Finite Group Databasebeta

← #201163

Subgroups — #201163

Order 512.

1249 conjugacy classes of subgroups.

The subgroup lattice has 467 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2983C211yesmaximal, characteristic, complemented
2#35178C211yesmaximal, characteristic
3#17675C212yesmaximal
4C2×#1467C211yesmaximal, characteristic, complemented
5#9448C218yesmaximal, complemented
6#7952C212yesmaximal, complemented
7C2×#536C412yes—
8C2×#532C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2983—24no—
11#2983—22no—
12#2983C2214yes—
13C2×#446—24no—
14C2×#432—21no—
15C2×#432C2212yes—
16C2×#432C2212yes—
17C2×#431—21no—
18#2854—21no—
19#2854—21no—
20#2618—21no—
21#2618—21no—
22#2548C412yes—
23#2546C412yes—
24#2545C2214yes—
25#2538—22no—
26#2536—22no—
27#2311C412yes—
28#2285C2211yescharacteristic
29#1533C2218yes—
30#1530—24no—
31#1530—24no—
32#1467—22no—
33#1467C2214yes—
34C2×#324—21no—
35C2×#324—21no—
36#1285C2218yes—
37C22×#41C2×C412yes—
38C23×Q8C2×C411yescharacteristic
39#536—48no—
40#532—22no—
41#532—22no—
42#532—22no—
43#532—22no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C414yes—
46C2×#127C2×C411yescharacteristic
47C2×#124C2×C414yes—
48C2×#124C2312yes—
49C2×#123—22no—
50C2×#123—22no—
51C2×C22⋊Q8—24no—
52C2×#121—21no—
53C2×#121—21no—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8C2×C412yes—
59C2×C4×Q8D412yes—
60C2×C4×Q8D412yes—
61C2×C4×Q8C2312yes—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×#117C2×C412yes—
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42C2311yescharacteristic
71#446—216no—
72#446—216no—
73#432—24no—
74#432—24no—
75#432—22no—
76#432—22no—
77#432—22no—
78#432—22no—
79#432—22no—
80#432D414yes—
81#432C2314yes—
82#432C2314yes—
83#431—24no—
84#431—24no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×#97—22no—
92C2×#97—22no—
93C2×#97D414yes—
94C2×C4×C8—22no—
95#398—22no—
96#398—22no—
97#396—24no—
98#396—24no—
99#394—22no—
100#394—22no—
101#391C2×C414yes—
102#390—22no—
103#390—22no—
104#390C2×C414yes—
105#389—22no—
106#389—22no—
107#388—24no—
108#388—24no—
109#386—22no—
110#386—22no—
111#384—21no—
112#382—21no—
113#380—21no—
114#380—21no—
115#380—24no—
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4C2312yes—
118#329—216no—
119#324—24no—
120#324—24no—
121C2×#41—24no—
122C2×#41—24no—
123C2×#41—44no—
124C2×#41—44no—
125C22×Q8—24no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—42no—
129C22×Q8C22⋊C412yes—
130C22×Q8C22×C412yes—
131C22×Q8C22×C412yes—
132C22×D4—22no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4C22×C412yes—
136C23×C4—22no—
137C23×C4C22⋊C411yescharacteristic
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C22×C8—24no—
145C22×C8—42no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—48no—
149#127—22no—
150#127—22no—
151#124—24no—
152#124—24no—
153#124—48no—
154#123—48no—
155#123—48no—
156C22⋊Q8—416no—
157#121—41no—
158#121—41no—
159#121—42no—
160#121—42no—
161#121—41no—
162#121—41no—
163C4×Q8—24no—
164C4×Q8—24no—
165C4×Q8—24no—
166C4×Q8—24no—
167C4×Q8—24no—
168C4×Q8—24no—
169C4×Q8—24no—
170C4×Q8—24no—
171C4×D4—42no—
172C4×D4—42no—
173C4×D4—42no—
174C4×D4—42no—
175#117—44no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C2×D412yes—
194C2×C4⋊C4C2×D412yes—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42C22⋊C412yes—
211C2×C42C22⋊C412yes—
212C2×C42C22×C414yes—
