Finite Group Databasebeta

← #200518

Subgroups — #200518

Order 512.

1446 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60230C211yesmaximal, characteristic
2C2×#2547C211yesmaximal, characteristic
3#20726C211yesmaximal, characteristic
4#17742C212yesmaximal, complemented
5#9401C214yesmaximal, complemented
6#9393C214yesmaximal, complemented
7#8098C212yesmaximal, complemented
8C22×C4⋊Q8C2211yescharacteristic
9C22×#124—21no—
10C22×C4×D4C2211yescharacteristic
11#2975C2212yes—
12#2974—24no—
13#2973—22no—
14#2972C2212yes—
15D4×M16—24no—
16#2920C2212yes—
17#2920C2212yes—
18C2×#407—22no—
19#2547—22no—
20#2547—21no—
21#2547—21no—
22#2547C414yes—
23#2547C2212yes—
24#2547C2212yes—
25C2×#397C414yes—
26C2×#391—21no—
27C2×#380—22no—
28C2×C4×C4⋊C4C2211yescharacteristic
29#1821—21no—
30#1800C2212yes—
31#1800C2212yes—
32#1783—21no—
33C4×#100—21no—
34C4×#100—21no—
35#1556—24no—
36#1549—24no—
37#1546C2218yescomplemented
38#1303C2218yescomplemented
39C23×Q8—21no—
40C23×D4D411yescharacteristic
41C24×C4D412yes—
42C22×M16—24no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8C2×C412yes—
50C2×C4⋊Q8C2312yes—
51C2×#124—24no—
52C2×#124—22no—
53C2×#124—22no—
54C2×#124—48no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4C2×C412yes—
60C2×C4×D4D412yes—
61C2×C4×D4D411yescharacteristic
62C2×C4×D4D411yescharacteristic
63C2×C4×D4C2311yescharacteristic
64C2×C4×D4C2311yescharacteristic
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4C2×C412yes—
68C22×C4⋊C4D411yescharacteristic
69C22×C22⋊C4—24no—
70C22×C22⋊C4C2×C412yes—
71C22×C42—21no—
72C22×C42C2311yescharacteristic
73#436—28no—
74#435—216no—
75C8×D4—28no—
76#432—22no—
77#432—22no—
78#432—22no—
79#432D414yes—
80#431—22no—
81#431—22no—
82#431—22no—
83#431C2314yes—
84#423D414yes—
85#423C2314yes—
86C2×#100—22no—
87C2×#100—22no—
88#407—24no—
89#407—24no—
90#407—22no—
91#407—24no—
92#407—22no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—24no—
95C4×M16—22no—
96C4×M16—22no—
97#397—24no—
98#397—24no—
99#397C2×C418yes—
100#397C2×C418yes—
101#391—24no—
102#391—22no—
103#391—22no—
104#380—24no—
105#380—24no—
106#380—24no—
107#380—24no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4C2×C412yes—
111C4×C4⋊C4C2312yes—
112C2×#95—24no—
113C2×#95C2×C412yes—
114#343—22no—
115#343—22no—
116#332—216no—
117C25C22⋊C412yes—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—44no—
121C22×Q8—44no—
122C22×Q8—42no—
123C22×Q8—42no—
124C22×D4—24no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—42no—
128C22×D4C22⋊C412yes—
129C22×D4C2×D411yescharacteristic
130C22×D4C2×D411yescharacteristic
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—48no—
138C23×C4C22⋊C412yes—
139C23×C4C22⋊C412yes—
140C23×C4C22⋊C412yes—
141C23×C4C22×C411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D412yes—
145C23×C4C2×D412yes—
146C2×M16—24no—
147C2×M16—28no—
148C2×M16—24no—
149C2×M16—88no—
150C22×C8—28no—
151C4⋊Q8—24no—
152C4⋊Q8—24no—
153C4⋊Q8—24no—
154C4⋊Q8—24no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157C4⋊Q8—42no—
158C4⋊Q8—42no—
159#124—416no—
160#124—48no—
161#124—48no—
162C4×D4—24no—
163C4×D4—24no—
164C4×D4—24no—
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—24no—
170C4×D4—22no—
171C4×D4—22no—
172C4×D4—22no—
173C4×D4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22×C414yes—
190C2×C4⋊C4C2×D411yescharacteristic
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—28no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—416no—
197C2×C22⋊C4C22⋊C412yes—
198C2×C22⋊C4C22⋊C412yes—
199C2×C22⋊C4C22×C412yes—
200C2×C22⋊C4C22×C412yes—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—44no—
204C2×C42C22×C411yescharacteristic
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42#4112yes—
210C2×C42#4112yes—
