Finite Group Databasebeta

← #219721

Subgroups — #219721

Order 512.

1354 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
1C2×#2354C211yesmaximal, characteristic
2#22599C212yesmaximal, complemented
3C2×#1458C212yesmaximal, complemented
4#9649C212yesmaximal
5#9080C214yesmaximal, complemented
6#8558C211yesmaximal, characteristic, complemented
7#8300C212yesmaximal, complemented
8C4×#324C211yesmaximal, characteristic, complemented
9C22×C4×D4C411yescharacteristic
10C22×#117C2212yes—
11C2×#431C2214yes—
12#2354—24no—
13#2354—22no—
14#2354—22no—
15C2×#390C411yescharacteristic
16C2×#390C2211yescharacteristic
17C2×#388C412yes—
18C2×#388C2212yes—
19C2×#384C411yescharacteristic
20C2×#380C412yes—
21C2×C4×C4⋊C4C411yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#1939—22no—
25#1916—24no—
26#1817C2214yes—
27#1797C2212yes—
28C8×C4⋊C4—22no—
29C4×C22⋊C8C2212yes—
30#1458—28no—
31C2×#324C2211yescharacteristic
32C2×#324C2212yes—
33C2×#324C2211yescharacteristic
34C2×#323C2214yes—
35C2×#323C2214yes—
36#1280C2212yes—
37#1280C2212yes—
38C23×D4C2×C411yescharacteristic
39C24×C4C2×C412yes—
40C2×C4×D4—22no—
41C2×C4×D4—21no—
42C2×C4×D4—21no—
43C2×C4×D4—24no—
44C2×C4×D4C818yescomplemented
45C2×#117—24no—
46C2×#117—24no—
47C2×#117—24no—
48C2×#117—24no—
49C2×#117—44no—
50C22×C4⋊C4C2×C412yes—
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C412yes—
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C22⋊C4C2×C412yes—
58C22×C22⋊C4C2×C412yes—
59C22×C22⋊C4C2×C412yes—
60C22×C22⋊C4C2×C412yes—
61C22×C22⋊C4C2312yes—
62C22×C42C2×C412yes—
63C22×C42C2×C411yescharacteristic
64C22×C42C2×C411yescharacteristic
65C22×C42C2312yes—
66C22×C42C2311yescharacteristic
67#431—28no—
68#431—44no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—24no—
71C2×C22⋊C8C2314yes—
72C2×C22⋊C8C2312yes—
73C2×C22⋊C8C2312yes—
74C2×C4×C8—24no—
75C2×C4×C8—22no—
76#390—22no—
77#390—22no—
78#390—22no—
79#390—22no—
80#388—24no—
81#388—24no—
82#388—24no—
83#388—24no—
84#384—22no—
85#384—22no—
86#380—24no—
87#380—24no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4D414yes—
94C4×C4⋊C4D412yes—
95C4×C4⋊C4D412yes—
96C2×#95C2×C412yes—
97C2×#95C2×C411yescharacteristic
98C2×#95C2×C412yes—
99C2×#95C2311yescharacteristic
100#336—22no—
101#336—24no—
102#336—24no—
103#336—22no—
104#324—24no—
105#324—22no—
106#324D414yescomplemented
107#323—28no—
108#323—44no—
109C25C22×C412yes—
110C22×D4—21no—
111C22×D4—21no—
112C22×D4—22no—
113C22×D4—42no—
114C22×D4—42no—
115C22×D4C2×C814yes—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—24no—
119C23×C4—24no—
120C23×C4—24no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4C2×C818yes—
124C23×C4C22×C412yes—
125C23×C4C22×C411yescharacteristic
126C23×C4C22×C411yescharacteristic
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C412yes—
129C23×C4C22×C412yes—
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C2411yescharacteristic
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—24no—
137C4×D4—216no—
138C4×D4—28no—
139C4×D4—28no—
140#117—48no—
141#117—48no—
142#117—48no—
143#117—48no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—82no—
158C2×C4⋊C4C22⋊C412yes—
159C2×C4⋊C4C22⋊C412yes—
160C2×C4⋊C4C22⋊C412yes—
161C2×C4⋊C4C22⋊C411yescharacteristic
162C2×C4⋊C4C22⋊C411yescharacteristic
163C2×C4⋊C4C2×C814yes—
164C2×C4⋊C4M1614yes—
165C2×C4⋊C4C2×D412yes—
166C2×C4⋊C4#4112yes—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—42no—
179C2×C22⋊C4—82no—
180C2×C22⋊C4—82no—
181C2×C22⋊C4C2×C818yes—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—24no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—24no—
193C2×C42—42no—
194C2×C42—42no—
195C2×C42—42no—
196C2×C42C2×C814yes—
197C2×C42C2×D414yes—
198C2×C42C2×D414yes—
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C4⋊C8—48no—
202C4⋊C8—44no—
203C22⋊C8—22no—
204C22⋊C8—24no—
205C22⋊C8—22no—
206C22⋊C8—22no—
207C22⋊C8—24no—
208C22⋊C8C2×D414yes—
209C4×C8—48no—
210C4×C8—44no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95C22⋊C414yes—
216#95C22⋊C414yes—
