Finite Group Databasebeta

← #506199

Subgroups — #506199

Order 512.

792 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2168C211yesmaximal, characteristic, complemented
2C2×#2140C211yesmaximal, characteristic, complemented
3#27660C211yesmaximal, characteristic, complemented
4#24267C211yesmaximal, characteristic
5#13476C211yesmaximal, characteristic, complemented
6#13474C211yesmaximal, characteristic, complemented
7#13381C211yesmaximal, characteristic, complemented
8#13380C211yesmaximal, characteristic, complemented
9#13149C211yesmaximal, characteristic, complemented
10#13149C211yesmaximal, characteristic, complemented
11#13059C211yesmaximal, characteristic, complemented
12#13059C211yesmaximal, characteristic, complemented
13#12868C211yesmaximal, characteristic, complemented
14#12836C211yesmaximal, characteristic, complemented
15#12731C211yesmaximal, characteristic, complemented
16C23×D8C2211yescharacteristic
17C22×M32C2211yescharacteristic
18C23×C16C2211yescharacteristic, complemented
19C2×#466C2211yescharacteristic
20C2×#466C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22C2×#427C2211yescharacteristic
23#2485—21no—
24#2408—21no—
25#2168—24no—
26C2×#367C2211yescharacteristic, complemented
27C2×#366C2211yescharacteristic, complemented
28C2×C4⋊C16C2211yescharacteristic, complemented
29#2140—24no—
30C2×#357C2211yescharacteristic
31C2×#357C2211yescharacteristic
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×C22⋊C16C2211yescharacteristic
39C2×C22⋊C16C2211yescharacteristic, complemented
40C2×#346C2211yescharacteristic, complemented
41#2088—21no—
42#2077—21no—
43#2054—21no—
44#1999—21no—
45#1970C2211yescharacteristic
46#1970C2211yescharacteristic
47#1942C2211yescharacteristic, complemented
48#1942C2211yescharacteristic, complemented
49#1825—21no—
50#1808C2211yescharacteristic
51#1808C2211yescharacteristic
52#1808C2211yescharacteristic, complemented
53#1808C2211yescharacteristic, complemented
54#1786—21no—
55#1380C2211yescharacteristic, complemented
56#1380C2211yescharacteristic, complemented
57#1380C2211yescharacteristic, complemented
58#1380C2211yescharacteristic, complemented
59#1380C2211yescharacteristic, complemented
60#1380C2211yescharacteristic, complemented
61C23×D4—21no—
62C22×D8—22no—
63C22×D8—22no—
64C22×D8—22no—
65C22×D8—21no—
66C22×D8—22no—
67C22×D8—41no—
68C22×D8—41no—
69C22×D8C2311yescharacteristic
70C22×D8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C2×#121—21no—
74C2×#121—21no—
75C2×#120—41no—
76C2×C4×D4D411yescharacteristic
77C2×#117—21no—
78C2×M32—41no—
79C2×M32—41no—
80C2×M32D414yescomplemented
81C22×C16—22no—
82C22×C16—41no—
83C22×C16—41no—
84C22×C16C2311yescharacteristic
85C22×C16C2311yescharacteristic
86C22×C16C2311yescharacteristic
87C22×C16C2311yescharacteristic
88#466—22no—
89#466—22no—
90#466—41no—
91#466—41no—
92C8×D4—22no—
93C8×D4—22no—
94#427—22no—
95#427—22no—
96C2×#107C2311yescharacteristic
97C2×#107C2311yescharacteristic
98C2×#107C2311yescharacteristic
99C2×#107C2311yescharacteristic
100C2×#107C2311yescharacteristic
101C2×#107C2311yescharacteristic
102C2×#106—21no—
103C2×C4⋊C8D411yescharacteristic
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×C22⋊C8D411yescharacteristic
109C2×C22⋊C8D411yescharacteristic
110C2×C4×C8C2311yescharacteristic
111#397—41no—
