Finite Group Databasebeta

← #127137

Subgroups — #127137

Order 512.

781 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21740C211yesmaximal, characteristic
2C2×C4×#100C211yesmaximal, characteristic
3#17362C211yesmaximal, characteristic
4#7203C214yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6C22×#100C412yes—
7C2×#407—21no—
8C2×#407C411yescharacteristic
9C22×C22⋊C8C411yescharacteristic
10C2×C4×M16—21no—
11#1870—21no—
12#1870—21no—
13#1869C412yes—
14#1869C2212yes—
15#1817—21no—
16#1779—21no—
17C4×#100—21no—
18C4×#100—21no—
19C4×#100C412yes—
20C4×#100C2212yes—
21C2×#343—21no—
22#1465—21no—
23#1465C412yes—
24#1463—21no—
25#1462C412yes—
26#1459—21no—
27#1459C2212yes—
28C23×D4C2×C411yescharacteristic
29C24×C4C2×C411yescharacteristic
30C24×C4D411yescharacteristic
31C22×M16—21no—
32C22×M16—22no—
33C22×M16—41no—
34C23×C8C2×C412yes—
35C2×C4×D4—41no—
36C2×C4×D4—41no—
37C2×C4×D4—41no—
38C2×C4×D4—41no—
39C2×C4×D4C2×C411yescharacteristic
40C2×C4×D4C2×C411yescharacteristic
41C2×C4×D4C2×C411yescharacteristic
42C2×C4×D4D411yescharacteristic
43C2×C4×D4D411yescharacteristic
44C2×C4×D4D411yescharacteristic
45C2×C4×D4Q811yescharacteristic
46C2×C4×D4C2311yescharacteristic
47C22×C4⋊C4D411yescharacteristic
48C22×C22⋊C4C2×C411yescharacteristic
49C22×C22⋊C4Q811yescharacteristic
50C22×C42D411yescharacteristic
51C2×#100—22no—
52C2×#100—28no—
53C2×#100C2×C414yes—
54#407—22no—
55#407—22no—
56#407—41no—
57#407—41no—
58#407—41no—
59#407—41no—
60C2×C22⋊C8—22no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—41no—
67C2×C22⋊C8—41no—
68C2×C22⋊C8—41no—
69C2×C22⋊C8C2×C412yes—
70C4×M16—22no—
71C4×M16—22no—
72C4×M16—41no—
73C4×M16—41no—
74C2×#97—41no—
75C2×C4×C8—41no—
76#343—22no—
77#343—22no—
78#336—22no—
79#336—22no—
80#324—22no—
81#324—22no—
82#323—22no—
83#323C2×C414yes—
84C25C4211yescharacteristic
85C25C4⋊C411yescharacteristic
86C22×D4—24no—
87C22×D4—42no—
88C22×D4—41no—
89C22×D4—41no—
90C22×D4C4211yescharacteristic
91C22×D4C22⋊C411yescharacteristic
92C22×D4C4⋊C411yescharacteristic
93C22×D4C22×C411yescharacteristic
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—82no—
103C23×C4—82no—
104C23×C4C4211yescharacteristic
105C23×C4C22⋊C411yescharacteristic
106C23×C4C22⋊C411yescharacteristic
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22⋊C411yescharacteristic
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22⋊C411yescharacteristic
111C23×C4C4⋊C411yescharacteristic
112C23×C4C4⋊C411yescharacteristic
113C23×C4C4⋊C411yescharacteristic
114C23×C4C4⋊C411yescharacteristic
115C23×C4C22×C411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C2×M16—28no—
118C2×M16—24no—
119C2×M16—44no—
120C2×M16—42no—
121C2×M16—42no—
122C2×M16—42no—
123C2×M16—42no—
124C2×M16—82no—
125C2×M16—84no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—42no—
129C22×C8—42no—
130C22×C8—42no—
131C22×C8—82no—
132C4×D4—41no—
133C4×D4—41no—
134C4×D4—41no—
135C4×D4—41no—
136C4×D4—41no—
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—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4C22⋊C411yescharacteristic
151C2×C4⋊C4C22⋊C411yescharacteristic
152C2×C4⋊C4C4⋊C411yescharacteristic
153C2×C4⋊C4C2×D411yescharacteristic
154C2×C22⋊C4—41no—
155C2×C22⋊C4—41no—
156C2×C22⋊C4—41no—
157C2×C22⋊C4—41no—
158C2×C22⋊C4C4211yescharacteristic
159C2×C22⋊C4C22⋊C411yescharacteristic
160C2×C22⋊C4C4⋊C411yescharacteristic
161C2×C22⋊C4C4⋊C411yescharacteristic
162C2×C22⋊C4C4⋊C411yescharacteristic
163C2×C22⋊C4C4⋊C411yescharacteristic
164C2×C22⋊C4C22×C411yescharacteristic
165C2×C22⋊C4C2×Q811yescharacteristic
