Finite Group Databasebeta

← #6490716

Subgroups — #6490716

Order 512.

1348 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35336C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic
3#33158C214yesmaximal
4#30104C214yesmaximal, complemented
5#29285C214yesmaximal, complemented
6#29223C212yesmaximal, complemented
7#28693C212yesmaximal, complemented
8#27966C211yesmaximal, characteristic
9#27965C211yesmaximal, characteristic
10#27622C214yesmaximal
11#25817C211yesmaximal, characteristic
12#25817C211yesmaximal, characteristic
13#22251C212yesmaximal, complemented
14#22250C212yesmaximal, complemented
15#20783C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C2×#501C2212yes—
18C2×#501C2212yes—
19#2859C2211yescharacteristic
20#2859C2211yescharacteristic
21#2851C2211yescharacteristic
22#2851C2211yescharacteristic
23#2647C2212yes—
24#2629C2212yes—
25#2590—22no—
26#2590—22no—
27#2587—22no—
28#2587—22no—
29#2360—22no—
30#2360—22no—
31#2355—22no—
32Q8×C4⋊C4—22no—
33#2330C2212yes—
34#2330C2212yes—
35#2330C2212yes—
36#2317C2212yes—
37#2314C2212yes—
38C4×C4⋊Q8C2212yes—
39C4×C4⋊Q8C2212yes—
40C4×#125C2211yescharacteristic
41C4×#125C2211yescharacteristic
42C2×#384C2211yescharacteristic
43#2274C2212yes—
44C4×#117C2211yescharacteristic
45#2100C2214yes—
46#2100C2214yes—
47#2099C2214yes—
48#2099C2214yes—
49#2085C2214yes—
50#2085C2214yes—
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#1854C2214yes—
70#1854C2214yes—
71#1840C2212yes—
72#1836C2212yes—
73#1836C2212yes—
74#1832C2212yes—
75#1824C2214yes—
76#1823C2214yes—
77#1818C2212yes—
78#1798C2212yes—
79#1767C2212yes—
80#1749C2211yescharacteristic
81#1749C2211yescharacteristic
82#1745C2212yes—
83#533—28no—
84C2×C4⋊Q8C2312yes—
85C2×#125C2311yescharacteristic
86C2×#125C2311yescharacteristic
87C2×#125C2312yes—
88C2×#125C2312yes—
89C2×C22⋊Q8—22no—
90C2×C4×Q8—22no—
91C2×C4×Q8—22no—
92C2×C4×Q8—22no—
93C2×#117C2312yes—
94C2×#117C2311yescharacteristic
95C2×#117C2311yescharacteristic
96C22×C4⋊C4—22no—
97C22×C4⋊C4C2312yes—
98C22×C42C2311yescharacteristic
99#501—28no—
100#501—28no—
101C2×#107C2314yes—
102C2×#107C2314yes—
103C2×#106C2314yes—
104C2×#106C2314yes—
105C2×C4⋊C8C2314yes—
106C2×C22⋊C8C2312yes—
107C2×C22⋊C8C2312yes—
108C2×#97C2312yes—
109C2×#97C2312yes—
110#400—22no—
111#400—22no—
112#398—22no—
113#398—22no—
114#398—22no—
115#398—22no—
116#396—22no—
117#396—22no—
118#391—22no—
119#391—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#389—24no—
130#389C2312yes—
131#389C2312yes—
132#389C2314yes—
133#389C2314yes—
134#389C2312yes—
135#389C2312yes—
136#388—22no—
137#388—22no—
138#387—22no—
139#387—22no—
140#387—22no—
141#387—22no—
142#387—22no—
143#387—22no—
144#387—22no—
145#387—22no—
146#387—24no—
147#387—24no—
148#385—22no—
149#385—22no—
150#385—22no—
151#385—22no—
152#384—22no—
153#384—22no—
154#384C2312yes—
155#384C2312yes—
156#384C2312yes—
157#384C2312yes—
158#382—22no—
159#382C2311yescharacteristic
160#382C2311yescharacteristic
161#382C2312yes—
162#380C2312yes—
163#380C2312yes—
164C4×C4⋊C4—24no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4—22no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4D412yes—
170C4×C4⋊C4D412yes—
171C4×C4⋊C4C2312yes—
172C4×C4⋊C4C2312yes—
173C4×C4⋊C4C2314yes—
174C4×C4⋊C4C2314yes—
175C4×C22⋊C4—24no—
176C4×C22⋊C4D412yes—
177C4×C22⋊C4D412yes—
178#376C2311yescharacteristic
179#376C2311yescharacteristic
180C43C2311yescharacteristic
181C43C2311yescharacteristic
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#336C2314yes—
195#336C2312yes—
196#336C2312yes—
197#336C2314yes—
198#336C2314yes—
199C22×Q8—22no—
200C23×C4—22no—
201C23×C4C2×D412yes—
202C23×C4C2411yescharacteristic
203C22×C8C2414yes—
204C22×C8C2414yes—
205C4⋊Q8—24no—
206C4⋊Q8#4118yes—
207#125—28no—
