Finite Group Databasebeta

← #412606

Subgroups — #412606

Order 512.

986 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23642C212yesmaximal
2C2×#1951C211yesmaximal, characteristic, complemented
3C2×#1810C211yesmaximal, characteristic
4C2×#1807C211yesmaximal, characteristic
5#20385C211yesmaximal, characteristic
6#11303C212yesmaximal, complemented
7#11290C212yesmaximal, complemented
8#11198C212yesmaximal
9#10869C211yesmaximal, characteristic, complemented
10#10869C211yesmaximal, characteristic, complemented
11#10869C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×#108C2211yescharacteristic
14C22×#108C2211yescharacteristic
15C2×#428C2212yes—
16C22×#106C2211yescharacteristic
17C22×#106C2211yescharacteristic
18C2×#407C2211yescharacteristic
19C2×#407C2212yes—
20C2×#405C2211yescharacteristic
21C2×#405C2211yescharacteristic
22#2280C2211yescharacteristic
23C2×#380C2212yes—
24#2077—22no—
25#2077—22no—
26#2076—22no—
27#2075—22no—
28#1951—24no—
29#1810—24no—
30#1807—22no—
31#1807—22no—
32#1752—22no—
33C8×C22⋊C4—22no—
34C2×#343C2212yes—
35C2×#343C2212yes—
36C2×#343C2212yes—
37C2×#343C2212yes—
38C2×#341C2212yes—
39#1736C2212yes—
40#1736C2212yes—
41#1736C2212yes—
42C2×#340C2212yes—
43#1729C2211yescharacteristic
44#1729C2211yescharacteristic
45C24×C4C2311yescharacteristic
46C22×M16C2312yes—
47C22×M16C2312yes—
48C22×M16C2311yescharacteristic
49C22×M16C2311yescharacteristic
50C23×C8—22no—
51C23×C8—22no—
52C23×C8—22no—
53C23×C8—22no—
54C23×C8—22no—
55C23×C8—22no—
56C23×C8D412yes—
57C23×C8D412yes—
58C23×C8C2311yescharacteristic
59C23×C8C2311yescharacteristic
60C2×#117C2311yescharacteristic
61C2×#117C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C22⋊C4—22no—
65C2×#108—22no—
66C2×#108—22no—
67C2×#108—22no—
68C2×#108—22no—
69C2×#108—22no—
70C2×#108D414yes—
71#428—24no—
72#428—42no—
73C2×#107—22no—
74C2×#106—21no—
75C2×#106—21no—
76C2×#106—21no—
77C2×#106—21no—
78C2×#106—22no—
79C2×#106—22no—
80C2×#106—22no—
81C2×#106—22no—
82C2×#106—22no—
83#407—22no—
84#407—22no—
85#407—24no—
86#407—42no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90#405—22no—
91#405—22no—
92#405—22no—
93#405—22no—
94C2×#97—21no—
95C2×#97—21no—
96C2×C4×C8—21no—
97C2×C4×C8—21no—
98#381—21no—
99#380—22no—
100#380—22no—
101#380—22no—
102#380—22no—
103#380D412yes—
104#380D412yes—
105#379—21no—
106C4×C22⋊C4—22no—
107C2×#95—22no—
108#343—24no—
109#343—24no—
110#343—24no—
111#343—42no—
112#341—24no—
113#340—24no—
114#340—22no—
115#340—22no—
116#340—22no—
117#340—22no—
118#340—22no—
119#340—22no—
120#336—22no—
121#336—22no—
122C25#4111yescharacteristic
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4C2×D412yes—
138C23×C4C2×D412yes—
139C23×C4#4111yescharacteristic
140C23×C4C2411yescharacteristic
141C2×M16—21no—
142C2×M16—21no—
143C2×M16—22no—
144C2×M16—21no—
145C2×M16—22no—
146C2×M16—21no—
147C2×M16—21no—
148C2×M16—21no—
149C2×M16—22no—
150C2×M16—22no—
151C2×M16—21no—
152C2×M16—21no—
153C2×M16—21no—
154C2×M16—21no—
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16C2×D414yes—
163C2×M16#4112yes—
164C2×M16#4112yes—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—24no—
175C22×C8—24no—
176C22×C8—24no—
177C22×C8—24no—
178C22×C8—24no—
179C22×C8—24no—
180C22×C8—22no—
181C22×C8—44no—
182C22×C8—44no—
183C22×C8—44no—
184C22×C8—44no—
185C22×C8C2×D412yes—
186C22×C8C2×D411yescharacteristic
187C22×C8C2×D411yescharacteristic
188C22×C8C2×D412yes—
189C22×C8#4112yes—
190C22×C8#4112yes—
191C22×C8#4112yes—
192C22×C8#4112yes—
193C22×C8#4112yes—
194#117—21no—
195#117—21no—
196#117—21no—
197#117—21no—
198#117—21no—
199#117—21no—
200#117—21no—
201#117—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2×D412yes—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—84no—
224C2×C42—21no—
225C2×C42—21no—
226#108—28no—
227#108—44no—
228#107—44no—
229#106—44no—
230#106—42no—
231#106—42no—
232#106—42no—
233#106—42no—
234C22⋊C8—28no—
235C22⋊C8—44no—
236C22⋊C8—44no—
237#97—42no—
238#97—42no—
239C4×C8—42no—
240C4×C8—42no—
241#95—22no—
242#95—22no—
243#95—42no—
244#95—42no—
245C24—21no—
246C24—21no—
247C24—24no—
248C24—21no—
249C24—21no—
250C24—21no—
251C24—21no—
252C24—42no—
253C24#12112yes—
254C24C2×#4111yescharacteristic
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4#12112yes—
