Finite Group Databasebeta

← #200555

Subgroups — #200555

Order 512.

1341 conjugacy classes of subgroups.

The subgroup lattice has 393 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2972C211yesmaximal, characteristic
2#35240C211yesmaximal, characteristic
3#20726C211yesmaximal, characteristic
4#17715C212yesmaximal, complemented
5#9395C214yesmaximal, complemented
6#9393C214yesmaximal, complemented
7#8096C212yesmaximal, complemented
8C2×#533C2211yescharacteristic
9C22×C4×D4C2211yescharacteristic
10#2972—24no—
11#2972—22no—
12#2972C2214yes—
13C2×C8×D4—24no—
14#2920C2212yes—
15C2×#431C2212yes—
16C22×C22⋊C8—22no—
17#2848—22no—
18#2633C414yes—
19#2547C2212yes—
20#2542C2212yes—
21#2536—22no—
22#2502—21no—
23#2482C414yes—
24#2435—21no—
25#2326—22no—
26#2283C2211yescharacteristic
27#1821—21no—
28#1800C2214yes—
29#1783—21no—
30C4×#100—22no—
31#1552—24no—
32#1549C2214yescomplemented
33#1546C2214yescomplemented
34#1544—24no—
35#1303C2218yescomplemented
36C23×D4D411yescharacteristic
37C24×C4D412yes—
38C23×C8—24no—
39#533—22no—
40#533—22no—
41#533—22no—
42#533—22no—
43C2×C4⋊Q8C2311yescharacteristic
44C2×C4⋊Q8C2311yescharacteristic
45C2×#125C2×C412yes—
46C2×#124—22no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—24no—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4C2×C412yes—
57C2×C4×D4D412yes—
58C2×C4×D4D412yes—
59C2×C4×D4C2312yes—
60C2×#117C2×C412yes—
61C22×C4⋊C4D411yescharacteristic
62C22×C22⋊C4C2×C412yes—
63C22×C42C2311yescharacteristic
64C8×D4—216no—
65C8×D4—216no—
66#432—22no—
67#432—22no—
68#431—22no—
69#431—22no—
70#431—22no—
71#431—22no—
72#431D414yes—
73#431C2314yes—
74#423C2314yes—
75C2×C4⋊C8D414yes—
76C2×#100—24no—
77#407—22no—
78C2×C22⋊C8—24no—
79C2×C22⋊C8—28no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—28no—
82C4×M16—22no—
83C2×C4×C8—22no—
84#399C2×C418yes—
85#398—24no—
86#397C2×C418yes—
87#394—24no—
88#391—22no—
89#390—22no—
90#388—28no—
91#385—28no—
92#385—24no—
93#384—22no—
94#383C2×C412yes—
95#382—21no—
96#381C2×C412yes—
97#380—21no—
98#380—24no—
99#379—24no—
100C4×C4⋊C4C2312yes—
101C4×C22⋊C4—21no—
102#376—21no—
103#343—24no—
104#332—28no—
105#332—28no—
106C25C22⋊C412yes—
107C22×Q8—21no—
108C22×Q8—21no—
109C22×D4—28no—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4C22⋊C412yes—
113C22×D4C2×D412yes—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—84no—
119C23×C4C22⋊C412yes—
120C23×C4C22⋊C414yes—
121C23×C4C22×C411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D414yes—
124C23×C4C2×D411yescharacteristic
125C2×M16—24no—
126C22×C8—24no—
127C22×C8—216no—
128C22×C8—88no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133#125—24no—
134#125—24no—
135#124—48no—
136C22⋊Q8—41no—
137C22⋊Q8—41no—
138C22⋊Q8—42no—
139C22⋊Q8—42no—
140C22⋊Q8—41no—
141C22⋊Q8—41no—
142C22⋊Q8—88no—
143C4×Q8—42no—
144C4×Q8—42no—
145C4×Q8—42no—
146C4×Q8—42no—
147C4×D4—24no—
148C4×D4—24no—
149C4×D4—24no—
150C4×D4—24no—
151C4×D4—24no—
152C4×D4—24no—
153C4×D4—24no—
154C4×D4—24no—
155#117—24no—
156#117—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4C22⋊C412yes—
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C22×C412yes—
168C2×C4⋊C4C2×D412yes—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—28no—
175C2×C22⋊C4C22⋊C414yes—
176C2×C22⋊C4C22×C414yes—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42C22×C411yescharacteristic
180C2×C42C2×D412yes—
181C2×C42C2×D412yes—
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42#4112yes—
185C2×C42#4112yes—
