Finite Group Databasebeta

← #363659

Subgroups — #363659

Order 512.

1202 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C8×C22⋊C4C211yesmaximal, characteristic
2#13875C212yesmaximal
3#11268C212yesmaximal, complemented
4#11095C212yesmaximal
5#10908C212yesmaximal, complemented
6#10861C211yesmaximal, characteristic, complemented
7#10842C211yesmaximal, characteristic, complemented
8#10820C211yesmaximal, characteristic, complemented
9#10721C211yesmaximal, characteristic
10#10721C211yesmaximal, characteristic
11C4×#340C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×#106C2212yes—
14C22×C22⋊C8C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17C2×#382C2212yes—
18C2×#380C2212yes—
19C2×#380C2212yes—
20C2×C4×C4⋊C4C2212yes—
21C2×C4×C22⋊C4C2211yescharacteristic
22C8×C22⋊C4C418yescomplemented
23C2×#340C2212yes—
24C2×#340C2211yescharacteristic
25C2×#340C2211yescharacteristic
26C2×#340C2212yes—
27C2×#340C2211yescharacteristic
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2211yescharacteristic
34#1443—22no—
35#1440—22no—
36#1437—22no—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39#1434—22no—
40C24×C4C2311yescharacteristic
41C23×C8—22no—
42C23×C8—22no—
43C23×C8—42no—
44C23×C8—42no—
45C23×C8C2×C414yes—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C22⋊C4—22no—
55C22×C22⋊C4—22no—
56C22×C22⋊C4D411yescharacteristic
57C22×C22⋊C4D411yescharacteristic
58C22×C42—22no—
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C2×#106—24no—
62C2×#106—24no—
63C2×#106—42no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8C2×C414yes—
67C2×C4×C8—42no—
68C2×C4×C8—42no—
69C2×C4×C8C2×C414yes—
70C2×C4×C8C2×C414yes—
71#382—24no—
72#382—42no—
73#380—24no—
74#380—24no—
75#380—42no—
76#380—42no—
77C4×C4⋊C4—24no—
78C4×C4⋊C4—42no—
79C4×C22⋊C4C2×C414yes—
80C4×C22⋊C4Q814yes—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95D411yescharacteristic
91C2×#95D411yescharacteristic
92#340—22no—
93#340—22no—
94#340—24no—
95#340—22no—
96#340—22no—
97#340—24no—
98#340—42no—
99#340—42no—
100#340—42no—
101#336C2×C414yes—
102#336C2×C414yes—
103C25C2×D411yescharacteristic
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4C22×C412yes—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×Q812yes—
123C23×C4C2411yescharacteristic
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—22no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—28no—
137C22×C8—44no—
138C22×C8—44no—
139C22×C8—48no—
140C22×C8—48no—
141C22×C8C22×C412yes—
142C22×C8C22×C412yes—
143C22×C8C22×C412yes—
144C22×C8C22×C412yes—
145C22×C8#4112yes—
146C22×C8#4112yes—
147C22×C8#4112yes—
148C22×C8#4112yes—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
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#4114yes—
172C2×C4⋊C4#4114yes—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—82no—
182C2×C22⋊C4—82no—
183C2×C22⋊C4C4⋊C414yes—
184C2×C22⋊C4C4⋊C414yes—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—44no—
190C2×C42—44no—
191C2×C42—82no—
192C2×C42C22×C412yes—
193C2×C42C22×C412yes—
194C2×C42C2×Q812yes—
195C2×C42C2×Q812yes—
196#106—48no—
197C22⋊C8—28no—
198C4×C8—44no—
199C4×C8—44no—
200#95—44no—
201#95—44no—
202#95—44no—
203#95—44no—
204#95—44no—
205#95—44no—
206#95—82no—
207#95—82no—
208#95—82no—
209#95C4⋊C414yes—
210#95C4⋊C414yes—
211C24—21no—
212C24—21no—
213C24—22no—
214C24—22no—
215C24—21no—
216C24—21no—
217C24—22no—
218C24—22no—
219C24C2×C4⋊C412yes—
220C24#13512yes—
221C24#13512yes—
222C24C22×D411yescharacteristic
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4C2×C4⋊C412yes—
288C22×C4C2×C4⋊C412yes—
289C22×C4C2×C4⋊C412yes—
