Finite Group Databasebeta

← #6484972

Subgroups — #6484972

Order 512.

1674 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35179C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic
3#33099C214yesmaximal
4#29909C214yesmaximal
5#26322C214yesmaximal, complemented
6#26282C214yesmaximal, complemented
7#25062C214yesmaximal, complemented
8#24996C214yesmaximal, complemented
9#21982C211yesmaximal, characteristic
10#21860C211yesmaximal, characteristic
11#21818C211yesmaximal, characteristic
12#21385C211yesmaximal, characteristic
13#21358C211yesmaximal, characteristic
14C2×#533C2211yescharacteristic
15C22×C4×Q8—21no—
16C22×C4×Q8C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19#2855C2214yes—
20#2627—22no—
21#2582C2212yes—
22#2582C2212yes—
23#2581—22no—
24#2568C2212yes—
25#2567C2212yes—
26#2565—22no—
27Q8×C4⋊C4—22no—
28Q8×C4⋊C4—22no—
29Q8×C4⋊C4C2214yes—
30Q8×C4⋊C4C2214yes—
31#2317—21no—
32#2310—21no—
33C2×#389—21no—
34C2×#389—22no—
35#2285C2211yescharacteristic
36C2×#384C2211yescharacteristic
37C2×#384C2212yes—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40#2100C2218yes—
41#2095C2218yes—
42#2083C2218yes—
43#2072C2218yes—
44#1990C2212yes—
45#1987C2212yes—
46#1984C2214yes—
47#1984C2214yes—
48#1981C2212yes—
49#1956C2212yes—
50#1947C2212yes—
51#1947C2212yes—
52#1947C2214yes—
53#1928C2218yes—
54#1928C2218yes—
55#1921C2218yes—
56#1921C2218yes—
57#1872C2212yes—
58#1872C2214yes—
59#1872C2212yes—
60#1852C2212yes—
61#1852C2214yes—
62#1852C2212yes—
63#1845C2212yes—
64#1834C2212yes—
65#1817C2214yes—
66#1803C2214yes—
67C4×#107C2214yes—
68C4×#103C2214yes—
69C23×Q8D411yescharacteristic
70#533—24no—
71#533—24no—
72C2×C4⋊Q8C2311yescharacteristic
73C2×C4⋊Q8C2311yescharacteristic
74C2×#125—21no—
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8C2312yes—
77C2×C4×Q8—24no—
78C2×C4×Q8—24no—
79C2×C4×Q8—21no—
80C2×C4×Q8—24no—
81C2×C4×Q8—24no—
82C2×C4×Q8—48no—
83C2×C4×Q8D412yes—
84C2×C4×Q8D412yes—
85C2×C4×Q8C2312yes—
86C2×C4×Q8C2312yes—
87C2×C4×Q8C2312yes—
88C2×#117—21no—
89C2×#117—21no—
90C22×C4⋊C4—22no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—22no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4D411yescharacteristic
95C22×C4⋊C4D412yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C42—22no—
100C22×C42—21no—
101C22×C42C2312yes—
102C22×C42C2311yescharacteristic
103#431—24no—
104#431—24no—
105C2×#107C2314yes—
106C2×#107C2314yes—
107C2×#107C2318yes—
108C2×C4⋊C8C2312yes—
109C2×C4⋊C8C2312yes—
110C2×#103C2314yes—
111C2×#103C2318yes—
112C2×#103C2314yes—
113C2×C22⋊C8C2312yes—
114C2×C22⋊C8C2312yes—
115C2×C4×C8C2312yes—
116C2×C4×C8C2312yes—
117#400—24no—
118#398—24no—
119#398—24no—
120#395—24no—
121#391—22no—
122#391—22no—
123#391—22no—
124#391—22no—
125#391C2314yes—
126#391C2314yes—
127#389—22no—
128#389—22no—
129#389—24no—
130#389—24no—
131#389—28no—
132#389—28no—
133#389C2314yes—
134#389C2314yes—
135#387—24no—
136#387—24no—
137#385—24no—
138#384—22no—
139#384—24no—
140#384—21no—
141#384—21no—
142#384C2312yes—
143#384C2311yescharacteristic
144#384C2311yescharacteristic
145#384C2318yes—
146#384C2312yes—
147#382—22no—
148#382—21no—
149#381—24no—
150#380—21no—
151#380C2312yes—
152C4×C4⋊C4—21no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4—21no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4C2312yes—
157C4×C4⋊C4C2318yes—
158C4×C4⋊C4C2312yes—
159C4×C4⋊C4C2312yes—
160C2×#95—22no—
161C2×#95—21no—
162#340C2318yes—
163#340C2314yes—
164#340C2318yes—
165#340C2314yes—
166#340C2314yes—
167#340C2314yes—
168#336C2318yes—
169#336C2318yes—
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—21no—
173C22×Q8Q1618yes—
174C22×Q8C2×D412yes—
175C22×Q8C2×D412yes—
176C22×Q8C2412yes—
177C23×C4—21no—
178C23×C4—22no—
179C23×C4—21no—
180C23×C4C2×D412yes—
181C23×C4C2×D412yes—
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2411yescharacteristic
185C22×C8C2414yes—
186C22×C8C2414yes—
187C4⋊Q8—22no—
188C4⋊Q8—22no—
189C4⋊Q8—22no—
190C4⋊Q8—22no—
191#125—44no—
192C22⋊Q8—24no—
193C22⋊Q8—24no—
194C22⋊Q8—44no—
195C4×Q8—28no—
196C4×Q8—28no—
197C4×Q8—24no—
198C4×Q8—24no—
199C4×Q8—24no—
200C4×Q8—24no—
201C4×Q8—24no—
202C4×Q8—24no—
203C4×Q8—416no—
204C4×Q8—416no—
205C4×Q8—44no—
206#117—44no—
