Finite Group Databasebeta

← #215439

Subgroups — #215439

Order 512.

1116 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#35011C211yesmaximal, characteristic
2#21902C211yesmaximal, characteristic
3#13948C211yesmaximal, characteristic
4#9714C212yesmaximal, complemented
5#9711C212yesmaximal, complemented
6#9679C212yesmaximal, complemented
7#9679C212yesmaximal, complemented
8#8329C212yesmaximal
9#8329C212yesmaximal
10C22×C22⋊Q8C2212yes—
11#2857C2212yes—
12#2857C2212yes—
13#2659—21no—
14#2659—21no—
15#2583—24no—
16#2502C414yescomplemented
17#2434C414yescomplemented
18#2315—21no—
19#2315—21no—
20C2×#388—21no—
21C2×C4×C22⋊C4C2211yescharacteristic
22#1923—24no—
23#1797C2212yes—
24#1779C2212yes—
25C4×C22⋊C8C2212yes—
26C2×#328C2214yes—
27C2×#328C2214yes—
28C2×#328C2214yes—
29C2×#328C2214yes—
30#1442—21no—
31#1439—21no—
32#1436—21no—
33C4⋊#95C2211yescharacteristic
34C4⋊#95C2211yescharacteristic
35#1434C2211yescharacteristic
36C22⋊#95C2211yescharacteristic
37C22⋊#95C2211yescharacteristic
38C4×#95C2211yescharacteristic
39C23×Q8C2×C412yes—
40C24×C4C2311yescharacteristic
41C2×C4⋊Q8C2×C412yes—
42C2×C4⋊Q8C2×C412yes—
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8—44no—
46C2×C22⋊Q8—44no—
47C2×C22⋊Q8C2×C414yes—
48C2×C22⋊Q8C2314yes—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4C2×C412yes—
53C22×C22⋊C4—22no—
54C22×C22⋊C4—21no—
55C22×C22⋊C4—21no—
56C22×C22⋊C4D411yescharacteristic
57C22×C22⋊C4D411yescharacteristic
58C22×C42—21no—
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C2×C4⋊C8—22no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8C2314yes—
65C2×C4×C8—22no—
66#398—24no—
67#398—24no—
68#395—24no—
69#395—24no—
70#391C2×C414yes—
71#391C2×C414yes—
72#389—22no—
73#389—22no—
74#389—22no—
75#389—22no—
76#388—44no—
77#385—24no—
78#385—24no—
79#385—24no—
80#385—24no—
81#384C2×C412yes—
82#384C2×C412yes—
83#380—22no—
84#380—22no—
85#380—22no—
86#380—22no—
87C4×C22⋊C4C2×C414yes—
88C4×C22⋊C4D414yes—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95D411yescharacteristic
97C2×#95D411yescharacteristic
98C2×#95C2311yescharacteristic
99C2×#95C2311yescharacteristic
100C2×#95C2311yescharacteristic
101C2×#95C2311yescharacteristic
102#336—22no—
103#336—22no—
104#328—44no—
105#328—44no—
106#328—44no—
107#328—44no—
108C25C2×D411yescharacteristic
109C22×Q8—22no—
110C22×Q8—44no—
111C22×Q8—82no—
112C22×Q8—82no—
113C22×Q8C22×C414yes—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—81no—
123C23×C4—81no—
124C23×C4C22×C412yes—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D412yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2411yescharacteristic
132C22×C8—24no—
133C4⋊Q8—82no—
134C4⋊Q8—82no—
135C22⋊Q8—24no—
136C22⋊Q8—24no—
137C22⋊Q8—24no—
138C22⋊Q8—24no—
139C22⋊Q8—44no—
140C22⋊Q8—44no—
141C22⋊Q8—48no—
142C22⋊Q8—44no—
143C22⋊Q8—44no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
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—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—48no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—81no—
162C2×C4⋊C4—81no—
163C2×C4⋊C4C22×C414yes—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—42no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—42no—
174C2×C22⋊C4—81no—
175C2×C22⋊C4—81no—
176C2×C22⋊C4C22⋊C414yes—
177C2×C22⋊C4C22⋊C414yes—
178C2×C22⋊C4SD1614yes—
179C2×C22⋊C4Q1614yes—
180C2×C42—42no—
181C2×C42—42no—
182C2×C42—81no—
183C2×C42—81no—
184C2×C42—81no—
185C2×C42C22×C412yes—
186C2×C42C22×C412yes—
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189C4⋊C8—82no—
190C22⋊C8—24no—
191C22⋊C8—24no—
