Finite Group Databasebeta

← #201381

Subgroups — #201381

Order 512.

1511 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60277C211yesmaximal, characteristic, complemented
2#34935C211yesmaximal, characteristic
3#20732C212yesmaximal, complemented
4#20632C211yesmaximal, characteristic, complemented
5#9474C214yesmaximal, complemented
6#9472C214yesmaximal, complemented
7#7981C212yesmaximal, complemented
8C2×#525—21no—
9C2×C4×#41C2211yescharacteristic
10C22×C4×D4C2211yescharacteristic
11#2989—24no—
12#2983—21no—
13#2975—21no—
14#2973C2214yes—
15#2965—24no—
16#2920—22no—
17C2×#431C2212yes—
18C2×#410—21no—
19#2538—21no—
20#2537—21no—
21#2536C412yescomplemented
22#2535C412yescomplemented
23#2534—22no—
24#2486C2212yes—
25#2408C2212yes—
26#2392C414yescomplemented
27C4×#121—22no—
28#2285C2212yes—
29#2285C2211yescharacteristic
30#2280—21no—
31#1823C2214yes—
32#1802—24no—
33#1799—22no—
34#1799C2214yes—
35#1786C2214yes—
36C4×#102—24no—
37#1740—21no—
38#1289C2218yes—
39C22×#41D411yescharacteristic
40C23×D4C2311yescharacteristic
41C24×C4—21no—
42#525—48no—
43C2×#123—22no—
44C2×C22⋊Q8—22no—
45C2×#121—22no—
46C2×#121—22no—
47C2×#120—22no—
48C4×#41—22no—
49C4×#41—22no—
50C4×#41—22no—
51C4×#41—22no—
52C2×C4×Q8C2×C411yescharacteristic
53C2×C4×Q8D411yescharacteristic
54C2×C4×D4—24no—
55C2×C4×D4—21no—
56C2×C4×D4—21no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4C2×C412yes—
62C2×C4×D4C2×C412yes—
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4D412yes—
65C2×C4×D4D411yescharacteristic
66C2×C4×D4C2312yes—
67C2×#117—21no—
68C2×#117—22no—
69C2×#117C2×C412yes—
70C2×#117D411yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C22⋊C4—21no—
73C22×C42D412yes—
74C22×C42C2311yescharacteristic
75#446—28no—
76#436—28no—
77#435—216no—
78#432—24no—
79#432—24no—
80#431—24no—
81#431—22no—
82#431—22no—
83#431—22no—
84#431—22no—
85#423—24no—
86C2×C4⋊C8D414yescomplemented
87C2×#102—28no—
88#410—44no—
89C2×#99—24no—
90#407—24no—
91C2×C22⋊C8C2314yes—
92C4×M16—24no—
93C2×C4×C8D414yescomplemented
94#398C2×C414yes—
95#397C2×C414yes—
96#396—24no—
97#394—24no—
98#394C2×C414yes—
99#392C2×C414yes—
100#390—22no—
101#389—22no—
102#388—24no—
103#386—24no—
104#384—21no—
105#384—22no—
106#382—22no—
107#382C2×C412yes—
108#381—22no—
109#380—22no—
110#380—21no—
111#380—22no—
112#380—22no—
113#380—22no—
114#380C2314yes—
115#380C2312yes—
116#379—22no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4C2×C412yes—
121C4×C22⋊C4—24no—
122#342—24no—
123#340—28no—
124C25—21no—
125C2×#41—22no—
126C2×#41—22no—
127C2×#41—24no—
128C2×#41—22no—
129C2×#41—22no—
130C22×Q8C22⋊C411yescharacteristic
131C22×D4—24no—
132C22×D4—22no—
133C22×D4—21no—
134C22×D4—22no—
135C22×D4—44no—
136C22×D4C22⋊C411yescharacteristic
137C22×D4C22×C412yes—
138C22×D4C2×D412yes—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—44no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4C22⋊C412yes—
148C23×C4C22⋊C412yes—
149C23×C4C2×D412yes—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2411yescharacteristic
153C2×M16—28no—
154C22×C8C2×D418yes—
155#123—48no—
156C22⋊Q8—48no—
157#121—48no—
158#121—48no—
159#120—48no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×D4—24no—
165C4×D4—24no—
166C4×D4—24no—
167C4×D4—24no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—24no—
171C4×D4—24no—
172C4×D4—24no—
173C4×D4—24no—
174C4×D4—22no—
175C4×D4—22no—
176C4×D4—416no—
177C4×D4—42no—
178C4×D4—41no—
179C4×D4—41no—
180C4×D4—41no—
181C4×D4—42no—
182C4×D4—41no—
183#117—22no—
184#117—22no—
185#117—24no—
186#117—24no—
187#117—44no—
188#117—44no—
189#117—44no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4C22⋊C412yes—
201C2×C4⋊C4C22⋊C411yescharacteristic
202C2×C4⋊C4C22⋊C411yescharacteristic
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C22×C412yes—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4C22⋊C412yes—
216C2×C22⋊C4C22×C412yes—
217C2×C22⋊C4C2×D412yes—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—42no—
224C2×C42C22⋊C412yes—
225C2×C42C22⋊C412yes—
226C2×C42C22×C411yescharacteristic
227C2×C42C22×C411yescharacteristic
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42#4112yes—
234C2×C42#4112yes—
235C4⋊C8—28no—
236C4⋊C8—24no—
237C4⋊C8—24no—
238#102—416no—
239#99—48no—
240C22⋊C8—28no—
241C22⋊C8—24no—
242C22⋊C8—24no—
243#97—28no—
244C4×C8—24no—
245C4×C8—24no—
246#95—24no—
247#95—24no—
248#95—22no—
249#95—24no—
250#95—42no—
