Finite Group Databasebeta

← #128043

Subgroups — #128043

Order 512.

955 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
1#20726C211yesmaximal, characteristic
2#20722C211yesmaximal, characteristic
3#13839C211yesmaximal, characteristic
4#7304C214yesmaximal
5C22×C4×D4C2211yescharacteristic
6C22×#117C411yescharacteristic, complemented
7#2920—21no—
8#2280—21no—
9C2×C4×C22⋊C4—21no—
10C2×C4×C22⋊C4C411yescharacteristic, complemented
11#1821—21no—
12#1819—21no—
13#1800—21no—
14#1800C412yescomplemented
15#1799—21no—
16#1799C412yescomplemented
17#1783C2212yes—
18#1781C2212yes—
19C4×#100—21no—
20C4×#100C412yescomplemented
21C4×#99—21no—
22C4×#99C412yescomplemented
23#1441—21no—
24#1440C412yescomplemented
25#1439—21no—
26#1438C2212yes—
27C4×#95—21no—
28C23×D4C2311yescharacteristic
29C24×C4C2×C411yescharacteristic
30C24×C4D411yescharacteristic
31C2×C4×D4—41no—
32C2×C4×D4—41no—
33C2×C4×D4—41no—
34C2×C4×D4—41no—
35C2×C4×D4C2×C411yescharacteristic
36C2×C4×D4C2×C411yescharacteristic
37C2×C4×D4C2×C411yescharacteristic
38C2×C4×D4C2×C411yescharacteristic
39C2×C4×D4D411yescharacteristic
40C2×C4×D4D411yescharacteristic
41C2×C4×D4Q811yescharacteristic
42C2×C4×D4Q811yescharacteristic
43C2×#117—44no—
44C2×#117—41no—
45C2×#117—41no—
46C2×#117—41no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4D411yescharacteristic
49C22×C22⋊C4—22no—
50C22×C22⋊C4C2×C412yes—
51C22×C22⋊C4C2×C412yes—
52C22×C22⋊C4C2×C411yescharacteristic
53C22×C22⋊C4D411yescharacteristic
54C22×C42—21no—
55C22×C42—21no—
56C22×C42D411yescharacteristic
57#432—41no—
58#432—41no—
59#431—41no—
60#431—41no—
61#423—22no—
62C2×#100—22no—
63C2×#100C2×C414yes—
64C2×#99—22no—
65C2×#99C2×C414yes—
66#407—22no—
67C4×M16—22no—
68#381—41no—
69#380—22no—
70#380—41no—
71#379—22no—
72C4×C22⋊C4—22no—
73C4×C22⋊C4—44no—
74C4×C22⋊C4—41no—
75C4×C22⋊C4—41no—
76C4×C22⋊C4—41no—
77C2×#95—22no—
78C2×#95—21no—
79C2×#95—22no—
80C2×#95—22no—
81#343—22no—
82#343—22no—
83#342—22no—
84#342—22no—
85C25C22×C411yescharacteristic
86C25C2×D411yescharacteristic
87C22×D4—42no—
88C22×D4—41no—
89C22×D4—41no—
90C22×D4—42no—
91C22×D4C22×C411yescharacteristic
92C22×D4C22×C411yescharacteristic
93C22×D4C2×D411yescharacteristic
94C22×D4C2×Q811yescharacteristic
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—42no—
106C23×C4—42no—
107C23×C4—42no—
108C23×C4—42no—
109C23×C4C4211yescharacteristic
110C23×C4C4211yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4C22⋊C411yescharacteristic
113C23×C4C22⋊C411yescharacteristic
114C23×C4C22⋊C411yescharacteristic
115C23×C4C22⋊C411yescharacteristic
116C23×C4C22⋊C411yescharacteristic
117C23×C4C4⋊C411yescharacteristic
118C23×C4C4⋊C411yescharacteristic
119C23×C4C4⋊C411yescharacteristic
120C23×C4C4⋊C411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C2×M16—24no—
123C4×D4—41no—
124C4×D4—41no—
125C4×D4—41no—
126C4×D4—41no—
127C4×D4—41no—
128C4×D4—41no—
129C4×D4—41no—
130C4×D4—41no—
131C4×D4—41no—
132C4×D4—41no—
133C4×D4—41no—
134C4×D4—41no—
135C4×D4—41no—
136C4×D4—41no—
137C4×D4—41no—
138C4×D4—41no—
139#117—48no—
140#117—44no—
141#117—44no—
142#117—82no—
143#117—82no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4C22⋊C411yescharacteristic
150C2×C4⋊C4C22⋊C411yescharacteristic
151C2×C4⋊C4C4⋊C411yescharacteristic
152C2×C4⋊C4C4⋊C411yescharacteristic
153C2×C22⋊C4—24no—
154C2×C22⋊C4—41no—
155C2×C22⋊C4—41no—
156C2×C22⋊C4—41no—
157C2×C22⋊C4—41no—
158C2×C22⋊C4—44no—
159C2×C22⋊C4—44no—
160C2×C22⋊C4—44no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—42no—
163C2×C22⋊C4—44no—
164C2×C22⋊C4—44no—
165C2×C22⋊C4—42no—
166C2×C22⋊C4—42no—
167C2×C22⋊C4C4211yescharacteristic
168C2×C22⋊C4C4211yescharacteristic
169C2×C22⋊C4C22⋊C411yescharacteristic
170C2×C22⋊C4C22⋊C411yescharacteristic
171C2×C22⋊C4C4⋊C411yescharacteristic
172C2×C22⋊C4C4⋊C411yescharacteristic
173C2×C22⋊C4C4⋊C411yescharacteristic
174C2×C22⋊C4C4⋊C411yescharacteristic
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—24no—
180C2×C42—42no—
181C2×C42—41no—
182C2×C42—41no—
183C2×C42—42no—
184C2×C42—42no—
185C2×C42—42no—
186C2×C42—41no—
187C2×C42—41no—
188C2×C42C22⋊C411yescharacteristic
189C2×C42C22⋊C411yescharacteristic
190C2×C42C4⋊C411yescharacteristic
