Finite Group Databasebeta

← #190655

Subgroups — #190655

Order 512.

1080 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34211C211yesmaximal, characteristic
2#21435C212yesmaximal, complemented
3#8721C211yesmaximal, characteristic, complemented
4#8721C211yesmaximal, characteristic, complemented
5#8583C211yesmaximal, characteristic, complemented
6#8581C212yesmaximal, complemented
7#8579C211yesmaximal, characteristic, complemented
8#8334C212yesmaximal
9#8226C212yesmaximal, complemented
10#8224C211yesmaximal, characteristic, complemented
11C4×#329C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C2×#519C411yescharacteristic
14C2×#518C411yescharacteristic
15#2367—22no—
16#2367—22no—
17#2342—22no—
18#2342—22no—
19#2326C411yescharacteristic
20#2326C411yescharacteristic
21#2326C411yescharacteristic
22#2326C411yescharacteristic
23#2310C411yescharacteristic
24#2309C411yescharacteristic
25C4×#124C2211yescharacteristic
26C4×#124C2211yescharacteristic
27C4×#124C2211yescharacteristic
28C4×#124C2211yescharacteristic
29#2285C2211yescharacteristic
30C4×#117C2211yescharacteristic
31#1924—22no—
32#1923—22no—
33#1917—22no—
34#1916—22no—
35#1767C2212yescomplemented
36#1767C2212yescomplemented
37#1749C2212yes—
38#1749C2212yes—
39#1745C2212yescomplemented
40C4×#97C2212yescomplemented
41C2×#329C2212yes—
42C2×#329C2212yes—
43C2×#329C2212yes—
44C2×#329C2212yes—
45#1295C2211yescharacteristic, complemented
46#1295C2211yescharacteristic, complemented
47#1295C2211yescharacteristic, complemented
48#1295C2211yescharacteristic, complemented
49#1295C2212yescomplemented
50#1295C2212yescomplemented
51C23×Q8C2×C411yescharacteristic
52C23×D4C2×C411yescharacteristic
53C2×#124—22no—
54C2×#124—22no—
55C2×#124—24no—
56C2×#124—22no—
57C2×#124C2×C411yescharacteristic
58C2×#124C2×C411yescharacteristic
59C2×#124C2311yescharacteristic
60C2×#124C2311yescharacteristic
61#519—44no—
62#518—44no—
63C2×C4×Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2×C411yescharacteristic
75C2×#117—21no—
76C2×#117—21no—
77C2×#117C2×C411yescharacteristic
78C2×#117C2×C411yescharacteristic
79C2×#117C2×C411yescharacteristic
80C2×#117C2×C411yescharacteristic
81C22×C22⋊C4—22no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C42C2311yescharacteristic
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×#97—22no—
90C2×#97C2312yes—
91C2×#97C2312yes—
92C2×#97C2312yes—
93C2×#97C2312yes—
94C2×C4×C8—22no—
95#391—22no—
96#391C2×C411yescharacteristic
97#391C2×C411yescharacteristic
98#391C2×C411yescharacteristic
99#391C2×C411yescharacteristic
100#390—22no—
101#390C2×C411yescharacteristic
102#390C2×C411yescharacteristic
103#390C2×C411yescharacteristic
104#390C2×C411yescharacteristic
105#388—22no—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#385—22no—
114#385—22no—
115#385—22no—
116#385—22no—
117#384C2×C411yescharacteristic
118#384C2×C411yescharacteristic
119#382C2311yescharacteristic
120#382C2311yescharacteristic
121#381—22no—
122#380—22no—
123#379—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4D412yes—
126C4×C4⋊C4D412yes—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4D412yes—
132C4×C22⋊C4D412yes—
133#376C2×C411yescharacteristic
134#376C2×C411yescharacteristic
135C43C2311yescharacteristic
136C43C2311yescharacteristic
137#336—22no—
138#336—22no—
139#336—22no—
140#336—22no—
141#336—22no—
142#336—22no—
143#329—24no—
144#329—24no—
145#329—24no—
146#329—24no—
147C25—21no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—21no—
151C22×Q8—42no—
152C22×Q8—42no—
153C22×Q8C22×C411yescharacteristic
154C22×Q8C22×C411yescharacteristic
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—21no—
158C22×D4—42no—
159C22×D4—42no—
160C22×D4C22×C411yescharacteristic
161C22×D4C22×C411yescharacteristic
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4C22×C411yescharacteristic
168C23×C4C2×D412yes—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173#124—28no—
174#124—28no—
175#124—22no—
176#124—22no—
177#124—22no—
178#124—22no—
179#124—21no—
180#124—21no—
181#124—22no—
182#124#4114yes—
183#124#4112yes—
184#124#4112yes—
185C4×Q8—42no—
186C4×Q8—42no—
187C4×Q8—42no—
188C4×Q8—42no—
189C4×Q8—44no—
190C4×Q8—44no—
191C4×D4—42no—
192C4×D4—42no—
193C4×D4—42no—
194C4×D4—42no—
195C4×D4—44no—
196C4×D4—44no—
197#117—24no—
198#117—42no—
199#117—44no—
200#117—44no—
201#117—42no—
202#117—42no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4C22⋊C411yescharacteristic
210C2×C4⋊C4C22⋊C412yes—
