Finite Group Databasebeta

← #190449

Subgroups — #190449

Order 512.

1838 conjugacy classes of subgroups.

The subgroup lattice has 515 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C4⋊C4C211yesmaximal, characteristic
2#20714C212yesmaximal
3#8698C212yesmaximal
4#8560C214yesmaximal, complemented
5#8299C212yesmaximal
6C4×#324C214yesmaximal, complemented
7C23×C4⋊C4C411yescharacteristic
8C23×C4⋊C4C2211yescharacteristic
9C23×C42C411yescharacteristic
10C23×C42C411yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#431—22no—
13C2×C4×C4⋊C4—22no—
14C2×C4×C4⋊C4—22no—
15C2×C4×C4⋊C4—22no—
16C2×C4×C4⋊C4—22no—
17C2×C4×C4⋊C4C412yes—
18C2×C4×C4⋊C4C412yes—
19C2×C4×C4⋊C4C2212yes—
20C2×C4×C4⋊C4C2212yes—
21C22×#95C411yescharacteristic
22C22×#95C2211yescharacteristic
23#1817—22no—
24#1797—22no—
25#1797C2214yes—
26#1779C2214yes—
27C4×C22⋊C8—22no—
28C4×C22⋊C8C2214yes—
29C2×#324C2218yes—
30#1280C2218yes—
31C24×C4C2×C411yescharacteristic
32C24×C4C2×C411yescharacteristic
33C24×C4C2×C411yescharacteristic
34C24×C4C2×C411yescharacteristic
35C24×C4C2×C411yescharacteristic
36C24×C4C2×C411yescharacteristic
37C24×C4C2311yescharacteristic
38C22×C4⋊C4—22no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—24no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—28no—
44C22×C4⋊C4C818yescomplemented
45C22×C4⋊C4C2×C412yes—
46C22×C4⋊C4C2×C412yes—
47C22×C4⋊C4C2×C412yes—
48C22×C4⋊C4C2312yes—
49C22×C42—21no—
50C22×C42—21no—
51C22×C42—21no—
52C22×C42—21no—
53C22×C42—28no—
54C22×C42—21no—
55C22×C42—21no—
56C22×C42—21no—
57C22×C42—21no—
58C22×C42—44no—
59C22×C42—44no—
60C22×C42C2×C412yes—
61C22×C42C2×C412yes—
62C22×C42C2×C412yes—
63C22×C42C2×C412yes—
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42D412yes—
67C22×C42D412yes—
68C22×C42D412yes—
69C22×C42D412yes—
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72#431—48no—
73C2×C4⋊C8—24no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8C2318yes—
77C2×C4×C8—24no—
78C4×C4⋊C4—24no—
79C4×C4⋊C4—24no—
80C4×C4⋊C4—24no—
81C4×C4⋊C4—24no—
82C4×C4⋊C4—24no—
83C4×C4⋊C4—24no—
84C4×C4⋊C4—24no—
85C4×C4⋊C4—24no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95C2×C412yes—
91C2×#95C2×C412yes—
92C2×#95C2×C412yes—
93C2×#95C2312yes—
94#336—24no—
95#336—24no—
96#324—216no—
97C25C22×C411yescharacteristic
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—22no—
101C23×C4—24no—
102C23×C4—24no—
103C23×C4—24no—
104C23×C4—24no—
105C23×C4—24no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—22no—
113C23×C4—24no—
114C23×C4—24no—
115C23×C4—24no—
116C23×C4—42no—
117C23×C4—42no—
118C23×C4—42no—
119C23×C4—42no—
120C23×C4—42no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4—42no—
124C23×C4—42no—
125C23×C4—42no—
126C23×C4C22⋊C411yescharacteristic
127C23×C4C22⋊C411yescharacteristic
128C23×C4C22⋊C412yes—
129C23×C4C22⋊C411yescharacteristic
130C23×C4C22⋊C411yescharacteristic
131C23×C4C22⋊C412yes—
132C23×C4C2×C814yes—
133C23×C4C2×C814yes—
134C23×C4C2×C814yes—
135C23×C4M1614yes—
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C411yescharacteristic
138C23×C4C22×C412yes—
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C412yes—
141C23×C4C22×C412yes—
142C23×C4C22×C412yes—
143C23×C4C22×C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2411yescharacteristic
148C22×C8—28no—
149C2×C4⋊C4—216no—
150C2×C4⋊C4—216no—
151C2×C4⋊C4—216no—
152C2×C4⋊C4—216no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4C2×C818yes—
163C2×C4⋊C4C2×C818yes—
164C2×C4⋊C4#4118yes—
165C2×C42—28no—
166C2×C42—28no—
167C2×C42—28no—
168C2×C42—28no—
169C2×C42—28no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—24no—
175C2×C42—24no—
176C2×C42—24no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—42no—
182C2×C42—42no—
183C2×C42—44no—
184C2×C42—44no—
185C2×C42—42no—
186C2×C42—42no—
187C2×C42—44no—
188C2×C42—44no—
189C2×C42—44no—
190C2×C42—44no—
191C2×C42—44no—
192C2×C42—44no—
193C2×C42—44no—
194C2×C42C22⋊C412yes—
195C2×C42C22⋊C412yes—
196C2×C42C22⋊C412yes—
197C2×C42C22⋊C412yes—
198C2×C42M1618yes—
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C4⋊C8—48no—
204C22⋊C8—216no—
