Finite Group Databasebeta

← #6528574

Subgroups — #6528574

Order 512.

1834 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35166C211yesmaximal, characteristic
2C2×#2570C211yesmaximal, characteristic, complemented
3#29334C212yesmaximal, complemented
4#29328C212yesmaximal, complemented
5#27982C212yesmaximal, complemented
6#27973C212yesmaximal, complemented
7#27760C218yesmaximal, complemented
8#27744C218yesmaximal, complemented
9#21865C212yesmaximal, complemented
10#21395C212yesmaximal, complemented
11#20534C211yesmaximal, characteristic, complemented
12C2×#537—21no—
13C2×#537—21no—
14C2×#532C2212yes—
15C2×#432C2212yes—
16#2856C2212yes—
17#2856C2212yes—
18#2656—21no—
19#2656—21no—
20#2570—24no—
21#2570C2218yes—
22#2542C2214yes—
23#2541C2214yes—
24#2540—24no—
25C2×#400—21no—
26C2×#398—21no—
27C2×#389C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#387—21no—
30C2×#387—21no—
31C2×#384C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36#2100C2214yes—
37#2099C2214yes—
38#2073C2218yes—
39#2071C2218yes—
40#2041C2214yes—
41#2041C2214yes—
42#1993C2214yes—
43#1987C2214yes—
44#1982C2214yes—
45#1961C2214yes—
46#1961C2214yes—
47#1957C2214yes—
48#1936C2218yes—
49#1934C2218yes—
50#1874C2214yes—
51#1874C2214yes—
52#1874C2214yes—
53#1874C2214yes—
54#1836C2214yes—
55#1834C2214yes—
56#1804C2218yes—
57#1771C2218yes—
58#1762C2218yes—
59#1749C2212yes—
60C4×C22⋊C8C2212yes—
61C22×#41—22no—
62#537—48no—
63#537—48no—
64#532—28no—
65#532—28no—
66C2×C4⋊Q8C2312yes—
67C2×#127C2312yes—
68C2×#127C2312yes—
69C2×#127C2312yes—
70C2×#124—22no—
71C2×C22⋊Q8—24no—
72C2×C22⋊Q8—24no—
73C2×#121—22no—
74C2×#121—22no—
75C2×#121—22no—
76C2×#121—22no—
77C2×#121—22no—
78C2×#121—24no—
79C2×#121—24no—
80C2×#121C2314yes—
81C2×C4×D4—22no—
82C2×C4×D4C2314yes—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4D411yescharacteristic
88C22×C4⋊C4D411yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C42—21no—
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96C22×C42C2311yescharacteristic
97#432—28no—
98C2×C4⋊C8C2314yes—
99C2×#102C2318yes—
100C2×#102C2318yes—
101C2×#102C2318yes—
102C2×#102C2318yes—
103C2×C22⋊C8C2314yes—
104C2×#97C2314yes—
105C2×#97C2314yes—
106C2×C4×C8C2314yes—
107#400—28no—
108#399—24no—
109#399—24no—
110#398—28no—
111#392—24no—
112#392—24no—
113#390—22no—
114#390—22no—
115#390—22no—
116#390—22no—
117#390C2314yes—
118#390C2314yes—
119#389—22no—
120#389—22no—
121#389C2314yes—
122#389C2314yes—
123#387—28no—
124#387—28no—
125#384—22no—
126#384C2314yes—
127#384C2312yes—
128#384C2312yes—
129#382—22no—
130#382—22no—
131#382C2314yes—
132#382C2314yes—
133C4×C4⋊C4—22no—
134C4×C4⋊C4D414yes—
135C4×C4⋊C4C2314yes—
136C4×C4⋊C4C2314yes—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95—21no—
140C2×#95—21no—
141C2×#95—21no—
142C2×#95—21no—
143C2×#95—21no—
144C2×#95D411yescharacteristic
145C2×#95D411yescharacteristic
146#340C2318yes—
147#340C2318yes—
148#340C2318yes—
149#340C2318yes—
150#336C2314yes—
151#336C2314yes—
152C2×#41—48no—
153C2×#41—48no—
154C2×#41—48no—
155C22×Q8—22no—
156C22×D4—24no—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4C2414yes—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—24no—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2411yescharacteristic
176C22×C8C2418yes—
177C4⋊Q8—24no—
178C4⋊Q8—24no—
179#127—24no—
180#127—24no—
181#127—44no—
182#127—44no—
183#124—48no—
184C22⋊Q8—416no—
185C22⋊Q8—416no—
186#121—28no—
187#121—28no—
188#121—416no—
189#121—416no—
190#121—48no—
191#121—48no—
192#121—48no—
193#121—48no—
194#121—48no—
195C4×D4—28no—
196C4×D4—28no—
197C4×D4—48no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4C2×D414yes—
222C2×C4⋊C4C2×D414yes—
223C2×C4⋊C4C2412yes—
224C2×C4⋊C4C2412yes—
225C2×C4⋊C4C2412yes—
226C2×C4⋊C4C2412yes—
227C2×C4⋊C4C2414yes—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C42—24no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42—82no—
