Finite Group Databasebeta

← #275405

Subgroups — #275405

Order 512.

617 conjugacy classes of subgroups.

The subgroup lattice has 399 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33149C211yesmaximal, characteristic
2#27961C211yesmaximal, characteristic
3#22629C211yesmaximal, characteristic
4C2×#1948C211yesmaximal, characteristic
5#10655C212yesmaximal, complemented
6#10597C212yesmaximal, complemented
7#9622C212yesmaximal, complemented
8#9575C212yesmaximal, complemented
9#9127C211yesmaximal, characteristic
10#8830C211yesmaximal, characteristic
11#8724C211yesmaximal, characteristic
12C22×#106—21no—
13C22×C4⋊C8C2211yescharacteristic
14#2857C2211yescharacteristic
15C2×#384C2211yescharacteristic
16C2×C4×C4⋊C4C2211yescharacteristic
17#2100—22no—
18#1990C2212yes—
19#1987C2211yescharacteristic
20#1987C2211yescharacteristic
21#1948—21no—
22#1948—21no—
23#1948C2212yes—
24#1948C2212yes—
25#1947C2211yescharacteristic
26#1947C2211yescharacteristic
27#1939C2212yes—
28#1921—21no—
29#1921—21no—
30#1917—22no—
31#1852—21no—
32#1834C2212yes—
33#1824—21no—
34#1824—21no—
35#1824—21no—
36#1797—21no—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39#1770C2212yes—
40#1749—21no—
41C4×C22⋊C8C2211yescharacteristic
42C2×#340—21no—
43C2×#334C2212yes—
44C2×#333C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C23×C8—21no—
50C2×C4⋊Q8C2311yescharacteristic
51C2×C22⋊Q8—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4D411yescharacteristic
57C22×C42—21no—
58C22×C42C2311yescharacteristic
59C2×#107—21no—
60C2×#107—21no—
61C2×#106—22no—
62C2×#106—22no—
63C2×#106—44no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—24no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2311yescharacteristic
69C2×C4⋊C8C2311yescharacteristic
70C2×C4⋊C8C2312yes—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8D411yescharacteristic
74C2×C22⋊C8D411yescharacteristic
75C2×#97—22no—
76C2×C4×C8C2312yes—
77#391—21no—
78#389—21no—
79#389—21no—
80#384—21no—
81#384—21no—
82#384—21no—
83#384C2311yescharacteristic
84#384C2312yes—
85#380—21no—
86#380—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4D411yescharacteristic
90C4×C4⋊C4D411yescharacteristic
91C4×C4⋊C4C2311yescharacteristic
92C4×C4⋊C4C2311yescharacteristic
93C2×#95—21no—
94#340—22no—
95#340—22no—
96#340—22no—
97#340—22no—
98#340—21no—
99#340—21no—
100#340—21no—
101#340—21no—
102#340—21no—
103#340—21no—
104#340—22no—
105#340—22no—
106#336—22no—
107#336—22no—
108#336—21no—
109#336—21no—
110#336D412yes—
111#334—24no—
112#333—24no—
113C22×Q8—21no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—41no—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4#4111yescharacteristic
120C22×C8—24no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—42no—
126C22×C8—42no—
127C22×C8C2×D412yes—
128C22×C8C2×D412yes—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C22⋊Q8—82no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—82no—
155C2×C4⋊C4C2×D411yescharacteristic
156C2×C4⋊C4C2×D411yescharacteristic
157C2×C4⋊C4C2×D411yescharacteristic
158C2×C4⋊C4C2×D411yescharacteristic
159C2×C22⋊C4—41no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—82no—
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42C2411yescharacteristic
171#107—42no—
172#107—42no—
173#106—44no—
174#106—44no—
175C4⋊C8—24no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—82no—
182C4⋊C8#4114yes—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—42no—
186C22⋊C8—42no—
187#97—82no—
188C4×C8—22no—
189C4×C8—22no—
190#95—21no—
191#95—21no—
192#95—21no—
193#95—21no—
194#95—41no—
195#95—41no—
196C24#12111yescharacteristic
197C2×Q8—42no—
198C2×Q8—81no—
199C2×Q8—81no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—41no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—42no—
219C22×C4—81no—
220C22×C4—81no—
221C22×C4—81no—
222C22×C4—82no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4#12111yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4#13512yes—
229C22×C4#13712yes—
230C22×C4#13712yes—
231C22×C4#13712yes—
232C22×C4C22×D411yescharacteristic
233C22×C4C22×D411yescharacteristic
234C22×C4C2×#4111yescharacteristic
235C2×C8—24no—
236C2×C8—22no—
237C2×C8—22no—
238C2×C8—22no—
239C2×C8—22no—
240C2×C8—22no—
241C2×C8—22no—
242C2×C8—48no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—42no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—82no—
274C4⋊C4—81no—
275C4⋊C4—82no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4AGL(1,ℤ/8)12yes—
280C22⋊C4—82no—
281C42—21no—
282C42—21no—
283C42—22no—
284C42—22no—
285C42—42no—
286C42—42no—
287C42—81no—
288C42—81no—
289C42C2×#4112yes—
290C422−1+412yes—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23#46412yes—
296C23#47012yes—
297C23#47112yes—
298C23C2×#12111yescharacteristic
299Q8—162no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—44no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—84no—
329C2×C4—84no—
330C2×C4—81no—
331C2×C4—82no—
332C2×C4—81no—
333C2×C4—82no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—82no—
338C2×C4—81no—
339C2×C4—164no—
340C2×C4#46412yes—
341C2×C4#47012yes—
342C2×C4#52511yescharacteristic
343C2×C4#53611yescharacteristic
344C2×C4#54712yes—
345C2×C4#55212yes—
346C2×C4C2×#13511yescharacteristic
347C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
348C2×C4C2×#13711yescharacteristic
349C2×C4C2×#13711yescharacteristic
350C2×C4C2×#13711yescharacteristic
351C2×C4#57511yescharacteristic
352C2×C4#57712yes—
353C8—42no—
354C8—42no—
355C8—44no—
356C8—82no—
357C8—81no—
358C8—81no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22#170414yes—
366C22#241311yescharacteristic
367C22C2×#46411yescharacteristic
368C22#304211yescharacteristic
369C22C2×#47011yescharacteristic
370C22#305511yescharacteristic
371C22C2×#47111yescharacteristic
372C22#306811yescharacteristic
373C4—22no—
374C4—22no—
375C4—22no—
376C4—42no—
377C4—41no—
378C4—81no—
379C4—81no—
380C4—81no—
381C4—161no—
382C4—161no—
383C4—162no—
384C4#311011yescharacteristic
385C4#311711yescharacteristic
386C4#323611yescharacteristic
387C4#323911yescharacteristic
388C4#333712yes—
389C4#334412yes—
390C2—24no—
391C2C2×#170411yescharacteristic
392C2#1917411yescharacteristic
393C2#1942911yescharacteristic
394C2#1946211yescharacteristic
395C2#2457911yescharacteristic
396C2#2460611yescharacteristic
397C2#2462311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10655C2semidirect
6. #10597C2semidirect
7. #9622C2semidirect
8. #9575C2semidirect

Maximal subgroups

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

subgroupindex
#331492
#279612
#226292
C2×#19482
#106552
#105972
#96222
#95752
#91272
#88302
#87242