Finite Group Databasebeta

← #206740

Subgroups — #206740

Order 512.

1006 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36640C211yesmaximal, characteristic
2C4×#423C211yesmaximal, characteristic
3C2×C4×C4⋊C8C211yesmaximal, characteristic
4#9867C214yesmaximal, complemented
5#9866C214yesmaximal, complemented
6#7897C212yesmaximal
7#7897C212yesmaximal
8C2×#423C2212yes—
9C22×C4⋊C8—21no—
10C2×C4×M16—21no—
11C22×C4×C8C2212yes—
12#2864—21no—
13#2864—21no—
14#2859C411yescharacteristic
15#2859C411yescharacteristic
16#2859C411yescharacteristic
17#2859C411yescharacteristic
18#2851—21no—
19#2851—21no—
20#2851C412yes—
21#2851C412yes—
22C4×#125—21no—
23C4×#125—21no—
24C4×#125C2212yes—
25C4×#125C2212yes—
26#2296—21no—
27#2296—21no—
28#2285C2211yescharacteristic
29#2285C2211yescharacteristic
30C2×C43C2211yescharacteristic
31#1767—21no—
32#1767C2212yes—
33C4×C4⋊C8—21no—
34C4×C4⋊C8—21no—
35C4×C4⋊C8—21no—
36C4×C4⋊C8C2212yes—
37C4×C4⋊C8C2212yes—
38C4×C4⋊C8C2212yes—
39#1728—21no—
40C2×#336—21no—
41#1279C2218yes—
42#1279C2218yes—
43C22×M16—22no—
44C23×C8—22no—
45C2×#125—21no—
46C2×#125—21no—
47C2×#125C2×C412yes—
48C2×#125C2×C412yes—
49C2×#117—21no—
50C2×#117—21no—
51C22×C42C2×C411yescharacteristic
52C22×C42C2×C411yescharacteristic
53C22×C42D411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42C2311yescharacteristic
58#423—24no—
59#423Q818yes—
60C2×C4⋊C8—28no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8C2314yes—
65C4×M16—44no—
66C2×#97—22no—
67C2×C4×C8—28no—
68C2×C4×C8—22no—
69C2×C4×C8—22no—
70C2×C4×C8C2314yes—
71#389—21no—
72#389—21no—
73#389—21no—
74#389—21no—
75#389—21no—
76#389—21no—
77#389—21no—
78#389—21no—
79#389—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#389C2×C412yes—
84#389C2×C412yes—
85#389C2×C412yes—
86#389C2×C412yes—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#385—22no—
96#385—22no—
97#385—22no—
98#385—22no—
99#384C2×C411yescharacteristic
100#384C2×C411yescharacteristic
101#384C2×C411yescharacteristic
102#384C2×C411yescharacteristic
103#382—21no—
104#382—21no—
105#382—21no—
106#382—21no—
107#382C2×C412yes—
108#382C2×C412yes—
109#380—21no—
110#380—21no—
111#380C2×C412yes—
112#380C2×C412yes—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4C2312yes—
120C4×C4⋊C4C2312yes—
121C43C2×C411yescharacteristic
122C43C2×C411yescharacteristic
123C43D411yescharacteristic
124C43D411yescharacteristic
125C43D411yescharacteristic
126C43D411yescharacteristic
127C43C2311yescharacteristic
128C43C2311yescharacteristic
129#336—22no—
130#336—22no—
131#336—22no—
132#336—22no—
133C23×C4C22⋊C411yescharacteristic
134C23×C4C22⋊C411yescharacteristic
135C23×C4C22⋊C411yescharacteristic
136C23×C4C22⋊C411yescharacteristic
137C23×C4C22×C411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C2×M16—28no—
141C2×M16—44no—
142C2×M16—44no—
143C22×C8—28no—
144C22×C8—24no—
145C22×C8—24no—
146C22×C8—44no—
147C22×C8—44no—
148#125—28no—
149#125—44no—
150#125—44no—
151#125—44no—
152#117—44no—
153#117—42no—
154#117—42no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C22×C412yes—
173C2×C4⋊C4C22×C412yes—
174C2×C4⋊C4C22×C412yes—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C42—24no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—24no—
181C2×C42—24no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—42no—
185C2×C42—42no—
186C2×C42C22⋊C411yescharacteristic
187C2×C42C22⋊C411yescharacteristic
188C2×C42C22⋊C411yescharacteristic
189C2×C42C22⋊C411yescharacteristic
190C2×C42C22⋊C411yescharacteristic