192C22⋊C8—82no—
193C22⋊C8—82no—
194C4×C8—82no—
195#95—24no—
196#95—24no—
197#95—24no—
198#95—24no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—42no—
204#95—42no—
205#95—42no—
206#95—42no—
207#95—81no—
208#95—81no—
209#95—81no—
210#95C22⋊C414yes—
211#95C22⋊C414yes—
212C24—21no—
213C24—21no—
214C24—21no—
215C24—21no—
216C24—41no—
217C24—41no—
218C24—41no—
219C24—41no—
220C24C2×C22⋊C412yes—
221C24C2×SD1612yes—
222C24C2×Q1612yes—
223C24C22×D411yescharacteristic
224C2×Q8—24no—
225C2×Q8—24no—
226C2×Q8—44no—
227C2×Q8—44no—
228C2×Q8—44no—
229C2×Q8—82no—
230C2×Q8—82no—
231C2×Q8—816no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
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—84no—
287C22×C4—84no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4C2×C22⋊C412yes—
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×SD1612yes—
298C22×C4C2×SD1612yes—
299C22×C4C2×Q1612yes—
300C22×C4C2×Q1612yes—
301C22×C4#13712yes—
302C22×C4#13712yes—
303C22×C4C23×C411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4H(5,𝔽2)12yes—
306C22×C4H(5,𝔽2)12yes—
307C2×C8—44no—
308C2×C8—44no—
309C2×C8—44no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—44no—
313C4⋊C4—48no—
314C4⋊C4—44no—
315C4⋊C4—44no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C4⋊C4—84no—
325C4⋊C4—84no—
326C22⋊C4—28no—
327C22⋊C4—48no—
328C22⋊C4—44no—
329C22⋊C4—44no—
330C22⋊C4—84no—
331C22⋊C4—84no—
332C22⋊C4#103116yes—
333C42—42no—
334C42—42no—
335C42—42no—
336C42—42no—
337C42—84no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—81no—
357C23—81no—
358C23—81no—
359C23—81no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23—81no—
364C23C2×#10314yes—
365C23C22×C22⋊C411yescharacteristic
366C23#53511yescharacteristic
367C23#53511yescharacteristic
368C23C22×SD1611yescharacteristic
369C23C22×Q1611yescharacteristic
370C23C2×#13711yescharacteristic
371C23C2×#13711yescharacteristic
372Q8—84no—
373Q8—84no—
374Q8—88no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—48no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—84no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—88no—
424C2×C4—88no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4C2×#10314yes—
442C2×C4C2×#10314yes—
443C2×C4#41914yes—
444C2×C4#51812yes—
445C2×C4#53512yes—
446C8—84no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—24no—
460C22—24no—
461C22—24no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22—82no—
467C22—82no—
468C22—82no—
469C22—82no—
470C22#230911yescharacteristic
471C22#288311yescharacteristic
472C22#288311yescharacteristic
473C22C22×#10311yescharacteristic
474C22C2×#41911yescharacteristic
475C22#290111yescharacteristic
476C22#290111yescharacteristic
477C22#320611yescharacteristic
478C22#320611yescharacteristic
479C22#320911yescharacteristic
480C22#320911yescharacteristic
481C4—22no—
482C4—22no—
483C4—24no—
484C4—24no—
485C4—24no—
486C4—42no—
487C4—42no—
488C4—82no—
489C4—82no—
490C4—84no—
491C4—84no—
492C4—161no—
493C4—161no—
494C2—21no—
495C2—21no—
496C2—21no—
497C2—21no—
498C2—44no—
499C2#1761411yescharacteristic
500C2#1761411yescharacteristic
501C2#1762111yescharacteristic
502C2#1762711yescharacteristic
503C2#2179511yescharacteristic
504C2#2180411yescharacteristic
505C2#2181711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9714C2semidirect
5. #9711C2semidirect
6. #9679C2semidirect
7. #9679C2semidirect
16. #2502C4semidirect
17. #2434C4semidirect

Maximal subgroups

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

subgroupindex
#350112
#219022
#139482
#97142
#97112
#96792
#83292