Finite Group Databasebeta

← #190956

Subgroups — #190956

Order 512.

851 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34085C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic
3#21430C211yesmaximal, characteristic
4#8711C212yesmaximal
5#8566C212yesmaximal, complemented
6#8564C212yesmaximal, complemented
7#8314C212yesmaximal
8#8178C212yesmaximal, complemented
9#8176C212yesmaximal, complemented
10C2×#533C2211yescharacteristic
11#2336—22no—
12#2332—22no—
13#2331C411yescharacteristic
14#2331C411yescharacteristic
15#2327C411yescharacteristic
16#2327C411yescharacteristic
17#2323C412yes—
18#2323C412yes—
19#2298—22no—
20#2295—22no—
21#2295C2212yes—
22#2283C2211yescharacteristic
23C42×Q8C2212yes—
24C4×#117C2211yescharacteristic
25#1924—22no—
26#1923—22no—
27#1917—22no—
28#1916—22no—
29#1842C2211yescharacteristic
30#1831C2211yescharacteristic
31#1797C2212yes—
32#1767C2212yes—
33C4×C4⋊C8C2212yes—
34C4×C22⋊C8C2212yes—
35#1744C2211yescharacteristic
36C42×C8C2211yescharacteristic
37C2×Q8⋊C8C2212yes—
38C2×Q8⋊C8C2212yes—
39C2×Q8⋊C8C2212yes—
40C2×Q8⋊C8C2212yes—
41#1279C2212yes—
42#1279C2212yes—
43#1279C2212yes—
44#1279C2212yes—
45#533—24no—
46#533—24no—
47C2×C4⋊Q8C2×C411yescharacteristic
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×#125—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8C2×C412yes—
52C2×C4×Q8—21no—
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8C2×C412yes—
55C2×C4×Q8C2312yes—
56C2×#117—21no—
57C2×#117C2×C411yescharacteristic
58C2×#117C2×C411yescharacteristic
59C22×C42C2311yescharacteristic
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2312yes—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—22no—
66C2×#97—22no—
67C2×C4×C8—22no—
68C2×C4×C8C2312yes—
69C2×C4×C8C2312yes—
70#391—22no—
71#391—22no—
72#391C2×C412yes—
73#391C2×C412yes—
74#389—22no—
75#389—22no—
76#389C2×C412yes—
77#389C2×C412yes—
78#388—22no—
79#388—22no—
80#388—22no—
81#388—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#383—21no—
93#382C2×C411yescharacteristic
94#382C2×C411yescharacteristic
95#381—21no—
96#380C2×C412yes—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4C2×C412yes—
102C4×C4⋊C4C2×C412yes—
103C4×C4⋊C4D412yes—
104C4×C4⋊C4D412yes—
105C4×C4⋊C4C2312yes—
106C4×C22⋊C4—21no—
107C4×C22⋊C4D412yes—
108C4×C22⋊C4D412yes—
109#376—21no—
110#376—22no—
111#376C2×C411yescharacteristic
112#376C2×C411yescharacteristic
113C43—22no—
114C43C2311yescharacteristic
115C43C2311yescharacteristic
116#336—22no—
117#336—22no—
118#336—22no—
119#336—22no—
120#336—22no—
121#336—22no—
122Q8⋊C8—24no—
123Q8⋊C8—24no—
124Q8⋊C8—24no—
125Q8⋊C8—24no—
126C22×Q8—21no—
127C22×Q8C22×C412yes—
128C23×C4C22×C411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C4⋊Q8—24no—
136C4⋊Q8—24no—
137#125—28no—
138C22⋊Q8—28no—
139C22⋊Q8—24no—
140C22⋊Q8—24no—
141C4×Q8—24no—
142C4×Q8—24no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—28no—
146C4×Q8—44no—
147C4×Q8#4114yes—
148C4×Q8#4114yes—
149#117—24no—
150#117—24no—
151#117—24no—
152#117—42no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4C22⋊C412yes—
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4C22×C412yes—
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C22×C412yes—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4C22⋊C412yes—
170C2×C22⋊C4C22⋊C412yes—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42C22×C411yescharacteristic
180C2×C42C22×C411yescharacteristic
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D412yes—
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189C2×C42C2×D412yes—
190C2×C42C2411yescharacteristic
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—44no—
196C4⋊C8—82no—
197C22⋊C8—44no—
198C22⋊C8—82no—
199#97—82no—
200C4×C8—22no—
201C4×C8—22no—
202C4×C8—22no—
203C4×C8—22no—
204C4×C8—44no—
205#95—22no—
206#95—21no—
207#95—21no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—21no—
213#95—21no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95C22⋊C412yes—
220#95C22⋊C412yes—
221#95C22⋊C412yes—
222#95C22⋊C412yes—
223C24C2×C22⋊C411yescharacteristic
224C2×Q8—24no—
225C2×Q8—22no—
226C2×Q8—24no—
227C2×Q8—24no—
228C2×Q8—82no—
229C2×Q8#11714yes—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—42no—
243C22×C4C2×C22⋊C412yes—
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4C23×C411yescharacteristic
251C22×C4C22×D411yescharacteristic
252C22×C4C22×D411yescharacteristic
253C2×C8—24no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—28no—
268C4⋊C4—24no—
269C4⋊C4—22no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—42no—
276C4⋊C4—44no—
277C4⋊C4—44no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C4⋊C4#11714yes—
281C22⋊C4—28no—
282C22⋊C4—24no—
283C22⋊C4—42no—
284C22⋊C4—82no—
285C42—22no—
286C42—21no—
287C42—21no—
288C42—22no—
289C42—24no—
290C42—24no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—42no—
299C42—42no—
300C42—44no—
301C42—42no—
302C42—42no—
303C42—44no—
304C42—82no—
305C42—82no—
306C42#13114yes—
307C42C2×#4112yes—
308C42C2×#4112yes—
309C23—21no—
310C23—21no—
311C23—22no—
312C23—22no—
313C23—21no—
314C23C22×C22⋊C411yescharacteristic
315Q8—28no—
316Q8—44no—
317Q8—84no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—82no—
354C2×C4#40814yes—
355C2×C4C2×#11712yes—
356C2×C4#52512yes—
357C2×C4#52511yescharacteristic
358C2×C4#52511yescharacteristic
359C2×C4C2×#13112yes—
360C2×C4#56812yes—
361C2×C4#57712yes—
362C2×C4#57812yes—
363C8—44no—
364C8—82no—
365C8—82no—
366C8—82no—
367C22—41no—
368C22—41no—
369C22—42no—
370C22—42no—
371C22—41no—
372C22#228011yescharacteristic
373C22C2×#40811yescharacteristic
374C22#287911yescharacteristic
375C22#289812yes—
376C22#290112yes—
377C4—22no—
378C4—24no—
379C4—41no—
380C4—41no—
381C4—44no—
382C4—42no—
383C4—42no—
384C4—42no—
385C4—41no—
386C4—44no—
387C4—82no—
388C4—82no—
389C4#292514yes—
390C4#312112yes—
391C4#312611yescharacteristic
392C4#312711yescharacteristic
393C2—81no—
394C2#1748412yes—
395C2#1750612yes—
396C2#2072111yescharacteristic
397C2#2074511yescharacteristic
398C2#2074911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. #8566C2semidirect
6. #8564C2semidirect
8. #8178C2semidirect
9. #8176C2semidirect

Maximal subgroups

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

subgroupindex
#340852
#219262
#214302
#87112
#85662
#85642
#83142
#81782
#81762