Finite Group Databasebeta

← #510305

Subgroups — #510305

Order 512.

535 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27839C211yesmaximal, characteristic, complemented
2#25412C211yesmaximal, characteristic
3#13516C211yesmaximal, characteristic, complemented
4#13480C211yesmaximal, characteristic, complemented
5#13476C211yesmaximal, characteristic, complemented
6#13408C211yesmaximal, characteristic, complemented
7#13382C211yesmaximal, characteristic, complemented
8#13380C211yesmaximal, characteristic
9#13254C211yesmaximal, characteristic, complemented
10#13112C211yesmaximal, characteristic, complemented
11#13060C211yesmaximal, characteristic, complemented
12#13059C211yesmaximal, characteristic
13#12936C211yesmaximal, characteristic, complemented
14#12912C211yesmaximal, characteristic, complemented
15#12812C211yesmaximal, characteristic, complemented
16C23×C16C2211yescharacteristic
17C2×#495C2211yescharacteristic
18C2×#467C2211yescharacteristic
19C2×#466C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21#2496—21no—
22#2426—21no—
23C2×#367C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×C4⋊C16C2211yescharacteristic
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×C22⋊C16C2211yescharacteristic
36#2099—21no—
37#2075—21no—
38#2042—21no—
39#2000—21no—
40#1970C2211yescharacteristic
41#1942C2211yescharacteristic
42#1942C2211yescharacteristic
43#1942C2211yescharacteristic
44#1918—21no—
45#1880—21no—
46#1845C2211yescharacteristic
47#1808C2211yescharacteristic
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×SD16—21no—
61C22×D8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C2×#126—41no—
64C2×#124—21no—
65C2×C22⋊Q8—21no—
66C2×#121—21no—
67C2×C4×Q8—21no—
68C22×C16—41no—
69C22×C16—41no—
70C22×C16—41no—
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75#495—22no—
76#495—41no—
77#467—22no—
78#467—41no—
79#466—22no—
80#466—41no—
81C4×Q16—22no—
82C4×Q16—41no—
83C2×#107—21no—
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#103—21no—
91C2×#103—21no—
92C2×#102—21no—
93C2×#102—21no—
94C2×C22⋊C8D411yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97#400—41no—
98#397—41no—
99#396—41no—
100#394—41no—
101#390—21no—
102#390—41no—
103#389—21no—
104#389—21no—
105#389—21no—
106#388—41no—
107#387—41no—
108#384—21no—
109#380—21no—
110#380—21no—
111C4×C4⋊C4—21no—
112C4×C22⋊C4D411yescharacteristic
113#367—41no—
114#366—22no—
115C4⋊C16—41no—
116C4⋊C16—41no—
117#358—22no—
118#358—41no—
119#358—41no—
120#358—41no—
121#357—41no—
122#357—41no—
123#357—41no—
124#357—41no—
125C22⋊C16—41no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#336D411yescharacteristic
133#336D411yescharacteristic
134C22×Q8—21no—
135C22×D4—21no—
136C22×D4—41no—
137C23×C4C2×D411yescharacteristic
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16#4112yes—
144C2×SD16—42no—
145C2×SD16—42no—
146C2×SD16—81no—
147C2×D8—21no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8—21no—
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
163#126—84no—
164#124—42no—
165#124—81no—
166C22⋊Q8—42no—
167C22⋊Q8—81no—
168#121—42no—
169#121—81no—
170C4×Q8—42no—
171C4×Q8—81no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊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×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
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—21no—
210C2×C16—21no—
211C2×C16—44no—
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—21no—
223#107—42no—
224#107#4112yes—
225#103—42no—
226#103—42no—
227#102—42no—
228#102—81no—
229C22⋊C8—41no—
230C4×C8—21no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8—21no—
234#95—41no—
235#95—41no—
236#95—41no—
237#95—41no—
238#95—41no—
239#95—41no—
240#95D811yescharacteristic
241#95D811yescharacteristic
242C24—41no—
243C24C2×D811yescharacteristic
244C2×Q8—22no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—82no—
257C2×D4—82no—
258C2×D4—161no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
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—
282SD16—84no—
283D8—41no—
284D8—41no—
285D8—41no—
286D8—41no—
287C2×C8—21no—
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—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
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—22no—
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—41no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C4⋊C4—161no—
325C22⋊C4—22no—
326C22⋊C4—22no—
327C22⋊C4—82no—
328C22⋊C4—82no—
329C22⋊C4—82no—
330C22⋊C4—82no—
331C22⋊C4—82no—
332C22⋊C4—82no—
333C22⋊C4—82no—
334C22⋊C4—161no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—82no—
340C42—82no—
341C16—41no—
342C16—41no—
343C16—41no—
344C16—41no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23—81no—
358C23#50912yes—
359C23C22×D811yescharacteristic
360Q8—42no—
361Q8—42no—
362Q8—81no—
363Q8—81no—
364D4—81no—
365D4—81no—
366D4—81no—
367D4—81no—
368D4—164no—
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—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
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#50812yes—
409C2×C4#53711yescharacteristic
410C2×C4#54611yescharacteristic
411C2×C4#54611yescharacteristic
412C2×C4#57511yescharacteristic
413C2×C4#57711yescharacteristic
414C2×C4#57711yescharacteristic
415C8—41no—
416C8—41no—
417C8—41no—
418C8#53911yescharacteristic
419C8#54211yescharacteristic
420C8#55311yescharacteristic
421C8#56011yescharacteristic
422C22—22no—
423C22—22no—
424C22—22no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—161no—
430C22—161no—
431C22—161no—
432C22—161no—
433C22—161no—
434C22—161no—
435C22—161no—
436C22#324511yescharacteristic
437C22#329411yescharacteristic
438C22#329411yescharacteristic
439C22C2×#50811yescharacteristic
440C22C2×#50911yescharacteristic
441C22#341611yescharacteristic
442C22#341711yescharacteristic
443C4—22no—
444C4—41no—
445C4—41no—
446C4—41no—
447C4—41no—
448C4—82no—
449C4—82no—
450C4—161no—
451C4—161no—
452C4—161no—
453C4—161no—
454C4—161no—
455C4#269111yescharacteristic
456C4#326011yescharacteristic
457C4#331111yescharacteristic
458C4#331711yescharacteristic
459C2—42no—
460C2—321no—
461C2#3148611yescharacteristic
462C2#3381311yescharacteristic
463C2#3382411yescharacteristic
464C2#3386711yescharacteristic
465C2#3387111yescharacteristic
466C2#3387311yescharacteristic
467C2#3387511yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
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. #27839C2semidirect
3. #13516C2semidirect
4. #13480C2semidirect
5. #13476C2semidirect
6. #13408C2semidirect
7. #13382C2semidirect
9. #13254C2semidirect
10. #13112C2semidirect
11. #13060C2semidirect
13. #12936C2semidirect
14. #12912C2semidirect
15. #12812C2semidirect

Maximal subgroups

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

subgroupindex
#278392
#254122
#135162
#134802
#134762
#134082
#133822
#133802
#132542
#131122
#130602
#130592
#129362
#129122
#128122