Finite Group Databasebeta

← #375491

Subgroups — #375491

Order 512.

1482 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic
2C2×#1729C211yesmaximal, characteristic
3#13777C211yesmaximal, characteristic
4#11384C212yesmaximal, complemented
5#11372C212yesmaximal, complemented
6#10885C212yesmaximal, complemented
7#10884C212yesmaximal, complemented
8#10722C212yesmaximal
9#10721C212yesmaximal
10C23×M16C2211yescharacteristic
11C24×C8C2211yescharacteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14#2628—24no—
15#2628C418yescomplemented
16C2×#400—21no—
17C2×#399—21no—
18C2×#395—21no—
19C2×#393—21no—
20C2×#389C2212yes—
21C2×#382C2211yescharacteristic
22C2×#380C2212yes—
23C2×#380C2212yes—
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34#1729—24no—
35C2×#336C2212yes—
36C2×#336C2212yes—
37#1437—21no—
38#1437—21no—
39#1436—21no—
40#1436—21no—
41C4⋊#95C2212yes—
42C23×Q8—21no—
43C24×C4C2311yescharacteristic
44C22×M16—22no—
45C22×M16—24no—
46C22×M16—24no—
47C22×M16—41no—
48C22×M16—41no—
49C23×C8—22no—
50C23×C8—24no—
51C23×C8—24no—
52C23×C8—41no—
53C23×C8—41no—
54C23×C8C2312yes—
55C23×C8C2312yes—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—41no—
58C2×C4⋊Q8—41no—
59C2×C4⋊Q8—41no—
60C2×C4⋊Q8—41no—
61C2×C4⋊Q8C2×C414yes—
62C2×#117—41no—
63C2×#117—41no—
64C2×#117—41no—
65C2×#117—41no—
66C2×#117C2×C414yes—
67C2×#117D414yes—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4D411yescharacteristic
79C22×C22⋊C4D411yescharacteristic
80C22×C42—21no—
81C22×C42—21no—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84#400—28no—
85#399—28no—
86#395—28no—
87#393—28no—
88#389—24no—
89#389C2×C418yes—
90#382—22no—
91#382C2×C414yes—
92#380—24no—
93#380—24no—
94#380C2×C418yes—
95#380Q818yescomplemented
96C4×C4⋊C4—41no—
97C4×C4⋊C4—41no—
98C4×C4⋊C4—41no—
99C4×C4⋊C4—41no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109#340—24no—
110#340—24no—
111#340—24no—
112#340—24no—
113#340—42no—
114#340—42no—
115#340—42no—
116#340—42no—
117#336—24no—
118#336—24no—
119C25C2×D411yescharacteristic
120C22×Q8—24no—
121C22×Q8—42no—
122C22×Q8—42no—
123C22×Q8—82no—
124C22×Q8—82no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4C22×C412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4C2×Q814yes—
143C23×C4#4114yes—
144C23×C4C2411yescharacteristic
145C2×M16—28no—
146C2×M16—48no—
147C2×M16—48no—
148C2×M16—44no—
149C2×M16—44no—
150C22×C8—28no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—48no—
164C22×C8—48no—
165C22×C8—44no—
166C22×C8—44no—
167C22×C8#4114yes—
168C22×C8#4114yes—
169C4⋊Q8—44no—
170C4⋊Q8—44no—
171C4⋊Q8—44no—
172C4⋊Q8—44no—
173#117—44no—
174#117—44no—
175#117—44no—
176#117—44no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4C22⋊C414yes—
208C2×C4⋊C4C22⋊C414yes—
209C2×C4⋊C4C22×C414yes—
210C2×C4⋊C4C22×C414yes—
211C2×C4⋊C4C2×Q814yes—
212C2×C4⋊C4C2×Q814yes—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—82no—
220C2×C22⋊C4—82no—
221C2×C22⋊C4—82no—
222C2×C22⋊C4—82no—
223C2×C22⋊C4C22⋊C414yes—
224C2×C22⋊C4C22⋊C414yes—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—42no—
234C2×C42—42no—
235C2×C42—82no—
236C2×C42—82no—
237C2×C42C22×C412yes—
238C2×C42C22×C412yes—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241#95—28no—
242#95—24no—
243#95—24no—
244#95—28no—
245#95—28no—
246#95—42no—
247#95—42no—
248#95—82no—
249#95—82no—
250C24—22no—
251C24—21no—
252C24—21no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24C2×C22⋊C412yes—