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42#4118yes—
242C2×C42C2412yes—
243C2×C42C2412yes—
244C2×C42C2412yes—
245C4⋊C8—28no—
246#102—48no—
247#102—48no—
248#102—48no—
249#102—48no—
250C22⋊C8—28no—
251#97—28no—
252#97—44no—
253C4×C8—44no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95C2×D414yes—
262#95C2×D414yes—
263C24—24no—
264C24C22×D411yescharacteristic
265#41—832no—
266C2×Q8—44no—
267C2×Q8—44no—
268C2×Q8—44no—
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—28no—
272C2×D4—28no—
273C2×D4—44no—
274C2×D4—44no—
275C2×D4—44no—
276C2×D4—44no—
277C2×D4—48no—
278C2×D4—48no—
279C2×D4—48no—
280C2×D4—44no—
281C2×D4—44no—
282C2×D4—44no—
283C2×D4—44no—
284C2×D4—82no—
285C2×D4—82no—
286C2×D4—82no—
287C2×D4—82no—
288C2×D4—82no—
289C2×D4—82no—
290C2×D4—82no—
291C2×D4—82no—
292C2×D4—82no—
293C2×D4—82no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—48no—
316C22×C4—48no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—48no—
330C22×C4#12414yes—
331C22×C4#12414yes—
332C22×C4AGL(1,ℤ/8)14yes—
333C22×C4AGL(1,ℤ/8)14yes—
334C22×C4C22×D412yes—
335C22×C4C22×D412yes—
336C22×C4C22×D412yes—
337C22×C4C22×D412yes—
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D412yes—
340C22×C4C22×D412yes—
341C22×C4C2×#4114yes—
342C22×C4C2511yescharacteristic
343C2×C8—28no—
344C2×C8—28no—
345C2×C8—28no—
346C4⋊C4—28no—
347C4⋊C4—28no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—48no—
362C4⋊C4—48no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—84no—
366C4⋊C4—84no—
367C22⋊C4—48no—
368C22⋊C4—48no—
369C22⋊C4—48no—
370C22⋊C4—48no—
371C22⋊C4—48no—
372C22⋊C4—48no—
373C42—24no—
374C42—24no—
375C42—42no—
376C42—42no—
377C42—84no—
378C42#12418yes—
379C42C2×#4114yes—
380C42C2×#4114yes—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—44no—
386C23—44no—
387C23—44no—
388C23—44no—
389C23—44no—
390C23—44no—
391C23—44no—
392C23C2×#12412yes—
393C23C2×AGL(1,ℤ/8)12yes—
394C23C2×AGL(1,ℤ/8)12yes—
395C23C23×D411yescharacteristic
396Q8—88no—
397D4—48no—
398D4—48no—
399D4—88no—
400D4—88no—
401D4—816no—
402D4—164no—
403D4—164no—
404D4—164no—
405D4—164no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—28no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—28no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—816no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—84no—
446C2×C4—84no—
447C2×C4—84no—
448C2×C4—84no—
449C2×C4C2×#12412yes—
450C2×C4C2×#12414yes—
451C2×C4C2×#12414yes—
452C2×C4C2×AGL(1,ℤ/8)14yes—
453C2×C4C2×AGL(1,ℤ/8)14yes—
454C2×C4C22×#4112yes—
455C2×C4#58512yes—
456C2×C4#58512yes—
457C8—48no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—84no—
465C22—84no—
466C22—84no—
467C22—84no—
468C22—84no—
469C22—84no—
470C22—84no—
471C22#313814yes—
472C22C22×#12411yescharacteristic
473C22#348511yescharacteristic
474C22#348711yescharacteristic
475C22C22×AGL(1,ℤ/8)11yescharacteristic
476C22C22×AGL(1,ℤ/8)11yescharacteristic
477C22#358511yescharacteristic
478C22#358611yescharacteristic
479C4—24no—
480C4—24no—
481C4—22no—
482C4—24no—
483C4—24no—
484C4—42no—
485C4—42no—
486C4—82no—
487C4—82no—
488C4—82no—
489C4—82no—
490C4—84no—
491C4#351214yes—
492C2—24no—
493C2—164no—
494C2#3564911yescharacteristic
495C2C2×#313811yescharacteristic
496C2#6062111yescharacteristic
497C2#6170411yescharacteristic
498C2#6170711yescharacteristic
499C2#6188811yescharacteristic
500C2#6189111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35166C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2570C2semidirect
3. #29334C2semidirect
4. #29328C2semidirect
5. #27982C2semidirect
6. #27973C2semidirect
7. #27760C2semidirect
8. #27744C2semidirect
9. #21865C2semidirect
10. #21395C2semidirect
11. #20534C2semidirect

Maximal subgroups

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

subgroupindex
#351662
C2×#25702
#293342
#293282
#279822
#279732
#277602
#277442
#218652
#213952
#205342