Finite Group Databasebeta

← #127821

Subgroups — #127821

Order 512.

809 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21740C211yesmaximal, characteristic
2#20722C211yesmaximal, characteristic
3C2×C4×#100C211yesmaximal, characteristic
4#7282C214yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6C22×#100—21no—
7C2×#407C411yescharacteristic, complemented
8C22×C22⋊C8C411yescharacteristic, complemented
9C2×C4×M16—21no—
10#2280—21no—
11#1870C412yescomplemented
12#1870C2212yes—
13#1869—21no—
14#1869—21no—
15#1819C412yes—
16#1817—21no—
17#1799—21no—
18#1799C412yes—
19#1781—21no—
20#1779—21no—
21C4×#100—21no—
22C4×#100—21no—
23C4×#100C412yescomplemented
24C4×#100C2212yes—
25C4×#99—21no—
26C4×#99C2212yes—
27C2×#343C412yescomplemented
28C23×D4D411yescharacteristic
29C24×C4C2×C411yescharacteristic
30C24×C4D411yescharacteristic
31C22×M16—22no—
32C22×M16C2×C412yes—
33C23×C8—21no—
34C2×C4×D4—41no—
35C2×C4×D4—41no—
36C2×C4×D4—41no—
37C2×C4×D4—41no—
38C2×C4×D4C2×C411yescharacteristic
39C2×C4×D4C2×C411yescharacteristic
40C2×C4×D4C2×C411yescharacteristic
41C2×C4×D4D411yescharacteristic
42C2×C4×D4D411yescharacteristic
43C2×C4×D4D411yescharacteristic
44C2×C4×D4Q811yescharacteristic
45C2×C4×D4C2311yescharacteristic
46C2×#117—41no—
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C22⋊C4C2×C411yescharacteristic
49C22×C22⋊C4Q811yescharacteristic
50C22×C42D411yescharacteristic
51C2×#100—28no—
52C2×#100—22no—
53C2×#100—22no—
54C2×#99—22no—
55C2×#99—22no—
56#407—41no—
57#407—41no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—21no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—41no—
64C2×C22⋊C8—41no—
65C2×C22⋊C8C2×C412yes—
66C4×M16—22no—
67C4×M16—22no—
68C4×M16—41no—
69C4×M16—41no—
70C2×#97—41no—
71C2×C4×C8—41no—
72#381—41no—
73#380—22no—
74#380—41no—
75#379—22no—
76C4×C22⋊C4—22no—
77C4×C22⋊C4—41no—
78#343—22no—
79#343C2×C414yes—
80#342—22no—
81#342C2×C414yes—
82#336—22no—
83#336—22no—
84C25C22⋊C411yescharacteristic
85C25C4⋊C411yescharacteristic
86C22×D4—24no—
87C22×D4—42no—
88C22×D4—41no—
89C22×D4—41no—
90C22×D4C22⋊C411yescharacteristic
91C22×D4C22⋊C411yescharacteristic
92C22×D4C4⋊C411yescharacteristic
93C22×D4C2×D411yescharacteristic
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—82no—
103C23×C4—82no—
104C23×C4C4211yescharacteristic
105C23×C4C4211yescharacteristic
106C23×C4C22⋊C411yescharacteristic
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22⋊C411yescharacteristic
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22⋊C411yescharacteristic
111C23×C4C4⋊C411yescharacteristic
112C23×C4C4⋊C411yescharacteristic
113C23×C4C4⋊C411yescharacteristic
114C23×C4C4⋊C411yescharacteristic
115C23×C4C22×C411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C2×M16—28no—
118C2×M16—24no—
119C2×M16—44no—
120C2×M16—42no—
121C2×M16—42no—
122C2×M16—82no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—42no—
126C22×C8—42no—
127C22×C8—82no—
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—
139C4×D4—41no—
140C4×D4—41no—
141C4×D4—41no—
142C4×D4—41no—
143C4×D4—41no—
144#117—84no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4C4211yescharacteristic
149C2×C4⋊C4C22⋊C411yescharacteristic
150C2×C4⋊C4C4⋊C411yescharacteristic
151C2×C4⋊C4C22×C411yescharacteristic
152C2×C22⋊C4—24no—
153C2×C22⋊C4—41no—
154C2×C22⋊C4—41no—
155C2×C22⋊C4—41no—
156C2×C22⋊C4—41no—
157C2×C22⋊C4—42no—
158C2×C22⋊C4—42no—
159C2×C22⋊C4C4211yescharacteristic
160C2×C22⋊C4C22⋊C411yescharacteristic
161C2×C22⋊C4C4⋊C411yescharacteristic
162C2×C22⋊C4C4⋊C411yescharacteristic
163C2×C22⋊C4C4⋊C411yescharacteristic
164C2×C22⋊C4C4⋊C411yescharacteristic
165C2×C22⋊C4C22×C411yescharacteristic
166C2×C22⋊C4C2×Q811yescharacteristic
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—42no—
