Finite Group Databasebeta

← #127242

Subgroups — #127242

Order 512.

805 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
1#20726C211yesmaximal, characteristic
2#17362C211yesmaximal, characteristic
3#13940C211yesmaximal, characteristic
4#7215C214yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6#2920—21no—
7C2×#407—21no—
8C2×#381C411yescharacteristic
9C2×C4×C22⋊C4C411yescharacteristic
10#1821C2212yes—
11#1800—21no—
12#1800C412yescomplemented
13#1783—21no—
14C4×#100—21no—
15C4×#100C412yescomplemented
16#1465—21no—
17#1465C412yescomplemented
18#1463C2212yes—
19#1462—21no—
20#1459—21no—
21#1459C412yescomplemented
22#1440—21no—
23#1438—21no—
24#1437C412yes—
25#1436C2212yes—
26#1434—21no—
27C22⋊#95—21no—
28C23×D4D411yescharacteristic
29C24×C4C2×C411yescharacteristic
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×D4C2×C411yescharacteristic
40C2×C4×D4D411yescharacteristic
41C2×C4×D4D411yescharacteristic
42C2×C4×D4Q811yescharacteristic
43C2×C4×D4Q811yescharacteristic
44C22×C4⋊C4C2311yescharacteristic
45C22×C22⋊C4—21no—
46C22×C22⋊C4—22no—
47C22×C22⋊C4—21no—
48C22×C22⋊C4C2×C411yescharacteristic
49C22×C22⋊C4D411yescharacteristic
50C22×C42C2×C412yes—
51C22×C42D411yescharacteristic
52#432—41no—
53#432—41no—
54#431—41no—
55#431—41no—
56#423—22no—
57C2×#100—22no—
58C2×#100—22no—
59#407—22no—
60#407—22no—
61#407—22no—
62#407—41no—
63#407—41no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—41no—
66C4×M16—22no—
67#381—41no—
68#381—41no—
69C4×C22⋊C4—41no—
70C4×C22⋊C4—41no—
71C2×#95—22no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95C2×C412yes—
77#343—22no—
78#343C2×C414yes—
79#324—22no—
80#324C2×C414yes—
81#323—22no—
82#323—22no—
83C25C22⋊C411yescharacteristic
84C25C22⋊C411yescharacteristic
85C22×D4—42no—
86C22×D4—41no—
87C22×D4—41no—
88C22×D4—42no—
89C22×D4C22⋊C411yescharacteristic
90C22×D4C22⋊C411yescharacteristic
91C22×D4C4⋊C411yescharacteristic
92C22×D4C4⋊C411yescharacteristic
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—82no—
104C23×C4—82no—
105C23×C4C4211yescharacteristic
106C23×C4C4211yescharacteristic
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
117C23×C4C2×D411yescharacteristic
118C2×M16—24no—
119C2×M16—42no—
120C2×M16—42no—
121C2×M16—84no—
122C22×C8—42no—
123C4×D4—41no—
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—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4C22×C411yescharacteristic
142C2×C4⋊C4C22×C411yescharacteristic
143C2×C4⋊C4C2×D411yescharacteristic
144C2×C4⋊C4C2×Q811yescharacteristic
145C2×C22⋊C4—41no—
146C2×C22⋊C4—41no—
147C2×C22⋊C4—41no—
148C2×C22⋊C4—41no—
149C2×C22⋊C4—44no—
150C2×C22⋊C4—42no—
151C2×C22⋊C4—44no—
152C2×C22⋊C4—44no—
153C2×C22⋊C4—42no—
154C2×C22⋊C4—82no—
155C2×C22⋊C4—82no—
156C2×C22⋊C4C4211yescharacteristic
157C2×C22⋊C4C4211yescharacteristic
158C2×C22⋊C4C22⋊C411yescharacteristic
159C2×C22⋊C4C22⋊C411yescharacteristic
160C2×C22⋊C4C4⋊C411yescharacteristic
161C2×C22⋊C4C4⋊C411yescharacteristic
162C2×C22⋊C4C4⋊C411yescharacteristic
163C2×C22⋊C4C4⋊C411yescharacteristic
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—42no—
169C2×C42—82no—
170C2×C42C22⋊C411yescharacteristic
171C2×C42C22⋊C411yescharacteristic
172C2×C42C4⋊C411yescharacteristic
173C2×C42C4⋊C411yescharacteristic
174C4⋊C8—42no—
175C4⋊C8—42no—
176#100—44no—
177#100—44no—
178C22⋊C8—24no—
179C22⋊C8—42no—
180C22⋊C8—42no—
181C22⋊C8—42no—
182C22⋊C8—42no—
183C22⋊C8—42no—
184#97—42no—
185C4×C8—42no—
186#95—42no—
187#95—44no—
188#95—44no—
189#95—41no—
190#95—41no—
191#95—41no—
192#95—41no—
193#95—82no—
194C24—22no—
195C24—21no—
196C24—21no—
197C24—21no—
