Finite Group Databasebeta

← #508574

Subgroups — #508574

Order 512.

651 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33215C211yesmaximal, characteristic, complemented
2#21230C211yesmaximal, characteristic
3#13529C211yesmaximal, characteristic, complemented
4#13528C211yesmaximal, characteristic, complemented
5#13477C211yesmaximal, characteristic, complemented
6#13476C211yesmaximal, characteristic, complemented
7#13475C211yesmaximal, characteristic, complemented
8#13474C211yesmaximal, characteristic, complemented
9#13364C211yesmaximal, characteristic, complemented
10#13363C211yesmaximal, characteristic, complemented
11#12920C211yesmaximal, characteristic, complemented
12#12919C211yesmaximal, characteristic, complemented
13#12784C211yesmaximal, characteristic, complemented
14#12733C211yesmaximal, characteristic
15#12733C211yesmaximal, characteristic
16C22×#135C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C2×#417—21no—
20C2×#405C2211yescharacteristic
21#2628—21no—
22#2311—21no—
23C2×#367C2211yescharacteristic
24C2×#367C2211yescharacteristic
25C2×#366C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×C4⋊C16C2211yescharacteristic
28C2×C4⋊C16C2211yescharacteristic
29C2×#358C2211yescharacteristic
30C2×#358C2211yescharacteristic
31C2×#358C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×C22⋊C16C2211yescharacteristic
38C2×C22⋊C16C2211yescharacteristic
39#1993—21no—
40#1973—21no—
41#1973—21no—
42#1970C2211yescharacteristic
43#1970C2211yescharacteristic
44#1969C2211yescharacteristic
45#1969C2211yescharacteristic
46#1944—21no—
47#1942C2211yescharacteristic
48#1942C2211yescharacteristic
49#1846C2211yescharacteristic
50#1809—21no—
51#1808C2211yescharacteristic
52#1808C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59C22×#41—21no—
60C2×#135—22no—
61C2×#135—41no—
62C2×#135—41no—
63C2×#135—41no—
64C2×#135—41no—
65C2×#135Q814yescomplemented
66C22×Q16C2311yescharacteristic
67C22×SD16—21no—
68C22×D8C2311yescharacteristic
69C22×M16D411yescharacteristic
70C23×C8C2311yescharacteristic
71C2×C4⋊Q8—21no—
72C2×#117D411yescharacteristic
73C22×C4⋊C4—21no—
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78#500—41no—
79#500—41no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—41no—
83C2×#107—41no—
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#400—41no—
103#399—41no—
104#395—41no—
105#393—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#382—21no—
115#380—21no—
116#380—21no—
117#380—41no—
118#380—41no—
119#367—41no—
120#367—41no—
121#366—41no—
122#366—41no—
123C4⋊C16—41no—
124C4⋊C16—41no—
125#358—41no—
126#358—41no—
127#358—41no—
128#358—41no—
129#357—41no—
130#357—41no—
131#357—41no—
132#357—41no—
133C22⋊C16—22no—
134C22⋊C16—22no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139C2×#41—24no—
140C2×#41—42no—
141C2×#41—42no—
142C2×#41—42no—
143C2×#41—42no—
144C2×#41—42no—
145C22×Q8—21no—
146C22×Q8—41no—
147C22×D4—21no—
148C22×D4—41no—
149C23×C4—41no—
150C23×C4—41no—
151C23×C4C2×D411yescharacteristic
152#135—44no—
153#135—44no—
154#135—44no—
155#135—44no—
156C2×Q16—21no—
157C2×Q16—21no—
158C2×Q16—21no—
159C2×Q16—21no—
160C2×Q16—21no—
161C2×Q16C2×Q812yes—
162C2×SD16—24no—
163C2×SD16—42no—
164C2×SD16—42no—
165C2×D8—21no—
166C2×D8—21no—
167C2×D8—21no—
168C2×D8—21no—
169C2×D8—21no—
170C2×D8C2×Q812yes—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—41no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×Q812yes—
182C22×C8#4112yes—
183C22×C8C2411yescharacteristic
184C4⋊Q8—42no—
185C4⋊Q8—42no—
186#117—41no—
187#117—41no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4D811yescharacteristic
201C2×C4⋊C4D811yescharacteristic
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4D811yescharacteristic
209C2×C22⋊C4D811yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
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—
222C2×C16—21no—
223C2×C16—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—21no—
236#107—42no—
237#107—42no—
238#106—42no—
239#103—42no—
240#103—42no—
241#102—42no—
242#102—42no—
243#97—41no—
