Finite Group Databasebeta

← #358408

Subgroups — #358408

Order 512.

1184 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1763C211yesmaximal, characteristic
2C2×C4×#103C211yesmaximal, characteristic
3C2×#1746C211yesmaximal, characteristic
4#13786C211yesmaximal, characteristic
5#13783C211yesmaximal, characteristic
6#11315C212yesmaximal
7#11293C212yesmaximal
8#11190C212yesmaximal
9#11067C212yesmaximal
10#10703C211yesmaximal, characteristic
11C4×#340C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×#103C2212yes—
14C22×#103C2211yescharacteristic
15C22×#103C2211yescharacteristic
16C22×#97C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C2×#391C2212yes—
19C2×#389C2212yes—
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×#376C2211yescharacteristic
24#1763—24no—
25C4×#103C418yescomplemented
26#1746—24no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#336C2212yes—
35C2×#336C2212yes—
36#1436—22no—
37#1436—22no—
38C4⋊#95—21no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41#1434—21no—
42#1434—21no—
43#1434—21no—
44C23×Q8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2312yes—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8C2×C414yes—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C42—21no—
62C22×C42—21no—
63C22×C42—21no—
64C22×C42—21no—
65C22×C42C2312yes—
66C22×C42C2311yescharacteristic
67C2×#103—24no—
68C2×#103—24no—
69C2×#103—24no—
70C2×#103—24no—
71C2×#103—24no—
72C2×#103C2×C414yes—
73C2×#103C2×C414yes—
74C2×#97—22no—
75C2×#97—22no—
76C2×#97D414yes—
77C2×C4×C8—22no—
78C2×C4×C8—22no—
79C2×C4×C8C2×C414yes—
80#391—24no—
81#391—24no—
82#389—24no—
83#389—24no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4C2×C414yes—
90#376—22no—
91#376—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95D411yescharacteristic
103C2×#95D411yescharacteristic
104C2×#95D412yes—
105#340—24no—
106#340—24no—
107#340—24no—
108#340—24no—
109#340—24no—
110#340C2×C414yes—
111#340C2×C414yes—
112#336—24no—
113#336—24no—
114C22×Q8—21no—
115C22×Q8—22no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8C22×C412yes—
123C22×Q8#4114yes—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2411yescharacteristic
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8C22×C412yes—
150C22×C8C22×C412yes—
151C22×C8C2×D414yes—
152C4×Q8—28no—
153C4×Q8—28no—
154C4×Q8—44no—
155C4×Q8—44no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4C22×C412yes—
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4#4114yes—
186C2×C4⋊C4#4114yes—
187C2×C4⋊C4#4114yes—
188C2×C42—24no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—24no—
196C2×C42—42no—
197C2×C42—42no—
198C2×C42—42no—
199C2×C42—42no—
200C2×C42—82no—
201C2×C42—81no—
202C2×C42—81no—
203C2×C42—81no—
204C2×C42—81no—
205C2×C42C22×C412yes—
206C2×C42C2×D412yes—
207C2×C42#4112yes—
208C2×C42#4112yes—
209C2×C42#4112yes—
210C2×C42#4112yes—
211#103—216no—
212#103—28no—
213#103—28no—
214#97—28no—
215C4×C8—28no—
216#95—28no—
217#95—28no—
218#95—28no—
219#95—24no—
220#95—22no—
221#95—22no—
222#95—24no—
223#95—42no—
224#95—42no—
225#95—82no—
226#95—82no—
227#95—82no—
228#95—82no—
229C24C22×D411yescharacteristic
230C2×Q8—22no—
231C2×Q8—24no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—24no—
237C2×Q8—24no—
238C2×Q8—24no—
