Finite Group Databasebeta

← #202853

Subgroups — #202853

Order 512.

946 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21964C211yesmaximal, characteristic, complemented
2#21803C211yesmaximal, characteristic
3#21803C211yesmaximal, characteristic
4#20538C211yesmaximal, characteristic, complemented
5C2×#1545C211yesmaximal, characteristic
6#9572C212yesmaximal, complemented
7#9571C212yesmaximal, complemented
8#9564C212yesmaximal, complemented
9#9563C212yesmaximal, complemented
10#8038C211yesmaximal, characteristic, complemented
11#7880C211yesmaximal, characteristic, complemented
12C22×#127C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#423C2211yescharacteristic
17#2285C2211yescharacteristic
18#2285C2211yescharacteristic
19#1982—21no—
20#1982—21no—
21#1982C2212yes—
22#1982C2212yes—
23#1957—21no—
24#1957—21no—
25#1957C2212yes—
26#1957C2212yes—
27#1948—21no—
28#1948C2212yescomplemented
29#1947—21no—
30#1947C2212yescomplemented
31#1809—22no—
32#1786—22no—
33#1786—22no—
34#1770—21no—
35#1770C2212yescomplemented
36C8×C4⋊C4—21no—
37C8×C4⋊C4C2212yescomplemented
38#1750—22no—
39#1545C418yescomplemented
40C2×#331C2212yes—
41C2×#331C2212yes—
42#1296C2211yescharacteristic, complemented
43#1296C2211yescharacteristic, complemented
44#1296C2211yescharacteristic, complemented
45#1296C2211yescharacteristic, complemented
46#1277C2212yescomplemented
47#1277C2212yescomplemented
48C23×D4—21no—
49C23×D4—21no—
50C23×D4—21no—
51C22×M16—21no—
52C2×#127—21no—
53C2×#127—21no—
54C2×#127—21no—
55C2×#127—22no—
56C2×#127—22no—
57C2×#127—22no—
58C2×#127C2×C414yes—
59C2×#127C2312yes—
60C2×#117—21no—
61C2×#117—21no—
62C22×C42C2311yescharacteristic
63#431—22no—
64#431—22no—
65#431—22no—
66#431—22no—
67#431—22no—
68#431C2×C414yes—
69C2×#107—22no—
70C2×#106—22no—
71#423—22no—
72#423C2×C414yes—
73C2×C4⋊C8—21no—
74C2×C4⋊C8D411yescharacteristic
75C2×C4⋊C8D411yescharacteristic
76C2×C4⋊C8C2311yescharacteristic
77C2×C4⋊C8C2311yescharacteristic
78C2×C4⋊C8C2311yescharacteristic
79C2×C4⋊C8C2311yescharacteristic
80C2×C4⋊C8C2312yes—
81C2×#102—22no—
82C2×#102—22no—
83C2×#102—22no—
84C2×#102—22no—
85C2×#102—22no—
86C2×#102—22no—
87C2×#102—22no—
88C2×#102—22no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8D412yes—
92C2×#97—22no—
93C2×C4×C8—21no—
94C2×C4×C8—21no—
95C2×C4×C8—22no—
96C2×C4×C8C2312yes—
97#384D411yescharacteristic
98#384D411yescharacteristic
99#384C2311yescharacteristic
100#384C2311yescharacteristic
101#382—21no—
102#382—21no—
103#380—21no—
104#380—21no—
105#380—21no—
106#380D412yes—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4C2312yes—
111#340—22no—
112#340—22no—
113#336—22no—
114#336—22no—
115#331C2×C418yes—
116#331C2×C418yes—
117C25—41no—
118C22×D4—24no—
119C22×D4—24no—
120C22×D4—24no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—41no—
125C22×D4—41no—
126C22×D4—41no—
127C22×D4—42no—
128C22×D4—42no—
129C22×D4—42no—
130C22×D4—81no—
131C22×D4—81no—
132C22×D4—81no—
133C22×D4—81no—
134C22×D4—81no—
135C22×D4—81no—
136C22×D4—81no—
137C22×D4—81no—
138C22×D4—82no—
139C22×D4—82no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4#4111yescharacteristic
143C2×M16—24no—
144C2×M16—81no—
145C2×M16—81no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8C2×D412yes—
152C22×C8C2×D412yes—
153#127—24no—
154#127—24no—
155#127—24no—
156#127—22no—
157#127—22no—
158#127—22no—
159#127—42no—
160#127—42no—
161#127—42no—
162#127C22×C414yes—
163#117—81no—
164#117—81no—
165#117—82no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4C2×D412yes—
177C2×C4⋊C4C2×D412yes—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—41no—
180C2×C42—22no—
181C2×C42—41no—
182C2×C42C22×C412yes—
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42#4112yes—
187C2×C42#4112yes—
188C2×C42#4112yes—
189C2×C42C2411yescharacteristic
190#107—82no—
191#106—82no—
192C4⋊C8—24no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8C22×C414yes—
203C4⋊C8C22×C412yes—
204C4⋊C8C22×C412yes—
205#102—82no—
206#102—82no—
207#102—82no—
208#102—82no—
209#102—82no—
210#102—82no—
211#102—82no—
212#102—82no—
213C22⋊C8—24no—
214C22⋊C8—24no—
215C22⋊C8—42no—
216#97—82no—
217C4×C8—24no—
218C4×C8—22no—
219C4×C8—22no—
220C4×C8—42no—
221C4×C8—82no—
222#95—22no—
223#95—22no—
