Finite Group Databasebeta

← #195051

Subgroups — #195051

Order 512.

1385 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60125C211yesmaximal, characteristic, complemented
2#34937C211yesmaximal, characteristic
3#17485C212yesmaximal, complemented
4#17349C211yesmaximal, characteristic, complemented
5#9036C218yesmaximal, complemented
6C2×#1303C212yesmaximal, complemented
7C2×#532—21no—
8C2×#532C2211yescharacteristic
9C2×C4×#41C2211yescharacteristic
10#2925—24no—
11#2925—24no—
12#2921C2214yes—
13#2920—22no—
14C2×#431C2212yes—
15C2×#423C2212yes—
16#2879—21no—
17#2873—21no—
18#2548C411yescharacteristic
19#2547C411yescharacteristic
20#2546C411yescharacteristic
21#2545—21no—
22#2544—21no—
23#2543C414yes—
24#2541C411yescharacteristic
25#2416—22no—
26#2392C2214yes—
27C4×#121—22no—
28#2297—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30#1492—24no—
31#1486C2218yes—
32#1480—24no—
33#1461—21no—
34#1460—21no—
35#1459—22no—
36#1458C2214yes—
37#1303C2218yes—
38C22×#41—21no—
39C22×#41D411yescharacteristic
40C22×M16—22no—
41#532—22no—
42#532—22no—
43#532—22no—
44#532—22no—
45#532—48no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8C2×C411yescharacteristic
48C2×#127—21no—
49C2×#127C2×C411yescharacteristic
50C2×#124—21no—
51C2×#124—21no—
52C2×#124—21no—
53C2×#121—24no—
54C2×#121—22no—
55C2×#121—21no—
56C2×#121C2×C412yes—
57C4×#41—22no—
58C4×#41—22no—
59C4×#41—22no—
60C4×#41—22no—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×D4—24no—
64C2×C4×D4—21no—
65C2×C4×D4C2×C412yes—
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4D412yes—
68C2×C4×D4D411yescharacteristic
69C2×C4×D4C2312yes—
70C2×#117D411yescharacteristic
71C2×#117C2312yes—
72C22×C4⋊C4—21no—
73C22×C42—21no—
74C22×C42D412yes—
75C22×C42C2311yescharacteristic
76#433—216no—
77#432—24no—
78#432—24no—
79#431—24no—
80#431—24no—
81#431—22no—
82#431—22no—
83#431—22no—
84#431C2314yes—
85#423—22no—
86#423—22no—
87#423D414yescomplemented
88#423C2314yes—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8D414yescomplemented
92#408—28no—
93#407—24no—
94C2×C22⋊C8—22no—
95#405—28no—
96C4×M16—24no—
97C2×C4×C8—22no—
98#398C2×C414yes—
99#397C2×C414yes—
100#396—24no—
101#394—24no—
102#394C2×C414yes—
103#392C2×C414yes—
104#391—21no—
105#391—21no—
106#390—21no—
107#390—21no—
108#390—22no—
109#389—22no—
110#388—24no—
111#386—24no—
112#386—24no—
113#380—22no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4C2312yes—
119C4×C22⋊C4—24no—
120C2×#95C2×C412yes—
121D4⋊C8—216no—
122#324—24no—
123#323—24no—
124C2×#41—22no—
125C2×#41—22no—
126C2×#41—24no—
127C2×#41—24no—
128C2×#41—42no—
129C2×#41—42no—
130C2×#41—44no—
131C2×#41—44no—
132C22×Q8—21no—
133C22×Q8C22⋊C411yescharacteristic
134C22×D4—22no—
135C22×D4—24no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4C22⋊C412yes—
139C22×D4C22⋊C411yescharacteristic
140C22×D4C22×C412yes—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4C22⋊C412yes—
144C23×C4C22⋊C412yes—
145C23×C4C22×C411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C2×M16—24no—
150C2×M16—24no—
151C2×M16—44no—
152C2×M16—44no—
153C22×C8—24no—
154C22×C8—42no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—44no—
158#127—22no—
159#127—22no—
160#127—44no—
161#124—44no—
162#124—44no—
163#124—42no—
164#124—42no—
165#121—24no—
166#121—24no—
167#121—48no—
168#121—416no—
169#121—42no—
170#121—42no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×Q8—22no—
175C4×D4—24no—
176C4×D4—24no—
177C4×D4—24no—
178C4×D4—24no—
179C4×D4—22no—
180C4×D4—22no—
181C4×D4—22no—
182C4×D4—22no—
183C4×D4—24no—
184C4×D4—24no—
185C4×D4—416no—
186C4×D4—41no—
187C4×D4—41no—
188C4×D4—41no—
189C4×D4—41no—
190#117—22no—
191#117—22no—
192#117—24no—
193#117—24no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4C22⋊C411yescharacteristic
201C2×C4⋊C4C22⋊C411yescharacteristic
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4C22⋊C412yes—
212C2×C22⋊C4C22×C412yes—
213C2×C22⋊C4C2×D412yes—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—22no—
