Finite Group Databasebeta

← #201251

Subgroups — #201251

Order 512.

1065 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic
2#35179C211yesmaximal, characteristic
3#17739C212yesmaximal
4#17365C211yesmaximal, characteristic
5#9456C214yesmaximal, complemented
6#9454C214yesmaximal, complemented
7#7953C212yesmaximal
8C2×#533C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2984—24no—
11#2983—21no—
12#2983C2214yes—
13#2982—21no—
14Q8×M16—24no—
15#2920—22no—
16#2920C2212yes—
17#2920C2212yes—
18#2855—22no—
19#2627—22no—
20#2582—21no—
21#2582—21no—
22#2581—21no—
23#2581—21no—
24#2568C2212yes—
25#2567C2212yes—
26#2565C414yes—
27#2317C412yes—
28#2310C412yes—
29#2285C2211yescharacteristic
30#1561—24no—
31#1556C2214yes—
32#1556C2214yes—
33#1544—24no—
34#1467C2214yes—
35#1463—22no—
36#1461—21no—
37#1461—21no—
38#1285C2218yes—
39C23×Q8C2×C411yescharacteristic
40#533—22no—
41#533—22no—
42#533—22no—
43#533—22no—
44C2×C4⋊Q8C2311yescharacteristic
45C2×C4⋊Q8C2311yescharacteristic
46C2×#125C2×C412yes—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8—21no—
52C2×C4×Q8—21no—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8C2×C412yes—
56C2×C4×Q8D411yescharacteristic
57C2×C4×Q8D411yescharacteristic
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×Q8C2312yes—
61C2×#117C2×C412yes—
62C2×#117C2×C412yes—
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4D412yes—
65C22×C42D412yes—
66C22×C42C2311yescharacteristic
67#446—28no—
68#446—216no—
69C8×Q8—28no—
70#432—24no—
71#432—24no—
72#432—22no—
73#432—22no—
74#432C2314yes—
75#432C2314yes—
76#431—24no—
77#431—24no—
78#431—22no—
79#431—22no—
80#431—22no—
81#431D414yes—
82#423—24no—
83#423—22no—
84#423D414yes—
85#407—22no—
86#407—22no—
87C4×M16—24no—
88C4×M16—22no—
89C4×M16—22no—
90#400—22no—
91#400—22no—
92#398—24no—
93#398—24no—
94#395—22no—
95#395—22no—
96#391—22no—
97#391—22no—
98#391—22no—
99#391—22no—
100#391C2×C414yes—
101#391C2×C414yes—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—22no—
106#387—24no—
107#387—24no—
108#385—24no—
109#384—21no—
110#384C2×C414yes—
111#382—21no—
112#382C2×C414yes—
113#381—24no—
114#380—21no—
115#380—21no—
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4C2312yes—
118#332—28no—
119#332—28no—
120#324—24no—
121#324—24no—
122C22×Q8—24no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—42no—
128C22×Q8C22⋊C411yescharacteristic
129C22×Q8C22⋊C411yescharacteristic
130C22×Q8C22×C412yes—
131C23×C4C22⋊C412yes—
132C23×C4C22⋊C412yes—
133C23×C4C22×C411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C2×M16—24no—
137C2×M16—42no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142#125—24no—
143#125—24no—
144C22⋊Q8—41no—
145C22⋊Q8—41no—
146C22⋊Q8—42no—
147C22⋊Q8—42no—
148C22⋊Q8—41no—
149C22⋊Q8—41no—
150C4×Q8—24no—
151C4×Q8—24no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—24no—
161C4×Q8—24no—
162C4×Q8—42no—
163C4×Q8—42no—
164C4×Q8—42no—
165C4×Q8—42no—
166#117—24no—
167#117—24no—
168#117—44no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4C22⋊C411yescharacteristic
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22⋊C412yes—
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C22×C412yes—
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42C22⋊C412yes—
199C2×C42C22⋊C412yes—
200C2×C42C22×C414yes—
201C2×C42C22×C411yescharacteristic
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42#4112yes—
209C2×C42#4112yes—
210C2×C42C2411yescharacteristic
211C4⋊C8—28no—
212C4⋊C8—28no—
213C4⋊C8—24no—
214C4⋊C8—24no—
215C4⋊C8—42no—
216C4⋊C8C2×D418yes—
217C22⋊C8—24no—
218C22⋊C8—24no—
219C22⋊C8—24no—
220C22⋊C8—42no—
221#97—28no—
222#97—24no—
223#97—24no—
224C4×C8—28no—
225C4×C8—24no—
226C4×C8—42no—
227#95—22no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—22no—
235C24C2×C22⋊C411yescharacteristic
236C2×Q8—24no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—24no—
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—42no—
244C2×Q8—48no—
245C2×Q8—48no—
246C2×Q8—44no—
247C2×Q8—41no—
248C2×Q8—42no—
249C2×Q8—41no—
250C2×Q8—82no—
251C2×Q8#10114yes—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—81no—
265C22×C4—82no—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C4×D412yes—
274C22×C4C4×D412yes—
275C22×C4#12011yescharacteristic
276C22×C4#12011yescharacteristic
277C22×C4#12112yes—
278C22×C4#12112yes—
279C22×C4#12311yescharacteristic
280C22×C4#12311yescharacteristic
281C22×C4C23×C411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C22×C4C22×D411yescharacteristic
284M16—48no—
285M16—84no—
286C2×C8—28no—
287C2×C8—44no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—44no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C4⋊C4—41no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—82no—
306C4⋊C4—84no—
307C4⋊C4—84no—
308C22⋊C4—44no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—44no—
320C42—44no—
321C42C4×D412yes—
322C42C4×D412yes—
323C42#12012yes—
324C42#12112yes—
325C42#12112yes—
326C42#12312yes—
327C42C22×D412yes—
328C42C2×#4112yes—
329C23—21no—
330C23—21no—
331C23—41no—
332C23—41no—
333C23#38611yescharacteristic
334C23#38611yescharacteristic
335C23C22×C22⋊C411yescharacteristic
336Q8—28no—
337Q8—48no—
338Q8—44no—
339Q8—82no—
340Q8—84no—
341Q8—82no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—82no—
377C2×C4#38611yescharacteristic
378C2×C4#38611yescharacteristic
379C2×C4#38612yes—
380C2×C4#38612yes—
381C2×C4C2×#10114yes—
382C2×C4C2×#10112yes—
383C2×C4#41312yes—
384C2×C4C2×C4×D412yes—
385C2×C4C2×#12011yescharacteristic
386C2×C4C2×#12112yes—
387C2×C4C2×#12311yescharacteristic
388C8—48no—
389C8—84no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—42no—
399C22—42no—
400C22C2×#38611yescharacteristic
401C22C22×#10111yescharacteristic
402C22C2×#41311yescharacteristic
403C4—22no—
404C4—22no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—22no—
409C4—24no—
410C4—42no—
411C4—42no—
412C4—42no—
413C4—84no—
414C4—82no—
415C4—82no—
416C4#190114yes—
417C4#302012yes—
418C2—21no—
419C2—21no—
420C2—41no—
421C2—41no—
422C2#2095411yescharacteristic
423C2C2×#190111yescharacteristic
424C2#2100911yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊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
5. #9456C2semidirect
6. #9454C2semidirect

Maximal subgroups

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

subgroupindex
#602482
#351792
#177392
#173652
#94562
#94542
#79532