Finite Group Databasebeta

← #223003

Subgroups — #223003

Order 512.

980 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34932C211yesmaximal, characteristic
2#22799C211yesmaximal, characteristic, complemented
3#21264C211yesmaximal, characteristic, complemented
4#9870C211yesmaximal, characteristic, complemented
5#9865C211yesmaximal, characteristic, complemented
6#9074C212yesmaximal, complemented
7#8943C212yesmaximal, complemented
8#8710C212yesmaximal, complemented
9#8563C211yesmaximal, characteristic, complemented
10C4×D4⋊C8C211yesmaximal, characteristic, complemented
11#7900C212yesmaximal, complemented
12C2×#529—21no—
13C2×#432C2212yes—
14C2×#431C2212yes—
15#2553—22no—
16#2552C411yescharacteristic
17#2547C411yescharacteristic
18#2542C411yescharacteristic
19#2537C411yescharacteristic
20#2496C412yes—
21#2489—21no—
22#2427—21no—
23#2422C412yes—
24#2395—22no—
25C4×C4⋊Q8C2211yescharacteristic
26C4×C22⋊Q8—21no—
27C4×#121C2212yes—
28#2285C2212yes—
29C42×D4C2211yescharacteristic
30C42×D4C2211yescharacteristic
31#1954—21no—
32#1954—21no—
33#1939—21no—
34#1924—22no—
35#1923—22no—
36#1842C2211yescharacteristic
37#1797C2212yes—
38#1770—21no—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41C4×C22⋊C8C2212yes—
42C42×C8C2211yescharacteristic
43C2×D4⋊C8C2211yescharacteristic
44C2×D4⋊C8C2211yescharacteristic
45C2×D4⋊C8C2212yes—
46#1279C2211yescharacteristic
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52#1279C2211yescharacteristic
53C2×C4⋊Q8C2×C411yescharacteristic
54#529—44no—
55#529—44no—
56C2×#126—22no—
57C2×#125—21no—
58C2×#124—21no—
59C2×#123—22no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×#121—21no—
63C2×#121C2×C412yes—
64C2×C4×Q8—21no—
65C2×C4×D4—22no—
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4C2×C411yescharacteristic
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4D411yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×#117—22no—
73C22×C42—21no—
74C22×C42C2312yes—
75#432—24no—
76#432—42no—
77#431—24no—
78#431—42no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2312yes—
85C2×C4⋊C8C2311yescharacteristic
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×#97—22no—
91C2×C4×C8—21no—
92C2×C4×C8—21no—
93C2×C4×C8—21no—
94C2×C4×C8C2312yes—
95C2×C4×C8C2311yescharacteristic
96C2×C4×C8C2311yescharacteristic
97#400—22no—
98#399—22no—
99#398—22no—
100#397—22no—
101#397—22no—
102#397—22no—
103#394—22no—
104#394—22no—
105#391—21no—
106#391—21no—
107#390C2×C412yes—
108#389—21no—
109#389—21no—
110#389C2×C412yes—
111#389C2×C412yes—
112#388—22no—
113#387—22no—
114#386—22no—
115#385—22no—
116#384C2×C411yescharacteristic
117#384C2×C412yes—
118#384C2×C412yes—
119#382—22no—
120#380—22no—
121#380—21no—
122#380C2×C412yes—
123#380C2×C412yes—
124#380C2×C412yes—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4C2×C411yescharacteristic
128C4×C4⋊C4C2×C411yescharacteristic
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4D411yescharacteristic
131C4×C4⋊C4D411yescharacteristic
132C4×C4⋊C4C2312yes—
133C4×C4⋊C4C2311yescharacteristic
134C4×C22⋊C4—21no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4—21no—
138C4×C22⋊C4C2×C412yes—
139C4×C22⋊C4D412yes—
140C4×C22⋊C4D412yes—
141C43C2311yescharacteristic
142#336—22no—
143#336—21no—
144#336—22no—
145#336—21no—
146D4⋊C8—22no—
147D4⋊C8—22no—
148D4⋊C8—42no—
149C22×Q8—21no—
150C22×D4—22no—
151C22×D4C22⋊C411yescharacteristic
152C22×D4C22×C411yescharacteristic
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4C22×C412yes—
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167#126—48no—
168#125—44no—
169#124—44no—
170#123—48no—
171C22⋊Q8—44no—
172C22⋊Q8—44no—
173#121—24no—
174#121—42no—
175#121—44no—
176C4×Q8—42no—
177C4×Q8—42no—
178C4×D4—22no—
179C4×D4—22no—
180C4×D4—22no—
181C4×D4—22no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—41no—
190C4×D4—42no—
191C4×D4—44no—
192C4×D4—44no—
193#117—44no—
194#117—44no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4C22⋊C411yescharacteristic
205C2×C4⋊C4C22⋊C411yescharacteristic
206C2×C4⋊C4C22⋊C411yescharacteristic
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C22×C412yes—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4C22⋊C412yes—
222C2×C22⋊C4C22⋊C412yes—
223C2×C22⋊C4C22⋊C412yes—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—22no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42C22×C411yescharacteristic
235C2×C42C22×C412yes—
236C2×C42C22×C411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D412yes—
