Finite Group Databasebeta

← #196844

Subgroups — #196844

Order 512.

1384 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35179C211yesmaximal, characteristic
2C2×#2351C211yesmaximal, characteristic
3#21818C211yesmaximal, characteristic
4C2×#1518C211yesmaximal, characteristic
5C2×#1503C211yesmaximal, characteristic
6#9152C214yesmaximal
7#9144C214yesmaximal
8#8026C211yesmaximal, characteristic
9#8014C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4×Q8—21no—
12C22×C4×Q8C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#2855C2214yes—
16#2627—22no—
17#2582C414yescomplemented
18#2581—22no—
19#2568C412yescomplemented
20#2567C412yescomplemented
21#2565—22no—
22#2351—24no—
23#2351—24no—
24#2317—21no—
25#2310—21no—
26C2×#389—21no—
27C2×#389—22no—
28#2285C2211yescharacteristic
29C2×#382C2212yes—
30C2×#382C2211yescharacteristic
31C2×C4×C4⋊C4C2212yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33#1984—22no—
34#1984—22no—
35#1817C2214yes—
36#1518—24no—
37#1503—24no—
38C2×#328C2212yes—
39C2×#328C2212yes—
40C2×#328C2214yes—
41#1294C2212yes—
42#1294C2212yes—
43#1294C2214yes—
44C23×Q8D411yescharacteristic
45#533—22no—
46#533—22no—
47#533—22no—
48#533—22no—
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×#125—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8C2312yes—
54C2×C4×Q8—28no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—21no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—44no—
61C2×C4×Q8—44no—
62C2×C4×Q8C2×C412yes—
63C2×C4×Q8C2×C412yes—
64C2×C4×Q8C2×C412yes—
65C2×C4×Q8D412yes—
66C2×C4×Q8D412yes—
67C2×#117—21no—
68C2×#117—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D412yes—
75C22×C42—22no—
76C22×C42—21no—
77C22×C42C2312yes—
78C22×C42C2311yescharacteristic
79#431—22no—
80#431—22no—
81#431—22no—
82#431—22no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C22⋊C8C2312yes—
86C2×C22⋊C8C2312yes—
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89#400—24no—
90#398—28no—
91#395—24no—
92#391—22no—
93#391—22no—
94#391—24no—
95#391C2×C418yes—
96#389—28no—
97#389—216no—
98#389—24no—
99#389C2×C418yes—
100#387—28no—
101#385—24no—
102#384—22no—
103#384C2×C412yes—
104#382—22no—
105#382—21no—
106#382—22no—
107#382—24no—
108#382—24no—
109#382D414yes—
110#381—24no—
111#380—21no—
112#380C2312yes—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—24no—
116C4×C4⋊C4—24no—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4D414yes—
119C2×#95—22no—
120C2×#95—21no—
121C2×#95C2311yescharacteristic
122C2×#95C2312yes—
123C2×#95C2311yescharacteristic
124#336—28no—
125#328—24no—
126#328—24no—
127#328—28no—
128C22×Q8—21no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8C22⋊C412yes—
134C22×Q8C22⋊C412yes—
135C22×Q8SD1614yes—
136C22×Q8Q1614yes—
137C22×Q8C22×C412yes—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2411yescharacteristic
146C22×C8—22no—
147C22×C8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152#125—42no—
153#125—42no—
154C22⋊Q8—22no—
155C22⋊Q8—22no—
156C22⋊Q8—22no—
157C22⋊Q8—22no—
158C22⋊Q8—42no—
159C22⋊Q8—42no—
160C4×Q8—24no—
161C4×Q8—24no—
162C4×Q8—24no—
163C4×Q8—24no—
164C4×Q8—24no—
165C4×Q8—24no—
166C4×Q8—24no—
167C4×Q8—24no—
168C4×Q8—24no—
169C4×Q8—24no—
170C4×Q8—432no—
171C4×Q8—42no—
172C4×Q8—42no—
173#117—42no—
174#117—42no—
175#117—42no—
176#117—42no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—48no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4C22⋊C412yes—
200C2×C4⋊C4C22⋊C412yes—
201C2×C4⋊C4C22⋊C414yes—
202C2×C4⋊C4C22⋊C414yes—
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C22×C414yes—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—28no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—24no—
215C2×C42—44no—
216C2×C42—44no—
217C2×C42—42no—
218C2×C42—42no—
219C2×C42C22×C414yes—
220C2×C42C22×C411yescharacteristic
221C2×C42C22×C411yescharacteristic
