Finite Group Databasebeta

← #363373

Subgroups — #363373

Order 512.

1327 conjugacy classes of subgroups.

The subgroup lattice has 507 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34185C211yesmaximal, characteristic
2C2×#1799C211yesmaximal, characteristic
3C2×C4×#99C211yesmaximal, characteristic
4#11166C214yesmaximal
5#10854C212yesmaximal, complemented
6#10846C212yesmaximal, complemented
7#10829C214yesmaximal, complemented
8C22×C4×D4C2211yescharacteristic
9C22×#99C2212yes—
10C22×#99C2212yes—
11#2353—22no—
12#2353—22no—
13#2352—22no—
14#2352—22no—
15#2313C412yescomplemented
16#2312C412yescomplemented
17#2308C2212yes—
18#2283C412yescomplemented
19#2280C412yescomplemented
20C2×#380C2212yes—
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23#2273C2212yes—
24C2×C4×C22⋊C4C2212yes—
25#1799—22no—
26#1799—22no—
27C4×#99—22no—
28C4×#99—22no—
29C2×#342C2212yes—
30C2×#342C2212yes—
31C2×#342C2218yes—
32C2×#342C2218yes—
33C23×D4C2311yescharacteristic
34C24×C4C2×C411yescharacteristic
35C24×C4C2311yescharacteristic
36C2×C4×D4—22no—
37C2×C4×D4—22no—
38C2×C4×D4—22no—
39C2×C4×D4—22no—
40C2×C4×D4D412yes—
41C2×C4×D4D412yes—
42C2×C4×D4D412yes—
43C2×C4×D4D412yes—
44C2×#117—22no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C22⋊C4C2×C411yescharacteristic
50C22×C22⋊C4C2314yes—
51C22×C22⋊C4C2312yes—
52C22×C22⋊C4C2312yes—
53C22×C22⋊C4C2314yes—
54C22×C22⋊C4C2311yescharacteristic
55C22×C42—22no—
56C22×C42C2×C411yescharacteristic
57C2×#99—24no—
58C2×#99—28no—
59C2×#99—24no—
60C2×#99—44no—
61#388—24no—
62#388—24no—
63#387—24no—
64#387—24no—
65#386—24no—
66#386—24no—
67#383—22no—
68#382—22no—
69#381—22no—
70#381C2×C414yes—
71#380—22no—
72#380—21no—
73#380—21no—
74#380—21no—
75#380—21no—
76#380—22no—
77#380—22no—
78#380—22no—
79#380—22no—
80#380C2×C412yes—
81#380C2×C412yes—
82#380C2×C412yes—
83#380C2×C412yes—
84#379C2×C414yes—
85C4×C4⋊C4—22no—
86C4×C22⋊C4—22no—
87C4×C22⋊C4—24no—
88C4×C22⋊C4—24no—
89C4×C22⋊C4—22no—
90C4×C22⋊C4—22no—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4C2×C414yes—
94C4×C22⋊C4C2×C414yes—
95#376—22no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100#342—24no—
101#342—24no—
102#342—216no—
103#342—48no—
104C25C22×C411yescharacteristic
105C25C2411yescharacteristic
106C22×D4—22no—
107C22×D4—22no—
108C22×D4—24no—
109C22×D4—22no—
110C22×D4—22no—
111C22×D4C2×D412yes—
112C22×D4C2×D412yes—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—24no—
120C23×C4—24no—
121C23×C4—82no—
122C23×C4—82no—
123C23×C4C22⋊C412yes—
124C23×C4C22⋊C412yes—
125C23×C4C22×C411yescharacteristic
126C23×C4C22×C411yescharacteristic
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4#4112yes—
135C23×C4#4112yes—
136C23×C4#4112yes—
137C23×C4#4112yes—
138C4×D4—24no—
139C4×D4—24no—
140C4×D4—24no—
141C4×D4—24no—
142C4×D4—24no—
143C4×D4—24no—
144C4×D4—24no—
145C4×D4—24no—
146#117—84no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—82no—
156C2×C4⋊C4—82no—
157C2×C4⋊C4—84no—
158C2×C4⋊C4C22⋊C412yes—
159C2×C4⋊C4C22⋊C412yes—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—28no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4C22⋊C412yes—
181C2×C22⋊C4C22⋊C412yes—
182C2×C22⋊C4C22×C414yes—
183C2×C22⋊C4C22×C414yes—
184C2×C22⋊C4C2×D412yes—
185C2×C22⋊C4C2×D412yes—
186C2×C22⋊C4#4118yes—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—24no—
192C2×C42—24no—
193C2×C42—42no—
194C2×C42—42no—
195C2×C42—84no—
196C2×C42C22⋊C412yes—
197C2×C42C22⋊C412yes—
198#99—216no—
199#99—48no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—24no—
205#95—24no—
206#95—24no—
207#95—24no—
208#95—42no—
209#95—42no—
210#95—42no—
211#95—42no—
212C24—21no—
213C24—21no—
214C24—22no—
215C24—22no—
216C24—81no—
217C24—81no—
218C24—81no—
219C24—81no—
220C24C2×C22⋊C412yes—
221C24C23×C411yescharacteristic
222C24C22×D411yescharacteristic
223C24C22×D411yescharacteristic
224C24C2×#4112yes—
225C24C2×#4112yes—
226C24C2×#4114yes—
227C24H(5,𝔽2)12yes—
228C242−1+412yes—
229C2×D4—24no—
230C2×D4—24no—
