Finite Group Databasebeta

← #510841

Subgroups — #510841

Order 512.

549 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28225C211yesmaximal, characteristic
2#27660C211yesmaximal, characteristic, complemented
3#13539C211yesmaximal, characteristic, complemented
4#13507C211yesmaximal, characteristic, complemented
5#13493C211yesmaximal, characteristic, complemented
6#13492C211yesmaximal, characteristic, complemented
7#13381C211yesmaximal, characteristic, complemented
8#13380C211yesmaximal, characteristic, complemented
9#13059C211yesmaximal, characteristic, complemented
10#12992C211yesmaximal, characteristic, complemented
11#12976C211yesmaximal, characteristic, complemented
12#12937C211yesmaximal, characteristic, complemented
13#12935C211yesmaximal, characteristic, complemented
14#12921C211yesmaximal, characteristic, complemented
15#12835C211yesmaximal, characteristic, complemented
16C23×C16C2211yescharacteristic
17C2×#466C2211yescharacteristic
18C2×#466C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C2×#427C2211yescharacteristic
22#2535—21no—
23#2485—21no—
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×C22⋊C16C2211yescharacteristic
33C2×C22⋊C16C2211yescharacteristic
34C2×C22⋊C16C2211yescharacteristic
35#2088—21no—
36#2077—21no—
37#2034—21no—
38#2034—21no—
39#1942C2211yescharacteristic
40#1942C2211yescharacteristic
41#1846C2211yescharacteristic
42#1825—21no—
43#1808C2211yescharacteristic
44#1808C2211yescharacteristic
45#1808C2211yescharacteristic
46#1808C2211yescharacteristic
47#1808C2211yescharacteristic
48#1804—21no—
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C2×#121—21no—
63C2×#121—21no—
64C2×C4×D4—21no—
65C2×C4×D4D411yescharacteristic
66C2×#117—21no—
67C22×C16—22no—
68C22×C16—41no—
69C22×C16—41no—
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74#466—22no—
75#466—22no—
76#466—41no—
77#466—41no—
78C4×D8—41no—
79C4×D8—41no—
80C8×D4—41no—
81C8×D4—41no—
82#427—41no—
83#427—41no—
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#106—21no—
92C2×#106—21no—
93C2×C4⋊C8D411yescharacteristic
94C2×#102—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×C22⋊C8D411yescharacteristic
98C2×C22⋊C8D411yescharacteristic
99C2×C4×C8C2311yescharacteristic
100#397—41no—
101#397—41no—
102#394—41no—
103#392—41no—
104#389—21no—
105#389—21no—
106#388—41no—
107#387—41no—
108#386—41no—
109#386—41no—
110#382—21no—
111#380—21no—
112#380—21no—
113#380—21no—
114#380—21no—
115#380—21no—
116#367—41no—
117#367—41no—
118#366—41no—
119#366—41no—
120#357—22no—
121#357—22no—
122#357—41no—
123#357—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×D4—21no—
135C22×D4—41no—
136C22×D4—41no—
137C22×D4D811yescharacteristic
138C23×C4—41no—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C2×D8—21no—
142C2×D8—21no—
143C2×D8—21no—
144C2×D8—21no—
145C2×D8—21no—
146C2×D8#4112yes—
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×C8—21no—
159C22×C8C2×D411yescharacteristic
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D411yescharacteristic
162C22×C8C2411yescharacteristic
163#121—42no—
164#121—42no—
165#121—81no—
166#121—81no—
167C4×D4—41no—
168C4×D4—41no—
169C4×D4—42no—
170C4×D4—42no—
171#117—42no—
172#117—42no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4D811yescharacteristic
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×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—21no—
223#106—42no—
224#106—81no—
225C4⋊C8—41no—
226#102—42no—
227#102—42no—
228#102—42no—
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—
242C24—41no—
243C24C2×D811yescharacteristic
244C24C2×D811yescharacteristic
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—82no—
255C2×D4—82no—
256C2×D4—82no—
257C2×D4—82no—
258C2×D4—161no—
259C2×D4—161no—
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—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—82no—
280C22×C4—161no—
281C22×C4—161no—
282C22×C4C2×D811yescharacteristic
283C22×C4C2×D811yescharacteristic
284C22×C4C2×D811yescharacteristic
285C22×C4C2×D811yescharacteristic
286C22×C4C22×D411yescharacteristic
287D8—22no—
288D8—22no—
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—21no—
299C2×C8—21no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8C2×#4111yescharacteristic
309C2×C8C2×#4111yescharacteristic
310C2×C8H(5,𝔽2)11yescharacteristic
311C2×C8H(5,𝔽2)11yescharacteristic
312C2×C8H(5,𝔽2)11yescharacteristic
313C2×C82−1+411yescharacteristic
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—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—82no—
330C4⋊C4—161no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C22⋊C4—82no—
338C22⋊C4—161no—
339C22⋊C4—161no—
340C22⋊C4—161no—
341C22⋊C4—161no—
342C42—21no—
343C42—21no—
344C42—82no—
345C16—22no—
346C16—41no—
347C16—41no—
348C16—41no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—42no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23#50812yes—
370C23C22×D811yescharacteristic
371D4—44no—
372D4—42no—
373D4—42no—
374D4—81no—
375D4—81no—
376D4—164no—
377D4—164no—
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—21no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
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—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—164no—
420C2×C4#53411yescharacteristic
421C2×C4#54611yescharacteristic
422C2×C4#54711yescharacteristic
423C2×C4#57511yescharacteristic
424C2×C4#57711yescharacteristic
425C2×C4#57811yescharacteristic
426C8—41no—
427C8—41no—
428C8—41no—
429C8#53811yescharacteristic
430C8#53911yescharacteristic
431C8#55311yescharacteristic
432C8#56211yescharacteristic
433C22—22no—
434C22—22no—
435C22—22no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—82no—
448C22—82no—
449C22—82no—
450C22—161no—
451C22—161no—
452C22—161no—
453C22—161no—
454C22#318611yescharacteristic
455C22#329411yescharacteristic
456C22#333411yescharacteristic
457C22C2×#50811yescharacteristic
458C22#341411yescharacteristic
459C22#341511yescharacteristic
460C22#341611yescharacteristic
461C4—41no—
462C4—41no—
463C4—41no—
464C4—41no—
465C4—161no—
466C4—161no—
467C4—161no—
468C4—161no—
469C4—161no—
470C4—161no—
471C4—161no—
472C4#270811yescharacteristic
473C4#320111yescharacteristic
474C4#331511yescharacteristic
475C4#334811yescharacteristic
476C2—42no—
477C2—81no—
478C2—162no—
479C2#3199411yescharacteristic
480C2#3375711yescharacteristic
481C2#3376411yescharacteristic
482C2#3386411yescharacteristic
483C2#3387811yescharacteristic
484C2#3390411yescharacteristic
485C2#3391411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#28225C2
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
2. #27660C2semidirect
3. #13539C2semidirect
4. #13507C2semidirect
5. #13493C2semidirect
6. #13492C2semidirect
7. #13381C2semidirect
8. #13380C2semidirect
9. #13059C2semidirect
10. #12992C2semidirect
11. #12976C2semidirect
12. #12937C2semidirect
13. #12935C2semidirect
14. #12921C2semidirect
15. #12835C2semidirect

Maximal subgroups

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

subgroupindex
#282252
#276602
#135392
#135072
#134932
#134922
#133812
#133802
#130592
#129922
#129762
#129372
#129352
#129212
#128352