Finite Group Databasebeta

← #229846

Subgroups — #229846

Order 512.

716 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22956C211yesmaximal, characteristic
2#22952C211yesmaximal, characteristic
3C4×#467C211yesmaximal, characteristic
4C2×#1587C211yesmaximal, characteristic, complemented
5C22×#334C211yesmaximal, characteristic
6#9664C211yesmaximal, characteristic, complemented
7#9599C211yesmaximal, characteristic, complemented
8#9547C211yesmaximal, characteristic, complemented
9#9543C211yesmaximal, characteristic, complemented
10#8937C211yesmaximal, characteristic
11#8937C211yesmaximal, characteristic
12#8850C211yesmaximal, characteristic, complemented
13#8846C211yesmaximal, characteristic, complemented
14#8288C211yesmaximal, characteristic, complemented
15#8145C211yesmaximal, characteristic
16C2×#467C411yescharacteristic
17C2×#446C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23C4×C22⋊Q8C2211yescharacteristic
24C4×C22⋊Q8C2211yescharacteristic
25#1984C2211yescharacteristic
26#1984C2211yescharacteristic
27#1969C411yescharacteristic
28#1942C411yescharacteristic
29#1940—21no—
30#1939—21no—
31#1939C2211yescharacteristic
32#1939C2211yescharacteristic
33#1933C411yescharacteristic
34#1933C411yescharacteristic
35#1917—21no—
36#1916—21no—
37#1892C411yescharacteristic
38#1892C411yescharacteristic
39#1850—21no—
40#1850—21no—
41#1808C411yescharacteristic
42C4×#107C2211yescharacteristic
43#1770C2211yescharacteristic
44C8×C4⋊C4C2211yescharacteristic
45C4×#103C2211yescharacteristic
46C4×#103C2211yescharacteristic
47#1752—21no—
48C8×C22⋊C4—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51#1587—24no—
52C2×#334—24no—
53C2×#334C2211yescharacteristic
54C2×#334C2211yescharacteristic
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×#322C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×Q16C2×C411yescharacteristic
70C23×C8—21no—
71C23×C8—21no—
72C23×C8C2×C411yescharacteristic
73C23×C8C2×C411yescharacteristic
74C2×C22⋊Q8C2×C411yescharacteristic
75C2×C22⋊Q8C2×C411yescharacteristic
76C2×C4×Q8C2311yescharacteristic
77C2×C4×Q8C2311yescharacteristic
78C22×C42C2311yescharacteristic
79#467—24no—
80#446—22no—
81#446—41no—
82C8×Q8—22no—
83C8×Q8—41no—
84C4×Q16—22no—
85C4×Q16—22no—
86C2×#107C2×C411yescharacteristic
87C2×#107C2×C411yescharacteristic
88C2×#107C2×C411yescharacteristic
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—41no—
98C2×C4⋊C8—41no—
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×C22⋊C8—21no—
110C2×C22⋊C8D411yescharacteristic
111C2×C22⋊C8D411yescharacteristic
112C2×#97—21no—
113C2×#97—21no—
114C2×C4×C8—22no—
115C2×C4×C8—22no—
116C2×C4×C8—21no—
117C2×C4×C8—21no—
118C2×C4×C8C2×C411yescharacteristic
119C2×C4×C8C2×C411yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126#391C2×C411yescharacteristic
127#391C2×C411yescharacteristic
128#389C2×C411yescharacteristic
129#389C2×C411yescharacteristic
130#387—21no—
131#387—21no—
132#385—21no—
133#385—21no—
134#380C2×C411yescharacteristic
135#380C2×C411yescharacteristic
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4C2311yescharacteristic
139C4×C4⋊C4C2311yescharacteristic
140C4×C22⋊C4—21no—
141C4×C22⋊C4—21no—
142#340C2×C411yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#336—21no—
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#336D411yescharacteristic
158#336D411yescharacteristic
159#334—22no—
160#334—22no—
161#334—22no—
162#334D414yes—
163Q8⋊C8—22no—
164Q8⋊C8—22no—
165Q8⋊C8—41no—
166Q8⋊C8—41no—
167#322—22no—
168#322—22no—
169C22×Q8C22×C411yescharacteristic
170C22×Q8C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4#4111yescharacteristic
174C2×Q16—22no—
175C2×Q16—22no—
176C2×Q16—41no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—22no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—41no—
189C22×C8—42no—
190C22×C8—42no—
191C22×C8—42no—
192C22×C8—42no—
193C22×C8—81no—
194C22×C8—81no—
195C22×C8C22×C411yescharacteristic
196C22×C8C22×C411yescharacteristic
197C22×C8C22×C411yescharacteristic
198C22×C8C22×C411yescharacteristic
199C22×C8C2×D411yescharacteristic
200C22×C8C2×D411yescharacteristic
201C22×C8C2×D411yescharacteristic
202C22×C8C2×D411yescharacteristic
203C22⋊Q8—22no—
204C22⋊Q8—22no—
205C22⋊Q8—22no—
206C22⋊Q8M1614yes—
207C4×Q8—21no—
208C4×Q8—21no—
209C4×Q8—21no—
210C4×Q8—21no—
211C4×Q8—21no—
212C4×Q8—21no—
213C4×Q8—21no—
214C4×Q8—21no—
