Finite Group Databasebeta

← #6850648

Subgroups — #6850648

Order 512.

1516 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic
2#36188C211yesmaximal, characteristic
3#33289C211yesmaximal, characteristic, complemented
4#32049C218yesmaximal, complemented
5#29808C212yesmaximal, complemented
6#29141C211yesmaximal, characteristic, complemented
7#28512C211yesmaximal, characteristic, complemented
8#27975C212yesmaximal, complemented
9#27889C212yesmaximal, complemented
10#24683C214yesmaximal, complemented
11#24123C214yesmaximal
12#21984C212yesmaximal, complemented
13C4×#494C211yesmaximal, characteristic, complemented
14#21264C211yesmaximal, characteristic
15C2×#556C2212yes—
16C2×#494C2211yescharacteristic
17C2×#475C2214yes—
18C2×C4×Q16C2214yes—
19C2×#431C2212yes—
20#2868C2211yescharacteristic
21#2868C2211yescharacteristic
22#2857C2212yes—
23#2857C2212yes—
24#2571—21no—
25#2568C2212yes—
26#2563—21no—
27#2547—21no—
28#2547—21no—
29#2538C2212yes—
30#2499C2214yes—
31#2497—22no—
32#2496—22no—
33#2496—22no—
34#2492C2214yes—
35#2439—22no—
36#2369—21no—
37#2365—21no—
38#2365—21no—
39#2355C2212yes—
40#2354—21no—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43#2285C2212yes—
44#2285C2212yes—
45C42×D4C2211yescharacteristic
46#2096C2214yes—
47#2074C2218yes—
48#2056C2218yes—
49#2035C2218yes—
50#2024C2214yes—
51#2001C2218yes—
52#1990C2214yes—
53#1990C2214yes—
54#1969C2214yes—
55#1956C2211yescharacteristic
56#1956C2212yes—
57#1956C2212yes—
58#1956C2211yescharacteristic
59#1954C2212yes—
60#1949C2218yes—
61#1939C2212yes—
62#1927C2218yes—
63#1923C2214yes—
64#1873C2218yes—
65#1854C2218yes—
66#1845C2211yescharacteristic
67#1844C2212yes—
68#1834C2212yes—
69#1834C2212yes—
70#1834C2212yes—
71#1803C2214yes—
72C4×C4⋊C8C2211yescharacteristic
73C4×C22⋊C8C2212yes—
74C42×C8C2211yescharacteristic
75C22×Q16C2314yes—
76#556—48no—
77C2×C4⋊Q8—21no—
78C2×C4⋊Q8—21no—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8C2311yescharacteristic
82C2×C4⋊Q8C2311yescharacteristic
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2312yes—
85C2×#126—24no—
86C2×#124—22no—
87C2×C22⋊Q8C2314yes—
88C2×C22⋊Q8C2314yes—
89C2×C4×Q8—22no—
90C2×C4×Q8C2312yes—
91C2×C4×Q8C2312yes—
92C2×C4×D4D411yescharacteristic
93C2×C4×D4D411yescharacteristic
94C2×C4×D4C2311yescharacteristic
95C2×#117—22no—
96C2×#117C2314yes—
97C22×C42C2312yes—
98#494—24no—
99#475—48no—
100C4×Q16—48no—
101#431—28no—
102C2×#107C2314yes—
103C2×#106C2318yes—
104C2×C4⋊C8C2312yes—
105C2×C4⋊C8C2314yes—
106C2×#103C2318yes—
107C2×#103C2318yes—
108C2×#103C2314yes—
109C2×#103C2314yes—
110C2×C22⋊C8C2314yes—
111C2×C22⋊C8C2314yes—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2312yes—
114C2×C4×C8C2312yes—
115C2×C4×C8C2312yes—
116C2×C4×C8C2311yescharacteristic
117#400—24no—
118#400—24no—
119#397—24no—
120#397—24no—
121#396—24no—
122#393—24no—
123#391—22no—
124#391—22no—
125#391—24no—
126#391C2312yes—
127#391C2312yes—
128#391C2314yes—
129#391C2314yes—
130#390—22no—
131#389—22no—
132#389—21no—
133#389—21no—
134#389—21no—
135#389—21no—
136#389—22no—
137#389—22no—
138#389—22no—
139#389—22no—
140#388—24no—
141#387—24no—
142#387—24no—
143#387—24no—
144#384—21no—
145#384—21no—
146#384—21no—
147#384—21no—
148#384C2311yescharacteristic
149#384C2311yescharacteristic
150#384C2312yes—
151#384C2312yes—
152#384C2312yes—
153#384C2312yes—
154#382—22no—
155#382—22no—
156#382C2314yes—
157#380—22no—
158#380—22no—
159#380—22no—
160#380—22no—
161#380—22no—
162#380—22no—
163#380—22no—
164#380C2314yes—
165C4×C4⋊C4—22no—
166C4×C4⋊C4—21no—
167C4×C4⋊C4—21no—
168C4×C4⋊C4—21no—
169C4×C4⋊C4—21no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4D411yescharacteristic
174C4×C4⋊C4D411yescharacteristic
175C4×C4⋊C4C2314yes—
176C4×C4⋊C4C2311yescharacteristic
177C4×C22⋊C4D412yes—
178C4×C22⋊C4D412yes—
179C4×C22⋊C4C2312yes—
180C43C2311yescharacteristic
181#340C2318yes—
182#340C2318yes—
183#340C2318yes—
184#336C2314yes—
185#336C2312yes—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8C2412yes—
189C22×Q8C2412yes—
190C22×D4C2×D411yescharacteristic
191C23×C4C2×D412yes—
192C23×C4C2×D412yes—
193C23×C4C2412yes—
194C2×Q16—28no—
195C2×Q16—44no—
196C2×Q16—44no—
197C22×C8C2412yes—
198C22×C8C2414yes—
199C22×C8C2412yes—
200C4⋊Q8—22no—
