Finite Group Databasebeta

← #6798078

Subgroups — #6798078

Order 512.

1552 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic
2#36134C211yesmaximal, characteristic
3#33225C211yesmaximal, characteristic, complemented
4#30634C218yesmaximal, complemented
5#29845C212yesmaximal, complemented
6#29206C211yesmaximal, characteristic, complemented
7#28748C211yesmaximal, characteristic, complemented
8#27908C212yesmaximal, complemented
9#26527C214yesmaximal, complemented
10#26231C214yesmaximal
11#25674C212yesmaximal, complemented
12#21864C212yesmaximal, complemented
13C4×#500C211yesmaximal, characteristic, complemented
14#21264C211yesmaximal, characteristic
15C2×#533C2212yes—
16C2×#500C2211yescharacteristic
17C2×#487C2212yes—
18C2×#487C2214yes—
19C2×#485C2218yes—
20C2×#431C2212yes—
21#2868C2211yescharacteristic
22#2868C2211yescharacteristic
23#2857C2212yes—
24#2857C2212yes—
25#2571C2212yes—
26#2563—21no—
27#2552—21no—
28#2542C2212yes—
29#2538—21no—
30#2538—21no—
31#2497—22no—
32#2492—22no—
33#2441—22no—
34#2439C2214yes—
35#2429C2214yes—
36#2423—22no—
37#2369—21no—
38#2365C2212yes—
39#2355—21no—
40#2354—21no—
41#2354C2212yes—
42Q8×C4⋊C4—21no—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45#2285C2212yes—
46C42×D4C2211yescharacteristic
47#2097C2218yes—
48#2066C2218yes—
49#2058C2214yes—
50#2043C2214yes—
51#2036C2218yes—
52#2035C2218yes—
53#1990C2212yes—
54#1990C2211yescharacteristic
55#1990C2211yescharacteristic
56#1981C2214yes—
57#1956C2212yes—
58#1956C2212yes—
59#1956C2212yes—
60#1956C2214yes—
61#1954C2212yes—
62#1948C2214yes—
63#1942C2214yes—
64#1939C2212yes—
65#1923C2214yes—
66#1873C2218yes—
67#1863C2218yes—
68#1845C2211yescharacteristic
69#1844C2212yes—
70#1805C2218yes—
71C4×#107C2214yes—
72C4×C4⋊C8C2211yescharacteristic
73C4×C22⋊C8C2212yes—
74C42×C8C2211yescharacteristic
75#533—28no—
76#533—44no—
77C2×C4⋊Q8—21no—
78C2×C4⋊Q8—21no—
79C2×C4⋊Q8C2312yes—
80C2×C4⋊Q8C2312yes—
81C2×C4⋊Q8C2312yes—
82C2×C4⋊Q8C2311yescharacteristic
83C2×C4⋊Q8C2311yescharacteristic
84C2×C4⋊Q8C2312yes—
85C2×#125—22no—
86C2×C22⋊Q8—22no—
87C2×C22⋊Q8—22no—
88C2×C22⋊Q8C2314yes—
89C2×C22⋊Q8C2314yes—
90C2×C4×Q8—21no—
91C2×C4×Q8—21no—
92C2×C4×Q8—22no—
93C2×C4×Q8C2314yes—
94C2×C4×D4D411yescharacteristic
95C2×C4×D4D411yescharacteristic
96C2×C4×D4C2311yescharacteristic
97C2×#117—22no—
98C2×#117C2314yes—
99C22×C42C2312yes—
100#500—24no—
101#487—48no—
102#487—44no—
103#485—416no—
104#431—44no—
105C2×#107C2314yes—
106C2×#107C2314yes—
107C2×#106C2318yes—
108C2×C4⋊C8C2312yes—
109C2×C4⋊C8C2314yes—
110C2×#103C2318yes—
111C2×#103C2318yes—
112C2×#103C2318yes—
113C2×#103C2318yes—
114C2×C22⋊C8C2314yes—
115C2×C22⋊C8C2314yes—
116C2×C4×C8C2312yes—
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2312yes—
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2312yes—
121#400—24no—
122#399—24no—
123#398—24no—
124#396—24no—
125#396—24no—
126#393—24no—
127#391—21no—
