Finite Group Databasebeta

← #6798311

Subgroups — #6798311

Order 512.

1540 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36188C211yesmaximal, characteristic
2#36134C211yesmaximal, characteristic
3#33289C211yesmaximal, characteristic, complemented
4#31691C218yesmaximal, complemented
5#29877C212yesmaximal, complemented
6#29219C211yesmaximal, characteristic, complemented
7#28687C211yesmaximal, characteristic, complemented
8#27878C212yesmaximal, complemented
9#26447C214yesmaximal, complemented
10#26303C214yesmaximal
11#25812C212yesmaximal, complemented
12#21835C212yesmaximal, complemented
13C4×#500C211yesmaximal, characteristic, complemented
14#21264C211yesmaximal, characteristic
15C2×#556—21no—
16C2×#533C2212yes—
17C2×#500C2211yescharacteristic
18C2×#498C2212yes—
19C2×#431C2212yes—
20#2868C2211yescharacteristic
21#2868C2211yescharacteristic
22#2857C2212yes—
23#2857C2212yes—
24#2568—21no—
25#2552C2212yes—
26#2547—21no—
27#2547C2212yes—
28#2542—21no—
29#2538—21no—
30#2499—22no—
31#2496—22no—
32#2496C2214yes—
33#2441C2214yes—
34#2429—22no—
35#2423—22no—
36#2365—21no—
37#2365—21no—
38#2365C2212yes—
39#2354C2212yes—
40Q8×C4⋊C4—21no—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43#2285C2212yes—
44C42×D4C2211yescharacteristic
45#2099C2214yes—
46#2088C2218yes—
47#2084C2214yes—
48#2077C2218yes—
49#2076C2218yes—
50#2059C2218yes—
51#1990C2212yes—
52#1990C2214yes—
53#1990C2212yes—
54#1990C2212yes—
55#1990C2212yes—
56#1990C2211yescharacteristic
57#1990C2211yescharacteristic
58#1989C2218yes—
59#1988C2214yes—
60#1954C2212yes—
61#1948C2214yes—
62#1942C2214yes—
63#1939C2212yes—
64#1923C2214yes—
65#1921C2218yes—
66#1854C2218yes—
67#1845C2211yescharacteristic
68#1834C2214yes—
69#1834C2212yes—
70#1825C2218yes—
71C4×#107C2214yes—
72C4×C4⋊C8C2211yescharacteristic
73C4×C22⋊C8C2212yes—
74C42×C8C2211yescharacteristic
75#556—48no—
76#533—28no—
77#533—44no—
78C2×C4⋊Q8—21no—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8C2311yescharacteristic
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2311yescharacteristic
86C2×#126—24no—
87C2×#125C2314yes—
88C2×#124—22no—
89C2×C22⋊Q8—22no—
90C2×C22⋊Q8—22no—
91C2×C22⋊Q8—22no—
92C2×C22⋊Q8—22no—
93C2×C4×Q8—22no—
94C2×C4×Q8—21no—
95C2×C4×Q8—21no—
96C2×C4×Q8—22no—
97C2×C4×Q8—22no—
98C2×C4×D4D411yescharacteristic
99C2×C4×D4D411yescharacteristic
100C2×C4×D4C2311yescharacteristic
101C2×#117—22no—
102C2×#117C2314yes—
103C22×C42C2312yes—
104#500—24no—
105#498—44no—
106#431—44no—
107C2×#107C2314yes—
108C2×#107C2314yes—
109C2×#106C2318yes—
110C2×C4⋊C8C2312yes—
111C2×C4⋊C8C2314yes—
112C2×C22⋊C8C2314yes—
113C2×C22⋊C8C2314yes—
114C2×C4×C8C2312yes—
115C2×C4×C8C2312yes—
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2312yes—
119#400—24no—
120#399—24no—
121#398—24no—
122#397—24no—
123#397—24no—
124#396—24no—
125#391—22no—
126#391—22no—
127#391—21no—
128#391—21no—
129#391—22no—
130#391—22no—
131#390—22no—
132#389—22no—
133#389—21no—
134#389—21no—
135#389—22no—
136#389—22no—
137#389—24no—
138#389C2312yes—
139#389C2312yes—
140#389C2314yes—
141#389C2314yes—
142#388—24no—
143#387—24no—
144#387—24no—
145#387—24no—
146#384—22no—
147#384—21no—
148#384—21no—
149#384—21no—
150#384—21no—
151#384C2311yescharacteristic
152#384C2311yescharacteristic
153#384C2312yes—
154#384C2312yes—
155#384C2312yes—
156#384C2312yes—
157#382—22no—
158#380—22no—
159#380—22no—
160#380—22no—
161#380C2314yes—
162#380C2314yes—
163#380C2314yes—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—21no—
166C4×C4⋊C4—21no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4D411yescharacteristic
169C4×C4⋊C4D411yescharacteristic
170C4×C4⋊C4C2312yes—
171C4×C4⋊C4C2312yes—
172C4×C4⋊C4C2314yes—
173C4×C4⋊C4C2311yescharacteristic
174C4×C22⋊C4D412yes—
175C4×C22⋊C4D412yes—
176C4×C22⋊C4C2312yes—
177C43C2311yescharacteristic
178#340C2318yes—
179#340C2318yes—
180#340C2318yes—
181#340C2318yes—
182#340C2314yes—
183#340C2318yes—
184#340C2314yes—
185#336C2314yes—
186#336C2312yes—
187C22×Q8—21no—
188C22×Q8—22no—
189C22×Q8—21no—
190C22×Q8—22no—
191C22×D4C2×D411yescharacteristic
192C23×C4C2×D412yes—
193C23×C4C2×D412yes—
194C23×C4C2412yes—
195C22×C8C2414yes—
196C22×C8C2412yes—
197C22×C8C2412yes—
198C4⋊Q8—24no—
199C4⋊Q8—22no—
200C4⋊Q8—22no—
201C4⋊Q8—24no—
202C4⋊Q8—22no—
203C4⋊Q8—28no—
204C4⋊Q8—44no—
205C4⋊Q8—44no—
206C4⋊Q8—44no—
207C4⋊Q8—82no—
208C4⋊Q8#4114yes—
209#126—416no—
210#125—28no—
211#125—44no—