231C2×D4—24no—
232C2×D4—24no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—24no—
236C2×D4—44no—
237C2×D4—44no—
238C2×D4—44no—
239C2×D4—44no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—28no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—44no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—44no—
267C22×C4—48no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—81no—
271C22×C4—81no—
272C22×C4—81no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4—81no—
279C22×C4—81no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4—84no—
285C22×C4—84no—
286C22×C4—84no—
287C22×C4—84no—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4#11711yescharacteristic
296C22×C4#11711yescharacteristic
297C22×C4#11711yescharacteristic
298C22×C4#11711yescharacteristic
299C22×C4#11712yes—
300C22×C4#11712yes—
301C22×C4#12312yes—
302C22×C4#12312yes—
303C22×C4#12312yes—
304C22×C4#12312yes—
305C22×C4#12312yes—
306C22×C4#12312yes—
307C22×C4#12312yes—
308C22×C4#12312yes—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C4⋊C4—84no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4—84no—
317C22⋊C4—24no—
318C22⋊C4—24no—
319C22⋊C4—24no—
320C22⋊C4—24no—
321C22⋊C4—28no—
322C22⋊C4—28no—
323C22⋊C4—28no—
324C22⋊C4—28no—
325C22⋊C4—44no—
326C22⋊C4—44no—
327C22⋊C4—44no—
328C22⋊C4—44no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—48no—
332C22⋊C4—84no—
333C22⋊C4—84no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—84no—
339C42—84no—
340C42—84no—
341C42—84no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—41no—
346C23—41no—
347C23—42no—
348C23—42no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—42no—
353C23—42no—
354C23—41no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—81no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23—84no—
364C23—82no—
365C23—82no—
366C23—84no—
367C23—84no—
368C23C22×C22⋊C411yescharacteristic
369C23C2×#11711yescharacteristic
370C23C2×#11711yescharacteristic
371C23C4×#4112yes—
372C23#52012yes—
373C23C2×#12312yes—
374C23C2×#12312yes—
375C23#52512yes—
376C23#53212yes—
377C23#53511yescharacteristic
378C23#53611yescharacteristic
379C23#53712yes—
380C23#55214yes—
381C23#55314yes—
382C23#55514yes—
383C23#55614yes—
384D4—48no—
385D4—44no—
386D4—44no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—48no—
422C2×C4—44no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—84no—
430C2×C4—84no—
431C2×C4—88no—
432C2×C4—88no—
433C2×C4—84no—
434C2×C4—84no—
435C2×C4—84no—
436C2×C4—164no—
437C2×C4—162no—
438C2×C4—162no—
439C2×C4—168no—
440C2×C4#38111yescharacteristic
441C2×C4#38111yescharacteristic
442C2×C4#38111yescharacteristic
443C2×C4#38112yes—
444C2×C4#38112yes—
445C2×C4#38111yescharacteristic
446C22—21no—
447C22—22no—
448C22—22no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—22no—
453C22—22no—
454C22—21no—
455C22—21no—
456C22—44no—
457C22—44no—
458C22—42no—
459C22—41no—
460C22—41no—
461C22—44no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22—44no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—82no—
471C22—82no—
472C22—84no—
473C22—84no—
474C22C2×#38111yescharacteristic
475C22#229711yescharacteristic
476C22#231111yescharacteristic
477C22#234612yes—
478C22#234712yes—
479C22#248514yes—
480C22#249912yes—
481C22#250012yes—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—22no—
486C4—22no—
487C4—42no—
488C4—42no—
489C4—42no—
490C4—84no—
491C4—84no—
492C4—88no—
493C4—164no—
494C4#182812yes—
495C2—21no—
496C2—21no—
497C2—24no—
498C2—22no—
499C2—22no—
500C2—22no—
501C2—22no—
502C2—82no—
503C2#1393911yescharacteristic
504C2C2×#182811yescharacteristic
505C2#2071011yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10854C2semidirect
6. #10846C2semidirect
7. #10829C2semidirect
15. #2313C4semidirect
16. #2312C4semidirect
18. #2283C4semidirect
19. #2280C4semidirect

Maximal subgroups

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

subgroupindex
#341852
C2×#17992
C2×C4×#992
#111662
#108542
#108462
#108292