Finite Group Databasebeta

← #4598446

Subgroups — #4598446

Order 512.

2058 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2822C211yesmaximal, characteristic, complemented
2C2×#2503C211yesmaximal, characteristic
3#17203C214yesmaximal, complemented
4#16629C212yesmaximal, complemented
5#16608C212yesmaximal, complemented
6#16523C214yesmaximal, complemented
7#16300C214yesmaximal
8#15945C212yesmaximal, complemented
9#15482C214yesmaximal
10#15430C212yesmaximal
11#14916C212yesmaximal
12#14821C212yesmaximal
13#14476C211yesmaximal, characteristic, complemented
14C22×#125C2212yes—
15C22×C22⋊Q8C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17#2822—28no—
18#2503—28no—
19C2×#400C2214yes—
20C2×#400C2214yes—
21C2×#399C2212yes—
22C2×#398C2212yes—
23C2×#398C2214yes—
24C2×#398C2212yes—
25C2×#398C2212yes—
26C2×#395C2214yes—
27C2×#391C2211yescharacteristic
28C2×#391C2211yescharacteristic
29C2×#385C2214yes—
30C2×#384C2212yes—
31C2×#382C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#381C2214yes—
34C2×#380C2212yes—
35C2×C4×C22⋊C4C2211yescharacteristic
36#1445C2212yes—
37#1444C2214yes—
38#1444C2212yes—
39#1444C2214yes—
40#1444C2214yes—
41#1443C2214yes—
42#1443C2214yes—
43#1443C2214yes—
44#1442C2212yes—
45#1442C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1441C2212yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2212yes—
57#1439C2212yes—
58#1439C2214yes—
59#1439C2212yes—
60#1439C2214yes—
61#1437C2212yes—
62#1436C2214yes—
63#1436C2212yes—
64#1434C2212yes—
65#1434C2214yes—
66#1434C2214yes—
67C22⋊#95C2212yes—
68C22⋊#95C2214yes—
69C23×Q8C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C24×C4C2312yes—
72C2×#125—28no—
73C2×#125—48no—
74C2×C22⋊Q8—24no—
75C2×C22⋊Q8—42no—
76C2×C22⋊Q8—42no—
77C22×C4⋊C4—24no—
78C22×C4⋊C4—42no—
79C22×C4⋊C4—42no—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C42C2312yes—
100C22×C42C2311yescharacteristic
101C22×C42C2311yescharacteristic
102#400—216no—
103#400—216no—
104#399—44no—
105#398—28no—
106#398—28no—
107#398—216no—
108#398—44no—
109#395—48no—
110#391—24no—
111#391—24no—
112#385—48no—
113#384—44no—
114#382—24no—
115#382D418yes—
116#381—216no—
117#380—44no—
118C4×C22⋊C4—24no—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2312yes—
123C2×#95C2312yes—
124C2×#95C2312yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C25C2411yescharacteristic
148C22×Q8—22no—
149C22×Q8—24no—
150C22×Q8—41no—
151C22×Q8—41no—
152C22×Q8—42no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—24no—
157C23×C4—28no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—44no—
163C23×C4C2414yes—
164C23×C4C2412yes—
165C23×C4C2414yes—
166C23×C4C2414yes—
167C23×C4C2412yes—
168C23×C4C2412yes—
169C23×C4C2412yes—
170C23×C4C2412yes—
171C23×C4C2412yes—
172C23×C4C2411yescharacteristic
173C23×C4C2414yes—
174C23×C4C2411yescharacteristic
175#125—432no—
176C22⋊Q8—416no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—416no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4C2×D414yes—
219C2×C4⋊C4C2×D414yes—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—28no—
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—22no—
234C2×C42—28no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42C2×D414yes—
238#95—24no—
239#95—28no—