186C2×C42C2411yescharacteristic
187C4⋊C8—24no—
188C4⋊C8—24no—
189C4⋊C8—24no—
190C4⋊C8C2×D418yes—
191#100—48no—
192C22⋊C8—216no—
193C22⋊C8—24no—
194C22⋊C8—24no—
195C22⋊C8—42no—
196C22⋊C8—42no—
197#97—42no—
198C4×C8—24no—
199C4×C8—24no—
200C4×C8—42no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—28no—
205#95C22×C414yes—
206C24—22no—
207C24—22no—
208C24—28no—
209C24—44no—
210C24—82no—
211C24C2×C22⋊C411yescharacteristic
212C24C2×C22⋊C414yes—
213C2×Q8—44no—
214C2×Q8—41no—
215C2×Q8—42no—
216C2×Q8—41no—
217C2×Q8—44no—
218C2×D4—216no—
219C2×D4—216no—
220C2×D4—28no—
221C2×D4—24no—
222C2×D4—24no—
223C2×D4—24no—
224C2×D4#13518yes—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—48no—
238C22×C4—48no—
239C22×C4—84no—
240C22×C4—84no—
241C22×C4—82no—
242C22×C4—84no—
243C22×C4—84no—
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C2×C22⋊C414yes—
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C4×D412yes—
249C22×C4C4×D412yes—
250C22×C4#12011yescharacteristic
251C22×C4#12011yescharacteristic
252C22×C4#12112yes—
253C22×C4#12112yes—
254C22×C4#12311yescharacteristic
255C22×C4#12311yescharacteristic
256C22×C4C23×C411yescharacteristic
257C22×C4C22×D411yescharacteristic
258C22×C4C22×D411yescharacteristic
259M16—48no—
260C2×C8—28no—
261C2×C8—48no—
262C2×C8—44no—
263C2×C8—816no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—44no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—44no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—88no—
276C22⋊C4—28no—
277C22⋊C4—28no—
278C22⋊C4—44no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—416no—
282C22⋊C4—88no—
283C42—44no—
284C42—44no—
285C42C4×D412yes—
286C42C4×D412yes—
287C42#12012yes—
288C42#12112yes—
289C42#12112yes—
290C42#12312yes—
291C42C22×D412yes—
292C42C2×#4112yes—
293C23—28no—
294C23—28no—
295C23—22no—
296C23—22no—
297C23—24no—
298C23—24no—
299C23—41no—
300C23—44no—
301C23—44no—
302C23—44no—
303C23—44no—
304C23—82no—
305C23—82no—
306C23—816no—
307C23#38611yescharacteristic
308C23#38611yescharacteristic
309C23#41618yes—
310C23C22×C22⋊C411yescharacteristic
311Q8—82no—
312Q8—82no—
313Q8—84no—
314D4—216no—
315D4—216no—
316C2×C4—22no—
317C2×C4—24no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—48no—
331C2×C4—44no—
332C2×C4—48no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—84no—
342C2×C4—84no—
343C2×C4—82no—
344C2×C4#38612yes—
345C2×C4#38612yes—
346C2×C4#38611yescharacteristic
347C2×C4#38611yescharacteristic
348C2×C4C2×C4×D412yes—
349C2×C4C2×#12011yescharacteristic
350C2×C4C2×#12112yes—
351C2×C4C2×#12311yescharacteristic
352C2×C4C2×#13514yes—
353C8—48no—
354C8—84no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—28no—
363C22—44no—
364C22—48no—
365C22—416no—
366C22—44no—
367C22—44no—
368C22—84no—
369C22—88no—
370C22#191512yes—
371C22C2×#38611yescharacteristic
372C22C2×#41612yes—
373C4—22no—
374C4—22no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—22no—
379C4—44no—
380C4—84no—
381C4—82no—
382C4—82no—
383C4#301912yes—
384C4#305014yes—
385C2—22no—
386C2—22no—
387C2—41no—
388C2—48no—
389C2#2095311yescharacteristic
390C2#2096711yescharacteristic
391C2C2×#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. #17715C2semidirect
5. #9395C2semidirect
6. #9393C2semidirect
7. #8096C2semidirect
32. #1549C22semidirect
33. #1546C22semidirect
35. #1303C22semidirect

Maximal subgroups

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

subgroupindex
C2×#29722
#352402
#207262
#177152
#93952
#93932
#80962