Finite Group Databasebeta

← #6498011

Subgroups — #6498011

Order 512.

1348 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35335C211yesmaximal, characteristic
2#35294C211yesmaximal, characteristic
3#33299C211yesmaximal, characteristic, complemented
4#33297C211yesmaximal, characteristic, complemented
5#33295C211yesmaximal, characteristic, complemented
6#33294C211yesmaximal, characteristic, complemented
7#30235C212yesmaximal, complemented
8#30235C212yesmaximal, complemented
9#29229C214yesmaximal, complemented
10#29228C214yesmaximal, complemented
11#27971C214yesmaximal
12#27970C214yesmaximal
13#21839C212yesmaximal
14#21838C212yesmaximal
15#20467C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C2×#432C2212yes—
19#2868C2211yescharacteristic
20#2859C2212yes—
21#2859C2211yescharacteristic
22#2859C2211yescharacteristic
23#2859C2211yescharacteristic
24#2859C2211yescharacteristic
25#2859C2211yescharacteristic
26#2660C2212yes—
27#2660C2212yes—
28#2629C2212yes—
29#2628C2212yes—
30#2584—22no—
31#2584—22no—
32#2583—22no—
33#2583—22no—
34#2570—22no—
35#2550—22no—
36#2441—22no—
37#2439—22no—
38#2331C2212yes—
39#2330C2212yes—
40#2317C2212yes—
41#2317C2212yes—
42#2285C2212yes—
43C4×#117C2211yescharacteristic
44#2101C2214yes—
45#2101C2214yes—
46#2101C2214yes—
47#2101C2214yes—
48#2100C2214yes—
49#2100C2214yes—
50#2099C2214yes—
51#2099C2214yes—
52#1993C2214yes—
53#1992C2212yes—
54#1992C2212yes—
55#1992C2212yes—
56#1992C2212yes—
57#1992C2212yes—
58#1992C2212yes—
59#1992C2212yes—
60#1992C2212yes—
61#1992C2214yes—
62#1991C2214yes—
63#1990C2214yes—
64#1989C2214yes—
65#1989C2214yes—
66#1988C2214yes—
67#1987C2214yes—
68#1836C2214yes—
69#1836C2214yes—
70#1835C2214yes—
71#1834C2214yes—
72#1825C2214yes—
73#1825C2214yes—
74#1824C2214yes—
75#1823C2214yes—
76#1768C2212yes—
77#1768C2212yes—
78#1750C2212yes—
79#1750C2212yes—
80#1745C2212yes—
81C4×#97C2211yescharacteristic
82C4×#97C2211yescharacteristic
83#533—28no—
84C2×C4⋊Q8—22no—
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2312yes—
87C2×#125—22no—
88C2×#125—22no—
89C2×#125—28no—
90C2×#125C2312yes—
91C2×#125C2311yescharacteristic
92C2×#125C2311yescharacteristic
93C2×#125C2311yescharacteristic
94C2×#125C2311yescharacteristic
95C2×#125C2312yes—
96C2×C22⋊Q8—22no—
97C2×C4×Q8—22no—
98C2×#117—22no—
99C2×#117C2312yes—
100C2×#117C2311yescharacteristic
101C2×#117C2311yescharacteristic
102C22×C4⋊C4—22no—
103C22×C4⋊C4C2312yes—
104C22×C42C2311yescharacteristic
105#432—28no—
106C2×C4⋊C8C2314yes—
107C2×C22⋊C8C2314yes—
108C2×#97C2312yes—
109C2×#97C2312yes—
110C2×#97C2312yes—
111C2×#97C2312yes—
112C2×C4×C8C2312yes—
113C2×C4×C8C2312yes—
114#400—22no—
115#399—22no—
116#398—24no—
117#398—22no—
118#398—22no—
119#398—22no—
120#398—22no—
121#396—24no—
122#395—22no—
123#395—22no—
124#395—22no—
125#395—22no—
126#395—22no—
127#393—22no—
128#391—24no—
129#389—22no—
130#389—22no—
131#389—22no—
132#389—22no—
133#389—22no—
134#389—22no—
135#389—22no—
136#389—22no—
137#389—22no—
138#389—22no—
139#389—22no—
140#389—22no—
141#389—22no—
142#389—22no—
143#389—22no—
144#389—22no—
145#389—24no—
146#389C2314yes—
147#389C2314yes—
148#387—24no—
149#387—24no—
150#385—22no—
151#385—22no—
152#385—22no—
153#385—22no—
154#384—22no—
155#384C2312yes—
156#384C2312yes—
157#384C2312yes—
158#384C2312yes—
159#384C2312yes—
160#384C2312yes—
161#384C2312yes—
162#382—22no—
163#382—22no—
164#382C2312yes—
165#382C2312yes—
166#382C2312yes—
167#382C2312yes—
168#382C2312yes—
169#380—22no—
170#380C2314yes—
171C4×C4⋊C4—22no—
172C4×C4⋊C4D412yes—
173C4×C4⋊C4D412yes—
174C4×C4⋊C4C2314yes—
175C4×C22⋊C4D412yes—
176C4×C22⋊C4D412yes—
177#376C2311yescharacteristic
178#376C2311yescharacteristic
179C43C2311yescharacteristic
180C43C2311yescharacteristic
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#336C2314yes—
198#336C2314yes—
199C22×Q8—22no—
200C22×Q8—22no—
201C23×C4—22no—
202C23×C4C2×D412yes—
203C23×C4C2411yescharacteristic
204C22×C8C2414yes—
205C22×C8C2414yes—
206C4⋊Q8—28no—
