Finite Group Databasebeta

← #128172

Subgroups — #128172

Order 512.

773 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17364C211yesmaximal, characteristic
2#17362C211yesmaximal, characteristic
3#14344C211yesmaximal, characteristic
4#7324C214yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6#2920—21no—
7C2×#407—21no—
8C2×#381C411yescharacteristic
9C2×#380C411yescharacteristic
10#1466—21no—
11#1466C412yescomplemented
12#1465—21no—
13#1465C412yescomplemented
14#1463—21no—
15#1462—21no—
16#1462C2212yes—
17#1462C2212yes—
18#1460—21no—
19#1460C412yescomplemented
20#1459—21no—
21#1459C412yescomplemented
22#1441—21no—
23#1440C2212yes—
24#1439—21no—
25#1438C412yes—
26#1434—21no—
27C22⋊#95—21no—
28C23×D4C2×C411yescharacteristic
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⋊C4—21no—
45C22×C4⋊C4D411yescharacteristic
46C22×C22⋊C4—22no—
47C22×C22⋊C4C2×C412yes—
48C22×C22⋊C4C2×C412yes—
49C22×C22⋊C4D411yescharacteristic
50C22×C22⋊C4C2311yescharacteristic
51C22×C42D411yescharacteristic
52#432—22no—
53#432—41no—
54#431—22no—
55#431—41no—
56#423—41no—
57#407—22no—
58#407—22no—
59#407—22no—
60#407—41no—
61#407—41no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—41no—
64C4×M16—41no—
65#381—41no—
66#381—41no—
67#380—41no—
68#380—41no—
69C2×#95—22no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—22no—
74C2×#95—22no—
75#324—22no—
76#324—22no—
77#323—22no—
78#323—22no—
79#323—22no—
80#323—22no—
81#323C2×C414yes—
82#323C2×C414yes—
83C25C22⋊C411yescharacteristic
84C25C22×C411yescharacteristic
85C22×D4—42no—
86C22×D4—41no—
87C22×D4—41no—
88C22×D4—42no—
89C22×D4C4211yescharacteristic
90C22×D4C4211yescharacteristic
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×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—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⋊C4—42no—
142C2×C4⋊C4—82no—
143C2×C4⋊C4C22⋊C411yescharacteristic
144C2×C4⋊C4C22⋊C411yescharacteristic
145C2×C4⋊C4C4⋊C411yescharacteristic
146C2×C4⋊C4C4⋊C411yescharacteristic
147C2×C22⋊C4—41no—
148C2×C22⋊C4—41no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—41no—
151C2×C22⋊C4—44no—
152C2×C22⋊C4—42no—
153C2×C22⋊C4—42no—
154C2×C22⋊C4—82no—
155C2×C22⋊C4—82no—
156C2×C22⋊C4—82no—
157C2×C22⋊C4—82no—
158C2×C22⋊C4C22⋊C411yescharacteristic
159C2×C22⋊C4C22⋊C411yescharacteristic
160C2×C22⋊C4C4⋊C411yescharacteristic
161C2×C22⋊C4C4⋊C411yescharacteristic
162C2×C22⋊C4C22×C411yescharacteristic
163C2×C22⋊C4C22×C411yescharacteristic
164C2×C22⋊C4C2×D411yescharacteristic
165C2×C22⋊C4C2×Q811yescharacteristic
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42C22⋊C411yescharacteristic
171C2×C42C22⋊C411yescharacteristic
172C2×C42C4⋊C411yescharacteristic
173C2×C42C4⋊C411yescharacteristic
174C4⋊C8—42no—
175C4⋊C8—42no—
176C22⋊C8—24no—
177C22⋊C8—24no—
178C22⋊C8—42no—
179C22⋊C8—42no—
180C22⋊C8—42no—
181C22⋊C8—42no—
182#97—42no—
183C4×C8—42no—
184#95—44no—
185#95—42no—
186#95—41no—
187#95—41no—
188#95—41no—
189#95—41no—
190#95—82no—
191#95—82no—
192C24—22no—
193C24—21no—
194C24—21no—
195C24—21no—
196C24—21no—
197C24—41no—
198C24—41no—
199C24—41no—
200C24—41no—
201C24—42no—
202C24—41no—
203C24—41no—
204C24—42no—
205C24—41no—
206C24#9511yescharacteristic
207C24#9511yescharacteristic
208C24C2×C4211yescharacteristic
209C24C2×C22⋊C411yescharacteristic
210C24C2×C4⋊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—42no—
251C22×C4—41no—
252C22×C4—44no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—41no—
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—42no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4—81no—
275C22×C4—81no—
276C22×C4—81no—
277C22×C4—81no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4—81no—
283C22×C4—81no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4#9511yescharacteristic
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×C4⋊C411yescharacteristic
299C22×C4C2×C4⋊C411yescharacteristic
300M16—82no—
301M16—82no—
302M16—84no—
303M16—84no—
304C2×C8—44no—
305C2×C8—44no—
306C2×C8—84no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—84no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—88no—
321C22⋊C4—84no—
322C22⋊C4—84no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42—21no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—21no—
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—21no—
346C23—21no—
347C23—42no—
348C23—42no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—44no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—44no—
358C23—42no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—42no—
364C23—42no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—81no—
370C23—81no—
371C23—82no—
372C23—82no—
373C23—82no—
374C23—82no—
375C23—82no—
376C23—82no—
377C23#34412yes—
378C23C2×#9511yescharacteristic
379C23C2×#9911yescharacteristic
380C23C2×#9911yescharacteristic
381D4—44no—
382D4—44no—
383D4—44no—
384D4—44no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
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—81no—
432C2×C4—81no—
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—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4#34211yescharacteristic
448C2×C4#34211yescharacteristic
449C2×C4#34211yescharacteristic
450C2×C4#34211yescharacteristic
451C2×C4#34412yes—
452C8—84no—
453C8—84no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
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—82no—
480C22—84no—
481C22—82no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22—82no—
488C22—84no—
489C22—82no—
490C22C2×#34211yescharacteristic
491C22C2×#34411yescharacteristic
492C22C2×#34411yescharacteristic
493C22#212112yes—
494C22#212211yescharacteristic
495C22#212311yescharacteristic
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—22no—
501C4—41no—
502C4—41no—
503C4—41no—
504C4—41no—
505C4—41no—
506C4—41no—
507C4—41no—
508C4—162no—
509C4—162no—
510C2—21no—
511C2—22no—
512C2—21no—
513C2—21no—
514C2—21no—
515C2—42no—
516C2—81no—
517C2—81no—
518C2—81no—
519C2#811211yescharacteristic
520C2#1257411yescharacteristic
521C2#1257511yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC24C2×C42
Frattini, cosocleC22×C22⋊C4C23
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7324C2semidirect
11. #1466C4semidirect
13. #1465C4semidirect
19. #1460C4semidirect
21. #1459C4semidirect

Maximal subgroups

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

subgroupindex
#173642
#173622
#143442
#73242