191C2×C42C22⋊C411yescharacteristic
192C2×C42C22⋊C411yescharacteristic
193C2×C42C22⋊C411yescharacteristic
194C2×C42C22⋊C411yescharacteristic
195C2×C42C22⋊C411yescharacteristic
196C2×C42C22⋊C411yescharacteristic
197C2×C42C22⋊C411yescharacteristic
198C2×C42C22×C411yescharacteristic
199C2×C42C22×C411yescharacteristic
200C2×C42C22×C411yescharacteristic
201C2×C42C22×C411yescharacteristic
202C2×C42C22×C411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×Q814yes—
214C2×C42#4114yes—
215C2×C42C2411yescharacteristic
216C4⋊C8—28no—
217C4⋊C8—28no—
218C4⋊C8—28no—
219C4⋊C8—24no—
220C4⋊C8C2×Q818yes—
221#97—44no—
222C4×C8—28no—
223C4×C8—44no—
224C4×C8—44no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237C24C2×C22⋊C411yescharacteristic
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C4×Q814yes—
270C22×C4C22⋊Q814yes—
271C22×C4#12412yes—
272C22×C4C4⋊Q812yes—
273C22×C4#13514yes—
274C22×C4AGL(1,ℤ/8)12yes—
275C22×C4#13712yes—
276C22×C4C23×C411yescharacteristic
277C22×C4C22×D411yescharacteristic
278C22×C4C22×D411yescharacteristic
279M16—416no—
280C2×C8—28no—
281C2×C8—28no—
282C2×C8—44no—
283C2×C8—416no—
284C2×C8—44no—
285C2×C8—44no—
286C2×C8—44no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C22⋊C4—44no—
308C22⋊C4—44no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—21no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—41no—
330C42—41no—
331C42—41no—
332C42—41no—
333C42—44no—
334C42—42no—
335C42—42no—
336C42#10414yes—
337C42C4×Q814yes—
338C42C22⋊Q814yes—
339C42#12412yes—
340C42C4⋊Q812yes—
341C42#13114yes—
342C42C22×Q812yes—
343C42C2×#4112yes—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23#39112yes—
349C23#41612yes—
350C23#41712yes—
351C23C22×C22⋊C411yescharacteristic
352C2×C4—28no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—28no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
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#39112yes—
396C2×C4#39112yes—
397C2×C4#39112yes—
398C2×C4#40812yes—
399C2×C4#40812yes—
400C2×C4#41612yes—
401C2×C4#41712yes—
402C2×C4C2×#10412yes—
403C2×C4C2×#10412yes—
404C2×C4C2×#10412yes—
405C2×C4#42112yes—
406C2×C4#47818yes—
407C2×C4#47918yes—
408C2×C4C2×C4×Q812yes—
409C2×C4C2×C22⋊Q812yes—
410C2×C4C2×#12411yescharacteristic
411C2×C4C2×C4⋊Q811yescharacteristic
412C2×C4C2×#13112yes—
413C2×C4#56812yes—
414C2×C4C2×#13512yes—
415C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
416C2×C4C2×#13711yescharacteristic
417C8—48no—
418C8—48no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22#198914yes—
426C22C2×#39111yescharacteristic
427C22C2×#40811yescharacteristic
428C22#287911yescharacteristic
429C22C2×#41611yescharacteristic
430C22C2×#41711yescharacteristic
431C22C22×#10411yescharacteristic
432C22C2×#42111yescharacteristic
433C4—22no—
434C4—21no—
435C4—21no—
436C4—22no—
437C4—22no—
438C4—22no—
439C4—22no—
440C4—41no—
441C4—41no—
442C4—41no—
443C4—41no—
444C4—82no—
445C4—82no—
446C4—82no—
447C4—82no—
448C4#198614yes—
449C4#198914yes—
450C4#296214yes—
451C4C2×#47812yes—
452C4C2×#47912yes—
453C2—24no—
454C2#1747311yescharacteristic
455C2#1748111yescharacteristic
456C2#1748911yescharacteristic
457C2#1750411yescharacteristic
458C2#2130211yescharacteristic
459C2C2×#198611yescharacteristic
460C2C2×#198911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9867C2semidirect
5. #9866C2semidirect

Maximal subgroups

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

subgroupindex
#366402
C4×#4232
C2×C4×C4⋊C82
#98672
#98662
#78972