Finite Group Databasebeta

← #503826

Subgroups — #503826

Order 512.

671 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2232C211yesmaximal, characteristic, complemented
2C22×#366C211yesmaximal, characteristic, complemented
3#24780C211yesmaximal, characteristic
4D4×#107C211yesmaximal, characteristic, complemented
5#13526C211yesmaximal, characteristic, complemented
6#13505C211yesmaximal, characteristic, complemented
7#13492C211yesmaximal, characteristic, complemented
8#13380C211yesmaximal, characteristic, complemented
9#13130C211yesmaximal, characteristic, complemented
10#13109C211yesmaximal, characteristic, complemented
11#13059C211yesmaximal, characteristic, complemented
12#12991C211yesmaximal, characteristic, complemented
13#12976C211yesmaximal, characteristic, complemented
14#12919C211yesmaximal, characteristic, complemented
15#12919C211yesmaximal, characteristic, complemented
16C23×C16C2211yescharacteristic
17C2×#466C2211yescharacteristic
18C2×#466C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#107C2211yescharacteristic
23#2416—21no—
24D4×C4⋊C4—21no—
25#2232—24no—
26C2×#366—24no—
27C2×#366C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×#366C2211yescharacteristic
31C2×#366C2211yescharacteristic
32C2×#366C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×C22⋊C16C2211yescharacteristic
38C2×C22⋊C16C2211yescharacteristic
39C2×C22⋊C16C2211yescharacteristic
40#2067—21no—
41#2033—21no—
42#1970C2211yescharacteristic
43#1947—21no—
44#1942C2211yescharacteristic
45#1874—21no—
46#1852—21no—
47#1852—21no—
48#1845C2211yescharacteristic
49#1808C2211yescharacteristic
50#1808C2211yescharacteristic
51#1808C2211yescharacteristic
52C4×#107C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×D8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C2×#121—21no—
65C2×#121—21no—
66C2×#121—41no—
67C2×C4×D4—41no—
68C2×C4×D4D411yescharacteristic
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C16—22no—
73C22×C16—22no—
74C22×C16—22no—
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78C22×C16C2311yescharacteristic
79#466—22no—
80#466—22no—
81#466—22no—
82#466—41no—
83C8×D4—22no—
84C8×D4—22no—
85C2×#107—22no—
86C2×#107—22no—
87C2×#107—22no—
88C2×#107—21no—
89C2×#107—21no—
90C2×#107—21no—
91C2×#107—21no—
92C2×#107—41no—
93C2×#107—41no—
94C2×#107—41no—
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#107C2311yescharacteristic
98C2×#107C2311yescharacteristic
99C2×#107C2311yescharacteristic
100C2×#107C2311yescharacteristic
101C2×#107C2311yescharacteristic
102C2×C4⋊C8D411yescharacteristic
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×C22⋊C8D411yescharacteristic
107C2×C22⋊C8D411yescharacteristic
108C2×C4×C8C2311yescharacteristic
109#396—41no—
110#394—41no—
111#390—21no—
112#389—21no—
113#389—41no—
114#386—41no—
115#385—41no—
116#385—41no—
117#384—21no—
118#380—21no—
119#380—21no—
120#380—21no—
121C4×C4⋊C4—21no—
122#366—22no—
123#366—22no—
124#366—22no—
125#366—22no—
126#366—41no—
127#366D414yescomplemented
128#357—22no—
129#357—22no—
130#357—22no—
131#357—41no—
132C22⋊C16—22no—
133C22⋊C16—22no—
134C22⋊C16—22no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140C22×D4—21no—
141C22×D4—41no—
142C22×D4—41no—
143C22×D4D811yescharacteristic
144C23×C4—41no—
145C23×C4—41no—
146C23×C4—41no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C2×D8—21no—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8—21no—
153C2×D8—21no—
154C2×D8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8#4112yes—
169C22×C8#4112yes—
170C22×C8C2411yescharacteristic
171#121—42no—
172#121—42no—
173#121—42no—
174#121—81no—
175#121—84no—
176C4×D4—22no—
177C4×D4—22no—
178C4×D4—84no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—81no—
196C2×C4⋊C4—81no—
197C2×C4⋊C4—81no—
198C2×C4⋊C4D811yescharacteristic
199C2×C22⋊C4—41no—
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×C42—41no—
209C2×C42C2×D411yescharacteristic
210C2×C16—28no—
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×C16C2×D412yes—
222C2×C16#4112yes—
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—21no—
235#107—42no—
236#107—42no—
