Finite Group Databasebeta

← #97571

Subgroups — #97571

Order 512.

1370 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C4⋊C4C211yesmaximal, characteristic
2#8558C211yesmaximal, characteristic, complemented
3#8310C212yesmaximal
4C4×#324C211yesmaximal, characteristic, complemented
5#7573C212yesmaximal, complemented
6C22×C4×D4C411yescharacteristic
7C22×C4×D4C411yescharacteristic
8C22×C4×D4C2211yescharacteristic
9C23×C4⋊C4—21no—
10D4×C4⋊C4—24no—
11D4×C4⋊C4—24no—
12C2×#389C411yescharacteristic
13C2×#389C2211yescharacteristic
14C2×#385—21no—
15C2×#385—21no—
16C2×#384C411yescharacteristic
17C2×#380—21no—
18C2×C4×C4⋊C4C2211yescharacteristic
19#1797C2212yes—
20C4×C22⋊C8C2212yes—
21C2×D4⋊C8—24no—
22C2×#324—21no—
23C2×#324—21no—
24#1280—21no—
25#1280—21no—
26#1271—24no—
27C23×D4C2×C411yescharacteristic
28C24×C4—21no—
29C24×C4—21no—
30C24×C4—21no—
31C2×C4×D4—22no—
32C2×C4×D4—22no—
33C2×C4×D4—22no—
34C2×C4×D4—22no—
35C2×C4×D4—22no—
36C2×C4×D4—22no—
37C2×C4×D4—22no—
38C2×C4×D4—22no—
39C2×C4×D4—44no—
40C2×C4×D4—44no—
41C2×C4×D4C814yescomplemented
42C2×C4×D4D414yes—
43C22×C4⋊C4—28no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—44no—
49C22×C4⋊C4—42no—
50C22×C4⋊C4—42no—
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D411yescharacteristic
56C22×C22⋊C4—21no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—21no—
59C22×C42C2×C411yescharacteristic
60C22×C42C2×C411yescharacteristic
61C22×C42C2311yescharacteristic
62C2×C4⋊C8—22no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—22no—
65C2×C4×C8—22no—
66#389—22no—
67#389—22no—
68#389—22no—
69#389D414yes—
70#385—28no—
71#385—28no—
72#384—22no—
73#384—22no—
74#380—28no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—22no—
77C2×#95—21no—
78C2×#95—21no—
79C2×#95D411yescharacteristic
80C2×#95D411yescharacteristic
81#336—22no—
82#336—22no—
83D4⋊C8—48no—
84#324—24no—
85#324—42no—
86C25—21no—
87C22×D4—24no—
88C22×D4—24no—
89C22×D4—21no—
90C22×D4—21no—
91C22×D4—21no—
92C22×D4C22⋊C412yes—
93C22×D4C2×C812yes—
94C22×D4M1612yes—
95C23×C4—24no—
96C23×C4—24no—
97C23×C4—21no—
98C23×C4—24no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—24no—
102C23×C4—24no—
103C23×C4—42no—
104C23×C4—42no—
105C23×C4—42no—
106C23×C4—42no—
107C23×C4—42no—
108C23×C4—44no—
109C23×C4—44no—
110C23×C4—42no—
111C23×C4—41no—
112C23×C4—41no—
113C23×C4—42no—
114C23×C4—82no—
115C23×C4—82no—
116C23×C4C22⋊C411yescharacteristic
117C23×C4C22⋊C411yescharacteristic
118C23×C4C22⋊C411yescharacteristic
119C23×C4C22⋊C411yescharacteristic
120C23×C4C22×C411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C22×C8—42no—
124C22×C8—42no—
125C4×D4—28no—
126C4×D4—28no—
127C4×D4—28no—
128C4×D4—28no—
129C4×D4—28no—
130C4×D4—44no—
131C4×D4—48no—
132C4×D4—48no—
133C4×D4—44no—
134C4×D4—44no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—416no—
150C2×C4⋊C4—416no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—82no—
161C2×C4⋊C4C22⋊C412yes—
162C2×C4⋊C4C22⋊C412yes—
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4C2×C812yes—
165C2×C4⋊C4M1612yes—
166C2×C4⋊C4D812yes—
167C2×C4⋊C4SD1612yes—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—44no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—42no—
182C2×C42—42no—
183C2×C42C2×C812yes—
184C2×C42M1612yes—
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C4⋊C8—44no—
188C22⋊C8—24no—
189C22⋊C8—42no—
190C22⋊C8—42no—
191C22⋊C8—42no—
192C4×C8—44no—
193#95—24no—
194#95—24no—
195#95—22no—
196#95—22no—
197C24—24no—
198C24—24no—
199C24—22no—
200C24—22no—
201C24—22no—
202C24—42no—
203C24—42no—
204C24—41no—
205C24—41no—
206C24—41no—
207C24—41no—
208C24C2×C22⋊C411yescharacteristic
209C2×D4—24no—
210C2×D4—24no—
211C2×D4—28no—
212C2×D4—28no—
213C2×D4—28no—
214C2×D4—28no—
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4C22⋊C814yes—
