Finite Group Databasebeta

← #2811120

Subgroups — #2811120

Order 512.

2212 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2582C211yesmaximal, characteristic, complemented
2C2×#2499C211yesmaximal, characteristic
3#16629C212yesmaximal, complemented
4#16528C214yesmaximal
5#16068C214yesmaximal, complemented
6#15945C212yesmaximal, complemented
7#15737C214yesmaximal
8#15430C212yesmaximal
9#14907C212yesmaximal
10#14881C212yesmaximal, complemented
11#14424C211yesmaximal, characteristic, complemented
12#14406C214yesmaximal, complemented
13#14172C212yesmaximal
14C22×C22⋊Q8C2211yescharacteristic
15C22×C4×Q8C2212yes—
16C22×#117C2211yescharacteristic
17#2582—28no—
18#2499—28no—
19C2×#399C2212yes—
20C2×#398C2212yes—
21C2×#398C2212yes—
22C2×#397C2214yes—
23C2×#397C2212yes—
24C2×#395C2214yes—
25C2×#393C2212yes—
26C2×#391C2211yescharacteristic
27C2×#391C2214yes—
28C2×#391C2211yescharacteristic
29C2×#389C2214yes—
30C2×#389C2212yes—
31C2×#387C2214yes—
32C2×#387C2214yes—
33C2×#385C2214yes—
34C2×#384C2212yes—
35C2×#384C2211yescharacteristic
36C2×#382C2211yescharacteristic
37C2×C4×C22⋊C4C2211yescharacteristic
38#1446C2214yes—
39#1446C2212yes—
40#1445C2212yes—
41#1445C2214yes—
42#1444C2214yes—
43#1444C2212yes—
44#1444C2214yes—
45#1444C2214yes—
46#1443C2214yes—
47#1443C2214yes—
48#1443C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1441C2214yes—
52#1441C2212yes—
53#1441C2214yes—
54#1440C2214yes—
55#1439C2212yes—
56#1439C2214yes—
57#1439C2212yes—
58#1437C2212yes—
59#1437C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2212yes—
65#1436C2214yes—
66#1434C2212yes—
67#1434C2212yes—
68#1434C2214yes—
69C23×Q8C2312yes—
70C23×Q8C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C2×C22⋊Q8—24no—
73C2×C22⋊Q8—42no—
74C2×C22⋊Q8—42no—
75C2×C4×Q8—28no—
76C2×C4×Q8—48no—
77C2×#117—24no—
78C2×#117—42no—
79C2×#117—42no—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C42C2312yes—
99C22×C42C2312yes—
100C22×C42C2311yescharacteristic
101C22×C42C2314yes—
102C22×C42C2311yescharacteristic
103#399—44no—
104#398—28no—
105#398—28no—
106#397—216no—
107#397—44no—
108#395—48no—
109#393—28no—
110#391—216no—
111#391—24no—
112#391—24no—
113#389—216no—
114#389—44no—
115#387—216no—
116#387—216no—
117#385—48no—
118#384—44no—
119#384D418yes—
120#382—24no—
121C4×C22⋊C4—24no—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2312yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C25C2411yescharacteristic
150C22×Q8—24no—
151C22×Q8—22no—
152C22×Q8—24no—
153C22×Q8—24no—
154C22×Q8—28no—
155C22×Q8—24no—
156C22×Q8—44no—
157C22×Q8—41no—
158C22×Q8—41no—
159C22×Q8—42no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—24no—
164C23×C4—42no—
165C23×C4C2414yes—
166C23×C4C2412yes—
167C23×C4C2414yes—
168C23×C4C2412yes—
169C23×C4C2412yes—
170C23×C4C2412yes—
171C23×C4C2412yes—
172C23×C4C2414yes—
173C23×C4C2411yescharacteristic
174C23×C4C2414yes—
175C23×C4C2412yes—
176C23×C4C2411yescharacteristic
177C22⋊Q8—416no—
178C4×Q8—432no—
179#117—416no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4C2×D4116yes—
215C2×C4⋊C4C2×D414yes—
216C2×C4⋊C4C2×D414yes—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—28no—
235C2×C42—22no—
236C2×C42—28no—
237C2×C42—24no—
238C2×C42—42no—
239C2×C42—44no—
240C2×C42—44no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42C2×D414yes—
244#95—24no—
245#95—28no—
246#95—28no—
247#95—28no—
248#95—28no—
249#95—28no—
250#95—24no—
251#95—24no—
252#95—28no—
