Finite Group Databasebeta

← #200398

Subgroups — #200398

Order 512.

1413 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60277C211yesmaximal, characteristic, complemented
2#34937C211yesmaximal, characteristic
3#20632C211yesmaximal, characteristic, complemented
4#17574C212yesmaximal, complemented
5#9386C218yesmaximal, complemented
6#8014C212yesmaximal, complemented
7C2×#532—21no—
8C2×#532C2211yescharacteristic
9C2×C4×#41C2211yescharacteristic
10#2989—24no—
11#2983—21no—
12#2975—21no—
13#2973C2214yes—
14#2965—24no—
15#2920—22no—
16C2×#431C2212yes—
17C2×#410—21no—
18#2548C411yescharacteristic, complemented
19#2547C411yescharacteristic, complemented
20#2546C411yescharacteristic, complemented
21#2545—21no—
22#2544—21no—
23#2543C2214yes—
24#2541C411yescharacteristic, complemented
25#2416—22no—
26#2392C414yescomplemented
27C4×#121—22no—
28#2297—21no—
29#2285C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31#1799—22no—
32#1799C2214yes—
33#1740—21no—
34#1519C2218yes—
35#1517—24no—
36#1502—24no—
37#1294C2218yes—
38C22×#41—21no—
39C22×#41D411yescharacteristic
40#532—22no—
41#532—22no—
42#532—22no—
43#532—22no—
44#532—48no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#127—21no—
48C2×#127C2×C411yescharacteristic
49C2×#124—21no—
50C2×#124—21no—
51C2×#124—21no—
52C2×#121—24no—
53C2×#121—22no—
54C2×#121—21no—
55C2×#121C2312yes—
56C4×#41—22no—
57C4×#41—22no—
58C4×#41—22no—
59C4×#41—22no—
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C2×C4×D4—24no—
63C2×C4×D4—21no—
64C2×C4×D4C2×C412yes—
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4D412yes—
67C2×C4×D4D411yescharacteristic
68C2×C4×D4C2312yes—
69C2×#117—22no—
70C2×#117C2×C412yes—
71C2×#117D411yescharacteristic
72C22×C4⋊C4—21no—
73C22×C42—21no—
74C22×C42D412yes—
75C22×C42C2311yescharacteristic
76#446—28no—
77#436—28no—
78#435—216no—
79#432—24no—
80#432—24no—
81#431—24no—
82#431—22no—
83#431—22no—
84#431D414yescomplemented
85#431D414yescomplemented
86#423—24no—
87C2×C4⋊C8—22no—
88#410—44no—
89C2×#99—24no—
90#407—24no—
91C2×C22⋊C8C2314yes—
92C4×M16—24no—
93C2×C4×C8—22no—
94#398C2×C414yes—
95#397C2×C414yes—
96#396—24no—
97#394—24no—
98#394C2×C414yes—
99#392C2×C414yes—
100#391—21no—
101#391—21no—
102#390—21no—
103#390—21no—
104#390—22no—
105#389—22no—
106#388—24no—
107#386—24no—
108#386—24no—
109#384—22no—
110#382—22no—
111#380—22no—
112#380—22no—
113#380C2314yes—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C22⋊C4—24no—
122C2×#95C2312yes—
123#342—24no—
124#327—216no—
125C2×#41—22no—
126C2×#41—22no—
127C2×#41—24no—
128C2×#41—22no—
129C2×#41—22no—
130C2×#41—42no—
131C2×#41—42no—
132C2×#41—44no—
133C2×#41—42no—
134C2×#41—42no—
135C22×Q8—21no—
136C22×Q8C22⋊C411yescharacteristic
137C22×D4—22no—
138C22×D4—24no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4C22⋊C411yescharacteristic
142C22×D4C22×C412yes—
143C22×D4C2×D412yes—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4C22⋊C412yes—
147C23×C4C22⋊C412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2411yescharacteristic
152C2×M16—28no—
153C22×C8—24no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—44no—
157#127—22no—
158#127—22no—
159#127—44no—
160#124—44no—
161#124—44no—
162#124—42no—
163#124—42no—
164#121—24no—
165#121—24no—
166#121—48no—
167#121—416no—
168#121—42no—
169#121—42no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×D4—24no—
175C4×D4—24no—
176C4×D4—24no—
177C4×D4—24no—
178C4×D4—22no—
179C4×D4—22no—
180C4×D4—24no—
181C4×D4—24no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—416no—
185C4×D4—41no—
186C4×D4—41no—
187C4×D4—41no—
188C4×D4—41no—
189#117—22no—
190#117—22no—
191#117—24no—
192#117—24no—
193#117—44no—
194#117—44no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4C22⋊C412yes—
205C2×C4⋊C4C22⋊C411yescharacteristic
206C2×C4⋊C4C22⋊C411yescharacteristic
207C2×C4⋊C4C22×C412yes—
208C2×C4⋊C4C22×C412yes—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4C22⋊C412yes—
217C2×C22⋊C4C22×C412yes—
218C2×C22⋊C4C2×D412yes—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—44no—
227C2×C42—42no—
228C2×C42C22⋊C412yes—
229C2×C42C22⋊C412yes—
230C2×C42C22×C411yescharacteristic
231C2×C42C22×C411yescharacteristic
