Finite Group Databasebeta

← #6850646

Subgroups — #6850646

Order 512.

1574 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic
2#36180C211yesmaximal, characteristic
3#33289C211yesmaximal, characteristic, complemented
4#32052C218yesmaximal, complemented
5#29808C212yesmaximal, complemented
6#29141C211yesmaximal, characteristic, complemented
7#28511C211yesmaximal, characteristic, complemented
8#27972C212yesmaximal, complemented
9#27892C212yesmaximal, complemented
10#24682C214yesmaximal, complemented
11#24122C214yesmaximal
12#21982C212yesmaximal, complemented
13C4×#494C211yesmaximal, characteristic, complemented
14#21264C211yesmaximal, characteristic
15C2×#554C2212yes—
16C2×#494C2211yescharacteristic
17C2×#477C2214yes—
18C2×C4×Q16C2214yes—
19C2×#431C2212yes—
20#2868C2211yescharacteristic
21#2868C2211yescharacteristic
22#2857C2212yes—
23#2857C2212yes—
24#2571—21no—
25#2567C2212yes—
26#2563—21no—
27#2552—21no—
28#2552—21no—
29#2538C2212yes—
30#2502C2214yes—
31#2497—22no—
32#2492C2214yes—
33#2440—22no—
34#2440—22no—
35#2439—22no—
36#2369—21no—
37#2355C2212yes—
38#2354—21no—
39D4×C4⋊C4—21no—
40D4×C4⋊C4—21no—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43#2285C2212yes—
44C2×#384C2212yes—
45C42×D4C2211yescharacteristic
46#2095C2214yes—
47#2072C2218yes—
48#2056C2218yes—
49#2036C2218yes—
50#2024C2214yes—
51#2001C2218yes—
52#1990C2214yes—
53#1990C2214yes—
54#1969C2214yes—
55#1956C2211yescharacteristic
56#1956C2212yes—
57#1956C2212yes—
58#1956C2211yescharacteristic
59#1954C2212yes—
60#1947C2218yes—
61#1939C2212yes—
62#1928C2218yes—
63#1923C2214yes—
64#1872C2218yes—
65#1852C2218yes—
66#1845C2212yes—
67#1845C2211yescharacteristic
68#1834C2212yes—
69#1834C2212yes—
70#1834C2212yes—
71#1803C2214yes—
72C4×C4⋊C8C2211yescharacteristic
73C4×C22⋊C8C2212yes—
74C42×C8C2211yescharacteristic
75C22×Q16C2314yes—
76#554—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×#125—22no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8C2314yes—
88C2×C22⋊Q8C2314yes—
89C2×C4×Q8C2312yes—
90C2×C4×Q8C2312yes—
91C2×C4×D4—22no—
92C2×C4×D4D411yescharacteristic
93C2×C4×D4D411yescharacteristic
94C2×C4×D4C2311yescharacteristic
95C2×#117—22no—
96C22×C4⋊C4—22no—
97C22×C4⋊C4—22no—
98C22×C4⋊C4C2314yes—
99C22×C42C2312yes—
100#494—24no—
101#477—48no—
102C4×Q16—48no—
103#431—28no—
104C2×#107C2318yes—
105C2×#107C2314yes—
106C2×C4⋊C8C2312yes—
107C2×C4⋊C8C2314yes—
108C2×#103C2318yes—
109C2×#103C2318yes—
110C2×#103C2314yes—
111C2×#103C2314yes—
112C2×C22⋊C8C2314yes—
113C2×C22⋊C8C2314yes—
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2312yes—
116C2×C4×C8C2312yes—
117C2×C4×C8C2312yes—
118C2×C4×C8C2311yescharacteristic
119#400—24no—
120#398—24no—
121#398—24no—
122#396—24no—
123#395—24no—
124#393—24no—
125#391—24no—
126#391C2312yes—
127#391C2312yes—
128#391C2314yes—
129#391C2314yes—
130#390—22no—
131#389—22no—
132#389—22no—
133#389—22no—
134#389—21no—
135#389—21no—
136#389—21no—
137#389—21no—
138#389—22no—
139#389—22no—
140#389—22no—
141#389—22no—
142#388—24no—
143#387—24no—
144#385—24no—
145#385—24no—
146#384—21no—
147#384—21no—
148#384—21no—
149#384—21no—
150#384—22no—
151#384—22no—
152#384C2311yescharacteristic
153#384C2311yescharacteristic
154#384C2314yes—
155#384C2312yes—
156#384C2312yes—
157#384C2312yes—
158#384C2312yes—
159#382—22no—
160#382—22no—
161#380—22no—
162#380—22no—
163#380—22no—
164#380—22no—
165#380—22no—
166#380C2314yes—
167C4×C4⋊C4—22no—
168C4×C4⋊C4—21no—
169C4×C4⋊C4—21no—
170C4×C4⋊C4—21no—
171C4×C4⋊C4—21no—
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×C4—24no—
192C23×C4C2×D412yes—
193C23×C4C2×D412yes—
194C23×C4C2412yes—
195C2×Q16—28no—
196C2×Q16—44no—
197C2×Q16—44no—
198C22×C8C2412yes—
199C22×C8C2414yes—
200C22×C8C2412yes—
201C4⋊Q8—22no—
202C4⋊Q8—22no—
203C4⋊Q8—28no—
204C4⋊Q8—22no—
205C4⋊Q8—22no—
206C4⋊Q8—44no—
207C4⋊Q8—44no—
208C4⋊Q8—82no—
209C4⋊Q8—82no—
210C4⋊Q8—82no—
211#125—48no—
212C22⋊Q8—416no—
213C22⋊Q8—48no—
214C22⋊Q8—48no—
215C4×Q8—28no—
216C4×Q8—28no—
217C4×D4—24no—
218C4×D4—24no—
219C4×D4—48no—
220C4×D4Q1618yescomplemented
221#117—84no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—48no—
248C2×C4⋊C4—84no—
249C2×C4⋊C4—84no—
250C2×C4⋊C4C2×D411yescharacteristic
251C2×C4⋊C4C2×D411yescharacteristic