213C2×C42C22×C411yescharacteristic
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42#4112yes—
221C2×C42#4112yes—
222C2×C42C2411yescharacteristic
223C4⋊C8—28no—
224C4⋊C8—28no—
225C4⋊C8—24no—
226C4⋊C8—24no—
227C4⋊C8—24no—
228C4⋊C8—42no—
229C22⋊C8—24no—
230C22⋊C8—24no—
231C22⋊C8—24no—
232C22⋊C8—42no—
233#97—28no—
234#97—24no—
235#97—24no—
236#97—42no—
237#97C2×D418yes—
238C4×C8—28no—
239#95—22no—
240#95—24no—
241#95—24no—
242#95—22no—
243C24—22no—
244C24C2×C22⋊C411yescharacteristic
245#41—88no—
246#41—88no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—48no—
255C2×Q8—48no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×D4—24no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—44no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—44no—
265C2×D4—44no—
266C2×D4—41no—
267C2×D4—42no—
268C2×D4—41no—
269C2×D4—82no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—81no—
290C22×C4—81no—
291C22×C4—81no—
292C22×C4#10014yes—
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C4×D412yes—
301C22×C4C4×D412yes—
302C22×C4#12011yescharacteristic
303C22×C4#12011yescharacteristic
304C22×C4#12112yes—
305C22×C4#12112yes—
306C22×C4#12311yescharacteristic
307C22×C4#12311yescharacteristic
308C22×C4C23×C411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C2×C8—28no—
312C2×C8—48no—
313C2×C8—44no—
314C2×C8—84no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—48no—
324C4⋊C4—41no—
325C4⋊C4—42no—
326C4⋊C4—41no—
327C4⋊C4—48no—
328C4⋊C4—82no—
329C22⋊C4—48no—
330C22⋊C4—44no—
331C22⋊C4—42no—
332C22⋊C4—44no—
333C22⋊C4—42no—
334C22⋊C4—48no—
335C22⋊C4—84no—
336C22⋊C4—84no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—44no—
346C42C4×D412yes—
347C42C4×D412yes—
348C42#12012yes—
349C42#12112yes—
350C42#12112yes—
351C42#12312yes—
352C42C22×D412yes—
353C42C2×#4112yes—
354C23—41no—
355C23—41no—
356C23—44no—
357C23—44no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23#38611yescharacteristic
362C23#38611yescharacteristic
363C23C2×#10012yes—
364C23#41312yes—
365C23C22×C22⋊C411yescharacteristic
366Q8—48no—
367Q8—48no—
368Q8—84no—
369D4—84no—
370D4—82no—
371D4—84no—
372D4—84no—
373D4—84no—
374D4—82no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—88no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4#38611yescharacteristic
416C2×C4#38612yes—
417C2×C4#38612yes—
418C2×C4#38611yescharacteristic
419C2×C4C2×#10014yes—
420C2×C4C2×C4×D412yes—
421C2×C4C2×#12011yescharacteristic
422C2×C4C2×#12112yes—
423C2×C4C2×#12311yescharacteristic
424C8—48no—
425C8—84no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—42no—
433C22—42no—
434C22—84no—
435C22—84no—
436C22—82no—
437C22—82no—
438C22—82no—
439C22#190012yes—
440C22#190012yes—
441C22C2×#38611yescharacteristic
442C22C22×#10011yescharacteristic
443C22C2×#41311yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—22no—
448C4—22no—
449C4—22no—
450C4—22no—
451C4—24no—
452C4—42no—
453C4—42no—
454C4—42no—
455C4—84no—
456C4—82no—
457C4#191212yes—
458C2—21no—
459C2—21no—
460C2—22no—
461C2—22no—
462C2—162no—
463C2#2094711yescharacteristic
464C2C2×#190011yescharacteristic
465C2C2×#191211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2983C2semidirect
4. C2×#1467C2semidirect
5. #9448C2semidirect
6. #7952C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29832
#351782
#176752
C2×#14672
#94482
#79522