212#124—48no—
213C22⋊Q8—48no—
214C22⋊Q8—48no—
215C22⋊Q8—48no—
216C22⋊Q8—84no—
217C4×Q8—48no—
218C4×Q8—48no—
219C4×Q8—48no—
220C4×Q8—44no—
221C4×Q8—44no—
222C4×D4—24no—
223C4×D4—24no—
224C4×D4—24no—
225#117—28no—
226#117—44no—
227#117—48no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4C2×D411yescharacteristic
251C2×C4⋊C4C2×D411yescharacteristic
252C2×C4⋊C4C2×D411yescharacteristic
253C2×C4⋊C4C2414yes—
254C2×C4⋊C4C2414yes—
255C2×C4⋊C4C2412yes—
256C2×C4⋊C4C2412yes—
257C2×C4⋊C4C2412yes—
258C2×C4⋊C4C2412yes—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4C2×D412yes—
264C2×C22⋊C4C2×D412yes—
265C2×C22⋊C4C2×D412yes—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2412yes—
279C2×C42C2411yescharacteristic
280C2×C42C2411yescharacteristic
281C2×C42C2411yescharacteristic
282#107—28no—
283#107—28no—
284#106—48no—
285C4⋊C8—24no—
286C4⋊C8—44no—
287C22⋊C8—28no—
288C22⋊C8—44no—
289C4×C8—24no—
290C4×C8—22no—
291C4×C8—22no—
292C4×C8—24no—
293C4×C8—24no—
294#95—24no—
295#95—24no—
296#95—22no—
297#95—22no—
298#95—24no—
299#95—22no—
300#95—22no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95C2×D412yes—
305#95C2×D412yes—
306#95C2×D412yes—
307C24C22×D412yes—
308C2×Q8—24no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8—42no—
312C2×Q8—44no—
313C2×Q8—44no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—44no—
317C2×Q8—44no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4C22×D412yes—
340C22×C4C22×D412yes—
341C22×C4C22×D412yes—
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C2511yescharacteristic
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—24no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C82−1+414yes—
358C4⋊C4—22no—
359C4⋊C4—24no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—28no—
370C4⋊C4—48no—
371C4⋊C4—48no—
372C4⋊C4—48no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—42no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—42no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—42no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—84no—
389C4⋊C4—82no—
390C4⋊C4—82no—
391C4⋊C4C2×#4114yes—
392C22⋊C4—24no—
393C22⋊C4—24no—
394C22⋊C4—24no—
395C22⋊C4—48no—
396C22⋊C4—48no—
397C22⋊C4—48no—
398C22⋊C4—84no—
399C42—24no—
400C42—24no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—48no—
407C42—48no—
408C42—44no—
409C42—44no—
410C42#13514yes—
411C42AGL(1,ℤ/8)14yes—
412C42C2×#4112yes—
413C42H(5,𝔽2)12yes—
414C422−1+414yes—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23C23×D411yescharacteristic
423Q8—48no—
424Q8—84no—
425Q8—88no—
426Q8—88no—
427D4—44no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—48no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—48no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4#53612yes—
466C2×C4#53612yes—
467C2×C4#54612yes—
468C2×C4#54612yes—
469C2×C4C2×#13512yes—
470C2×C4C2×#13512yes—
471C2×C4C2×#13512yes—
472C2×C4C2×AGL(1,ℤ/8)12yes—
473C2×C4C2×AGL(1,ℤ/8)12yes—
474C2×C4C2×AGL(1,ℤ/8)12yes—
475C2×C4#57712yes—
476C2×C4#57812yes—
477C2×C4C22×#4111yescharacteristic
478C2×C4C2×H(5,𝔽2)11yescharacteristic
479C2×C4C2×2−1+412yes—
480C2×C4C2×2−1+412yes—
481C8—24no—
482C8—42no—
483C8—44no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22C2×#53611yescharacteristic
492C22C2×#54611yescharacteristic
493C22#349911yescharacteristic
494C22C22×#13511yescharacteristic
495C22C22×AGL(1,ℤ/8)11yescharacteristic
496C22C2×#57711yescharacteristic
497C22C2×#57811yescharacteristic
498C4—22no—
499C4—22no—
500C4—22no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—44no—
505C4—82no—
506C4—82no—
507C4—84no—
508C4—84no—
509C4—82no—
510C4#322814yes—
511C4#330414yes—
512C4#333014yes—
513C4#352312yes—
514C4#355012yes—
515C2—82no—
516C2#3895411yescharacteristic
517C2C2×#322811yescharacteristic
518C2#6087811yescharacteristic
519C2C2×#330411yescharacteristic
520C2C2×#333011yescharacteristic
521C2#6247311yescharacteristic
522C2#6247711yescharacteristic
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. #31691C2semidirect
5. #29877C2semidirect
6. #29219C2semidirect
7. #28687C2semidirect
8. #27878C2semidirect
9. #26447C2semidirect
11. #25812C2semidirect
12. #21835C2semidirect
13. C4×#500C2semidirect

Maximal subgroups

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

subgroupindex
#361882
#361342
#332892
#316912
#298772
#292192
#286872
#278782
#264472
#263032
#258122
#218352
C4×#5002
#212642