207#117—44no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—48no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—82no—
236C2×C4⋊C4—84no—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×D414yes—
240C2×C4⋊C4C2×D414yes—
241C2×C4⋊C4C2412yes—
242C2×C4⋊C4C2414yes—
243C2×C4⋊C4C2412yes—
244C2×C4⋊C4C2412yes—
245C2×C4⋊C4C2412yes—
246C2×C4⋊C4C2412yes—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—28no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—48no—
257C2×C42—44no—
258C2×C42C2×D414yes—
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42#4114yes—
262C2×C42#4114yes—
263C2×C42C2414yes—
264C2×C42C2411yescharacteristic
265C2×C42C2411yescharacteristic
266#107—28no—
267#107—48no—
268#107—44no—
269C4⋊C8—24no—
270C4⋊C8—24no—
271#103—28no—
272#103—44no—
273#103—48no—
274C22⋊C8—24no—
275C22⋊C8—24no—
276C4×C8—24no—
277C4×C8—24no—
278#95—28no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—24no—
289C24C22×D411yescharacteristic
290C2×Q8—24no—
291C2×Q8—24no—
292C2×Q8—24no—
293C2×Q8—24no—
294C2×Q8—232no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—42no—
300C2×Q8—81no—
301C2×Q8—81no—
302C2×Q8C2×Q1618yes—
303C2×Q8C2×Q1618yes—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—42no—
327C22×C4—44no—
328C22×C4—42no—
329C22×C4#12314yes—
330C22×C4#12314yes—
331C22×C4C2×Q1618yes—
332C22×C4C2×Q1614yes—
333C22×C4AGL(1,ℤ/8)14yes—
334C22×C4C22×D414yes—
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D414yes—
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C2×#4112yes—
342C22×C4C2×#4112yes—
343C22×C4C2511yescharacteristic
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C4⋊C4—24no—
351C4⋊C4—28no—
352C4⋊C4—28no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—28no—
361C4⋊C4—24no—
362C4⋊C4—22no—
363C4⋊C4—42no—
364C4⋊C4—44no—
365C4⋊C4—416no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—42no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—81no—
376C4⋊C4—81no—
377C4⋊C4—84no—
378C4⋊C4—84no—
379C4⋊C4—82no—
380C4⋊C4—82no—
381C4⋊C4—88no—
382C4⋊C42−1+414yes—
383C22⋊C4—44no—
384C22⋊C4—44no—
385C42—28no—
386C42—24no—
387C42—24no—
388C42—416no—
389C42—44no—
390C42—44no—
391C42—44no—
392C42—44no—
393C42#12314yes—
394C42#12314yes—
395C42C2×#4112yes—
396C42C2×#4112yes—
397C42C2×#4112yes—
398C42C2×#4112yes—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23C2×#12312yes—
404C23C22×Q1612yes—
405C23C2×AGL(1,ℤ/8)12yes—
406C23C23×D411yescharacteristic
407Q8—216no—
408Q8—216no—
409Q8—44no—
410Q8—44no—
411Q8—84no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—216no—
417C2×C4—24no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—44no—
428C2×C4—48no—
429C2×C4—48no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—88no—
451C2×C4—84no—
452C2×C4#484116yes—
453C2×C4C2×#12312yes—
454C2×C4C2×#12312yes—
455C2×C4C2×#12312yes—
456C2×C4C2×#12312yes—
457C2×C4C2×#12312yes—
458C2×C4C22×Q1618yes—
459C2×C4C22×Q1612yes—
460C2×C4C22×Q1612yes—
461C2×C4C2×AGL(1,ℤ/8)12yes—
462C2×C4C2×AGL(1,ℤ/8)12yes—
463C2×C4C22×#4111yescharacteristic
464C2×C4C22×#4111yescharacteristic
465C2×C4C2×2−1+412yes—
466C2×C4#58512yes—
467C8—44no—
468C8—44no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22C2×#48414yes—
476C22C22×#12311yescharacteristic
477C22#348611yescharacteristic
478C22#348711yescharacteristic
479C22C23×Q1611yescharacteristic
480C22C22×AGL(1,ℤ/8)11yescharacteristic
481C22#358511yescharacteristic
482C22#358611yescharacteristic
483C4—28no—
484C4—22no—
485C4—22no—
486C4—22no—
487C4—44no—
488C4—44no—
489C4—82no—
490C4—82no—
491C4—82no—
492C4—82no—
493C4—82no—
494C4—84no—
495C4C2×#48414yes—
496C4C2×#48414yes—
497C4#346112yes—
498C4#350712yes—
499C2—24no—
500C2#3518211yescharacteristic
501C2C22×#48411yescharacteristic
502C2#6051711yescharacteristic
503C2#6182111yescharacteristic
504C2#6182711yescharacteristic
505C2#6188411yescharacteristic
506C2#6189011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #26322C2semidirect
6. #26282C2semidirect
7. #25062C2semidirect
8. #24996C2semidirect

Maximal subgroups

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

subgroupindex
#351792
C2×Q8×C4⋊C42
#330992
#299092
#263222
#262822
#250622
#249962
#219822
#218602
#218182
#213852
#213582