Finite Group Databasebeta

← #355135

Subgroups — #355135

Order 512.

1570 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2316C211yesmaximal, characteristic
2C2×#1824C211yesmaximal, characteristic
3C2×#1823C211yesmaximal, characteristic
4C2×#1750C211yesmaximal, characteristic
5#13777C211yesmaximal, characteristic
6#10885C212yesmaximal, complemented
7#10884C212yesmaximal, complemented
8#10881C212yesmaximal, complemented
9#10880C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic
11#10731C211yesmaximal, characteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C22×#97C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19#2316—24no—
20#2316C418yescomplemented
21C2×#389—22no—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#380—22no—
25C2×#380C2211yescharacteristic
26C2×#380C2211yescharacteristic
27C2×C4×C4⋊C4C2212yes—
28#1824C418yescomplemented
29#1823C418yescomplemented
30#1750—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41#1437—22no—
42#1436—22no—
43C4⋊#95C2212yes—
44C24×C4—21no—
45C24×C4C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×#117—22no—
51C2×#117—42no—
52C2×#117—42no—
53C2×#117—42no—
54C2×#117—42no—
55C2×#117C2×C414yes—
56C2×#117C2×C414yes—
57C2×#117D414yes—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4C2×C414yes—
65C22×C4⋊C4C2×C414yes—
66C22×C4⋊C4C2×C414yes—
67C22×C4⋊C4D412yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4D412yes—
76C22×C42—21no—
77C22×C42—22no—
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8C2×C414yes—
85C2×C22⋊C8C2×C414yes—
86C2×#97—22no—
87C2×#97—42no—
88C2×C4×C8—22no—
89C2×C4×C8—42no—
90#389—216no—
91#389—24no—
92#389—24no—
93#389C2×C418yes—
94#389C2×C418yes—
95#380—22no—
96#380—22no—
97#380—216no—
98#380C2×C414yes—
99#380C2×C414yes—
100C4×C4⋊C4—42no—
101C4×C4⋊C4—42no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109#340—42no—
110#340—42no—
111#340—42no—
112#340—42no—
113#340C2×C418yes—
114#340C2×C418yes—
115#340C2×C418yes—
116#340C2×C418yes—
117#336—22no—
118#336—22no—
119C25C2×D411yescharacteristic
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—24no—
126C23×C4—24no—
127C23×C4—24no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—82no—
132C23×C4—82no—
133C23×C4C22×C412yes—
134C23×C4C22×C412yes—
135C23×C4C22×C412yes—
136C23×C4C2×D412yes—
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C23×C4#4112yes—
141C23×C4#4112yes—
142C23×C4C2411yescharacteristic
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
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×C8C22×C412yes—
160C22×C8C22×C412yes—
161C22×C8C22×C412yes—
162C22×C8C22×C412yes—
163#117—48no—
164#117—44no—
165#117—44no—
166#117—48no—
167#117—44no—
168#117—44no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
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—28no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4C42116yescomplemented
202C2×C4⋊C4C22⋊C418yes—
203C2×C4⋊C4C22×C414yes—
204C2×C4⋊C4C22×C414yes—
205C2×C4⋊C4C22×C414yes—
206C2×C4⋊C4C22×C414yes—
207C2×C4⋊C4C22×C414yes—
208C2×C4⋊C4C22×C414yes—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—82no—
216C2×C22⋊C4—82no—
217C2×C22⋊C4C22⋊C418yes—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—24no—
223C2×C42—44no—
224C2×C42—44no—
225C2×C42—82no—
226C2×C42—82no—
227C2×C42C22×C412yes—
228C2×C42C22×C412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C22⋊C8—28no—
232C22⋊C8—28no—
233#97—44no—
234C4×C8—44no—
235#95—28no—
236#95—28no—
237#95—24no—
238#95—24no—
239#95—44no—
240#95—82no—
241#95—82no—
242C24—22no—
243C24—22no—
244C24—22no—
245C24—22no—
246C24—22no—
247C24—22no—
248C24C2×C22⋊C412yes—
249C24C4×D412yes—
250C24C4×D412yes—
251C24C22×D411yescharacteristic
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—216no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