201C4⋊Q8—22no—
202C4⋊Q8—28no—
203C4⋊Q8—22no—
204C4⋊Q8—22no—
205C4⋊Q8—44no—
206C4⋊Q8—44no—
207C4⋊Q8—44no—
208C4⋊Q8—44no—
209C4⋊Q8—82no—
210#126—416no—
211#124—48no—
212C22⋊Q8—48no—
213C22⋊Q8—48no—
214C4×Q8—28no—
215C4×Q8—28no—
216C4×Q8—48no—
217C4×D4—24no—
218C4×D4—24no—
219C4×D4—24no—
220#117—48no—
221#117—84no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C4⋊C4C2×D411yescharacteristic
246C2×C4⋊C4C2×D411yescharacteristic
247C2×C4⋊C4C2412yes—
248C2×C4⋊C4C2414yes—
249C2×C4⋊C4C2412yes—
250C2×C4⋊C4C2414yes—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4C2×D412yes—
256C2×C22⋊C4C2×D412yes—
257C2×C22⋊C4C2×D412yes—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—24no—
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2411yescharacteristic
271C2×C42C2412yes—
272C2×C42C2411yescharacteristic
273C2×C42C2411yescharacteristic
274#107—44no—
275#106—48no—
276C4⋊C8—28no—
277C4⋊C8—24no—
278#103—48no—
279#103—44no—
280#103—44no—
281#103—48no—
282C22⋊C8—28no—
283C22⋊C8—44no—
284C4×C8—24no—
285C4×C8—24no—
286C4×C8—22no—
287C4×C8—42no—
288C4×C8#4114yes—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95C2×D412yes—
302#95C2×D412yes—
303#95C2×D412yes—
304C24C22×D412yes—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—44no—
314C2×Q8—44no—
315C2×Q8—44no—
316C2×Q8—44no—
317C2×Q8—82no—
318C2×Q8—82no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4C22×D412yes—
339C22×C4C22×D412yes—
340C22×C4C22×D412yes—
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C2511yescharacteristic
348Q16—416no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—24no—
352C2×C8—22no—
353C2×C8—24no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—24no—
357C2×C8C2×#4114yes—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—24no—
362C4⋊C4—28no—
363C4⋊C4—24no—
364C4⋊C4—28no—
365C4⋊C4—48no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—44no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—44no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—84no—
385C4⋊C4—82no—
386C4⋊C4—82no—
387C22⋊C4—24no—
388C22⋊C4—24no—
389C22⋊C4—24no—
390C22⋊C4—48no—
391C22⋊C4—48no—
392C22⋊C4—84no—
393C22⋊C4—84no—
394C42—24no—
395C42—24no—
396C42—22no—
397C42—22no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—48no—
402C42—44no—
403C42—44no—
404C42—84no—
405C42#13514yes—
406C42#13714yes—
407C42C2×#4112yes—
408C42H(5,𝔽2)14yes—
409C422−1+412yes—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23C23×D411yescharacteristic
418Q8—44no—
419Q8—44no—
420Q8—44no—
421Q8—44no—
422Q8—88no—
423Q8—164no—
424D4—44no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—48no—
441C2×C4—48no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4#53512yes—
463C2×C4#53512yes—
464C2×C4#54712yes—
465C2×C4#54712yes—
466C2×C4C2×#13512yes—
467C2×C4C2×#13512yes—
468C2×C4C2×#13512yes—
469C2×C4C2×#13712yes—
470C2×C4C2×#13712yes—
471C2×C4C2×#13712yes—
472C2×C4#57712yes—
473C2×C4#57812yes—
474C2×C4C22×#4111yescharacteristic
475C2×C4C2×H(5,𝔽2)12yes—
476C2×C4C2×2−1+411yescharacteristic
477C2×C4#58512yes—
478C8—24no—
479C8—44no—
480C8—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22C2×#53511yescharacteristic
489C22C2×#54711yescharacteristic
490C22#350011yescharacteristic
491C22C22×#13511yescharacteristic
492C22C22×#13711yescharacteristic
493C22C2×#57711yescharacteristic
494C22C2×#57811yescharacteristic
495C4—22no—
496C4—22no—
497C4—22no—
498C4—42no—
499C4—42no—
500C4—42no—
501C4—82no—
502C4—82no—
503C4—84no—
504C4—84no—
505C4—82no—
506C4—82no—
507C4#321014yes—
508C4#334214yes—
509C4#335614yes—
510C4#350612yes—
511C4#354712yes—
512C2—82no—
513C2#3949111yescharacteristic
514C2C2×#321011yescharacteristic
515C2#6082411yescharacteristic
516C2C2×#334211yescharacteristic
517C2C2×#335611yescharacteristic
518C2#6253111yescharacteristic
519C2#6253611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33289C2semidirect
4. #32049C2semidirect
5. #29808C2semidirect
6. #29141C2semidirect
7. #28512C2semidirect
8. #27975C2semidirect
9. #27889C2semidirect
10. #24683C2semidirect
12. #21984C2semidirect
13. C4×#494C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#361882
#332892
#320492
#298082
#291412
#285122
#279752
#278892
#246832
#241232
#219842
C4×#4942
#212642