Finite Group Databasebeta

← #364359

Subgroups — #364359

Order 512.

1302 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1750C211yesmaximal, characteristic
2#14340C211yesmaximal, characteristic
3#13795C211yesmaximal, characteristic
4#11268C212yesmaximal, complemented
5#11267C212yesmaximal, complemented
6#10821C212yesmaximal, complemented
7#10820C212yesmaximal, complemented
8#10731C212yesmaximal
9#10730C212yesmaximal
10C22×#117C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C22×C22⋊C8C2212yes—
14C22×#97C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C2×#384C2211yescharacteristic
17C2×#382C2211yescharacteristic
18C2×#380C2212yes—
19#1750C418yescomplemented
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#336C2212yes—
29#1441—21no—
30#1441—21no—
31#1439—21no—
32#1439—21no—
33C4⋊#95—21no—
34C4⋊#95—21no—
35C4⋊#95C2212yes—
36C4⋊#95C2212yes—
37C4⋊#95C2212yes—
38C22⋊#95—21no—
39C22⋊#95—21no—
40C24×C4—21no—
41C24×C4C2311yescharacteristic
42C23×C8C2312yes—
43C23×C8C2312yes—
44C2×#117—24no—
45C2×#117—24no—
46C2×#117—41no—
47C2×#117—41no—
48C2×#117—41no—
49C2×#117—41no—
50C2×#117—41no—
51C2×#117—41no—
52C2×#117C2×C414yes—
53C2×#117Q814yes—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—24no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—41no—
60C22×C4⋊C4—41no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4D411yescharacteristic
70C22×C22⋊C4D411yescharacteristic
71C22×C42—21no—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C2×C22⋊C8—24no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8C2×C418yes—
77C2×#97—41no—
78C2×#97—41no—
79C2×#97C2×C414yescomplemented
80C2×C4×C8—41no—
81C2×C4×C8—41no—
82C2×C4×C8C2×C414yescomplemented
83#384—41no—
84#384—41no—
85#382—41no—
86#382—41no—
87#380—24no—
88#380—24no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—42no—
103#340—42no—
104#340—42no—
105#340—42no—
106#336C2×C418yescomplemented
107C25C2×D411yescharacteristic
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—24no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—42no—
120C23×C4—42no—
121C23×C4—82no—
122C23×C4—82no—
123C23×C4C22×C412yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×Q812yes—
130C23×C4#4114yes—
131C23×C4#4114yes—
132C23×C4C2411yescharacteristic
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8C22×C414yes—
146C22×C8C22×C414yes—
147#117—44no—
148#117—44no—
149#117—44no—
150#117—44no—
151#117—48no—
152#117—48no—
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—24no—
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—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—48no—
177C2×C4⋊C4—48no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4C4⋊C414yes—
189C2×C4⋊C4C4⋊C414yes—
190C2×C4⋊C4#4114yes—
191C2×C4⋊C4#4114yes—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—82no—
201C2×C22⋊C4—82no—
202C2×C22⋊C4C4⋊C414yes—
203C2×C22⋊C4C4⋊C414yes—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—24no—
213C2×C42—42no—
214C2×C42—42no—
215C2×C42C22×C412yes—
216C2×C42C22×C412yes—
217C2×C42C2×Q812yes—
218C2×C42C2×Q812yes—
219C22⋊C8—216no—
220#97—44no—
221C4×C8—44no—
222#95—42no—
223#95—42no—
224#95—44no—
225#95—82no—
226#95—82no—
227#95—82no—
228#95—82no—
229#95—82no—
230#95—82no—
231C24—21no—
232C24—21no—
233C24—21no—
234C24—21no—
235C24—22no—
236C24—22no—
237C24—22no—
238C24—22no—
239C24C2×C4⋊C412yes—
240C24#12412yes—
241C24#12412yes—
242C24C22×D411yescharacteristic
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—48no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—88no—
307C22×C4—88no—
