Finite Group Databasebeta

← #507380

Subgroups — #507380

Order 512.

652 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33216C211yesmaximal, characteristic, complemented
2#21230C211yesmaximal, characteristic
3#13481C211yesmaximal, characteristic, complemented
4#13480C211yesmaximal, characteristic, complemented
5#13366C211yesmaximal, characteristic, complemented
6#13365C211yesmaximal, characteristic, complemented
7#13150C211yesmaximal, characteristic, complemented
8#13149C211yesmaximal, characteristic, complemented
9#13148C211yesmaximal, characteristic, complemented
10#13147C211yesmaximal, characteristic, complemented
11#12867C211yesmaximal, characteristic, complemented
12#12866C211yesmaximal, characteristic, complemented
13#12733C211yesmaximal, characteristic
14#12733C211yesmaximal, characteristic
15#12600C211yesmaximal, characteristic, complemented
16C22×#135C2211yescharacteristic
17C2×#499C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C2×#417—21no—
20C2×#405C2211yescharacteristic
21#2629—21no—
22#2311—21no—
23C2×#358C2211yescharacteristic
24C2×#358C2211yescharacteristic
25C2×#358C2211yescharacteristic
26C2×#358C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×C22⋊C16C2211yescharacteristic
32C2×C22⋊C16C2211yescharacteristic
33C2×#346C2211yescharacteristic
34C2×C4×C16C2211yescharacteristic
35#1992—21no—
36#1973—21no—
37#1973—21no—
38#1970C2211yescharacteristic
39#1970C2211yescharacteristic
40#1969C2211yescharacteristic
41#1969C2211yescharacteristic
42#1943—21no—
43#1942C2211yescharacteristic
44#1942C2211yescharacteristic
45#1845C2211yescharacteristic
46#1809—21no—
47#1808C2211yescharacteristic
48#1808C2211yescharacteristic
49#1380C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×#41—21no—
60C2×#135—22no—
61C2×#135—22no—
62C2×#135—22no—
63C2×#135—22no—
64C2×#135—41no—
65C2×#135—41no—
66C22×Q16C2311yescharacteristic
67C22×SD16—21no—
68C22×D8C2311yescharacteristic
69C22×M16D411yescharacteristic
70C23×C8C2311yescharacteristic
71C2×#125—21no—
72C2×#117D411yescharacteristic
73C22×C4⋊C4—21no—
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78#499—41no—
79#499—41no—
80C2×#107—22no—
81C2×#107—22no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#106—21no—
91#417—44no—
92C2×#103—21no—
93C2×#103—21no—
94C2×#102—21no—
95C2×#102—21no—
96#405—41no—
97#405—41no—
98C2×#97D411yescharacteristic
99C2×#97D411yescharacteristic
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102#398—41no—
103#398—41no—
104#396—41no—
105#396—41no—
106#391—21no—
107#391—21no—
108#390—21no—
109#390—21no—
110#389—21no—
111#389—21no—
112#386—41no—
113#386—41no—
114#384—21no—
115#380—21no—
116#380—21no—
117#380—41no—
118#380—41no—
119#358—22no—
120#358—22no—
121#358—41no—
122#358—41no—
123#357—22no—
124#357—22no—
125#357—41no—
126#357—41no—
127C22⋊C16—22no—
128C22⋊C16—22no—
129#346—41no—
130C4×C16—41no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135C2×#41—24no—
136C2×#41—42no—
137C2×#41—42no—
138C2×#41—42no—
139C2×#41—42no—
140C2×#41—42no—
141C22×Q8—21no—
142C22×D4—21no—
143C22×D4—41no—
144C23×C4—41no—
145C23×C4—41no—
146C23×C4C2×D411yescharacteristic
147#135—44no—
148#135—44no—
149#135—44no—
150#135—44no—
151C2×Q16—21no—
152C2×Q16—21no—
153C2×Q16—21no—
154C2×Q16—21no—
155C2×Q16—21no—
156C2×Q16#4112yes—
157C2×SD16—24no—
158C2×SD16—42no—
159C2×SD16—42no—
160C2×D8—21no—
161C2×D8—21no—
162C2×D8—21no—
163C2×D8—21no—
164C2×D8—21no—
165C2×D8#4112yes—
166C2×M16—41no—
167C2×M16—41no—
168C2×M16SD1614yes—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8#4112yes—
177C22×C8#4112yes—
178C22×C8C2411yescharacteristic
179#125—42no—
180#125—42no—
181#117—41no—
182#117—41no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4D811yescharacteristic
199C2×C4⋊C4D811yescharacteristic
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4D811yescharacteristic
207C2×C22⋊C4D811yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C16—21no—
211C2×C16—21no—
212C2×C16—21no—
213C2×C16—21no—
214C2×C16—21no—
215C2×C16—21no—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16—21no—
219C2×C16—21no—
220C2×C16—21no—
221C2×C16—21no—
222#107—21no—
223#107—21no—
224#107—21no—
