Finite Group Databasebeta

← #190219

Subgroups — #190219

Order 512.

1119 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4⋊C4×#41C211yesmaximal, characteristic
2#20723C211yesmaximal, characteristic
3#17363C211yesmaximal, characteristic
4#8761C214yesmaximal, complemented
5#8760C214yesmaximal, complemented
6#8021C212yesmaximal, complemented
7#7984C212yesmaximal, complemented
8C2×C4×#41—21no—
9C2×C4×#41C2211yescharacteristic
10C2×#432C2212yes—
11#2351C412yescomplemented
12Q8×C4⋊C4—21no—
13#2349C412yescomplemented
14#2349C2212yes—
15#2349C2212yes—
16D4×C4⋊C4—21no—
17D4×C4⋊C4—21no—
18D4×C4⋊C4—21no—
19#2296—21no—
20#2296—21no—
21#2296—21no—
22#2285C411yescharacteristic, complemented
23#2285C411yescharacteristic, complemented
24#2285C2211yescharacteristic
25#2283C2212yes—
26C2×C4×C4⋊C4C411yescharacteristic, complemented
27C2×C4×C4⋊C4C411yescharacteristic, complemented
28C2×C4×C4⋊C4C2211yescharacteristic
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
44C4×#41—22no—
45C4×#41—22no—
46C4×#41—22no—
47C4×#41—22no—
48C4×#41—48no—
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×C42—21no—
71C22×C42—21no—
72C22×C42—21no—
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42C2311yescharacteristic
76#432—22no—
77#432—22no—
78#432—22no—
79#432—22no—
80C2×C4⋊C8—22no—
81C2×#99—22no—
82C2×#99—22no—
83C2×C22⋊C8C2314yes—
84C2×#97—22no—
85#389—22no—
86#389—22no—
87#389—22no—
88#389—22no—
89#389—22no—
90#389—22no—
91#385—22no—
92#385—22no—
93#385—22no—
94#385—22no—
95#385—22no—
96#385—22no—
97#384—21no—
98#384—21no—
99#384—21no—
100#383—22no—
101#382—22no—
102#382C2×C412yes—
103#382C2×C412yes—
104#382C2×C412yes—
105#381—22no—
106#380—21no—
107#380—21no—
108#380—21no—
109#380C2×C412yes—
110#380C2×C412yes—
111#380C2314yes—
112#380C2312yes—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4C2×C412yes—
126C4×C4⋊C4C2×C411yescharacteristic
127C4×C4⋊C4C2×C411yescharacteristic
128C4×C4⋊C4C2×C411yescharacteristic
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C22⋊C4—22no—
131#376—22no—
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—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
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—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—21no—
239C2×C42—44no—
240C2×C42—44no—
241C2×C42—44no—
242C2×C42—42no—
243C2×C42C22⋊C411yescharacteristic
244C2×C42C22⋊C411yescharacteristic
245C2×C42C22⋊C411yescharacteristic
246C2×C42C22⋊C411yescharacteristic
247C2×C42C22×C411yescharacteristic
248C2×C42C22×C411yescharacteristic
249C2×C42C22×C411yescharacteristic
250C2×C42C22×C411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42#4112yes—
254C2×C42#4112yes—
255C2×C42#4112yes—
256C2×C42#4112yes—
257C4⋊C8—42no—
258C4⋊C8—42no—
259#99—44no—
260#99—44no—
261C22⋊C8—24no—
262C22⋊C8—24no—
263#97—42no—
264#97—42no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—24no—
272#95—42no—
273#95—42no—
274#95—42no—
275#95C22×C412yes—
276#95C22×C412yes—
277#95C22×C412yes—
278#95C22×C412yes—
279C24C2×C22⋊C411yescharacteristic
280C24C2×C22⋊C411yescharacteristic
281C24C22×D411yescharacteristic
282#41—216no—
283#41—216no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4AGL(1,ℤ/8)14yes—
295C2×D4#13714yes—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—81no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4—82no—
332C22×C4—81no—
333C22×C4—81no—
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×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4#12312yes—
345C22×C4#12312yes—
346C22×C4#12312yes—
347C22×C4#12312yes—
348C22×C4C23×C411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C2×#4112yes—
351C22×C4C2×#4112yes—
352C2×C8—44no—
353C2×C8—84no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—82no—
371C4⋊C4—82no—
372C4⋊C4—82no—
373C4⋊C4—84no—
374C22⋊C4—22no—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C22⋊C4—44no—
381C22⋊C4—44no—
382C22⋊C4—44no—
383C22⋊C4—44no—
384C22⋊C4—44no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—42no—
398C42—42no—
399C42—42no—
400C42—42no—
401C42—44no—
402C42—44no—
403C42—44no—
404C42—84no—
405C42#11711yescharacteristic
406C42#11711yescharacteristic
407C42#11711yescharacteristic
408C42#11711yescharacteristic
409C42#11712yes—
410C42#11712yes—
411C42#12312yes—
412C42#12312yes—
413C42#12312yes—
414C42#12312yes—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23#41812yes—
426C23#41912yes—
427C23C22×C22⋊C411yescharacteristic
428C23C2×#12312yes—
429C23C2×AGL(1,ℤ/8)12yes—
430C23C2×#13712yes—
431Q8—28no—
432D4—28no—
433D4—28no—
434D4—28no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—88no—
486C2×C4—88no—
487C2×C4—88no—
488C2×C4—84no—
489C2×C4—84no—
490C2×C4—84no—
491C2×C4#38111yescharacteristic
492C2×C4#38111yescharacteristic
493C2×C4#38111yescharacteristic
494C2×C4#38112yes—
495C2×C4#38112yes—
496C2×C4#38111yescharacteristic
497C2×C4#41812yes—
498C2×C4#41912yes—
499C2×C4C2×#11711yescharacteristic
500C2×C4C2×#11711yescharacteristic
501C2×C4C2×#12312yes—
502C8—84no—
503C22—21no—
504C22—22no—
505C22—21no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22C2×#38111yescharacteristic
520C22C2×#41811yescharacteristic
521C22C2×#41911yescharacteristic
522C22#309214yes—
523C4—21no—
524C4—21no—
525C4—22no—
526C4—22no—
527C4—22no—
528C4—21no—
529C4—22no—
530C4—22no—
531C4—21no—
532C4—21no—
533C4—44no—
534C4—41no—
535C4—41no—
536C4—41no—
537C4—41no—
538C4—82no—
539C4—82no—
540C4—82no—
541C4—164no—
542C4#182712yes—
543C2—21no—
544C2—24no—
545C2—21no—
546C2—42no—
547C2—42no—
548C2#2070011yescharacteristic
549C2C2×#182711yescharacteristic
550C2#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. #8021C2semidirect
7. #7984C2semidirect
11. #2351C4semidirect
13. #2349C4semidirect
22. #2285C4semidirect
23. #2285C4semidirect
26. C2×C4×C4⋊C4C4semidirect
27. C2×C4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×#412
#207232
#173632
#87612
#87602
#80212
#79842