237#107—42no—
238#107—81no—
239#107C2×D412yes—
240#107C2×D412yes—
241C4⋊C8—22no—
242#102—42no—
243#102—42no—
244#102—81no—
245C22⋊C8—22no—
246C22⋊C8—22no—
247C4×C8—21no—
248C4×C8—21no—
249#95—41no—
250#95—41no—
251#95—41no—
252#95—41no—
253#95—41no—
254C24—41no—
255C24C2×D811yescharacteristic
256C24C2×D811yescharacteristic
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—82no—
266C2×D4—82no—
267C2×D4—82no—
268C2×D4—82no—
269C2×D4—82no—
270C2×D4—161no—
271C2×D4D1614yes—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4—161no—
296C22×C4—161no—
297C22×C4—161no—
298C22×C4#12312yes—
299C22×C4C2×D811yescharacteristic
300C22×C4C2×D811yescharacteristic
301C22×C4C2×D811yescharacteristic
302C22×C4C2×D811yescharacteristic
303C22×C4#13712yes—
304C22×C4C22×D411yescharacteristic
305D8—41no—
306D8—41no—
307D8—41no—
308D8—41no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8#12312yes—
326C2×C8C22×D411yescharacteristic
327C2×C8C2×#4111yescharacteristic
328C2×C8C2×#4111yescharacteristic
329C2×C8C2×#4111yescharacteristic
330C2×C8H(5,𝔽2)11yescharacteristic
331C2×C82−1+411yescharacteristic
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
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—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—82no—
348C4⋊C4—82no—
349C4⋊C4—82no—
350C4⋊C4—82no—
351C4⋊C4—82no—
352C4⋊C4—161no—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—82no—
356C22⋊C4—82no—
357C22⋊C4—82no—
358C22⋊C4—82no—
359C22⋊C4—82no—
360C22⋊C4—82no—
361C22⋊C4—161no—
362C42—21no—
363C42—21no—
364C42—82no—
365C16—22no—
366C16—22no—
367C16—41no—
368C16—41no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—81no—
380C23—81no—
381C23—81no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23#48012yes—
387C23C2×D1612yes—
388C23C2×D1612yes—
389C23#51012yes—
390C23C22×D811yescharacteristic
391D4—28no—
392D4—81no—
393D4—81no—
394D4—81no—
395D4—81no—
396D4—164no—
397D4—164no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
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—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—164no—
438C2×C4—164no—
439C2×C4—164no—
440C2×C4#48012yes—
441C2×C4C2×D1612yes—
442C2×C4C2×#12311yescharacteristic
443C2×C4D4211yescharacteristic
444C2×C4#54711yescharacteristic
445C2×C4C2×#13711yescharacteristic
446C2×C4#57611yescharacteristic
447C2×C4#57811yescharacteristic
448C8—22no—
449C8—41no—
450C8—41no—
451C8#52511yescharacteristic
452C8#53911yescharacteristic
453C8#54811yescharacteristic
454C8#55311yescharacteristic
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—161no—
469C22—161no—
470C22—161no—
471C22—161no—
472C22—161no—
473C22—161no—
474C22—161no—
475C22#223214yescomplemented
476C22C2×#48011yescharacteristic
477C22D4×D811yescharacteristic
478C22#333411yescharacteristic
479C22C22×D1611yescharacteristic
480C22C2×#51011yescharacteristic
481C22#341511yescharacteristic
482C22#341711yescharacteristic
483C4—22no—
484C4—22no—
485C4—41no—
486C4—41no—
487C4—82no—
488C4—82no—
489C4—161no—
490C4—161no—
491C4—161no—
492C4—161no—
493C4—161no—
494C4#244311yescharacteristic
495C4#309811yescharacteristic
496C4#327011yescharacteristic
497C4#334611yescharacteristic
498C2—24no—
499C2—42no—
500C2—321no—
501C2#2612611yescharacteristic
502C2C2×#223211yescharacteristic
503C2#3365211yescharacteristic
504C2#3383611yescharacteristic
505C2D4×D1611yescharacteristic
506C2#3390111yescharacteristic
507C2#3391311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#24780C2
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
1. C2×#2232C2semidirect
2. C22×#366C2semidirect
4. D4×#107C2semidirect
5. #13526C2semidirect
6. #13505C2semidirect
7. #13492C2semidirect
8. #13380C2semidirect
9. #13130C2semidirect
10. #13109C2semidirect
11. #13059C2semidirect
12. #12991C2semidirect
13. #12976C2semidirect
14. #12919C2semidirect
15. #12919C2semidirect
127. #366D4semidirect
475. C22#2232semidirect

Maximal subgroups

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

subgroupindex
C2×#22322
C22×#3662
#247802
D4×#1072
#135262
#135052
#134922
#133802
#131302
#131092
#130592
#129912
#129762
#129192