Finite Group Databasebeta

← #204563

Subgroups — #204563

Order 512.

1194 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34946C211yesmaximal, characteristic
2#21350C211yesmaximal, characteristic, complemented
3#20470C211yesmaximal, characteristic, complemented
4C2×#1497C211yesmaximal, characteristic, complemented
5C2×#1485C211yesmaximal, characteristic, complemented
6C2×#1490C211yesmaximal, characteristic
7C2×#1478C211yesmaximal, characteristic
8#9676C214yesmaximal, complemented
9#9675C214yesmaximal, complemented
10C2×#533C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16#2563C412yes—
17#2562—22no—
18#2561—22no—
19#2547C412yes—
20#2503C412yes—
21#2435C412yes—
22#2398—21no—
23#2398—21no—
24C4×#124C2214yes—
25#2280—22no—
26C4×#117C2211yescharacteristic
27#1940—22no—
28#1851—22no—
29#1767C2212yescomplemented
30C4×C4⋊C8C2212yescomplemented
31C4×C4⋊C8C2214yescomplemented
32#1750—22no—
33#1746—21no—
34#1726—21no—
35#1497—24no—
36#1490—24no—
37#1485—24no—
38#1478—24no—
39C2×Q8⋊C8C2212yescomplemented
40C2×Q8⋊C8C2212yescomplemented
41C2×Q8⋊C8C2214yescomplemented
42C2×D4⋊C8C2212yes—
43C2×D4⋊C8C2212yes—
44C2×D4⋊C8C2214yes—
45C23×D4C2×C411yescharacteristic
46C24×C4—21no—
47C22×M16—21no—
48#533—22no—
49#533—22no—
50#533—22no—
51#533—22no—
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#125—21no—
55C2×#124C2×C414yes—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8C2×C412yes—
58C2×C4×Q8—21no—
59C2×C4×Q8C2312yes—
60C2×C4×D4—21no—
61C2×C4×D4—21no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4C2×C412yes—
67C2×C4×D4C2312yes—
68C2×#117—21no—
69C2×#117D412yes—
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C22⋊C4—21no—
72C22×C42C2311yescharacteristic
73#432—22no—
74#432D414yes—
75#431—22no—
76#431—22no—
77#431—22no—
78#431D414yes—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8C2311yescharacteristic
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2312yes—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C4×M16—22no—
88C4×M16—22no—
89C2×#97—21no—
90C2×#97—21no—
91C2×#97—22no—
92C2×C4×C8—21no—
93C2×C4×C8—22no—
94C2×C4×C8C2312yes—
95C2×C4×C8C2312yes—
96#400—22no—
97#400—22no—
98#397—22no—
99#397—22no—
100#394—22no—
101#394—22no—
102#393—22no—
103#393—22no—
104#391C2×C414yes—
105#390C2×C414yes—
106#388—24no—
107#388—24no—
108#382C2×C414yes—
109#381—24no—
110#380—24no—
111#380—24no—
112#379—24no—
113C4×C4⋊C4C2×C412yes—
114C4×C4⋊C4D412yes—
115C4×C22⋊C4—24no—
116C4×C22⋊C4—24no—
117C4×C22⋊C4C2×C412yes—
118C4×C22⋊C4D412yes—
119#376D411yescharacteristic
120#376D411yescharacteristic
121C43C2312yes—
122#336—24no—
123#336—22no—
124#336—22no—
125Q8⋊C8—24no—
126Q8⋊C8—24no—
127Q8⋊C8—28no—
128D4⋊C8—24no—
129D4⋊C8—24no—
130D4⋊C8—28no—
131C25—21no—
132C22×Q8—21no—
133C22×Q8C22×C412yes—
134C22×D4—24no—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—42no—
139C22×D4C22×C412yes—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—44no—
144C23×C4—44no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4C22×C411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C2×M16—44no—
151C2×M16—42no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161#125—42no—
162#125—42no—
163#124—28no—
164#124—44no—
165C22⋊Q8—24no—
166C22⋊Q8—24no—
167C22⋊Q8—42no—
168C22⋊Q8—42no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—42no—
174C4×Q8—42no—
175C4×D4—24no—
176C4×D4—24no—
177C4×D4—22no—
178C4×D4—22no—
179C4×D4—22no—
180C4×D4—22no—
181C4×D4—42no—
182C4×D4—42no—
183C4×D4—42no—
184C4×D4—42no—
185#117—42no—
186#117—42no—
187#117—42no—
188#117—42no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4C22⋊C414yes—
198C2×C4⋊C4C22×C412yes—
199C2×C4⋊C4C22×C412yes—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4C22⋊C414yes—
210C2×C42—22no—
211C2×C42C22×C414yes—
212C2×C42C22×C411yescharacteristic
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42C2×D414yes—
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42#4112yes—
221C2×C42#4112yes—
222C2×C42C2411yescharacteristic