128#391—21no—
129#391—22no—
130#391—24no—
131#391C2314yes—
132#390—22no—
133#390—22no—
134#389—22no—
135#389—22no—
136#389—21no—
137#389—21no—
138#389—22no—
139#389—22no—
140#389—22no—
141#389—24no—
142#389C2312yes—
143#389C2312yes—
144#389C2314yes—
145#388—24no—
146#388—24no—
147#387—24no—
148#387—24no—
149#384—22no—
150#384—21no—
151#384—21no—
152#384—21no—
153#384—21no—
154#384—21no—
155#384—21no—
156#384C2311yescharacteristic
157#384C2311yescharacteristic
158#384C2312yes—
159#384C2312yes—
160#382—22no—
161#382C2314yes—
162#380—22no—
163#380—22no—
164#380—22no—
165#380—22no—
166#380—22no—
167#380C2314yes—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4—21no—
171C4×C4⋊C4—21no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4D411yescharacteristic
175C4×C4⋊C4D411yescharacteristic
176C4×C4⋊C4C2312yes—
177C4×C4⋊C4C2312yes—
178C4×C4⋊C4C2311yescharacteristic
179C4×C22⋊C4D412yes—
180C4×C22⋊C4D412yes—
181C4×C22⋊C4C2312yes—
182C43C2311yescharacteristic
183#340C2318yes—
184#340C2314yes—
185#340C2314yes—
186#336C2312yes—
187#336C2314yes—
188C22×Q8—21no—
189C22×Q8—21no—
190C22×Q8—22no—
191C22×Q8C2414yes—
192C22×D4C2×D411yescharacteristic
193C23×C4C2×D412yes—
194C23×C4C2×D412yes—
195C23×C4C2412yes—
196C22×C8C2414yes—
197C22×C8C2412yes—
198C22×C8C2412yes—
199C4⋊Q8—22no—
200C4⋊Q8—24no—
201C4⋊Q8—22no—
202C4⋊Q8—22no—
203C4⋊Q8—24no—
204C4⋊Q8—28no—
205C4⋊Q8—44no—
206C4⋊Q8—44no—
207C4⋊Q8—44no—
208C4⋊Q8—82no—
209C4⋊Q8#4114yes—
210#125—48no—
211C22⋊Q8—28no—
212C22⋊Q8—44no—
213C22⋊Q8—48no—
214C22⋊Q8—48no—
215C22⋊Q8—84no—
216C4×Q8—28no—
217C4×Q8—44no—
218C4×Q8—48no—
219C4×Q8—44no—
220C4×Q8—44no—
221C4×D4—24no—
222C4×D4—24no—
223C4×D4—24no—
224#117—48no—
225#117—48no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4C2×D411yescharacteristic
250C2×C4⋊C4C2×D411yescharacteristic
251C2×C4⋊C4C2×D411yescharacteristic
252C2×C4⋊C4C2412yes—
253C2×C4⋊C4C2414yes—
254C2×C4⋊C4C2412yes—
255C2×C4⋊C4C2412yes—
256C2×C4⋊C4C2412yes—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4C2×D412yes—
262C2×C22⋊C4C2×D412yes—
263C2×C22⋊C4C2×D412yes—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—24no—
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2412yes—
277C2×C42C2411yescharacteristic
278C2×C42C2411yescharacteristic
279C2×C42C2411yescharacteristic
280#107—28no—
281#107—28no—
282#106—48no—
283C4⋊C8—24no—
284C4⋊C8—44no—
285#103—216no—
286#103—48no—
287#103—48no—
288#103—48no—
289C22⋊C8—28no—
290C22⋊C8—44no—
291C4×C8—24no—
292C4×C8—22no—
293C4×C8—22no—
294C4×C8—24no—
295C4×C8—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—24no—
305#95—24no—
306#95C2×D412yes—
307#95C2×D412yes—
308#95C2×D412yes—
309C24C22×D412yes—
310C2×Q8—24no—
311C2×Q8—24no—
312C2×Q8—28no—
313C2×Q8—24no—
314C2×Q8—28no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—44no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—82no—
322C2×Q8—82no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4C22×D412yes—
