Finite Group Databasebeta

← #4465862

Subgroups — #4465862

Order 512.

2208 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2734C211yesmaximal, characteristic, complemented
2C2×#2621C211yesmaximal, characteristic
3#17106C214yesmaximal
4#16993C212yesmaximal, complemented
5#16615C214yesmaximal, complemented
6#16522C214yesmaximal, complemented
7#15945C212yesmaximal, complemented
8#15905C214yesmaximal
9#15762C212yesmaximal, complemented
10#15430C212yesmaximal
11#15265C212yesmaximal
12#14907C212yesmaximal
13#14489C211yesmaximal, characteristic, complemented
14C22×C4⋊Q8C2211yescharacteristic
15C22×C4⋊Q8C2212yes—
16C22×C22⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×#117C2211yescharacteristic
19#2734—28no—
20#2621—28no—
21C2×#399C2212yes—
22C2×#398C2214yes—
23C2×#398C2212yes—
24C2×#398C2212yes—
25C2×#397C2214yes—
26C2×#397C2212yes—
27C2×#395C2214yes—
28C2×#395C2214yes—
29C2×#395C2212yes—
30C2×#395C2214yes—
31C2×#393C2212yes—
32C2×#391C2211yescharacteristic
33C2×#391C2211yescharacteristic
34C2×#389C2212yes—
35C2×#387C2214yes—
36C2×#385C2214yes—
37C2×#384C2212yes—
38C2×#382C2211yescharacteristic
39#1446C2212yes—
40#1445C2212yes—
41#1444C2214yes—
42#1444C2214yes—
43#1444C2214yes—
44#1444C2214yes—
45#1444C2214yes—
46#1443C2214yes—
47#1443C2214yes—
48#1443C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1441C2214yes—
54#1441C2212yes—
55#1441C2214yes—
56#1440C2214yes—
57#1439C2212yes—
58#1439C2214yes—
59#1439C2214yes—
60#1437C2212yes—
61#1437C2212yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2212yes—
65#1436C2212yes—
66#1434C2214yes—
67#1434C2214yes—
68#1434C2212yes—
69C23×Q8C2311yescharacteristic
70C23×Q8C2311yescharacteristic
71C23×Q8C2312yes—
72C24×C4C2311yescharacteristic
73C2×C4⋊Q8—24no—
74C2×C4⋊Q8—28no—
75C2×C4⋊Q8—44no—
76C2×C4⋊Q8—48no—
77C2×C22⋊Q8—24no—
78C2×C22⋊Q8—24no—
79C2×C22⋊Q8—42no—
80C2×C22⋊Q8—42no—
81C2×C22⋊Q8—42no—
82C2×C22⋊Q8—42no—
83C2×#117—24no—
84C2×#117—44no—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C42C2311yescharacteristic
105C22×C42C2312yes—
106C22×C42C2312yes—
107C22×C42C2311yescharacteristic
108#399—44no—
109#398—28no—
110#398—216no—
111#398—44no—
112#397—216no—
113#397—44no—
114#395—216no—
115#395—48no—
116#395—48no—
117#395—44no—
118#393—44no—
119#391—24no—
120#391D418yes—
121#389—28no—
122#387—216no—
123#385—48no—
124#384—44no—
125#382—24no—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C25C2411yescharacteristic
153C22×Q8—22no—
154C22×Q8—24no—
155C22×Q8—24no—
156C22×Q8—24no—
157C22×Q8—41no—
158C22×Q8—41no—
159C22×Q8—41no—
160C22×Q8—41no—
161C22×Q8—42no—
162C22×Q8—42no—
163C22×Q8—44no—
164C22×Q8—42no—
165C22×Q8—42no—
166C22×Q8—44no—
167C22×Q8C2×D414yes—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—24no—
172C23×C4—42no—
173C23×C4C2412yes—
174C23×C4C2414yes—
175C23×C4C2412yes—
176C23×C4C2414yes—
177C23×C4C2412yes—
178C23×C4C2412yes—
179C23×C4C2412yes—
180C23×C4C2412yes—
181C23×C4C2411yescharacteristic
182C23×C4C2414yes—
183C23×C4C2414yes—
184C23×C4C2411yescharacteristic
185C4⋊Q8—416no—
186C4⋊Q8—432no—
187C22⋊Q8—416no—
188C22⋊Q8—416no—
189#117—416no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4C2×D414yes—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C42—22no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—28no—
250C2×C42—42no—
251C2×C42—44no—
252C2×C42—42no—
253C2×C42C2×D414yes—
254#95—28no—
255#95—24no—
256#95—28no—
