Finite Group Databasebeta

← #440941

Subgroups — #440941

Order 512.

1306 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2C2×#2026C211yesmaximal, characteristic
3C2×#1992C211yesmaximal, characteristic
4#15218C211yesmaximal, characteristic
5#11384C212yesmaximal
6#11373C211yesmaximal, characteristic
7#11207C212yesmaximal
8#11067C212yesmaximal
9#10928C211yesmaximal, characteristic
10#10875C211yesmaximal, characteristic
11#10789C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#97C2211yescharacteristic
18#2567—24no—
19#2567—24no—
20C2×#395—22no—
21C2×#391C2211yescharacteristic
22C2×#391C2211yescharacteristic
23C2×#389—22no—
24C2×#389C2212yes—
25C2×#384C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#2026—24no—
29#1992—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39#1444—22no—
40#1441—22no—
41C4⋊#95C2212yes—
42C4⋊#95C2211yescharacteristic
43C23×Q8—21no—
44C23×Q8C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—24no—
50C2×C4⋊Q8—42no—
51C2×#125—24no—
52C2×#125—22no—
53C2×#125—22no—
54C2×#125—42no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—24no—
58C2×C4×Q8—42no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4D412yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C42—21no—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C2×#103—24no—
76C2×#103—24no—
77C2×#103—42no—
78C2×#103—42no—
79C2×#103—42no—
80C2×#103Q818yescomplemented
81C2×#97—22no—
82C2×#97—42no—
83#395—216no—
84#391—22no—
85#391—22no—
86#391—22no—
87#391D414yes—
88#389—216no—
89#389—24no—
90#389—42no—
91#384—22no—
92#384—22no—
93#384—22no—
94#384D414yes—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95D412yes—
104#340—24no—
105#340—24no—
106#340—24no—
107#340—42no—
108#340—42no—
109#340—42no—
110#336—22no—
111#336—24no—
112#336—42no—
113C22×Q8—24no—
114C22×Q8—21no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—42no—
121C22×Q8—42no—
122C22×Q8—82no—
123C22×Q8—82no—
124C22×Q8C2×D412yes—
125C22×Q8C2×Q814yes—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2411yescharacteristic
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8C2×Q814yes—
151C4⋊Q8—48no—
152C4⋊Q8—48no—
153#125—28no—
154#125—48no—
155#125—44no—
156C4×Q8—48no—
157C4×Q8—48no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×Q814yes—
202C2×C4⋊C4#4114yes—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—24no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—42no—
214C2×C42—42no—
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42#4114yes—
218#103—48no—
219#103—48no—
220#97—44no—
221#95—28no—
222#95—28no—
223#95—24no—
224#95—44no—
225#95—44no—
226#95—82no—
227#95—82no—
228C24C22×D411yescharacteristic
229C2×Q8—22no—
230C2×Q8—24no—
231C2×Q8—24no—
232C2×Q8—24no—
233C2×Q8—24no—
234C2×Q8—24no—
235C2×Q8—24no—
236C2×Q8—24no—
237C2×Q8—48no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—88no—
246C2×Q8—88no—
247C2×Q8—84no—
248C2×Q8—84no—
249C2×Q8C4⋊Q814yes—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4#12014yes—
304C22×C4#12412yes—
305C22×C4#12412yes—
306C22×C4#13714yes—
307C22×C4#13714yes—
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C22×Q812yes—
311C22×C4C2×#4112yes—
312C22×C4C2×#4112yes—
313C22×C4H(5,𝔽2)12yes—
314C22×C4H(5,𝔽2)12yes—
315C22×C42−1+412yes—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C4⋊C4—216no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—48no—
339C4⋊C4—48no—
340C4⋊C4—416no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—88no—
349C4⋊C4—88no—
350C4⋊C4—88no—
351C42—22no—
352C42—22no—
353C42—24no—
354C42—24no—
355C42—24no—
356C42—48no—
357C42—41no—
358C42—41no—
359C42—42no—
360C23C2×#12011yescharacteristic
361C23C2×#12411yescharacteristic
362C23#53511yescharacteristic
363C23#54612yes—
364C23D4×Q812yes—
365C23C2×#13712yes—
366C23C2×#13712yes—
367C23#57612yes—
368C23#57712yes—
369Q8—28no—
370Q8—44no—
371Q8—44no—
372Q8—44no—
373Q8—816no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—22no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—81no—
421C2×C4—82no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4#49114yes—
432C2×C4#52511yescharacteristic
433C2×C4C2×C4⋊Q811yescharacteristic
434C2×C4#53312yes—
435C2×C4#53511yescharacteristic
436C2×C4#53611yescharacteristic
437C2×C4#54212yes—
438C2×C4#54212yes—
439C2×C4#54312yes—
440C2×C4#54612yes—
441C2×C4#54612yes—
442C2×C4#54612yes—
443C2×C4D4×Q812yes—
444C2×C4#55412yes—
445C2×C4#55412yes—
446C2×C4#55612yes—
447C2×C4#56212yes—
448C8—44no—
449C8—82no—
450C22#240411yescharacteristic
451C22#243411yescharacteristic
452C22#243511yescharacteristic
453C22#249212yes—
454C22#269212yes—
455C22#300412yes—
456C22#301212yes—
457C22C2×#49111yescharacteristic
458C22#313612yes—
459C22#313911yescharacteristic
460C22#321812yes—
461C22#322011yescharacteristic
462C22#322211yescharacteristic
463C22#331412yes—
464C22#331612yes—
465C22#333112yes—
466C22#333312yes—
467C22#338512yes—
468C22#338512yes—
469C22#338712yes—
470C22#338712yes—
471C4—24no—
472C4—42no—
473C4—44no—
474C4—44no—
475C4—44no—
476C4—82no—
477C4—82no—
478C4—82no—
479C4—84no—
480C4—84no—
481C4#246512yes—
482C4#254912yes—
483C4#269212yes—
484C4#270012yes—
485C2#1567111yescharacteristic
486C2#2398112yes—
487C2#2584712yes—
488C2#2586111yescharacteristic
489C2#2586711yescharacteristic
490C2#2782412yes—
491C2#2783212yes—
492C2#3153412yes—
493C2#3153612yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
80. C2×#103Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#20262
C2×#19922
#152182
#113842
#113732
#112072
#110672
#109282
#108752
#107892