Finite Group Databasebeta

← #200524

Subgroups — #200524

Order 512.

1292 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic
2C2×#2546C211yesmaximal, characteristic
3#20727C211yesmaximal, characteristic
4#17742C212yesmaximal
5#9402C214yesmaximal, complemented
6#9394C214yesmaximal, complemented
7#8098C212yesmaximal
8C22×C4⋊Q8—21no—
9C22×C4⋊Q8C2211yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11#2984—24no—
12#2983—22no—
13#2983C2212yes—
14#2982C2212yes—
15Q8×M16—24no—
16#2920—22no—
17#2920C2212yes—
18#2920C2212yes—
19#2546—21no—
20#2546—21no—
21#2546—22no—
22#2546C414yes—
23#2546C2212yes—
24#2546C2212yes—
25C2×#398C414yes—
26C2×#391—21no—
27C2×#389—22no—
28C2×C4×C4⋊C4C2211yescharacteristic
29#1821—21no—
30#1801C2212yes—
31#1801C2212yes—
32#1784—21no—
33C4×#101—21no—
34C4×#101—21no—
35#1556—24no—
36#1549—24no—
37#1546C2218yes—
38#1303C2218yes—
39C23×Q8—21no—
40C23×Q8D411yescharacteristic
41C2×C4⋊Q8—24no—
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—48no—
51C2×C4⋊Q8C2×C412yes—
52C2×C4⋊Q8C2312yes—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8C2×C412yes—
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×Q8D412yes—
61C2×C4×Q8C2311yescharacteristic
62C2×C4×Q8C2311yescharacteristic
63C22×C4⋊C4—21no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2×C412yes—
67C22×C4⋊C4C2×C412yes—
68C22×C4⋊C4D411yescharacteristic
69C22×C42—21no—
70C22×C42D412yes—
71C22×C42C2311yescharacteristic
72#446—216no—
73#446—28no—
74C8×Q8—28no—
75#432—24no—
76#432—24no—
77#432—22no—
78#432—22no—
79#432—22no—
80#432D414yes—
81#431—24no—
82#431—24no—
83#431—22no—
84#431—22no—
85#431—22no—
86#431C2314yes—
87#423—24no—
88#423D414yes—
89#423C2314yes—
90C2×#101—22no—
91C2×#101—22no—
92#407—22no—
93#407—22no—
94C4×M16—24no—
95C4×M16—22no—
96C4×M16—22no—
97#398—24no—
98#398—24no—
99#398C2×C418yes—
100#398C2×C418yes—
101#391—24no—
102#391—22no—
103#391—22no—
104#389—24no—
105#389—24no—
106#389—24no—
107#389—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—
117C22×Q8—24no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—42no—
123C22×Q8—44no—
124C22×Q8—44no—
125C22×Q8—42no—
126C22×Q8—42no—
127C22×Q8C22⋊C412yes—
128C22×Q8C2×D411yescharacteristic
129C22×Q8C2×D411yescharacteristic
130C23×C4—22no—
131C23×C4—22no—
132C23×C4C22⋊C412yes—
133C23×C4C22⋊C412yes—
134C23×C4C22×C411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C2×M16—24no—
138C2×M16—24no—
139C4⋊Q8—24no—
140C4⋊Q8—24no—
141C4⋊Q8—24no—
142C4⋊Q8—24no—
143C4⋊Q8—416no—
144C4⋊Q8—48no—
145C4⋊Q8—48no—
146C4⋊Q8—42no—
147C4⋊Q8—42no—
148C4⋊Q8—42no—
149C4⋊Q8—42no—
150C4×Q8—24no—
151C4×Q8—24no—
152C4×Q8—24no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—24no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—22no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—416no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22×C414yes—
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C4⋊C4C2×D411yescharacteristic
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—44no—
196C2×C42C22⋊C412yes—
197C2×C42C22⋊C412yes—
198C2×C42C22×C411yescharacteristic
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42#4112yes—
206C2×C42#4112yes—
207C2×C42C2411yescharacteristic
208C4⋊C8—28no—
209C4⋊C8—28no—
210C4⋊C8—24no—
211C4⋊C8—24no—
212C4⋊C8—24no—
213C4⋊C8C2×D418yes—
214#101—44no—
215#101—44no—
216C22⋊C8—24no—
217C22⋊C8—24no—
218C22⋊C8—42no—
219C22⋊C8—42no—
220#97—28no—
221#97—24no—
222#97—42no—
223C4×C8—28no—
224C4×C8—24no—
225C4×C8—42no—
226#95—28no—
227#95—28no—
228#95—22no—
229#95C22×C414yes—
230C24C2×C22⋊C411yescharacteristic
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—24no—
237C2×Q8—24no—
238C2×Q8—42no—
239C2×Q8—48no—
240C2×Q8—48no—
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×Q8AGL(1,ℤ/8)14yes—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—28no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4C2×C22⋊C412yes—
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C4×D412yes—
278C22×C4C4×D412yes—
279C22×C4#12011yescharacteristic
280C22×C4#12011yescharacteristic
281C22×C4#12112yes—
282C22×C4#12112yes—
283C22×C4#12311yescharacteristic
284C22×C4#12311yescharacteristic
285C22×C4C23×C411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288M16—48no—
289M16—44no—
290M16—44no—
291C2×C8—28no—
292C2×C8—44no—
293C4⋊C4—28no—
294C4⋊C4—28no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—28no—
303C4⋊C4—416no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—44no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—416no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—44no—
319C42—44no—
320C42C4×D412yes—
321C42C4×D412yes—
322C42#12012yes—
323C42#12112yes—
324C42#12112yes—
325C42#12312yes—
326C42C22×D412yes—
327C42C2×#4112yes—
328C23—21no—
329C23—21no—
330C23—22no—
331C23—22no—
332C23#38611yescharacteristic
333C23#38611yescharacteristic
334C23C22×C22⋊C411yescharacteristic
335Q8—28no—
336Q8—48no—
337Q8—44no—
338Q8—88no—
339Q8—88no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—28no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—416no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—82no—
373C2×C4—84no—
374C2×C4—84no—
375C2×C4#38611yescharacteristic
376C2×C4#38612yes—
377C2×C4#38612yes—
378C2×C4#38611yescharacteristic
379C2×C4#41814yes—
380C2×C4#45614yes—
381C2×C4C2×C4×D412yes—
382C2×C4C2×#12011yescharacteristic
383C2×C4C2×#12112yes—
384C2×C4C2×#12311yescharacteristic
385C2×C4C2×AGL(1,ℤ/8)12yes—
386C2×C4#57512yes—
387C8—48no—
388C8—84no—
389C22—24no—
390C22—24no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22#188612yes—
398C22#191512yes—
399C22C2×#38611yescharacteristic
400C22C2×#41811yescharacteristic
401C22#289611yescharacteristic
402C4—22no—
403C4—22no—
404C4—21no—
405C4—21no—
406C4—22no—
407C4—24no—
408C4—48no—
409C4—42no—
410C4—42no—
411C4—42no—
412C4—84no—
413C4#188612yes—
414C4C2×#45612yes—
415C4#305514yes—
416C2—21no—
417C2—21no—
418C2—22no—
419C2—22no—
420C2C2×#188611yescharacteristic
421C2#2097311yescharacteristic
422C2C2×#191511yescharacteristic
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. #9402C2semidirect
6. #9394C2semidirect

Maximal subgroups

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

subgroupindex
#602482
C2×#25462
#207272
#177422
#94022
#93942
#80982