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⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42C22×C411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42#4111yescharacteristic
242C2×C42C2411yescharacteristic
243#107—22no—
244#107—22no—
245#107—41no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—42no—
253C4⋊C8—42no—
254C4⋊C8—42no—
255C4⋊C8—42no—
256C4⋊C8—81no—
257C4⋊C8—81no—
258C4⋊C8C2×D412yes—
259C4⋊C8C2×D412yes—
260#103—22no—
261#103—22no—
262#103—22no—
263#103—22no—
264#103—22no—
265#103—22no—
266C22⋊C8—24no—
267C22⋊C8—22no—
268C22⋊C8—22no—
269#97—42no—
270#97—81no—
271C4×C8—22no—
272C4×C8—22no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—21no—
283C4×C8—42no—
284C4×C8—81no—
285C4×C8C2×D412yes—
286C4×C8#4112yes—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297C24C4×D411yescharacteristic
298C2×Q8—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×Q8C2×M1612yes—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—41no—
330C22×C4C4×D411yescharacteristic
331C22×C4C4×D411yescharacteristic
332C22×C4C4×D411yescharacteristic
333C22×C4C2×M1612yes—
334C22×C4#13112yes—
335C22×C4#13712yes—
336C22×C4#13712yes—
337C22×C4C23×C411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C2×#4111yescharacteristic
340Q16—44no—
341C2×C8—28no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—44no—
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—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—84no—
383C2×C8—84no—
384C2×C8C4×D412yes—
385C2×C8C4×D412yes—
386C2×C8#13112yes—
387C4⋊C4—21no—
388C4⋊C4—24no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—42no—
404C4⋊C4—81no—
405C4⋊C4—81no—
406C4⋊C4C2×M1612yes—
407C22⋊C4—24no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—41no—
418C42—41no—
419C42—41no—
420C42—41no—
421C42—41no—
422C42—41no—
423C42—41no—
424C42—41no—
425C42C22×D411yescharacteristic
426C42C2×#4111yescharacteristic
427C42H(5,𝔽2)11yescharacteristic
428C422−1+411yescharacteristic
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23#43512yes—
434C23#44112yes—
435C23#44312yes—
436C23C2×C4×D411yescharacteristic
437Q8—22no—
438Q8—22no—
439Q8—41no—
440Q8—41no—
441Q8—41no—
442Q8—41no—
443Q8—41no—
444Q8—41no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
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×C4—41no—
487C2×C4—41no—
488C2×C4#43512yes—
489C2×C4#44112yes—
490C2×C4C2×C4×D411yescharacteristic
491C2×C4#52011yescharacteristic
492C2×C4D4211yescharacteristic
493C2×C4#54711yescharacteristic
494C2×C4#54811yescharacteristic
495C2×C4#55311yescharacteristic
496C2×C4C22×M1611yescharacteristic
497C2×C4C2×#13111yescharacteristic
498C2×C4C2×#13111yescharacteristic
499C2×C4#56811yescharacteristic
500C2×C4C2×#13711yescharacteristic
501C2×C4C2×#13711yescharacteristic
502C2×C4#57711yescharacteristic
503C2×C4#57711yescharacteristic
504C8—22no—
505C8—22no—
506C8—22no—
507C8—42no—
508C8—42no—
509C8—41no—
510C8—41no—
511C8—41no—
512C8—41no—
513C8—41no—
514C8—42no—
515C8—42no—
516C8—81no—
517C8—81no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22#158714yescomplemented
525C22#236111yescharacteristic
526C22C2×#43511yescharacteristic
527C22#293511yescharacteristic
528C22C2×#44111yescharacteristic
529C22#295211yescharacteristic
530C22C2×#44311yescharacteristic
531C22#295711yescharacteristic
532C4—22no—
533C4—21no—
534C4—21no—
535C4—22no—
536C4—42no—
537C4—42no—
538C4—41no—
539C4—81no—
540C4—81no—
541C4—81no—
542C4—81no—
543C4—81no—
544C4—81no—
545C4#292911yescharacteristic
546C4D4×M1611yescharacteristic
547C4#299011yescharacteristic
548C4#299211yescharacteristic
549C4#326611yescharacteristic
550C4#326611yescharacteristic
551C4#334411yescharacteristic
552C4#334411yescharacteristic
553C2—24no—
554C2C2×#158711yescharacteristic
555C2#1788211yescharacteristic
556C2#1807211yescharacteristic
557C2#1808811yescharacteristic
558C2#2298711yescharacteristic
559C2#2301811yescharacteristic
560C2#2302811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1587C2semidirect
6. #9664C2semidirect
7. #9599C2semidirect
8. #9547C2semidirect
9. #9543C2semidirect
12. #8850C2semidirect
13. #8846C2semidirect
14. #8288C2semidirect
524. C22#1587semidirect

Maximal subgroups

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

subgroupindex
#229562
#229522
C4×#4672
C2×#15872
C22×#3342
#96642
#95992
#95472
#95432
#89372
#88502
#88462
#82882
#81452