Finite Group Databasebeta

← #238680

Subgroups — #238680

Order 512.

1110 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2582C211yesmaximal, characteristic
2#21981C211yesmaximal, characteristic
3#20734C211yesmaximal, characteristic
4#20647C211yesmaximal, characteristic
5C2×#1487C211yesmaximal, characteristic
6C2×#1479C211yesmaximal, characteristic
7#10438C214yesmaximal, complemented
8#10437C214yesmaximal, complemented
9#8043C211yesmaximal, characteristic
10C22×C4×Q8C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C2×#419—21no—
16#2582—22no—
17#2582—22no—
18#2582—22no—
19#2582C2214yes—
20C2×#398—22no—
21C2×#391—21no—
22C2×#391—21no—
23C2×#389—21no—
24C2×#389—21no—
25#2285C2211yescharacteristic
26C2×#384C2211yescharacteristic
27#1948—22no—
28#1853—22no—
29#1844C2212yes—
30#1834C2212yes—
31#1824—21no—
32#1805—22no—
33#1803C2212yes—
34#1803C2212yes—
35#1803C2212yes—
36#1803C2212yes—
37#1787—21no—
38C4×#103—22no—
39#1736—21no—
40#1487—24no—
41#1479—24no—
42C2×Q8⋊C8C2214yes—
43C2×Q8⋊C8C2214yes—
44#1296C2214yes—
45C23×Q8D411yescharacteristic
46C23×Q8D411yescharacteristic
47C22×M16—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C2×C4×Q8—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8D412yes—
61C2×C4×Q8D412yes—
62C2×C4×Q8C2312yes—
63C2×C4×Q8C2312yes—
64C2×#117—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D412yes—
73C22×C42—21no—
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76#431—22no—
77#431—22no—
78#431—22no—
79#431D414yes—
80C2×#106—24no—
81#423—22no—
82#423—22no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8C2312yes—
86C2×C4⋊C8C2312yes—
87#419—44no—
88C2×#103—22no—
89C2×#103—22no—
90C2×#103—22no—
91C2×#103—22no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C4×C8—21no—
95C2×C4×C8C2312yes—
96#398—28no—
97#398—28no—
98#391—24no—
99#391—24no—
100#391—24no—
101#391—24no—
102#389—24no—
103#389—24no—
104#389—24no—
105#389—24no—
106#384—21no—
107#384—21no—
108#384D412yes—
109#384C2311yescharacteristic
110#384C2311yescharacteristic
111#384C2312yes—
112#382—21no—
113#380—21no—
114#380—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120#340—22no—
121#340—22no—
122#340—22no—
123#340—22no—
124#340—24no—
125Q8⋊C8—28no—
126Q8⋊C8—28no—
127C22×Q8—24no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—42no—
138C22×Q8SD1614yes—
139C22×Q8C2×D412yes—
140C22×Q8C2×D412yes—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C2×M16—42no—
150C2×M16—42no—
151C2×M16—42no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C4×Q8—24no—
156C4×Q8—24no—
157C4×Q8—24no—
158C4×Q8—24no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—42no—
168C4×Q8—42no—
169C4×Q8—42no—
170C4×Q8—42no—
171C4×Q8—42no—
172C4×Q8—42no—
173C4×Q8—42no—
174C4×Q8—42no—
175#117—42no—
176#117—42no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4—84no—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D414yes—
211C2×C22⋊C4—41no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—42no—
217C2×C42—42no—
218C2×C42—42no—
219C2×C42—42no—
220C2×C42—41no—
221C2×C42C2×D412yes—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42#4112yes—
226C2×C42#4112yes—
227C2×C42#4112yes—
228C2×C42C2411yescharacteristic
229#106—84no—
230C4⋊C8—24no—
231C4⋊C8—22no—
232C4⋊C8—22no—
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8—42no—
236#103—44no—
237#103—44no—
238#103—44no—
239#103—44no—
240C22⋊C8—24no—
241C22⋊C8—42no—
242C4×C8—22no—
243C4×C8—42no—
244C4×C8C2×D414yes—
245#95—28no—
246#95—28no—
247#95—28no—
248#95—41no—
249#95—41no—
250C24#12011yescharacteristic