198C24—21no—
199C24—41no—
200C24—41no—
201C24—41no—
202C24—42no—
203C24—42no—
204C24—42no—
205C24—42no—
206C24#9511yescharacteristic
207C24#9511yescharacteristic
208C24#9511yescharacteristic
209C24#9511yescharacteristic
210C24C2×C22⋊C411yescharacteristic
211C2×D4—22no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—48no—
224C2×D4—48no—
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—24no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
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—42no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—81no—
274C22×C4—81no—
275C22×C4—81no—
276C22×C4—81no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4—81no—
282C22×C4—81no—
283C22×C4—81no—
284C22×C4—81no—
285C22×C4#9511yescharacteristic
286C22×C4#9511yescharacteristic
287C22×C4#9511yescharacteristic
288C22×C4#9511yescharacteristic
289C22×C4#9911yescharacteristic
290C22×C4#9911yescharacteristic
291C22×C4#9911yescharacteristic
292C22×C4#9911yescharacteristic
293C22×C4C2×C4211yescharacteristic
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C4⋊C411yescharacteristic
297C22×C4C2×C4⋊C411yescharacteristic
298C22×C4C2×C4⋊C411yescharacteristic
299M16—44no—
300M16—44no—
301M16—84no—
302M16—84no—
303C2×C8—44no—
304C2×C8—44no—
305C2×C8—84no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—48no—
319C22⋊C4—88no—
320C22⋊C4—84no—
321C42—21no—
322C42—21no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42—21no—
327C42—21no—
328C42—21no—
329C42—84no—
330C23—22no—
331C23—22no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—42no—
346C23—42no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—42no—
352C23—42no—
353C23—44no—
354C23—42no—
355C23—42no—
356C23—44no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—81no—
366C23—81no—
367C23—82no—
368C23—82no—
369C23—82no—
370C23—82no—
371C23—82no—
372C23—82no—
373C23H(3,ℤ/4)12yes—
374C23C2×#9511yescharacteristic
375C23C2×#9911yescharacteristic
376C23C2×#9911yescharacteristic
377D4—44no—
378D4—44no—
379D4—44no—
380D4—44no—
381C2×C4—22no—
382C2×C4—22no—
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—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
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—82no—
441C2×C4—82no—
442C2×C4#34211yescharacteristic
443C2×C4#34211yescharacteristic
444C2×C4#34211yescharacteristic
445C2×C4#34211yescharacteristic
446C2×C4#42112yes—
447C8—84no—
448C8—84no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—22no—
460C22—42no—
461C22—42no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—81no—
474C22—84no—
475C22—84no—
476C22—81no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22—81no—
481C22—84no—
482C22—84no—
483C22C2×H(3,ℤ/4)11yescharacteristic
484C22#173311yescharacteristic
485C22C2×#34211yescharacteristic
486C22#212112yes—
487C22#213212yes—
488C4—22no—
489C4—21no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—41no—
494C4—41no—
495C4—41no—
496C4—41no—
497C4—41no—
498C4—41no—
499C4—41no—
500C4—84no—
501C4—162no—
502C2—22no—
503C2—22no—
504C2—21no—
505C2—21no—
506C2—42no—
507C2—81no—
508C2—81no—
509C2—81no—
510C2#794611yescharacteristic
511C2#1257411yescharacteristic
512C2#1258511yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC22×C4⋊C4C23
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ 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. #7215C2semidirect
12. #1800C4semidirect
15. C4×#100C4semidirect
17. #1465C4semidirect
21. #1459C4semidirect

Maximal subgroups

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

subgroupindex
#207262
#173622
#139402
#72152