Finite Group Databasebeta

← #127145

Subgroups — #127145

Order 512.

834 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#21788C211yesmaximal, characteristic
2C2×C4×#99C211yesmaximal, characteristic
3#17362C211yesmaximal, characteristic
4#7205C214yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6C2×#432C411yescharacteristic, complemented
7C2×#431C411yescharacteristic, complemented
8C22×#99C412yes—
9C2×#407—21no—
10C2×C4×C22⋊C4—21no—
11#1955—21no—
12#1955—21no—
13#1954C412yescomplemented
14#1954C2212yes—
15#1779—21no—
16#1779—21no—
17C4×#99—21no—
18C4×#99—21no—
19C4×#99C412yes—
20C4×#99C2212yes—
21C2×#342—21no—
22#1465—21no—
23#1465C412yescomplemented
24#1463C412yescomplemented
25#1462—21no—
26#1459—21no—
27#1459C2212yes—
28C23×D4C2×C411yescharacteristic
29C24×C4D411yescharacteristic
30C24×C4D411yescharacteristic
31C22×M16—41no—
32C2×C4×D4—41no—
33C2×C4×D4—41no—
34C2×C4×D4—41no—
35C2×C4×D4—41no—
36C2×C4×D4C2×C411yescharacteristic
37C2×C4×D4C2×C411yescharacteristic
38C2×C4×D4C2×C411yescharacteristic
39C2×C4×D4D411yescharacteristic
40C2×C4×D4D411yescharacteristic
41C2×C4×D4D411yescharacteristic
42C2×C4×D4Q811yescharacteristic
43C2×C4×D4C2311yescharacteristic
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C22⋊C4—22no—
46C22×C22⋊C4D411yescharacteristic
47C22×C22⋊C4Q811yescharacteristic
48C22×C42—41no—
49C22×C42C2×C411yescharacteristic
50#432—41no—
51#432—41no—
52#431—41no—
53#431—41no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8C2×C412yescomplemented
56C2×#99—22no—
57C2×#99—28no—
58C2×#99C2×C414yes—
59#407—22no—
60#407—22no—
61#407—41no—
62#407—41no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—41no—
71C2×#97—21no—
72C2×C4×C8C2×C412yescomplemented
73C4×C22⋊C4—22no—
74C4×C22⋊C4—22no—
75C4×C22⋊C4—41no—
76C4×C22⋊C4—41no—
77C2×#95—41no—
78#342—22no—
79#342—22no—
80#324—22no—
81#324C2×C414yescomplemented
82#323—22no—
83#323—22no—
84C25C22⋊C411yescharacteristic
85C25C4⋊C411yescharacteristic
86C22×D4—24no—
87C22×D4—41no—
88C22×D4—41no—
89C22×D4—42no—
90C22×D4C4211yescharacteristic
91C22×D4C22⋊C411yescharacteristic
92C22×D4C4⋊C411yescharacteristic
93C22×D4C22×C411yescharacteristic
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—42no—
103C23×C4—82no—
104C23×C4—82no—
105C23×C4C4211yescharacteristic
106C23×C4C22⋊C411yescharacteristic
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22⋊C411yescharacteristic
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4C4⋊C411yescharacteristic
113C23×C4C4⋊C411yescharacteristic
114C23×C4C4⋊C411yescharacteristic
115C23×C4C4⋊C411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C2×M16—42no—
119C2×M16—42no—
120C2×M16—84no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—42no—
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—
139C4×D4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4C4211yescharacteristic
143C2×C4⋊C4C22⋊C411yescharacteristic
144C2×C4⋊C4C4⋊C411yescharacteristic
145C2×C4⋊C4C22×C411yescharacteristic
146C2×C22⋊C4—28no—
147C2×C22⋊C4—24no—
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⋊C4C22⋊C411yescharacteristic
155C2×C22⋊C4C22⋊C411yescharacteristic
156C2×C22⋊C4C4⋊C411yescharacteristic
157C2×C22⋊C4C4⋊C411yescharacteristic
158C2×C22⋊C4C4⋊C411yescharacteristic
159C2×C22⋊C4C4⋊C411yescharacteristic
160C2×C22⋊C4C2×D411yescharacteristic
161C2×C22⋊C4C2×Q811yescharacteristic
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—42no—
167C2×C42—42no—
168C2×C42—84no—
169C2×C42C4211yescharacteristic
170C2×C42C22⋊C411yescharacteristic
171C2×C42C4⋊C411yescharacteristic
172C2×C42C22×C411yescharacteristic
173C4⋊C8—42no—
174C4⋊C8—42no—
175#99—28no—
176#99—24no—
177#99M1618yescomplemented