257#95—28no—
258#95—28no—
259#95—28no—
260#95—28no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—28no—
265#95—28no—
266#95—28no—
267#95—28no—
268#95—44no—
269#95—42no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—42no—
274#95—44no—
275#95—44no—
276#95—42no—
277#95—42no—
278#95—42no—
279#95C2×D4116yes—
280C24—21no—
281C24—22no—
282C24—21no—
283C24—22no—
284C24—24no—
285C24—24no—
286C24H(5,𝔽2)12yes—
287C24C2511yescharacteristic
288C2×Q8—28no—
289C2×Q8—44no—
290C2×Q8—44no—
291C2×Q8—44no—
292C2×Q8—44no—
293C2×Q8—48no—
294C2×Q8—44no—
295C2×Q8—84no—
296C2×Q8—84no—
297C2×Q8—84no—
298C2×Q8—84no—
299C2×Q8—84no—
300C2×Q8—84no—
301C2×Q8—88no—
302C2×Q8—88no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—28no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—48no—
364C22×C4—48no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4C22×D414yes—
368C22×C4C22×D414yes—
369C22×C4C22×D414yes—
370C22×C4C22×D412yes—
371C22×C4H(5,𝔽2)14yes—
372C22×C4H(5,𝔽2)12yes—
373C22×C4H(5,𝔽2)14yes—
374C22×C42−1+418yes—
375C22×C42−1+418yes—
376C4⋊C4—216no—
377C4⋊C4—44no—
378C4⋊C4—48no—
379C4⋊C4—48no—
380C4⋊C4—48no—
381C4⋊C4—416no—
382C4⋊C4—44no—
383C4⋊C4—416no—
384C4⋊C4—416no—
385C4⋊C4—84no—
386C4⋊C4—84no—
387C4⋊C4—88no—
388C4⋊C4—88no—
389C4⋊C4—88no—
390C22⋊C4—416no—
391C22⋊C4—48no—
392C22⋊C4—48no—
393C22⋊C4—48no—
394C22⋊C4—88no—
395C42—28no—
396C42—44no—
397C42—48no—
398C42—48no—
399C23—22no—
400C23—24no—
401C23—24no—
402C23—24no—
403C23—41no—
404C23—41no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—44no—
409C23—42no—
410C23—44no—
411C23—42no—
412C23—44no—
413C23—44no—
414C23C23×D411yescharacteristic
415C23C2×H(5,𝔽2)12yes—
416C23C2×H(5,𝔽2)11yescharacteristic
417C23C2×H(5,𝔽2)11yescharacteristic
418C23C2×H(5,𝔽2)12yes—
419C23C2×2−1+414yes—
420C23C2×2−1+414yes—
421Q8—88no—
422Q8—88no—
423Q8—816no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—28no—
437C2×C4—28no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—28no—
441C2×C4—28no—
442C2×C4—28no—
443C2×C4—28no—
444C2×C4—28no—
445C2×C4—28no—
446C2×C4—416no—
447C2×C4—44no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—44no—
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—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4#53514yes—
493C22—42no—
494C22—48no—
495C22—42no—
496C22—42no—
497C22—82no—
498C22—84no—
499C22—82no—
500C22C2×#53511yescharacteristic
501C22#348411yescharacteristic
502C22#348411yescharacteristic
503C22#348612yes—
504C22#349912yes—
505C22#353712yes—
506C22#353712yes—
507C22#353814yes—
508C22#353812yes—
509C22#354414yes—
510C22#354412yes—
511C22#354412yes—
512C22#355012yes—
513C22#355214yes—
514C22#355214yes—
515C4—24no—
516C4—48no—
517C4—48no—
518C4—44no—
519C4—44no—
520C4—44no—
521C4—42no—
522C4—44no—
523C4—84no—
524C4—82no—
525C4—84no—
526C4—82no—
527C2—82no—
528C2#3671711yescharacteristic
529C2#4279212yes—
530C2#4642612yes—
531C2#5154211yescharacteristic
532C2#5207211yescharacteristic
533C2#5512114yes—
534C2#5898314yes—
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×#2734C2semidirect
4. #16993C2semidirect
5. #16615C2semidirect
6. #16522C2semidirect
7. #15945C2semidirect
9. #15762C2semidirect
13. #14489C2semidirect

Maximal subgroups

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

subgroupindex
C2×#27342
C2×#26212
#171062
#169932
#166152
#165222
#159452
#159052
#157622
#154302
#152652
#149072
#144892