Finite Group Databasebeta

← #188107

Subgroups — #188107

Order 512.

2596 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#553C211yesmaximal, characteristic
2#20726C212yesmaximal, complemented
3#17394C212yesmaximal, complemented
4#17364C212yesmaximal, complemented
5#8556C214yesmaximal, complemented
6#8549C214yesmaximal, complemented
7C2×#553—28no—
8C2×#553—24no—
9C2×#553—24no—
10C2×#553—24no—
11C2×#553—24no—
12C22×#126C2212yes—
13C22×#125C2211yescharacteristic
14C22×#123C412yes—
15C22×#121C411yescharacteristic
16C22×#121C411yescharacteristic
17C22×#121C2212yes—
18C22×C4×D4C412yes—
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×#117C411yescharacteristic
22C23×C4⋊C4C411yescharacteristic
23#2920C2214yes—
24#1821C2214yes—
25#1800—24no—
26#1783C2214yes—
27C4×#100—24no—
28#1470—28no—
29#1466—24no—
30#1463C2218yes—
31#1462C2218yes—
32#1460—24no—
33C23×D4C2×C412yes—
34C23×D4C2×C412yes—
35C23×D4C2×C411yescharacteristic
36C23×D4C2311yescharacteristic
37C24×C4C2×C412yes—
38C24×C4C2×C411yescharacteristic
39C24×C4C2×C411yescharacteristic
40C24×C4C2×C412yes—
41#553—232no—
42#553—232no—
43#553—232no—
44#553—232no—
45C2×#126—28no—
46C2×#126—28no—
47C2×#126—28no—
48C2×#125—22no—
49C2×#125—22no—
50C2×#125—22no—
51C2×#125—22no—
52C2×#125D418yes—
53C2×#123—28no—
54C2×#123—28no—
55C2×#123—24no—
56C2×#123—24no—
57C2×#121—22no—
58C2×#121—22no—
59C2×#121—22no—
60C2×#121—22no—
61C2×#121—28no—
62C2×#121—24no—
63C2×#121—24no—
64C2×#121—24no—
65C2×#121—24no—
66C2×#121—28no—
67C2×#121—28no—
68C2×C4×D4—28no—
69C2×C4×D4—28no—
70C2×C4×D4—24no—
71C2×C4×D4—24no—
72C2×C4×D4—24no—
73C2×C4×D4—24no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4—22no—
78C2×C4×D4D414yes—
79C2×C4×D4D414yes—
80C2×C4×D4D414yes—
81C2×C4×D4D414yes—
82C2×#117—24no—
83C2×#117—24no—
84C2×#117—22no—
85C2×#117—22no—
86C22×C4⋊C4—24no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4—22no—
90C22×C4⋊C4C2×C412yes—
91C22×C4⋊C4C2×C411yescharacteristic
92C22×C4⋊C4C2×C411yescharacteristic
93C22×C4⋊C4C2×C411yescharacteristic
94C22×C4⋊C4C2×C412yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C22⋊C4C2×C412yes—
98C22×C22⋊C4C2×C412yes—
99C22×C22⋊C4C2×C412yes—
100C22×C22⋊C4C2×C411yescharacteristic
101C22×C22⋊C4C2×C411yescharacteristic
102C22×C22⋊C4C2312yes—
103C22×C42C2×C412yes—
104C22×C42C2311yescharacteristic
105#432—24no—
106#432—24no—
107#431—24no—
108#431—24no—
109#423—24no—
110C2×#100—28no—
111#407C2318yes—
112C4×M16—24no—
113#343—28no—
114#324—216no—
115#323—216no—
116C25C22×C411yescharacteristic
117C25C22×C411yescharacteristic
118C25C22×C412yes—
119C22×D4—24no—
120C22×D4—24no—
121C22×D4—24no—
122C22×D4—24no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—24no—
126C22×D4—24no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—44no—
130C22×D4—44no—
131C22×D4—48no—
132C22×D4—84no—
133C22×D4C22⋊C412yes—
134C22×D4C22⋊C412yes—
135C22×D4C2×D414yes—
136C23×C4—24no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—44no—
149C23×C4—44no—
150C23×C4—84no—
151C23×C4—84no—
152C23×C4C22⋊C414yes—
153C23×C4C22⋊C414yes—
154C23×C4C22⋊C418yes—
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C412yes—
159C23×C4C22×C412yes—
160C23×C4C22×C412yes—
161C23×C4C22×C411yescharacteristic
162C23×C4C2411yescharacteristic
163C2×M16—28no—
164#126—264no—
165#125—28no—
166#125—28no—
167#125—28no—
168#125—28no—
169#123—232no—
170#123—232no—
171#121—264no—
172#121—216no—
173#121—216no—
174#121—216no—
175#121—216no—
176C4×D4—232no—
177C4×D4—232no—
178C4×D4—28no—
179C4×D4—28no—
180C4×D4—28no—
181C4×D4—28no—
182C4×D4—28no—
183C4×D4—28no—
184C4×D4—28no—
185C4×D4—28no—
186#117—216no—
187#117—216no—
188C2×C4⋊C4—216no—
189C2×C4⋊C4—216no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4C22⋊C412yes—
209C2×C4⋊C4C22⋊C412yes—
210C2×C4⋊C4C22⋊C414yes—
211C2×C4⋊C4C22⋊C414yes—
212C2×C4⋊C4C22⋊C418yes—
213C2×C4⋊C4C2×D414yes—
214C2×C4⋊C4C2×D418yes—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4C22⋊C412yes—
231C2×C22⋊C4C22⋊C412yes—
232C2×C22⋊C4C22⋊C412yes—
233C2×C22⋊C4C22⋊C412yes—
234C2×C22⋊C4C2×D418yes—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42C2×D414yes—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C4⋊C8—44no—
