Finite Group Databasebeta

← #201204

Subgroups — #201204

Order 512.

1554 conjugacy classes of subgroups.

The subgroup lattice has 463 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60230C211yesmaximal, characteristic
2C2×#2542C211yesmaximal, characteristic
3#17742C212yesmaximal, complemented
4#17364C211yesmaximal, characteristic
5#9455C218yesmaximal, complemented
6#7950C212yesmaximal, complemented
7C22×C4⋊Q8C2211yescharacteristic
8C22×C22⋊Q8—22no—
9C22×C4×D4C2211yescharacteristic
10#2975—21no—
11#2974—24no—
12#2973C2214yes—
13#2972—21no—
14D4×M16—24no—
15#2920C2212yes—
16#2920C2212yes—
17C2×#407—22no—
18#2542—21no—
19#2542—21no—
20#2542—21no—
21#2542—21no—
22#2542C414yes—
23#2542C2214yes—
24C2×#399—22no—
25C2×#390C412yes—
26C2×#384C412yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28#1556C2218yescomplemented
29#1549—24no—
30#1546—24no—
31#1466C2214yes—
32#1462—22no—
33#1460—21no—
34#1460—21no—
35#1285C2218yescomplemented
36C23×Q8—21no—
37C23×D4C2×C411yescharacteristic
38C24×C4D412yes—
39C22×M16—24no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8C2×C412yes—
47C2×C4⋊Q8C2312yes—
48C2×C22⋊Q8—24no—
49C2×C22⋊Q8—24no—
50C2×C22⋊Q8—24no—
51C2×C22⋊Q8—24no—
52C2×C22⋊Q8—416no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4C2×C412yes—
58C2×C4×D4D411yescharacteristic
59C2×C4×D4D411yescharacteristic
60C2×C4×D4D411yescharacteristic
61C2×C4×D4D411yescharacteristic
62C2×C4×D4C2312yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2×C412yes—
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C22⋊C4—24no—
69C22×C22⋊C4D412yes—
70C22×C42C2×C412yes—
71C22×C42C2311yescharacteristic
72#436—28no—
73#435—216no—
74C8×D4—28no—
75#432—22no—
76#432—22no—
77#432C2314yes—
78#432C2314yes—
79#431—22no—
80#431—22no—
81#431—22no—
82#431D414yes—
83#423—22no—
84#423D414yes—
85#407—24no—
86#407—24no—
87#407—22no—
88#407—24no—
89#407—22no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—24no—
92C4×M16—22no—
93C4×M16—22no—
94#399—24no—
95#399—24no—
96#399—24no—
97#399—24no—
98#390—22no—
99#390—22no—
100#390C2×C418yes—
101#384—22no—
102#384—22no—
103#384C2×C418yes—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4C2×C412yes—
107C4×C4⋊C4C2312yes—
108C2×#95—21no—
109#332—216no—
110#323—24no—
111#323—24no—
112C25C22⋊C412yes—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—48no—
116C22×Q8—41no—
117C22×Q8—41no—
118C22×Q8—41no—
119C22×Q8—41no—
120C22×D4—24no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—42no—
124C22×D4C22⋊C411yescharacteristic
125C22×D4C22⋊C411yescharacteristic
126C22×D4C22×C412yes—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—48no—
134C23×C4C22⋊C412yes—
135C23×C4C22⋊C412yes—
136C23×C4C22⋊C412yes—
137C23×C4C22×C411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C2×M16—24no—
143C2×M16—28no—
144C2×M16—42no—
145C2×M16—88no—
146C22×C8—28no—
147C4⋊Q8—24no—
148C4⋊Q8—24no—
149C4⋊Q8—24no—
150C4⋊Q8—24no—
151C4⋊Q8—42no—
152C4⋊Q8—42no—
153C4⋊Q8—42no—
154C4⋊Q8—42no—
155C22⋊Q8—416no—
156C22⋊Q8—416no—
157C22⋊Q8—416no—
158C22⋊Q8—416no—
159C4×D4—24no—
160C4×D4—24no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—22no—
164C4×D4—22no—
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—24no—
170C4×D4—24no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—416no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22×C414yes—
189C2×C4⋊C4C22×C412yes—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—28no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—416no—
195C2×C22⋊C4C22⋊C412yes—
196C2×C22⋊C4C22⋊C412yes—
197C2×C22⋊C4C2×D412yes—
198C2×C22⋊C4C2×D412yes—
199C2×C42—22no—
200C2×C42—44no—
201C2×C42C22×C414yes—
202C2×C42C22×C411yescharacteristic
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42#4112yes—
208C2×C42#4112yes—
209C2×C42C2411yescharacteristic
210C4⋊C8—24no—
211C4⋊C8—24no—
212C4⋊C8—42no—
213C4⋊C8C2×D418yes—
214C22⋊C8—28no—