308C22×C4—84no—
309C22×C4—84no—
310C22×C4C2×C4⋊C412yes—
311C22×C4C2×C4⋊C412yes—
312C22×C4C2×C4⋊C412yes—
313C22×C4C2×C4⋊C412yes—
314C22×C4C2×C4⋊C412yes—
315C22×C4#11718yes—
316C22×C4#12412yes—
317C22×C4#12412yes—
318C22×C4#12518yes—
319C22×C4C23×C411yescharacteristic
320C22×C4C22×Q811yescharacteristic
321C22×C4C2×#4112yes—
322C22×C4C2×#4112yes—
323C22×C4C2×#4112yes—
324C22×C4C2×#4112yes—
325C22×C4H(5,𝔽2)11yescharacteristic
326C22×C4H(5,𝔽2)11yescharacteristic
327C22×C4H(5,𝔽2)11yescharacteristic
328C22×C4H(5,𝔽2)11yescharacteristic
329C2×C8—24no—
330C2×C8—24no—
331C2×C8—24no—
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—44no—
348C4⋊C4—44no—
349C4⋊C4—48no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—84no—
363C4⋊C4—84no—
364C22⋊C4—44no—
365C22⋊C4—44no—
366C22⋊C4—48no—
367C22⋊C4—88no—
368C42—22no—
369C42—22no—
370C42—22no—
371C42—22no—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—48no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—24no—
383C23—24no—
384C23—24no—
385C23—24no—
386C23—24no—
387C23—24no—
388C23—22no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—42no—
398C23—42no—
399C23#38214yes—
400C23C22×C4⋊C411yescharacteristic
401C23#52511yescharacteristic
402C23#52511yescharacteristic
403C23C2×#12411yescharacteristic
404C23C2×#12411yescharacteristic
405C23#53411yescharacteristic
406C23#53411yescharacteristic
407C23#57612yes—
408C23#57712yes—
409C23#57712yes—
410C23#57812yes—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—28no—
418C2×C4—28no—
419C2×C4—28no—
420C2×C4—28no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—816no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4#38214yes—
471C2×C4C2×#11712yes—
472C2×C4C4×#4112yes—
473C2×C4#51811yescharacteristic
474C2×C4#51811yescharacteristic
475C2×C4C2×#12512yes—
476C2×C4#53312yes—
477C2×C4#54311yescharacteristic
478C2×C4#54311yescharacteristic
479C2×C4#55112yes—
480C2×C4#55112yes—
481C2×C4#55112yes—
482C2×C4#55112yes—
483C2×C4#55312yes—
484C2×C4#55312yes—
485C2×C4#55312yes—
486C2×C4#55312yes—
487C8—44no—
488C8—44no—
489C22—28no—
490C22—24no—
491C22—44no—
492C22—44no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—44no—
500C22C2×#38211yescharacteristic
501C22#229611yescharacteristic
502C22#231511yescharacteristic
503C22#256111yescharacteristic
504C22#256111yescharacteristic
505C22#259111yescharacteristic
506C22#259111yescharacteristic
507C22#291212yes—
508C22#291312yes—
509C22#311812yes—
510C22#311912yes—
511C22#312012yes—
512C22#312112yes—
513C22#313612yes—
514C22#313612yes—
515C22#313712yes—
516C22#313712yes—
517C22#319911yescharacteristic
518C22#320011yescharacteristic
519C22#320011yescharacteristic
520C22#320211yescharacteristic
521C22#320312yes—
522C22#320412yes—
523C4—24no—
524C4—42no—
525C4—42no—
526C4—42no—
527C4—42no—
528C4—84no—
529C4—84no—
530C4—162no—
531C4—162no—
532C4#235312yes—
533C4#235312yes—
534C4#256912yes—
535C4#256912yes—
536C2—44no—
537C2#1396611yescharacteristic
538C2#2076312yes—
539C2#2136712yes—
540C2#2194611yescharacteristic
541C2#2194711yescharacteristic
542C2#2911611yescharacteristic
543C2#2911811yescharacteristic
544C2#2911912yes—
545C2#2934212yes—
546C2#2934311yescharacteristic
547C2#2934411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11268C2semidirect
5. #11267C2semidirect
6. #10821C2semidirect
7. #10820C2semidirect
19. #1750C4semidirect
79. C2×#97C2×C4semidirect
82. C2×C4×C8C2×C4semidirect
106. #336C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#17502
#143402
#137952
#112682
#112672
#108212
#108202
#107312
#107302