Finite Group Databasebeta

← #127143

Subgroups — #127143

Order 512.

891 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21190C211yesmaximal, characteristic
2C2×C4×#99C211yesmaximal, characteristic
3#17362C211yesmaximal, characteristic
4#7205C214yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6C2×#423C411yescharacteristic, complemented
7C22×#99C412yes—
8C2×#407—21no—
9C2×#407—21no—
10C2×#407—21no—
11C2×C4×M16C411yescharacteristic, complemented
12C2×C4×C22⋊C4—21no—
13#1955—21no—
14#1955—21no—
15#1954C412yescomplemented
16#1954C2212yes—
17C4×#99—21no—
18C4×#99—21no—
19C4×#99C412yes—
20C4×#99C2212yes—
21C2×#342—21no—
22#1465—21no—
23#1465C412yescomplemented
24#1463C412yescomplemented
25#1462—21no—
26#1459—21no—
27#1459C2212yes—
28C23×D4C2×C411yescharacteristic
29C24×C4D411yescharacteristic
30C24×C4D411yescharacteristic
31C22×M16—21no—
32C22×M16—21no—
33C22×M16—41no—
34C2×C4×D4—41no—
35C2×C4×D4—41no—
36C2×C4×D4—41no—
37C2×C4×D4—41no—
38C2×C4×D4C2×C411yescharacteristic
39C2×C4×D4C2×C411yescharacteristic
40C2×C4×D4C2×C411yescharacteristic
41C2×C4×D4D411yescharacteristic
42C2×C4×D4D411yescharacteristic
43C2×C4×D4D411yescharacteristic
44C2×C4×D4Q811yescharacteristic
45C2×C4×D4C2311yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C22⋊C4—22no—
48C22×C22⋊C4D411yescharacteristic
49C22×C22⋊C4Q811yescharacteristic
50C22×C42—41no—
51C22×C42C2×C411yescharacteristic
52#423—41no—
53#423—41no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8C2×C412yescomplemented
56C2×#99—22no—
57C2×#99—28no—
58C2×#99C2×C414yes—
59#407—22no—
60#407—22no—
61#407—44no—
62#407—44no—
63#407—41no—
64#407—41no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—41no—
71C4×M16—41no—
72C4×M16—41no—
73C2×#97—21no—
74C2×C4×C8C2×C412yescomplemented
75C4×C22⋊C4—22no—
76C4×C22⋊C4—22no—
77C4×C22⋊C4—41no—
78C4×C22⋊C4—41no—
79C2×#95—41no—
80#342—22no—
81#342—22no—
82#324—22no—
83#324C2×C414yescomplemented
84#323—22no—
85#323—22no—
86C25C22⋊C411yescharacteristic
87C25C4⋊C411yescharacteristic
88C22×D4—24no—
89C22×D4—41no—
90C22×D4—41no—
91C22×D4—42no—
92C22×D4C4211yescharacteristic
93C22×D4C22⋊C411yescharacteristic
94C22×D4C4⋊C411yescharacteristic
95C22×D4C22×C411yescharacteristic
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—42no—
105C23×C4—42no—
106C23×C4—42no—
107C23×C4—42no—
108C23×C4—42no—
109C23×C4C4211yescharacteristic
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4C22⋊C411yescharacteristic
113C23×C4C22⋊C411yescharacteristic
114C23×C4C22⋊C411yescharacteristic
115C23×C4C22⋊C411yescharacteristic
116C23×C4C4⋊C411yescharacteristic
117C23×C4C4⋊C411yescharacteristic
118C23×C4C4⋊C411yescharacteristic
119C23×C4C4⋊C411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C2×M16—24no—
123C2×M16—42no—
124C2×M16—41no—
125C2×M16—41no—
126C2×M16—42no—
127C2×M16—42no—
128C2×M16—42no—
129C2×M16—42no—
130C2×M16—41no—
131C2×M16—41no—
132C2×M16—82no—
133C2×M16—82no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—42no—
137C4×D4—41no—
138C4×D4—41no—
139C4×D4—41no—
140C4×D4—41no—
141C4×D4—41no—
142C4×D4—41no—
143C4×D4—41no—
144C4×D4—41no—
145C4×D4—41no—
146C4×D4—41no—
147C4×D4—41no—
148C4×D4—41no—
149C4×D4—41no—
150C4×D4—41no—
151C4×D4—41no—
152C4×D4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4C4211yescharacteristic
156C2×C4⋊C4C22⋊C411yescharacteristic
157C2×C4⋊C4C4⋊C411yescharacteristic
158C2×C4⋊C4C22×C411yescharacteristic
159C2×C22⋊C4—28no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—41no—
162C2×C22⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—44no—
166C2×C22⋊C4—42no—
167C2×C22⋊C4C22⋊C411yescharacteristic
168C2×C22⋊C4C22⋊C411yescharacteristic
169C2×C22⋊C4C4⋊C411yescharacteristic
170C2×C22⋊C4C4⋊C411yescharacteristic
171C2×C22⋊C4C4⋊C411yescharacteristic
172C2×C22⋊C4C4⋊C411yescharacteristic
173C2×C22⋊C4C2×D411yescharacteristic
174C2×C22⋊C4C2×Q811yescharacteristic
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—42no—
180C2×C42—42no—
181C2×C42—82no—
182C2×C42—82no—
183C2×C42C4211yescharacteristic
184C2×C42C22⋊C411yescharacteristic
