Finite Group Databasebeta

← #196853

Subgroups — #196853

Order 512.

1638 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35165C211yesmaximal, characteristic
2C2×#2357C211yesmaximal, characteristic, complemented
3#21786C211yesmaximal, characteristic, complemented
4C2×#1517C211yesmaximal, characteristic, complemented
5C2×#1502C211yesmaximal, characteristic, complemented
6#9158C214yesmaximal, complemented
7#9150C214yesmaximal, complemented
8#8026C211yesmaximal, characteristic, complemented
9#8014C211yesmaximal, characteristic, complemented
10C2×#535—21no—
11C2×#532C2211yescharacteristic
12C22×C4×D4C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#2847C2214yes—
16#2617—22no—
17#2547C412yescomplemented
18#2544—22no—
19#2541C412yescomplemented
20#2537C414yescomplemented
21#2535—22no—
22#2357—24no—
23#2357—24no—
24#2311—21no—
25C2×#390—21no—
26C2×#388—22no—
27#2285C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×#381C2212yes—
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2212yes—
33#1954—22no—
34#1954—22no—
35#1779C2214yes—
36#1517—24no—
37#1502—24no—
38C2×#327C2212yes—
39C2×#327C2212yes—
40C2×#327C2214yes—
41#1294C2212yes—
42#1294C2212yes—
43#1294C2214yes—
44C22×#41—21no—
45C23×D4D411yescharacteristic
46C24×C4C2312yes—
47#535—44no—
48#535—44no—
49#532—22no—
50#532—22no—
51#532—22no—
52#532—22no—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#127—22no—
55C2×#127C2×C411yescharacteristic
56C2×#124—22no—
57C2×#124—21no—
58C2×#121—24no—
59C2×#121—21no—
60C2×#121C2×C418yes—
61C2×#121C2312yes—
62C2×#120—24no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—21no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4C2×C412yes—
69C2×C4×D4C2×C412yes—
70C2×C4×D4C2×C412yes—
71C2×C4×D4D412yes—
72C2×C4×D4D412yes—
73C2×#117—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4D411yescharacteristic
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4D412yes—
80C22×C42—21no—
81C22×C42—22no—
82C22×C42C2311yescharacteristic
83#431—22no—
84#431—22no—
85#431—22no—
86#431—22no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C22⋊C8—28no—
90C2×C22⋊C8C2312yes—
91C2×C22⋊C8C2312yes—
92C2×C4×C8—21no—
93C2×C4×C8—21no—
94#397—24no—
95#394—28no—
96#392—24no—
97#391—22no—
98#390—28no—
99#390—22no—
100#388—216no—
101#386—24no—
102#386—28no—
103#384C2×C412yes—
104#382—22no—
105#382—21no—
106#382D414yescomplemented
107#381—24no—
108#381—24no—
109#380—21no—
110#380—24no—
111#380C2×C418yes—
112#380C2312yes—
113C4×C4⋊C4—28no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4D414yescomplemented
118C4×C22⋊C4—24no—
119C4×C22⋊C4—24no—
120C2×#95—22no—
121C2×#95—21no—
122C2×#95C2311yescharacteristic
123C2×#95C2312yes—
124C2×#95C2311yescharacteristic
125#327—24no—
126#327—24no—
127#327—28no—
128C25C2×D412yes—
129C2×#41—42no—
130C2×#41—42no—
131C2×#41—42no—
132C2×#41—42no—
133C2×#41—42no—
134C2×#41—42no—
135C22×Q8—21no—
136C22×D4—22no—
137C22×D4—28no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4C22⋊C412yes—
147C22×D4C22⋊C412yes—
148C22×D4C22×C412yes—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—84no—
158C23×C4C22×C414yes—
159C23×C4C2×D412yes—
160C23×C4C2×D414yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2411yescharacteristic
164C22×C8—22no—
165C22×C8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168#127—22no—
169#127—22no—
170#127—48no—
171#124—48no—
172#124—42no—
173#124—42no—
174#121—22no—
175#121—22no—
176#121—22no—
177#121—22no—
178#121—416no—
179#121—42no—
180#121—42no—
181#121—88no—
182#120—48no—
183#120—48no—
184C4×D4—24no—
185C4×D4—24no—
186C4×D4—24no—
187C4×D4—24no—
188C4×D4—24no—
189C4×D4—24no—
190C4×D4—24no—
191C4×D4—24no—
192C4×D4—24no—
193C4×D4—24no—
194C4×D4—42no—
195C4×D4—42no—
196#117—42no—
197#117—42no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4C22⋊C412yes—
209C2×C4⋊C4C22⋊C412yes—
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C22×C414yes—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—48no—
222C2×C22⋊C4—84no—
223C2×C22⋊C4C22⋊C414yes—
224C2×C22⋊C4C22⋊C414yes—
225C2×C42—22no—
226C2×C42—24no—
227C2×C42—28no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42—84no—
231C2×C42C22×C411yescharacteristic
232C2×C42C22×C411yescharacteristic
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42#4112yes—
238C2×C42#4112yes—
239C4⋊C8—42no—
240C4⋊C8—42no—
241C22⋊C8—22no—
242C22⋊C8—22no—
243C22⋊C8—22no—
244C22⋊C8—22no—
245C22⋊C8—88no—
246C4×C8—42no—
247C4×C8—42no—
248#95—28no—
249#95—22no—
