Finite Group Databasebeta

← #360402

Subgroups — #360402

Order 512.

1110 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1805C211yesmaximal, characteristic
2C2×#1770C211yesmaximal, characteristic
3C2×C4×#103C211yesmaximal, characteristic
4#13955C211yesmaximal, characteristic
5#13942C211yesmaximal, characteristic
6#11316C212yesmaximal
7#11190C212yesmaximal
8#10928C212yesmaximal
9#10850C211yesmaximal, characteristic
10#10842C211yesmaximal, characteristic
11#10821C212yesmaximal
12C22×C4×Q8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#97C2212yes—
18C22×C4×C8C2211yescharacteristic
19C2×#391C2212yes—
20C2×#382C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#1805—24no—
25#1770—24no—
26C4×#103C418yescomplemented
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#336C2212yes—
35#1442—22no—
36#1441—22no—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40#1434—21no—
41#1434—21no—
42#1434—21no—
43#1434—21no—
44C23×Q8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8C2×C414yes—
54C22×C4⋊C4—21no—
55C22×C4⋊C4D412yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C42—21no—
62C22×C42—21no—
63C22×C42C2312yes—
64C22×C42C2311yescharacteristic
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8D414yes—
68C2×#103—22no—
69C2×#103—22no—
70C2×#103—22no—
71C2×#103—22no—
72C2×#103—24no—
73C2×#103—24no—
74C2×#103—42no—
75C2×#103C2×C414yes—
76C2×#103C2×C414yes—
77C2×#97—24no—
78C2×#97—24no—
79C2×#97—42no—
80C2×C4×C8—22no—
81C2×C4×C8—22no—
82C2×C4×C8C2×C414yes—
83#391—24no—
84#391—42no—
85#382—22no—
86#382—22no—
87#382—22no—
88#382—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4C2×C414yes—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95D411yescharacteristic
107C2×#95D411yescharacteristic
108#340—24no—
109#340—24no—
110#340—42no—
111#340—42no—
112#340—42no—
113#340C2×C414yes—
114#340C2×C414yes—
115#336—24no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8C22×C412yes—
128C22×Q8#4112yes—
129C22×Q8#4112yes—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4C2×D412yes—
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2411yescharacteristic
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8C22×C412yes—
158C22×C8C22×C412yes—
159C22×C8C2×D414yes—
160C22×C8#4112yes—
161C22×C8#4112yes—
162C4×Q8—28no—
163C4×Q8—28no—
164C4×Q8—44no—
165C4×Q8—44no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
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—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4C22×C412yes—
200C2×C4⋊C4C22×C412yes—
201C2×C4⋊C4C22×C412yes—
202C2×C42—24no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—24no—
210C2×C42—42no—
211C2×C42—42no—
212C2×C42—82no—
213C2×C42C22×C412yes—
214C2×C42C2×D412yes—
215C2×C42#4112yes—
216C2×C42#4112yes—
217C2×C42#4112yes—
218C2×C42#4112yes—
219C4⋊C8—28no—
220#103—28no—
221#103—28no—
222#103—48no—
223#97—48no—
224C4×C8—28no—
225#95—24no—
226#95—22no—
227#95—22no—
228#95—44no—
229#95—44no—
230#95—44no—
231#95—44no—
232#95—44no—
233#95—44no—
234#95—42no—
235#95—42no—
236#95—42no—
237#95—42no—
238#95—82no—
239#95—82no—
240C24C22×D411yescharacteristic
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8#11714yes—
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—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
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×C4C4×D412yes—
324C22×C4C4×D412yes—
325C22×C4#12112yes—
326C22×C4#12112yes—
327C22×C4#12412yes—
328C22×C4#12412yes—
