Finite Group Databasebeta

← #203118

Subgroups — #203118

Order 512.

740 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21382C211yesmaximal, characteristic
2#21370C211yesmaximal, characteristic, complemented
3#17700C211yesmaximal, characteristic
4#17473C211yesmaximal, characteristic
5C2×C8×M16C211yesmaximal, characteristic, complemented
6#9590C214yesmaximal, complemented
7#9589C214yesmaximal, complemented
8#8092C211yesmaximal, characteristic, complemented
9#7890C211yesmaximal, characteristic, complemented
10#3444C2211yescharacteristic
11#2920C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C4×#117C2211yescharacteristic
17#1986C2212yes—
18#1980C2212yes—
19#1950C2214yes—
20#1871—22no—
21#1855—22no—
22C4×#108—22no—
23C4×#104—22no—
24#1544C412yescomplemented
25#1543C412yescomplemented
26#1542C411yescharacteristic, complemented
27#1542C411yescharacteristic, complemented
28#1542C2212yes—
29#1540C411yescharacteristic, complemented
30#1540C411yescharacteristic, complemented
31#1540C2212yes—
32#1494—22no—
33#1493C2212yes—
34#1492C2212yes—
35#1476—21no—
36#1476—21no—
37C8×M16—22no—
38C8×M16C2214yes—
39C2×#322—21no—
40C2×C82—21no—
41#1303C2214yes—
42#1279C2214yes—
43C22×M16D412yes—
44C23×C8—21no—
45#533C2×C411yescharacteristic
46#533C2×C411yescharacteristic
47#533C2311yescharacteristic
48#532C2×C411yescharacteristic
49#532C2×C411yescharacteristic
50#532C2311yescharacteristic
51#529—21no—
52#529—21no—
53#529—21no—
54#529—21no—
55#529—21no—
56#529—21no—
57#525—21no—
58#525—21no—
59#525—21no—
60C4×#41—21no—
61C4×#41—21no—
62C4×#41—21no—
63C2×#117—21no—
64C22×C42C2311yescharacteristic
65#432—21no—
66#432—21no—
67#431—21no—
68#431—21no—
69#431—21no—
70#431—21no—
71#431—21no—
72#431—21no—
73#431C2×C411yescharacteristic
74#431C2×C411yescharacteristic
75#431C2×C412yes—
76#431C2×C412yes—
77#431C2312yes—
78C2×#108—24no—
79#423C2312yes—
80C2×C4⋊C8C2×C411yescharacteristic
81C2×C4⋊C8C2×C411yescharacteristic
82C2×C4⋊C8C2312yes—
83C2×#104—24no—
84#407—21no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C4×M16—21no—
88C4×M16—21no—
89C4×M16—21no—
90C4×M16D412yes—
91C4×M16C2312yes—
92C2×#97—21no—
93C2×C4×C8—21no—
94C2×C4×C8—21no—
95C2×C4×C8—21no—
96C2×C4×C8—21no—
97C2×C4×C8—21no—
98C2×C4×C8—21no—
99C2×C4×C8D411yescharacteristic
100C2×C4×C8D411yescharacteristic
101C2×C4×C8D412yes—
102C2×C4×C8C2312yes—
103C4×C4⋊C4—21no—
104C4×C4⋊C4C2312yes—
105C4×C22⋊C4—21no—
106C4×C22⋊C4—21no—
107#376—21no—
108C43—21no—
109#341—24no—
110#339—24no—
111#333C2×C412yes—
112#333C2×C412yes—
113#332C2×C412yes—
114#332C2×C412yes—
115#332C2×C412yes—
116#332C2×C412yes—
117#331C2×C412yes—
118#331C2×C412yes—
119Q8⋊C8—22no—
120Q8⋊C8—22no—
121D4⋊C8—22no—
122D4⋊C8—22no—
123#322—24no—
124C82—24no—
125C2×#41—21no—
126C2×#41—21no—
127C2×#41—21no—
128C23×C4C2×D411yescharacteristic
129C23×C4#4111yescharacteristic
130C23×C4#4111yescharacteristic
131C2×M16—22no—
132C2×M16—28no—
133C2×M16—22no—
134C2×M16C2×D414yes—
135C22×C8—22no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—22no—
140C22×C8—21no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—44no—
144C4⋊Q8C22×C411yescharacteristic
145C4⋊Q8C22×C411yescharacteristic
146C4⋊Q8C22×C411yescharacteristic
147#127C22×C411yescharacteristic
148#126—41no—
149#126—41no—
150#125—21no—
151#125—21no—
152#125—21no—
153#124—21no—
154#124—21no—
155#124—21no—
156#123—41no—
157#123—41no—
158#123—41no—
159C22⋊Q8—21no—
160C22⋊Q8—21no—
161C22⋊Q8—21no—
162C22⋊Q8—21no—
163C22⋊Q8—21no—
164C22⋊Q8—21no—
165#121—21no—
166#121—21no—
167#121—21no—
168#121—21no—
169#121—21no—
170#121—21no—
171#120—41no—
172C4×Q8—21no—
173C4×Q8—21no—
174C4×Q8—21no—
175C4×Q8—21no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—41no—
185C4×D4—41no—
186C4×D4—41no—
187C4×D4—41no—
188C4×D4—41no—
189C4×D4—41no—
190#117—24no—
191#117—21no—
192#117—21no—
193#117—21no—
194#117—42no—
195#117—41no—
196#117—41no—
197#117—41no—
198C2×C4⋊C4—22no—
199C2×C22⋊C4—41no—
200C2×C42—22no—
201C2×C42—41no—
202C2×C42—41no—
203C2×C42—41no—
204C2×C42C22×C411yescharacteristic
205C2×C42C22×C411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42#4111yescharacteristic
210C2×C42#4111yescharacteristic
211C2×C42#4111yescharacteristic
212C2×C42#4111yescharacteristic
213C2×C42#4111yescharacteristic
214C2×C42#4111yescharacteristic
215C2×C42C2411yescharacteristic
216#108—48no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8C22×C412yes—
