Finite Group Databasebeta

← #200520

Subgroups — #200520

Order 512.

1566 conjugacy classes of subgroups.

The subgroup lattice has 471 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60230C211yesmaximal, characteristic
2C2×#2542C211yesmaximal, characteristic
3#20726C211yesmaximal, characteristic
4#17742C212yesmaximal, complemented
5#9403C214yesmaximal, complemented
6#9395C214yesmaximal, complemented
7#8098C212yesmaximal, complemented
8C22×C4⋊Q8C2211yescharacteristic
9C22×C22⋊Q8—22no—
10C22×C4×D4C2211yescharacteristic
11#2975C2212yes—
12#2974—24no—
13#2973—22no—
14#2972C2212yes—
15D4×M16—24no—
16#2920C2212yes—
17#2920C2212yes—
18C2×#407—22no—
19#2542—22no—
20#2542—21no—
21#2542—21no—
22#2542C414yes—
23#2542C2212yes—
24#2542C2212yes—
25C2×#399C414yes—
26C2×#390—21no—
27C2×#384—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29#1821—21no—
30#1800C2212yes—
31#1800C2212yes—
32#1783—21no—
33C4×#100—21no—
34C4×#100—21no—
35#1556—24no—
36#1549C2218yescomplemented
37#1546—24no—
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×C22⋊Q8—24no—
52C2×C22⋊Q8—24no—
53C2×C22⋊Q8—24no—
54C2×C22⋊Q8—24no—
55C2×C22⋊Q8—416no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4C2×C412yes—
61C2×C4×D4D412yes—
62C2×C4×D4D411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×C4×D4C2311yescharacteristic
65C2×C4×D4C2311yescharacteristic
66C22×C4⋊C4—21no—
67C22×C4⋊C4—24no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4C2×C412yes—
70C22×C4⋊C4D411yescharacteristic
71C22×C22⋊C4—24no—
72C22×C22⋊C4C2×C412yes—
73C22×C42—21no—
74C22×C42C2311yescharacteristic
75#436—28no—
76#435—216no—
77C8×D4—28no—
78#432—22no—
79#432—22no—
80#432—22no—
81#432D414yes—
82#431—22no—
83#431—22no—
84#431—22no—
85#431C2314yes—
86#423D414yes—
87#423C2314yes—
88C2×#100—22no—
89C2×#100—22no—
90#407—24no—
91#407—24no—
92#407—22no—
93#407—24no—
94#407—22no—
95C2×C22⋊C8—24no—
96C2×C22⋊C8—24no—
97C4×M16—22no—
98C4×M16—22no—
99#399—24no—
100#399—24no—
101#399C2×C418yes—
102#399C2×C418yes—
103#390—24no—
104#390—22no—
105#390—22no—
106#384—24no—
107#384—22no—
108#384—22no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4C2×C412yes—
112C4×C4⋊C4C2312yes—
113C2×#95C2×C412yes—
114#343—22no—
115#343—22no—
116#332—216no—
117C25C22⋊C412yes—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—48no—
121C22×Q8—41no—
122C22×Q8—41no—
123C22×Q8—41no—
124C22×Q8—41no—
125C22×D4—24no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—42no—
129C22×D4C22⋊C412yes—
130C22×D4C2×D411yescharacteristic
131C22×D4C2×D411yescharacteristic
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—48no—
139C23×C4C22⋊C412yes—
140C23×C4C22⋊C412yes—
141C23×C4C22⋊C412yes—
142C23×C4C22×C411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C2×M16—24no—
148C2×M16—28no—
149C2×M16—24no—
150C2×M16—88no—
151C22×C8—28no—
152C4⋊Q8—24no—
153C4⋊Q8—24no—
154C4⋊Q8—24no—
155C4⋊Q8—24no—
156C4⋊Q8—42no—
157C4⋊Q8—42no—
158C4⋊Q8—42no—
159C4⋊Q8—42no—
160C22⋊Q8—416no—
161C22⋊Q8—416no—
162C22⋊Q8—416no—
163C22⋊Q8—416no—
164C4×D4—24no—
165C4×D4—24no—
166C4×D4—24no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—22no—
171C4×D4—24no—
172C4×D4—22no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4—22no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—416no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22×C414yes—
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—416no—
200C2×C22⋊C4C22⋊C412yes—
201C2×C22⋊C4C22⋊C412yes—
202C2×C22⋊C4C22×C412yes—
203C2×C22⋊C4C22×C412yes—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—44no—
207C2×C42C22×C411yescharacteristic
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42#4112yes—
213C2×C42#4112yes—
214C2×C42C2411yescharacteristic
215C4⋊C8—24no—
216C4⋊C8—24no—
217C4⋊C8—24no—
218C4⋊C8C2×D418yes—
219#100—44no—
220#100—44no—
221C22⋊C8—28no—
222C22⋊C8—28no—