217C24—22no—
218C24—22no—
219C24—22no—
220C24—22no—
221C24—22no—
222C24—22no—
223C24—42no—
224C24—42no—
225C24—42no—
226C24—42no—
227C24C22×C814yes—
228C24C23×C411yescharacteristic
229C2×D4—24no—
230C2×D4—24no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—48no—
234C2×D4—48no—
235C2×D4—42no—
236C2×D4—42no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—28no—
256C22×C4—28no—
257C22×C4—216no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—41no—
265C22×C4—48no—
266C22×C4—41no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—48no—
272C22×C4—48no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C414yes—
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C4×D412yes—
292C22×C4C4×D412yes—
293C22×C4C22×C812yes—
294C22×C4C22×C814yes—
295C22×C4C22×C812yes—
296C22×C4C22×C812yes—
297C22×C4C2×M1612yes—
298C22×C4C2×M1614yes—
299C22×C4#13112yes—
300C22×C4C22×D412yes—
301C22×C4C22×D411yescharacteristic
302C22×C4C2×#4111yescharacteristic
303C22×C4H(5,𝔽2)12yes—
304C2×C8—24no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—44no—
309C2×C8—42no—
310C2×C8—44no—
311C2×C8—44no—
312C2×C8—42no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—42no—
317C4⋊C4—44no—
318C4⋊C4—48no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—84no—
322C4⋊C4C22⋊C818yes—
323C4⋊C4C22⋊C814yes—
324C4⋊C4C22⋊C814yes—
325C22⋊C4—28no—
326C22⋊C4—28no—
327C22⋊C4—48no—
328C22⋊C4—44no—
329C22⋊C4—44no—
330C22⋊C4—84no—
331C22⋊C4—84no—
332C42—24no—
333C42—24no—
334C42—24no—
335C42—24no—
336C42—24no—
337C42—24no—
338C42—24no—
339C42—42no—
340C42—44no—
341C42—44no—
342C42—42no—
343C42—42no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—24no—
349C23—24no—
350C23—24no—
351C23—44no—
352C23—44no—
353C23—44no—
354C23—44no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—82no—
364C23—82no—
365C23—82no—
366C23—82no—
367C23—82no—
368C23—82no—
369C23—82no—
370C23—82no—
371C23C22×C22⋊C411yescharacteristic
372C23C2×C4×D411yescharacteristic
373C23#51811yescharacteristic
374C23C23×C811yescharacteristic
375C23C22×M1611yescharacteristic
376C23C2×#13111yescharacteristic
377C23#56811yescharacteristic
378D4—48no—
379D4—48no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—24no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—416no—
412C2×C4—416no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4C2×C22⋊C812yes—
436C2×C4C2×C22⋊C814yes—
437C2×C4C2×C22⋊C814yes—
438C2×C4C2×C22⋊C812yes—
439C2×C4#41012yes—
440C2×C4#41312yes—
441C2×C4C8×D414yes—
442C2×C4D4212yes—
443C2×C4#54612yes—
444C8—44no—
445C8—84no—
446C8—82no—
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—44no—
460C22—44no—
461C22—44no—
462C22—84no—
463C22—84no—
464C22—84no—
465C22—84no—
466C22—82no—
467C22—82no—
468C22—82no—
469C22—82no—
470C22D4×C22⋊C411yescharacteristic
471C22C22×C22⋊C811yescharacteristic
472C22#288011yescharacteristic
473C22C2×#41011yescharacteristic
474C22#288411yescharacteristic
475C22C2×#41311yescharacteristic
476C22#288811yescharacteristic
477C22C2×C8×D411yescharacteristic
478C22D4×M1611yescharacteristic
479C22#297211yescharacteristic
480C22#297411yescharacteristic
481C4—24no—
482C4—24no—
483C4—24no—
484C4—22no—
485C4—22no—
486C4—22no—
487C4—42no—
488C4—42no—
489C4—44no—
490C4—42no—
491C4—81no—
492C4—82no—
493C4#145814yescomplemented
494C2—21no—
495C2—21no—
496C2—21no—
497C2—21no—
498C2—84no—
499C2C2×#145811yescharacteristic
500C2#1735311yescharacteristic
501C2#1741811yescharacteristic
502C2#1742811yescharacteristic
503C2D4×C22⋊C811yescharacteristic
504C2#2240611yescharacteristic
505C2#2241211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×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
3. C2×#1458C2semidirect
5. #9080C2semidirect
6. #8558C2semidirect
7. #8300C2semidirect
8. C4×#324C2semidirect
44. C2×C4×D4C8semidirect
106. #324D4semidirect
493. C4#1458semidirect

Maximal subgroups

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

subgroupindex
C2×#23542
#225992
C2×#14582
#96492
#90802
#85582
#83002
C4×#3242