224#95—41no—
225#95—41no—
226C24—42no—
227C24—44no—
228C24—82no—
229C24—82no—
230C24—82no—
231C24—82no—
232C24—82no—
233C24—82no—
234C24#12111yescharacteristic
235C2×D4—24no—
236C2×D4—24no—
237C2×D4—24no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—44no—
250C2×D4—44no—
251C2×D4—44no—
252C2×D4—84no—
253C2×D4—84no—
254C2×D4—84no—
255C2×D4—84no—
256C2×D4—82no—
257C2×D4—82no—
258C2×D4—82no—
259C2×D4—82no—
260C2×D4—82no—
261C2×D4—82no—
262C2×D4—84no—
263C2×D4—84no—
264C2×D4—84no—
265C2×D4—84no—
266C2×D4—88no—
267C2×D4—88no—
268C2×D4—82no—
269C2×D4—82no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4#11712yes—
285C22×C4C4×D412yes—
286C22×C4C4×D412yes—
287C22×C4#12111yescharacteristic
288C22×C4#12111yescharacteristic
289C22×C4#12111yescharacteristic
290C22×C4#12312yes—
291C22×C4#12412yes—
292C22×C4#12612yes—
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C2×#4111yescharacteristic
296M16—84no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—24no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—81no—
333C4⋊C4—81no—
334C4⋊C4—81no—
335C4⋊C4—81no—
336C22⋊C4—82no—
337C22⋊C4—82no—
338C42—42no—
339C42—42no—
340C42—82no—
341C42#11712yes—
342C42C4×D412yes—
343C42C4×D412yes—
344C42#12312yes—
345C42#12412yes—
346C42#12612yes—
347C42C23×C411yescharacteristic
348C42C2×#4111yescharacteristic
349C42C2×#4111yescharacteristic
350C42C2×#4111yescharacteristic
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—44no—
358C23—88no—
359C23—88no—
360C23—88no—
361C23—82no—
362C23—82no—
363C23—82no—
364C23—82no—
365C23—82no—
366C23—82no—
367C23—164no—
368C23—164no—
369C23—164no—
370C23—164no—
371C23#38812yes—
372C23C2×#12111yescharacteristic
373D4—44no—
374D4—44no—
375D4—44no—
376D4—44no—
377D4—84no—
378D4—88no—
379D4—84no—
380D4—88no—
381D4—82no—
382D4—82no—
383D4—82no—
384D4—82no—
385D4—88no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4#38812yes—
415C2×C4#38812yes—
416C2×C4#38812yes—
417C2×C4#47112yes—
418C2×C4#47212yes—
419C2×C4C2×#11711yescharacteristic
420C2×C4C2×C4×D411yescharacteristic
421C2×C4C2×C4×D411yescharacteristic
422C2×C4C2×#12311yescharacteristic
423C2×C4C2×#12411yescharacteristic
424C2×C4C2×#12611yescharacteristic
425C2×C4#57611yescharacteristic
426C2×C4#57611yescharacteristic
427C2×C4#57611yescharacteristic
428C2×C4#57611yescharacteristic
429C2×C4#57711yescharacteristic
430C2×C4#57711yescharacteristic
431C2×C4#57711yescharacteristic
432C2×C4#57711yescharacteristic
433C8—42no—
434C8—42no—
435C8—42no—
436C8—42no—
437C8—44no—
438C8—81no—
439C8—81no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—84no—
448C22—84no—
449C22—84no—
450C22—1616no—
451C22—162no—
452C22—162no—
453C22—162no—
454C22—162no—
455C22C2×#38811yescharacteristic
456C22#304111yescharacteristic
457C22#304311yescharacteristic
458C22#306211yescharacteristic
459C22#306211yescharacteristic
460C22C2×#47111yescharacteristic
461C22C2×#47211yescharacteristic
462C4—22no—
463C4—21no—
464C4—21no—
465C4—22no—
466C4—82no—
467C4—82no—
468C4—161no—
469C4—161no—
470C4#193114yes—
471C4#294611yescharacteristic
472C4#294611yescharacteristic
473C4#294811yescharacteristic
474C4#294911yescharacteristic
475C4#309611yescharacteristic
476C4#309611yescharacteristic
477C4#313611yescharacteristic
478C4#313611yescharacteristic
479C2—42no—
480C2—164no—
481C2#1955211yescharacteristic
482C2#1955411yescharacteristic
483C2#1956411yescharacteristic
484C2#1956611yescharacteristic
485C2#2106311yescharacteristic
486C2C2×#193111yescharacteristic
487C2#2108911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#127C22
LCSG ▷ 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
1. #21964C2semidirect
4. #20538C2semidirect
6. #9572C2semidirect
7. #9571C2semidirect
8. #9564C2semidirect
9. #9563C2semidirect
10. #8038C2semidirect
11. #7880C2semidirect
28. #1948C22semidirect
30. #1947C22semidirect
35. #1770C22semidirect
37. C8×C4⋊C4C22semidirect
39. #1545C4semidirect
42. #1296C22semidirect
43. #1296C22semidirect
44. #1296C22semidirect
45. #1296C22semidirect
46. #1277C22semidirect
47. #1277C22semidirect

Maximal subgroups

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

subgroupindex
#219642
#218032
#205382
C2×#15452
#95722
#95712
#95642
#95632
#80382
#78802