Finite Group Databasebeta

← #238192

Subgroups — #238192

Order 512.

888 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35181C211yesmaximal, characteristic
2#21885C211yesmaximal, characteristic
3#21400C211yesmaximal, characteristic
4#21312C211yesmaximal, characteristic
5#10356C212yesmaximal, complemented
6#10355C212yesmaximal, complemented
7#10353C212yesmaximal, complemented
8#10352C212yesmaximal, complemented
9#8045C211yesmaximal, characteristic
10#7897C211yesmaximal, characteristic
11C2×#1279C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#2851—22no—
17#2646—22no—
18#2571—21no—
19#2571C2212yes—
20#2570—22no—
21#2570—22no—
22#2552—21no—
23#2552C2212yes—
24#2317—21no—
25#2317—21no—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1989—22no—
32#1988C2212yes—
33#1987C2212yes—
34#1987C2212yes—
35#1981C2212yes—
36#1956C2211yescharacteristic
37#1956C2211yescharacteristic
38#1892—22no—
39#1892—22no—
40#1803—21no—
41#1803—21no—
42C4×#103—22no—
43#1736—21no—
44#1736—21no—
45#1296C2212yes—
46#1296C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51C22×M16—21no—
52C23×C8—21no—
53C23×C8—21no—
54#533—22no—
55#533—22no—
56#533—22no—
57#533Q814yes—
58C2×C4⋊Q8C2311yescharacteristic
59C2×C4⋊Q8C2311yescharacteristic
60C2×#125—22no—
61C2×#125—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×#117—21no—
67C2×#117—21no—
68C2×#117—21no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C42—21no—
72C22×C42C2311yescharacteristic
73#423—22no—
74#423—22no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8C2312yes—
80C2×C4⋊C8C2312yes—
81C2×#103—22no—
82C2×#103—22no—
83C2×#103—22no—
84C2×#103—22no—
85C2×C4×C8—22no—
86C2×C4×C8—22no—
87C2×C4×C8—21no—
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2311yescharacteristic
90#400—24no—
91#400—24no—
92#398—24no—
93#398—24no—
94#389—22no—
95#389—22no—
96#389—22no—
97#389—22no—
98#389—22no—
99#389—22no—
100#387—24no—
101#387—24no—
102#385—24no—
103#385—24no—
104#384—22no—
105#384—22no—
106#384D411yescharacteristic
107#384D411yescharacteristic
108#384C2311yescharacteristic
109#384C2311yescharacteristic
110#382—22no—
111#382—22no—
112#382—22no—
113#382—21no—
114#382—21no—
115#380—21no—
116#380—21no—
117#380D412yes—
118#380D412yes—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4D412yes—
124C4×C4⋊C4D412yes—
125C4×C4⋊C4C2312yes—
126C4×C4⋊C4C2312yes—
127C2×#95—21no—
128#340—22no—
129#340—22no—
130#340—22no—
131#340—22no—
132#340—22no—
133#340—22no—
134#340—22no—
135#340—22no—
136C22×Q8—21no—
137C22×Q8—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C2×M16—42no—
144C2×M16—42no—
145C2×M16—42no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—42no—
151C22×C8—42no—
152C22×C8—42no—
153C22×C8—42no—
154C22×C8—81no—
155C22×C8—81no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8C2×Q812yes—
163C4⋊Q8C2×Q812yes—
164#125—24no—
165#125—42no—
166#125—82no—
167#125—82no—
168C22⋊Q8—24no—
169C22⋊Q8—24no—
170C22⋊Q8—42no—
171C22⋊Q8—42no—
172C4×Q8—24no—
173C4×Q8—24no—
174C4×Q8—42no—
175C4×Q8—42no—
176#117—24no—
177#117—42no—
178#117—42no—
179#117—42no—
180#117—42no—
181#117—42no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
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—21no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—41no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—41no—
220C2×C42—81no—
221C2×C42—81no—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×Q812yes—
226C2×C42#4112yes—
227C2×C42#4112yes—
228C2×C42#4112yes—
229C2×C42C2411yescharacteristic
230C4⋊C8—22no—
231C4⋊C8—22no—
232C4⋊C8—22no—
233C4⋊C8—22no—
234C4⋊C8—42no—
235C4⋊C8—42no—
236#103—44no—
