Finite Group Databasebeta

← #190225

Subgroups — #190225

Order 512.

1058 conjugacy classes of subgroups.

The subgroup lattice has 545 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34184C211yesmaximal, characteristic
2#20723C211yesmaximal, characteristic
3#17363C211yesmaximal, characteristic
4#8761C214yesmaximal, complemented
5#8760C214yesmaximal, complemented
6#8017C212yesmaximal, complemented
7#7984C212yesmaximal, complemented
8C2×#520—21no—
9C2×C4×#41C2211yescharacteristic
10C2×#432C2212yes—
11#2351—21no—
12#2351C412yescomplemented
13#2349—21no—
14#2349—21no—
15#2349—21no—
16#2349C412yescomplemented
17#2349C2212yes—
18#2349C2212yes—
19#2317—21no—
20#2317—21no—
21#2317—21no—
22#2285C411yescharacteristic, complemented
23#2285C411yescharacteristic, complemented
24#2285C2211yescharacteristic
25#2283C2212yes—
26C2×#382C411yescharacteristic, complemented
27C2×#382C411yescharacteristic, complemented
28C2×#382C2211yescharacteristic
29#1820—21no—
30#1799C2212yes—
31#1799C2212yes—
32#1782—21no—
33C4×#99—21no—
34C4×#99—21no—
35#1467—21no—
36#1466—21no—
37#1464—21no—
38#1464—21no—
39#1459C2212yes—
40#1459C2212yes—
41#1294C2218yes—
42#1289C2218yes—
43C22×#41D411yescharacteristic
44#520—48no—
45C4×#41—22no—
46C4×#41—22no—
47C4×#41—22no—
48C4×#41—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8C2×C411yescharacteristic
51C2×C4×Q8D411yescharacteristic
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4C2×C411yescharacteristic
56C2×C4×D4D411yescharacteristic
57C2×C4×D4D411yescharacteristic
58C2×C4×D4D411yescharacteristic
59C2×C4×D4C2311yescharacteristic
60C2×C4×D4C2311yescharacteristic
61C2×#117—21no—
62C2×#117—21no—
63C2×#117—21no—
64C2×#117D411yescharacteristic
65C2×#117D411yescharacteristic
66C2×#117D411yescharacteristic
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C42C2×C411yescharacteristic
71C22×C42C2×C411yescharacteristic
72C22×C42C2311yescharacteristic
73#432—22no—
74#432—22no—
75#432—22no—
76#432—22no—
77C2×C4⋊C8—22no—
78C2×#99—22no—
79C2×#99—22no—
80C2×C22⋊C8C2314yes—
81C2×#97—22no—
82#389—22no—
83#389—22no—
84#389—22no—
85#389—22no—
86#389—22no—
87#389—22no—
88#385—22no—
89#385—22no—
90#385—22no—
91#385—22no—
92#385—22no—
93#385—22no—
94#384—21no—
95#384—21no—
96#384—21no—
97#383—22no—
98#382—21no—
99#382—21no—
100#382—21no—
101#382—21no—
102#382—21no—
103#382—21no—
104#382—22no—
105#382C2×C412yes—
106#382C2×C412yes—
107#382C2×C412yes—
108#382C2×C411yescharacteristic
109#382C2×C411yescharacteristic
110#382C2×C411yescharacteristic
111#382C2×C411yescharacteristic
112#381—22no—
113#380—21no—
114#380—21no—
115#380—21no—
116#380C2×C412yes—
117#380C2×C412yes—
118#380C2314yes—
119#380C2312yes—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4C2×C412yes—
126C4×C4⋊C4C2×C412yes—
127C4×C22⋊C4—22no—
128#376—22no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95C2×C412yes—
133C2×#95C2×C412yes—
134C2×#95C2312yes—
135#342—22no—
136#342—22no—
137#324—22no—
138#324—22no—
139#323—22no—
140#323—22no—
141C2×#41—24no—
142C2×#41—24no—
143C2×#41—22no—
144C2×#41—22no—
145C22×Q8C22⋊C411yescharacteristic
146C22×D4C22⋊C411yescharacteristic
147C22×D4C2×D411yescharacteristic
148C22×D4C2×D411yescharacteristic
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4C22⋊C411yescharacteristic
153C23×C4C22⋊C411yescharacteristic
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2411yescharacteristic
161C22×C8—24no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—48no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—22no—
171C4×D4—22no—
172C4×D4—22no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4—22no—
176C4×D4—22no—
177C4×D4—22no—
178C4×D4—22no—
179C4×D4—48no—
180C4×D4—48no—
181C4×D4—48no—
182#117—22no—
183#117—22no—
184#117—22no—
185#117—22no—
186#117—22no—
187#117—22no—
188#117—44no—
189#117—44no—
190#117—44no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4C22⋊C411yescharacteristic
209C2×C4⋊C4C22⋊C411yescharacteristic
210C2×C4⋊C4C22⋊C411yescharacteristic
211C2×C4⋊C4C22⋊C411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4C22⋊C411yescharacteristic