207C4⋊Q8—28no—
208C4⋊Q8—84no—
209#125—24no—
210#125—24no—
211#125—28no—
212#125—24no—
213#125—24no—
214#125—28no—
215#125—48no—
216#125—84no—
217C22⋊Q8—48no—
218C4×Q8—48no—
219#117—24no—
220#117—24no—
221#117—28no—
222#117—84no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—22no—
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—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—48no—
250C2×C4⋊C4C2×D411yescharacteristic
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4C2×D411yescharacteristic
253C2×C4⋊C4C2412yes—
254C2×C4⋊C4C2412yes—
255C2×C4⋊C4C2412yes—
256C2×C4⋊C4C2412yes—
257C2×C4⋊C4C2414yes—
258C2×C4⋊C4C2414yes—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4C2×D411yescharacteristic
262C2×C22⋊C4C2×D412yes—
263C2×C22⋊C4C2×D411yescharacteristic
264C2×C42—22no—
265C2×C42—24no—
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42#4118yes—
272C2×C42C2411yescharacteristic
273C2×C42C2411yescharacteristic
274C2×C42C2411yescharacteristic
275C2×C42C2411yescharacteristic
276C2×C42C2411yescharacteristic
277C2×C42C2411yescharacteristic
278C4⋊C8—28no—
279C22⋊C8—28no—
280#97—24no—
281#97—24no—
282#97—24no—
283#97—24no—
284C4×C8—24no—
285C4×C8—24no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95C2×D412yes—
301#95C2×D412yes—
302#95C2×D412yes—
303#95C2×D412yes—
304C24C22×D411yescharacteristic
305C2×Q8—44no—
306C2×Q8—44no—
307C2×Q8—82no—
308C2×Q8—82no—
309C2×Q8—84no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—44no—
321C22×C4—82no—
322C22×C4—82no—
323C22×C4#12418yes—
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D412yes—
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D412yes—
329C22×C4C22×D412yes—
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D412yes—
333C22×C4C2×#4114yes—
334C22×C4C2511yescharacteristic
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—24no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—28no—
349C4⋊C4—28no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—48no—
363C4⋊C4—44no—
364C4⋊C4—82no—
365C4⋊C4—82no—
366C4⋊C4—82no—
367C4⋊C4—82no—
368C4⋊C4—82no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4—82no—
372C4⋊C4—84no—
373C4⋊C4—84no—
374C4⋊C4—84no—
375C22⋊C4—24no—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—44no—
379C22⋊C4—84no—
380C42—22no—
381C42—22no—
382C42—42no—
383C42—42no—
384C42—42no—
385C42—42no—
386C42—44no—
387C42—84no—
388C42#12418yes—
389C42C2×#4114yes—
390C42C2×#4114yes—
391C422−1+414yes—
392C422−1+414yes—
393C23—22no—
394C23—22no—
395C23—21no—
396C23—21no—
397C23C2×#12412yes—
398C23C23×D411yescharacteristic
399Q8—88no—
400Q8—164no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—28no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—48no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—88no—
439C2×C4C2×#12412yes—
440C2×C4C2×#12414yes—
441C2×C4C2×#12414yes—
442C2×C4#53612yes—
443C2×C4#53614yes—
444C2×C4#53612yes—
445C2×C4C22×#4112yes—
446C2×C4C2×2−1+412yes—
447C2×C4C2×2−1+412yes—
448C8—44no—
449C8—44no—
450C22—24no—
451C22—22no—
452C22—41no—
453C22—41no—
454C22—42no—
455C22C22×#12411yescharacteristic
456C22C2×#53611yescharacteristic
457C22C2×#53611yescharacteristic
458C22#358512yes—
459C22#358612yes—
460C4—24no—
461C4—24no—
462C4—22no—
463C4—22no—
464C4—24no—
465C4—22no—
466C4—41no—
467C4—41no—
468C4—42no—
469C4—82no—
470C4—82no—
471C4—82no—
472C4—82no—
473C4—84no—
474C4—84no—
475C4#350914yes—
476C2—42no—
477C2#3532011yescharacteristic
478C2#6062112yes—
479C2#6088311yescharacteristic
480C2#6088411yescharacteristic
481C2#6088612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33299C2semidirect
4. #33297C2semidirect
5. #33295C2semidirect
6. #33294C2semidirect
7. #30235C2semidirect
8. #30235C2semidirect
9. #29229C2semidirect
10. #29228C2semidirect

Maximal subgroups

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

subgroupindex
#353352
#352942
#332992
#332972
#332952
#332942
#302352
#292292
#292282
#279712
#279702
#218392
#218382
#204672