Finite Group Databasebeta

← #239618

Subgroups — #239618

Order 512.

1062 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2568C211yesmaximal, characteristic
2#21365C211yesmaximal, characteristic
3#21324C211yesmaximal, characteristic
4#20740C211yesmaximal, characteristic
5C2×#1556C211yesmaximal, characteristic
6C2×#1491C211yesmaximal, characteristic
7#10423C214yesmaximal, complemented
8#10422C214yesmaximal, complemented
9#7904C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15#2568—22no—
16#2568—22no—
17#2568—22no—
18#2568C2214yes—
19C2×#400—22no—
20C2×#391—21no—
21C2×#391—21no—
22C2×#391—21no—
23#2285C2211yescharacteristic
24C2×#382—21no—
25C2×C4×C4⋊C4C2211yescharacteristic
26#1993—22no—
27#1990C2212yes—
28#1990C2212yes—
29#1948C2212yes—
30#1947C2212yes—
31#1853—21no—
32#1852—21no—
33#1823—21no—
34#1803C2212yes—
35#1803C2212yes—
36#1787—21no—
37#1776—22no—
38#1763—22no—
39#1736—22no—
40#1556—24no—
41C2×#332C2214yes—
42#1491—24no—
43C2×Q8⋊C8C2214yes—
44#1279C2214yes—
45C23×Q8—21no—
46C23×Q8D411yescharacteristic
47C22×M16—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8D412yes—
55C2×C4⋊Q8C2312yes—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8D412yes—
63C2×C4×Q8C2312yes—
64C2×#117—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4D412yes—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C42—21no—
72C22×C42—21no—
73C22×C42C2311yescharacteristic
74#432—22no—
75#432—22no—
76#432—22no—
77#432D414yes—
78C2×#107—22no—
79C2×#106—22no—
80C2×C4⋊C8C2312yes—
81C2×C4⋊C8C2312yes—
82C2×#103—22no—
83C2×#103—22no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C4×M16—22no—
87C4×M16—22no—
88C2×#97—21no—
89C2×#97—21no—
90C2×#97—21no—
91C2×C4×C8C2312yes—
92#400—28no—
93#400—28no—
94#391—24no—
95#391—24no—
96#391—24no—
97#391—24no—
98#391—24no—
99#391—24no—
100#384C2311yescharacteristic
101#384C2311yescharacteristic
102#382—24no—
103#382—24no—
104#382—21no—
105#380—21no—
106#380—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4D412yes—
112C4×C4⋊C4C2312yes—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—21no—
118#340—22no—
119#340—22no—
120#340—22no—
121#340—22no—
122#340—24no—
123#340—24no—
124#332—28no—
125Q8⋊C8—28no—
126C22×Q8—24no—
127C22×Q8—21no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—42no—
133C22×Q8—44no—
134C22×Q8—44no—
135C22×Q8—42no—
136C22×Q8—42no—
137C22×Q8C2×D412yes—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C2×M16—24no—
147C2×M16—42no—
148C2×M16—42no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C4⋊Q8—24no—
153C4⋊Q8—24no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—42no—
159C4⋊Q8—42no—
160C4⋊Q8—42no—
161C4⋊Q8—42no—
162C4×Q8—24no—
163C4×Q8—24no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—42no—
169C4×Q8—42no—
170C4×Q8—42no—
171C4×Q8—42no—
172#117—44no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D414yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D412yes—
201C2×C22⋊C4—41no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—44no—
207C2×C42—42no—
208C2×C42—42no—
209C2×C42—41no—
210C2×C42C2×D412yes—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42#4112yes—
215C2×C42#4112yes—
216C2×C42#4112yes—
217C2×C42C2411yescharacteristic
218#107—44no—
219#106—44no—
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—22no—
223C4⋊C8C2×D414yes—