208#125—24no—
209#125—24no—
210#125—28no—
211C22⋊Q8—216no—
212C4×Q8—216no—
213C4×Q8—84no—
214C4×Q8—84no—
215#117—24no—
216#117—28no—
217#117—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—84no—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C4⋊C4C2414yes—
244C2×C4⋊C4C2412yes—
245C2×C4⋊C4C2412yes—
246C2×C4⋊C4C2412yes—
247C2×C4⋊C4C2412yes—
248C2×C4⋊C4C2414yes—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4C2×D412yes—
253C2×C22⋊C4C2×D411yescharacteristic
254C2×C22⋊C4C2×D411yescharacteristic
255C2×C42—22no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42C2×D412yes—
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2411yescharacteristic
267C2×C42C2411yescharacteristic
268C2×C42C2411yescharacteristic
269C2×C42C2411yescharacteristic
270C2×C42C2411yescharacteristic
271C2×C42C2411yescharacteristic
272#107—28no—
273#107—28no—
274#106—28no—
275#106—28no—
276C4⋊C8—28no—
277C22⋊C8—24no—
278C22⋊C8—24no—
279#97—24no—
280#97—24no—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95C2×D412yes—
298#95C2×D412yes—
299#95C2×D412yes—
300#95C2×D412yes—
301C24C22×D411yescharacteristic
302C2×Q8—28no—
303C2×Q8—44no—
304C2×Q8—44no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—44no—
317C22×C4—84no—
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D412yes—
321C22×C4C22×D412yes—
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D412yes—
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D412yes—
327C22×C42−1+414yes—
328C22×C4C2511yescharacteristic
329C2×C8—24no—
330C2×C8—24no—
331C2×C8—24no—
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—24no—
335C4⋊C4—216no—
336C4⋊C4—28no—
337C4⋊C4—24no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—28no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—44no—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—82no—
356C4⋊C4—82no—
357C4⋊C4—82no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C4⋊C4—82no—
362C4⋊C4—82no—
363C4⋊C4—84no—
364C4⋊C4—84no—
365C4⋊C4—84no—
366C4⋊C4C2×#4118yes—
367C22⋊C4—28no—
368C22⋊C4—22no—
369C22⋊C4—22no—
370C22⋊C4—24no—
371C22⋊C4—82no—
372C22⋊C4—82no—
373C42—28no—
374C42—24no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—42no—
380C42—42no—
381C42—42no—
382C42—42no—
383C42—82no—
384C42—82no—
385C42—84no—
386C42—84no—
387C42C2×#4114yes—
388C422−1+414yes—
389C23—22no—
390C23—22no—
391C23—21no—
392C23—21no—
393C23#53612yes—
394C23C23×D411yescharacteristic
395Q8—416no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—28no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—28no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—48no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—164no—
437C2×C4#52514yes—
438C2×C4#53614yes—
439C2×C4#53612yes—
440C2×C4C22×#4112yes—
441C2×C4C2×2−1+412yes—
442C2×C4C2×2−1+412yes—
443C8—44no—
444C8—44no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—42no—
449C22—42no—
450C22C2×#52511yescharacteristic
451C22C2×#53611yescharacteristic
452C22C2×#53611yescharacteristic
453C22#358512yes—
454C22#358612yes—
455C4—24no—
456C4—22no—
457C4—24no—
458C4—42no—
459C4—48no—
460C4—41no—
461C4—41no—
462C4—41no—
463C4—41no—
464C4—42no—
465C4—82no—
466C4—82no—
467C4—84no—
468C4—82no—
469C4—82no—
470C4#352314yes—
471C2—42no—
472C2#3524811yescharacteristic
473C2#6056512yes—
474C2#6088311yescharacteristic
475C2#6088411yescharacteristic
476C2#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
4. #30104C2semidirect
5. #29285C2semidirect
6. #29223C2semidirect
7. #28693C2semidirect
13. #22251C2semidirect
14. #22250C2semidirect

Maximal subgroups

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

subgroupindex
#353362
#342142
#331582
#301042
#292852
#292232
#286932
#279662
#279652
#276222
#258172
#222512
#222502
#207832