Finite Group Databasebeta

← #215399

Subgroups — #215399

Order 512.

1420 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic
3#14685C211yesmaximal, characteristic
4#9714C214yesmaximal, complemented
5#9679C214yesmaximal, complemented
6#8329C214yesmaximal
7C22×C4⋊Q8C411yescharacteristic, complemented
8C22×C22⋊Q8C2212yes—
9C23×C42C2211yescharacteristic
10C2×#431—21no—
11#2857—21no—
12#2857—21no—
13#2857—21no—
14#2857—21no—
15#2857C412yescomplemented
16#2857C412yescomplemented
17#2857C2212yes—
18#2857C2212yes—
19C2×#391C412yescomplemented
20C2×#389—21no—
21C2×#389—21no—
22C2×#384C411yescharacteristic, complemented
23C2×#381—21no—
24C2×#380—21no—
25C2×#380—21no—
26C2×#380—21no—
27#1817—21no—
28#1797—21no—
29#1797C2212yes—
30#1779C2212yes—
31C4×C22⋊C8—21no—
32C4×C22⋊C8C2212yes—
33C2×#328C22116yes—
34C4⋊#95—21no—
35C4⋊#95C2212yes—
36#1434—21no—
37#1434C2212yes—
38C22⋊#95C2212yes—
39C23×Q8C2×C412yes—
40C24×C4D411yescharacteristic
41C24×C4D411yescharacteristic
42C24×C4C2311yescharacteristic
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—44no—
46C2×C4⋊Q8C2×C412yes—
47C2×C4⋊Q8C2×C412yes—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—48no—
51C2×C22⋊Q8C2×C414yes—
52C2×C22⋊Q8C2314yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4C2×C412yes—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C42—44no—
63C22×C42C2×C411yescharacteristic
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75#431—44no—
76C2×C4⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8C2314yes—
80C2×C4×C8—22no—
81#391—22no—
82#391—22no—
83#391C2×C414yes—
84#391C2×C414yes—
85#389—22no—
86#389—22no—
87#389—22no—
88#389—22no—
89#389—22no—
90#389—22no—
91#389—22no—
92#389—22no—
93#384—21no—
94#384—21no—
95#384C2×C412yes—
96#384C2×C412yes—
97#381—44no—
98#380—22no—
99#380—22no—
100#380—22no—
101#380—22no—
102#380—22no—
103#380—22no—
104#380—22no—
105#380—22no—
106#380—44no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95C2314yes—
114#336—22no—
115#336—22no—
116#328—416no—
117C25C2×D411yescharacteristic
118C22×Q8—22no—
119C22×Q8—44no—
120C22×Q8—44no—
121C22×Q8—44no—
122C22×Q8C22×C414yes—
123C23×C4—24no—
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—24no—
129C23×C4—42no—
130C23×C4—42no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4C22⋊C411yescharacteristic
135C23×C4C22⋊C411yescharacteristic
136C23×C4C22⋊C411yescharacteristic
137C23×C4C22⋊C411yescharacteristic
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C411yescharacteristic
142C23×C4SD1614yes—
143C23×C4Q1614yes—
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2411yescharacteristic
154C22×C8—24no—
155C4⋊Q8—44no—
156C4⋊Q8—44no—
157C4⋊Q8—44no—
158C4⋊Q8—44no—
159C22⋊Q8—216no—
160C22⋊Q8—48no—
161C22⋊Q8—48no—
162C22⋊Q8—48no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—48no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4C22×C414yes—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—48no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C42—24no—
191C2×C42—24no—
192C2×C42—42no—
193C2×C42—42no—
194C2×C42—42no—
195C2×C42—42no—
196C2×C42—42no—
197C2×C42—42no—
198C2×C42—42no—
199C2×C42—42no—
200C2×C42—42no—
201C2×C42—42no—
202C2×C42C22⋊C411yescharacteristic
203C2×C42C22⋊C411yescharacteristic
204C2×C42C22⋊C411yescharacteristic
205C2×C42C22⋊C411yescharacteristic
206C2×C42C22⋊C411yescharacteristic
207C2×C42C22⋊C411yescharacteristic
208C2×C42C22⋊C411yescharacteristic
209C2×C42C22⋊C411yescharacteristic
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C411yescharacteristic
