Finite Group Databasebeta

← #219720

Subgroups — #219720

Order 512.

1316 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2344C211yesmaximal, characteristic
2#22599C212yesmaximal, complemented
3C2×#1458C212yesmaximal
4#9650C212yesmaximal, complemented
5#9082C214yesmaximal, complemented
6#8557C211yesmaximal, characteristic
7#8300C212yesmaximal
8C4×#323C211yesmaximal, characteristic
9C22×#117C2212yes—
10C23×C4⋊C4C411yescharacteristic
11C2×#431C2214yes—
12#2344—24no—
13#2344—22no—
14#2344—22no—
15C2×#389C411yescharacteristic
16C2×#389C2211yescharacteristic
17C2×#387C412yes—
18C2×#387C2212yes—
19C2×#381C412yes—
20C2×#380C411yescharacteristic
21C2×C4×C4⋊C4C411yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C22⋊C4C2211yescharacteristic
24#1939—22no—
25#1916—24no—
26#1817C2214yes—
27#1797C2212yes—
28C8×C4⋊C4—22no—
29C4×C22⋊C8C2212yes—
30#1458—28no—
31C2×#324C2214yes—
32C2×#324C2214yes—
33C2×#323C2211yescharacteristic
34C2×#323C2212yes—
35C2×#323C2211yescharacteristic
36#1280C2212yes—
37#1280C2212yes—
38C24×C4C2×C412yes—
39C24×C4C2×C411yescharacteristic
40C2×#117—24no—
41C2×#117—24no—
42C2×#117—24no—
43C2×#117—24no—
44C2×#117—44no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—24no—
49C22×C4⋊C4C818yescomplemented
50C22×C4⋊C4C2×C412yes—
51C22×C4⋊C4C2×C412yes—
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C22⋊C4C2×C412yes—
58C22×C22⋊C4C2×C411yescharacteristic
59C22×C22⋊C4C2311yescharacteristic
60C22×C42C2×C412yes—
61C22×C42C2×C411yescharacteristic
62C22×C42C2312yes—
63C22×C42C2311yescharacteristic
64#431—28no—
65#431—44no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—24no—
68C2×C22⋊C8C2314yes—
69C2×C22⋊C8C2312yes—
70C2×C22⋊C8C2312yes—
71C2×C4×C8—24no—
72C2×C4×C8—22no—
73#389—22no—
74#389—22no—
75#389—22no—
76#389—22no—
77#387—24no—
78#387—24no—
79#387—24no—
80#387—24no—
81#381—24no—
82#381—24no—
83#380—22no—
84#380—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4D414yes—
89C4×C4⋊C4D412yes—
90C4×C4⋊C4D412yes—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93C2×#95C2×C411yescharacteristic
94C2×#95C2×C412yes—
95C2×#95C2×C412yes—
96C2×#95C2×C412yes—
97C2×#95C2×C412yes—
98C2×#95C2×C412yes—
99C2×#95C2311yescharacteristic
100#336—22no—
101#336—24no—
102#336—24no—
103#336—22no—
104#324—28no—
105#324—44no—
106#323—24no—
107#323—22no—
108#323D414yes—
109C25C22×C411yescharacteristic
110C23×C4—24no—
111C23×C4—24no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—24no—
117C23×C4—22no—
118C23×C4—42no—
119C23×C4—42no—
120C23×C4—42no—
121C23×C4—42no—
122C23×C4C2×C818yes—
123C23×C4C2×C814yes—
124C23×C4C22×C412yes—
125C23×C4C22×C411yescharacteristic
126C23×C4C22×C412yes—
127C23×C4C22×C412yes—
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C412yes—
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C2411yescharacteristic
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—24no—
137#117—48no—
138#117—48no—
139#117—48no—
140#117—48no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—216no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4C22⋊C412yes—
162C2×C4⋊C4C22⋊C412yes—
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4C22⋊C411yescharacteristic
165C2×C4⋊C4C22⋊C411yescharacteristic
166C2×C4⋊C4C2×C814yes—
167C2×C4⋊C4C2×C818yes—
168C2×C4⋊C4C2×C814yes—
169C2×C4⋊C4M1614yes—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—42no—
178C2×C22⋊C4C2×D412yes—
179C2×C22⋊C4#4112yes—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—24no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—24no—
188C2×C42—42no—
189C2×C42—42no—
190C2×C42—42no—
191C2×C42C2×D414yes—
192C2×C42C2×D414yes—
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C4⋊C8—48no—
196C4⋊C8—44no—
197C22⋊C8—22no—