251C2×Q8—24no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—24no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—28no—
264C2×Q8—28no—
265C2×Q8—48no—
266C2×Q8—48no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—44no—
272C2×Q8—44no—
273C2×Q8C2×SD1614yes—
274C2×Q8#13714yes—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—44no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4#12011yescharacteristic
302C22×C4#12011yescharacteristic
303C22×C4#12011yescharacteristic
304C22×C4#12112yes—
305C22×C4#12112yes—
306C22×C4#12112yes—
307C22×C4#12312yes—
308C22×C4#12312yes—
309C22×C4#12412yes—
310C22×C4C2×SD1612yes—
311C22×C4#13512yes—
312C22×C4#13712yes—
313C22×C4#13712yes—
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317M16—84no—
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—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—44no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—44no—
345C4⋊C4—44no—
346C4⋊C4—44no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—42no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—82no—
354C4⋊C4—88no—
355C22⋊C4—82no—
356C42—42no—
357C42—42no—
358C42—42no—
359C42—42no—
360C42—44no—
361C42—44no—
362C42—82no—
363C42#12112yes—
364C42#12112yes—
365C42#12112yes—
366C42#12312yes—
367C42#12312yes—
368C42#12412yes—
369C42C22×D411yescharacteristic
370C42C2×#4111yescharacteristic
371C42C2×#4111yescharacteristic
372C42C2×#4111yescharacteristic
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23#39412yes—
378C23#44812yes—
379C23#45212yes—
380C23C2×#12011yescharacteristic
381Q8—24no—
382Q8—24no—
383Q8—24no—
384Q8—24no—
385Q8—48no—
386Q8—42no—
387Q8—42no—
388Q8—42no—
389Q8—42no—
390Q8—48no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—48no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—48no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—88no—
419C2×C4—84no—
420C2×C4—84no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4#39412yes—
428C2×C4#39412yes—
429C2×C4#39412yes—
430C2×C4#44812yes—
431C2×C4#45212yes—
432C2×C4#46514yes—
433C2×C4#47014yes—
434C2×C4#48314yes—
435C2×C4#48514yes—
436C2×C4C2×#12111yescharacteristic
437C2×C4C2×#12111yescharacteristic
438C2×C4C2×#12111yescharacteristic
439C2×C4C2×#12311yescharacteristic
440C2×C4C2×#12311yescharacteristic
441C2×C4C2×#12411yescharacteristic
442C2×C4C22×SD1611yescharacteristic
443C2×C4C2×#13511yescharacteristic
444C2×C4C2×#13711yescharacteristic
445C2×C4C2×#13711yescharacteristic
446C2×C4C2×#13712yes—
447C2×C4#57512yes—
448C8—44no—
449C8—82no—
450C8—82no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22#203514yes—
458C22C2×#39411yescharacteristic
459C22C2×#44811yescharacteristic
460C22C2×#45211yescharacteristic
461C22#300411yescharacteristic
462C22#300511yescharacteristic
463C22#301911yescharacteristic
464C22#302011yescharacteristic
465C4—22no—
466C4—21no—
467C4—21no—
468C4—22no—
469C4—44no—
470C4—44no—
471C4—84no—
472C4—82no—
473C4—82no—
474C4—161no—
475C4—161no—
476C4#203514yes—
477C4C2×#46511yescharacteristic
478C4C2×#47011yescharacteristic
479C4#305612yes—
480C4C2×#48311yescharacteristic
481C4C2×#48511yescharacteristic
482C4#309512yes—
483C2—24no—
484C2#1875111yescharacteristic
485C2#1876111yescharacteristic
486C2#1876811yescharacteristic
487C2#1877811yescharacteristic
488C2C2×#203511yescharacteristic
489C2#2348111yescharacteristic
490C2#2349911yescharacteristic
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. #10438C2semidirect
8. #10437C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25822
#219812
#207342
#206472
C2×#14872
C2×#14792
#104382
#104372
#80432