Finite Group Databasebeta

← #505342

Subgroups — #505342

Order 512.

619 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#25414C211yesmaximal, characteristic
2#22611C211yesmaximal, characteristic, complemented
3#13514C211yesmaximal, characteristic, complemented
4#13407C211yesmaximal, characteristic, complemented
5#13380C211yesmaximal, characteristic, complemented
6#13380C211yesmaximal, characteristic, complemented
7#13252C211yesmaximal, characteristic, complemented
8#13147C211yesmaximal, characteristic, complemented
9#13130C211yesmaximal, characteristic, complemented
10#13120C211yesmaximal, characteristic, complemented
11#13109C211yesmaximal, characteristic, complemented
12#12991C211yesmaximal, characteristic, complemented
13#12991C211yesmaximal, characteristic, complemented
14#12935C211yesmaximal, characteristic, complemented
15C4×#357C211yesmaximal, characteristic, complemented
16C2×#493C2211yescharacteristic
17C2×#466C2211yescharacteristic
18C2×#466C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20C22×#107C2211yescharacteristic
21#2428—21no—
22#2344—21no—
23C2×#366C2211yescharacteristic
24C2×#366C2211yescharacteristic
25C2×C4⋊C16C2211yescharacteristic
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2211yescharacteristic
29C2×#357C2211yescharacteristic
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×C22⋊C16C2211yescharacteristic
35C2×C22⋊C16C2211yescharacteristic
36C2×C4×C16C2211yescharacteristic
37#2041—21no—
38#1999—21no—
39#1970C2211yescharacteristic
40#1942C2211yescharacteristic
41#1942C2211yescharacteristic
42#1942C2211yescharacteristic
43#1918—21no—
44#1918—21no—
45#1879—21no—
46#1823—21no—
47#1808C2211yescharacteristic
48C4×#107C2211yescharacteristic
49C4×#107C2211yescharacteristic
50C8×C22⋊C4C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×D8—21no—
60C22×D8C2311yescharacteristic
61C22×D8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C2×#127—21no—
64C2×#121—21no—
65C2×#121—21no—
66C2×#121—41no—
67C2×C4×D4—21no—
68C2×#117—41no—
69C22×C4⋊C4—21no—
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74#493—22no—
75#493—41no—
76#466—22no—
77#466—22no—
78#466—41no—
79#466—41no—
80C4×D8—22no—
81C4×D8—41no—
82C2×#107—21no—
83C2×#107—22no—
84C2×#107—22no—
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#107C2311yescharacteristic
88C2×#107C2311yescharacteristic
89C2×#107C2311yescharacteristic
90C2×#107C2311yescharacteristic
91C2×#102—21no—
92C2×#102—21no—
93C2×#102—21no—
94C2×#102—21no—
95C2×C22⋊C8D411yescharacteristic
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98#394—41no—
99#392—41no—
100#390—21no—
101#389—21no—
102#389—21no—
103#389—21no—
104#387—41no—
105#387—41no—
106#386—41no—
107#381—41no—
108#380—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C22⋊C4D411yescharacteristic
112#366—22no—
113#366—41no—
114C4⋊C16—41no—
115#357—22no—
116#357—22no—
117#357—22no—
118#357—22no—
119#357—22no—
120#357—41no—
121#357—41no—
122#357—41no—
123C22⋊C16—41no—
124C22⋊C16—41no—
125C4×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×D4—21no—
135C22×D4—21no—
136C22×D4—41no—
137C22×D4—41no—
138C22×D4—41no—
139C22×D4—41no—
140C23×C4—41no—
141C23×C4—41no—
142C23×C4C2×D411yescharacteristic
143C2×D8—21no—
144C2×D8—21no—
145C2×D8—21no—
146C2×D8—21no—
147C2×D8—21no—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8—21no—
153C2×D8—21no—
154C2×D8—42no—
155C2×D8—42no—
156C2×D8—81no—
157C2×D8#4112yes—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2411yescharacteristic
168#127—42no—
169#127—81no—
170#121—42no—
171#121—42no—
172#121—81no—
173#121—81no—
174#121—84no—
175C4×D4—42no—
176C4×D4—81no—
177#117—84no—
178C2×C4⋊C4—24no—
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—42no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4D811yescharacteristic
197C2×C22⋊C4D811yescharacteristic
198C2×C42—41no—
199C2×C42—41no—
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
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—21no—
212C2×C16—21no—
213C2×C16—21no—
214#107—24no—
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—21no—