253#95—28no—
254#95—28no—
255#95—28no—
256#95—28no—
257#95—28no—
258#95—28no—
259#95—28no—
260#95—24no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—42no—
268#95—44no—
269#95—42no—
270#95#4118yes—
271C24—21no—
272C24—22no—
273C24—21no—
274C24—22no—
275C24—22no—
276C24—24no—
277C24—22no—
278C24H(5,𝔽2)12yes—
279C24C2511yescharacteristic
280C2×Q8—216no—
281C2×Q8—416no—
282C2×Q8—416no—
283C2×Q8—48no—
284C2×Q8—44no—
285C2×Q8—44no—
286C2×Q8—44no—
287C2×Q8—48no—
288C2×Q8—84no—
289C2×Q8—84no—
290C22×C4—24no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—28no—
296C22×C4—24no—
297C22×C4—21no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
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—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—48no—
350C22×C4—48no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4C22×D414yes—
354C22×C4C22×D414yes—
355C22×C4C22×D412yes—
356C22×C4C22×D414yes—
357C22×C4C2×#4118yes—
358C22×C4C2×#4114yes—
359C22×C4H(5,𝔽2)12yes—
360C22×C4H(5,𝔽2)14yes—
361C22×C42−1+418yes—
362C22×C42−1+414yes—
363C4⋊C4—232no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—48no—
367C4⋊C4—48no—
368C4⋊C4—44no—
369C4⋊C4—48no—
370C4⋊C4—416no—
371C4⋊C4—84no—
372C4⋊C4—84no—
373C4⋊C4—88no—
374C4⋊C4—88no—
375C4⋊C4—88no—
376C22⋊C4—416no—
377C22⋊C4—48no—
378C22⋊C4—48no—
379C22⋊C4—48no—
380C22⋊C4—88no—
381C42—28no—
382C42—48no—
383C42—48no—
384C42—44no—
385C42—416no—
386C23—22no—
387C23—24no—
388C23—24no—
389C23—24no—
390C23—41no—
391C23—42no—
392C23—41no—
393C23—42no—
394C23—41no—
395C23—44no—
396C23—44no—
397C23—42no—
398C23—44no—
399C23—42no—
400C23—41no—
401C23—44no—
402C23C23×D411yescharacteristic
403C23C22×#4112yes—
404C23C2×H(5,𝔽2)12yes—
405C23C2×H(5,𝔽2)11yescharacteristic
406C23C2×H(5,𝔽2)11yescharacteristic
407C23C2×2−1+412yes—
408C23C2×2−1+414yes—
409C23#58512yes—
410Q8—432no—
411Q8—88no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—28no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—28no—
420C2×C4—28no—
421C2×C4—28no—
422C2×C4—28no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—28no—
427C2×C4—28no—
428C2×C4—28no—
429C2×C4—28no—
430C2×C4—24no—
431C2×C4—28no—
432C2×C4—416no—
433C2×C4—44no—
434C2×C4—41no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4#53514yes—
480C2×C4#54718yes—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—48no—
485C22—82no—
486C22—82no—
487C22—82no—
488C22—82no—
489C22C2×#53511yescharacteristic
490C22C2×#54712yes—
491C22#348111yescharacteristic
492C22#348411yescharacteristic
493C22#348612yes—
494C22#348711yescharacteristic
495C22#350714yes—
496C22#352014yes—
497C22#352212yes—
498C22#353412yes—
499C22#353412yes—
500C22#353814yes—
501C22#354412yes—
502C22#355012yes—
503C22#355011yescharacteristic
504C22#355514yes—
505C4—28no—
506C4—24no—
507C4—48no—
508C4—42no—
509C4—44no—
510C4—44no—
511C4—48no—
512C4—82no—
513C4—84no—
514C4—82no—
515C4—82no—
516C4—84no—
517C2—82no—
518C2#3671011yescharacteristic
519C2#3891712yes—
520C2#4346212yes—
521C2#4539412yes—
522C2#5169414yes—
523C2#5207211yescharacteristic
524C2#5646312yes—
525C2#5907011yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2582C2semidirect
3. #16629C2semidirect
5. #16068C2semidirect
6. #15945C2semidirect
10. #14881C2semidirect
11. #14424C2semidirect
12. #14406C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25822
C2×#24992
#166292
#165282
#160682
#159452
#157372
#154302
#149072
#148812
#144242
#144062
#141722