205C22⋊C8—48no—
206C22⋊C8—48no—
207C4×C8—48no—
208#95—24no—
209#95—24no—
210#95—24no—
211#95—24no—
212C24—22no—
213C24—22no—
214C24—22no—
215C24—22no—
216C24—22no—
217C24—22no—
218C24C2×C22⋊C411yescharacteristic
219C24C2×C22⋊C411yescharacteristic
220C24C22×C812yes—
221C24C2×M1612yes—
222C24C23×C411yescharacteristic
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—216no—
228C22×C4—232no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—216no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—42no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—48no—
278C22×C4—48no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—416no—
287C22×C4—416no—
288C22×C4—416no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—48no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—48no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4C22⋊C818yes—
308C22×C4C22⋊C818yes—
309C22×C4#9914yes—
310C22×C4#10114yes—
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4#11714yes—
319C22×C4#11714yes—
320C22×C4C22×C814yes—
321C22×C4C22×C814yes—
322C22×C4C22×C812yes—
323C22×C4C22×C812yes—
324C22×C4C2×M1614yes—
325C22×C4C2×M1614yes—
326C22×C4C2×M1614yes—
327C22×C4C2×M1612yes—
328C22×C4C2×M1612yes—
329C22×C4C22×D412yes—
330C22×C4C2×#4114yes—
331C2×C8—48no—
332C2×C8—48no—
333C2×C8—48no—
334C4⋊C4—216no—
335C4⋊C4—216no—
336C4⋊C4—216no—
337C4⋊C4—216no—
338C42—28no—
339C42—28no—
340C42—28no—
341C42—28no—
342C42—24no—
343C42—24no—
344C42—24no—
345C42—24no—
346C42—42no—
347C42—42no—
348C42—42no—
349C42—42no—
350C42—44no—
351C42—44no—
352C42—44no—
353C42—44no—
354C23—28no—
355C23—28no—
356C23—28no—
357C23—28no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—24no—
369C23—24no—
370C23—24no—
371C23—24no—
372C23—24no—
373C23—24no—
374C23—21no—
375C23—21no—
376C23C2×C22⋊C812yes—
377C23C2×C22⋊C812yes—
378C23C2×#9912yes—
379C23#41012yes—
380C23C2×#10112yes—
381C23#41312yes—
382C23C22×C22⋊C411yescharacteristic
383C23C2×#11712yes—
384C23C23×C811yescharacteristic
385C23C22×M1611yescharacteristic
386C23C22×M1612yes—
387C2×C4—28no—
388C2×C4—28no—
389C2×C4—28no—
390C2×C4—216no—
391C2×C4—216no—
392C2×C4—24no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—24no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—216no—
410C2×C4—24no—
411C2×C4—28no—
412C2×C4—28no—
413C2×C4—28no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—48no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—48no—
427C2×C4—48no—
428C2×C4—44no—
429C2×C4—48no—
430C2×C4—44no—
431C2×C4—48no—
432C2×C4—48no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—48no—
436C2×C4—48no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—48no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4C2×C22⋊C814yes—
448C2×C4C2×C22⋊C814yes—
449C2×C4#40712yes—
450C2×C4#40712yes—
451C2×C4C2×#9912yes—
452C2×C4C2×#9912yes—
453C2×C4C2×#10112yes—
454C2×C4C2×#10112yes—
455C2×C4#43118yescomplemented
456C2×C4#52514yes—
457C8—88no—
458C22—216no—
459C22—216no—
460C22—21no—
461C22—21no—
462C22—22no—
463C22—21no—
464C22—21no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22#145914yes—
481C22#146114yes—
482C22#228011yescharacteristic
483C22C22×C22⋊C811yescharacteristic
484C22C2×#40711yescharacteristic
485C22C22×#9911yescharacteristic
486C22C2×#41011yescharacteristic
487C22C22×#10111yescharacteristic
488C22C2×#41311yescharacteristic
489C22C2×#43112yes—
490C22#292012yes—
491C4—24no—
492C4—24no—
493C4—22no—
494C4—22no—
495C4—22no—
496C4—28no—
497C4—24no—
498C4—44no—
499C4—44no—
500C4—42no—
501C4—44no—
502C4#146314yes—
503C2—22no—
504C2—22no—
505C2—24no—
506C2—24no—
507C2C2×#145911yescharacteristic
508C2C2×#146111yescharacteristic
509C2C2×#146311yescharacteristic
510C2#1735711yescharacteristic
511C2#2071411yescharacteristic
512C2#2072211yescharacteristic
513C2#2072711yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC25C22×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8560C2semidirect
6. C4×#324C2semidirect
44. C22×C4⋊C4C8semidirect
455. C2×C4#431semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C4⋊C42
#207142
#86982
#85602
#82992
C4×#3242