Finite Group Databasebeta

← #503966

Subgroups — #503966

Order 512.

672 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2233C211yesmaximal, characteristic, complemented
2C2×#2235C211yesmaximal, characteristic, complemented
3#28484C211yesmaximal, characteristic
4D4×#107C211yesmaximal, characteristic, complemented
5#13383C211yesmaximal, characteristic, complemented
6#13382C211yesmaximal, characteristic, complemented
7#13381C211yesmaximal, characteristic, complemented
8#13380C211yesmaximal, characteristic, complemented
9#13254C211yesmaximal, characteristic, complemented
10#13205C211yesmaximal, characteristic, complemented
11#13190C211yesmaximal, characteristic, complemented
12#13177C211yesmaximal, characteristic, complemented
13#12993C211yesmaximal, characteristic, complemented
14#12991C211yesmaximal, characteristic, complemented
15#12674C211yesmaximal, characteristic, complemented
16C2×#495C2211yescharacteristic
17C2×#467C2211yescharacteristic
18C2×#467C2211yescharacteristic
19C2×#466C2211yescharacteristic
20C2×#466C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#107C2211yescharacteristic
24#2540—21no—
25D4×C4⋊C4—21no—
26#2235—24no—
27#2233—24no—
28C2×#367C2211yescharacteristic
29C2×#367C2211yescharacteristic
30C2×#367C2211yescharacteristic
31C2×#366C2211yescharacteristic
32C2×C4⋊C16C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#358C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×#357C2211yescharacteristic
40C2×#357C2211yescharacteristic
41C2×C22⋊C16C2211yescharacteristic
42C2×C22⋊C16C2211yescharacteristic
43C2×C22⋊C16C2211yescharacteristic
44C2×C22⋊C16C2211yescharacteristic
45C2×#346C2211yescharacteristic
46#2000—21no—
47#2000—21no—
48#1983—21no—
49#1947—21no—
50#1852—21no—
51#1852—21no—
52#1845C2211yescharacteristic
53#1808C2211yescharacteristic
54#1808C2211yescharacteristic
55C4×#107C2211yescharacteristic
56C4×#107C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×SD16—21no—
63C22×D8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C2×#124—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—21no—
69C2×#121—21no—
70C2×#121—21no—
71C2×C4×D4—41no—
72C2×C4×D4D411yescharacteristic
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C16C2311yescharacteristic
76C22×C16C2311yescharacteristic
77C22×C16C2311yescharacteristic
78C22×C16C2311yescharacteristic
79#495—22no—
80#495—22no—
81#467—22no—
82#467—22no—
83#467—41no—
84#467—41no—
85#466—22no—
86#466—22no—
87#466—41no—
88#466—41no—
89C8×D4—22no—
90C8×D4—22no—
91C2×#107—22no—
92C2×#107—22no—
93C2×#107—21no—
94C2×#107—21no—
95C2×#107—21no—
96C2×#107—41no—
97C2×#107—41no—
98C2×#107C2311yescharacteristic
99C2×#107C2311yescharacteristic
100C2×#107C2311yescharacteristic
101C2×#107C2311yescharacteristic
102C2×#107C2311yescharacteristic
103C2×#107C2311yescharacteristic
104C2×C4⋊C8D411yescharacteristic
105C2×#103—21no—
106C2×#103—21no—
107C2×#102—21no—
108C2×#102—21no—
109C2×C22⋊C8D411yescharacteristic
110C2×C22⋊C8D411yescharacteristic
111C2×C4×C8C2311yescharacteristic
112#399—41no—
113#392—41no—
114#390—41no—
115#390—41no—
116#389—41no—
117#385—41no—
118#385—41no—
119#384—21no—
120#380—21no—
121#380—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124#367—22no—
125#367—22no—
126#367D414yescomplemented
127#366—22no—
128C4⋊C16—22no—
129#358—22no—
130#358—22no—
131#358—41no—
132#358—41no—
133#357—22no—
134#357—22no—
135#357—41no—
136#357—41no—
137C22⋊C16—22no—
138C22⋊C16—22no—
139C22⋊C16—41no—
140C22⋊C16—41no—
141#346—22no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146C22×Q8—21no—
147C22×D4—21no—
148C22×D4—41no—
149C22×D4—41no—
150C22×D4D811yescharacteristic
151C23×C4—41no—
152C23×C4—41no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C2×Q16—21no—
156C2×Q16—21no—
157C2×Q16—21no—
158C2×Q16—21no—
159C2×Q16—21no—
160C2×Q16—21no—
161C2×SD16—42no—
162C2×SD16—81no—
163C2×SD16—81no—
164C2×D8—21no—
165C2×D8—21no—
166C2×D8—21no—
167C2×D8—21no—
168C2×D8—21no—
169C2×D8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C22×C8C2411yescharacteristic
186#124—42no—
187#124—42no—
188C22⋊Q8—42no—
189C22⋊Q8—42no—
190C22⋊Q8—81no—
191C22⋊Q8—81no—
192#121—42no—
193#121—42no—
194#121—81no—
195#121—81no—
196C4×D4—22no—
197C4×D4—22no—
198C4×D4—84no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—81no—
214C2×C4⋊C4—81no—
215C2×C4⋊C4D811yescharacteristic
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4D811yescharacteristic
