Finite Group Databasebeta

← #6797949

Subgroups — #6797949

Order 512.

1686 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36180C211yesmaximal, characteristic
2#36134C211yesmaximal, characteristic
3#33289C211yesmaximal, characteristic, complemented
4#31936C218yesmaximal, complemented
5#29844C212yesmaximal, complemented
6#28717C211yesmaximal, characteristic, complemented
7#28511C211yesmaximal, characteristic, complemented
8#27950C212yesmaximal, complemented
9#25600C212yesmaximal, complemented
10#24817C214yesmaximal, complemented
11#24122C214yesmaximal
12#21860C212yesmaximal, complemented
13C4×#494C211yesmaximal, characteristic, complemented
14#21264C211yesmaximal, characteristic
15C2×#554C2212yes—
16C2×#533C2212yes—
17C2×#494C2211yescharacteristic
18C2×#487C2212yes—
19C2×#477C2214yes—
20C2×C4×Q16C2214yes—
21C2×#431C2212yes—
22#2868C2211yescharacteristic
23#2868C2211yescharacteristic
24#2857C2212yes—
25#2857C2212yes—
26#2567C2212yes—
27#2552—21no—
28#2552—21no—
29#2552—21no—
30#2542—21no—
31#2538C2212yes—
32#2502C2214yes—
33#2441—22no—
34#2440—22no—
35#2440—22no—
36#2429—22no—
37#2423C2214yes—
38#2365—21no—
39#2354—21no—
40Q8×C4⋊C4C2212yes—
41D4×C4⋊C4—21no—
42D4×C4⋊C4—21no—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C2×#384C2212yes—
46C42×D4C2211yescharacteristic
47#2095C2214yes—
48#2083C2218yes—
49#2072C2214yes—
50#2068C2218yes—
51#2036C2218yes—
52#2001C2218yes—
53#1990C2212yes—
54#1990C2214yes—
55#1990C2212yes—
56#1990C2212yes—
57#1969C2214yes—
58#1956C2212yes—
59#1956C2211yescharacteristic
60#1956C2211yescharacteristic
61#1954C2212yes—
62#1939C2212yes—
63#1923C2214yes—
64#1878C2218yes—
65#1872C2218yes—
66#1852C2218yes—
67#1845C2211yescharacteristic
68#1845C2212yes—
69#1834C2214yes—
70#1803C2214yes—
71#1803C2218yes—
72C4×C4⋊C8C2211yescharacteristic
73C4×C22⋊C8C2212yes—
74C42×C8C2211yescharacteristic
75C22×Q16C2314yes—
76#554—48no—
77#533—28no—
78#533—44no—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8C2311yescharacteristic
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2312yes—
87C2×#125—22no—
88C2×#125—22no—
89C2×C22⋊Q8—22no—
90C2×C22⋊Q8—22no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8C2314yes—
93C2×C22⋊Q8C2314yes—
94C2×C4×Q8—22no—
95C2×C4×Q8C2312yes—
96C2×C4×Q8C2312yes—
97C2×C4×Q8C2314yes—
98C2×C4×D4—22no—
99C2×C4×D4D411yescharacteristic
100C2×C4×D4D411yescharacteristic
101C2×C4×D4C2311yescharacteristic
102C2×#117—22no—
103C22×C4⋊C4—22no—
104C22×C4⋊C4—22no—
105C22×C4⋊C4C2314yes—
106C22×C42C2312yes—
107#494—24no—
108#487—44no—
109#477—48no—
110C4×Q16—48no—
111#431—44no—
112C2×#107C2318yes—
113C2×#107C2314yes—
114C2×C4⋊C8C2312yes—
115C2×C4⋊C8C2314yes—
116C2×#103C2318yes—
117C2×#103C2314yes—
118C2×#103C2318yes—
119C2×#103C2314yes—
120C2×C22⋊C8C2314yes—
121C2×C22⋊C8C2314yes—
122C2×C4×C8C2312yes—
123C2×C4×C8C2312yes—
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2312yes—
127#399—24no—
128#398—24no—
129#398—24no—
130#398—24no—
131#396—24no—
132#395—24no—
133#391—22no—
134#391C2312yes—
135#391C2312yes—
136#391C2314yes—
137#390—22no—
138#389—22no—
139#389—22no—