172C2×C42C22⋊C411yescharacteristic
173C2×C42C22⋊C411yescharacteristic
174C2×C42C4⋊C411yescharacteristic
175C2×C42C2×D411yescharacteristic
176#100—28no—
177#100—28no—
178#99—44no—
179#99—44no—
180C22⋊C8—42no—
181C22⋊C8—41no—
182C22⋊C8—41no—
183C22⋊C8—41no—
184C22⋊C8—41no—
185C22⋊C8—42no—
186C22⋊C8—82no—
187C22⋊C8—82no—
188#97—42no—
189#97—42no—
190C4×C8—42no—
191C4×C8—42no—
192#95—42no—
193#95—42no—
194C24—21no—
195C24—21no—
196C24—21no—
197C24—21no—
198C24—22no—
199C24—22no—
200C24—41no—
201C24—41no—
202C24—42no—
203C24—42no—
204C24—81no—
205C24—81no—
206C24#9511yescharacteristic
207C24#9511yescharacteristic
208C24#9511yescharacteristic
209C24C2×C22⋊C411yescharacteristic
210C24C2×C4⋊C411yescharacteristic
211C2×D4—22no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—22no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—48no—
223C2×D4—48no—
224C2×D4#10414yes—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
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—42no—
252C22×C4—44no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—44no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—81no—
270C22×C4—81no—
271C22×C4—81no—
272C22×C4#9511yescharacteristic
273C22×C4#9511yescharacteristic
274C22×C4#9511yescharacteristic
275C22×C4#9511yescharacteristic
276C22×C4#9511yescharacteristic
277C22×C4#9911yescharacteristic
278C22×C4#9911yescharacteristic
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
286M16—28no—
287M16—48no—
288M16—44no—
289M16—84no—
290C2×C8—44no—
291C2×C8—48no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—84no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—84no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—44no—
313C22⋊C4—44no—
314C22⋊C4—84no—
315C22⋊C4—84no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—21no—
324C42—84no—
325C23—22no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—41no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—42no—
350C23—42no—
351C23—41no—
352C23—41no—
353C23—42no—
354C23—42no—
355C23—81no—
356C23—81no—
357C23—81no—
358C23—81no—
359C23—84no—
360C23—84no—
361C23—84no—
362C23—84no—
363C23#33912yes—
364C23#34211yescharacteristic
365C23#34211yescharacteristic
366C23C2×#9511yescharacteristic
367C23C2×#10412yes—
368D4—44no—
369D4—44no—
370D4—44no—
371D4—44no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—84no—
419C2×C4—84no—
420C2×C4—84no—
421C2×C4#33912yes—
422C2×C4#34211yescharacteristic
423C2×C4#34211yescharacteristic
424C2×C4C2×#9911yescharacteristic
425C2×C4C2×#9911yescharacteristic
426C2×C4#42112yes—
427C8—48no—
428C8—82no—
429C8—82no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—22no—
438C22—22no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—81no—
456C22—84no—
457C22—84no—
458C22—81no—
459C22—81no—
460C22—81no—
461C22—84no—
462C22—84no—
463C22#139512yes—
464C22C2×#33911yescharacteristic
465C22#173311yescharacteristic
466C22C2×#34211yescharacteristic
467C4—21no—
468C4—21no—
469C4—21no—
470C4—22no—
471C4—22no—
472C4—41no—
473C4—41no—
474C4—41no—
475C4—41no—
476C4—41no—
477C4—41no—
478C4—164no—
479C4#212112yes—
480C2—21no—
481C2—21no—
482C2—22no—
483C2—42no—
484C2—41no—
485C2—42no—
486C2—81no—
487C2—81no—
488C2#805111yescharacteristic
489C2#1257411yescharacteristic
490C2C2×#139511yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
socleC22C2×#342
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ 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. #7282C2semidirect
7. C2×#407C4semidirect
8. C22×C22⋊C8C4semidirect
11. #1870C4semidirect
23. C4×#100C4semidirect
27. C2×#343C4semidirect

Maximal subgroups

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

subgroupindex
#217402
#207222
C2×C4×#1002
#72822