Finite Group Databasebeta

← #201822

Subgroups — #201822

Order 512.

1300 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60124C211yesmaximal, characteristic
2C2×#2484C211yesmaximal, characteristic
3#20694C211yesmaximal, characteristic
4C2×#1463C212yesmaximal
5#9514C218yesmaximal, complemented
6#8114C212yesmaximal
7C22×#117C2211yescharacteristic
8C23×C4⋊C4C2211yescharacteristic
9#2924—24no—
10#2924—24no—
11#2921C2214yes—
12#2920—22no—
13#2905—21no—
14C2×#407C2212yes—
15C2×#407C2212yes—
16#2873—21no—
17#2484—22no—
18#2484—21no—
19#2484—21no—
20#2484C412yes—
21#2484C412yes—
22#2484C2214yes—
23C2×#400—21no—
24C2×#398—21no—
25C2×#387—22no—
26C2×#385C412yes—
27C2×#381C412yes—
28C2×#381C2211yescharacteristic
29#1822C2214yes—
30#1821—21no—
31#1821—21no—
32#1742—22no—
33#1463—24no—
34#1463C2218yes—
35C2×#324—24no—
36#1306C2218yes—
37C24×C4C2×C411yescharacteristic
38C24×C4C2×C411yescharacteristic
39C24×C4C2311yescharacteristic
40C22×M16—22no—
41C22×M16—22no—
42C2×#117—22no—
43C2×#117—22no—
44C2×#117—22no—
45C2×#117—22no—
46C2×#117C2×C411yescharacteristic
47C2×#117C2×C411yescharacteristic
48C2×#117D412yes—
49C2×#117D411yescharacteristic
50C2×#117D411yescharacteristic
51C2×#117C2312yes—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4C2×C412yes—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C22⋊C4C2×C412yes—
66C22×C22⋊C4C2×C412yes—
67C22×C42D412yes—
68#433—216no—
69#432—24no—
70#432—24no—
71#431—24no—
72#431—24no—
73#424—28no—
74#423—24no—
75#407—22no—
76#407—22no—
77#407—22no—
78#407—22no—
79#407—22no—
80#407D414yes—
81#407C2314yes—
82#407C2314yes—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8D414yes—
87#405—28no—
88C4×M16—24no—
89#400—22no—
90#400—22no—
91#400—24no—
92#398—22no—
93#398—22no—
94#398—24no—
95#387—24no—
96#387—24no—
97#387—24no—
98#387—24no—
99#385—22no—
100#385—22no—
101#385C2×C414yes—
102#385C2×C414yes—
103#381—21no—
104#381—21no—
105#381—22no—
106#381—22no—
107#381C2×C412yes—
108#381C2×C414yes—
109#381C2×C414yes—
110#381C2312yes—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95—21no—
116#343—24no—
117#343—24no—
118#324—216no—
119C25C22×C411yescharacteristic
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—44no—
130C23×C4—44no—
131C23×C4—44no—
132C23×C4C22⋊C412yes—
133C23×C4C22⋊C412yes—
134C23×C4C22×C412yes—
135C23×C4C22×C412yes—
136C23×C4C22×C411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4#4112yes—
144C23×C4#4112yes—
145C23×C4C2411yescharacteristic
146C2×M16—24no—
147C2×M16—24no—
148C2×M16—24no—
149C2×M16—48no—
150C2×M16—42no—
151C2×M16—48no—
152C22×C8—24no—
153C22×C8—42no—
154#117—22no—
155#117—22no—
156#117—22no—
157#117—22no—
158#117—22no—
159#117—22no—
160#117—24no—
161#117—24no—
162#117—22no—
163#117—22no—
164#117—24no—
165#117—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—48no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×D412yes—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4C22⋊C412yes—
202C2×C22⋊C4C22⋊C412yes—
203C2×C22⋊C4C22×C412yes—
204C2×C22⋊C4C22×C412yes—
205C2×C22⋊C4C22×C414yes—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42C22⋊C412yes—
211C2×C42C22⋊C412yes—
212C2×C42C2×D412yes—
213C2×C42C2×D412yes—
214C4⋊C8—28no—
