Finite Group Databasebeta

← #414917

Subgroups — #414917

Order 512.

1188 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#501C211yesmaximal, characteristic
2C2×#1991C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic
4C2×#1768C211yesmaximal, characteristic
5#14283C211yesmaximal, characteristic
6#14282C211yesmaximal, characteristic
7#11257C212yesmaximal
8#11256C212yesmaximal
9#11232C212yesmaximal
10#11231C212yesmaximal
11#10843C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C2×#501—24no—
15C2×#501—24no—
16C2×#501—24no—
17C22×#107C2212yes—
18C22×#106C2212yes—
19C22×C4⋊C8C2212yes—
20C22×#97C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C2×#376C2211yescharacteristic
25#1991—24no—
26#1990—24no—
27#1768—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35#1446—22no—
36#1445—22no—
37#1441—22no—
38#1441—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45C23×Q8—21no—
46C23×C8C2312yes—
47C23×C8C2312yes—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8D414yes—
54C2×#125—22no—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125—22no—
58C2×#125—22no—
59C2×#125D414yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74#501—232no—
75C2×#107—24no—
76C2×#107—24no—
77C2×#107—24no—
78C2×#106—24no—
79C2×#106—24no—
80C2×#106—24no—
81C2×C4⋊C8—24no—
82C2×C4⋊C8—24no—
83C2×C4⋊C8Q818yescomplemented
84C2×#97—22no—
85C2×#97—22no—
86C2×#97D414yes—
87C2×C4×C8—22no—
88C2×C4×C8—22no—
89C2×C4×C8—22no—
90#384—22no—
91#384—22no—
92#382—22no—
93#382—22no—
94#376—22no—
95#376—22no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95D412yes—
107C2×#95D412yes—
108#340—24no—
109#340—24no—
110#340—24no—
111#340—24no—
112#340—24no—
113#340—24no—
114#336—24no—
115C22×Q8—24no—
116C22×Q8—42no—
117C22×Q8—42no—
118C22×Q8—82no—
119C22×Q8—82no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4C2×D412yes—
127C23×C4C2×D412yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2411yescharacteristic
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8C2×D414yes—
142C22×C8C2×Q814yes—
143C22×C8C2×Q814yes—
144C22×C8#4114yes—
145C4⋊Q8—28no—
146C4⋊Q8—28no—
147C4⋊Q8—44no—
148C4⋊Q8—44no—
149#125—28no—
150#125—28no—
151#125—44no—
152#125—44no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D414yes—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—22no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208C2×C42C2×Q814yes—
209C2×C42#4112yes—
210C2×C42#4112yes—
211C2×C42#4112yes—
212#107—216no—
213#106—216no—
214C4⋊C8—216no—
215#97—28no—
216C4×C8—44no—
217#95—24no—
218#95—24no—
219#95—42no—
220#95—42no—
221#95—82no—
222#95—82no—
223#95—82no—
224#95—82no—
225#95—82no—
226#95—82no—
227#95—82no—
228#95—82no—
229C24C22×D411yescharacteristic
230C2×Q8—48no—
231C2×Q8—84no—
232C2×Q8—84no—
233C2×Q8—88no—
234C2×Q8—88no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4#12112yes—
287C22×C4#12112yes—
288C22×C4#12112yes—
289C22×C4#12112yes—
290C22×C4#12112yes—
291C22×C4#12112yes—
292C22×C4C4⋊Q814yes—
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×Q812yes—
297C22×C4C2×#4112yes—
298C22×C4C2×#4111yescharacteristic
299C22×C4C2×#4111yescharacteristic
300C22×C4C2×#4111yescharacteristic
301C22×C42−1+412yes—
302C22×C42−1+412yes—
303C2×C8—24no—
304C2×C8—24no—
305C2×C8—24no—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8C22⋊Q818yes—
317C4⋊C4—22no—
318C4⋊C4—24no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—24no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—42no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—42no—
329C4⋊C4—48no—
330C4⋊C4—48no—
331C4⋊C4—48no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—84no—
341C4⋊C4—84no—
342C42—24no—
343C42—24no—
344C42—42no—
345C42—42no—
346C42#12014yes—
347C42#12314yes—
348C42#12314yes—
349C42#12414yes—
350C23C2×#12111yescharacteristic
351C23C2×#12111yescharacteristic
352C23C2×#12111yescharacteristic
353C23C2×C4⋊Q811yescharacteristic
354C23#53211yescharacteristic
355C23#53611yescharacteristic
356C23#53611yescharacteristic
357C23#57612yes—
358C23#57712yes—
359C23#57712yes—
360C23#57812yes—
361Q8—816no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—82no—
418C2×C4#39414yes—
419C2×C4#39414yes—
420C2×C4C2×#12011yescharacteristic
421C2×C4C2×C22⋊Q812yes—
422C2×C4C2×#12311yescharacteristic
423C2×C4C2×#12311yescharacteristic
424C2×C4C2×#12411yescharacteristic
425C2×C4#53312yes—
426C2×C4#53612yes—
427C2×C4#54112yes—
428C2×C4#54812yes—
429C2×C4#54812yes—
430C2×C4D4×Q812yes—
431C2×C4D4×Q812yes—
432C2×C4#55612yes—
433C2×C4#55612yes—
434C2×C4#55712yes—
435C2×C4#55712yes—
436C8—28no—
437C8—44no—
438C22C2×#39411yescharacteristic
439C22#254511yescharacteristic
440C22#254511yescharacteristic
441C22#254611yescharacteristic
442C22#254611yescharacteristic
443C22#262111yescharacteristic
444C22#285711yescharacteristic
445C22#304111yescharacteristic
446C22#304211yescharacteristic
447C22#304311yescharacteristic
448C22#304411yescharacteristic
449C22#304512yes—
450C22#304612yes—
451C22#306212yes—
452C22#306312yes—
453C22#316312yes—
454C22#316412yes—
455C22#317812yes—
456C22#317912yes—
457C22#324112yes—
458C22#324212yes—
459C22#324312yes—
460C22#324412yes—
461C4—24no—
462C4—24no—
463C4—44no—
464C4—82no—
465C4—82no—
466C4—84no—
467C4—82no—
468C4—161no—
469C4—161no—
470C4—162no—
471C4#240912yes—
472C4#244612yes—
473C4#244612yes—
474C4#256812yes—
475C2#1518011yescharacteristic
476C2#2354411yescharacteristic
477C2#2354511yescharacteristic
478C2#2866611yescharacteristic
479C2#2866711yescharacteristic
480C2#2866812yes—
481C2#2869611yescharacteristic
482C2#2869711yescharacteristic
483C2#2869812yes—
484C2#2988512yes—
485C2#3321912yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
83. C2×C4⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5012
C2×#19912
C2×#19902
C2×#17682
#142832
#142822
#112572
#112562
#112322
#112312
#108432