252C2×C4⋊C4C2×D411yescharacteristic
253C2×C4⋊C4C2412yes—
254C2×C4⋊C4C2414yes—
255C2×C4⋊C4C2412yes—
256C2×C4⋊C4C2414yes—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4C2×D412yes—
261C2×C22⋊C4C2×D412yes—
262C2×C22⋊C4C2×D412yes—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2411yescharacteristic
275C2×C42C2412yes—
276C2×C42C2411yescharacteristic
277C2×C42C2411yescharacteristic
278#107—44no—
279#107—48no—
280C4⋊C8—28no—
281C4⋊C8—24no—
282#103—48no—
283#103—44no—
284#103—44no—
285#103—48no—
286C22⋊C8—28no—
287C22⋊C8—44no—
288C4×C8—24no—
289C4×C8—24no—
290C4×C8—22no—
291C4×C8—42no—
292C4×C8#4114yes—
293#95—24no—
294#95—24no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95C2×D412yes—
304#95C2×D412yes—
305#95C2×D412yes—
306C24C22×D412yes—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—24no—
314C2×Q8—24no—
315C2×Q8—44no—
316C2×Q8—44no—
317C2×Q8—82no—
318C2×Q8—82no—
319C2×Q8—82no—
320C2×Q8—82no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4C2×Q1614yes—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—48no—
340C22×C4—84no—
341C22×C4—84no—
342C22×C4C2×Q1618yes—
343C22×C4C22×D412yes—
344C22×C4C22×D412yes—
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D412yes—
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C2511yescharacteristic
353Q16—416no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—24no—
357C2×C8—22no—
358C2×C8—24no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—24no—
362C2×C8C2×#4114yes—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—28no—
368C4⋊C4—28no—
369C4⋊C4—48no—
370C4⋊C4—48no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—42no—
381C4⋊C4—44no—
382C4⋊C4—42no—
383C4⋊C4—44no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4—84no—
388C4⋊C4—84no—
389C4⋊C4—84no—
390C4⋊C4—82no—
391C4⋊C4—82no—
392C4⋊C4—82no—
393C4⋊C4—82no—
394C4⋊C4C2×Q1614yes—
395C22⋊C4—24no—
396C22⋊C4—24no—
397C22⋊C4—48no—
398C22⋊C4—84no—
399C22⋊C4—84no—
400C22⋊C4C2×Q1618yes—
401C42—24no—
402C42—24no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—44no—
409C42—44no—
410C42—84no—
411C42C2×Q1614yes—
412C42#13714yes—
413C42C2×#4112yes—
414C42H(5,𝔽2)14yes—
415C422−1+412yes—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23C22×Q1614yes—
423C23C23×D411yescharacteristic
424Q8—44no—
425Q8—44no—
426Q8—44no—
427Q8—44no—
428Q8—164no—
429Q8—164no—
430D4—28no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—216no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—48no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—168no—
468C2×C4#53512yes—
469C2×C4#53512yes—
470C2×C4#54712yes—
471C2×C4#54712yes—
472C2×C4C22×Q1612yes—
473C2×C4C22×Q1614yes—
474C2×C4C22×Q1612yes—
475C2×C4C22×Q1612yes—
476C2×C4C2×#13712yes—
477C2×C4C2×#13712yes—
478C2×C4C2×#13712yes—
479C2×C4#57612yes—
480C2×C4#57712yes—
481C2×C4C22×#4111yescharacteristic
482C2×C4C2×H(5,𝔽2)12yes—
483C2×C4C2×2−1+411yescharacteristic
484C2×C4#58512yes—
485C8—24no—
486C8—44no—
487C8—42no—
488C22—24no—
489C22—24no—
490C22—24no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22C2×#53511yescharacteristic
496C22C2×#54711yescharacteristic
497C22#350011yescharacteristic
498C22C23×Q1611yescharacteristic
499C22C22×#13711yescharacteristic
500C22C2×#57611yescharacteristic
501C22C2×#57711yescharacteristic
502C4—24no—
503C4—22no—
504C4—22no—
505C4—22no—
506C4—42no—
507C4—42no—
508C4—82no—
509C4—82no—
510C4—84no—
511C4—82no—
512C4—82no—
513C4—84no—
514C4#320914yes—
515C4#333814yes—
516C4#335614yes—
517C4#350612yes—
518C4#354712yes—
519C2—44no—
520C2#3949111yescharacteristic
521C2C2×#320911yescharacteristic
522C2#6082211yescharacteristic
523C2C2×#333811yescharacteristic
524C2C2×#335611yescharacteristic
525C2#6253011yescharacteristic
526C2#6253711yescharacteristic
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. #32052C2semidirect
5. #29808C2semidirect
6. #29141C2semidirect
7. #28511C2semidirect
8. #27972C2semidirect
9. #27892C2semidirect
10. #24682C2semidirect
12. #21982C2semidirect
13. C4×#494C2semidirect
220. C4×D4Q16semidirect

Maximal subgroups

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

subgroupindex
#361942
#361802
#332892
#320522
#298082
#291412
#285112
#279722
#278922
#246822
#241222
#219822
C4×#4942
#212642