178C22⋊C8—24no—
179C22⋊C8—42no—
180C22⋊C8—42no—
181C22⋊C8—42no—
182C22⋊C8—41no—
183C22⋊C8—41no—
184C22⋊C8—41no—
185C22⋊C8—41no—
186C22⋊C8—82no—
187C22⋊C8—82no—
188C22⋊C8—82no—
189C22⋊C8—82no—
190#97—42no—
191C4×C8—42no—
192#95—42no—
193#95—42no—
194C24—21no—
195C24—21no—
196C24—22no—
197C24—22no—
198C24—21no—
199C24—21no—
200C24—22no—
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×D4C2×M1614yes—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
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—22no—
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—42no—
251C22×C4—41no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—44no—
260C22×C4—48no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4—81no—
266C22×C4—81no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—84no—
274C22×C4—84no—
275C22×C4#9511yescharacteristic
276C22×C4#9511yescharacteristic
277C22×C4#9511yescharacteristic
278C22×C4#9511yescharacteristic
279C22×C4#9511yescharacteristic
280C22×C4#9911yescharacteristic
281C22×C4#9911yescharacteristic
282C22×C4#9911yescharacteristic
283C22×C4#9911yescharacteristic
284C22×C4C2×C4211yescharacteristic
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C4⋊C411yescharacteristic
288C22×C4C2×C4⋊C411yescharacteristic
289M16—84no—
290M16—84no—
291C2×C8—44no—
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—
303C22⋊C4—28no—
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—48no—
313C22⋊C4—44no—
314C42—21no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—84no—
323C42—84no—
324C23—24no—
325C23—24no—
326C23—22no—
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—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23—84no—
358C23—84no—
359C23—84no—
360C23—84no—
361C23#34211yescharacteristic
362C23#34211yescharacteristic
363C23#34412yes—
364C23C2×#9511yescharacteristic
365C23#40712yes—
366C23#42312yes—
367D4—44no—
368D4—44no—
369D4—44no—
370D4—44no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
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—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—48no—
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—81no—
419C2×C4—84no—
420C2×C4—816no—
421C2×C4—84no—
422C2×C4#34211yescharacteristic
423C2×C4#34211yescharacteristic
424C2×C4C4×M1612yes—
425C2×C4C2×#9911yescharacteristic
426C2×C4C2×#9911yescharacteristic
427C8—82no—
428C8—84no—
429C8—82no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—22no—
437C22—21no—
438C22—22no—
439C22—22no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—48no—
446C22—42no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—84no—
460C22—81no—
461C22—84no—
462C22—84no—
463C22#139512yes—
464C22#172811yescharacteristic
465C22C2×#34211yescharacteristic
466C22C2×#34411yescharacteristic
467C4—21no—
468C4—21no—
469C4—21no—
470C4—22no—
471C4—41no—
472C4—41no—
473C4—41no—
474C4—41no—
475C4—41no—
476C4—41no—
477C4—41no—
478C4—88no—
479C4#212712yes—
480C2—22no—
481C2—21no—
482C2—21no—
483C2—42no—
484C2—41no—
485C2—42no—
486C2—42no—
487C2—81no—
488C2#792911yescharacteristic
489C2#1258211yescharacteristic
490C2C2×#139511yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
socleC22C2×#342
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ 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. #7205C2semidirect
6. C2×#432C4semidirect
7. C2×#431C4semidirect
13. #1954C4semidirect
23. #1465C4semidirect
24. #1463C4semidirect
55. C2×C4⋊C8C2×C4semidirect
72. C2×C4×C8C2×C4semidirect
81. #324C2×C4semidirect
177. #99M16semidirect

Maximal subgroups

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

subgroupindex
#217882
C2×C4×#992
#173622
#72052