223C4⋊C8C22×C412yes—
224C4⋊C8C22×C412yes—
225C4⋊C8C22×C412yes—
226#104—48no—
227C22⋊C8—22no—
228C22⋊C8—21no—
229C22⋊C8—21no—
230C22⋊C8—22no—
231C22⋊C8—22no—
232C22⋊C8—41no—
233C22⋊C8—41no—
234#97—22no—
235#97—22no—
236#97—41no—
237C4×C8—22no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—22no—
241C4×C8—22no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—22no—
245C4×C8—22no—
246C4×C8—22no—
247C4×C8—22no—
248C4×C8—41no—
249C4×C8C2×D412yes—
250C4×C8C2×D412yes—
251#95—22no—
252#95—41no—
253C24#12411yescharacteristic
254#41—44no—
255#41—42no—
256#41—42no—
257#41—82no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4#11711yescharacteristic
276C22×C4#11711yescharacteristic
277C22×C4C4×D411yescharacteristic
278C22×C4C4×D411yescharacteristic
279C22×C4C4×D411yescharacteristic
280C22×C4C4×D411yescharacteristic
281C22×C4#12111yescharacteristic
282C22×C4#12111yescharacteristic
283C22×C4#12311yescharacteristic
284C22×C4#12311yescharacteristic
285C22×C4#12411yescharacteristic
286C22×C4#12411yescharacteristic
287C22×C4#12411yescharacteristic
288C22×C4#12611yescharacteristic
289C22×C4#12611yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4C2×#4111yescharacteristic
292C22×C4C2×#4111yescharacteristic
293M16—28no—
294M16—24no—
295M16—44no—
296M16#13518yes—
297C2×C8—24no—
298C2×C8—28no—
299C2×C8—24no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—44no—
307C2×C8—44no—
308C2×C8—42no—
309C2×C8—44no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C4⋊C4—24no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—42no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C22⋊C4—44no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C42—44no—
338C42—41no—
339C42—41no—
340C42—41no—
341C42—44no—
342C42—82no—
343C42#11711yescharacteristic
344C42#11711yescharacteristic
345C42C4×D411yescharacteristic
346C42C4×D411yescharacteristic
347C42C4×D411yescharacteristic
348C42C4×D411yescharacteristic
349C42#12111yescharacteristic
350C42#12111yescharacteristic
351C42#12311yescharacteristic
352C42#12311yescharacteristic
353C42#12611yescharacteristic
354C42#12611yescharacteristic
355C42C23×C411yescharacteristic
356C42C22×D411yescharacteristic
357C42C2×#4111yescharacteristic
358C42C2×#4111yescharacteristic
359C23—22no—
360C23—22no—
361C23—41no—
362C23—41no—
363C23#38811yescharacteristic
364C23#38811yescharacteristic
365C23C2×#12411yescharacteristic
366Q8—42no—
367Q8—41no—
368Q8—41no—
369Q8—81no—
370D4—42no—
371D4—41no—
372D4—41no—
373D4—81no—
374D4—82no—
375D4—82no—
376D4—82no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—44no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4#38811yescharacteristic
405C2×C4#38811yescharacteristic
406C2×C4#38811yescharacteristic
407C2×C4#38811yescharacteristic
408C2×C4#38811yescharacteristic
409C2×C4#38811yescharacteristic
410C2×C4#44312yes—
411C2×C4#44412yes—
412C2×C4C2×#11711yescharacteristic
413C2×C4C2×C4×D411yescharacteristic
414C2×C4C2×C4×D411yescharacteristic
415C2×C4C2×#12111yescharacteristic
416C2×C4C2×#12311yescharacteristic
417C2×C4C2×#12611yescharacteristic
418C2×C4C2×#13514yes—
419C8—24no—
420C8—28no—
421C8—42no—
422C8—42no—
423C8—42no—
424C8—42no—
425C8—42no—
426C8—82no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—44no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22#192912yes—
439C22C2×#38811yescharacteristic
440C22#313112yes—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—84no—
449C4—81no—
450C4—81no—
451C4—81no—
452C4#192912yes—
453C4#193212yes—
454C4C4×#13512yes—
455C4C2×#44311yescharacteristic
456C4C2×#44411yescharacteristic
457C4#305012yes—
458C2—22no—
459C2—22no—
460C2—41no—
461C2—161no—
462C2C2×#192911yescharacteristic
463C2C2×#193211yescharacteristic
464C2#2108311yescharacteristic
subgroupG/H
centerC42#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#388
prime radical#3444C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C42 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21370C2semidirect
5. C2×C8×M16C2semidirect
6. #9590C2semidirect
7. #9589C2semidirect
8. #8092C2semidirect
9. #7890C2semidirect
24. #1544C4semidirect
25. #1543C4semidirect
26. #1542C4semidirect
27. #1542C4semidirect
29. #1540C4semidirect
30. #1540C4semidirect

Maximal subgroups

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

subgroupindex
#213822
#213702
#177002
#174732
C2×C8×M162
#95902
#95892
#80922
#78902