223C22⋊C8—24no—
224C22⋊C8—24no—
225C22⋊C8—42no—
226C22⋊C8—42no—
227#97—24no—
228#97—42no—
229C4×C8—24no—
230C4×C8—42no—
231#95—22no—
232#95C22×C414yes—
233C24—24no—
234C24—22no—
235C24—22no—
236C24—42no—
237C24—44no—
238C24—44no—
239C24C2×C22⋊C411yescharacteristic
240C24C2×C22⋊C412yes—
241C24C2×C22⋊C412yes—
242C2×Q8—48no—
243C2×Q8—48no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—84no—
249C2×Q8—84no—
250C2×Q8—84no—
251C2×Q8—84no—
252C2×D4—24no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—24no—
259C2×D4—48no—
260C2×D4—48no—
261C2×D4—42no—
262C2×D4#13714yes—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—28no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—42no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—48no—
287C22×C4—44no—
288C22×C4—48no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—84no—
292C22×C4—84no—
293C22×C4—84no—
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C4×D412yes—
300C22×C4C4×D412yes—
301C22×C4#12011yescharacteristic
302C22×C4#12011yescharacteristic
303C22×C4#12112yes—
304C22×C4#12112yes—
305C22×C4#12311yescharacteristic
306C22×C4#12311yescharacteristic
307C22×C4C23×C411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D411yescharacteristic
310M16—48no—
311M16—44no—
312M16—44no—
313M16—88no—
314C2×C8—28no—
315C2×C8—44no—
316C2×C8—88no—
317C4⋊C4—28no—
318C4⋊C4—28no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—24no—
322C4⋊C4—28no—
323C4⋊C4—416no—
324C4⋊C4—416no—
325C4⋊C4—44no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C22⋊C4—24no—
329C22⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—24no—
332C22⋊C4—416no—
333C22⋊C4—416no—
334C42—44no—
335C42—44no—
336C42C4×D412yes—
337C42C4×D412yes—
338C42#12012yes—
339C42#12112yes—
340C42#12112yes—
341C42#12312yes—
342C42C22×D412yes—
343C42C2×#4112yes—
344C23—21no—
345C23—21no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—44no—
354C23—44no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—416no—
360C23—44no—
361C23—44no—
362C23—82no—
363C23—82no—
364C23—88no—
365C23#38611yescharacteristic
366C23#38611yescharacteristic
367C23#41914yes—
368C23C22×C22⋊C411yescharacteristic
369Q8—84no—
370Q8—84no—
371Q8—84no—
372Q8—84no—
373D4—28no—
374D4—48no—
375D4—44no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—24no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—28no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—416no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—48no—
403C2×C4—44no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—84no—
410C2×C4—84no—
411C2×C4#38612yes—
412C2×C4#38611yescharacteristic
413C2×C4#38612yes—
414C2×C4#38611yescharacteristic
415C2×C4#45314yes—
416C2×C4C2×C4×D412yes—
417C2×C4C2×#12011yescharacteristic
418C2×C4C2×#12112yes—
419C2×C4C2×#12311yescharacteristic
420C2×C4C2×#13712yes—
421C2×C4#57512yes—
422C8—48no—
423C8—84no—
424C22—24no—
425C22—24no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—24no—
433C22—44no—
434C22—44no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22—82no—
441C22—88no—
442C22—88no—
443C22—82no—
444C22#188712yes—
445C22#191512yes—
446C22C2×#38611yescharacteristic
447C22C2×#41911yescharacteristic
448C22#289611yescharacteristic
449C4—22no—
450C4—22no—
451C4—21no—
452C4—21no—
453C4—22no—
454C4—48no—
455C4—42no—
456C4—42no—
457C4—84no—
458C4#188712yes—
459C4C2×#45312yes—
460C4#305614yes—
461C2—21no—
462C2—21no—
463C2—22no—
464C2—22no—
465C2—44no—
466C2—82no—
467C2C2×#188711yescharacteristic
468C2#2097211yescharacteristic
469C2C2×#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. #9403C2semidirect
6. #9395C2semidirect
7. #8098C2semidirect
36. #1549C22semidirect
38. #1303C22semidirect

Maximal subgroups

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

subgroupindex
#602302
C2×#25422
#207262
#177422
#94032
#93952
#80982