112#394—41no—
113#394—41no—
114#392—41no—
115#390—21no—
116#390—21no—
117#389—21no—
118#389—21no—
119#388—41no—
120#387—41no—
121#380—21no—
122#380—21no—
123#380—21no—
124#380—21no—
125#379—41no—
126#367—41no—
127#366—41no—
128C4⋊C16—22no—
129#357—22no—
130#357—22no—
131#357—22no—
132#357—22no—
133#357—22no—
134#357—22no—
135#357—41no—
136#357—41no—
137C22⋊C16—22no—
138C22⋊C16—22no—
139#346—22no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146C25—41no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—42no—
150C22×D4—42no—
151C22×D4—41no—
152C22×D4—41no—
153C22×D4—42no—
154C22×D4—41no—
155C22×D4—42no—
156C22×D4—81no—
157C22×D4—81no—
158C22×D4D811yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C2×D8—28no—
162C2×D8—21no—
163C2×D8—21no—
164C2×D8—21no—
165C2×D8—21no—
166C2×D8—21no—
167C2×D8—21no—
168C2×D8—21no—
169C2×D8—21no—
170C2×D8—21no—
171C2×D8—21no—
172C2×D8—44no—
173C2×D8—44no—
174C2×D8—42no—
175C2×D8—42no—
176C2×D8—44no—
177C2×D8—42no—
178C2×D8#4112yes—
179C2×D8#4112yes—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8C2×D411yescharacteristic
191C22×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D412yes—
194C22×C8#4112yes—
195C22×C8C2411yescharacteristic
196#121—42no—
197#121—42no—
198#121—81no—
199#121—81no—
200#120—84no—
201C4×D4—22no—
202C4×D4—22no—
203#117—42no—
204#117—42no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4D811yescharacteristic
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
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⋊C4—41no—
225C2×C22⋊C4D811yescharacteristic
226C2×C22⋊C4D811yescharacteristic
227C2×C42—41no—
228C2×C42C2×D411yescharacteristic
229M32—44no—
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×C16—44no—
241C2×C16C2×D412yes—
242C2×C16C2×D412yes—
243#107—21no—
244#107—21no—
245#107—21no—
246#107—21no—
247#107—21no—
248#107—21no—
249#107—21no—
250#107—21no—
251#107—21no—
252#107—21no—
253#107—21no—
254#107—21no—
255#106—42no—
256C4⋊C8—22no—
257#102—42no—
258#102—42no—
259#102—81no—
260#102—81no—
261C22⋊C8—22no—
262C22⋊C8—22no—
263C4×C8—21no—
264C4×C8—21no—
265#95—41no—
266#95—41no—
267#95—41no—
268#95—41no—
269#95—41no—
270#95—41no—
271C24—41no—
272C24—41no—
273C24—82no—
274C24—82no—
275C24—82no—
276C24—82no—
277C24—82no—
278C24—161no—
279C24—161no—
280C24C2×D811yescharacteristic
281C24C2×D811yescharacteristic
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—41no—
287C2×D4—48no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—84no—
299C2×D4—84no—
300C2×D4—82no—
301C2×D4—82no—
302C2×D4—84no—
303C2×D4—82no—
304C2×D4—82no—
305C2×D4—82no—
306C2×D4—82no—
307C2×D4—82no—
308C2×D4—161no—
309C2×D4—161no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4#12112yes—
327C22×C4C2×D811yescharacteristic
328C22×C4C2×D811yescharacteristic
329C22×C4C2×D811yescharacteristic
330C22×C4C2×D811yescharacteristic
331C22×C4#13512yes—
332C22×C4C22×D411yescharacteristic
333D8—22no—
334D8—22no—
335D8—22no—
336D8—22no—
337D8—48no—
338D8—41no—
339D8—41no—
340D8—41no—
341D8—41no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8#12112yes—
359C2×C8C22×D411yescharacteristic
360C2×C8C2×#4111yescharacteristic
361C2×C8C2×#4111yescharacteristic
362C2×C8C2×#4111yescharacteristic
363C2×C8H(5,𝔽2)11yescharacteristic
364C2×C8H(5,𝔽2)11yescharacteristic