244C4⋊C8—44no—
245#100—416no—
246C22⋊C8—28no—
247C22⋊C8—28no—
248#97—44no—
249C4×C8—44no—
250C24—21no—
251C24—21no—
252C24—24no—
253C24—21no—
254C24—22no—
255C24—22no—
256C24—21no—
257C24—42no—
258C24—44no—
259C24—42no—
260C24—42no—
261C24—42no—
262C24—44no—
263C24—44no—
264C24—42no—
265C24C2×C22⋊C412yes—
266C24C2×C22⋊C414yes—
267C24C2×C22⋊C412yes—
268C24C23×C411yescharacteristic
269C2×D4—216no—
270C2×D4—28no—
271C2×D4—216no—
272C2×D4—28no—
273C2×D4—24no—
274C2×D4—24no—
275C2×D4—28no—
276C2×D4—44no—
277C2×D4—44no—
278C2×D4—48no—
279C2×D4—48no—
280C2×D4—48no—
281C2×D4—48no—
282C2×D4—416no—
283C2×D4—48no—
284C2×D4—416no—
285C2×D4—48no—
286C2×D4—88no—
287C2×D4—88no—
288C2×D4—88no—
289C22×C4—24no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—216no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—28no—
306C22×C4—28no—
307C22×C4—42no—
308C22×C4—44no—
309C22×C4—42no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—48no—
313C22×C4—48no—
314C22×C4—48no—
315C22×C4—48no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—88no—
323C22×C4—88no—
324C22×C4—84no—
325C22×C4—84no—
326C22×C4—88no—
327C22×C4—88no—
328C22×C4—88no—
329C22×C4C2×C22⋊C414yes—
330C22×C4C2×C22⋊C412yes—
331C22×C4C2×C22⋊C412yes—
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4C2×C22⋊C412yes—
334C22×C4C2×C22⋊C412yes—
335C22×C4C2×C22⋊C414yes—
336C22×C4C2×C22⋊C414yes—
337C22×C4C2×C22⋊C412yes—
338C22×C4C2×C22⋊C412yes—
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C22×D412yes—
343C22×C4C22×D412yes—
344C22×C4C22×D412yes—
345M16—416no—
346C2×C8—48no—
347C4⋊C4—216no—
348C4⋊C4—28no—
349C4⋊C4—216no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—28no—
357C4⋊C4—28no—
358C22⋊C4—216no—
359C22⋊C4—216no—
360C22⋊C4—216no—
361C22⋊C4—216no—
362C22⋊C4—24no—
363C22⋊C4—24no—
364C22⋊C4—24no—
365C22⋊C4—24no—
366C42—216no—
367C42#12014yes—
368C42#12014yes—
369C42#12014yes—
370C42#12014yes—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—24no—
375C23—24no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—24no—
380C23—24no—
381C23—28no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—416no—
387C23—42no—
388C23—44no—
389C23—42no—
390C23—42no—
391C23—44no—
392C23—44no—
393C23—44no—
394C23—42no—
395C23—42no—
396C23—44no—
397C23—44no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—44no—
402C23—42no—
403C23—44no—
404C23—42no—
405C23—88no—
406C23—816no—
407C23#41014yes—
408C23C22×C22⋊C411yescharacteristic
409C23C22×C22⋊C411yescharacteristic
410C23C22×C22⋊C411yescharacteristic
411D4—416no—
412D4—48no—
413D4—432no—
414D4—48no—
415D4—816no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—28no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—48no—
433C2×C4—48no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—48no—
445C2×C4—48no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—88no—
451C2×C4—88no—
452C2×C4—816no—
453C2×C4#37912yes—
454C2×C4#37912yes—
455C2×C4#37912yes—
456C2×C4#37912yes—
457C2×C4#41314yes—
458C2×C4C2×#12014yes—
459C8—88no—
460C22—28no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—24no—
468C22—44no—
469C22—44no—
470C22—48no—
471C22—48no—
472C22—42no—
473C22—42no—
474C22—42no—
475C22—44no—
476C22—42no—
477C22—44no—
478C22—44no—
479C22—42no—
480C22—84no—
481C22—88no—
482C22—88no—
483C22—84no—
484C22—84no—
485C22—88no—
486C22C2×#37911yescharacteristic
487C22C2×#41012yes—
488C22C2×#41312yes—
489C22#288812yes—
490C4—24no—
491C4—22no—
492C4—22no—
493C4—22no—
494C4—44no—
495C4—42no—
496C4—42no—
497C4—44no—
498C4—44no—
499C4#178514yes—
500C2—22no—
501C2—22no—
502C2—22no—
503C2—44no—
504C2—82no—
505C2—84no—
506C2#2058411yescharacteristic
507C2C2×#178511yescharacteristic
508C2#2059211yescharacteristic
subgroupG/H
center, socleC22C2×#379
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#553C2
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20726C2semidirect
3. #17394C2semidirect
4. #17364C2semidirect
5. #8556C2semidirect
6. #8549C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5532
#207262
#173942
#173642
#85562
#85492