290C22×C4C2×C4⋊C412yes—
291C22×C4C2×C4⋊C412yes—
292C22×C4#13512yes—
293C22×C4#13512yes—
294C22×C4C23×C411yescharacteristic
295C22×C4C22×Q811yescharacteristic
296C22×C4C2×#4111yescharacteristic
297C22×C4C2×#4111yescharacteristic
298C22×C4C2×#4111yescharacteristic
299C22×C4C2×#4111yescharacteristic
300C22×C4C2×#4112yes—
301C22×C4C2×#4112yes—
302C22×C4H(5,𝔽2)12yes—
303C22×C4H(5,𝔽2)12yes—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—416no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8#11714yes—
324C2×C8#11714yes—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C22⋊C4—28no—
342C22⋊C4—28no—
343C22⋊C4—88no—
344C22⋊C4—88no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—24no—
350C42—24no—
351C42—88no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—24no—
362C23—24no—
363C23—24no—
364C23—24no—
365C23—22no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23#42714yes—
375C23C22×C4⋊C411yescharacteristic
376C23#53211yescharacteristic
377C23#53211yescharacteristic
378C23#54612yes—
379C23#54612yes—
380C23C2×#13511yescharacteristic
381C23C2×#13511yescharacteristic
382C23C2×#13511yescharacteristic
383C23C2×#13511yescharacteristic
384C23#57712yes—
385C23#57712yes—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—22no—
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—22no—
401C2×C4—22no—
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—44no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4#42714yes—
445C2×C4C2×#11711yescharacteristic
446C2×C4C2×#11711yescharacteristic
447C2×C4C4×#4111yescharacteristic
448C2×C4C4×#4111yescharacteristic
449C2×C4#51811yescharacteristic
450C2×C4#51811yescharacteristic
451C2×C4#52912yes—
452C2×C4#52912yes—
453C2×C4#53311yescharacteristic
454C2×C4#53311yescharacteristic
455C2×C4#53812yes—
456C2×C4#53812yes—
457C2×C4#55112yes—
458C2×C4#55312yes—
459C2×C4#55312yes—
460C2×C4#55412yes—
461C2×C4#55412yes—
462C2×C4#55912yes—
463C8—24no—
464C8—24no—
465C8—42no—
466C8—42no—
467C22—24no—
468C22—24no—
469C22—24no—
470C22—44no—
471C22—44no—
472C22—42no—
473C22—44no—
474C22—44no—
475C22—42no—
476C22—42no—
477C22—42no—
478C22#228511yescharacteristic
479C22#234911yescharacteristic
480C22#234911yescharacteristic
481C22#255712yes—
482C22#255912yes—
483C22C2×#42711yescharacteristic
484C22C2×#42711yescharacteristic
485C22#291211yescharacteristic
486C22#291211yescharacteristic
487C22#316811yescharacteristic
488C22#316811yescharacteristic
489C22#316811yescharacteristic
490C22#316811yescharacteristic
491C22#317312yes—
492C22#317412yes—
493C22#329612yes—
494C22#329612yes—
495C22#329612yes—
496C22#330112yes—
497C22#330112yes—
498C22#330112yes—
499C22#330312yes—
500C22#330412yes—
501C4—24no—
502C4—42no—
503C4—42no—
504C4—42no—
505C4—42no—
506C4—84no—
507C4—84no—
508C4—162no—
509C4—162no—
510C4#229811yescharacteristic
511C4#231211yescharacteristic
512C4#235311yescharacteristic
513C4#235711yescharacteristic
514C4#258712yes—
515C4#258912yes—
516C2—44no—
517C2#1394811yescharacteristic
518C2#2081811yescharacteristic
519C2#2082011yescharacteristic
520C2#2270611yescharacteristic
521C2#2270611yescharacteristic
522C2#2270611yescharacteristic
523C2#2270811yescharacteristic
524C2#2898012yes—
525C2#2898212yes—
526C2#2904412yes—
527C2#2904612yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11268C2semidirect
5. #10908C2semidirect
6. #10861C2semidirect
7. #10842C2semidirect
8. #10820C2semidirect
11. C4×#340C2semidirect
22. C8×C22⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
C2×C8×C22⋊C42
#138752
#112682
#110952
#109082
#108612
#108422
#108202
#107212
C4×#3402