Finite Group Databasebeta

← #195027

Subgroups — #195027

Order 512.

1455 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60125C211yesmaximal, characteristic, complemented
2#34935C211yesmaximal, characteristic
3C2×#1486C212yesmaximal, complemented
4#17349C211yesmaximal, characteristic, complemented
5#9036C218yesmaximal, complemented
6#8095C212yesmaximal, complemented
7C2×#525—21no—
8C2×C4×#41C2211yescharacteristic
9C22×C4×D4C2211yescharacteristic
10#2925—24no—
11#2925—24no—
12#2921C2214yescomplemented
13#2920—22no—
14C2×#431C2212yes—
15C2×#423C2212yes—
16#2879—21no—
17#2873—21no—
18#2538—21no—
19#2537—21no—
20#2536C412yes—
21#2535C412yes—
22#2534—22no—
23#2486C412yes—
24#2408C412yes—
25#2392C2214yes—
26C4×#121—22no—
27#2285C2211yescharacteristic
28#2280—21no—
29#1486—24no—
30#1486C2218yes—
31C2×D4⋊C8—24no—
32#1461—21no—
33#1460—21no—
34#1459—22no—
35#1458C2214yescomplemented
36#1303C2218yescomplemented
37C22×#41D411yescharacteristic
38C23×D4C2×C411yescharacteristic
39C24×C4—21no—
40C22×M16—22no—
41#525—48no—
42C2×#123—22no—
43C2×C22⋊Q8—22no—
44C2×#121—22no—
45C2×#121—22no—
46C2×#120—22no—
47C4×#41—22no—
48C4×#41—22no—
49C4×#41—22no—
50C4×#41—22no—
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×Q8D411yescharacteristic
53C2×C4×D4—24no—
54C2×C4×D4—21no—
55C2×C4×D4—21no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4C2×C412yes—
61C2×C4×D4C2×C412yes—
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4D412yes—
64C2×C4×D4D411yescharacteristic
65C2×C4×D4C2312yes—
66C2×#117—21no—
67C2×#117D411yescharacteristic
68C2×#117C2312yes—
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C22⋊C4—21no—
71C22×C42D412yes—
72C22×C42C2311yescharacteristic
73#433—216no—
74#432—24no—
75#432—24no—
76#431—24no—
77#431—24no—
78#431—22no—
79#431—22no—
80#431—22no—
81#431C2314yes—
82#423—22no—
83#423—22no—
84#423D414yescomplemented
85#423C2314yes—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8D414yescomplemented
89#408—28no—
90#407—24no—
91C2×C22⋊C8—22no—
92#405—28no—
93C4×M16—24no—
94C2×C4×C8—22no—
95#398C2×C414yes—
96#397C2×C414yes—
97#396—24no—
98#394—24no—
99#394C2×C414yes—
100#392C2×C414yes—
101#390—22no—
102#389—22no—
103#388—24no—
104#386—24no—
105#384—21no—
106#382C2×C412yes—
107#381—22no—
108#380—22no—
109#380—21no—
110#380—22no—
111#380—22no—
112#380C2×C412yes—
113#379—22no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4C2312yes—
116C4×C22⋊C4—24no—
117D4⋊C8—216no—
118#324—24no—
119#323—24no—
120C25—21no—
121C2×#41—22no—
122C2×#41—22no—
123C2×#41—24no—
124C2×#41—24no—
125C22×Q8C22⋊C411yescharacteristic
126C22×D4—24no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—44no—
131C22×D4C22⋊C412yes—
132C22×D4C22⋊C411yescharacteristic
133C22×D4C22×C412yes—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—42no—
138C23×C4—42no—
139C23×C4—44no—
140C23×C4—44no—
141C23×C4C22⋊C412yes—
142C23×C4C22⋊C412yes—
143C23×C4C22×C411yescharacteristic
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C2×M16—24no—
148C2×M16—24no—
149C2×M16—44no—
150C2×M16—44no—
151C22×C8—24no—
152C22×C8—42no—
153#123—48no—
154C22⋊Q8—48no—
155#121—48no—
156#121—48no—
157#120—48no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×D4—24no—
163C4×D4—24no—
164C4×D4—24no—
165C4×D4—24no—
166C4×D4—22no—
167C4×D4—22no—
168C4×D4—24no—
169C4×D4—24no—
170C4×D4—22no—
171C4×D4—22no—
172C4×D4—24no—
173C4×D4—24no—
174C4×D4—416no—
175C4×D4—41no—
176C4×D4—41no—
177C4×D4—42no—
178C4×D4—42no—
179C4×D4—41no—
180C4×D4—41no—
181#117—22no—
182#117—22no—
183#117—24no—
184#117—24no—
185#117—44no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C22×C412yes—
196C2×C4⋊C4C22×C412yes—
197C2×C4⋊C4C2×D412yes—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4C22⋊C412yes—
208C2×C22⋊C4C22×C412yes—
209C2×C22⋊C4C2×D412yes—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42C22⋊C412yes—
216C2×C42C22⋊C412yes—
217C2×C42C22×C411yescharacteristic
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42#4112yes—
224C2×C42#4112yes—
225C2×C42C2411yescharacteristic