211C2×C42C2411yescharacteristic
212C4⋊C8—24no—
213C4⋊C8—24no—
214C4⋊C8—24no—
215C4⋊C8C2×D418yes—
216#100—44no—
217#100—44no—
218C22⋊C8—28no—
219C22⋊C8—28no—
220C22⋊C8—24no—
221C22⋊C8—24no—
222C22⋊C8—42no—
223C22⋊C8—42no—
224#97—24no—
225#97—42no—
226C4×C8—24no—
227C4×C8—42no—
228#95—28no—
229#95—28no—
230#95—22no—
231#95C22×C414yes—
232C24—24no—
233C24—22no—
234C24—22no—
235C24—42no—
236C24—44no—
237C24—44no—
238C24C2×C22⋊C411yescharacteristic
239C24C2×C22⋊C412yes—
240C24C2×C22⋊C412yes—
241C2×Q8—44no—
242C2×Q8—44no—
243C2×Q8—44no—
244C2×Q8—44no—
245C2×Q8—44no—
246C2×Q8—44no—
247C2×Q8—88no—
248C2×Q8—88no—
249C2×D4—24no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—24no—
256C2×D4—48no—
257C2×D4—48no—
258C2×D4—42no—
259C2×D4AGL(1,ℤ/8)14yes—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—42no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—48no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—84no—
290C22×C4—84no—
291C22×C4—84no—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C4×D412yes—
298C22×C4C4×D412yes—
299C22×C4#12011yescharacteristic
300C22×C4#12011yescharacteristic
301C22×C4#12112yes—
302C22×C4#12112yes—
303C22×C4#12311yescharacteristic
304C22×C4#12311yescharacteristic
305C22×C4C23×C411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D411yescharacteristic
308M16—48no—
309M16—44no—
310M16—44no—
311M16—88no—
312C2×C8—28no—
313C2×C8—44no—
314C2×C8—88no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—24no—
318C4⋊C4—28no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—44no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C22⋊C4—24no—
327C22⋊C4—24no—
328C22⋊C4—24no—
329C22⋊C4—24no—
330C22⋊C4—416no—
331C22⋊C4—416no—
332C42—44no—
333C42—44no—
334C42C4×D412yes—
335C42C4×D412yes—
336C42#12012yes—
337C42#12112yes—
338C42#12112yes—
339C42#12312yes—
340C42C22×D412yes—
341C42C2×#4112yes—
342C23—21no—
343C23—21no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—44no—
352C23—44no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—416no—
358C23—44no—
359C23—44no—
360C23—82no—
361C23—82no—
362C23—88no—
363C23#38611yescharacteristic
364C23#38611yescharacteristic
365C23#41814yes—
366C23C22×C22⋊C411yescharacteristic
367Q8—88no—
368Q8—88no—
369D4—28no—
370D4—48no—
371D4—44no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—24no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—88no—
406C2×C4—84no—
407C2×C4—84no—
408C2×C4#38612yes—
409C2×C4#38611yescharacteristic
410C2×C4#38612yes—
411C2×C4#38611yescharacteristic
412C2×C4#45514yes—
413C2×C4C2×C4×D412yes—
414C2×C4C2×#12011yescharacteristic
415C2×C4C2×#12112yes—
416C2×C4C2×#12311yescharacteristic
417C2×C4C2×AGL(1,ℤ/8)12yes—
418C2×C4#57512yes—
419C8—48no—
420C8—84no—
421C22—24no—
422C22—24no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—24no—
430C22—44no—
431C22—44no—
432C22—42no—
433C22—42no—
434C22—42no—
435C22—42no—
436C22—42no—
437C22—82no—
438C22—88no—
439C22—88no—
440C22—82no—
441C22#188512yes—
442C22#191512yes—
443C22C2×#38611yescharacteristic
444C22C2×#41811yescharacteristic
445C22#289611yescharacteristic
446C4—22no—
447C4—22no—
448C4—21no—
449C4—21no—
450C4—22no—
451C4—42no—
452C4—42no—
453C4—84no—
454C4—84no—
455C4#188512yes—
456C4C2×#45512yes—
457C4#305514yes—
458C2—21no—
459C2—21no—
460C2—22no—
461C2—22no—
462C2—44no—
463C2—82no—
464C2C2×#188511yescharacteristic
465C2#2097311yescharacteristic
466C2C2×#191511yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #17742C2semidirect
5. #9401C2semidirect
6. #9393C2semidirect
7. #8098C2semidirect
37. #1546C22semidirect
38. #1303C22semidirect

Maximal subgroups

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

subgroupindex
#602302
C2×#25472
#207262
#177422
#94012
#93932
#80982