Finite Group Databasebeta

← #365685

Subgroups — #365685

Order 512.

1140 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34213C211yesmaximal, characteristic
2#21322C211yesmaximal, characteristic
3C2×#1986C211yesmaximal, characteristic
4#20764C211yesmaximal, characteristic
5#13911C211yesmaximal, characteristic
6#11276C214yesmaximal
7#10892C214yesmaximal
8#10851C211yesmaximal, characteristic
9C4×#343C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13#2370—22no—
14#2369—21no—
15#2365—21no—
16#2355—21no—
17#2355—22no—
18Q8×C4⋊C4—21no—
19#2329—22no—
20#2317C2212yes—
21C4×C22⋊Q8—22no—
22#2285C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#382C2212yes—
25#2274C2212yes—
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1992—22no—
29#1990—21no—
30#1990—21no—
31#1986C418yescomplemented
32#1836—22no—
33#1834—21no—
34#1834—21no—
35C2×#343C2212yes—
36C2×#343C2212yes—
37#1736C2214yes—
38#1736C2214yes—
39C2×#339C2214yes—
40C2×#339C2214yes—
41#1442—22no—
42C4⋊#95—22no—
43C4⋊#95C2212yes—
44C22×M16C2312yes—
45C22×M16C2312yes—
46#533—22no—
47#533—22no—
48#533C2×C414yes—
49#533Q814yes—
50C2×C4⋊Q8D411yescharacteristic
51C2×C4⋊Q8D411yescharacteristic
52C2×#125D412yes—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C4×Q8—22no—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×#117C2312yes—
59C2×#117C2312yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C42D412yes—
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69C4×M16—22no—
70C4×M16—22no—
71C4×M16—22no—
72C4×M16C2×C414yes—
73C2×#97—21no—
74C2×#97—21no—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77#391—24no—
78#391—22no—
79#389—24no—
80#389—22no—
81#389—22no—
82#389—22no—
83#387—24no—
84#387—24no—
85#387—24no—
86#385—24no—
87#384—22no—
88#384—21no—
89#384—21no—
90#384—21no—
91#384—21no—
92#384D411yescharacteristic
93#384D411yescharacteristic
94#382—22no—
95#382—28no—
96#382D412yes—
97#380—21no—
98#380—21no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—24no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4C2×C414yes—
113C4×C22⋊C4—24no—
114C2×#95—21no—
115C2×#95—24no—
116C2×#95—24no—
117C2×#95D412yes—
118#343—24no—
119#343—24no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—24no—
124#339—28no—
125#339C2×C4116yes—
126C22×Q8—21no—
127C22×Q8—21no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D412yes—
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2411yescharacteristic
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16C22×C414yes—
146C2×M16#4114yes—
147C22×C8—22no—
148C22×C8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8C4⋊C414yes—
152C4⋊Q8C4⋊C414yes—
153#125—24no—
154#125C4⋊C418yes—
155C22⋊Q8—24no—
156C22⋊Q8—24no—
157C22⋊Q8—24no—
158C22⋊Q8—24no—
159C4×Q8—24no—
160C4×Q8—24no—
161C4×Q8—24no—
162C4×Q8—24no—
163C4×Q8—84no—
164#117—22no—
165#117—22no—
166#117—24no—
167#117—24no—
168#117C22×C414yes—
169#117C2×Q814yes—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4#4114yes—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C42—22no—
208C2×C42—24no—
209C2×C42—24no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—24no—
214C2×C42—24no—
215C2×C42—22no—
216C2×C42—44no—
217C2×C42C22×C414yes—
218C2×C42C22×C412yes—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×Q812yes—
224C2×C42#4112yes—
225C2×C42#4112yes—
226#97—24no—
227#97—42no—
228C4×C8—24no—
229C4×C8—42no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—48no—
240#95—84no—
241C24C22×D411yescharacteristic
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—81no—