198C22⋊C8—24no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—24no—
202C22⋊C8C2×D414yes—
203C4×C8—48no—
204C4×C8—44no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—24no—
209#95—24no—
210#95—24no—
211#95—22no—
212#95C22⋊C414yes—
213#95C22⋊C414yes—
214C24—21no—
215C24—21no—
216C24—21no—
217C24—21no—
218C24—41no—
219C24—41no—
220C24—41no—
221C24—41no—
222C24C4×D412yes—
223C24C22×C812yes—
224C24#13112yes—
225C24C23×C411yescharacteristic
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—216no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—28no—
248C22×C4—28no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—41no—
258C22×C4—48no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—48no—
269C22×C4—48no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—48no—
273C22×C4—48no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C414yes—
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C4×D412yes—
291C22×C4C22×C814yes—
292C22×C4C22×C814yes—
293C22×C4C22×C812yes—
294C22×C4C22×C812yes—
295C22×C4C2×M1614yes—
296C22×C4C2×M1612yes—
297C22×C4C22×D412yes—
298C22×C4C22×D411yescharacteristic
299C22×C4C2×#4111yescharacteristic
300C22×C4H(5,𝔽2)12yes—
301C2×C8—24no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—44no—
306C2×C8—42no—
307C2×C8—44no—
308C2×C8—44no—
309C2×C8—42no—
310C4⋊C4—28no—
311C4⋊C4—28no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—42no—
316C4⋊C4—48no—
317C4⋊C4—42no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—42no—
321C4⋊C4C22⋊C818yes—
322C4⋊C4C22⋊C814yes—
323C4⋊C4C22⋊C814yes—
324C22⋊C4—24no—
325C22⋊C4—24no—
326C22⋊C4—44no—
327C22⋊C4—48no—
328C42—24no—
329C42—24no—
330C42—24no—
331C42—24no—
332C42—24no—
333C42—42no—
334C42—44no—
335C42—44no—
336C42—42no—
337C42—42no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23C8×D414yes—
365C23C22×C22⋊C411yescharacteristic
366C23C2×C4×D411yescharacteristic
367C23#51811yescharacteristic
368C23C23×C811yescharacteristic
369C23C22×M1611yescharacteristic
370C23C2×#13111yescharacteristic
371C23#56811yescharacteristic
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—416no—
401C2×C4—416no—
402C2×C4—48no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—48no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4C2×C22⋊C814yes—
431C2×C4C2×C22⋊C814yes—
432C2×C4C2×C22⋊C812yes—
433C2×C4C2×C22⋊C812yes—
434C2×C4#41012yes—
435C2×C4#41312yes—
436C2×C4D4212yes—
437C2×C4#54612yes—
438C8—44no—
439C8—84no—
440C8—82no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—24no—
454C22—44no—
455C22—44no—
456C22—44no—
457C22—44no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22D4×C22⋊C411yescharacteristic
465C22C22×C22⋊C811yescharacteristic
466C22#288011yescharacteristic
467C22C2×#41011yescharacteristic
468C22#288411yescharacteristic
469C22C2×#41311yescharacteristic
470C22#288911yescharacteristic
471C22C2×C8×D411yescharacteristic
472C22D4×M1611yescharacteristic
473C22#297211yescharacteristic
474C22#297411yescharacteristic
475C4—24no—
476C4—24no—
477C4—22no—
478C4—22no—
479C4—22no—
480C4—42no—
481C4—44no—
482C4—44no—
483C4—42no—
484C4—42no—
485C4—81no—
486C4—82no—
487C4#145814yescomplemented
488C2—21no—
489C2—21no—
490C2—21no—
491C2—21no—
492C2—44no—
493C2C2×#145811yescharacteristic
494C2#1735211yescharacteristic
495C2#1741311yescharacteristic
496C2#1742311yescharacteristic
497C2D4×C22⋊C811yescharacteristic
498C2#2240611yescharacteristic
499C2#2241111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22599C2semidirect
4. #9650C2semidirect
5. #9082C2semidirect
49. C22×C4⋊C4C8semidirect
487. C4#1458semidirect

Maximal subgroups

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

subgroupindex
C2×#23442
#225992
C2×#14582
#96502
#90822
#85572
#83002
C4×#3232