140#389—22no—
141#389—21no—
142#389—21no—
143#389—21no—
144#389—21no—
145#389—22no—
146#389—22no—
147#389—22no—
148#389—24no—
149#389C2314yes—
150#388—24no—
151#387—24no—
152#385—24no—
153#385—24no—
154#384—22no—
155#384—21no—
156#384—21no—
157#384—21no—
158#384—21no—
159#384—22no—
160#384—22no—
161#384C2311yescharacteristic
162#384C2311yescharacteristic
163#384C2314yes—
164#384C2312yes—
165#384C2312yes—
166#384C2312yes—
167#384C2312yes—
168#380—22no—
169#380—22no—
170#380—22no—
171#380C2314yes—
172C4×C4⋊C4—22no—
173C4×C4⋊C4—21no—
174C4×C4⋊C4—21no—
175C4×C4⋊C4—21no—
176C4×C4⋊C4—21no—
177C4×C4⋊C4D411yescharacteristic
178C4×C4⋊C4D411yescharacteristic
179C4×C4⋊C4C2311yescharacteristic
180C4×C22⋊C4D412yes—
181C4×C22⋊C4D412yes—
182C4×C22⋊C4C2312yes—
183C43C2311yescharacteristic
184#340C2318yes—
185#340C2318yes—
186#340C2318yes—
187#336C2312yes—
188#336C2314yes—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8C2412yes—
192C22×Q8C2412yes—
193C22×D4C2×D411yescharacteristic
194C23×C4—24no—
195C23×C4C2×D412yes—
196C23×C4C2×D412yes—
197C23×C4C2412yes—
198C2×Q16—28no—
199C2×Q16—28no—
200C2×Q16—44no—
201C22×C8C2414yes—
202C22×C8C2412yes—
203C22×C8C2412yes—
204C4⋊Q8—24no—
205C4⋊Q8—22no—
206C4⋊Q8—22no—
207C4⋊Q8—24no—
208C4⋊Q8—28no—
209C4⋊Q8—22no—
210C4⋊Q8—44no—
211C4⋊Q8—82no—
212C4⋊Q8—82no—
213C4⋊Q8—82no—
214C4⋊Q8#4114yes—
215#125—48no—
216#125—48no—
217C22⋊Q8—28no—
218C22⋊Q8—416no—
219C22⋊Q8—44no—
220C22⋊Q8—48no—
221C22⋊Q8—48no—
222C22⋊Q8—84no—
223C4×Q8—28no—
224C4×Q8—28no—
225C4×Q8—28no—
226C4×Q8—44no—
227C4×Q8—48no—
228C4×D4—24no—
229C4×D4—24no—
230C4×D4—48no—
231C4×D4Q1618yescomplemented
232#117—48no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—48no—
261C2×C4⋊C4—84no—
262C2×C4⋊C4—84no—
263C2×C4⋊C4C2×D411yescharacteristic
264C2×C4⋊C4C2×D411yescharacteristic
265C2×C4⋊C4C2×D411yescharacteristic
266C2×C4⋊C4C2412yes—
267C2×C4⋊C4C2414yes—
268C2×C4⋊C4C2414yes—
269C2×C4⋊C4C2412yes—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4C2×D412yes—
274C2×C22⋊C4C2×D412yes—
275C2×C22⋊C4C2×D412yes—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2412yes—
288C2×C42C2411yescharacteristic
289C2×C42C2411yescharacteristic
290C2×C42C2411yescharacteristic
291#107—44no—
292#107—48no—
293C4⋊C8—24no—
294C4⋊C8—44no—
295#103—44no—
296#103—44no—
297#103—48no—
298#103—48no—
299C22⋊C8—28no—
300C22⋊C8—44no—
301C4×C8—24no—
302C4×C8—22no—
303C4×C8—22no—
304C4×C8—24no—
305C4×C8—24no—
306#95—24no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95C2×D412yes—
315#95C2×D412yes—
316#95C2×D412yes—
317C24C22×D412yes—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—24no—
321C2×Q8—22no—
322C2×Q8—24no—
323C2×Q8—24no—
324C2×Q8—24no—
325C2×Q8—44no—
326C2×Q8—44no—
327C2×Q8—44no—
328C2×Q8—44no—
329C2×Q8—82no—
330C2×Q8—82no—
331C2×Q8C2×#4114yes—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4C2×Q1614yes—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—48no—
351C22×C4—84no—
352C22×C4—84no—
353C22×C4C2×Q1618yes—
354C22×C4C22×D412yes—
355C22×C4C22×D412yes—
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D412yes—
