Finite Group Databasebeta

← #210669

Subgroups — #210669

Order 512.

720 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23677C211yesmaximal, characteristic
2#21714C211yesmaximal, characteristic, complemented
3#20495C211yesmaximal, characteristic
4C2×#1590C211yesmaximal, characteristic
5C2×#1583C211yesmaximal, characteristic
6#9673C211yesmaximal, characteristic, complemented
7#9602C211yesmaximal, characteristic, complemented
8#9547C211yesmaximal, characteristic, complemented
9#9547C211yesmaximal, characteristic, complemented
10#8945C211yesmaximal, characteristic
11#8945C211yesmaximal, characteristic
12#8567C211yesmaximal, characteristic
13#8262C211yesmaximal, characteristic, complemented
14#8211C211yesmaximal, characteristic, complemented
15C8×#103C211yesmaximal, characteristic, complemented
16C2×#446C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C4×SD16C2211yescharacteristic
19C2×C4×SD16C2211yescharacteristic
20C2×#436C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22C2×#405—21no—
23C4×C4⋊Q8C2211yescharacteristic
24#2293C2211yescharacteristic
25#2084C411yescharacteristic
26#2071C411yescharacteristic
27#2066C411yescharacteristic
28#2066C411yescharacteristic
29#2055C411yescharacteristic
30#2055C411yescharacteristic
31#2043C411yescharacteristic
32#2013C411yescharacteristic
33#1984C2211yescharacteristic
34#1984C2211yescharacteristic
35#1940—21no—
36#1940—21no—
37#1842C2211yescharacteristic
38#1771—21no—
39#1771—21no—
40#1770C2211yescharacteristic
41C8×C4⋊C4C2211yescharacteristic
42C4×C4⋊C8C2211yescharacteristic
43C4×C4⋊C8C2211yescharacteristic
44C4×C4⋊C8C2211yescharacteristic
45C4×#103C2211yescharacteristic
46C4×#103C2211yescharacteristic
47#1753—21no—
48#1752—21no—
49#1750—21no—
50C4×#97C2211yescharacteristic
51#1590—24no—
52#1583—24no—
53C2×#334C2211yescharacteristic
54C2×#334C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×C82C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×SD16C2×C411yescharacteristic
70C22×SD16C2×C411yescharacteristic
71C22×M16—21no—
72C23×C8—21no—
73C2×C4⋊Q8C2×C411yescharacteristic
74C2×C4×Q8C2311yescharacteristic
75C2×C4×Q8C2311yescharacteristic
76C2×C4×D4C2311yescharacteristic
77C2×#117—21no—
78#446—22no—
79#446—22no—
80C8×Q8—22no—
81C8×Q8—22no—
82C4×SD16—22no—
83C4×SD16—22no—
84C4×SD16—22no—
85C4×SD16—22no—
86#436—22no—
87#436D414yes—
88C8×D4—22no—
89C8×D4—22no—
90C2×#106C2×C411yescharacteristic
91C2×#106C2×C411yescharacteristic
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8C2×C411yescharacteristic
98C2×C4⋊C8C2×C411yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×#103C2×C411yescharacteristic
104C2×#103C2×C411yescharacteristic
105C2×#103C2×C411yescharacteristic
106C2×#103C2×C411yescharacteristic
107C2×#103C2×C411yescharacteristic
108C2×#103C2×C411yescharacteristic
109C2×#102C2×C411yescharacteristic
110C2×#102C2×C411yescharacteristic
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113#405—44no—
114C2×#97—21no—
115C2×#97—21no—
116C2×#97D411yescharacteristic
117C2×#97D411yescharacteristic
118C2×#97D411yescharacteristic
119C2×#97D411yescharacteristic
120C2×C4×C8—21no—
121C2×C4×C8—21no—
122C2×C4×C8—21no—
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129#391C2×C411yescharacteristic
130#391C2×C411yescharacteristic
131#390C2×C411yescharacteristic
132#390C2×C411yescharacteristic
133#389—21no—
134#389—21no—
135#389C2×C411yescharacteristic
136#389C2×C411yescharacteristic
137#388—21no—
138#388—21no—
139#384C2×C411yescharacteristic
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4C2311yescharacteristic
143C4×C22⋊C4—21no—
144C43C2311yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#340C2×C411yescharacteristic
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336C2×C411yescharacteristic
158#336C2×C411yescharacteristic
159#334—22no—
160#334—22no—
161Q8⋊C8—22no—
162Q8⋊C8—22no—
163Q8⋊C8—22no—
164Q8⋊C8—22no—
165D4⋊C8—22no—
166D4⋊C8—22no—
167#322—22no—
168C82—22no—
169C22×Q8C22×C411yescharacteristic
170C22×Q8C22×C411yescharacteristic
171C22×D4C22×C411yescharacteristic
172C23×C4—21no—
173C2×SD16—22no—
174C2×SD16—22no—
175C2×SD16—22no—
176C2×SD16—22no—
177C2×SD16—41no—
178C2×SD16—41no—
179C2×M16—24no—
180C2×M16—42no—
181C2×M16—42no—
182C22×C8—24no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—42no—
190C22×C8—42no—
191C22×C8C22×C411yescharacteristic
192C22×C8C22×C411yescharacteristic
193C22×C8C22×C411yescharacteristic
194C22×C8C22×C411yescharacteristic
195C22×C8C2×D411yescharacteristic
196C22×C8C2×D411yescharacteristic
197C22×C8C2×D411yescharacteristic
198C22×C8C2×D411yescharacteristic
199C4⋊Q8—22no—
200C4⋊Q8—41no—
201C4×Q8—21no—
202C4×Q8—21no—
203C4×Q8—21no—
204C4×Q8—21no—
205C4×Q8—21no—
206C4×Q8—21no—
207C4×Q8—21no—