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—22no—
226C4⋊C8—48no—
227C4⋊C8—42no—
228C4⋊C8C2×D412yes—
229C4⋊C8C2×D412yes—
230C4⋊C8C2×D414yes—
231C22⋊C8—24no—
232C22⋊C8—24no—
233C22⋊C8—42no—
234#97—24no—
235#97—44no—
236#97—42no—
237C4×C8—24no—
238C4×C8—22no—
239C4×C8—22no—
240C4×C8—22no—
241C4×C8—22no—
242C4×C8—44no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95C22⋊C412yes—
248#95C22⋊C412yes—
249#95C22⋊C412yes—
250#95C22⋊C412yes—
251C24—22no—
252C24—44no—
253C24—44no—
254C24—42no—
255C24—82no—
256C24C2×C22⋊C411yescharacteristic
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—42no—
262C2×Q8—82no—
263C2×D4—28no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—48no—
270C2×D4—44no—
271C2×D4—44no—
272C2×D4—42no—
273C2×D4#13114yes—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—42no—
282C22×C4—48no—
283C22×C4—44no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—84no—
291C22×C4—84no—
292C22×C4—88no—
293C22×C4C2×C22⋊C414yes—
294C22×C4C2×C22⋊C414yes—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C4×D412yes—
299C22×C4C4×D412yes—
300C22×C4#12112yes—
301C22×C4#12112yes—
302C22×C4#12412yes—
303C22×C4#12712yes—
304C22×C4C23×C411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D411yescharacteristic
307M16—48no—
308C2×C8—24no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—24no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—44no—
319C2×C8—42no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—44no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—44no—
331C4⋊C4—42no—
332C22⋊C4—44no—
333C22⋊C4—48no—
334C22⋊C4—48no—
335C22⋊C4—44no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—44no—
339C22⋊C4—84no—
340C22⋊C4—84no—
341C42—24no—
342C42—24no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—44no—
348C42—42no—
349C42—42no—
350C42C4×D412yes—
351C42C4×D412yes—
352C42#12112yes—
353C42#12112yes—
354C42#12412yes—
355C42#12712yes—
356C42C22×D411yescharacteristic
357C42C22×D411yescharacteristic
358C42C2×#4111yescharacteristic
359C42C2×#4111yescharacteristic
360C23—22no—
361C23—22no—
362C23—22no—
363C23—24no—
364C23—48no—
365C23—42no—
366C23—44no—
367C23—44no—
368C23—42no—
369C23—84no—
370C23—84no—
371C23—816no—
372C23#39012yes—
373C23C22×C22⋊C411yescharacteristic
374Q8—42no—
375Q8—42no—
376Q8—44no—
377Q8—84no—
378D4—24no—
379D4—24no—
380D4—48no—
381D4—42no—
382D4—42no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—88no—
409C2×C4—84no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—84no—
413C2×C4#39012yes—
414C2×C4#39012yes—
415C2×C4#39012yes—
416C2×C4C2×C4×D411yescharacteristic
417C2×C4C2×C4×D411yescharacteristic
418C2×C4C2×#12111yescharacteristic
419C2×C4C2×#12111yescharacteristic
420C2×C4C2×#12411yescharacteristic
421C2×C4C2×#12711yescharacteristic
422C2×C4C2×#13112yes—
423C2×C4#56812yes—
424C2×C4#57512yes—
425C2×C4#57512yes—
426C8—44no—
427C8—42no—
428C8—42no—
429C8—44no—
430C8—82no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—44no—
435C22—48no—
436C22—44no—
437C22—816no—
438C22—82no—
439C22—82no—
440C22—84no—
441C22C2×#39011yescharacteristic
442C22#288012yes—
443C22#289612yes—
444C22#290112yes—
445C4—21no—
446C4—21no—
447C4—22no—
448C4—22no—
449C4—22no—
450C4—44no—
451C4—44no—
452C4—82no—
453C4—82no—
454C4—82no—
455C4#292911yescharacteristic
456C4#293011yescharacteristic
457C4#293312yes—
458C4#304011yescharacteristic
459C4#304011yescharacteristic
460C4#304012yes—
461C2—42no—
462C2—84no—
463C2#1751314yes—
464C2#2119111yescharacteristic
465C2#2125311yescharacteristic
466C2#2126111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21350C2semidirect
3. #20470C2semidirect
4. C2×#1497C2semidirect
5. C2×#1485C2semidirect
8. #9676C2semidirect
9. #9675C2semidirect
29. #1767C22semidirect
30. C4×C4⋊C8C22semidirect
31. C4×C4⋊C8C22semidirect
39. C2×Q8⋊C8C22semidirect
40. C2×Q8⋊C8C22semidirect
41. C2×Q8⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#349462
#213502
#204702
C2×#14972
C2×#14852
C2×#14902
C2×#14782
#96762
#96752