251#95C22×C414yes—
252C24—22no—
253C24—44no—
254C24—42no—
255C24—42no—
256C24—41no—
257C24—41no—
258C24—82no—
259C24C2×C22⋊C412yes—
260C24C22×D411yescharacteristic
261#41—44no—
262#41—44no—
263#41—44no—
264#41—44no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×D4—28no—
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—22no—
272C2×D4—24no—
273C2×D4—22no—
274C2×D4—24no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—44no—
279C2×D4—42no—
280C2×D4—44no—
281C2×D4—48no—
282C2×D4—48no—
283C2×D4—48no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—84no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—42no—
298C22×C4—44no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—44no—
305C22×C4—42no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—81no—
312C22×C4—82no—
313C22×C4—84no—
314C22×C4—84no—
315C22×C4—84no—
316C22×C4—84no—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C412yes—
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4#12112yes—
325C22×C4#12112yes—
326C22×C4#12112yes—
327C22×C4#12112yes—
328C22×C4AGL(1,ℤ/8)12yes—
329C22×C4AGL(1,ℤ/8)14yes—
330C22×C4#13712yes—
331C22×C4C23×C411yescharacteristic
332C22×C4C22×D412yes—
333C22×C4C22×D411yescharacteristic
334C22×C4C2×#4112yes—
335M16—88no—
336C2×C8—28no—
337C2×C8—48no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—48no—
344C4⋊C4—44no—
345C4⋊C4—48no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—44no—
349C4⋊C4—82no—
350C4⋊C4—84no—
351C4⋊C4—84no—
352C22⋊C4—24no—
353C22⋊C4—24no—
354C22⋊C4—24no—
355C22⋊C4—48no—
356C22⋊C4—48no—
357C22⋊C4—48no—
358C22⋊C4—44no—
359C22⋊C4—42no—
360C22⋊C4—42no—
361C22⋊C4—44no—
362C22⋊C4—44no—
363C22⋊C4—44no—
364C22⋊C4—84no—
365C42—22no—
366C42—22no—
367C42—22no—
368C42—22no—
369C42—22no—
370C42—22no—
371C42—44no—
372C42—84no—
373C42C4×D412yes—
374C42C4×D412yes—
375C42C4×D412yes—
376C42C4×D412yes—
377C42#12012yes—
378C42#12011yescharacteristic
379C42#12011yescharacteristic
380C42#12311yescharacteristic
381C42#12312yes—
382C42#12311yescharacteristic
383C23—24no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—41no—
388C23—41no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—48no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—84no—
399C23—88no—
400C23—84no—
401C23—84no—
402C23—82no—
403C23—82no—
404C23C22×C22⋊C411yescharacteristic
405C23C2×#12112yes—
406C23C2×AGL(1,ℤ/8)12yes—
407C23C2×AGL(1,ℤ/8)11yescharacteristic
408C23C2×#13711yescharacteristic
409Q8—44no—
410D4—28no—
411D4—44no—
412D4—44no—
413D4—44no—
414D4—44no—
415D4—44no—
416D4—44no—
417D4—816no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—84no—
448C2×C4—84no—
449C2×C4—88no—
450C2×C4—84no—
451C2×C4—84no—
452C2×C4—84no—
453C2×C4#38611yescharacteristic
454C2×C4#38611yescharacteristic
455C2×C4#38612yes—
456C2×C4#38611yescharacteristic
457C2×C4#38612yes—
458C2×C4#38611yescharacteristic
459C2×C4#41712yes—
460C2×C4#41712yes—
461C2×C4#41812yes—
462C2×C4#41812yes—
463C2×C4#46814yes—
464C2×C4#46814yes—
465C2×C4#46914yes—
466C2×C4#46914yes—
467C2×C4C2×C4×D412yes—
468C2×C4C2×#12011yescharacteristic
469C2×C4C2×#12311yescharacteristic
470C8—48no—
471C22—21no—
472C22—22no—
473C22—21no—
474C22—22no—
475C22—42no—
476C22—41no—
477C22—41no—
478C22—44no—
479C22—44no—
480C22—44no—
481C22—42no—
482C22—42no—
483C22—44no—
484C22—42no—
485C22—82no—
486C22—82no—
487C22—82no—
488C22—88no—
489C22—88no—
490C22—82no—
491C22—82no—
492C22C2×#38611yescharacteristic
493C22C2×#41811yescharacteristic
494C22C2×#41711yescharacteristic
495C22C2×#46812yes—
496C22C2×#46912yes—
497C4—21no—
498C4—21no—
499C4—22no—
500C4—21no—
501C4—22no—
502C4—42no—
503C4—42no—
504C4—42no—
505C4—84no—
506C4—82no—
507C4—164no—
508C4#189512yes—
509C4#189512yes—
510C4#189514yes—
511C4#302512yes—
512C2—21no—
513C2—22no—
514C2—21no—
515C2—22no—
516C2—82no—
517C2—84no—
518C2#2095911yescharacteristic
519C2C2×#189511yescharacteristic
520C2#2100911yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#386
prime radical#34935C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60277C2semidirect
3. #20732C2semidirect
4. #20632C2semidirect
5. #9474C2semidirect
6. #9472C2semidirect
7. #7981C2semidirect
21. #2536C4semidirect
22. #2535C4semidirect
26. #2392C4semidirect
86. C2×C4⋊C8D4semidirect
93. C2×C4×C8D4semidirect

Maximal subgroups

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

subgroupindex
#602772
#349352
#207322
#206322
#94742
#94722
#79812