219C2×C22⋊C4C22⋊C411yescharacteristic
220C2×C22⋊C4C2×D411yescharacteristic
221C2×C22⋊C4C2×D411yescharacteristic
222C2×C22⋊C4C2×D411yescharacteristic
223C2×C22⋊C4C2×D411yescharacteristic
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—42no—
236C2×C42C22⋊C411yescharacteristic
237C2×C42C22⋊C411yescharacteristic
238C2×C42C22⋊C411yescharacteristic
239C2×C42C22⋊C411yescharacteristic
240C2×C42C22×C411yescharacteristic
241C2×C42C22×C411yescharacteristic
242C2×C42C22×C411yescharacteristic
243C2×C42C22×C411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42#4112yes—
247C2×C42#4112yes—
248C2×C42#4112yes—
249C2×C42#4112yes—
250C4⋊C8—42no—
251C4⋊C8—42no—
252#99—44no—
253#99—44no—
254C22⋊C8—24no—
255C22⋊C8—24no—
256#97—42no—
257#97—42no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—21no—
267#95—22no—
268#95—24no—
269#95—42no—
270#95—42no—
271#95—42no—
272#95C22×C412yes—
273#95C22×C412yes—
274#95C22×C412yes—
275#95C22×C412yes—
276C24C2×C22⋊C411yescharacteristic
277C24C2×C22⋊C411yescharacteristic
278C24C22×D411yescharacteristic
279#41—216no—
280#41—216no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4AGL(1,ℤ/8)14yes—
292C2×D4#13714yes—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—81no—
320C22×C4—81no—
321C22×C4—81no—
322C22×C4—81no—
323C22×C4—81no—
324C22×C4—81no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4—82no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4#12312yes—
342C22×C4#12312yes—
343C22×C4#12312yes—
344C22×C4#12312yes—
345C22×C4C23×C411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C2×#4112yes—
348C22×C4C2×#4112yes—
349C2×C8—44no—
350C2×C8—84no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—42no—
363C4⋊C4—44no—
364C4⋊C4—82no—
365C4⋊C4—82no—
366C4⋊C4—82no—
367C4⋊C4—82no—
368C4⋊C4—82no—
369C4⋊C4—82no—
370C4⋊C4—84no—
371C22⋊C4—22no—
372C22⋊C4—22no—
373C22⋊C4—22no—
374C22⋊C4—22no—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C22⋊C4—44no—
378C22⋊C4—44no—
379C22⋊C4—44no—
380C22⋊C4—44no—
381C22⋊C4—44no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—44no—
395C42—44no—
396C42—44no—
397C42—84no—
398C42#11711yescharacteristic
399C42#11711yescharacteristic
400C42#11711yescharacteristic
401C42#11711yescharacteristic
402C42#11712yes—
403C42#11712yes—
404C42#12312yes—
405C42#12312yes—
406C42#12312yes—
407C42#12312yes—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23#41812yes—
419C23#41912yes—
420C23C22×C22⋊C411yescharacteristic
421C23C2×#12312yes—
422C23C2×AGL(1,ℤ/8)12yes—
423C23C2×#13712yes—
424Q8—28no—
425D4—28no—
426D4—28no—
427D4—28no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
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—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—84no—
479C2×C4—84no—
480C2×C4—84no—
481C2×C4—164no—
482C2×C4—164no—
483C2×C4—164no—
484C2×C4#38111yescharacteristic
485C2×C4#38111yescharacteristic
486C2×C4#38111yescharacteristic
487C2×C4#38111yescharacteristic
488C2×C4#38112yes—
489C2×C4#38112yes—
490C2×C4#41812yes—
491C2×C4#41912yes—
492C2×C4C2×#11711yescharacteristic
493C2×C4C2×#11711yescharacteristic
494C2×C4C2×#12312yes—
495C8—84no—
496C22—21no—
497C22—22no—
498C22—21no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22C2×#38111yescharacteristic
513C22C2×#41811yescharacteristic
514C22C2×#41911yescharacteristic
515C22#309214yes—
516C4—21no—
517C4—21no—
518C4—22no—
519C4—22no—
520C4—22no—
521C4—21no—
522C4—21no—
523C4—22no—
524C4—22no—
525C4—21no—
526C4—21no—
527C4—41no—
528C4—41no—
529C4—41no—
530C4—41no—
531C4—82no—
532C4—82no—
533C4—82no—
534C4—82no—
535C4—164no—
536C2—21no—
537C2—24no—
538C2—21no—
539C2—42no—
540C2—42no—
541C2#2070011yescharacteristic
542C2#2070811yescharacteristic
543C2#2071011yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8761C2semidirect
5. #8760C2semidirect
6. #8017C2semidirect
7. #7984C2semidirect
12. #2351C4semidirect
16. #2349C4semidirect
22. #2285C4semidirect
23. #2285C4semidirect
26. C2×#382C4semidirect
27. C2×#382C4semidirect

Maximal subgroups

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

subgroupindex
#341842
#207232
#173632
#87612
#87602
#80172
#79842