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42#4112yes—
238C2×C42#4112yes—
239C4⋊C8—28no—
240C4⋊C8—24no—
241C4⋊C8—24no—
242#99—48no—
243C22⋊C8—28no—
244C22⋊C8—24no—
245C22⋊C8C2×D418yes—
246#97—28no—
247C4×C8—24no—
248C4×C8—24no—
249#95—24no—
250#95—24no—
251#95—22no—
252#95—24no—
253#95—42no—
254#95C22×C412yes—
255#95C22×C412yes—
256C24—22no—
257C24C2×C22⋊C412yes—
258C24C22×D411yescharacteristic
259#41—44no—
260#41—44no—
261#41—44no—
262#41—44no—
263#41—84no—
264#41—84no—
265#41—84no—
266#41—84no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×D4—24no—
274C2×D4—22no—
275C2×D4—24no—
276C2×D4—22no—
277C2×D4—24no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—44no—
282C2×D4—48no—
283C2×D4—48no—
284C2×D4—44no—
285C2×D4—44no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—84no—
294C2×D4—82no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—42no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—81no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C412yes—
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4#12112yes—
326C22×C4#12112yes—
327C22×C4#12112yes—
328C22×C4#12112yes—
329C22×C4AGL(1,ℤ/8)12yes—
330C22×C4AGL(1,ℤ/8)14yes—
331C22×C4#13712yes—
332C22×C4C23×C411yescharacteristic
333C22×C4C22×D412yes—
334C22×C4C22×D411yescharacteristic
335C22×C4C2×#4112yes—
336M16—88no—
337C2×C8—28no—
338C2×C8—84no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—42no—
346C4⋊C4—44no—
347C4⋊C4—48no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—84no—
351C4⋊C4—84no—
352C4⋊C4—84no—
353C22⋊C4—24no—
354C22⋊C4—24no—
355C22⋊C4—24no—
356C22⋊C4—48no—
357C22⋊C4—48no—
358C22⋊C4—48no—
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—42no—
373C42—42no—
374C42—84no—
375C42C4×D412yes—
376C42C4×D412yes—
377C42C4×D412yes—
378C42C4×D412yes—
379C42#12012yes—
380C42#12011yescharacteristic
381C42#12011yescharacteristic
382C42#12311yescharacteristic
383C42#12312yes—
384C42#12311yescharacteristic
385C23—22no—
386C23—22no—
387C23—22no—
388C23—41no—
389C23—41no—
390C23—42no—
391C23—42no—
392C23—44no—
393C23—44no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23C22×C22⋊C411yescharacteristic
398C23C2×#12112yes—
399C23C2×AGL(1,ℤ/8)12yes—
400C23C2×AGL(1,ℤ/8)11yescharacteristic
401C23C2×#13711yescharacteristic
402Q8—44no—
403Q8—84no—
404D4—44no—
405D4—44no—
406D4—44no—
407D4—84no—
408D4—82no—
409D4—816no—
410D4—84no—
411D4—84no—
412D4—82no—
413D4—84no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—88no—
445C2×C4—84no—
446C2×C4—84no—
447C2×C4—84no—
448C2×C4—84no—
449C2×C4—84no—
450C2×C4#38611yescharacteristic
451C2×C4#38612yes—
452C2×C4#38611yescharacteristic
453C2×C4#38611yescharacteristic
454C2×C4#38612yes—
455C2×C4#38611yescharacteristic
456C2×C4#41712yes—
457C2×C4#41712yes—
458C2×C4#41812yes—
459C2×C4#41812yes—
460C2×C4#46814yes—
461C2×C4#46814yes—
462C2×C4#46914yes—
463C2×C4#46914yes—
464C2×C4C2×C4×D412yes—
465C2×C4C2×#12011yescharacteristic
466C2×C4C2×#12311yescharacteristic
467C8—48no—
468C22—21no—
469C22—22no—
470C22—21no—
471C22—22no—
472C22—42no—
473C22—41no—
474C22—41no—
475C22—44no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22—82no—
480C22—84no—
481C22—84no—
482C22—82no—
483C22—82no—
484C22—82no—
485C22C2×#38611yescharacteristic
486C22C2×#41811yescharacteristic
487C22C2×#41711yescharacteristic
488C22C2×#46812yes—
489C22C2×#46912yes—
490C4—21no—
491C4—21no—
492C4—22no—
493C4—21no—
494C4—22no—
495C4—44no—
496C4—42no—
497C4—42no—
498C4—42no—
499C4—82no—
500C4—82no—
501C4—164no—
502C4#188412yes—
503C4#189512yes—
504C4#189512yes—
505C4#189514yes—
506C2—21no—
507C2—22no—
508C2—21no—
509C2—22no—
510C2—82no—
511C2—162no—
512C2C2×#188411yescharacteristic
513C2C2×#189511yescharacteristic
514C2#2101411yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#386
prime radical#34937C2
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. #20632C2semidirect
4. #17574C2semidirect
5. #9386C2semidirect
6. #8014C2semidirect
18. #2548C4semidirect
19. #2547C4semidirect
20. #2546C4semidirect
24. #2541C4semidirect
26. #2392C4semidirect
84. #431D4semidirect
85. #431D4semidirect

Maximal subgroups

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

subgroupindex
#602772
#349372
#206322
#175742
#93862
#80142