215C4⋊C8—28no—
216C22⋊C8—24no—
217C22⋊C8—24no—
218C22⋊C8—24no—
219C22⋊C8—24no—
220C22⋊C8—24no—
221C22⋊C8—42no—
222C22⋊C8—42no—
223C22⋊C8C2×D418yes—
224#97—28no—
225C4×C8—28no—
226#95—22no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—22no—
232#95—24no—
233#95—24no—
234#95—22no—
235#95—22no—
236C24—24no—
237C24—42no—
238C24—42no—
239C24C2×C22⋊C411yescharacteristic
240C24C2×C22⋊C411yescharacteristic
241C24C4×D412yes—
242C24C4×D412yes—
243C24C23×C411yescharacteristic
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—28no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—44no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—42no—
266C22×C4—44no—
267C22×C4—48no—
268C22×C4—42no—
269C22×C4—44no—
270C22×C4—42no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—84no—
285C22×C4—84no—
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C412yes—
292C22×C4C4×D412yes—
293C22×C4C4×D412yes—
294C22×C4#12012yes—
295C22×C4#12011yescharacteristic
296C22×C4#12011yescharacteristic
297C22×C4#12112yes—
298C22×C4#12112yes—
299C22×C4#12112yes—
300C22×C4#12112yes—
301C22×C4#12312yes—
302C22×C4#12311yescharacteristic
303C22×C4#12311yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D412yes—
307C22×C4C2×#4112yes—
308M16—48no—
309M16—48no—
310M16—48no—
311M16—84no—
312C2×C8—28no—
313C2×C8—48no—
314C2×C8—44no—
315C2×C8—84no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—48no—
323C4⋊C4—48no—
324C4⋊C4—42no—
325C4⋊C4—44no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—44no—
330C22⋊C4—24no—
331C22⋊C4—24no—
332C22⋊C4—24no—
333C22⋊C4—24no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—24no—
349C23—24no—
350C23—24no—
351C23—24no—
352C23—41no—
353C23—42no—
354C23—41no—
355C23—41no—
356C23—42no—
357C23—41no—
358C23—416no—
359C23#38611yescharacteristic
360C23#38612yes—
361C23#38611yescharacteristic
362C23#41014yes—
363C23C22×C22⋊C411yescharacteristic
364C23C2×C4×D412yes—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—28no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—48no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—88no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—84no—
412C2×C4—84no—
413C2×C4#38611yescharacteristic
414C2×C4#38611yescharacteristic
415C2×C4#38612yes—
416C2×C4C2×#12011yescharacteristic
417C2×C4C2×#12112yes—
418C2×C4C2×#12311yescharacteristic
419C8—48no—
420C8—84no—
421C22—28no—
422C22—21no—
423C22—21no—
424C22—22no—
425C22—21no—
426C22—21no—
427C22—22no—
428C22—21no—
429C22—21no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—44no—
434C22—48no—
435C22—48no—
436C22—44no—
437C22#189112yes—
438C22#189914yes—
439C22#191512yes—
440C22C2×#38611yescharacteristic
441C22C2×#41012yes—
442C4—21no—
443C4—22no—
444C4—21no—
445C4—22no—
446C4—21no—
447C4—21no—
448C4—42no—
449C4—42no—
450C4—44no—
451C4—42no—
452C4—42no—
453C4—84no—
454C4—82no—
455C2—22no—
456C2—22no—
457C2—22no—
458C2—24no—
459C2—42no—
460C2C2×#189111yescharacteristic
461C2C2×#189911yescharacteristic
462C2C2×#191511yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #9514C2semidirect

Maximal subgroups

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

subgroupindex
#601242
C2×#24842
#206942
C2×#14632
#95142
#81142