224#107#4112yes—
225#107#4112yes—
226#107#4112yes—
227#102—42no—
228#102—42no—
229#102—42no—
230#102—81no—
231C22⋊C8—41no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236#95—41no—
237#95—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95D811yescharacteristic
243#95D811yescharacteristic
244C24—41no—
245C24—41no—
246C24C2×D811yescharacteristic
247C2×D4—22no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—82no—
260C2×D4—82no—
261C2×D4—82no—
262C2×D4—82no—
263C2×D4—82no—
264C2×D4—82no—
265C2×D4—82no—
266C2×D4—82no—
267C2×D4—161no—
268C2×D4—161no—
269C2×D4—161no—
270C2×D4—161no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—44no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4—161no—
289C22×C4C2×D811yescharacteristic
290C22×C4C2×D811yescharacteristic
291C22×C4C2×D811yescharacteristic
292C22×C4C2×D811yescharacteristic
293C22×C4C2×D811yescharacteristic
294C22×C4C22×D411yescharacteristic
295D8—22no—
296D8—22no—
297D8—41no—
298D8—41no—
299D8—41no—
300D8—41no—
301D8—41no—
302D8—41no—
303D8—84no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8C2×#4111yescharacteristic
318C2×C8C2×#4111yescharacteristic
319C2×C8C2×#4111yescharacteristic
320C2×C8C2×#4111yescharacteristic
321C2×C8H(5,𝔽2)11yescharacteristic
322C2×C8H(5,𝔽2)11yescharacteristic
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—44no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C4⋊C4—82no—
339C22⋊C4—41no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C22⋊C4—161no—
346C22⋊C4D1614yes—
347C42—21no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—82no—
352C42—82no—
353C16—41no—
354C16—41no—
355C16—41no—
356C16—41no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—42no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23—81no—
377C23C2×D1612yes—
378C23C22×D811yescharacteristic
379D4—42no—
380D4—42no—
381D4—81no—
382D4—81no—
383D4—81no—
384D4—81no—
385D4—81no—
386D4—81no—
387D4—164no—
388D4—164no—
389D4—164no—
390D4—164no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—88no—
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—164no—
428C2×C4C2×D1612yes—
429C2×C4C2×D1612yes—
430C2×C4#52511yescharacteristic
431C2×C4#54611yescharacteristic
432C2×C4#54611yescharacteristic
433C2×C4#57511yescharacteristic
434C2×C4#57711yescharacteristic
435C2×C4#57711yescharacteristic
436C8—41no—
437C8—41no—
438C8—41no—
439C8#52911yescharacteristic
440C8#54011yescharacteristic
441C8#55311yescharacteristic
442C8#55511yescharacteristic
443C22—22no—
444C22—22no—
445C22—22no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—82no—
451C22—82no—
452C22—82no—
453C22—161no—
454C22—161no—
455C22—161no—
456C22—161no—
457C22—161no—
458C22—161no—
459C22—161no—
460C22—161no—
461C22—161no—
462C22—161no—
463C22—161no—
464C22#310011yescharacteristic
465C22#329411yescharacteristic
466C22#329411yescharacteristic
467C22C22×D1611yescharacteristic
468C22#341411yescharacteristic
469C22#341611yescharacteristic
470C22#341611yescharacteristic
471C4—22no—
472C4—22no—
473C4—41no—
474C4—41no—
475C4—41no—
476C4—82no—
477C4—82no—
478C4—82no—
479C4—161no—
480C4—161no—
481C4—161no—
482C4#247611yescharacteristic
483C4#312611yescharacteristic
484C4#331711yescharacteristic
485C4#332311yescharacteristic
486C2—42no—
487C2—162no—
488C2—321no—
489C2#2739011yescharacteristic
490C2#3366011yescharacteristic
491C2#3368611yescharacteristic
492C2#3386111yescharacteristic
493C2#3386111yescharacteristic
494C2#3388411yescharacteristic
495C2#3389011yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#25414C2
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
2. #22611C2semidirect
3. #13514C2semidirect
4. #13407C2semidirect
5. #13380C2semidirect
6. #13380C2semidirect
7. #13252C2semidirect
8. #13147C2semidirect
9. #13130C2semidirect
10. #13120C2semidirect
11. #13109C2semidirect
12. #12991C2semidirect
13. #12991C2semidirect
14. #12935C2semidirect
15. C4×#357C2semidirect

Maximal subgroups

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

subgroupindex
#254142
#226112
#135142
#134072
#133802
#132522
#131472
#131302
#131202
#131092
#129912
#129352
C4×#3572