208C4×Q8—21no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4C2×D412yes—
212C4×D4#4112yes—
213#117—42no—
214#117—42no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42C22×C411yescharacteristic
237C2×C42C22×C411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42#4111yescharacteristic
241C2×C42#4111yescharacteristic
242C2×C42C2411yescharacteristic
243#106—22no—
244#106—22no—
245C4⋊C8—22no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—41no—
253C4⋊C8—42no—
254C4⋊C8—42no—
255C4⋊C8—42no—
256C4⋊C8—42no—
257C4⋊C8—42no—
258C4⋊C8C2×D412yes—
259C4⋊C8#4112yes—
260#103—22no—
261#103—22no—
262#103—22no—
263#103—22no—
264#103—41no—
265#103—41no—
266#102—22no—
267#102—22no—
268C22⋊C8—24no—
269C22⋊C8—24no—
270#97—22no—
271#97—22no—
272#97—41no—
273#97—42no—
274#97—42no—
275#97SD1614yescomplemented
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8—42no—
287C4×C8—42no—
288C4×C8—42no—
289C4×C8C2×D412yes—
290C4×C8#4112yes—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298C24—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8#13112yes—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4C4×D412yes—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4C4×D411yescharacteristic
331C22×C4C4×D411yescharacteristic
332C22×C4C4×D411yescharacteristic
333C22×C4C4×D411yescharacteristic
334C22×C4C23×C411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C2×#4111yescharacteristic
337SD16—44no—
338SD16—44no—
339M16—48no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—22no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—22no—
351C2×C8—21no—
352C2×C8—22no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—48no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8#12112yes—
378C2×C8#12112yes—
379C2×C8#13112yes—
380C2×C8C2×SD1612yes—
381C2×C8C2×SD1612yes—
382C2×C8#13712yes—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—81no—
401C4⋊C4—81no—
402C4⋊C4C4×D412yes—
403C22⋊C4—22no—
404C22⋊C4—22no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—81no—
408C22⋊C4—81no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—41no—
419C42—41no—
420C42—41no—
421C42—41no—
422C42—41no—
423C42C2×SD1612yes—
424C42#13512yes—
425C42C22×D411yescharacteristic
426C42C2×#4111yescharacteristic
427C42C2×#4111yescharacteristic
428C42C2×#4111yescharacteristic
429C23—22no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23C2×C4×D411yescharacteristic
437Q8—22no—
438Q8—22no—
439Q8—41no—
440Q8—41no—
441Q8—41no—
442Q8—41no—
443Q8—41no—
444Q8—41no—
445D4—22no—
446D4—22no—
447D4—41no—
448D4—41no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4C4×SD1612yes—
487C2×C4C4×SD1612yes—
488C2×C4C2×C4×D411yescharacteristic
489C2×C4C4×#4111yescharacteristic
490C2×C4C2×#12111yescharacteristic
491C2×C4#52511yescharacteristic
492C2×C4#52911yescharacteristic
493C2×C4#53211yescharacteristic
494C2×C4C2×#13111yescharacteristic
495C2×C4C2×#13111yescharacteristic
496C2×C4#56811yescharacteristic
497C2×C4#56811yescharacteristic
498C2×C4C22×SD1611yescharacteristic
499C2×C4C2×#13511yescharacteristic
500C2×C4C2×#13711yescharacteristic
501C2×C4#57511yescharacteristic
502C8—22no—
503C8—22no—
504C8—22no—
505C8—42no—
506C8—42no—
507C8—41no—
508C8—41no—
509C8—41no—
510C8—41no—
511C8—42no—
512C8—81no—
513C8—81no—
514C8—81no—
515C8#46114yes—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—81no—
520C22—81no—
521C22—81no—
522C22—81no—
523C22C4×#12111yescharacteristic
524C22#293311yescharacteristic
525C22#293511yescharacteristic
526C22C2×C4×SD1611yescharacteristic
527C22C4×#13711yescharacteristic
528C22#295611yescharacteristic
529C22#295711yescharacteristic
530C4—21no—
531C4—21no—
532C4—22no—
533C4—22no—
534C4—41no—
535C4—41no—
536C4—42no—
537C4—42no—
538C4—81no—
539C4—81no—
540C4—81no—
541C4—81no—
542C4—81no—
543C4#293311yescharacteristic
544C4#293311yescharacteristic
545C4#296611yescharacteristic
546C4#296711yescharacteristic
547C4C2×#46111yescharacteristic
548C4#303511yescharacteristic
549C4#310111yescharacteristic
550C4#310711yescharacteristic
551C2—82no—
552C2#1799711yescharacteristic
553C2#1801211yescharacteristic
554C2#1806711yescharacteristic
555C2#1807911yescharacteristic
556C2#2149611yescharacteristic
557C2C4×#46111yescharacteristic
558C2#2151911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#97 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21714C2semidirect
6. #9673C2semidirect
7. #9602C2semidirect
8. #9547C2semidirect
9. #9547C2semidirect
13. #8262C2semidirect
14. #8211C2semidirect
15. C8×#103C2semidirect
275. #97SD16semidirect

Maximal subgroups

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

subgroupindex
#236772
#217142
#204952
C2×#15902
C2×#15832
#96732
#96022
#95472
#89452
#85672
#82622
#82112
C8×#1032