185C2×C42C4⋊C411yescharacteristic
186C2×C42C22×C411yescharacteristic
187C4⋊C8—42no—
188C4⋊C8—42no—
189#99—28no—
190#99—24no—
191#99M1618yescomplemented
192C22⋊C8—24no—
193C22⋊C8—44no—
194C22⋊C8—44no—
195C22⋊C8—44no—
196C22⋊C8—44no—
197C22⋊C8—42no—
198C22⋊C8—42no—
199C22⋊C8—42no—
200#97—42no—
201C4×C8—42no—
202#95—42no—
203#95—42no—
204C24—21no—
205C24—21no—
206C24—22no—
207C24—22no—
208C24—21no—
209C24—21no—
210C24—22no—
211C24—41no—
212C24—42no—
213C24—42no—
214C24—81no—
215C24—81no—
216C24#9511yescharacteristic
217C24#9511yescharacteristic
218C24#9511yescharacteristic
219C24C2×C22⋊C411yescharacteristic
220C24C2×C4⋊C411yescharacteristic
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—48no—
233C2×D4—48no—
234C2×D4C2×M1614yes—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—44no—
282C22×C4—48no—
283C22×C4—81no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4—81no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—81no—
292C22×C4—84no—
293C22×C4—84no—
294C22×C4#9511yescharacteristic
295C22×C4#9511yescharacteristic
296C22×C4#9511yescharacteristic
297C22×C4#9511yescharacteristic
298C22×C4#9511yescharacteristic
299C22×C4#9911yescharacteristic
300C22×C4#9911yescharacteristic
301C22×C4#9911yescharacteristic
302C22×C4#9911yescharacteristic
303C22×C4C2×C4211yescharacteristic
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C4⋊C411yescharacteristic
307C22×C4C2×C4⋊C411yescharacteristic
308M16—44no—
309M16—44no—
310M16—82no—
311M16—84no—
312M16—84no—
313M16—82no—
314C2×C8—24no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—44no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—82no—
322C2×C8—82no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C22⋊C4—28no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—48no—
337C22⋊C4—44no—
338C42—21no—
339C42—21no—
340C42—21no—
341C42—21no—
342C42—21no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—82no—
347C42—82no—
348C42—84no—
349C23—24no—
350C23—24no—
351C23—22no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—41no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—81no—
379C23—81no—
380C23—81no—
381C23—81no—
382C23—84no—
383C23—84no—
384C23—84no—
385C23—84no—
386C23#34211yescharacteristic
387C23#34211yescharacteristic
388C23#34412yes—
389C23C2×#9511yescharacteristic
390C23#40712yes—
391C23#42312yes—
392D4—44no—
393D4—44no—
394D4—44no—
395D4—44no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—48no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—84no—
450C2×C4—816no—
451C2×C4—84no—
452C2×C4#34211yescharacteristic
453C2×C4#34211yescharacteristic
454C2×C4#35312yes—
455C2×C4#35412yes—
456C2×C4C4×M1612yes—
457C2×C4C2×#9911yescharacteristic
458C2×C4C2×#9911yescharacteristic
459C8—44no—
460C8—84no—
461C8—82no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—22no—
469C22—21no—
470C22—22no—
471C22—22no—
472C22—42no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22—48no—
478C22—42no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—81no—
489C22—81no—
490C22—81no—
491C22—84no—
492C22—81no—
493C22—84no—
494C22—84no—
495C22#139312yes—
496C22#172811yescharacteristic
497C22C2×#34211yescharacteristic
498C22C2×#34411yescharacteristic
499C4—21no—
500C4—21no—
501C4—22no—
502C4—41no—
503C4—41no—
504C4—41no—
505C4—41no—
506C4—41no—
507C4—41no—
508C4—41no—
509C4—88no—
510C4#139312yes—
511C4C2×#35311yescharacteristic
512C4C2×#35411yescharacteristic
513C2—22no—
514C2—21no—
515C2—21no—
516C2—42no—
517C2—41no—
518C2—42no—
519C2—42no—
520C2—81no—
521C2#792911yescharacteristic
522C2C2×#139311yescharacteristic
523C2#1258911yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
socleC22C2×#342
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7205C2semidirect
6. C2×#423C4semidirect
11. C2×C4×M16C4semidirect
15. #1954C4semidirect
23. #1465C4semidirect
24. #1463C4semidirect
55. C2×C4⋊C8C2×C4semidirect
74. C2×C4×C8C2×C4semidirect
83. #324C2×C4semidirect
191. #99M16semidirect

Maximal subgroups

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

subgroupindex
#211902
C2×C4×#992
#173622
#72052