224#103—44no—
225#103—44no—
226C22⋊C8—24no—
227C22⋊C8—42no—
228#97—24no—
229#97—42no—
230#97—42no—
231C4×C8—22no—
232C4×C8—22no—
233#95—28no—
234#95—28no—
235#95—28no—
236#95—28no—
237#95—22no—
238#95—22no—
239#95—41no—
240#95—41no—
241C24#12011yescharacteristic
242C2×Q8—24no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—48no—
249C2×Q8—48no—
250C2×Q8—42no—
251C2×Q8—44no—
252C2×Q8—44no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—44no—
256C2×Q8—44no—
257C2×Q8—42no—
258C2×Q8—44no—
259C2×Q8—88no—
260C2×Q8—88no—
261C2×Q8#13714yes—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—42no—
283C22×C4—44no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4#12011yescharacteristic
291C22×C4#12011yescharacteristic
292C22×C4#12011yescharacteristic
293C22×C4#12112yes—
294C22×C4#12112yes—
295C22×C4#12112yes—
296C22×C4#12312yes—
297C22×C4#12312yes—
298C22×C4#12412yes—
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D411yescharacteristic
302M16—48no—
303C2×C8—24no—
304C2×C8—24no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C4⋊C4—22no—
313C4⋊C4—24no—
314C4⋊C4—22no—
315C4⋊C4—24no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C22⋊C4—82no—
337C42—44no—
338C42—42no—
339C42—42no—
340C42—44no—
341C42—44no—
342C42—82no—
343C42#12112yes—
344C42#12112yes—
345C42#12112yes—
346C42#12312yes—
347C42#12312yes—
348C42#12412yes—
349C42C22×D411yescharacteristic
350C42C2×#4111yescharacteristic
351C42C2×#4111yescharacteristic
352C42C2×#4111yescharacteristic
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23#39412yes—
358C23#45412yes—
359C23#45612yes—
360C23C2×#12011yescharacteristic
361Q8—24no—
362Q8—24no—
363Q8—48no—
364Q8—42no—
365Q8—42no—
366Q8—88no—
367Q8—84no—
368Q8—84no—
369C2×C4—21no—
370C2×C4—21no—
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—24no—
383C2×C4—24no—
384C2×C4—48no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—84no—
400C2×C4—84no—
401C2×C4—88no—
402C2×C4—84no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4#39412yes—
410C2×C4#39412yes—
411C2×C4#39412yes—
412C2×C4C2×#12111yescharacteristic
413C2×C4C2×#12111yescharacteristic
414C2×C4C2×#12111yescharacteristic
415C2×C4C2×#12311yescharacteristic
416C2×C4C2×#12311yescharacteristic
417C2×C4C2×#12411yescharacteristic
418C2×C4C2×#13712yes—
419C2×C4#57512yes—
420C2×C4#57512yes—
421C2×C4#57512yes—
422C8—44no—
423C8—44no—
424C8—82no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22#203814yes—
432C22C2×#39411yescharacteristic
433C22#300411yescharacteristic
434C22#300511yescharacteristic
435C22#300511yescharacteristic
436C22#300511yescharacteristic
437C22C2×#45411yescharacteristic
438C22C2×#45611yescharacteristic
439C4—22no—
440C4—21no—
441C4—21no—
442C4—22no—
443C4—22no—
444C4—44no—
445C4—82no—
446C4—82no—
447C4—82no—
448C4—84no—
449C4—161no—
450C4—161no—
451C4#304012yes—
452C4#305612yes—
453C4#309512yes—
454C4#314012yes—
455C2—24no—
456C2#1853511yescharacteristic
457C2#1853611yescharacteristic
458C2#1854111yescharacteristic
459C2#1854211yescharacteristic
460C2#2348111yescharacteristic
461C2C2×#203811yescharacteristic
462C2#2350611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #10423C2semidirect
8. #10422C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25682
#213652
#213242
#207402
C2×#15562
C2×#14912
#104232
#104222
#79042