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C2511yescharacteristic
364Q16—416no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—24no—
369C2×C8—22no—
370C2×C8—24no—
371C2×C8—22no—
372C2×C8—24no—
373C2×C82−1+414yes—
374C4⋊C4—22no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—22no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—28no—
383C4⋊C4—48no—
384C4⋊C4—48no—
385C4⋊C4—48no—
386C4⋊C4—48no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—42no—
396C4⋊C4—44no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—44no—
401C4⋊C4—84no—
402C4⋊C4—84no—
403C4⋊C4—84no—
404C4⋊C4—82no—
405C4⋊C4—82no—
406C4⋊C4—82no—
407C4⋊C4—82no—
408C4⋊C4C2×Q1614yes—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—48no—
412C22⋊C4—48no—
413C22⋊C4—84no—
414C22⋊C4C2×Q1618yes—
415C42—24no—
416C42—24no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—22no—
422C42—48no—
423C42—44no—
424C42—44no—
425C42C2×Q1614yes—
426C42#13714yes—
427C42C2×#4112yes—
428C42H(5,𝔽2)14yes—
429C42H(5,𝔽2)12yes—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23C22×Q1614yes—
437C23C23×D411yescharacteristic
438Q8—28no—
439Q8—44no—
440Q8—44no—
441Q8—44no—
442Q8—88no—
443Q8—164no—
444D4—28no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—216no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—48no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—168no—
482C2×C4#53512yes—
483C2×C4#53512yes—
484C2×C4#54612yes—
485C2×C4#54612yes—
486C2×C4C22×Q1612yes—
487C2×C4C22×Q1614yes—
488C2×C4C22×Q1612yes—
489C2×C4C22×Q1612yes—
490C2×C4C2×#13712yes—
491C2×C4C2×#13712yes—
492C2×C4C2×#13712yes—
493C2×C4#57612yes—
494C2×C4#57712yes—
495C2×C4C22×#4111yescharacteristic
496C2×C4C2×H(5,𝔽2)12yes—
497C2×C4C2×H(5,𝔽2)11yescharacteristic
498C2×C4C2×2−1+412yes—
499C8—24no—
500C8—42no—
501C8—44no—
502C22—24no—
503C22—24no—
504C22—24no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22C2×#53511yescharacteristic
510C22C2×#54611yescharacteristic
511C22#349911yescharacteristic
512C22C23×Q1611yescharacteristic
513C22C22×#13711yescharacteristic
514C22C2×#57611yescharacteristic
515C22C2×#57711yescharacteristic
516C4—22no—
517C4—22no—
518C4—22no—
519C4—24no—
520C4—42no—
521C4—44no—
522C4—42no—
523C4—82no—
524C4—82no—
525C4—82no—
526C4—84no—
527C4—84no—
528C4#320914yes—
529C4#329914yes—
530C4#333314yes—
531C4#352012yes—
532C4#354812yes—
533C2—44no—
534C2#3895111yescharacteristic
535C2C2×#320911yescharacteristic
536C2#6082211yescharacteristic
537C2C2×#329911yescharacteristic
538C2C2×#333311yescharacteristic
539C2#6247211yescharacteristic
540C2#6247811yescharacteristic
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. #31936C2semidirect
5. #29844C2semidirect
6. #28717C2semidirect
7. #28511C2semidirect
8. #27950C2semidirect
9. #25600C2semidirect
10. #24817C2semidirect
12. #21860C2semidirect
13. C4×#494C2semidirect
231. C4×D4Q16semidirect

Maximal subgroups

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

subgroupindex
#361802
#361342
#332892
#319362
#298442
#287172
#285112
#279502
#256002
#248172
#241222
#218602
C4×#4942
#212642