237#103—44no—
238#103—82no—
239#103—82no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—42no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255C24#12011yescharacteristic
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—81no—
283C22×C4—81no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4#12011yescharacteristic
287C22×C4#12011yescharacteristic
288C22×C4#12011yescharacteristic
289C22×C4C22⋊Q812yes—
290C22×C4C22⋊Q812yes—
291C22×C4C22⋊Q812yes—
292C22×C4#12412yes—
293C22×C4#12412yes—
294C22×C4#12412yes—
295C22×C4#13512yes—
296C22×C4#13512yes—
297C22×C4#13712yes—
298C22×C4#13712yes—
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D411yescharacteristic
302M16—84no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—48no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—84no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—81no—
350C4⋊C4—81no—
351C4⋊C4—84no—
352C4⋊C4—84no—
353C4⋊C4—84no—
354C4⋊C4—84no—
355C22⋊C4—44no—
356C22⋊C4—44no—
357C22⋊C4—44no—
358C42—42no—
359C42—42no—
360C42—42no—
361C42—42no—
362C42—44no—
363C42—44no—
364C42—82no—
365C42C22⋊Q812yes—
366C42C22⋊Q812yes—
367C42C22⋊Q812yes—
368C42#12412yes—
369C42#12412yes—
370C42#12412yes—
371C42C22×Q811yescharacteristic
372C42C2×#4111yescharacteristic
373C42C2×#4111yescharacteristic
374C42C2×#4111yescharacteristic
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23#39312yes—
380C23#45212yes—
381C23#45212yes—
382C23C2×#12011yescharacteristic
383Q8—44no—
384Q8—42no—
385Q8—42no—
386Q8—82no—
387Q8—81no—
388Q8—81no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—84no—
422C2×C4—84no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4#39312yes—
428C2×C4#39312yes—
429C2×C4#39312yes—
430C2×C4#45212yes—
431C2×C4#45212yes—
432C2×C4#47914yes—
433C2×C4#47914yes—
434C2×C4#48814yes—
435C2×C4#49114yes—
436C2×C4C2×C22⋊Q811yescharacteristic
437C2×C4C2×C22⋊Q811yescharacteristic
438C2×C4C2×C22⋊Q811yescharacteristic
439C2×C4C2×#12411yescharacteristic
440C2×C4C2×#12411yescharacteristic
441C2×C4C2×#12411yescharacteristic
442C2×C4C2×#13511yescharacteristic
443C2×C4C2×#13511yescharacteristic
444C2×C4C2×#13711yescharacteristic
445C2×C4C2×#13711yescharacteristic
446C2×C4#57611yescharacteristic
447C2×C4#57711yescharacteristic
448C2×C4#57711yescharacteristic
449C2×C4#57811yescharacteristic
450C8—42no—
451C8—42no—
452C8—82no—
453C8—82no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22#202614yes—
461C22C2×#39311yescharacteristic
462C22C2×#45211yescharacteristic
463C22C2×#45211yescharacteristic
464C22#301211yescharacteristic
465C22#301311yescharacteristic
466C22#301811yescharacteristic
467C22#301811yescharacteristic
468C4—22no—
469C4—21no—
470C4—21no—
471C4—22no—
472C4—82no—
473C4—82no—
474C4—82no—
475C4—82no—
476C4—82no—
477C4—82no—
478C4—82no—
479C4—82no—
480C4#202614yes—
481C4C2×#47911yescharacteristic
482C4C2×#47911yescharacteristic
483C4#308311yescharacteristic
484C4#308411yescharacteristic
485C4C2×#48811yescharacteristic
486C4C2×#49111yescharacteristic
487C4#313611yescharacteristic
488C4#313711yescharacteristic
489C2—24no—
490C2#1866711yescharacteristic
491C2#1866911yescharacteristic
492C2#1867911yescharacteristic
493C2#1868111yescharacteristic
494C2C2×#202611yescharacteristic
495C2#2345811yescharacteristic
496C2#2346111yescharacteristic
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
5. #10356C2semidirect
6. #10355C2semidirect
7. #10353C2semidirect
8. #10352C2semidirect

Maximal subgroups

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

subgroupindex
#351812
#218852
#214002
#213122
#103562
#103552
#103532
#103522
#80452
#78972
C2×#12792