Finite Group Databasebeta

← #127619

Subgroups — #127619

Order 512.

1044 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1800C211yesmaximal, characteristic
2C2×#1459C211yesmaximal, characteristic
3#13872C211yesmaximal, characteristic
4#7251C214yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6C23×C4⋊C4C411yescharacteristic
7C2×#423—21no—
8C22×#100—21no—
9C22×C22⋊C8—21no—
10C2×#386—21no—
11C2×#380—21no—
12C2×#380C411yescharacteristic
13#1800—22no—
14#1800C414yescomplemented
15C2×#343C2212yes—
16#1459—22no—
17#1459C414yescomplemented
18C2×#324C2212yes—
19C2×#323—21no—
20#1441C2212yes—
21#1439C412yes—
22C4⋊#95—21no—
23C22⋊#95—21no—
24C23×D4D411yescharacteristic
25C24×C4—21no—
26C24×C4C2×C411yescharacteristic
27C24×C4D411yescharacteristic
28C22×M16—22no—
29C23×C8—41no—
30C2×C4×D4—41no—
31C2×C4×D4—41no—
32C2×C4×D4—42no—
33C2×C4×D4C2×C412yes—
34C2×C4×D4C2×C412yes—
35C2×C4×D4D412yes—
36C2×C4×D4Q812yes—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—44no—
40C22×C4⋊C4—41no—
41C22×C4⋊C4—41no—
42C22×C4⋊C4C2×C412yes—
43C22×C4⋊C4C2311yescharacteristic
44C22×C22⋊C4—21no—
45C22×C22⋊C4—22no—
46C22×C22⋊C4C2×C411yescharacteristic
47C22×C22⋊C4D411yescharacteristic
48C22×C42D411yescharacteristic
49#423—24no—
50#423—42no—
51C2×C4⋊C8—41no—
52C2×C4⋊C8—41no—
53C2×#100—24no—
54C2×#100—42no—
55C2×#100—42no—
56C2×C22⋊C8—24no—
57C2×C22⋊C8—22no—
58C2×C22⋊C8—42no—
59C2×C22⋊C8—41no—
60#386—42no—
61#386—42no—
62#380—44no—
63#380—42no—
64C2×#95—22no—
65C2×#95—22no—
66C2×#95—22no—
67C2×#95C2×C412yes—
68#343—22no—
69#343C2×C414yescomplemented
70#324—22no—
71#324C2×C414yescomplemented
72#323—24no—
73C25C22⋊C411yescharacteristic
74C25C22⋊C411yescharacteristic
75C22×D4—41no—
76C22×D4—41no—
77C22×D4—41no—
78C22×D4—41no—
79C22×D4C22⋊C412yes—
80C22×D4C4⋊C412yes—
81C22×D4D812yes—
82C22×D4SD1612yes—
83C22×D4SD1612yes—
84C22×D4Q1612yes—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—22no—
89C23×C4—21no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—42no—
94C23×C4—42no—
95C23×C4—42no—
96C23×C4—42no—
97C23×C4—44no—
98C23×C4—42no—
99C23×C4—42no—
100C23×C4—42no—
101C23×C4—82no—
102C23×C4C4212yes—
103C23×C4C22⋊C411yescharacteristic
104C23×C4C22⋊C412yes—
105C23×C4C22⋊C411yescharacteristic
106C23×C4C4⋊C412yes—
107C23×C4C4⋊C412yes—
108C23×C4C22×C411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C2×M16—24no—
112C2×M16—42no—
113C2×M16—42no—
114C2×M16—42no—
115C2×M16—42no—
116C2×M16—42no—
117C22×C8—42no—
118C22×C8—44no—
119C22×C8—42no—
120C22×C8—82no—
121C22×C8—82no—
122C4×D4—42no—
123C4×D4—42no—
124C4×D4—42no—
125C4×D4—42no—
126C4×D4—42no—
127C4×D4—42no—
128C4×D4—42no—
129C4×D4—42no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—48no—
136C2×C4⋊C4—44no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—44no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4C22×C411yescharacteristic
145C2×C4⋊C4C22×C411yescharacteristic
146C2×C4⋊C4C2×D411yescharacteristic
147C2×C4⋊C4C2×Q811yescharacteristic
148C2×C22⋊C4—41no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—41no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4—44no—
153C2×C22⋊C4—42no—
154C2×C22⋊C4—42no—
155C2×C22⋊C4—44no—
156C2×C22⋊C4—42no—
157C2×C22⋊C4—82no—
158C2×C22⋊C4C4212yes—
159C2×C22⋊C4C22⋊C412yes—
160C2×C22⋊C4C4⋊C412yes—
161C2×C22⋊C4C4⋊C412yes—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—22no—
165C2×C42C22⋊C412yes—
166C2×C42C4⋊C412yes—
167C4⋊C8—44no—
168C4⋊C8—44no—
169#100—48no—
170C22⋊C8—24no—
171C22⋊C8—42no—
172C22⋊C8—44no—
173#95—42no—
174#95—44no—
175#95—42no—
176#95—42no—
177#95—82no—
178C24—21no—
179C24—21no—
180C24—21no—
181C24—21no—
182C24—21no—