215C22⋊C8—28no—
216C22⋊C8—24no—
217C22⋊C8—24no—
218C22⋊C8—24no—
219C22⋊C8—42no—
220#97—24no—
221#97—24no—
222C4×C8—24no—
223C4×C8—42no—
224#95—22no—
225#95—22no—
226C24—24no—
227C24—22no—
228C24—22no—
229C24—42no—
230C24—44no—
231C24—44no—
232C24C2×C22⋊C411yescharacteristic
233C24C2×C22⋊C412yes—
234C24C2×C22⋊C412yes—
235C2×Q8—48no—
236C2×Q8—48no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—84no—
242C2×Q8—84no—
243C2×Q8—84no—
244C2×Q8—84no—
245C2×D4—24no—
246C2×D4—24no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—48no—
253C2×D4—48no—
254C2×D4—42no—
255C2×D4#10114yes—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—28no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
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—24no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—48no—
280C22×C4—44no—
281C22×C4—48no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—84no—
285C22×C4—84no—
286C22×C4—84no—
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C4×D412yes—
293C22×C4C4×D412yes—
294C22×C4#12011yescharacteristic
295C22×C4#12011yescharacteristic
296C22×C4#12112yes—
297C22×C4#12112yes—
298C22×C4#12311yescharacteristic
299C22×C4#12311yescharacteristic
300C22×C4C23×C411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D411yescharacteristic
303M16—48no—
304M16—88no—
305M16—84no—
306C2×C8—28no—
307C2×C8—44no—
308C2×C8—88no—
309C4⋊C4—28no—
310C4⋊C4—28no—
311C4⋊C4—24no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—28no—
315C4⋊C4—416no—
316C4⋊C4—416no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C22⋊C4—24no—
321C22⋊C4—24no—
322C22⋊C4—24no—
323C22⋊C4—24no—
324C22⋊C4—416no—
325C22⋊C4—416no—
326C42—44no—
327C42—44no—
328C42C4×D412yes—
329C42C4×D412yes—
330C42#12012yes—
331C42#12112yes—
332C42#12112yes—
333C42#12312yes—
334C42C22×D412yes—
335C42C2×#4112yes—
336C23—21no—
337C23—21no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—44no—
346C23—44no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—416no—
352C23—44no—
353C23—44no—
354C23—82no—
355C23—82no—
356C23—88no—
357C23#38611yescharacteristic
358C23#38611yescharacteristic
359C23C2×#10114yes—
360C23C22×C22⋊C411yescharacteristic
361Q8—84no—
362Q8—84no—
363Q8—84no—
364Q8—84no—
365D4—28no—
366D4—48no—
367D4—44no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—28no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—416no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—48no—
395C2×C4—44no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—84no—
402C2×C4—84no—
403C2×C4#38612yes—
404C2×C4#38611yescharacteristic
405C2×C4#38611yescharacteristic
406C2×C4#38612yes—
407C2×C4C2×#10112yes—
408C2×C4#41312yes—
409C2×C4#45314yes—
410C2×C4C2×C4×D412yes—
411C2×C4C2×#12011yescharacteristic
412C2×C4C2×#12112yes—
413C2×C4C2×#12311yescharacteristic
414C8—48no—
415C8—84no—
416C22—24no—
417C22—24no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—24no—
425C22—44no—
426C22—44no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—82no—
433C22—88no—
434C22—88no—
435C22—82no—
436C22#188712yes—
437C22#190912yes—
438C22C2×#38611yescharacteristic
439C22C22×#10111yescharacteristic
440C22C2×#41311yescharacteristic
441C4—22no—
442C4—22no—
443C4—21no—
444C4—21no—
445C4—22no—
446C4—48no—
447C4—42no—
448C4—42no—
449C4—84no—
450C4#188712yes—
451C4#190114yes—
452C4C2×#45312yes—
453C2—21no—
454C2—21no—
455C2—22no—
456C2—22no—
457C2—44no—
458C2—82no—
459C2C2×#188711yescharacteristic
460C2C2×#190111yescharacteristic
461C2C2×#190911yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×C4
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
3. #17742C2semidirect
5. #9455C2semidirect
6. #7950C2semidirect
28. #1556C22semidirect
35. #1285C22semidirect

Maximal subgroups

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

subgroupindex
#602302
C2×#25422
#177422
#173642
#94552
#79502