Finite Group Databasebeta

← #355136

Subgroups — #355136

Order 512.

1570 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2317C211yesmaximal, characteristic
2C2×#1825C212yesmaximal
3C2×#1750C211yesmaximal, characteristic
4#13777C211yesmaximal, characteristic
5#10885C212yesmaximal, complemented
6#10884C212yesmaximal, complemented
7#10881C212yesmaximal, complemented
8#10880C212yesmaximal, complemented
9#10730C212yesmaximal
10C22×#117C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C22×C22⋊C8C2212yes—
14C22×#97C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#2317—24no—
17#2317C418yescomplemented
18C2×#389C2212yes—
19C2×#389C2212yes—
20C2×#385—21no—
21C2×#385—21no—
22C2×#384—21no—
23C2×#382—21no—
24C2×#380C2212yes—
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1825C4116yescomplemented
28#1750—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38#1437—21no—
39#1437—21no—
40#1436—21no—
41#1436—21no—
42C4⋊#95C2212yes—
43C24×C4—21no—
44C24×C4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×#117—22no—
48C2×#117—24no—
49C2×#117—41no—
50C2×#117—41no—
51C2×#117—41no—
52C2×#117—41no—
53C2×#117C2×C414yes—
54C2×#117C2×C418yes—
55C2×#117C2×C414yes—
56C2×#117D414yes—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—41no—
62C22×C4⋊C4—41no—
63C22×C4⋊C4—41no—
64C22×C4⋊C4—41no—
65C22×C4⋊C4C2×C414yes—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4D411yescharacteristic
75C22×C22⋊C4D411yescharacteristic
76C22×C42—21no—
77C22×C42—21no—
78C22×C42—21no—
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C2×C22⋊C8—24no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8C2×C418yes—
84C2×#97—22no—
85C2×#97—41no—
86C2×#97—41no—
87C2×C4×C8—22no—
88C2×C4×C8—41no—
89C2×C4×C8—41no—
90#389—24no—
91#389—24no—
92#389C2×C418yes—
93#389C2×C418yes—
94#385—28no—
95#385—28no—
96#384—28no—
97#382—28no—
98#380—24no—
99#380C2×C418yes—
100C4×C4⋊C4—41no—
101C4×C4⋊C4—41no—
102C4×C4⋊C4—41no—
103C4×C4⋊C4—41no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—22no—
111#340—42no—
112#340—42no—
113#340—42no—
114#340—42no—
115#340C2×C418yescomplemented
116#340C2×C418yescomplemented
117#340C2×C418yescomplemented
118#340C2×C418yescomplemented
119#336—24no—
120C25C2×D411yescharacteristic
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—24no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—82no—
135C23×C4—82no—
136C23×C4C22×C412yes—
137C23×C4C22×C414yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4#4114yes—
145C23×C4C2411yescharacteristic
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8C22×C414yes—
159C22×C8C22×C414yes—
160#117—28no—
161#117—216no—
162#117—44no—
163#117—44no—
164#117—44no—
165#117—44no—
166#117C42116yescomplemented
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—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4C22⋊C414yes—
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×C4⋊C4C22×C414yes—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—82no—
218C2×C22⋊C4—82no—
219C2×C22⋊C4C22⋊C414yes—
220C2×C22⋊C4C22⋊C414yes—
221C2×C42—24no—
222C2×C42—28no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42—82no—
234C2×C42—82no—
235C2×C42C22×C412yes—
236C2×C42C22×C412yes—
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C22⋊C8—216no—
240#97—44no—
241C4×C8—44no—
242#95—28no—
243#95—28no—
244#95—28no—
245#95—42no—
246#95—42no—
247#95—82no—
248#95—82no—
249C24—21no—
250C24—21no—
251C24—22no—
252C24—22no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24C2×C22⋊C412yes—
257C24C4×D414yes—
258C24C22×D411yescharacteristic
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—48no—
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—88no—
323C22×C4—88no—