225#107—21no—
226#107—21no—
227#107—21no—
228#107—21no—
229#107—21no—
230#107—21no—
231#107—21no—
232#107—21no—
233#107—21no—
234#107—42no—
235#107—42no—
236#106—42no—
237#103—42no—
238#103—42no—
239#102—42no—
240#102—42no—
241#97—41no—
242#97—41no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251#95—41no—
252#95—41no—
253#95—41no—
254#95—41no—
255C24—41no—
256C24C2×D811yescharacteristic
257#41—48no—
258#41—48no—
259#41—84no—
260#41—84no—
261C2×Q8—22no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×D4—22no—
268C2×D4—44no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—82no—
275C2×D4—82no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—44no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—161no—
292C22×C4—161no—
293C22×C4#12412yes—
294C22×C4C2×D811yescharacteristic
295C22×C4C2×D811yescharacteristic
296C22×C4C2×D811yescharacteristic
297C22×C4C2×D811yescharacteristic
298C22×C4C2×D811yescharacteristic
299C22×C4C2×SD1612yes—
300C22×C4C22×D411yescharacteristic
301Q16—22no—
302Q16—22no—
303Q16—41no—
304Q16—41no—
305SD16—48no—
306D8—22no—
307D8—22no—
308D8—41no—
309D8—41no—
310M16—44no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8#12412yes—
322C2×C8C2×SD1612yes—
323C2×C8C2×SD1612yes—
324C2×C8C2×#4111yescharacteristic
325C2×C8C2×#4111yescharacteristic
326C2×C8C2×#4111yescharacteristic
327C2×C8C2×#4111yescharacteristic
328C2×C8H(5,𝔽2)11yescharacteristic
329C2×C8H(5,𝔽2)11yescharacteristic
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C4⋊C4—82no—
347C4⋊C4—161no—
348C4⋊C4—161no—
349C4⋊C4—161no—
350C4⋊C4—161no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—161no—
354C22⋊C4—161no—
355C22⋊C4—161no—
356C22⋊C4—161no—
357C22⋊C4—161no—
358C22⋊C4—161no—
359C42—21no—
360C42—21no—
361C42—21no—
362C42—21no—
363C16—41no—
364C16—41no—
365C16—41no—
366C16—41no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—42no—
374C23—81no—
375C23—81no—
376C23—81no—
377C23—81no—
378C23—81no—
379C23—81no—
380C23#48612yes—
381C23C22×D811yescharacteristic
382Q8—42no—
383Q8—42no—
384Q8—81no—
385Q8—81no—
386D4—42no—
387D4—42no—
388D4—88no—
389D4—81no—
390D4—81no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—42no—
410C2×C4—88no—
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—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—164no—
432C2×C4#48612yes—
433C2×C4#52511yescharacteristic
434C2×C4C2×#12411yescharacteristic
435C2×C4#53411yescharacteristic
436C2×C4C22×SD1611yescharacteristic
437C2×C4#57511yescharacteristic
438C2×C4#57511yescharacteristic
439C8—22no—
440C8—22no—
441C8—22no—
442C8#55111yescharacteristic
443C8#55111yescharacteristic
444C8#55111yescharacteristic
445C8#55111yescharacteristic
446C22—22no—
447C22—22no—
448C22—22no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—82no—
454C22—82no—
455C22—82no—
456C22—161no—
457C22—161no—
458C22—161no—
459C22—161no—
460C22#310011yescharacteristic
461C22C2×#48611yescharacteristic
462C22#318611yescharacteristic
463C22#341511yescharacteristic
464C22#341611yescharacteristic
465C22#341611yescharacteristic
466C22#341711yescharacteristic
467C4—22no—
468C4—41no—
469C4—41no—
470C4—41no—
471C4—41no—
472C4—82no—
473C4—161no—
474C4—161no—
475C4—161no—
476C4—161no—
477C4—161no—
478C4—161no—
479C4#256111yescharacteristic
480C4#310111yescharacteristic
481C4#314011yescharacteristic
482C4#318711yescharacteristic
483C2—42no—
484C2—162no—
485C2#2910311yescharacteristic
486C2#3367911yescharacteristic
487C2#3368011yescharacteristic
488C2#3370211yescharacteristic
489C2#3370311yescharacteristic
490C2#3376911yescharacteristic
491C2#3377011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×#135C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33216C2semidirect
3. #13481C2semidirect
4. #13480C2semidirect
5. #13366C2semidirect
6. #13365C2semidirect
7. #13150C2semidirect
8. #13149C2semidirect
9. #13148C2semidirect
10. #13147C2semidirect
11. #12867C2semidirect
12. #12866C2semidirect
15. #12600C2semidirect

Maximal subgroups

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

subgroupindex
#332162
#212302
#134812
#134802
#133662
#133652
#131502
#131492
#131482
#131472
#128672
#128662
#127332
#126002