Finite Group Databasebeta

← #204152

Subgroups — #204152

Order 512.

1123 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34972C211yesmaximal, characteristic
2#21976C211yesmaximal, characteristic, complemented
3#20366C211yesmaximal, characteristic, complemented
4C2×#1494C211yesmaximal, characteristic
5C2×#1483C211yesmaximal, characteristic
6C2×#1491C211yesmaximal, characteristic, complemented
7C22×Q8⋊C8C211yesmaximal, characteristic, complemented
8#9645C212yesmaximal, complemented
9#9643C212yesmaximal, complemented
10#9641C212yesmaximal, complemented
11#9640C212yesmaximal, complemented
12C2×#529C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18#2604—22no—
19#2565C412yes—
20#2545C412yes—
21#2503C411yescharacteristic
22#2501C411yescharacteristic
23#2436C411yescharacteristic
24#2434C411yescharacteristic
25#2413—22no—
26#2396—22no—
27C4×C4⋊Q8C2212yes—
28C4×#124C2212yes—
29C4×#123—22no—
30C4×#117C2211yescharacteristic
31#1940—22no—
32#1850—22no—
33#1842C2212yes—
34#1831C2212yes—
35#1746—22no—
36#1744C2212yes—
37#1744C2212yes—
38#1729—22no—
39#1494—24no—
40#1491—24no—
41#1483—24no—
42C2×Q8⋊C8—24no—
43C2×Q8⋊C8C2212yes—
44C2×Q8⋊C8C2212yes—
45C2×Q8⋊C8C2212yes—
46C2×Q8⋊C8C2212yes—
47C2×Q8⋊C8C2212yes—
48C2×Q8⋊C8C2211yescharacteristic
49C2×Q8⋊C8C2211yescharacteristic
50C2×D4⋊C8C2212yes—
51C2×D4⋊C8C2211yescharacteristic
52C2×D4⋊C8C2211yescharacteristic
53C23×Q8C2×C411yescharacteristic
54C22×M16—21no—
55C23×C8—21no—
56C23×C8—21no—
57C2×C4⋊Q8C2×C412yes—
58#529—22no—
59#529—22no—
60#529—22no—
61#529—22no—
62C2×#126—21no—
63C2×#125C2×C411yescharacteristic
64C2×#124C2×C411yescharacteristic
65C2×#124C2×C412yes—
66C2×#123—21no—
67C2×#123—22no—
68C2×C22⋊Q8C2×C411yescharacteristic
69C2×#121C2×C411yescharacteristic
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—22no—
73C2×C4×Q8—22no—
74C2×C4×Q8—22no—
75C2×C4×Q8—22no—
76C2×C4×Q8C2×C412yes—
77C2×C4×Q8C2311yescharacteristic
78C2×C4×Q8C2312yes—
79C2×C4×D4—21no—
80C2×C4×D4C2311yescharacteristic
81C2×#117—21no—
82C2×#117D412yes—
83C22×C4⋊C4—21no—
84C22×C4⋊C4C2×C411yescharacteristic
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87#432—22no—
88#432—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—24no—
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2312yes—
95C2×C4⋊C8C2311yescharacteristic
96C2×C22⋊C8—21no—
97C4×M16—22no—
98C4×M16D414yes—
99C2×#97—21no—
100C2×#97—21no—
101C2×C4×C8—21no—
102C2×C4×C8—22no—
103C2×C4×C8D414yes—
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2312yes—
107#400—22no—
108#399—22no—
109#398—24no—
110#396—24no—
111#395—22no—
112#393—22no—
113#391C2×C412yes—
114#391C2×C412yes—
115#391C2×C412yes—
116#390C2×C412yes—
117#389—24no—
118#388—24no—
119#388—24no—
120#387—24no—
121#386—22no—
122#385—22no—
123#384C2×C412yes—
124#382C2×C412yes—
125#381—22no—
126C4×C4⋊C4—24no—
127C4×C4⋊C4C2×C412yes—
128C4×C4⋊C4D412yes—
129C4×C22⋊C4—24no—
130C4×C22⋊C4C2×C411yescharacteristic
131C4×C22⋊C4C2×C411yescharacteristic
132C4×C22⋊C4D412yes—
133#376D412yes—
134C43C2312yes—
135#336—24no—
136#336—24no—
137#336—24no—
138Q8⋊C8—24no—
139Q8⋊C8—24no—
140Q8⋊C8—24no—
141Q8⋊C8—24no—
142Q8⋊C8—24no—
143Q8⋊C8—22no—
144Q8⋊C8—22no—
145D4⋊C8—24no—
146D4⋊C8—22no—
147D4⋊C8—22no—
148C22×Q8—21no—
149C22×Q8—24no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—42no—
153C22×Q8C22×C411yescharacteristic
154C22×Q8C22×C412yes—
155C22×D4—21no—
156C22×D4C22×C411yescharacteristic
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4C22×C411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C2×M16—24no—
164C2×M16—44no—
165C22×C8—24no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—44no—
173C22×C8—42no—
174C22×C8—82no—
175C4⋊Q8—24no—
176C4⋊Q8—42no—
177#126—42no—
178#126—42no—
179#125—22no—
180#125—22no—
181#124—24no—
182#124—22no—
183#124—22no—
184#124—42no—
185#123—48no—
186#123—42no—
187#123—42no—
188C22⋊Q8—22no—
189C22⋊Q8—22no—
190#121—22no—
191#121—22no—
192C4×Q8—24no—
193C4×Q8—24no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8—21no—
197C4×Q8—21no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×Q8—42no—
203C4×Q8—42no—
204C4×Q8—42no—
205C4×Q8—42no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—21no—
210C4×D4—42no—
211C4×D4—42no—
212#117—44no—
213#117—42no—
214#117—42no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4C22⋊C414yes—
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4C22⋊C412yes—
