Finite Group Databasebeta

← #192476

Subgroups — #192476

Order 512.

987 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60231C211yesmaximal, characteristic
2#21948C211yesmaximal, characteristic, complemented
3#21935C211yesmaximal, characteristic, complemented
4#21774C211yesmaximal, characteristic
5#17572C211yesmaximal, characteristic
6#8875C214yesmaximal, complemented
7#8874C214yesmaximal, complemented
8#8049C211yesmaximal, characteristic, complemented
9#7925C211yesmaximal, characteristic, complemented
10C2×#525C2211yescharacteristic
11#2978—22no—
12#2977—22no—
13#2976—22no—
14#2976C2214yes—
15D4×M16—22no—
16D4×M16—22no—
17#2921—21no—
18#2879—21no—
19C2×#407C2211yescharacteristic
20C2×#407C2211yescharacteristic
21C2×#407C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23#2280C2211yescharacteristic
24#2280C2211yescharacteristic
25#1897—22no—
26#1871C2212yes—
27#1871C2212yes—
28#1855—22no—
29#1855C2214yescomplemented
30#1851—22no—
31#1851C2214yescomplemented
32#1826—21no—
33#1810—22no—
34#1798—22no—
35#1789—21no—
36#1520C412yescomplemented
37#1519C412yescomplemented
38#1516C414yescomplemented
39#1501C2214yes—
40#1297C2212yescomplemented
41#1297C2212yescomplemented
42#1282C2212yescomplemented
43#1282C2212yescomplemented
44C22×#41D411yescharacteristic
45C24×C4C2311yescharacteristic
46C22×M16—22no—
47C22×M16D412yes—
48C22×M16D412yes—
49C22×M16C2312yes—
50C23×C8—21no—
51#525—21no—
52#525—21no—
53#525—21no—
54#525—21no—
55#525C2×C412yes—
56#525D412yes—
57#525Q812yes—
58#525C2312yes—
59C2×#123—21no—
60C2×C22⋊Q8C2×C411yescharacteristic
61C2×C22⋊Q8Q811yescharacteristic
62C2×#121C2×C411yescharacteristic
63C2×#121Q811yescharacteristic
64C2×#120—21no—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×#117—21no—
68C2×#117—21no—
69C2×#117D411yescharacteristic
70#436—24no—
71#436—24no—
72#435—24no—
73#435—24no—
74#435—24no—
75#435—24no—
76C8×D4—24no—
77C8×D4—24no—
78#433—24no—
79#433—24no—
80C2×#108—24no—
81#423—22no—
82C2×C4⋊C8—24no—
83C2×#104—22no—
84C2×#104—22no—
85#408—24no—
86#408—24no—
87#407—22no—
88#407—21no—
89#407—21no—
90#407—21no—
91#407—21no—
92#407—21no—
93#407—21no—
94#407—21no—
95#407—21no—
96#407—21no—
97#407—21no—
98#407D412yes—
99#407C2312yes—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8C2×C412yes—
108C2×C22⋊C8C2×C412yes—
109C2×C22⋊C8D412yes—
110C2×C22⋊C8C2312yes—
111C2×C22⋊C8C2312yes—
112C4×M16—22no—
113#381—21no—
114#381C2×C412yes—
115#380—21no—
116#380—21no—
117#380—21no—
118#380C2×C412yes—
119#379—21no—
120#379—21no—
121C4×C22⋊C4—21no—
122C4×C22⋊C4—21no—
123C4×C22⋊C4C2311yescharacteristic
124C4×C22⋊C4C2311yescharacteristic
125C4×C22⋊C4C2312yes—
126#341—24no—
127#339—22no—
128#339—22no—
129#336—24no—
130#328C2×C418yes—
131#327C2×C418yes—
132C25#4111yescharacteristic
133C2×#41—21no—
134C2×#41—21no—
135C2×#41—82no—
136C2×#41C4⋊C412yes—
137C2×#41C2×D412yes—
138C22×Q8C4⋊C411yescharacteristic
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4C4⋊C411yescharacteristic
143C23×C4—21no—
144C23×C4—82no—
