C42—41no—
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189Q32—44no—
190C2×C16—21no—
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—44no—
201C2×C16#4112yes—
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—81no—
217#103—42no—
218#103—42no—
219#103—42no—
220#103—42no—
221#103—42no—
222#103—81no—
223C22⋊C8—22no—
224C4×C8—21no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8—21no—
228#95—41no—
229#95—41no—
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#95D811yescharacteristic
241#95D811yescharacteristic
242C24C2×D811yescharacteristic
243C2×Q8—22no—
244C2×Q8—41no—
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—82no—
256C2×Q8—82no—
257C2×Q8—161no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4C2×D811yescharacteristic
273C22×C4C2×D811yescharacteristic
274C22×C4C2×D811yescharacteristic
275C22×C4C2×D811yescharacteristic
276C22×C4C2×D811yescharacteristic
277C22×C4C22×D411yescharacteristic
278Q16—22no—
279Q16—22no—
280Q16—41no—
281Q16—41no—
282Q16—41no—
283Q16—41no—
284Q16—41no—
285Q16—41no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8#13512yes—
299C2×C8C2×#4111yescharacteristic
300C2×C8C2×#4111yescharacteristic
301C2×C8C2×#4111yescharacteristic
302C2×C8C2×#4111yescharacteristic
303C2×C8H(5,𝔽2)11yescharacteristic
304C2×C8H(5,𝔽2)11yescharacteristic
305C4⋊C4—22no—
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—41no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C22⋊C4—22no—
323C22⋊C4—41no—
324C22⋊C4—82no—
325C22⋊C4—82no—
326C22⋊C4—82no—
327C22⋊C4—161no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—82no—
333C42—161no—
334C16—22no—
335C16—22no—
336C16—41no—
337C16—41no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23#51012yes—
345C23C22×D811yescharacteristic
346Q8—42no—
347Q8—42no—
348Q8—81no—
349Q8—81no—
350Q8—81no—
351Q8—81no—
352Q8—81no—
353Q8—81no—
354Q8—164no—
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—21no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
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—81no—
398C2×C4#53211yescharacteristic
399C2×C4#54611yescharacteristic
400C2×C4#54611yescharacteristic
401C2×C4C2×#13511yescharacteristic
402C2×C4#57711yescharacteristic
403C2×C4#57711yescharacteristic
404C8—22no—
405C8—41no—
406C8—41no—
407C8#52911yescharacteristic
408C8#53811yescharacteristic
409C8#55111yescharacteristic
410C8#55911yescharacteristic
411C22—22no—
412C22—22no—
413C22—22no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22#316911yescharacteristic
419C22#329411yescharacteristic
420C22#329411yescharacteristic
421C22C2×#51011yescharacteristic
422C22#341411yescharacteristic
423C22#341611yescharacteristic
424C22#341711yescharacteristic
425C4—41no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—41no—
430C4—82no—
431C4—82no—
432C4—161no—
433C4—161no—
434C4—161no—
435C4—161no—
436C4—161no—
437C4—161no—
438C4#255711yescharacteristic
439C4#317311yescharacteristic
440C4#330111yescharacteristic
441C4#330311yescharacteristic
442C2—42no—
443C2#2897511yescharacteristic
444C2#3374511yescharacteristic
445C2#3374711yescharacteristic
446C2#3387811yescharacteristic
447C2#3387911yescharacteristic
448C2#3388011yescharacteristic
449C2#3388311yescharacteristic
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
4. #13483C2semidirect
5. #13364C2semidirect
7. #13283C2semidirect
8. #13240C2semidirect
9. #13150C2semidirect
10. #13111C2semidirect
13. #13015C2semidirect
14. #12638C2semidirect

Maximal subgroups

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

subgroupindex
#277922
#250332
#134942
#134832
#133642
#133062
#132832
#132402
#131502
#131112
#130602
#130152
#126382
#126112