Finite Group Databasebeta

← #128107

Subgroups — #128107

Order 512.

839 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1870C211yesmaximal, characteristic
2#20722C211yesmaximal, characteristic
3#17364C211yesmaximal, characteristic
4#7320C214yesmaximal, complemented
5C23×M16C411yescharacteristic, complemented
6C22×C4×D4C2211yescharacteristic
7#2920—21no—
8C22×C22⋊C8—21no—
9C22×C22⋊C8C411yescharacteristic, complemented
10#2280—21no—
11#1870—21no—
12#1870—21no—
13#1870C412yescomplemented
14#1870C2212yes—
15#1819—21no—
16#1799—21no—
17#1799C2212yes—
18#1781C412yes—
19C4×#99—21no—
20C4×#99C412yes—
21C2×#336—21no—
22#1466—21no—
23#1466C2212yes—
24#1462—21no—
25#1462C412yescomplemented
26#1460—21no—
27#1460C412yescomplemented
28C23×D4C2×C411yescharacteristic
29C24×C4C2×C411yescharacteristic
30C24×C4D411yescharacteristic
31C22×M16—21no—
32C22×M16—44no—
33C22×M16—41no—
34C22×M16—41no—
35C22×M16C2×C412yes—
36C23×C8—21no—
37C23×C8—21no—
38C2×C4×D4—41no—
39C2×C4×D4—41no—
40C2×C4×D4—41no—
41C2×C4×D4—41no—
42C2×C4×D4C2×C411yescharacteristic
43C2×C4×D4C2×C411yescharacteristic
44C2×C4×D4C2×C411yescharacteristic
45C2×C4×D4D411yescharacteristic
46C2×C4×D4D411yescharacteristic
47C2×C4×D4D411yescharacteristic
48C2×C4×D4Q811yescharacteristic
49C2×C4×D4C2311yescharacteristic
50C2×#117—41no—
51C22×C4⋊C4Q811yescharacteristic
52C22×C22⋊C4C2×C411yescharacteristic
53C22×C22⋊C4D411yescharacteristic
54C22×C42D411yescharacteristic
55#432—22no—
56#432—41no—
57#431—22no—
58#431—41no—
59#423—41no—
60C2×#99—22no—
61C2×#99C2×C414yes—
62#407—22no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—44no—
67C2×C22⋊C8—41no—
68C2×C22⋊C8—41no—
69C2×C22⋊C8C2×C412yes—
70C4×M16—41no—
71#381—41no—
72#380—22no—
73#380—41no—
74#379—22no—
75C4×C22⋊C4—22no—
76C4×C22⋊C4—41no—
77#342—22no—
78#342—22no—
79#336—22no—
80#336—22no—
81#323—22no—
82#323—22no—
83#323—22no—
84#323C2×C414yes—
85C25C4211yescharacteristic
86C25C22⋊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×C4C4⋊C411yescharacteristic
114C23×C4C4⋊C411yescharacteristic
115C23×C4C4⋊C411yescharacteristic
116C23×C4C4⋊C411yescharacteristic
117C23×C4C4⋊C411yescharacteristic
118C23×C4C22×C411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C2×M16—44no—
121C2×M16—44no—
122C2×M16—42no—
123C2×M16—48no—
124C2×M16—44no—
125C2×M16—42no—
126C2×M16—42no—
127C2×M16—44no—
128C22×C8—24no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—42no—
132C22×C8—41no—
133C22×C8—41no—
134C22×C8—42no—
135C22×C8—42no—
136C22×C8—41no—
137C22×C8—41no—
138C4×D4—41no—
139C4×D4—41no—
140C4×D4—41no—
141C4×D4—41no—
142C4×D4—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—
154#117—82no—
155#117—82no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4C4⋊C411yescharacteristic
160C2×C4⋊C4C4⋊C411yescharacteristic
161C2×C4⋊C4C4⋊C411yescharacteristic
162C2×C4⋊C4C2×Q811yescharacteristic
163C2×C22⋊C4—24no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—42no—
169C2×C22⋊C4—42no—
170C2×C22⋊C4C4211yescharacteristic
171C2×C22⋊C4C22⋊C411yescharacteristic
172C2×C22⋊C4C22⋊C411yescharacteristic
173C2×C22⋊C4C22⋊C411yescharacteristic
174C2×C22⋊C4C4⋊C411yescharacteristic
175C2×C22⋊C4C4⋊C411yescharacteristic
176C2×C22⋊C4C22×C411yescharacteristic
177C2×C22⋊C4C2×D411yescharacteristic
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—42no—
183C2×C42C22⋊C411yescharacteristic
184C2×C42C22⋊C411yescharacteristic