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D412yes—
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2411yescharacteristic
244C4⋊C8—21no—
245C4⋊C8—22no—
246C4⋊C8—21no—
247C4⋊C8—22no—
248C4⋊C8—21no—
249C4⋊C8—44no—
250C4⋊C8—42no—
251C4⋊C8—42no—
252C4⋊C8—81no—
253C4⋊C8#4112yes—
254C22⋊C8—44no—
255C22⋊C8—44no—
256C22⋊C8—44no—
257C22⋊C8—82no—
258#97—44no—
259C4×C8—21no—
260C4×C8—22no—
261C4×C8—21no—
262C4×C8—22no—
263C4×C8—21no—
264C4×C8—42no—
265C4×C8—42no—
266C4×C8—42no—
267C4×C8#4112yes—
268#95—21no—
269#95—21no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—21no—
276#95—22no—
277#95C22⋊C412yes—
278#95C22⋊C412yes—
279#95C22⋊C412yes—
280C24—21no—
281C24C2×C22⋊C412yes—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—44no—
294C2×D4—44no—
295C2×D4—82no—
296C2×D4#13112yes—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—81no—
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C2×C22⋊C412yes—
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C412yes—
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C23×C411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—42no—
356C2×C8—41no—
357C2×C8—42no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—22no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—44no—
378C4⋊C4—81no—
379C4⋊C4—82no—
380C22⋊C4—24no—
381C22⋊C4—24no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—44no—
385C22⋊C4—44no—
386C22⋊C4—44no—
387C22⋊C4—44no—
388C22⋊C4—44no—
389C22⋊C4—44no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—42no—
393C42—22no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—22no—
398C42—22no—
399C42—21no—
400C42—21no—
401C42—21no—
402C42—21no—
403C42—22no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—42no—
408C42—42no—
409C42—42no—
410C42—42no—
411C42#13112yes—
412C42#13112yes—
413C42AGL(1,ℤ/8)12yes—
414C42AGL(1,ℤ/8)12yes—
415C42C2×#4111yescharacteristic
416C42C2×#4111yescharacteristic
417C42H(5,𝔽2)11yescharacteristic
418C422−1+411yescharacteristic
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23C22×C22⋊C411yescharacteristic
434Q8—84no—
435D4—22no—
436D4—22no—
437D4—41no—
438D4—41no—
439D4—44no—
440D4—88no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—84no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—84no—
490C2×C4#40812yes—
491C2×C4#40812yes—
492C2×C4#41812yes—
493C2×C4#41812yes—
494C2×C4#42112yes—
495C2×C4C4×#4111yescharacteristic
496C2×C4#52011yescharacteristic
497C2×C4#54611yescharacteristic
498C2×C4#54611yescharacteristic
499C2×C4#54711yescharacteristic
500C2×C4#54711yescharacteristic
501C2×C4C2×#13111yescharacteristic
502C2×C4C2×#13111yescharacteristic
503C2×C4C2×#13111yescharacteristic
504C2×C4#56811yescharacteristic
505C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
506C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
507C2×C4#57711yescharacteristic
508C2×C4#57811yescharacteristic
509C8—42no—
510C8—42no—
511C8—42no—
512C8—82no—
513C8—82no—
514C8—81no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—81no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22#233911yescharacteristic
530C22C2×#40811yescharacteristic
531C22#288011yescharacteristic
532C22C2×#41811yescharacteristic
533C22#289811yescharacteristic
534C22C2×#42111yescharacteristic
535C22#290111yescharacteristic
536C4—22no—
537C4—22no—
538C4—21no—
539C4—21no—
540C4—41no—
541C4—42no—
542C4—42no—
543C4—42no—
544C4—81no—
545C4—82no—
546C4—81no—
547C4—82no—
548C4—81no—
549C4#149614yes—
550C4#296611yescharacteristic
551C4#296911yescharacteristic
552C4#299311yescharacteristic
553C4#299411yescharacteristic
554C4#333011yescharacteristic
555C4#333211yescharacteristic
556C4#335311yescharacteristic
557C4#335411yescharacteristic
558C2—82no—
559C2—82no—
560C2C2×#149611yescharacteristic
561C2#1748411yescharacteristic
562C2#1751111yescharacteristic
563C2#1752311yescharacteristic
564C2#2247411yescharacteristic
565C2#2250011yescharacteristic
566C2#2250711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radical#2537C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22799C2semidirect
3. #21264C2semidirect
4. #9870C2semidirect
5. #9865C2semidirect
6. #9074C2semidirect
7. #8943C2semidirect
8. #8710C2semidirect
9. #8563C2semidirect
10. C4×D4⋊C8C2semidirect
11. #7900C2semidirect

Maximal subgroups

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

subgroupindex
#349322
#227992
#212642
#98702
#98652
#90742
#89432
#87102
#85632
C4×D4⋊C82
#79002