329C22×C4C23×C411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4H(5,𝔽2)11yescharacteristic
340C22×C4H(5,𝔽2)11yescharacteristic
341C22×C42−1+411yescharacteristic
342C22×C42−1+411yescharacteristic
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8#11714yes—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—48no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—48no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C42—42no—
387C42—42no—
388C42—42no—
389C42—42no—
390C42—48no—
391C42—84no—
392C42#11714yes—
393C42C4×D414yes—
394C42#12314yes—
395C42#12614yes—
396C23C2×C4×D411yescharacteristic
397C23C2×#12111yescharacteristic
398C23C2×#12411yescharacteristic
399C23#54611yescharacteristic
400C23#54711yescharacteristic
401C23#54811yescharacteristic
402C23#55011yescharacteristic
403C23#57512yes—
404C23#57512yes—
405C23#57712yes—
406C23#57812yes—
407Q8—24no—
408Q8—24no—
409Q8—42no—
410Q8—42no—
411Q8—42no—
412Q8—42no—
413Q8—42no—
414Q8—42no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
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—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4#38814yes—
474C2×C4#38814yes—
475C2×C4C2×#11711yescharacteristic
476C2×C4C2×#11711yescharacteristic
477C2×C4C2×#11711yescharacteristic
478C2×C4C2×C4×D411yescharacteristic
479C2×C4#52011yescharacteristic
480C2×C4#52011yescharacteristic
481C2×C4C2×#12311yescharacteristic
482C2×C4C2×#12611yescharacteristic
483C2×C4#52911yescharacteristic
484C2×C4#52911yescharacteristic
485C2×C4#52911yescharacteristic
486C2×C4#52911yescharacteristic
487C2×C4#54011yescharacteristic
488C2×C4#54111yescharacteristic
489C2×C4#54211yescharacteristic
490C2×C4#54211yescharacteristic
491C2×C4#54611yescharacteristic
492C2×C4#54611yescharacteristic
493C2×C4#54811yescharacteristic
494C2×C4#54811yescharacteristic
495C2×C4#55211yescharacteristic
496C2×C4#55211yescharacteristic
497C2×C4#55211yescharacteristic
498C2×C4#55311yescharacteristic
499C2×C4#55311yescharacteristic
500C2×C4#55311yescharacteristic
501C2×C4#55511yescharacteristic
502C2×C4#55611yescharacteristic
503C8—24no—
504C8—42no—
505C8—44no—
506C22C2×#38811yescharacteristic
507C22#235211yescharacteristic
508C22#235411yescharacteristic
509C22#242611yescharacteristic
510C22#242711yescharacteristic
511C22#248911yescharacteristic
512C22#249011yescharacteristic
513C22#294311yescharacteristic
514C22#294311yescharacteristic
515C22#294911yescharacteristic
516C22#295011yescharacteristic
517C22#305412yes—
518C22#306312yes—
519C22#313512yes—
520C22#313712yes—
521C22#332612yes—
522C22#332712yes—
523C22#335112yes—
524C22#335212yes—
525C22#337112yes—
526C22#337712yes—
527C22#340112yes—
528C22#340212yes—
529C4—24no—
530C4—24no—
531C4—44no—
532C4—82no—
533C4—82no—
534C4—82no—
535C4—82no—
536C4—82no—
537C4—162no—
538C4—162no—
539C4#228311yescharacteristic
540C4#231411yescharacteristic
541C4#235211yescharacteristic
542C4#235211yescharacteristic
543C4#242011yescharacteristic
544C4#242111yescharacteristic
545C4#249411yescharacteristic
546C4#249511yescharacteristic
547C2#1387511yescharacteristic
548C2#2108611yescharacteristic
549C2#2108811yescharacteristic
550C2#2276111yescharacteristic
551C2#2276511yescharacteristic
552C2#2281211yescharacteristic
553C2#2281511yescharacteristic
554C2#2529912yes—
555C2#2534512yes—
556C2#2773412yes—
557C2#2775412yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
26. C4×#103C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18052
C2×#17702
C2×C4×#1032
#139552
#139422
#113162
#111902
#109282
#108502
#108422
#108212