239C2×Q8—24no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8#11714yes—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4C4×D412yes—
305C22×C4C4×D412yes—
306C22×C4#12112yes—
307C22×C4#12112yes—
308C22×C4#12412yes—
309C22×C4#12412yes—
310C22×C4C23×C411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C22×C4C2×#4112yes—
314C22×C4C2×#4111yescharacteristic
315C22×C4C2×#4112yes—
316C22×C4C2×#4111yescharacteristic
317C22×C4C2×#4112yes—
318C22×C4C2×#4112yes—
319C22×C4C2×#4111yescharacteristic
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—24no—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C4⋊C4—22no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—48no—
355C4⋊C4—88no—
356C4⋊C4#11714yes—
357C42—42no—
358C42—42no—
359C42—42no—
360C42—42no—
361C42—48no—
362C42—84no—
363C42—84no—
364C42—84no—
365C42#11714yes—
366C42C4×D414yes—
367C42#12314yes—
368C42#12614yes—
369C23C2×C4×D411yescharacteristic
370C23C2×#12111yescharacteristic
371C23#52512yes—
372C23#52512yes—
373C23C2×#12411yescharacteristic
374C23#57611yescharacteristic
375C23#57712yes—
376C23#57711yescharacteristic
377C23#57711yescharacteristic
378C23#57812yes—
379C23#57811yescharacteristic
380Q8—24no—
381Q8—24no—
382Q8—44no—
383Q8—44no—
384Q8—42no—
385Q8—42no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—81no—
440C2×C4—82no—
441C2×C4—81no—
442C2×C4—82no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—82no—
446C2×C4#38814yes—
447C2×C4#38814yes—
448C2×C4C2×#11711yescharacteristic
449C2×C4C2×#11711yescharacteristic
450C2×C4C2×#11711yescharacteristic
451C2×C4C2×C4×D411yescharacteristic
452C2×C4C4×#4111yescharacteristic
453C2×C4C4×#4111yescharacteristic
454C2×C4C2×#12311yescharacteristic
455C2×C4#52512yes—
456C2×C4C2×#12611yescharacteristic
457C2×C4#52912yes—
458C2×C4#52912yes—
459C2×C4#52912yes—
460C2×C4#52912yes—
461C2×C4#52912yes—
462C2×C4#52912yes—
463C2×C4#52912yes—
464C2×C4#52912yes—
465C2×C4#53212yes—
466C8—44no—
467C8—42no—
468C8—42no—
469C22C2×#38811yescharacteristic
470C22#229011yescharacteristic
471C22#229011yescharacteristic
472C22#239212yes—
473C22#239812yes—
474C22#294911yescharacteristic
475C22#295011yescharacteristic
476C22#295111yescharacteristic
477C22#295211yescharacteristic
478C22#306211yescharacteristic
479C22#306311yescharacteristic
480C22#306312yes—
481C22#311912yes—
482C22#312012yes—
483C22#312112yes—
484C22#312112yes—
485C22#312312yes—
486C22#312412yes—
487C22#312512yes—
488C22#312712yes—
489C22#313611yescharacteristic
490C22#313711yescharacteristic
491C22#313712yes—
492C4—24no—
493C4—24no—
494C4—44no—
495C4—44no—
496C4—84no—
497C4—82no—
498C4—82no—
499C4—82no—
500C4—82no—
501C4—84no—
502C4#228311yescharacteristic
503C4#229011yescharacteristic
504C4#229311yescharacteristic
505C4#229811yescharacteristic
506C4#239512yes—
507C4#240112yes—
508C2#1384211yescharacteristic
509C2#2108711yescharacteristic
510C2#2108811yescharacteristic
511C2#2113211yescharacteristic
512C2#2113311yescharacteristic
513C2#2113511yescharacteristic
514C2#2113611yescharacteristic
515C2#2371312yes—
516C2#2371412yes—
517C2#2382912yes—
518C2#2383212yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
25. C4×#103C4semidirect

Maximal subgroups

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

subgroupindex
C2×#17632
C2×C4×#1032
C2×#17462
#137862
#137832
#113152
#112932
#111902
#110672
#107032
C4×#3402