250#95—28no—
251#95—24no—
252#95—21no—
253#95—22no—
254#95—21no—
255#95—22no—
256#95C2×D412yes—
257#95C2×D414yes—
258#95C2×D412yes—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—44no—
267C24—44no—
268C24C2×C22⋊C414yes—
269C24C22×D411yescharacteristic
270#41—84no—
271#41—84no—
272#41—84no—
273#41—84no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—24no—
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—24no—
285C2×D4—44no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—416no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—44no—
298C2×D4—44no—
299C2×D4—44no—
300C2×D4—44no—
301C2×D4—82no—
302C2×D4—88no—
303C2×D4—88no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—44no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—48no—
320C22×C4—48no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—84no—
336C22×C4—84no—
337C22×C4—84no—
338C22×C4—84no—
339C22×C4C2×C22⋊C414yes—
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4#12012yes—
345C22×C4#12012yes—
346C22×C4#12312yes—
347C22×C4#12312yes—
348C22×C4#13512yes—
349C22×C4#13512yes—
350C22×C4AGL(1,ℤ/8)12yes—
351C22×C4AGL(1,ℤ/8)12yes—
352C22×C4C23×C411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C2×C8—44no—
359C2×C8—44no—
360C2×C8—42no—
361C2×C8—42no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—44no—
367C4⋊C4—48no—
368C4⋊C4—42no—
369C4⋊C4—44no—
370C4⋊C4—48no—
371C4⋊C4—82no—
372C22⋊C4—28no—
373C22⋊C4—28no—
374C22⋊C4—416no—
375C22⋊C4—416no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—44no—
381C22⋊C4—88no—
382C22⋊C4—88no—
383C42—44no—
384C42—48no—
385C42—88no—
386C42C4×D412yes—
387C42C4×D412yes—
388C42C4×D412yes—
389C42C4×D412yes—
390C42#12112yes—
391C42#12112yes—
392C42#12112yes—
393C42#12112yes—
394C23—22no—
395C23—22no—
396C23—28no—
397C23—24no—
398C23—24no—
399C23—28no—
400C23—24no—
401C23—44no—
402C23—44no—
403C23—44no—
404C23—44no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—44no—
410C23—42no—
411C23—42no—
412C23—44no—
413C23—42no—
414C23—84no—
415C23—84no—
416C23—84no—
417C23—84no—
418C23C22×C22⋊C411yescharacteristic
419C23C2×#12011yescharacteristic
420C23C2×#12311yescharacteristic
421C23C2×#13511yescharacteristic
422C23C2×#13511yescharacteristic
423C23C2×AGL(1,ℤ/8)11yescharacteristic
424C23C2×AGL(1,ℤ/8)11yescharacteristic
425Q8—84no—
426D4—48no—
427D4—48no—
428D4—82no—
429D4—82no—
430D4—84no—
431D4—82no—
432D4—82no—
433D4—82no—
434D4—82no—
435D4—84no—
436D4—1616no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—28no—
443C2×C4—24no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—84no—
469C2×C4—84no—
470C2×C4—84no—
471C2×C4—84no—
472C2×C4—82no—
473C2×C4—84no—
474C2×C4—82no—
475C2×C4—84no—
476C2×C4—88no—
477C2×C4#38612yes—
478C2×C4#38612yes—
479C2×C4#38612yes—
480C2×C4#38612yes—
481C2×C4#41612yes—
482C2×C4#41612yes—
483C2×C4#41812yes—
484C2×C4#41812yes—
485C2×C4#44914yes—
486C2×C4#44914yes—
487C2×C4#48214yes—
488C2×C4#48214yes—
489C2×C4C2×C4×D411yescharacteristic
490C2×C4C2×C4×D411yescharacteristic
491C2×C4C2×#12111yescharacteristic
492C2×C4C2×#12111yescharacteristic
493C8—82no—
494C8—82no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—44no—
504C22—48no—
505C22—48no—
506C22—44no—
507C22—44no—
508C22—84no—
509C22—84no—
510C22—84no—
511C22—84no—
512C22—82no—
513C22—84no—
514C22—84no—
515C22—82no—
516C22—82no—
517C22C2×#38611yescharacteristic
518C22#288312yes—
519C22C2×#41611yescharacteristic
520C22C2×#41811yescharacteristic
521C22#290112yes—
522C22C2×#44911yescharacteristic
523C22C2×#44911yescharacteristic
524C22C2×#48211yescharacteristic
525C22C2×#48211yescharacteristic
526C4—22no—
527C4—22no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—44no—
533C4—44no—
534C4—82no—
535C4—84no—
536C4—82no—
537C4—84no—
538C4#187514yes—
539C4#187514yes—
540C2—21no—
541C2—21no—
542C2—21no—
543C2—21no—
544C2—84no—
545C2—162no—
546C2#1761712yes—
547C2#1762612yes—
548C2C2×#187511yescharacteristic
549C2#2098011yescharacteristic
550C2#2098511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#35165C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2357C2semidirect
3. #21786C2semidirect
4. C2×#1517C2semidirect
5. C2×#1502C2semidirect
6. #9158C2semidirect
7. #9150C2semidirect
8. #8026C2semidirect
9. #8014C2semidirect
17. #2547C4semidirect
19. #2541C4semidirect
20. #2537C4semidirect
106. #382D4semidirect
117. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
#351652
C2×#23572
#217862
C2×#15172
C2×#15022
#91582
#91502
#80262
#80142