213C2×C42C22×C411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C4⋊C8—44no—
219C22⋊C8—28no—
220C22⋊C8—44no—
221C22⋊C8—44no—
222C4×C8—44no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—24no—
227#95—28no—
228#95—44no—
229#95—44no—
230#95—44no—
231#95—44no—
232C24—22no—
233C24—22no—
234C24—22no—
235C24—22no—
236C24—41no—
237C24—41no—
238C24C2×C22⋊C411yescharacteristic
239C24C2×C22⋊C411yescharacteristic
240C24C2×SD1612yes—
241C24C2×Q1612yes—
242C24C22×D411yescharacteristic
243C2×Q8—28no—
244C2×Q8—44no—
245C2×Q8—44no—
246C2×Q8—44no—
247C2×Q8—44no—
248C2×Q8—44no—
249C2×Q8—816no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—48no—
292C22×C4—48no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—416no—
300C22×C4—416no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4#9914yes—
315C22×C4#10314yes—
316C22×C4#10314yes—
317C22×C4#10314yes—
318C22×C4#10314yes—
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×SD1612yes—
330C22×C4C2×SD1612yes—
331C22×C4C2×Q1612yes—
332C22×C4C2×Q1612yes—
333C22×C4#13714yes—
334C22×C4C23×C411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4H(5,𝔽2)14yes—
337C2×C8—44no—
338C2×C8—44no—
339C2×C8—44no—
340C4⋊C4—28no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—48no—
346C4⋊C4—44no—
347C4⋊C4—44no—
348C4⋊C4—48no—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—88no—
353C22⋊C4—48no—
354C22⋊C4—48no—
355C22⋊C4—88no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42—41no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—24no—
387C23—24no—
388C23—24no—
389C23—24no—
390C23—42no—
391C23—42no—
392C23—44no—
393C23—44no—
394C23—44no—
395C23—44no—
396C23C2×#9912yes—
397C23C2×#10312yes—
398C23C2×#10312yes—
399C23#42112yes—
400C23C22×C22⋊C411yescharacteristic
401C23#53512yes—
402C23C22×SD1611yescharacteristic
403C23C22×Q1611yescharacteristic
404C23C2×#13712yes—
405Q8—88no—
406Q8—88no—
407C2×C4—28no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—28no—
427C2×C4—28no—
428C2×C4—28no—
429C2×C4—28no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—88no—
458C2×C4—88no—
459C2×C4—84no—
460C2×C4—84no—
461C2×C4—84no—
462C2×C4—84no—
463C2×C4—84no—
464C2×C4—84no—
465C2×C4—84no—
466C2×C4C2×#9912yes—
467C2×C4C2×#9912yes—
468C2×C4#41012yes—
469C2×C4C2×#10312yes—
470C2×C4C2×#10312yes—
471C2×C4C2×#10312yes—
472C2×C4C2×#10312yes—
473C2×C4#41912yes—
474C2×C4#41912yes—
475C2×C4#42112yes—
476C2×C4#51812yes—
477C2×C4#53512yes—
478C8—84no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—48no—
497C22—48no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22#152014yes—
503C22#230911yescharacteristic
504C22C22×#9911yescharacteristic
505C22C2×#41011yescharacteristic
506C22C22×#10311yescharacteristic
507C22C2×#41911yescharacteristic
508C22C2×#42111yescharacteristic
509C22C2×#42111yescharacteristic
510C22#320612yes—
511C22#320912yes—
512C4—22no—
513C4—22no—
514C4—22no—
515C4—22no—
516C4—22no—
517C4—22no—
518C4—22no—
519C4—22no—
520C4—22no—
521C4—41no—
522C4—41no—
523C4—84no—
524C4—84no—
525C4—164no—
526C4#151114yes—
527C2—22no—
528C2—22no—
529C2—24no—
530C2—42no—
531C2C2×#151111yescharacteristic
532C2C2×#152011yescharacteristic
533C2#1757511yescharacteristic
534C2#1758411yescharacteristic
535C2#2179311yescharacteristic
536C2#2180411yescharacteristic
537C2#2181611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9714C2semidirect
5. #9679C2semidirect
7. C22×C4⋊Q8C4semidirect
15. #2857C4semidirect
16. #2857C4semidirect
19. C2×#391C4semidirect
22. C2×#384C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
#207142
#146852
#97142
#96792
#83292