240#95—28no—
241#95—28no—
242#95—28no—
243#95—28no—
244#95—24no—
245#95—24no—
246#95—28no—
247#95—28no—
248#95—28no—
249#95—28no—
250#95—42no—
251#95—44no—
252#95—44no—
253#95—42no—
254#95—44no—
255#95—44no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—44no—
261#95—44no—
262#95—42no—
263#95—44no—
264#95—42no—
265#95C2×D4116yes—
266C24—21no—
267C24—22no—
268C24—21no—
269C24—24no—
270C24—22no—
271C24—22no—
272C24H(5,𝔽2)12yes—
273C242−1+414yes—
274C24C2511yescharacteristic
275C2×Q8—44no—
276C2×Q8—44no—
277C2×Q8—44no—
278C2×Q8—84no—
279C2×Q8—84no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—28no—
284C22×C4—21no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—48no—
343C22×C4—48no—
344C22×C4—44no—
345C22×C4—48no—
346C22×C4—416no—
347C22×C4—44no—
348C22×C4—48no—
349C22×C4—84no—
350C22×C4—88no—
351C22×C4C22×D414yes—
352C22×C4C22×D414yes—
353C22×C4C22×D414yes—
354C22×C4C22×D412yes—
355C22×C4H(5,𝔽2)12yes—
356C22×C4H(5,𝔽2)14yes—
357C22×C42−1+418yes—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—48no—
361C4⋊C4—48no—
362C4⋊C4—44no—
363C4⋊C4—48no—
364C4⋊C4—416no—
365C4⋊C4—48no—
366C4⋊C4—416no—
367C4⋊C4—84no—
368C4⋊C4—84no—
369C4⋊C4—88no—
370C4⋊C4—88no—
371C4⋊C4—84no—
372C4⋊C4—88no—
373C22⋊C4—48no—
374C22⋊C4—416no—
375C22⋊C4—48no—
376C22⋊C4—88no—
377C42—28no—
378C42—48no—
379C42—44no—
380C23—22no—
381C23—24no—
382C23—24no—
383C23—28no—
384C23—24no—
385C23—22no—
386C23—24no—
387C23—28no—
388C23—41no—
389C23—41no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—44no—
394C23—44no—
395C23#56314yes—
396C23C23×D411yescharacteristic
397C23C2×H(5,𝔽2)12yes—
398C23C2×H(5,𝔽2)11yescharacteristic
399C23C2×H(5,𝔽2)11yescharacteristic
400C23C2×2−1+412yes—
401C23C2×2−1+414yes—
402C23#58512yes—
403C23#58512yes—
404Q8—88no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—28no—
415C2×C4—28no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—28no—
419C2×C4—28no—
420C2×C4—28no—
421C2×C4—416no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
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—816no—
474C2×C4#53514yes—
475C22—24no—
476C22—28no—
477C22—42no—
478C22—42no—
479C22—44no—
480C22—44no—
481C22—48no—
482C22—44no—
483C22C2×#53511yescharacteristic
484C22#348411yescharacteristic
485C22#348711yescharacteristic
486C22#348912yes—
487C22#349912yes—
488C22C2×#56311yescharacteristic
489C22#353412yes—
490C22#353412yes—
491C22#353812yes—
492C22#353814yes—
493C22#354412yes—
494C22#354511yescharacteristic
495C22#354714yes—
496C22#355314yes—
497C22#356112yes—
498C22#356314yes—
499C4—24no—
500C4—44no—
501C4—44no—
502C4—42no—
503C4—48no—
504C4—44no—
505C4—82no—
506C4—84no—
507C4—82no—
508C4—82no—
509C4—84no—
510C4—84no—
511C2—44no—
512C2#3671011yescharacteristic
513C2#4299412yes—
514C2#4613611yescharacteristic
515C2#5089512yes—
516C2#5154211yescharacteristic
517C2#5256812yes—
518C2#5397612yes—
519C2#5731514yes—
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×#2822C2semidirect
3. #17203C2semidirect
4. #16629C2semidirect
5. #16608C2semidirect
6. #16523C2semidirect
8. #15945C2semidirect
13. #14476C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28222
C2×#25032
#172032
#166292
#166082
#165232
#163002
#159452
#154822
#154302
#149162
#148212
#144762