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×Q811yescharacteristic
154C23×C4C2×Q811yescharacteristic
155C23×C4#4111yescharacteristic
156C23×C4#4111yescharacteristic
157C23×C4#4111yescharacteristic
158C23×C4C2411yescharacteristic
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—24no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—82no—
166C2×M16—82no—
167C2×M16—84no—
168C2×M16—82no—
169C2×M16C2×D414yes—
170C22×C8—22no—
171C22×C8—24no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—82no—
177C22×C8C2×D414yes—
178#123—22no—
179#123—22no—
180#123—22no—
181#123—22no—
182C22⋊Q8—21no—
183C22⋊Q8—21no—
184C22⋊Q8—21no—
185C22⋊Q8—21no—
186C22⋊Q8C4⋊C412yes—
187C22⋊Q8C4⋊C412yes—
188C22⋊Q8C22×C412yes—
189C22⋊Q8C2×Q812yes—
190#121—21no—
191#121—21no—
192#121—21no—
193#121—21no—
194#121C4⋊C412yes—
195#121C4⋊C412yes—
196#121C22×C412yes—
197#121C2×Q812yes—
198#120—22no—
199#120—22no—
200#120—22no—
201#120—22no—
202C4×D4—22no—
203C4×D4—22no—
204C4×D4—22no—
205C4×D4—22no—
206C4×D4—22no—
207C4×D4—22no—
208C4×D4—22no—
209C4×D4—22no—
210#117—21no—
211#117—21no—
212#117—82no—
213#117—82no—
214#117C4⋊C412yes—
215#117C2×D412yes—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4C4⋊C411yescharacteristic
221C2×C4⋊C4C4⋊C411yescharacteristic
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—21no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C42—21no—
235C2×C42—22no—
236C2×C42—21no—
237C2×C42—21no—
238#108—84no—
239C4⋊C8—24no—
240C4⋊C8—24no—
241C4⋊C8—84no—
242#104—82no—
243#104—82no—
244C22⋊C8—24no—
245C22⋊C8—24no—
246C22⋊C8—22no—
247C22⋊C8—22no—
248C22⋊C8—22no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—22no—
252C22⋊C8—22no—
253C22⋊C8—22no—
254C22⋊C8—22no—
255C22⋊C8—22no—
256C22⋊C8—22no—
257C22⋊C8—41no—
258C22⋊C8—41no—
259C22⋊C8—41no—
260C22⋊C8—41no—
261C22⋊C8C22×C414yes—
262#97—24no—
263C4×C8—24no—
264#95—22no—
265#95—21no—
266#95—21no—
267#95—41no—
268#95C22×C414yes—
269C24—21no—
270C24—42no—
271C24—81no—
272C24—81no—
273C24C4×D411yescharacteristic
274C24C4×D411yescharacteristic
275C24C22⋊Q811yescharacteristic
276C24C22⋊Q811yescharacteristic
277C24#12311yescharacteristic
278C24#12311yescharacteristic
279C24C2×#4111yescharacteristic
280#41—42no—
281#41—42no—
282#41—82no—
283#41—82no—
284#41—82no—
285C2×Q8—21no—
286C2×Q8—81no—
287C2×Q8C2×C4⋊C412yes—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—81no—
296C2×D4—82no—
297C2×D4—82no—
298C2×D4—82no—
299C2×D4C2×C4⋊C412yes—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—44no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—81no—
317C22×C4—81no—
318C22×C4—81no—
319C22×C4—81no—
320C22×C4—81no—
321C22×C4—81no—
322C22×C4—81no—
323C22×C4C2×C4⋊C411yescharacteristic
324C22×C4C2×C4⋊C411yescharacteristic
325C22×C4C2×C4⋊C411yescharacteristic
326C22×C4C2×C4⋊C411yescharacteristic
327C22×C4C4×D411yescharacteristic
328C22×C4C4×D411yescharacteristic
329C22×C4C4×D411yescharacteristic
330C22×C4C4×D411yescharacteristic
331C22×C4C4×D411yescharacteristic
332C22×C4C4×D411yescharacteristic
333C22×C4#12011yescharacteristic
334C22×C4#12011yescharacteristic
