Finite Group Databasebeta

← #128111

Subgroups — #128111

Order 512.

819 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1870C211yesmaximal, characteristic
2#20726C211yesmaximal, characteristic
3#17362C211yesmaximal, characteristic
4#7321C214yesmaximal, complemented
5C23×M16C411yescharacteristic
6C22×C4×D4C2211yescharacteristic
7#2920—21no—
8C2×#407—21no—
9C22×C22⋊C8—21no—
10C22×C22⋊C8C411yescharacteristic
11#1870—21no—
12#1870—21no—
13#1870C412yes—
14#1870C2212yes—
15#1821—21no—
16#1800—21no—
17#1800C2212yes—
18#1783C412yes—
19C4×#100—21no—
20C4×#100C412yes—
21C2×#336—21no—
22#1465—21no—
23#1465C2212yes—
24#1463—21no—
25#1462C412yes—
26#1459—21no—
27#1459C412yes—
28C23×D4C2×C411yescharacteristic
29C24×C4C2×C411yescharacteristic
30C24×C4D411yescharacteristic
31C22×M16—21no—
32C22×M16—44no—
33C22×M16—41no—
34C22×M16—41no—
35C22×M16—41no—
36C22×M16C2×C412yes—
37C23×C8—21no—
38C23×C8—21no—
39C2×C4×D4—41no—
40C2×C4×D4—41no—
41C2×C4×D4—41no—
42C2×C4×D4—41no—
43C2×C4×D4C2×C411yescharacteristic
44C2×C4×D4C2×C411yescharacteristic
45C2×C4×D4C2×C411yescharacteristic
46C2×C4×D4D411yescharacteristic
47C2×C4×D4D411yescharacteristic
48C2×C4×D4D411yescharacteristic
49C2×C4×D4Q811yescharacteristic
50C2×C4×D4C2311yescharacteristic
51C22×C4⋊C4D411yescharacteristic
52C22×C22⋊C4C2×C411yescharacteristic
53C22×C22⋊C4Q811yescharacteristic
54C22×C42D411yescharacteristic
55#432—41no—
56#432—41no—
57#431—41no—
58#431—41no—
59#423—22no—
60C2×#100—22no—
61C2×#100C2×C414yes—
62#407—22no—
63#407—22no—
64#407—22no—
65#407—41no—
66#407—41no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—44no—
72C2×C22⋊C8—41no—
73C2×C22⋊C8—41no—
74C2×C22⋊C8—41no—
75C2×C22⋊C8C2×C412yes—
76C4×M16—22no—
77#343—22no—
78#343—22no—
79#336—22no—
80#336—22no—
81#324—22no—
82#324—22no—
83#323—22no—
84#323C2×C414yes—
85C25C4211yescharacteristic
86C25C4⋊C411yescharacteristic
87C22×D4—42no—
88C22×D4—41no—
89C22×D4—41no—
90C22×D4—42no—
91C22×D4C4211yescharacteristic
92C22×D4C22⋊C411yescharacteristic
93C22×D4C4⋊C411yescharacteristic
94C22×D4C22×C411yescharacteristic
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—42no—
104C23×C4—42no—
105C23×C4—42no—
106C23×C4—42no—
107C23×C4C4211yescharacteristic
108C23×C4C22⋊C411yescharacteristic
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4C22⋊C411yescharacteristic
113C23×C4C22⋊C411yescharacteristic
114C23×C4C4⋊C411yescharacteristic
115C23×C4C4⋊C411yescharacteristic
116C23×C4C4⋊C411yescharacteristic
117C23×C4C4⋊C411yescharacteristic
118C23×C4C22×C411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C2×M16—24no—
121C2×M16—44no—
122C2×M16—44no—
123C2×M16—42no—
124C2×M16—48no—
125C2×M16—44no—
126C2×M16—42no—
127C2×M16—42no—
128C2×M16—42no—
129C2×M16—44no—
130C2×M16—82no—
131C2×M16—82no—
132C22×C8—24no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—42no—
136C22×C8—41no—
137C22×C8—41no—
138C22×C8—42no—
139C22×C8—42no—
140C22×C8—42no—
141C22×C8—41no—
142C22×C8—41no—
143C4×D4—41no—
144C4×D4—41no—
145C4×D4—41no—
146C4×D4—41no—
147C4×D4—41no—
148C4×D4—41no—
149C4×D4—41no—
150C4×D4—41no—
151C4×D4—41no—
152C4×D4—41no—
153C4×D4—41no—
154C4×D4—41no—
155C4×D4—41no—
156C4×D4—41no—
157C4×D4—41no—
158C4×D4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4C22⋊C411yescharacteristic
162C2×C4⋊C4C22⋊C411yescharacteristic
163C2×C4⋊C4C4⋊C411yescharacteristic
164C2×C4⋊C4C2×D411yescharacteristic
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4C4211yescharacteristic
170C2×C22⋊C4C22⋊C411yescharacteristic
171C2×C22⋊C4C4⋊C411yescharacteristic
172C2×C22⋊C4C4⋊C411yescharacteristic
173C2×C22⋊C4C4⋊C411yescharacteristic
174C2×C22⋊C4C4⋊C411yescharacteristic
175C2×C22⋊C4C22×C411yescharacteristic
176C2×C22⋊C4C2×Q811yescharacteristic
