Finite Group Databasebeta

← #201550

Subgroups — #201550

Order 512.

1180 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic
2C2×#2581C211yesmaximal, characteristic
3#20727C211yesmaximal, characteristic
4#17449C212yesmaximal
5#9488C214yesmaximal, complemented
6#9484C214yesmaximal, complemented
7#7959C212yesmaximal
8C22×C4×Q8C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#2984—24no—
11#2983—22no—
12#2983C2212yes—
13#2982C2212yes—
14Q8×M16—24no—
15#2920—22no—
16#2920C2212yes—
17#2920C2212yes—
18#2581—21no—
19#2581—21no—
20#2581—22no—
21#2581C414yes—
22#2581C2212yes—
23#2581C2212yes—
24C2×#400C412yes—
25C2×#395C412yes—
26C2×#391—21no—
27C2×#389—21no—
28C2×#387—22no—
29C2×#382C2211yescharacteristic
30#1821—21no—
31#1801C2212yes—
32#1801C2212yes—
33#1784—21no—
34C4×#101—21no—
35C4×#101—21no—
36#1491—24no—
37#1487C2218yes—
38#1479—24no—
39#1286C2218yes—
40C23×Q8C2×C411yescharacteristic
41C23×Q8D411yescharacteristic
42C2×C4×Q8—22no—
43C2×C4×Q8—22no—
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—22no—
47C2×C4×Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8C2×C412yes—
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8D411yescharacteristic
55C2×C4×Q8D411yescharacteristic
56C2×C4×Q8D412yes—
57C2×C4×Q8C2312yes—
58C2×C4×Q8C2311yescharacteristic
59C2×C4×Q8C2311yescharacteristic
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C412yes—
65C22×C4⋊C4D411yescharacteristic
66C22×C42—21no—
67C22×C42D412yes—
68C22×C42C2311yescharacteristic
69#446—216no—
70#446—28no—
71C8×Q8—28no—
72#432—24no—
73#432—24no—
74#432—22no—
75#432—22no—
76#432—22no—
77#432—22no—
78#431—24no—
79#431—24no—
80#431—22no—
81#431—22no—
82#431D414yes—
83#431C2314yes—
84#423—24no—
85#423—22no—
86#423C2314yes—
87C2×#101—22no—
88C2×#101—22no—
89#407—22no—
90#407—22no—
91C4×M16—24no—
92C4×M16—22no—
93C4×M16D414yes—
94#400—22no—
95#400—22no—
96#400C2×C414yes—
97#400C2×C414yes—
98#395—22no—
99#395—22no—
100#395C2×C414yes—
101#395C2×C414yes—
102#391—24no—
103#391—22no—
104#391—22no—
105#389—24no—
106#389—22no—
107#389—22no—
108#387—24no—
109#387—24no—
110#387—24no—
111#387—24no—
112#382—21no—
113#382—21no—
114#382C2×C412yes—
115#382C2312yes—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95C2×C412yes—
121#343—22no—
122#343—22no—
123Q8⋊C8—216no—
124C22×Q8—24no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—44no—
131C22×Q8—42no—
132C22×Q8C22⋊C412yes—
133C22×Q8C22×C412yes—
134C22×Q8C2×D411yescharacteristic
135C22×Q8C2×D411yescharacteristic
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4C22⋊C412yes—
140C23×C4C22⋊C412yes—
141C23×C4C22×C411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C2×M16—24no—
145C2×M16—24no—
146C4×Q8—24no—
147C4×Q8—24no—
148C4×Q8—24no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—24no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—24no—
159C4×Q8—24no—
160C4×Q8—24no—
161C4×Q8—24no—
162C4×Q8—42no—
163C4×Q8—42no—
164C4×Q8—42no—
165C4×Q8—42no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—48no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4C22⋊C412yes—
186C2×C4⋊C4C22⋊C412yes—
187C2×C4⋊C4C22⋊C412yes—
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C412yes—
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—42no—
200C2×C42—42no—
201C2×C42C22⋊C412yes—
202C2×C42C22⋊C412yes—
203C2×C42C22×C411yescharacteristic
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42#4112yes—
211C2×C42#4112yes—
212C2×C42C2411yescharacteristic
213C4⋊C8—28no—
214C4⋊C8—28no—
215C4⋊C8—24no—
216C4⋊C8—24no—
217C4⋊C8—24no—
218C4⋊C8—24no—
219#101—44no—
220#101—44no—
221C22⋊C8—24no—
222C22⋊C8—24no—
223C22⋊C8—42no—
224C22⋊C8—42no—
225#97—28no—
226#97—24no—
227#97—42no—
228C4×C8—28no—
229C4×C8—42no—
230C4×C8C2×D418yes—
231#95—22no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—22no—
239#95—22no—
240#95C22×C414yes—
241C24C2×C22⋊C411yescharacteristic
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—28no—
250C2×Q8—24no—
251C2×Q8—42no—
252C2×Q8—44no—
253C2×Q8—48no—
254C2×Q8—48no—
255C2×Q8—44no—
256C2×Q8—44no—
257C2×Q8—48no—
258C2×Q8—42no—
259C2×Q8#13714yes—
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—24no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C4×D412yes—
292C22×C4C4×D412yes—
293C22×C4#12011yescharacteristic
294C22×C4#12011yescharacteristic
295C22×C4#12112yes—
296C22×C4#12112yes—
297C22×C4#12311yescharacteristic
298C22×C4#12311yescharacteristic
299C22×C4C23×C411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D411yescharacteristic
302M16—48no—
303M16—44no—
304M16—44no—
305C2×C8—28no—
306C2×C8—44no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—44no—
316C4⋊C4—48no—
317C4⋊C4—48no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—44no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
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#38611yescharacteristic
347C23#38611yescharacteristic
348C23C22×C22⋊C411yescharacteristic
349Q8—28no—
350Q8—28no—
351Q8—48no—
352Q8—44no—
353Q8—48no—
354Q8—44no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—82no—
392C2×C4—88no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4#38611yescharacteristic
396C2×C4#38611yescharacteristic
397C2×C4#38612yes—
398C2×C4#38612yes—
399C2×C4#41914yes—
400C2×C4C2×C4×D412yes—
401C2×C4C2×#12011yescharacteristic
402C2×C4C2×#12112yes—
403C2×C4C2×#12311yescharacteristic
404C2×C4C2×#13712yes—
405C2×C4#57512yes—
406C8—48no—
407C8—84no—
408C22—24no—
409C22—24no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22#189112yes—
417C22#191212yes—
418C22C2×#38611yescharacteristic
419C22C2×#41911yescharacteristic
420C22#289611yescharacteristic
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—22no—
426C4—22no—
427C4—22no—
428C4—24no—
429C4—44no—
430C4—42no—
431C4—42no—
432C4—42no—
433C4—84no—
434C4—82no—
435C4#305614yes—
436C2—21no—
437C2—21no—
438C2—22no—
439C2—22no—
440C2C2×#189111yescharacteristic
441C2#2097211yescharacteristic
442C2C2×#191211yescharacteristic
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. #9488C2semidirect
6. #9484C2semidirect

Maximal subgroups

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

subgroupindex
#602482
C2×#25812
#207272
#174492
#94882
#94842
#79592