240C2×C22⋊C4C22⋊C412yes—
241C2×C42—22no—
242C2×C42—21no—
243C2×C42—22no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42C22×C412yes—
247C2×C42C22×C412yes—
248C2×C42C22×C411yescharacteristic
249C2×C42C2×D414yes—
250C2×C42C2×D414yes—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42#4112yes—
256C2×C42#4112yes—
257C2×C42C2411yescharacteristic
258C4⋊C8—22no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—22no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—42no—
265C4⋊C8—84no—
266C22⋊C8—42no—
267#97—24no—
268#97—42no—
269C4×C8—24no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—22no—
273C4×C8C2×D412yes—
274C4×C8C2×D412yes—
275C4×C8C2×D414yes—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95C22⋊C412yes—
281#95C22⋊C412yes—
282#95C22⋊C414yes—
283C24—21no—
284C24C2×C22⋊C411yescharacteristic
285C2×Q8—22no—
286C2×Q8—28no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—44no—
296C2×Q8—48no—
297C2×Q8—44no—
298C2×Q8—42no—
299C2×Q8#13114yes—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—42no—
305C2×D4—44no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—44no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—44no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4C2×C22⋊C414yes—
325C22×C4C2×C22⋊C414yes—
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C4×D412yes—
330C22×C4C4×D412yes—
331C22×C4#12112yes—
332C22×C4#12112yes—
333C22×C4#12412yes—
334C22×C4#12712yes—
335C22×C4C23×C411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338M16—48no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—24no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—48no—
345C2×C8—44no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—44no—
351C2×C8—42no—
352C2×C8—84no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—44no—
360C4⋊C4—48no—
361C4⋊C4—48no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—44no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C22⋊C4—24no—
372C22⋊C4—24no—
373C22⋊C4—48no—
374C22⋊C4—48no—
375C22⋊C4—42no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—24no—
386C42—24no—
387C42—44no—
388C42—42no—
389C42—41no—
390C42—42no—
391C42—41no—
392C42—44no—
393C42C4×D412yes—
394C42C4×D412yes—
395C42#12112yes—
396C42#12112yes—
397C42#12412yes—
398C42#12712yes—
399C42C22×D411yescharacteristic
400C42C22×D411yescharacteristic
401C42C2×#4111yescharacteristic
402C42C2×#4111yescharacteristic
403C23—22no—
404C23—22no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—41no—
409C23#39012yes—
410C23#42114yes—
411C23C22×C22⋊C411yescharacteristic
412Q8—24no—
413Q8—24no—
414Q8—48no—
415Q8—42no—
416Q8—42no—
417Q8—41no—
418Q8—41no—
419Q8—42no—
420D4—41no—
421D4—41no—
422D4—42no—
423D4—84no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—48no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—84no—
455C2×C4—82no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4#39012yes—
459C2×C4#39012yes—
460C2×C4#39012yes—
461C2×C4C2×C4×D411yescharacteristic
462C2×C4C2×C4×D411yescharacteristic
463C2×C4C2×#12111yescharacteristic
464C2×C4C2×#12111yescharacteristic
465C2×C4C2×#12411yescharacteristic
466C2×C4C2×#12711yescharacteristic
467C2×C4C2×#13112yes—
468C2×C4#56812yes—
469C2×C4#57712yes—
470C2×C4#57812yes—
471C8—42no—
472C8—42no—
473C8—44no—
474C8—81no—
475C8—82no—
476C8—81no—
477C22—28no—
478C22—22no—
479C22—22no—
480C22—84no—
481C22—81no—
482C22—81no—
483C22—81no—
484C22#196314yes—
485C22C2×#39011yescharacteristic
486C22#288012yes—
487C22#289812yes—
488C22C2×#42112yes—
489C4—22no—
490C4—22no—
491C4—21no—
492C4—21no—
493C4—22no—
494C4—44no—
495C4—44no—
496C4—44no—
497C4—81no—
498C4—81no—
499C4—82no—
500C4—81no—
501C4#292911yescharacteristic
502C4#293011yescharacteristic
503C4#293312yes—
504C4#313712yes—
505C4#315411yescharacteristic
506C4#315611yescharacteristic
507C2—24no—
508C2—161no—
509C2#1752414yes—
510C2#2119111yescharacteristic
511C2#2120511yescharacteristic
512C2C2×#196311yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#121C2×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. #21976C2semidirect
3. #20366C2semidirect
6. C2×#1491C2semidirect
7. C22×Q8⋊C8C2semidirect
8. #9645C2semidirect
9. #9643C2semidirect
10. #9641C2semidirect
11. #9640C2semidirect

Maximal subgroups

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

subgroupindex
#349722
#219762
#203662
C2×#14942
C2×#14832
C2×#14912
C22×Q8⋊C82
#96452
#96432
#96412
#96402