324C22×C4—84no—
325C22×C4—84no—
326C22×C4C2×C4214yes—
327C22×C4C2×C22⋊C412yes—
328C22×C4C2×C22⋊C412yes—
329C22×C4C2×C22⋊C412yes—
330C22×C4C2×C22⋊C412yes—
331C22×C4C2×C22⋊C412yes—
332C22×C4#11714yes—
333C22×C4C4×D418yes—
334C22×C4C4×D414yes—
335C22×C4C23×C411yescharacteristic
336C22×C4C23×C412yes—
337C22×C4C22×D411yescharacteristic
338C22×C4C2×#4112yes—
339C22×C4H(5,𝔽2)11yescharacteristic
340C22×C4H(5,𝔽2)11yescharacteristic
341C22×C4H(5,𝔽2)11yescharacteristic
342C22×C4H(5,𝔽2)11yescharacteristic
343C22×C42−1+412yes—
344C22×C42−1+412yes—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—48no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—88no—
378C4⋊C4—88no—
379C4⋊C4C2×C4218yes—
380C22⋊C4—216no—
381C22⋊C4—44no—
382C22⋊C4—44no—
383C22⋊C4—88no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42—216no—
393C42—84no—
394C42—84no—
395C23—22no—
396C23—22no—
397C23—24no—
398C23—24no—
399C23—24no—
400C23—24no—
401C23—24no—
402C23—24no—
403C23—24no—
404C23—24no—
405C23—42no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—42no—
410C23—41no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23C4×C22⋊C414yes—
416C23C22×C22⋊C411yescharacteristic
417C23C2×C4×D412yes—
418C23#53411yescharacteristic
419C23#53411yescharacteristic
420C23#53612yes—
421C23#57612yes—
422C23#57712yes—
423C23#57712yes—
424C23#57812yes—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—28no—
432C2×C4—28no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—28no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—41no—
458C2×C4—41no—
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—44no—
470C2×C4—44no—
471C2×C4—816no—
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×C4C4×C22⋊C414yes—
485C2×C4C22×C4211yescharacteristic
486C2×C4C2×#11711yescharacteristic
487C2×C4C2×C4×D412yes—
488C2×C4#51811yescharacteristic
489C2×C4#51811yescharacteristic
490C2×C4#51911yescharacteristic
491C2×C4#51911yescharacteristic
492C2×C4#52012yes—
493C2×C4#52012yes—
494C2×C4#52012yes—
495C2×C4#52012yes—
496C2×C4#53411yescharacteristic
497C2×C4#53411yescharacteristic
498C2×C4#53612yes—
499C2×C4#53912yes—
500C2×C4#53912yes—
501C2×C4#53912yes—
502C2×C4#53912yes—
503C8—44no—
504C8—44no—
505C22—28no—
506C22—24no—
507C22—42no—
508C22—42no—
509C22—44no—
510C22—44no—
511C22—44no—
512C22—44no—
513C22—44no—
514C22C2×C4×C22⋊C411yescharacteristic
515C22#230811yescharacteristic
516C22#231011yescharacteristic
517C22#232212yes—
518C22#232212yes—
519C22#289712yes—
520C22#289812yes—
521C22#294612yes—
522C22#294712yes—
523C22#295112yes—
524C22#295212yes—
525C22#319812yes—
526C22#319911yescharacteristic
527C22#320011yescharacteristic
528C22#320011yescharacteristic
529C22#320112yes—
530C22#320211yescharacteristic
531C22#324112yes—
532C22#324212yes—
533C22#324312yes—
534C22#324412yes—
535C4—24no—
536C4—42no—
537C4—42no—
538C4—42no—
539C4—42no—
540C4—48no—
541C4—84no—
542C4—84no—
543C4—84no—
544C4#227411yescharacteristic
545C4#227411yescharacteristic
546C4#231311yescharacteristic
547C4#231311yescharacteristic
548C4#232212yes—
549C4#232212yes—
550C2—44no—
551C2#1377611yescharacteristic
552C2#2041512yes—
553C2#2177011yescharacteristic
554C2#2177111yescharacteristic
555C2#2184312yes—
556C2#2208812yes—
557C2#2209012yes—
558C2#2209212yes—
559C2#2209412yes—
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
5. #10885C2semidirect
6. #10884C2semidirect
7. #10881C2semidirect
8. #10880C2semidirect
17. #2317C4semidirect
27. #1825C4semidirect
115. #340C2×C4semidirect
116. #340C2×C4semidirect
117. #340C2×C4semidirect
118. #340C2×C4semidirect
166. #117C42semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
C2×#18252
C2×#17502
#137772
#108852
#108842
#108812
#108802
#107302