211C2×C4⋊C4C22⋊C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4C22⋊C411yescharacteristic
235C2×C22⋊C4C22⋊C411yescharacteristic
236C2×C22⋊C4C22⋊C412yes—
237C2×C42—22no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42C22×C411yescharacteristic
247C2×C42C22×C411yescharacteristic
248C2×C42C22×C411yescharacteristic
249C2×C42C22×C411yescharacteristic
250C2×C42C22×C411yescharacteristic
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42C2411yescharacteristic
257C4⋊C8—44no—
258C4⋊C8—44no—
259C22⋊C8—44no—
260C22⋊C8—44no—
261#97—22no—
262#97—22no—
263#97—22no—
264#97—22no—
265#97—22no—
266#97—22no—
267#97—22no—
268#97—22no—
269#97—44no—
270C4×C8—44no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95C22⋊C412yes—
278#95C22⋊C412yes—
279#95C22⋊C412yes—
280#95C22⋊C412yes—
281C24—24no—
282C24—21no—
283C24—21no—
284C24—42no—
285C24—42no—
286C24—82no—
287C24—82no—
288C24C2×C22⋊C411yescharacteristic
289C2×Q8—24no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—48no—
293C2×Q8—48no—
294C2×Q8—42no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—42no—
300C2×Q8#11712yes—
301C2×Q8#11712yes—
302C2×D4—24no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—48no—
310C2×D4—48no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4#11712yes—
314C2×D4#11712yes—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—82no—
337C22×C4—82no—
338C22×C4—82no—
339C22×C4—82no—
340C22×C4C2×C22⋊C412yes—
341C22×C4C2×C22⋊C412yes—
342C22×C4C2×C22⋊C412yes—
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C23×C411yescharacteristic
349C22×C4C22×D412yes—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C4⋊C4—22no—
363C4⋊C4—42no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—82no—
373C4⋊C4—82no—
374C4⋊C4#10014yes—
375C22⋊C4—28no—
376C22⋊C4—28no—
377C22⋊C4—24no—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C22⋊C4—24no—
381C22⋊C4—24no—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—82no—
391C22⋊C4—82no—
392C22⋊C4—82no—
393C22⋊C4—82no—
394C42—21no—
395C42—21no—
396C42—22no—
397C42—22no—
398C42—22no—
399C42—22no—
400C42—42no—
401C42—42no—
402C42—42no—
403C42—42no—
404C42—41no—
405C42—41no—
406C42—42no—
407C42—42no—
408C42—42no—
409C42—42no—
410C42—42no—
411C42—42no—
412C42C2×#4112yes—
413C42C2×#4111yescharacteristic
414C42C2×#4111yescharacteristic
415C23—21no—
416C23—22no—
417C23—22no—
418C23—21no—
419C23—22no—
420C23—22no—
421C23—48no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—88no—
431C23—164no—
432C23—162no—
433C23—162no—
434C23#41312yes—
435C23C22×C22⋊C411yescharacteristic
436Q8—42no—
437Q8—42no—
438Q8—42no—
439Q8—42no—
440Q8—48no—
441D4—42no—
442D4—42no—
443D4—42no—
444D4—42no—
445D4—48no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—48no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—162no—
487C2×C4—162no—
488C2×C4—162no—
489C2×C4—162no—
490C2×C4C2×#10012yes—
491C2×C4C2×#10012yes—
492C2×C4C2×#10012yes—
493C2×C4C2×#11711yescharacteristic
494C2×C4C2×#11711yescharacteristic
495C2×C4#52512yes—
496C2×C4#52512yes—
497C8—44no—
498C8—44no—
499C8—82no—
500C8—82no—
501C22—24no—
502C22—22no—
503C22—42no—
504C22—41no—
505C22—41no—
506C22—44no—
507C22—44no—
508C22—42no—
509C22—42no—
510C22—81no—
511C22—81no—
512C22—82no—
513C22—82no—
514C22—81no—
515C22—81no—
516C22—168no—
517C22#228011yescharacteristic
518C22C22×#10011yescharacteristic
519C22C2×#41311yescharacteristic
520C22#290112yes—
521C22#290112yes—
522C4—22no—
523C4—22no—
524C4—22no—
525C4—22no—
526C4—21no—
527C4—21no—
528C4—22no—
529C4—42no—
530C4—42no—
531C4—42no—
532C4—41no—
533C4—41no—
534C4—42no—
535C4—42no—
536C4—81no—
537C4—81no—
538C4—81no—
539C4—81no—
540C2—42no—
541C2—82no—
542C2—82no—
543C2#1768511yescharacteristic
544C2#1768612yes—
545C2#1768811yescharacteristic
546C2#2072611yescharacteristic
547C2#2074912yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21435C2semidirect
3. #8721C2semidirect
4. #8721C2semidirect
5. #8583C2semidirect
6. #8581C2semidirect
7. #8579C2semidirect
9. #8226C2semidirect
10. #8224C2semidirect
11. C4×#329C2semidirect
35. #1767C22semidirect
36. #1767C22semidirect
39. #1745C22semidirect
40. C4×#97C22semidirect
45. #1295C22semidirect
46. #1295C22semidirect
47. #1295C22semidirect
48. #1295C22semidirect
49. #1295C22semidirect
50. #1295C22semidirect

Maximal subgroups

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

subgroupindex
#342112
#214352
#87212
#85832
#85812
#85792
#83342
#82262
#82242
C4×#3292