247C2×Q8—81no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—41no—
261C22×C4—41no—
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—42no—
271C22×C4—42no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4C4×D412yes—
278C22×C4C4×D412yes—
279C22×C4#12011yescharacteristic
280C22×C4#12011yescharacteristic
281C22×C4#12011yescharacteristic
282C22×C4#12011yescharacteristic
283C22×C4C22⋊Q812yes—
284C22×C4C22⋊Q812yes—
285C22×C4#12414yes—
286C22×C4C23×C411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×Q811yescharacteristic
290C22×C4C2×#4111yescharacteristic
291C22×C4C2×#4112yes—
292C22×C4C2×#4112yes—
293C22×C4C2×#4111yescharacteristic
294C22×C42−1+412yes—
295C22×C42−1+412yes—
296M16—24no—
297M16—24no—
298M16—24no—
299M16—24no—
300M16—42no—
301M16—42no—
302M16—42no—
303M16—42no—
304C2×C8—24no—
305C2×C8—24no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—28no—
315C4⋊C4—28no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—48no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—44no—
326C4⋊C4—44no—
327C4⋊C4—81no—
328C4⋊C4—81no—
329C4⋊C4—82no—
330C4⋊C4—82no—
331C4⋊C4—82no—
332C4⋊C4—84no—
333C4⋊C4—84no—
334C4⋊C4—82no—
335C4⋊C4C2×C4⋊C414yes—
336C4⋊C4C2×C4⋊C414yes—
337C22⋊C4—24no—
338C22⋊C4—24no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C42—24no—
342C42—24no—
343C42—28no—
344C42—44no—
345C42—44no—
346C42—44no—
347C42—82no—
348C42—84no—
349C42—84no—
350C42C2×C4⋊C412yes—
351C42C2×C4⋊C412yes—
352C42C4×D412yes—
353C42C4×D412yes—
354C42C22⋊Q812yes—
355C42C22⋊Q812yes—
356C42#12312yes—
357C42#12312yes—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23C2×C4×D411yescharacteristic
363C23C2×#12011yescharacteristic
364C23C2×C22⋊Q811yescharacteristic
365C23C2×#12411yescharacteristic
366C23#53211yescharacteristic
367C23#53611yescharacteristic
368C23#53611yescharacteristic
369Q8—48no—
370Q8—84no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—84no—
411C2×C4—84no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4#38512yes—
416C2×C4#38512yes—
417C2×C4#38512yes—
418C2×C4#38512yes—
419C2×C4#45412yes—
420C2×C4#45512yes—
421C2×C4C22×C4⋊C411yescharacteristic
422C2×C4C2×C4×D411yescharacteristic
423C2×C4C4×#4112yes—
424C2×C4#51912yes—
425C2×C4C2×C22⋊Q811yescharacteristic
426C2×C4C2×#12311yescharacteristic
427C2×C4#52912yes—
428C2×C4#54112yes—
429C2×C4#54712yes—
430C2×C4#54812yes—
431C2×C4#54812yes—
432C2×C4#54812yes—
433C2×C4#55212yes—
434C2×C4#55612yes—
435C2×C4#55712yes—
436C2×C4#55712yes—
437C8—44no—
438C8—44no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22C2×#38511yescharacteristic
452C22C4×#12411yescharacteristic
453C22#232711yescharacteristic
454C22#243211yescharacteristic
455C22#243611yescharacteristic
456C22#256511yescharacteristic
457C22#256811yescharacteristic
458C22C2×#45511yescharacteristic
459C22C2×#45411yescharacteristic
460C22#302111yescharacteristic
461C22#302211yescharacteristic
462C4—22no—
463C4—22no—
464C4—22no—
465C4—44no—
466C4—44no—
467C4—44no—
468C4—84no—
469C4—82no—
470C4—82no—
471C4—82no—
472C4—162no—
473C4#186014yes—
474C4#235112yes—
475C4#236212yes—
476C4#242112yes—
477C4#245312yes—
478C2—21no—
479C2—21no—
480C2—21no—
481C2—21no—
482C2#1397811yescharacteristic
483C2C2×#186011yescharacteristic
484C2#2086511yescharacteristic
485C2#2569311yescharacteristic
486C2#2569611yescharacteristic
487C2#2591711yescharacteristic
488C2#2592011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
31. #1986C4semidirect

Maximal subgroups

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

subgroupindex
#342132
#213222
C2×#19862
#207642
#139112
#112762
#108922
#108512
C4×#3432