226C4⋊C8—24no—
227C4⋊C8—24no—
228C4⋊C8—24no—
229C4⋊C8—24no—
230C4⋊C8—42no—
231C4⋊C8C2×D418yes—
232C22⋊C8—28no—
233C22⋊C8—28no—
234C22⋊C8—24no—
235C22⋊C8—42no—
236#97—28no—
237C4×C8—28no—
238C4×C8—24no—
239C4×C8—42no—
240#95—24no—
241#95—24no—
242#95—22no—
243#95C22×C414yes—
244C24—22no—
245C24—44no—
246C24—42no—
247C24—42no—
248C24—42no—
249C24—82no—
250C24C2×C22⋊C412yes—
251C24C2×C22⋊C411yescharacteristic
252#41—44no—
253#41—44no—
254#41—48no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×D4—28no—
259C2×D4—24no—
260C2×D4—24no—
261C2×D4—24no—
262C2×D4—24no—
263C2×D4—24no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—42no—
268C2×D4—44no—
269C2×D4—44no—
270C2×D4—48no—
271C2×D4—48no—
272C2×D4—48no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—84no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—81no—
299C22×C4—82no—
300C22×C4—84no—
301C22×C4—84no—
302C22×C4—88no—
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C4×D412yes—
310C22×C4C4×D412yes—
311C22×C4#12012yes—
312C22×C4#12112yes—
313C22×C4#12112yes—
314C22×C4#12312yes—
315C22×C4#13514yes—
316C22×C4AGL(1,ℤ/8)14yes—
317C22×C4C23×C411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320M16—44no—
321M16—44no—
322M16—48no—
323C2×C8—28no—
324C2×C8—48no—
325C2×C8—44no—
326C2×C8—84no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—24no—
332C4⋊C4—48no—
333C4⋊C4—44no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—44no—
337C4⋊C4—82no—
338C22⋊C4—24no—
339C22⋊C4—24no—
340C22⋊C4—24no—
341C22⋊C4—48no—
342C22⋊C4—48no—
343C22⋊C4—48no—
344C22⋊C4—44no—
345C22⋊C4—44no—
346C22⋊C4—42no—
347C22⋊C4—42no—
348C22⋊C4—84no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—44no—
356C42C4×D412yes—
357C42C4×D412yes—
358C42#12011yescharacteristic
359C42#12011yescharacteristic
360C42#12112yes—
361C42#12112yes—
362C42#12311yescharacteristic
363C42#12311yescharacteristic
364C42C22×D412yes—
365C42C2×#4112yes—
366C23—24no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—41no—
371C23—41no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—48no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—84no—
382C23—88no—
383C23—88no—
384C23—84no—
385C23#38612yes—
386C23#41612yes—
387C23#41812yes—
388C23C22×C22⋊C411yescharacteristic
389Q8—44no—
390D4—28no—
391D4—44no—
392D4—44no—
393D4—48no—
394D4—44no—
395D4—44no—
396D4—816no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—84no—
425C2×C4—84no—
426C2×C4—88no—
427C2×C4—84no—
428C2×C4#38611yescharacteristic
429C2×C4#38611yescharacteristic
430C2×C4#38611yescharacteristic
431C2×C4#38612yes—
432C2×C4#38611yescharacteristic
433C2×C4#41612yes—
434C2×C4#41812yes—
435C2×C4#44914yes—
436C2×C4#44914yes—
437C2×C4#48218yes—
438C2×C4C2×C4×D412yes—
439C2×C4C2×#12011yescharacteristic
440C2×C4C2×#12112yes—
441C2×C4C2×#12311yescharacteristic
442C2×C4C2×#13512yes—
443C2×C4C2×AGL(1,ℤ/8)12yes—
444C8—48no—
445C8—84no—
446C22—21no—
447C22—22no—
448C22—21no—
449C22—22no—
450C22—42no—
451C22—42no—
452C22—44no—
453C22—44no—
454C22—44no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—44no—
459C22—82no—
460C22—82no—
461C22—82no—
462C22—816no—
463C22—82no—
464C22—82no—
465C22#187514yes—
466C22C2×#38611yescharacteristic
467C22C2×#41611yescharacteristic
468C22C2×#41811yescharacteristic
469C4—22no—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—22no—
474C4—42no—
475C4—42no—
476C4—42no—
477C4—84no—
478C4—82no—
479C4#187514yes—
480C4#189112yes—
481C4C2×#44912yes—
482C4C2×#48212yes—
483C2—21no—
484C2—21no—
485C2—24no—
486C2—82no—
487C2—84no—
488C2C2×#187511yescharacteristic
489C2#2094711yescharacteristic
490C2C2×#189111yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radical#34935C2
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. C2×#1486C2semidirect
4. #17349C2semidirect
5. #9036C2semidirect
6. #8095C2semidirect
12. #2921C22semidirect
35. #1458C22semidirect
36. #1303C22semidirect
84. #423D4semidirect
88. C2×C4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
#601252
#349352
C2×#14862
#173492
#90362
#80952