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42C22⋊C411yescharacteristic
171C2×C42C22⋊C411yescharacteristic
172C2×C42C4⋊C411yescharacteristic
173C2×C42C2×D411yescharacteristic
174#100—28no—
175#100—24no—
176#100M1618yescomplemented
177C22⋊C8—24no—
178C22⋊C8—42no—
179C22⋊C8—42no—
180C22⋊C8—42no—
181C22⋊C8—42no—
182C22⋊C8—41no—
183C22⋊C8—41no—
184C22⋊C8—41no—
185C22⋊C8—41no—
186C22⋊C8—42no—
187C22⋊C8—82no—
188C22⋊C8—82no—
189#97—42no—
190#97—42no—
191C4×C8—42no—
192C4×C8—42no—
193C24—21no—
194C24—21no—
195C24—21no—
196C24—21no—
197C24—22no—
198C24—22no—
199C24—41no—
200C24—41no—
201C24—42no—
202C24—42no—
203C24—81no—
204C24—81no—
205C24#9511yescharacteristic
206C24#9511yescharacteristic
207C24#9511yescharacteristic
208C24C2×C4211yescharacteristic
209C24C2×C4⋊C411yescharacteristic
210C2×D4—22no—
211C2×D4—22no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—48no—
222C2×D4—48no—
223C2×D4C2×M1614yes—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
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—21no—
250C22×C4—42no—
251C22×C4—44no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4—81no—
270C22×C4#9511yescharacteristic
271C22×C4#9511yescharacteristic
272C22×C4#9511yescharacteristic
273C22×C4#9511yescharacteristic
274C22×C4#9511yescharacteristic
275C22×C4#9911yescharacteristic
276C22×C4#9911yescharacteristic
277C22×C4#9911yescharacteristic
278C22×C4#9911yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C4⋊C411yescharacteristic
283C22×C4C2×C4⋊C411yescharacteristic
284M16—28no—
285M16—48no—
286M16—44no—
287M16—84no—
288M16—84no—
289M16—84no—
290C2×C8—44no—
291C2×C8—48no—
292C2×C8—44no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—84no—
300C2×C8—84no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—21no—
321C23—22no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—41no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—42no—
346C23—42no—
347C23—41no—
348C23—41no—
349C23—42no—
350C23—42no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23—84no—
356C23—84no—
357C23—84no—
358C23—84no—
359C23#34211yescharacteristic
360C23#34211yescharacteristic
361C23C2×#9511yescharacteristic
362C23C4×M1612yes—
363C23#42312yes—
364D4—44no—
365D4—44no—
366D4—44no—
367D4—44no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4#34211yescharacteristic
415C2×C4#34211yescharacteristic
416C2×C4#34412yes—
417C2×C4#40712yes—
418C2×C4C2×#9911yescharacteristic
419C2×C4C2×#9911yescharacteristic
420C8—48no—
421C8—82no—
422C8—84no—
423C8—82no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—22no—
432C22—22no—
433C22—42no—
434C22—42no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—81no—
450C22—84no—
451C22—84no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22—84no—
456C22—84no—
457C22#139212yes—
458C22#172811yescharacteristic
459C22C2×#34211yescharacteristic
460C22C2×#34411yescharacteristic
461C4—21no—
462C4—21no—
463C4—21no—
464C4—22no—
465C4—22no—
466C4—41no—
467C4—41no—
468C4—41no—
469C4—41no—
470C4—41no—
471C4—41no—
472C4#212112yes—
473C2—21no—
474C2—21no—
475C2—22no—
476C2—41no—
477C2—42no—
478C2—42no—
479C2—81no—
480C2—81no—
481C2#792911yescharacteristic
482C2#1257411yescharacteristic
483C2C2×#139211yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC24C2×C42
Frattini, cosocleC2×C4×D4C23
socleC22C2×#342
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ 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. #7203C2semidirect
176. #100M16semidirect

Maximal subgroups

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

subgroupindex
#217402
C2×C4×#1002
#173622
#72032