Finite Group Databasebeta

← #362866

Subgroups — #362866

Order 512.

1493 conjugacy classes of subgroups.

The subgroup lattice has 515 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34181C211yesmaximal, characteristic
2C2×#1799C211yesmaximal, characteristic
3C2×C4×#99C211yesmaximal, characteristic
4#11351C214yesmaximal, complemented
5#10854C212yesmaximal, complemented
6#10846C212yesmaximal, complemented
7#10768C214yesmaximal
8C2×#518C2212yes—
9C22×C4×D4C2211yescharacteristic
10C22×#99C2212yes—
11C22×#99C2212yes—
12#2349—22no—
13#2349—22no—
14D4×C4⋊C4—22no—
15D4×C4⋊C4—22no—
16#2316C412yescomplemented
17#2315C412yescomplemented
18#2315C2212yes—
19#2285C412yescomplemented
20C2×#380C412yescomplemented
21C2×#380C2212yes—
22C2×#380C2211yescharacteristic
23C2×#380C2211yescharacteristic
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
36#518—48no—
37C2×C4×D4—22no—
38C2×C4×D4—22no—
39C2×C4×D4—22no—
40C2×C4×D4—22no—
41C2×C4×D4—24no—
42C2×C4×D4—24no—
43C2×C4×D4D412yes—
44C2×C4×D4D412yes—
45C2×C4×D4D412yes—
46C2×C4×D4D412yes—
47C2×#117C2×C414yes—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C22⋊C4C2×C411yescharacteristic
54C22×C22⋊C4C2×C414yes—
55C22×C22⋊C4C2314yes—
56C22×C22⋊C4C2312yes—
57C22×C22⋊C4C2312yes—
58C22×C22⋊C4C2314yes—
59C22×C22⋊C4C2311yescharacteristic
60C22×C42—22no—
61C22×C42C2×C411yescharacteristic
62C2×#99—24no—
63C2×#99—24no—
64C2×#99—24no—
65C2×#99—28no—
66C2×#99Q818yes—
67#389—24no—
68#389—24no—
69#385—24no—
70#385—24no—
71#385—24no—
72#385—24no—
73#384—22no—
74#382—22no—
75#380—22no—
76#380—22no—
77#380—21no—
78#380—21no—
79#380—21no—
80#380—21no—
81#380—22no—
82#380—22no—
83#380—22no—
84#380—22no—
85#380C2×C414yes—
86#380C2×C412yes—
87#380C2×C412yes—
88#380C2×C414yes—
89#380C2×C412yes—
90#380C2×C412yes—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—22no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—22no—
102#342—24no—
103#342—24no—
104#342—48no—
105#342—48no—
106C25C22×C411yescharacteristic
107C25C2411yescharacteristic
108C22×D4—22no—
109C22×D4—22no—
110C22×D4—24no—
111C22×D4C2×D412yes—
112C22×D4C2×D412yes—
113C22×D4C2×Q814yes—
114C22×D4#4114yes—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—24no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—24no—
123C23×C4—24no—
124C23×C4—82no—
125C23×C4—82no—
126C23×C4C22⋊C412yes—
127C23×C4C22⋊C412yes—
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C411yescharacteristic
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4#4112yes—
138C23×C4#4112yes—
139C23×C4#4112yes—
140C23×C4#4112yes—
141C4×D4—24no—
142C4×D4—24no—
143C4×D4—24no—
144C4×D4—24no—
145C4×D4—24no—
146C4×D4—24no—
147C4×D4—24no—
148C4×D4—24no—
149C4×D4—416no—
150C4×D4—416no—
151#117—48no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4—84no—
166C2×C4⋊C4—84no—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22⋊C412yes—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—28no—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—48no—
188C2×C22⋊C4C22⋊C412yes—
189C2×C22⋊C4C22⋊C412yes—
190C2×C22⋊C4C22×C414yes—
191C2×C22⋊C4C22×C414yes—
192C2×C22⋊C4C2×D412yes—
193C2×C22⋊C4C2×D412yes—
194C2×C22⋊C4C2×Q818yes—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—24no—
200C2×C42—42no—
201C2×C42—42no—
202C2×C42—84no—
203C2×C42C22⋊C412yes—
204C2×C42C22⋊C412yes—
205#99—216no—
206#99—48no—
207#95—22no—
208#95—22no—
209#95—24no—
210#95—22no—
211#95—22no—
212#95—24no—
213#95—42no—
214#95—42no—
215#95—42no—
216#95—42no—
217C24—21no—
218C24—21no—
219C24—22no—
220C24—81no—
221C24—81no—
222C24—81no—
223C24—81no—
224C24C2×C22⋊C412yes—
225C24C4×Q814yes—
226C24C23×C411yescharacteristic
227C24C22×D411yescharacteristic
228C24C22×D411yescharacteristic
229C24C22×Q812yes—
230C24C2×#4112yes—
231C24C2×#4112yes—
232C24C2×#4112yes—
233C24H(5,𝔽2)14yes—
234C2×D4—24no—
235C2×D4—24no—
236C2×D4—24no—
237C2×D4—24no—
238C2×D4—28no—
239C2×D4—28no—
240C2×D4—28no—
241C2×D4—28no—
242C2×D4C22⋊Q818yes—
243C2×D4#12414yes—
244C2×D4C4⋊Q814yes—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—28no—
