Finite Group Databasebeta

← #401680

Subgroups — #401680

Order 512.

1442 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#424C211yesmaximal, characteristic
2C2×#2087C211yesmaximal, characteristic, complemented
3C2×#2086C211yesmaximal, characteristic, complemented
4C2×#1809C211yesmaximal, characteristic
5#14264C212yesmaximal
6#11231C211yesmaximal, characteristic, complemented
7#11231C211yesmaximal, characteristic, complemented
8#11228C212yesmaximal, complemented
9#11227C212yesmaximal, complemented
10#10722C211yesmaximal, characteristic
11#10721C212yesmaximal
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C2×#424—24no—
18C2×#424—24no—
19C2×#424—24no—
20C22×C4⋊C8C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#380C2212yes—
27C2×#380C2212yes—
28#2087—24no—
29#2086—24no—
30#1809—24no—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39#1444—22no—
40#1442—22no—
41#1439—22no—
42#1439—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C24×C4C2311yescharacteristic
46C22×M16—24no—
47C22×M16—22no—
48C22×M16—22no—
49C22×M16—42no—
50C22×M16D414yes—
51C23×C8—22no—
52C23×C8—22no—
53C23×C8—22no—
54C23×C8—24no—
55C23×C8—42no—
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C23×C8C2311yescharacteristic
60C2×#117—22no—
61C2×#117—22no—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117—22no—
65C2×#117—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4D411yescharacteristic
77C22×C22⋊C4D411yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×#107—22no—
81C2×#107—22no—
82C2×#107—22no—
83C2×#106—22no—
84C2×#106—42no—
85#424—232no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8D414yes—
97C2×C4⋊C8D414yes—
98#389—24no—
99#389—24no—
100#389—24no—
101#389—24no—
102#380—24no—
103#380—24no—
104#380—24no—
105#380Q818yes—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114#340—24no—
115#340—22no—
116#340—24no—
117#340—22no—
118#340—24no—
119#340—42no—
120#340—42no—
121#340—42no—
122C25C2×D411yescharacteristic
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D412yes—
139C23×C4C2×Q814yes—
140C23×C4#4112yes—
141C23×C4#4114yes—
142C23×C4C2411yescharacteristic
143C2×M16—28no—
144C2×M16—28no—
145C2×M16—44no—
146C2×M16—48no—
147C2×M16—48no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—28no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
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×C8C2×D412yes—
168C22×C8C2×D412yes—
169C22×C8C2×D412yes—
170C22×C8C2×D412yes—
171C22×C8C2×D412yes—
172C22×C8C2×D412yes—
173#117—28no—
174#117—28no—
175#117—28no—
176#117—28no—
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—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4C2×Q814yes—
212C2×C4⋊C4C2×Q814yes—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—82no—
222C2×C22⋊C4—82no—
223C2×C22⋊C4—82no—
224C2×C22⋊C4—82no—
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×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42#4112yes—
236C2×C42#4112yes—
237#107—44no—
238#106—44no—
239C4⋊C8—28no—
240C4⋊C8—28no—
241C4⋊C8—28no—
242C4⋊C8—28no—
243#95—28no—
244#95—44no—
245#95—44no—
246#95—44no—
247#95—82no—
248#95—82no—
249#95—82no—
250#95—82no—
251C24—21no—
252C24—21no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—22no—
258C24#12112yes—
259C24C22⋊Q814yes—
260C24C22×D411yescharacteristic
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
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—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#12112yes—
324C22×C4#12112yes—
325C22×C4#12112yes—
326C22×C4#12112yes—
327C22×C4#12112yes—
328C22×C4C22⋊Q814yes—
329C22×C4#12314yes—
330C22×C4#12518yes—
331C22×C4C4⋊Q814yes—
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C22×Q812yes—
336C22×C4C2×#4112yes—
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4111yescharacteristic
340C22×C4H(5,𝔽2)12yes—
341C22×C42−1+412yes—
342M16—416no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—416no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8#12014yes—
364C4⋊C4—28no—
365C4⋊C4—28no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
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—42no—
382C4⋊C4—88no—
383C4⋊C4—88no—
384C22⋊C4—28no—
385C22⋊C4—28no—
386C22⋊C4—88no—
387C22⋊C4—88no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—22no—
396C23—24no—
397C23—24no—
398C23—24no—
399C23—24no—
400C23—22no—
401C23—22no—
402C23—24no—
403C23—24no—
404C23—24no—
405C23—24no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—42no—
415C23—42no—
416C23#39814yes—
417C23C2×#12111yescharacteristic
418C23C2×#12111yescharacteristic
419C23C2×#12111yescharacteristic
420C23C2×C22⋊Q812yes—
421C23#53712yes—
422C23#57612yes—
423C23#57712yes—
424C23#57712yes—
425C23#57812yes—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—28no—
434C2×C4—28no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
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—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#39814yes—
485C2×C4C2×#12011yescharacteristic
486C2×C4C2×#12311yescharacteristic
487C2×C4#52511yescharacteristic
488C2×C4#52511yescharacteristic
489C2×C4#52511yescharacteristic
490C2×C4C2×#12512yes—
491C2×C4C2×C4⋊Q811yescharacteristic
492C2×C4#53411yescharacteristic
493C2×C4#53611yescharacteristic
494C2×C4#53912yes—
495C2×C4#54612yes—
496C2×C4#54612yes—
497C2×C4D4×Q812yes—
498C2×C4D4×Q812yes—
499C2×C4#55212yes—
500C2×C4#55212yes—
501C2×C4#55412yes—
502C2×C4#55412yes—
503C8—24no—
504C8—42no—
505C8—42no—
506C8—42no—
507C22—28no—
508C22—24no—
509C22—44no—
510C22—44no—
511C22—44no—
512C22—44no—
513C22—44no—
514C22—42no—
515C22—42no—
516C22C2×#39811yescharacteristic
517C22#241712yes—
518C22#241712yes—
519C22#261811yescharacteristic
520C22#284711yescharacteristic
521C22#305811yescharacteristic
522C22#305911yescharacteristic
523C22#306011yescharacteristic
524C22#306011yescharacteristic
525C22#306212yes—
526C22#306212yes—
527C22#306312yes—
528C22#306312yes—
529C22#308112yes—
530C22#308212yes—
531C22#308312yes—
532C22#308412yes—
533C22#325712yes—
534C22#325812yes—
535C22#326012yes—
536C22#326212yes—
537C4—24no—
538C4—42no—
539C4—42no—
540C4—42no—
541C4—42no—
542C4—84no—
543C4—84no—
544C4—162no—
545C4—162no—
546C4#240511yescharacteristic
547C4#243411yescharacteristic
548C4#247312yes—
549C4#247312yes—
550C4#248411yescharacteristic
551C4#258311yescharacteristic
552C2—44no—
553C2#1482111yescharacteristic
554C2#2367112yes—
555C2#2484712yes—
556C2#2484912yes—
557C2#2485212yes—
558C2#2485412yes—
559C2#2978712yes—
560C2#3293911yescharacteristic
561C2#3294011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×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
2. C2×#2087C2semidirect
3. C2×#2086C2semidirect
6. #11231C2semidirect
7. #11231C2semidirect
8. #11228C2semidirect
9. #11227C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4242
C2×#20872
C2×#20862
C2×#18092
#142642
#112312
#112282
#112272
#107222
#107212