222C2×C42C2×D414yes—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42#4112yes—
228C2×C42#4112yes—
229C4⋊C8—42no—
230C4⋊C8—42no—
231C22⋊C8—22no—
232C22⋊C8—22no—
233C22⋊C8—22no—
234C22⋊C8—22no—
235C4×C8—42no—
236C4×C8—42no—
237#95—28no—
238#95—22no—
239#95—28no—
240#95—28no—
241#95—28no—
242#95—24no—
243#95—21no—
244#95—21no—
245#95—22no—
246#95—22no—
247#95C2×D412yes—
248#95C2×D414yes—
249#95C2×D412yes—
250C24C22×D411yescharacteristic
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—24no—
254C2×Q8—24no—
255C2×Q8—28no—
256C2×Q8—28no—
257C2×Q8—28no—
258C2×Q8—28no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—44no—
262C2×Q8—42no—
263C2×Q8—82no—
264C2×Q8#10318yes—
265C2×Q8#10318yes—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—42no—
280C22×C4—44no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4C2×C22⋊C414yes—
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C414yes—
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4#12012yes—
300C22×C4#12012yes—
301C22×C4#12312yes—
302C22×C4#12312yes—
303C22×C4C2×SD1614yes—
304C22×C4C2×SD1612yes—
305C22×C4C2×Q1614yes—
306C22×C4C2×Q1612yes—
307C22×C4AGL(1,ℤ/8)12yes—
308C22×C4#13712yes—
309C22×C4C23×C411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C2×#4111yescharacteristic
314C22×C4C2×#4111yescharacteristic
315C2×C8—44no—
316C2×C8—44no—
317C2×C8—42no—
318C2×C8—42no—
319C4⋊C4—28no—
320C4⋊C4—28no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—44no—
326C4⋊C4—48no—
327C4⋊C4—416no—
328C4⋊C4—416no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—44no—
334C4⋊C4—42no—
335C4⋊C4—44no—
336C4⋊C4—48no—
337C4⋊C4—82no—
338C4⋊C4—88no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C42—24no—
344C42—24no—
345C42—24no—
346C42—24no—
347C42—416no—
348C42—44no—
349C42—44no—
350C42—48no—
351C42C4×D412yes—
352C42C4×D412yes—
353C42C4×D412yes—
354C42C4×D412yes—
355C42#12112yes—
356C42#12112yes—
357C42#12112yes—
358C42#12112yes—
359C23—22no—
360C23—22no—
361C23C22×C22⋊C411yescharacteristic
362C23C2×#12011yescharacteristic
363C23C2×#12311yescharacteristic
364C23C22×SD1611yescharacteristic
365C23C22×Q1611yescharacteristic
366C23C2×AGL(1,ℤ/8)11yescharacteristic
367C23C2×#13711yescharacteristic
368Q8—216no—
369Q8—216no—
370Q8—82no—
371Q8—82no—
372Q8—84no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—28no—
385C2×C4—24no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—84no—
412C2×C4—82no—
413C2×C4—84no—
414C2×C4—82no—
415C2×C4#38612yes—
416C2×C4#38612yes—
417C2×C4#38612yes—
418C2×C4#38612yes—
419C2×C4C2×#10318yes—
420C2×C4C2×#10312yes—
421C2×C4C2×#10312yes—
422C2×C4#41712yes—
423C2×C4#41712yes—
424C2×C4#45014yes—
425C2×C4#45114yes—
426C2×C4#48314yes—
427C2×C4#48414yes—
428C2×C4C2×C4×D411yescharacteristic
429C2×C4C2×C4×D411yescharacteristic
430C2×C4C2×#12111yescharacteristic
431C2×C4C2×#12111yescharacteristic
432C8—82no—
433C8—82no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22C2×#38611yescharacteristic
443C22#288412yes—
444C22C22×#10311yescharacteristic
445C22C2×#41711yescharacteristic
446C22#290112yes—
447C22C2×#45011yescharacteristic
448C22C2×#45111yescharacteristic
449C22C2×#48311yescharacteristic
450C22C2×#48411yescharacteristic
451C4—28no—
452C4—22no—
453C4—22no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—44no—
459C4—44no—
460C4—84no—
461C4—82no—
462C4—84no—
463C4—82no—
464C4—82no—
465C4#187214yes—
466C4#187214yes—
467C2—21no—
468C2—21no—
469C2—21no—
470C2—21no—
471C2#1761912yes—
472C2#1762812yes—
473C2C2×#187211yescharacteristic
474C2#2098011yescharacteristic
475C2#2098611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #2582C4semidirect
19. #2568C4semidirect
20. #2567C4semidirect

Maximal subgroups

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

subgroupindex
#351792
C2×#23512
#218182
C2×#15182
C2×#15032
#91522
#91442
#80262
#80142