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C2×#4111yescharacteristic
307C22×C4C2×#4111yescharacteristic
308C22×C4C2×#4111yescharacteristic
309C22×C4C2×#4111yescharacteristic
310C22×C4C2×#4111yescharacteristic
311C22×C4C2×#4111yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C22×C4H(5,𝔽2)11yescharacteristic
314C22×C42−1+411yescharacteristic
315C22×C42−1+411yescharacteristic
316C22×C42−1+411yescharacteristic
317M16—24no—
318M16—24no—
319M16—22no—
320M16—22no—
321M16—22no—
322M16—22no—
323M16—42no—
324M16—41no—
325M16—42no—
326M16—41no—
327M16—41no—
328M16—41no—
329M16—42no—
330M16—42no—
331M16—42no—
332M16—42no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—24no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—42no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—42no—
349C2×C8—48no—
350C2×C8—48no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8#12112yes—
355C2×C8#12112yes—
356C2×C8#12312yes—
357C2×C8#12312yes—
358C2×C8#12312yes—
359C2×C8#12312yes—
360C2×C8#12314yes—
361C2×C8#12414yes—
362C2×C8#12614yes—
363C2×C8#12614yes—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C22⋊C4—44no—
379C22⋊C4—41no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4—41no—
383C22⋊C4—44no—
384C22⋊C4—84no—
385C22⋊C4—84no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—41no—
390C23—22no—
391C23—22no—
392C23—24no—
393C23—24no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—44no—
401C23—44no—
402C23—44no—
403C23—44no—
404C23—42no—
405C23—42no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23#47114yes—
411C23C2×#12111yescharacteristic
412C23#52911yescharacteristic
413C23#52911yescharacteristic
414C23#54811yescharacteristic
415C23#55211yescharacteristic
416C23#55211yescharacteristic
417C23#55311yescharacteristic
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—162no—
447C2×C4—162no—
448C2×C4—162no—
449C2×C4—162no—
450C2×C4C2×#12111yescharacteristic
451C2×C4C2×#12311yescharacteristic
452C2×C4C2×#12311yescharacteristic
453C2×C4C2×#12311yescharacteristic
454C2×C4#52511yescharacteristic
455C2×C4C2×#12411yescharacteristic
456C2×C4C2×#12611yescharacteristic
457C2×C4C2×#12611yescharacteristic
458C2×C4#53211yescharacteristic
459C2×C4#53211yescharacteristic
460C2×C4#53611yescharacteristic
461C2×C4#53711yescharacteristic
462C2×C4#54111yescharacteristic
463C2×C4#54111yescharacteristic
464C2×C4#54211yescharacteristic
465C2×C4#54211yescharacteristic
466C2×C4D4211yescharacteristic
467C2×C4#54711yescharacteristic
468C2×C4#54711yescharacteristic
469C2×C4#54711yescharacteristic
470C2×C4#54711yescharacteristic
471C2×C4#54811yescharacteristic
472C2×C4#55211yescharacteristic
473C2×C4#55211yescharacteristic
474C2×C4#55311yescharacteristic
475C2×C4#55511yescharacteristic
476C2×C4#55611yescharacteristic
477C2×C4#55611yescharacteristic
478C8—42no—
479C8—42no—
480C8—44no—
481C8—42no—
482C8—82no—
483C8#39714yes—
484C8#39714yes—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—24no—
494C22—24no—
495C22—24no—
496C22—44no—
497C22—44no—
498C22—44no—
499C22—44no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22#241611yescharacteristic
505C22#242111yescharacteristic
506C22#248611yescharacteristic
507C22#248911yescharacteristic
508C22#249411yescharacteristic
509C22#249511yescharacteristic
510C22#285411yescharacteristic
511C22C2×#47112yes—
512C22#306712yes—
513C4—42no—
514C4—81no—
515C4—81no—
516C4—82no—
517C4—162no—
518C4—162no—
519C4C2×#39711yescharacteristic
520C4#241511yescharacteristic
521C4#242211yescharacteristic
522C4#248511yescharacteristic
523C4#249111yescharacteristic
524C4#249611yescharacteristic
525C4#249611yescharacteristic
526C4#262511yescharacteristic
527C2—21no—
528C2—21no—
529C2—21no—
530C2—21no—
531C2—44no—
532C2#1513211yescharacteristic
533C2#2480212yes—
534C2#2767211yescharacteristic
535C2#2767311yescharacteristic
536C2#3312212yes—
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#1951C2semidirect
6. #11303C2semidirect
7. #11290C2semidirect
9. #10869C2semidirect
10. #10869C2semidirect
11. #10869C2semidirect

Maximal subgroups

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

subgroupindex
#236422
C2×#19512
C2×#18102
C2×#18072
#203852
#113032
#112902
#111982
#108692