221C2×C42—44no—
222C2×C42C22⋊C412yes—
223C2×C42C22⋊C412yes—
224C2×C42C22×C411yescharacteristic
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42#4112yes—
231C2×C42#4112yes—
232C2×C42C2411yescharacteristic
233C4⋊C8—24no—
234C4⋊C8—24no—
235C4⋊C8—24no—
236C4⋊C8—24no—
237C4⋊C8—42no—
238C4⋊C8C2×D418yes—
239C22⋊C8—28no—
240C22⋊C8—28no—
241C22⋊C8—24no—
242C22⋊C8—42no—
243#97—28no—
244C4×C8—28no—
245C4×C8—24no—
246C4×C8—42no—
247#95—22no—
248#95—24no—
249#95—24no—
250#95C22×C412yes—
251#95C22×C412yes—
252C24—22no—
253C24C2×C22⋊C412yes—
254C24C2×C22⋊C411yescharacteristic
255#41—44no—
256#41—44no—
257#41—48no—
258#41—84no—
259#41—84no—
260#41—88no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—44no—
275C2×D4—48no—
276C2×D4—48no—
277C2×D4—44no—
278C2×D4—44no—
279C2×D4—44no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—82no—
286C2×D4—84no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—42no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—81no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C4×D412yes—
316C22×C4C4×D412yes—
317C22×C4#12012yes—
318C22×C4#12112yes—
319C22×C4#12112yes—
320C22×C4#12312yes—
321C22×C4#13514yes—
322C22×C4AGL(1,ℤ/8)14yes—
323C22×C4C23×C411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326M16—44no—
327M16—44no—
328M16—48no—
329C2×C8—28no—
330C2×C8—48no—
331C2×C8—44no—
332C2×C8—84no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—24no—
339C4⋊C4—42no—
340C4⋊C4—44no—
341C4⋊C4—48no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C22⋊C4—24no—
345C22⋊C4—24no—
346C22⋊C4—24no—
347C22⋊C4—48no—
348C22⋊C4—48no—
349C22⋊C4—48no—
350C22⋊C4—44no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C22⋊C4—84no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—44no—
361C42—42no—
362C42—42no—
363C42C4×D412yes—
364C42C4×D412yes—
365C42#12011yescharacteristic
366C42#12011yescharacteristic
367C42#12112yes—
368C42#12112yes—
369C42#12311yescharacteristic
370C42#12311yescharacteristic
371C42C22×D412yes—
372C42C2×#4112yes—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—41no—
377C23—41no—
378C23—42no—
379C23—42no—
380C23—44no—
381C23—44no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23#38612yes—
386C23#41612yes—
387C23#41812yes—
388C23C22×C22⋊C411yescharacteristic
389Q8—44no—
390Q8—84no—
391D4—48no—
392D4—48no—
393D4—44no—
394D4—82no—
395D4—816no—
396D4—88no—
397D4—84no—
398D4—82no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—88no—
430C2×C4—84no—
431C2×C4—84no—
432C2×C4#38611yescharacteristic
433C2×C4#38611yescharacteristic
434C2×C4#38611yescharacteristic
435C2×C4#38612yes—
436C2×C4#38611yescharacteristic
437C2×C4#41612yes—
438C2×C4#41812yes—
439C2×C4#44914yes—
440C2×C4#44914yes—
441C2×C4#48218yes—
442C2×C4C2×C4×D412yes—
443C2×C4C2×#12011yescharacteristic
444C2×C4C2×#12112yes—
445C2×C4C2×#12311yescharacteristic
446C2×C4C2×#13512yes—
447C2×C4C2×AGL(1,ℤ/8)12yes—
448C8—48no—
449C8—84no—
450C22—21no—
451C22—22no—
452C22—21no—
453C22—22no—
454C22—42no—
455C22—42no—
456C22—44no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—82no—
461C22—84no—
462C22—84no—
463C22—82no—
464C22—82no—
465C22—82no—
466C22#187514yes—
467C22C2×#38611yescharacteristic
468C22C2×#41611yescharacteristic
469C22C2×#41811yescharacteristic
470C4—22no—
471C4—21no—
472C4—21no—
473C4—21no—
474C4—22no—
475C4—44no—
476C4—42no—
477C4—42no—
478C4—42no—
479C4—82no—
480C4—82no—
481C4#187514yes—
482C4#188412yes—
483C4C2×#44912yes—
484C4C2×#48212yes—
485C2—21no—
486C2—21no—
487C2—24no—
488C2—82no—
489C2—162no—
490C2C2×#187511yescharacteristic
491C2C2×#188411yescharacteristic
492C2#2096411yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radical#34937C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60125C2semidirect
3. #17485C2semidirect
4. #17349C2semidirect
5. #9036C2semidirect
6. C2×#1303C2semidirect
87. #423D4semidirect
91. C2×C4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
#601252
#349372
#174852
#173492
#90362
C2×#13032