244#97—41no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249#95—41no—
250#95—41no—
251#95—41no—
252#95—41no—
253#95—41no—
254#95—41no—
255#95—41no—
256#95—41no—
257#95—41no—
258#95—41no—
259C24—41no—
260C24C2×D811yescharacteristic
261#41—48no—
262#41—48no—
263#41—84no—
264#41—84no—
265C2×Q8—22no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—82no—
272C2×Q8—161no—
273C2×Q8—161no—
274C2×D4—22no—
275C2×D4—44no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—82no—
282C2×D4—82no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—41no—
288C22×C4—44no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4—161no—
299C22×C4—161no—
300C22×C4C22⋊Q812yes—
301C22×C4C2×D811yescharacteristic
302C22×C4C2×D811yescharacteristic
303C22×C4C2×D811yescharacteristic
304C22×C4C2×D811yescharacteristic
305C22×C4C2×D811yescharacteristic
306C22×C4#13712yes—
307C22×C4C22×D411yescharacteristic
308Q16—22no—
309Q16—22no—
310Q16—41no—
311Q16—41no—
312SD16—48no—
313D8—22no—
314D8—22no—
315D8—41no—
316D8—41no—
317M16—44no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8C22⋊Q812yes—
330C2×C8AGL(1,ℤ/8)12yes—
331C2×C8C22×Q811yescharacteristic
332C2×C8C2×#4111yescharacteristic
333C2×C8H(5,𝔽2)11yescharacteristic
334C2×C8H(5,𝔽2)11yescharacteristic
335C2×C8H(5,𝔽2)11yescharacteristic
336C2×C8H(5,𝔽2)11yescharacteristic
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—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—82no—
352C4⋊C4—82no—
353C4⋊C4—161no—
354C4⋊C4—161no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—161no—
358C22⋊C4—161no—
359C22⋊C4—161no—
360C22⋊C4—161no—
361C22⋊C4—161no—
362C22⋊C4—161no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—21no—
367C16—41no—
368C16—41no—
369C16—41no—
370C16—41no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—42no—
378C23—81no—
379C23—81no—
380C23—81no—
381C23—81no—
382C23—81no—
383C23—81no—
384C23#47412yes—
385C23C22×D811yescharacteristic
386Q8—42no—
387Q8—42no—
388Q8—81no—
389Q8—81no—
390Q8—164no—
391D4—42no—
392D4—42no—
393D4—88no—
394D4—81no—
395D4—81no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—42no—
415C2×C4—88no—
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—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—164no—
437C2×C4#47412yes—
438C2×C4C2×C22⋊Q811yescharacteristic
439C2×C4#53411yescharacteristic
440C2×C4#53511yescharacteristic
441C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
442C2×C4C2×#13711yescharacteristic
443C2×C4#57511yescharacteristic
444C8—22no—
445C8—22no—
446C8—41no—
447C8#53811yescharacteristic
448C8#53811yescharacteristic
449C8#54311yescharacteristic
450C8#54311yescharacteristic
451C22—22no—
452C22—22no—
453C22—22no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—82no—
459C22—82no—
460C22—82no—
461C22—161no—
462C22—161no—
463C22—161no—
464C22—161no—
465C22C2×#47411yescharacteristic
466C22#318611yescharacteristic
467C22#320511yescharacteristic
468C22#341511yescharacteristic
469C22#341611yescharacteristic
470C22#341611yescharacteristic
471C22#341711yescharacteristic
472C4—22no—
473C4—41no—
474C4—41no—
475C4—41no—
476C4—41no—
477C4—82no—
478C4—161no—
479C4—161no—
480C4—161no—
481C4—161no—
482C4—161no—
483C4—161no—
484C4#262611yescharacteristic
485C4#307711yescharacteristic
486C4#319311yescharacteristic
487C4#321511yescharacteristic
488C2—42no—
489C2—162no—
490C2#2995011yescharacteristic
491C2#3363311yescharacteristic
492C2#3363411yescharacteristic
493C2#3376411yescharacteristic
494C2#3376711yescharacteristic
495C2#3379011yescharacteristic
496C2#3379111yescharacteristic
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. #33215C2semidirect
3. #13529C2semidirect
4. #13528C2semidirect
5. #13477C2semidirect
6. #13476C2semidirect
7. #13475C2semidirect
8. #13474C2semidirect
9. #13364C2semidirect
10. #13363C2semidirect
11. #12920C2semidirect
12. #12919C2semidirect
13. #12784C2semidirect
65. C2×#135Q8semidirect

Maximal subgroups

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

subgroupindex
#332152
#212302
#135292
#135282
#134772
#134762
#134752
#134742
#133642
#133632
#129202
#129192
#127842
#127332