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—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—88no—
310C22×C4—88no—
311C22×C4—84no—
312C22×C4—84no—
313C22×C4C2×C4214yes—
314C22×C4C2×C22⋊C414yes—
315C22×C4C2×C22⋊C414yes—
316C22×C4C2×C22⋊C412yes—
317C22×C4#11714yes—
318C22×C4C4×D414yes—
319C22×C4C4×D414yes—
320C22×C4C4×D412yes—
321C22×C4C4×D412yes—
322C22×C4C23×C411yescharacteristic
323C22×C4C23×C411yescharacteristic
324C22×C4C23×C411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C2×#4111yescharacteristic
327C22×C4C2×#4111yescharacteristic
328C22×C4H(5,𝔽2)12yes—
329C22×C4H(5,𝔽2)12yes—
330C22×C42−1+412yes—
331C22×C42−1+412yes—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—216no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—48no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—88no—
370C4⋊C4—88no—
371C4⋊C4C2×C4218yes—
372C22⋊C4—48no—
373C22⋊C4—48no—
374C22⋊C4—84no—
375C22⋊C4—84no—
376C42—24no—
377C42—24no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—48no—
383C42—88no—
384C23—22no—
385C23—22no—
386C23—24no—
387C23—24no—
388C23—24no—
389C23—24no—
390C23—24no—
391C23—24no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23C4×C22⋊C414yes—
405C23C22×C22⋊C411yescharacteristic
406C23C2×C4×D411yescharacteristic
407C23C2×C4×D411yescharacteristic
408C23#53412yes—
409C23#53612yes—
410C23#57612yes—
411C23#57712yes—
412C23#57712yes—
413C23#57812yes—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—28no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—28no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—28no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—816no—
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×C4C4×C22⋊C414yes—
468C2×C4C22×C4211yescharacteristic
469C2×C4C2×#11711yescharacteristic
470C2×C4C2×C4×D411yescharacteristic
471C2×C4C2×C4×D411yescharacteristic
472C2×C4#51812yes—
473C2×C4#51912yes—
474C2×C4#52012yes—
475C2×C4#52012yes—
476C2×C4#52012yes—
477C2×C4#52012yes—
478C2×C4#53411yescharacteristic
479C2×C4#53411yescharacteristic
480C2×C4#53611yescharacteristic
481C2×C4#53611yescharacteristic
482C2×C4#53912yes—
483C2×C4#53912yes—
484C2×C4#53912yes—
485C2×C4#53912yes—
486C8—42no—
487C8—42no—
488C8—42no—
489C8—42no—
490C22—24no—
491C22—24no—
492C22—24no—
493C22—44no—
494C22—44no—
495C22—44no—
496C22—44no—
497C22—44no—
498C22—44no—
499C22C2×C4×C22⋊C411yescharacteristic
500C22#230811yescharacteristic
501C22#231011yescharacteristic
502C22#232212yes—
503C22#232212yes—
504C22#289712yes—
505C22#289812yes—
506C22#294612yes—
507C22#294712yes—
508C22#294811yescharacteristic
509C22#294911yescharacteristic
510C22#294911yescharacteristic
511C22#295011yescharacteristic
512C22#319812yes—
513C22#320112yes—
514C22#320312yes—
515C22#320412yes—
516C22#324112yes—
517C22#324212yes—
518C22#324312yes—
519C22#324412yes—
520C4—24no—
521C4—44no—
522C4—44no—
523C4—48no—
524C4—84no—
525C4—84no—
526C4—84no—
527C4#227412yes—
528C4#231312yes—
529C4#232211yescharacteristic
530C4#232211yescharacteristic
531C4#232211yescharacteristic
532C4#232211yescharacteristic
533C2—44no—
534C2#1377611yescharacteristic
535C2#2041611yescharacteristic
536C2#2041711yescharacteristic
537C2#2177212yes—
538C2#2184312yes—
539C2#2208712yes—
540C2#2208912yes—
541C2#2209112yes—
542C2#2209312yes—
subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C2×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
6. #10885C2semidirect
7. #10884C2semidirect
8. #10881C2semidirect
9. #10880C2semidirect
20. #2316C4semidirect
28. #1824C4semidirect
29. #1823C4semidirect
201. C2×C4⋊C4C42semidirect

Maximal subgroups

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

subgroupindex
C2×#23162
C2×#18242
C2×#18232
C2×#17502
#137772
#108852
#108842
#108812
#108802
#107312