Finite Group Databasebeta

← #6490713

Subgroups — #6490713

Order 512.

1340 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35284C211yesmaximal, characteristic
2#34217C211yesmaximal, characteristic
3#33157C212yesmaximal
4#33156C212yesmaximal
5#30103C214yesmaximal, complemented
6#29285C214yesmaximal, complemented
7#29227C212yesmaximal, complemented
8#28694C212yesmaximal, complemented
9#27968C212yesmaximal
10#27621C212yesmaximal
11#27620C212yesmaximal
12#25819C212yesmaximal
13#22251C212yesmaximal, complemented
14#22250C212yesmaximal, complemented
15#20783C211yesmaximal, characteristic
16C22×#125C2211yescharacteristic
17C2×#501C2212yes—
18C2×#501C2212yes—
19#2868C2211yescharacteristic
20#2859C2211yescharacteristic
21#2857C2212yes—
22#2647C2212yes—
23#2629C2212yes—
24#2507—22no—
25#2497—22no—
26#2446—22no—
27#2429—22no—
28#2370—22no—
29#2370—22no—
30#2345—22no—
31#2345—22no—
32#2330C2212yes—
33#2330C2212yes—
34#2330C2212yes—
35#2316C2211yescharacteristic
36#2316C2211yescharacteristic
37#2314C2212yes—
38C4×C4⋊Q8C2212yes—
39C4×#125C2212yes—
40C4×#125C2212yes—
41#2285C2211yescharacteristic
42#2274C2211yescharacteristic
43#2274C2211yescharacteristic
44C4×#117C2211yescharacteristic
45#2101C2214yes—
46#2101C2214yes—
47#2100C2214yes—
48#2099C2214yes—
49#2084C2214yes—
50#2083C2214yes—
51#2059C2214yes—
52#2059C2214yes—
53#1993C2212yes—
54#1993C2212yes—
55#1992C2212yes—
56#1992C2212yes—
57#1989C2214yes—
58#1989C2214yes—
59#1948C2214yes—
60#1947C2214yes—
61#1921C2214yes—
62#1921C2214yes—
63#1920C2214yes—
64#1920C2214yes—
65#1919C2214yes—
66#1919C2214yes—
67#1918C2214yes—
68#1918C2214yes—
69#1853C2214yes—
70#1852C2214yes—
71#1840C2212yes—
72#1836C2212yes—
73#1836C2212yes—
74#1832C2212yes—
75#1825C2214yes—
76#1825C2214yes—
77#1818C2212yes—
78#1798C2212yes—
79#1767C2212yes—
80#1749C2212yes—
81#1745C2211yescharacteristic
82#1745C2211yescharacteristic
83C2×C4⋊Q8—22no—
84C2×C4⋊Q8C2312yes—
85C2×#125—28no—
86C2×#125—21no—
87C2×#125—21no—
88C2×#125C2312yes—
89C2×#125C2312yes—
90C2×#125C2312yes—
91C2×C22⋊Q8—22no—
92C2×C4×Q8—22no—
93C2×#117C2311yescharacteristic
94C2×#117C2311yescharacteristic
95C2×#117C2311yescharacteristic
96C2×#117C2311yescharacteristic
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C42C2311yescharacteristic
102#501—28no—
103#501—28no—
104C2×#107C2314yes—
105C2×#107C2314yes—
106C2×#106C2314yes—
107C2×#106C2314yes—
108C2×C4⋊C8C2314yes—
109C2×C22⋊C8C2314yes—
110C2×#97C2312yes—
111C2×#97C2312yes—
112#400—24no—
113#398—24no—
114#398—24no—
115#396—24no—
116#391—22no—
117#391—22no—
118#389—22no—
119#389—22no—
120#389—22no—
121#389—22no—
122#389—22no—
123#389—22no—
124#389—22no—
125#389—22no—
126#389—22no—
127#389—22no—
128#389—24no—
129#389C2312yes—
130#389C2312yes—
131#389C2312yes—
132#389C2312yes—
133#389C2314yes—
134#389C2314yes—
135#388—24no—
136#387—24no—
137#387—22no—
138#387—22no—
139#387—22no—
140#387—22no—
141#387—22no—
142#387—22no—
143#387—22no—
144#387—22no—
145#387—24no—
146#384—22no—
147#384C2312yes—
148#384C2312yes—
149#384C2312yes—
150#384C2312yes—
151#382C2312yes—
152#382C2312yes—
153#380—21no—
154#380—21no—
155#380—22no—
156#380—22no—
157#380—22no—
158#380C2314yes—
159C4×C4⋊C4—22no—
160C4×C4⋊C4—22no—
161C4×C4⋊C4—22no—
162C4×C4⋊C4—22no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4—21no—
167C4×C4⋊C4—21no—
168C4×C4⋊C4D412yes—
169C4×C4⋊C4D412yes—
170C4×C4⋊C4C2312yes—
171C4×C4⋊C4C2312yes—
172C4×C4⋊C4C2312yes—
173C4×C4⋊C4C2312yes—
174C4×C4⋊C4C2314yes—
175C4×C22⋊C4—22no—
176C4×C22⋊C4—22no—
177C4×C22⋊C4D412yes—
178C4×C22⋊C4D412yes—
179#376C2311yescharacteristic
180#376C2311yescharacteristic
181C43C2311yescharacteristic
182C43C2311yescharacteristic
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#336C2312yes—
196#336C2312yes—
197#336C2314yes—
198#336C2314yes—
199#336C2312yes—
200#336C2312yes—
201C22×Q8—22no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2411yescharacteristic