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42C22⋊C411yescharacteristic
182C2×C42C22⋊C411yescharacteristic
183C2×C42C4⋊C411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C4⋊C8—42no—
186C4⋊C8—42no—
187#100—44no—
188#100—44no—
189C22⋊C8—24no—
190C22⋊C8—44no—
191C22⋊C8—44no—
192C22⋊C8—42no—
193C22⋊C8—42no—
194C22⋊C8—42no—
195C22⋊C8—42no—
196C22⋊C8—42no—
197C22⋊C8—42no—
198C22⋊C8—42no—
199#97—42no—
200C4×C8—42no—
201C24—21no—
202C24—21no—
203C24—22no—
204C24—21no—
205C24—21no—
206C24—41no—
207C24—41no—
208C24—41no—
209C24—42no—
210C24—42no—
211C24—42no—
212C24—42no—
213C24#9511yescharacteristic
214C24#9511yescharacteristic
215C24#9511yescharacteristic
216C24C2×C4211yescharacteristic
217C24C2×C4⋊C411yescharacteristic
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—48no—
231C2×D4—48no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
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—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—41no—
260C22×C4—42no—
261C22×C4—44no—
262C22×C4—42no—
263C22×C4—44no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—81no—
284C22×C4—81no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4#9511yescharacteristic
288C22×C4#9511yescharacteristic
289C22×C4#9511yescharacteristic
290C22×C4#9511yescharacteristic
291C22×C4#9511yescharacteristic
292C22×C4#9911yescharacteristic
293C22×C4#9911yescharacteristic
294C22×C4#9911yescharacteristic
295C22×C4#9911yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C4⋊C411yescharacteristic
300C22×C4C2×C4⋊C411yescharacteristic
301M16—48no—
302M16—44no—
303M16—44no—
304M16—44no—
305M16—44no—
306M16—84no—
307M16—82no—
308M16—82no—
309C2×C8—24no—
310C2×C8—42no—
311C2×C8—44no—
312C2×C8—44no—
313C2×C8—42no—
314C2×C8—44no—
315C2×C8—44no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—82no—
320C2×C8—82no—
321C2×C8—84no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C42—21no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42—21no—
341C42—21no—
342C23—22no—
343C23—22no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—42no—
358C23—42no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—48no—
367C23—42no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—81no—
377C23—81no—
378C23—84no—
379C23—84no—
380C23—84no—
381C23#34211yescharacteristic
382C23#34211yescharacteristic
383C23#34412yes—
384C23C2×#9511yescharacteristic
385D4—44no—
386D4—44no—
387D4—44no—
388D4—44no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4#34211yescharacteristic
442C2×C4#34211yescharacteristic
443C2×C4AΣL(1,𝔽16)12yes—
444C2×C4#35212yes—
445C2×C4C2×#9911yescharacteristic
446C2×C4C2×#9911yescharacteristic
447C2×C4#41312yes—
448C8—44no—
449C8—82no—
450C8—84no—
451C8—84no—
452C22—24no—
453C22—24no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—22no—
461C22—42no—
462C22—42no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22—81no—
475C22—84no—
476C22—84no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22—81no—
481C22—81no—
482C22—84no—
483C22—84no—
484C22#139012yes—
485C22#139212yes—
486C22C2×#34211yescharacteristic
487C22#174211yescharacteristic
488C22C2×#34411yescharacteristic
489C4—21no—
490C4—21no—
491C4—22no—
492C4—41no—
493C4—41no—
494C4—41no—
495C4—41no—
496C4—41no—
497C4—41no—
498C4—41no—
499C4#139012yes—
500C4C2×AΣL(1,𝔽16)11yescharacteristic
501C4C2×#35211yescharacteristic
502C2—21no—
503C2—22no—
504C2—22no—
505C2—21no—
506C2—42no—
507C2—81no—
508C2—81no—
509C2—81no—
510C2#810411yescharacteristic
511C2C2×#139011yescharacteristic
512C2C2×#139211yescharacteristic
subgroupG/H
centerC2×C4#342
commutatorC24C2×C42
Frattini, cosocleC2×C4×D4C23
socleC22C2×#342
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7321C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18702
#207262
#173622
#73212