221C2×D4#9914yes—
222C2×D4#10114yes—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—216no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—48no—
249C22×C4—48no—
250C22×C4—48no—
251C22×C4—48no—
252C22×C4—48no—
253C22×C4—48no—
254C22×C4—48no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—48no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—48no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—84no—
278C22×C4—84no—
279C22×C4—84no—
280C22×C4—84no—
281C22×C4—88no—
282C22×C4—88no—
283C22×C4—88no—
284C22×C4C22⋊C812yes—
285C22×C4C22⋊C812yes—
286C22×C4#10212yes—
287C22×C4#10212yes—
288C22×C4#10412yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4#12012yes—
292C22×C4#12012yes—
293C22×C4C22×C811yescharacteristic
294C22×C4C2×M1611yescharacteristic
295C22×C4C2×M1611yescharacteristic
296C22×C4C2×M1611yescharacteristic
297C22×C4C2×D811yescharacteristic
298C22×C4C2×SD1611yescharacteristic
299C22×C4AGL(1,ℤ/8)12yes—
300C2×C8—44no—
301C2×C8—82no—
302C2×C8—82no—
303C2×C8—82no—
304C2×C8—82no—
305C2×C8—82no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—44no—
313C4⋊C4—42no—
314C4⋊C4—48no—
315C4⋊C4—48no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—84no—
321C4⋊C4C22⋊C814yes—
322C22⋊C4—216no—
323C22⋊C4—48no—
324C22⋊C4—48no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—24no—
330C42—42no—
331C42—42no—
332C42—42no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—28no—
338C23—28no—
339C23—24no—
340C23—24no—
341C23—24no—
342C23—44no—
343C23—44no—
344C23—44no—
345C23—44no—
346C23—44no—
347C23—44no—
348C23—44no—
349C23—44no—
350C23—44no—
351C23—44no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23#37911yescharacteristic
361C23C2×C22⋊C811yescharacteristic
362C23#40711yescharacteristic
363C23C2×#10211yescharacteristic
364C23#41811yescharacteristic
365C23C2×#10411yescharacteristic
366C23#42111yescharacteristic
367D4—28no—
368D4—28no—
369D4—24no—
370D4—24no—
371D4—24no—
372D4#32418yescomplemented
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—432no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—88no—
421C2×C4—88no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—816no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4D4⋊C814yescomplemented
432C2×C4C2×C22⋊C811yescharacteristic
433C2×C4C2×C22⋊C811yescharacteristic
434C2×C4#40711yescharacteristic
435C2×C4#40711yescharacteristic
436C2×C4C2×#9912yes—
437C2×C4C2×#10112yes—
438C2×C4#44712yes—
439C2×C4#45012yes—
440C8—84no—
441C8—162no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—28no—
455C22—48no—
456C22—48no—
457C22—48no—
458C22—48no—
459C22—44no—
460C22—44no—
461C22—44no—
462C22—44no—
463C22—44no—
464C22—44no—
465C22C2×D4⋊C811yescharacteristic
466C22#147811yescharacteristic
467C22#148611yescharacteristic
468C22#149011yescharacteristic
469C22#177911yescharacteristic
470C22#178111yescharacteristic
471C22#178411yescharacteristic
472C22#178611yescharacteristic
473C22#178911yescharacteristic
474C22#179511yescharacteristic
475C22#179611yescharacteristic
476C4—21no—
477C4—21no—
478C4—24no—
479C4—22no—
480C4—22no—
481C4—44no—
482C4—42no—
483C4—42no—
484C4—42no—
485C4—81no—
486C4—82no—
487C4—82no—
488C4C2×#32412yes—
489C4#146312yes—
490C2—21no—
491C2—21no—
492C2—21no—
493C2—21no—
494C2—48no—
495C2#829911yescharacteristic
496C2#831011yescharacteristic
497C2#832411yescharacteristic
498C2#836411yescharacteristic
499C2#837211yescharacteristic
500C2#837511yescharacteristic
501C2#838811yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #8558C2semidirect
4. C4×#324C2semidirect
5. #7573C2semidirect
41. C2×C4×D4C8semidirect
372. D4#324semidirect
431. C2×C4D4⋊C8semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C4⋊C42
#85582
#83102
C4×#3242
#75732