185C2×C42C4⋊C411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C4⋊C8—42no—
188C4⋊C8—42no—
189#99—44no—
190#99—44no—
191C22⋊C8—24no—
192C22⋊C8—44no—
193C22⋊C8—44no—
194C22⋊C8—42no—
195C22⋊C8—42no—
196C22⋊C8—42no—
197#97—42no—
198C4×C8—42no—
199#95—42no—
200#95—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×C22⋊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—22no—
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—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—44no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4#9511yescharacteristic
289C22×C4#9511yescharacteristic
290C22×C4#9511yescharacteristic
291C22×C4#9511yescharacteristic
292C22×C4#9511yescharacteristic
293C22×C4#9911yescharacteristic
294C22×C4#9911yescharacteristic
295C22×C4#9911yescharacteristic
296C22×C4#9911yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C4⋊C411yescharacteristic
300C22×C4C2×C4⋊C411yescharacteristic
301C22×C4C2×C4⋊C411yescharacteristic
302M16—48no—
303M16—44no—
304M16—44no—
305M16—84no—
306C2×C8—24no—
307C2×C8—42no—
308C2×C8—44no—
309C2×C8—44no—
310C2×C8—42no—
311C2×C8—44no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—82no—
316C2×C8—82no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—44no—
332C22⋊C4—44no—
333C22⋊C4—84no—
334C22⋊C4—84no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42—21no—
341C42—21no—
342C42—21no—
343C42—84no—
344C23—22no—
345C23—22no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—42no—
360C23—42no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—48no—
369C23—42no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—81no—
379C23—81no—
380C23—84no—
381C23—84no—
382C23—84no—
383C23#34211yescharacteristic
384C23#34211yescharacteristic
385C23C2×#9511yescharacteristic
386C23#41312yes—
387D4—44no—
388D4—44no—
389D4—44no—
390D4—44no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
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—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—84no—
444C2×C4—84no—
445C2×C4—84no—
446C2×C4#34211yescharacteristic
447C2×C4#34211yescharacteristic
448C2×C4#34412yes—
449C2×C4AΣL(1,𝔽16)12yes—
450C2×C4#35212yes—
451C2×C4C2×#9911yescharacteristic
452C2×C4C2×#9911yescharacteristic
453C8—44no—
454C8—82no—
455C8—84no—
456C22—24no—
457C22—24no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—22no—
465C22—42no—
466C22—42no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—81no—
479C22—84no—
480C22—84no—
481C22—81no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22—84no—
487C22—84no—
488C22#139012yes—
489C22#139112yes—
490C22C2×#34211yescharacteristic
491C22#174211yescharacteristic
492C22C2×#34411yescharacteristic
493C4—21no—
494C4—21no—
495C4—22no—
496C4—41no—
497C4—41no—
498C4—41no—
499C4—41no—
500C4—41no—
501C4—41no—
502C4—41no—
503C4—164no—
504C4#139012yes—
505C4C2×AΣL(1,𝔽16)11yescharacteristic
506C4C2×#35211yescharacteristic
507C2—21no—
508C2—22no—
509C2—22no—
510C2—21no—
511C2—42no—
512C2—81no—
513C2—81no—
514C2—81no—
515C2#810411yescharacteristic
516C2C2×#139011yescharacteristic
517C2C2×#139111yescharacteristic
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. #7320C2semidirect
5. C23×M16C4semidirect
9. C22×C22⋊C8C4semidirect
13. #1870C4semidirect
25. #1462C4semidirect
27. #1460C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18702
#207222
#173642
#73202