335C22×C4#12011yescharacteristic
336C22×C4#12011yescharacteristic
337C22×C4C22⋊Q811yescharacteristic
338C22×C4C22⋊Q811yescharacteristic
339C22×C4C22⋊Q811yescharacteristic
340C22×C4C22⋊Q811yescharacteristic
341C22×C4C22⋊Q811yescharacteristic
342C22×C4C22⋊Q811yescharacteristic
343C22×C4#12311yescharacteristic
344C22×C4#12311yescharacteristic
345C22×C4C23×C411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C22×Q811yescharacteristic
350C22×C4C2×#4111yescharacteristic
351M16—44no—
352M16—44no—
353M16—44no—
354M16—42no—
355M16—42no—
356M16—44no—
357M16—82no—
358M16—84no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—44no—
363C2×C8—44no—
364C2×C8—44no—
365C2×C8—42no—
366C2×C8—82no—
367C2×C8—82no—
368C2×C8—82no—
369C2×C8—84no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—82no—
377C4⋊C4—82no—
378C4⋊C4C2×C4⋊C412yes—
379C4⋊C4C2×C4⋊C412yes—
380C22⋊C4—22no—
381C22⋊C4—22no—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—82no—
393C22⋊C4—82no—
394C22⋊C4—82no—
395C22⋊C4—82no—
396C22⋊C4—82no—
397C42—22no—
398C42—22no—
399C42—44no—
400C42—42no—
401C42—42no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—42no—
425C23—81no—
426C23—81no—
427C23—88no—
428C23#38511yescharacteristic
429C23#38511yescharacteristic
430C23#38511yescharacteristic
431C23#38511yescharacteristic
432C23C2×C4×D411yescharacteristic
433C23C2×C22⋊Q811yescharacteristic
434C23C2×#12311yescharacteristic
435Q8—42no—
436Q8—81no—
437D4—42no—
438D4—44no—
439D4—44no—
440D4—81no—
441D4—82no—
442D4—82no—
443D4—82no—
444D4—82no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4#38511yescharacteristic
492C2×C4#38511yescharacteristic
493C2×C4#38511yescharacteristic
494C2×C4#38511yescharacteristic
495C2×C4C22×C4⋊C411yescharacteristic
496C2×C4C2×C4×D411yescharacteristic
497C2×C4C2×#12011yescharacteristic
498C2×C4C2×C22⋊Q811yescharacteristic
499C8—44no—
500C8—44no—
501C8—44no—
502C8—82no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—44no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22—82no—
532C22—82no—
533C22—162no—
534C22C2×#38511yescharacteristic
535C22#295712yes—
536C4—21no—
537C4—21no—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—42no—
544C4—42no—
545C4—42no—
546C4—84no—
547C4—161no—
548C4—161no—
549C4#302612yes—
550C2—22no—
551C2—41no—
552C2—41no—
553C2—41no—
554C2—41no—
555C2—41no—
556C2—41no—
557C2—82no—
558C2#2086711yescharacteristic
559C2#2088611yescharacteristic
560C2#2088911yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#385
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21948C2semidirect
3. #21935C2semidirect
6. #8875C2semidirect
7. #8874C2semidirect
8. #8049C2semidirect
9. #7925C2semidirect
29. #1855C22semidirect
31. #1851C22semidirect
36. #1520C4semidirect
37. #1519C4semidirect
38. #1516C4semidirect
40. #1297C22semidirect
41. #1297C22semidirect
42. #1282C22semidirect
43. #1282C22semidirect

Maximal subgroups

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

subgroupindex
#602312
#219482
#219352
#217742
#175722
#88752
#88742
#80492
#79252