258C24C22⋊Q814yes—
259C24C22×D411yescharacteristic
260C2×Q8—48no—
261C2×Q8—88no—
262C2×Q8—88no—
263C2×Q8—84no—
264C2×Q8—84no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4C2×C22⋊C412yes—
328C22×C4C2×C22⋊C412yes—
329C22×C4C2×C22⋊C412yes—
330C22×C4C2×C22⋊C412yes—
331C22×C4C2×C22⋊C412yes—
332C22×C4C4×Q818yes—
333C22×C4C22⋊Q814yes—
334C22×C4#12414yes—
335C22×C4C4⋊Q814yes—
336C22×C4C23×C411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×Q812yes—
339C22×C4C2×#4112yes—
340C22×C4C2×#4112yes—
341C22×C4C2×#4112yes—
342C22×C4H(5,𝔽2)11yescharacteristic
343C22×C4H(5,𝔽2)11yescharacteristic
344C22×C4H(5,𝔽2)11yescharacteristic
345C22×C4H(5,𝔽2)11yescharacteristic
346M16—416no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—416no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—48no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—88no—
382C22⋊C4—44no—
383C22⋊C4—44no—
384C22⋊C4—88no—
385C22⋊C4—88no—
386C42—22no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—22no—
394C42—84no—
395C42—84no—
396C23—22no—
397C23—24no—
398C23—24no—
399C23—24no—
400C23—24no—
401C23—24no—
402C23—24no—
403C23—24no—
404C23—24no—
405C23—22no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—42no—
415C23—41no—
416C23#39114yes—
417C23C22×C22⋊C411yescharacteristic
418C23C2×C22⋊Q812yes—
419C23#53212yes—
420C23#53411yescharacteristic
421C23#53411yescharacteristic
422C23#57612yes—
423C23#57712yes—
424C23#57712yes—
425C23#57812yes—
426Q8—816no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—28no—
436C2×C4—28no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4#39114yes—
486C2×C4C2×C4×Q812yes—
487C2×C4C4×#4112yes—
488C2×C4#51811yescharacteristic
489C2×C4#51811yescharacteristic
490C2×C4#52511yescharacteristic
491C2×C4#52511yescharacteristic
492C2×C4C2×#12411yescharacteristic
493C2×C4C2×C4⋊Q811yescharacteristic
494C2×C4#53411yescharacteristic
495C2×C4#53411yescharacteristic
496C2×C4#55112yes—
497C2×C4#55112yes—
498C2×C4#55112yes—
499C2×C4#55112yes—
500C2×C4#55412yes—
501C2×C4#55412yes—
502C2×C4#55412yes—
503C2×C4#55412yes—
504C8—44no—
505C8—44no—
506C22—28no—
507C22—24no—
508C22—42no—
509C22—42no—
510C22—44no—
511C22—44no—
512C22—44no—
513C22—44no—
514C22—44no—
515C22C2×#39111yescharacteristic
516C22#229711yescharacteristic
517C22#230811yescharacteristic
518C22#256412yes—
519C22#256412yes—
520C22#289712yes—
521C22#289812yes—
522C22#308112yes—
523C22#308212yes—
524C22#308312yes—
525C22#308412yes—
526C22#317312yes—
527C22#317512yes—
528C22#317812yes—
529C22#317912yes—
530C22#319812yes—
531C22#319911yescharacteristic
532C22#320011yescharacteristic
533C22#320011yescharacteristic
534C22#320112yes—
535C22#320211yescharacteristic
536C4—24no—
537C4—42no—
538C4—42no—
539C4—42no—
540C4—42no—
541C4—84no—
542C4—84no—
543C4—84no—
544C4—84no—
545C4#236812yes—
546C4#236812yes—
547C4#256111yescharacteristic
548C4#256111yescharacteristic
549C4#258311yescharacteristic
550C4#258311yescharacteristic
551C2—44no—
552C2#1415911yescharacteristic
553C2#2131512yes—
554C2#2141212yes—
555C2#2177011yescharacteristic
556C2#2177111yescharacteristic
557C2#2917012yes—
558C2#2917112yes—
559C2#2917412yes—
560C2#2917512yes—
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. #11384C2semidirect
5. #11372C2semidirect
6. #10885C2semidirect
7. #10884C2semidirect
15. #2628C4semidirect
95. #380Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
C2×#17292
#137772
#113842
#113722
#108852
#108842
#107222
#107212