183C24—21no—
184C24—21no—
185C24—21no—
186C24—41no—
187C24—41no—
188C24—41no—
189C24—41no—
190C24—41no—
191C24—41no—
192C24—41no—
193C24—41no—
194C24#9512yes—
195C24#9512yes—
196C24#10212yes—
197C24#10212yes—
198C24#10312yes—
199C24#10312yes—
200C24C2×C22⋊C411yescharacteristic
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4#10214yes—
214C2×D4#10314yes—
215C2×D4#10614yes—
216C2×D4#10714yes—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—48no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—48no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—84no—
270C22×C4—81no—
271C22×C4—81no—
272C22×C4—88no—
273C22×C4—88no—
274C22×C4—84no—
275C22×C4—84no—
276C22×C4#9512yes—
277C22×C4#9512yes—
278C22×C4#9912yes—
279C22×C4#9911yescharacteristic
280C22×C4#9911yescharacteristic
281C22×C4C2×C4211yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C4⋊C411yescharacteristic
285C22×C4C2×C4⋊C411yescharacteristic
286C22×C4C2×C4⋊C411yescharacteristic
287C22×C4C2×D811yescharacteristic
288C22×C4C2×SD1611yescharacteristic
289C22×C4C2×SD1611yescharacteristic
290C22×C4C2×Q1611yescharacteristic
291M16—44no—
292M16—44no—
293M16—82no—
294M16—82no—
295C2×C8—44no—
296C2×C8—44no—
297C2×C8—82no—
298C2×C8—82no—
299C2×C8—82no—
300C2×C8—82no—
301C2×C8—88no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—44no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—84no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C22⋊C4—88no—
316C22⋊C4—84no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—42no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—81no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23#34014yes—
370C23#34014yes—
371C23C2×#9511yescharacteristic
372C23C2×#9911yescharacteristic
373C23C2×#9911yescharacteristic
374C23C2×#10211yescharacteristic
375C23C2×#10311yescharacteristic
376C23#42111yescharacteristic
377C23#42111yescharacteristic
378D4—44no—
379D4—44no—
380D4—44no—
381D4—44no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—816no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4#34212yes—
441C2×C4#34212yes—
442C2×C4AΣL(1,𝔽16)12yes—
443C2×C4#35212yes—
444C2×C4C2×#10211yescharacteristic
445C2×C4C2×#10311yescharacteristic
446C2×C4C2×#10611yescharacteristic
447C2×C4C2×#10711yescharacteristic
448C8—84no—
449C8—84no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—24no—
463C22—24no—
464C22—44no—
465C22—44no—
466C22—44no—
467C22—44no—
468C22—44no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—82no—
482C22—82no—
483C22—82no—
484C22#151911yescharacteristic
485C22#151911yescharacteristic
486C22#152011yescharacteristic
487C22#152011yescharacteristic
488C22#173311yescharacteristic
489C22C2×#34011yescharacteristic
490C22C2×#34211yescharacteristic
491C22C2×AΣL(1,𝔽16)11yescharacteristic
492C22C2×#35211yescharacteristic
493C22#213411yescharacteristic
494C22#213511yescharacteristic
495C4—22no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—41no—
501C4—41no—
502C4—41no—
503C4—41no—
504C4—42no—
505C4—84no—
506C4—162no—
507C4#139014yes—
508C2—21no—
509C2—21no—
510C2—21no—
511C2—21no—
512C2—44no—
513C2—44no—
514C2#802311yescharacteristic
515C2#1188011yescharacteristic
516C2#1188111yescharacteristic
517C2#1190311yescharacteristic
518C2#1190411yescharacteristic
519C2C2×#139011yescharacteristic
520C2#1258711yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC22×C4⋊C4C23
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7251C2semidirect
14. #1800C4semidirect
17. #1459C4semidirect
69. #343C2×C4semidirect
71. #324C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18002
C2×#14592
#138722
#72512