343C22×C4C22×D412yes—
344C22×C4C22×D412yes—
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C2511yescharacteristic
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—24no—
356C2×C8—22no—
357C2×C8—24no—
358C2×C8—22no—
359C2×C8—24no—
360C2×C8H(5,𝔽2)14yes—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—28no—
368C4⋊C4—22no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—48no—
372C4⋊C4—48no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—42no—
378C4⋊C4—44no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—84no—
388C4⋊C4—84no—
389C4⋊C4—82no—
390C4⋊C4—82no—
391C4⋊C4—82no—
392C4⋊C4—82no—
393C4⋊C4C2×#4114yes—
394C22⋊C4—24no—
395C22⋊C4—24no—
396C22⋊C4—24no—
397C22⋊C4—48no—
398C22⋊C4—84no—
399C22⋊C4—84no—
400C22⋊C4—84no—
401C42—24no—
402C42—24no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—48no—
409C42—44no—
410C42—44no—
411C42—84no—
412C42#13514yes—
413C42#13714yes—
414C42C2×#4112yes—
415C42H(5,𝔽2)12yes—
416C422−1+414yes—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23C23×D411yescharacteristic
425Q8—48no—
426Q8—48no—
427Q8—48no—
428Q8—84no—
429Q8—164no—
430D4—44no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—48no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—48no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4#53612yes—
469C2×C4#53612yes—
470C2×C4#54612yes—
471C2×C4#54612yes—
472C2×C4C2×#13512yes—
473C2×C4C2×#13512yes—
474C2×C4C2×#13512yes—
475C2×C4C2×#13712yes—
476C2×C4C2×#13712yes—
477C2×C4C2×#13712yes—
478C2×C4#57712yes—
479C2×C4#57812yes—
480C2×C4C22×#4111yescharacteristic
481C2×C4C2×H(5,𝔽2)12yes—
482C2×C4C2×H(5,𝔽2)11yescharacteristic
483C2×C4C2×2−1+412yes—
484C8—24no—
485C8—42no—
486C8—44no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22C2×#53611yescharacteristic
495C22C2×#54611yescharacteristic
496C22#349811yescharacteristic
497C22C22×#13511yescharacteristic
498C22C22×#13711yescharacteristic
499C22C2×#57711yescharacteristic
500C22C2×#57811yescharacteristic
501C4—22no—
502C4—22no—
503C4—22no—
504C4—42no—
505C4—42no—
506C4—44no—
507C4—42no—
508C4—84no—
509C4—82no—
510C4—82no—
511C4—82no—
512C4—84no—
513C4#323114yes—
514C4#330314yes—
515C4#332814yes—
516C4#352212yes—
517C4#354812yes—
518C2—82no—
519C2#3895211yescharacteristic
520C2C2×#323111yescharacteristic
521C2#6087811yescharacteristic
522C2C2×#330311yescharacteristic
523C2C2×#332811yescharacteristic
524C2#6241811yescharacteristic
525C2#6242211yescharacteristic
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. #33225C2semidirect
4. #30634C2semidirect
5. #29845C2semidirect
6. #29206C2semidirect
7. #28748C2semidirect
8. #27908C2semidirect
9. #26527C2semidirect
11. #25674C2semidirect
12. #21864C2semidirect
13. C4×#500C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#361342
#332252
#306342
#298452
#292062
#287482
#279082
#265272
#262312
#256742
#218642
C4×#5002
#212642