259C22×C4—44no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—44no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—48no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—48no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4—81no—
275C22×C4—81no—
276C22×C4—81no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4—81no—
283C22×C4—81no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—84no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4—84no—
290C22×C4—84no—
291C22×C4—84no—
292C22×C4—84no—
293C22×C4—84no—
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4#11711yescharacteristic
302C22×C4#11711yescharacteristic
303C22×C4#11711yescharacteristic
304C22×C4#11711yescharacteristic
305C22×C4#11712yes—
306C22×C4#11712yes—
307C22×C4C4×Q814yes—
308C22×C4#12312yes—
309C22×C4#12312yes—
310C22×C4#12312yes—
311C22×C4#12312yes—
312C22×C4#12312yes—
313C22×C4#12312yes—
314C22×C4#12312yes—
315C22×C4#12312yes—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—48no—
321C4⋊C4—84no—
322C4⋊C4—84no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C4⋊C4—84no—
326C4⋊C4—84no—
327C22⋊C4—24no—
328C22⋊C4—24no—
329C22⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—28no—
332C22⋊C4—28no—
333C22⋊C4—28no—
334C22⋊C4—28no—
335C22⋊C4—48no—
336C22⋊C4—44no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C22⋊C4—44no—
340C22⋊C4—44no—
341C22⋊C4—44no—
342C22⋊C4—48no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—48no—
348C42—84no—
349C42—84no—
350C23—22no—
351C23—22no—
352C23—24no—
353C23—24no—
354C23—41no—
355C23—41no—
356C23—42no—
357C23—42no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—42no—
362C23—42no—
363C23—41no—
364C23—42no—
365C23—42no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23—81no—
370C23—82no—
371C23—82no—
372C23—84no—
373C23—84no—
374C23—84no—
375C23#39114yes—
376C23C22×C22⋊C411yescharacteristic
377C23C2×#11711yescharacteristic
378C23C2×#11711yescharacteristic
379C23C2×C4×Q812yes—
380C23#51812yes—
381C23C2×C22⋊Q812yes—
382C23C2×#12312yes—
383C23C2×#12312yes—
384C23C2×#12411yescharacteristic
385C23C2×C4⋊Q811yescharacteristic
386C23#53512yes—
387C23#53512yes—
388C23#55314yes—
389C23#55314yes—
390C23#55414yes—
391C23#55414yes—
392D4—216no—
393D4—28no—
394D4—28no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—48no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—84no—
437C2×C4—84no—
438C2×C4—88no—
439C2×C4—88no—
440C2×C4—84no—
441C2×C4—84no—
442C2×C4—84no—
443C2×C4—164no—
444C2×C4—164no—
445C2×C4—168no—
446C2×C4—168no—
447C2×C4#38111yescharacteristic
448C2×C4#38111yescharacteristic
449C2×C4#38111yescharacteristic
450C2×C4#38112yes—
451C2×C4#38112yes—
452C2×C4#38111yescharacteristic
453C2×C4#39114yes—
454C22—21no—
455C22—22no—
456C22—22no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—22no—
461C22—22no—
462C22—21no—
463C22—21no—
464C22—24no—
465C22—44no—
466C22—44no—
467C22—42no—
468C22—42no—
469C22—44no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—41no—
474C22—41no—
475C22—44no—
476C22—44no—
477C22—44no—
478C22—42no—
479C22—42no—
480C22—84no—
481C22—84no—
482C22C2×#38111yescharacteristic
483C22C2×#39111yescharacteristic
484C22#230911yescharacteristic
485C22#234512yes—
486C22#234512yes—
487C22#248714yes—
488C22#250212yes—
489C22#250412yes—
490C4—24no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—22no—
495C4—22no—
496C4—42no—
497C4—42no—
498C4—84no—
499C4—84no—
500C4—88no—
501C4—164no—
502C4#182812yes—
503C2—21no—
504C2—24no—
505C2—22no—
506C2—21no—
507C2—22no—
508C2—22no—
509C2—22no—
510C2—44no—
511C2#1392411yescharacteristic
512C2C2×#182811yescharacteristic
513C2#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
4. #11351C2semidirect
5. #10854C2semidirect
6. #10846C2semidirect
16. #2316C4semidirect
17. #2315C4semidirect
19. #2285C4semidirect
20. C2×#380C4semidirect

Maximal subgroups

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

subgroupindex
#341812
C2×#17992
C2×C4×#992
#113512
#108542
#108462
#107682