191C2×C42C4⋊C411yescharacteristic
192C4⋊C8—42no—
193C4⋊C8—42no—
194#100—44no—
195#100—44no—
196#99—44no—
197#99—44no—
198C22⋊C8—42no—
199C22⋊C8—42no—
200#97—42no—
201C4×C8—42no—
202#95—44no—
203#95—44no—
204#95—42no—
205#95—44no—
206#95—42no—
207#95—42no—
208C24—21no—
209C24—21no—
210C24—22no—
211C24—21no—
212C24—21no—
213C24—41no—
214C24—41no—
215C24—41no—
216C24—41no—
217C24—42no—
218C24—41no—
219C24—41no—
220C24—42no—
221C24—41no—
222C24C2×C4211yescharacteristic
223C24C2×C22⋊C411yescharacteristic
224C24C2×C22⋊C411yescharacteristic
225C24C2×C4⋊C411yescharacteristic
226C24C2×C4⋊C411yescharacteristic
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—48no—
240C2×D4—48no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—41no—
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—44no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4—81no—
308C22×C4—81no—
309C22×C4#9511yescharacteristic
310C22×C4#9511yescharacteristic
311C22×C4#9511yescharacteristic
312C22×C4#9511yescharacteristic
313C22×C4#9511yescharacteristic
314C22×C4#9511yescharacteristic
315C22×C4#9511yescharacteristic
316C22×C4#9511yescharacteristic
317C22×C4#9911yescharacteristic
318C22×C4#9911yescharacteristic
319C22×C4#9911yescharacteristic
320C22×C4#9911yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C4⋊C411yescharacteristic
323M16—44no—
324M16—44no—
325C2×C8—44no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—48no—
331C4⋊C4—84no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—416no—
341C22⋊C4—44no—
342C22⋊C4—44no—
343C22⋊C4—44no—
344C22⋊C4—44no—
345C22⋊C4—44no—
346C22⋊C4—44no—
347C22⋊C4—84no—
348C22⋊C4—82no—
349C22⋊C4—82no—
350C42—21no—
351C42—21no—
352C42—21no—
353C42—21no—
354C42—21no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—44no—
359C42—44no—
360C42—82no—
361C42—82no—
362C42—84no—
363C23—21no—
364C23—21no—
365C23—22no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—42no—
380C23—42no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—44no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—44no—
390C23—42no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—42no—
400C23—42no—
401C23—81no—
402C23—81no—
403C23—82no—
404C23—82no—
405C23—82no—
406C23—82no—
407C23—82no—
408C23—82no—
409C23C2×#9511yescharacteristic
410C23C2×#9911yescharacteristic
411C23C2×#9911yescharacteristic
412C23#41012yes—
413D4—44no—
414D4—44no—
415D4—44no—
416D4—44no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
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—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
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—84no—
481C2×C4—84no—
482C2×C4—84no—
483C2×C4#34211yescharacteristic
484C2×C4#34211yescharacteristic
485C2×C4#34211yescharacteristic
486C2×C4#34211yescharacteristic
487C2×C4AΣL(1,𝔽16)12yes—
488C2×C4#35212yes—
489C2×C4#41012yes—
490C8—84no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—22no—
502C22—42no—
503C22—42no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—81no—
516C22—82no—
517C22—84no—
518C22—82no—
519C22—81no—
520C22—81no—
521C22—81no—
522C22—81no—
523C22—81no—
524C22—82no—
525C22—84no—
526C22—82no—
527C22C2×#34211yescharacteristic
528C22#174011yescharacteristic
529C22#174011yescharacteristic
530C22C2×AΣL(1,𝔽16)11yescharacteristic
531C22C2×#35211yescharacteristic
532C22#212012yes—
533C4—21no—
534C4—21no—
535C4—22no—
536C4—41no—
537C4—41no—
538C4—41no—
539C4—41no—
540C4—41no—
541C4—41no—
542C4—41no—
543C4—84no—
544C4—84no—
545C4—164no—
546C4#139012yes—
547C4#139012yes—
548C2—21no—
549C2—22no—
550C2—21no—
551C2—21no—
552C2—21no—
553C2—42no—
554C2—81no—
555C2—81no—
556C2—81no—
557C2#810011yescharacteristic
558C2C2×#139011yescharacteristic
559C2#1257311yescharacteristic
subgroupG/H
centerC2×C4#342
commutatorC24C2×C42
Frattini, prime radical, cosocleC23×D4C23
socleC22C2×#342
LCSG ▷ C24 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C22×#117C4semidirect
10. C2×C4×C22⋊C4C4semidirect
14. #1800C4semidirect
16. #1799C4semidirect
20. C4×#100C4semidirect
22. C4×#99C4semidirect
24. #1440C4semidirect

Maximal subgroups

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

subgroupindex
#207262
#207222
#138392
#73042