225C2×C22⋊C4D811yescharacteristic
226C2×C42—41no—
227C2×C42—41no—
228C2×C42C2×D411yescharacteristic
229C2×C16—21no—
230C2×C16—21no—
231C2×C16—21no—
232C2×C16—21no—
233C2×C16—21no—
234C2×C16—21no—
235C2×C16—21no—
236C2×C16—21no—
237C2×C16—21no—
238C2×C16—21no—
239C2×C16—21no—
240C2×C16C2×D412yes—
241#107—21no—
242#107—21no—
243#107—21no—
244#107—21no—
245#107—21no—
246#107—21no—
247#107—21no—
248#107—21no—
249#107—21no—
250#107—42no—
251#107—42no—
252#107—81no—
253#107C2×D412yes—
254#107C2×D412yes—
255#107#4112yes—
256C4⋊C8—22no—
257#103—42no—
258#103—42no—
259#102—42no—
260#102—42no—
261C22⋊C8—22no—
262C22⋊C8—22no—
263C4×C8#4112yes—
264C4×C8#4112yes—
265#95—41no—
266#95—41no—
267#95—41no—
268#95—41no—
269C24—41no—
270C24C2×D811yescharacteristic
271C24C2×D811yescharacteristic
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—82no—
288C2×D4—82no—
289C2×D4—82no—
290C2×D4—82no—
291C2×D4—161no—
292C2×D4—161no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—82no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4—161no—
317C22×C4—161no—
318C22×C4C2×D811yescharacteristic
319C22×C4C2×D811yescharacteristic
320C22×C4C2×D811yescharacteristic
321C22×C4C2×D811yescharacteristic
322C22×C4C22×D411yescharacteristic
323Q16—41no—
324Q16—41no—
325Q16—41no—
326Q16—41no—
327SD16—84no—
328D8—41no—
329D8—41no—
330D8—41no—
331D8—41no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8#12312yes—
349C2×C8C22×D411yescharacteristic
350C2×C8C2×#4111yescharacteristic
351C2×C8C2×#4111yescharacteristic
352C2×C8C2×#4111yescharacteristic
353C2×C8H(5,𝔽2)11yescharacteristic
354C2×C8H(5,𝔽2)11yescharacteristic
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—82no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4—82no—
372C4⋊C4—82no—
373C4⋊C4—161no—
374C22⋊C4—22no—
375C22⋊C4—22no—
376C22⋊C4—82no—
377C22⋊C4—82no—
378C22⋊C4—82no—
379C22⋊C4—82no—
380C22⋊C4—82no—
381C22⋊C4—82no—
382C22⋊C4—82no—
383C22⋊C4—82no—
384C42—82no—
385C42—161no—
386C42#12312yes—
387C42#13712yes—
388C16—22no—
389C16—41no—
390C16—41no—
391C16—41no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—81no—
405C23—81no—
406C23—81no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23—81no—
411C23#50912yes—
412C23#51012yes—
413C23C22×D811yescharacteristic
414Q8—81no—
415Q8—81no—
416Q8—81no—
417Q8—81no—
418D4—28no—
419D4—81no—
420D4—81no—
421D4—81no—
422D4—81no—
423D4—164no—
424D4—164no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—164no—
465C2×C4—164no—
466C2×C4#48012yes—
467C2×C4#48012yes—
468C2×C4C2×#12311yescharacteristic
469C2×C4D4211yescharacteristic
470C2×C4#54611yescharacteristic
471C2×C4C2×#13711yescharacteristic
472C2×C4#57611yescharacteristic
473C2×C4#57711yescharacteristic
474C8—22no—
475C8—41no—
476C8—41no—
477C8#53211yescharacteristic
478C8#54011yescharacteristic
479C8#55011yescharacteristic
480C8#55911yescharacteristic
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—41no—
488C22—41no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—161no—
496C22—161no—
497C22—161no—
498C22—161no—
499C22—161no—
500C22—161no—
501C22—161no—
502C22C2×#48011yescharacteristic
503C22D4×D811yescharacteristic
504C22#329411yescharacteristic
505C22C2×#50911yescharacteristic
506C22C2×#51011yescharacteristic
507C22#341611yescharacteristic
508C22#341611yescharacteristic
509C4—22no—
510C4—22no—
511C4—41no—
512C4—41no—
513C4—82no—
514C4—82no—
515C4—82no—
516C4—161no—
517C4—161no—
518C4—161no—
519C4—161no—
520C4#244511yescharacteristic
521C4#309611yescharacteristic
522C4#327811yescharacteristic
523C4#333311yescharacteristic
524C2—42no—
525C2—42no—
526C2—321no—
527C2#2623011yescharacteristic
528C2#3364611yescharacteristic
529C2#3365311yescharacteristic
530C2#3383111yescharacteristic
531C2#3383211yescharacteristic
532C2#3387311yescharacteristic
533C2#3388311yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#28484C2
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×#2233C2semidirect
2. C2×#2235C2semidirect
4. D4×#107C2semidirect
5. #13383C2semidirect
6. #13382C2semidirect
7. #13381C2semidirect
8. #13380C2semidirect
9. #13254C2semidirect
10. #13205C2semidirect
11. #13190C2semidirect
12. #13177C2semidirect
13. #12993C2semidirect
14. #12991C2semidirect
15. #12674C2semidirect
126. #367D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22332
C2×#22352
#284842
D4×#1072
#133832
#133822
#133812
#133802
#132542
#132052
#131902
#131772
#129932
#129912
#126742