365C4⋊C4—22no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—82no—
379C4⋊C4—161no—
380C22⋊C4—22no—
381C22⋊C4—22no—
382C22⋊C4—82no—
383C22⋊C4—82no—
384C22⋊C4—82no—
385C22⋊C4—82no—
386C22⋊C4—82no—
387C22⋊C4—82no—
388C22⋊C4—82no—
389C22⋊C4—161no—
390C22⋊C4—161no—
391C22⋊C4—161no—
392C22⋊C4—161no—
393C42—21no—
394C42—21no—
395C42—82no—
396C16—22no—
397C16—22no—
398C16—41no—
399C16—41no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—42no—
413C23—42no—
414C23—88no—
415C23—81no—
416C23—81no—
417C23—81no—
418C23—81no—
419C23—81no—
420C23—81no—
421C23—81no—
422C23—81no—
423C23—81no—
424C23—81no—
425C23—81no—
426C23—81no—
427C23—164no—
428C23—164no—
429C23—164no—
430C23—164no—
431C23—164no—
432C23—164no—
433C23#46612yes—
434C23#50912yes—
435C23C22×D811yescharacteristic
436D4—44no—
437D4—42no—
438D4—42no—
439D4—42no—
440D4—42no—
441D4—88no—
442D4—81no—
443D4—81no—
444D4—81no—
445D4—81no—
446D4—164no—
447D4—164no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4#46612yes—
486C2×C4C2×#12111yescharacteristic
487C2×C4#54611yescharacteristic
488C2×C4#54611yescharacteristic
489C2×C4C2×#13511yescharacteristic
490C2×C4#57711yescharacteristic
491C2×C4#57711yescharacteristic
492C8—22no—
493C8—41no—
494C8—41no—
495C8#52511yescharacteristic
496C8#53411yescharacteristic
497C8#55111yescharacteristic
498C8#55111yescharacteristic
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22—82no—
519C22—1616no—
520C22—161no—
521C22—161no—
522C22—161no—
523C22—161no—
524C22—161no—
525C22—161no—
526C22—161no—
527C22—161no—
528C22C2×#46611yescharacteristic
529C22#329411yescharacteristic
530C22#329411yescharacteristic
531C22C2×#50911yescharacteristic
532C22#341411yescharacteristic
533C22#341511yescharacteristic
534C22#341611yescharacteristic
535C4—22no—
536C4—41no—
537C4—41no—
538C4—41no—
539C4—161no—
540C4—161no—
541C4—161no—
542C4—161no—
543C4—161no—
544C4—161no—
545C4#253411yescharacteristic
546C4#305711yescharacteristic
547C4#329611yescharacteristic
548C4#329611yescharacteristic
549C2—42no—
550C2—42no—
551C2—162no—
552C2—162no—
553C2#2809411yescharacteristic
554C2#3360511yescharacteristic
555C2#3361211yescharacteristic
556C2#3387711yescharacteristic
557C2#3387811yescharacteristic
558C2#3388511yescharacteristic
559C2#3388611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#24267C2
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×#2168C2semidirect
2. C2×#2140C2semidirect
3. #27660C2semidirect
5. #13476C2semidirect
6. #13474C2semidirect
7. #13381C2semidirect
8. #13380C2semidirect
9. #13149C2semidirect
10. #13149C2semidirect
11. #13059C2semidirect
12. #13059C2semidirect
13. #12868C2semidirect
14. #12836C2semidirect
15. #12731C2semidirect
18. C23×C16C22semidirect
26. C2×#367C22semidirect
27. C2×#366C22semidirect
28. C2×C4⋊C16C22semidirect
39. C2×C22⋊C16C22semidirect
40. C2×#346C22semidirect
47. #1942C22semidirect
48. #1942C22semidirect
52. #1808C22semidirect
53. #1808C22semidirect
55. #1380C22semidirect
56. #1380C22semidirect
57. #1380C22semidirect
58. #1380C22semidirect
59. #1380C22semidirect
60. #1380C22semidirect
80. C2×M32D4semidirect

Maximal subgroups

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

subgroupindex
C2×#21682
C2×#21402
#276602
#242672
#134762
#134742
#133812
#133802
#131492
#130592
#128682
#128362
#127312