207C22×C8C2414yes—
208C22×C8C2414yes—
209C4⋊Q8—24no—
210C4⋊Q8—24no—
211C4⋊Q8—84no—
212#125—28no—
213#125—28no—
214#125—24no—
215#125—24no—
216#125—24no—
217#125—28no—
218#125#4118yes—
219C22⋊Q8—84no—
220C4×Q8—84no—
221#117—24no—
222#117—24no—
223#117—24no—
224#117—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
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—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4C2×D412yes—
250C2×C4⋊C4C2×D412yes—
251C2×C4⋊C4C2412yes—
252C2×C4⋊C4C2412yes—
253C2×C4⋊C4C2414yes—
254C2×C4⋊C4C2414yes—
255C2×C4⋊C4C2412yes—
256C2×C4⋊C4C2412yes—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4C2×D412yes—
261C2×C22⋊C4C2×D412yes—
262C2×C42—24no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—24no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42C2411yescharacteristic
277C2×C42C2411yescharacteristic
278C2×C42C2411yescharacteristic
279C2×C42C2411yescharacteristic
280C2×C42C2411yescharacteristic
281C2×C42C2411yescharacteristic
282#107—28no—
283#107—28no—
284#106—28no—
285#106—28no—
286C4⋊C8—28no—
287C22⋊C8—28no—
288#97—24no—
289#97—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#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95C2×D412yes—
306#95C2×D412yes—
307#95C2×D412yes—
308#95C2×D412yes—
309C24C22×D411yescharacteristic
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—44no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—82no—
328C22×C4—82no—
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D412yes—
331C22×C4C22×D412yes—
332C22×C4C22×D412yes—
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D412yes—
338C22×C42−1+414yes—
339C22×C4C2511yescharacteristic
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—24no—
345C2×C8—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—28no—
349C4⋊C4—28no—
350C4⋊C4—28no—
351C4⋊C4—28no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—28no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—48no—
365C4⋊C4—82no—
366C4⋊C4—82no—
367C4⋊C4—82no—
368C4⋊C4—82no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4—84no—
372C4⋊C4—84no—
373C4⋊C4—84no—
374C4⋊C4—84no—
375C4⋊C4C2×#4114yes—
376C4⋊C4C2×#4114yes—
377C22⋊C4—24no—
378C22⋊C4—24no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—84no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—42no—
389C42—42no—
390C42—42no—
391C42—42no—
392C42—42no—
393C42—42no—
394C42—82no—
395C42—82no—
396C42—82no—
397C42—82no—
398C42—84no—
399C42C2×#4114yes—
400C422−1+414yes—
401C23—22no—
402C23—22no—
403C23—21no—
404C23—21no—
405C23#53612yes—
406C23C23×D411yescharacteristic
407Q8—88no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—28no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—84no—
449C2×C4—84no—
450C2×C4#52512yes—
451C2×C4#52512yes—
452C2×C4#53612yes—
453C2×C4#53614yes—
454C2×C4C22×#4112yes—
455C2×C4C2×2−1+412yes—
456C2×C4C2×2−1+412yes—
457C8—44no—
458C8—44no—
459C22—22no—
460C22—24no—
461C22—41no—
462C22—41no—
463C22—42no—
464C22C2×#52511yescharacteristic
465C22C2×#53611yescharacteristic
466C22C2×#53611yescharacteristic
467C22#358512yes—
468C22#358612yes—
469C4—22no—
470C4—22no—
471C4—22no—
472C4—24no—
473C4—41no—
474C4—44no—
475C4—41no—
476C4—44no—
477C4—42no—
478C4—42no—
479C4—42no—
480C4—84no—
481C4—84no—
482C4—82no—
483C4—82no—
484C4#352314yes—
485C2—42no—
486C2#3524811yescharacteristic
487C2#6056811yescharacteristic
488C2#6056911yescharacteristic
489C2#6088612yes—
490C2#6088712yes—
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
5. #30103C2semidirect
6. #29285C2semidirect
7. #29227C2semidirect
8. #28694C2semidirect
13. #22251C2semidirect
14. #22250C2semidirect

Maximal subgroups

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

subgroupindex
#352842
#342172
#331572
#331562
#301032
#292852
#292272
#286942
#279682
#276212
#276202
#258192
#222512
#222502
#207832