Finite Group Databasebeta

← #275414

Subgroups — #275414

Order 512.

639 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33148C211yesmaximal, characteristic
2#27948C211yesmaximal, characteristic
3#22603C211yesmaximal, characteristic
4C2×#1948C211yesmaximal, characteristic
5#10654C212yesmaximal, complemented
6#10601C212yesmaximal, complemented
7#9618C212yesmaximal, complemented
8#9575C212yesmaximal, complemented
9#9127C211yesmaximal, characteristic
10#8829C211yesmaximal, characteristic
11#8724C211yesmaximal, characteristic
12C2×#432—21no—
13C22×#106—21no—
14C22×C4⋊C8C2211yescharacteristic
15#2857C2211yescharacteristic
16C2×#384C2211yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18#2084—22no—
19#1990C2212yes—
20#1987C2211yescharacteristic
21#1987C2211yescharacteristic
22#1948—21no—
23#1948—21no—
24#1948C2212yes—
25#1948C2212yes—
26#1948C2211yescharacteristic
27#1948C2211yescharacteristic
28#1939C2212yes—
29#1920—21no—
30#1920—21no—
31#1917—22no—
32#1853—21no—
33#1853—21no—
34#1853—21no—
35#1844C2212yes—
36#1823—21no—
37#1817—21no—
38#1797C2211yescharacteristic
39#1797C2211yescharacteristic
40#1770C2212yes—
41C4×C22⋊C8C2211yescharacteristic
42C2×#340—21no—
43C2×#335C2212yes—
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
59#432—44no—
60C2×#106—21no—
61C2×#106—21no—
62C2×#106—22no—
63C2×#106—22no—
64C2×#106—22no—
65C2×#106—22no—
66C2×#106—44no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—24no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8C2312yes—
71C2×C4⋊C8C2311yescharacteristic
72C2×C4⋊C8C2311yescharacteristic
73C2×C4⋊C8C2312yes—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8D411yescharacteristic
77C2×C22⋊C8D411yescharacteristic
78C2×#97—22no—
79C2×C4×C8C2312yes—
80#391—21no—
81#389—21no—
82#389—21no—
83#384—21no—
84#384—21no—
85#384—21no—
86#384C2311yescharacteristic
87#384C2312yes—
88#380—21no—
89#380—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4D411yescharacteristic
93C4×C4⋊C4D411yescharacteristic
94C4×C4⋊C4C2311yescharacteristic
95C4×C4⋊C4C2311yescharacteristic
96C2×#95—21no—
97#340—22no—
98#340—22no—
99#340—21no—
100#340—21no—
101#340—21no—
102#340—21no—
103#340—21no—
104#340—21no—
105#340—22no—
106#340—22no—
107#336—22no—
108#336—22no—
109#336—21no—
110#336—21no—
111#336D412yes—
112#335—24no—
113#333—24no—
114C22×Q8—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—41no—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4#4111yescharacteristic
121C22×C8—24no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—42no—
127C22×C8—42no—
128C22×C8C2×D412yes—
129C22×C8C2×D412yes—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C22⋊Q8—82no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—82no—
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—42no—
167C2×C42—42no—
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42C2411yescharacteristic
172#106—42no—
173#106—44no—
174#106—44no—
175#106—42no—
176#106—82no—
177#106—82no—
178C4⋊C8—24no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—44no—
185C4⋊C8#4114yes—
186C22⋊C8—22no—
187C22⋊C8—42no—
188C22⋊C8—42no—
189C22⋊C8SD1614yes—
190#97—44no—
191C4×C8—22no—
192C4×C8—22no—
193#95—21no—
194#95—21no—
195#95—21no—
196#95—21no—
197#95—41no—
198#95—41no—
199C24#12111yescharacteristic
200C2×Q8—42no—
201C2×Q8—81no—
202C2×Q8—81no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—81no—
225C22×C4—82no—
226C22×C4—82no—
227C22×C4—82no—
228C22×C4#12111yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4C2×SD1612yes—
232C22×C4AGL(1,ℤ/8)12yes—
233C22×C4#13712yes—
234C22×C4#13712yes—
235C22×C4C22×D411yescharacteristic
236C22×C4C22×D411yescharacteristic
237C22×C4C2×#4111yescharacteristic
238C2×C8—22no—
239C2×C8—22no—
240C2×C8—24no—
241C2×C8—22no—
242C2×C8—22no—
243C2×C8—22no—
244C2×C8—48no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—42no—
256C2×C8C2×SD1614yes—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—82no—
277C4⋊C4—81no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C4⋊C4#13712yes—
283C22⋊C4—82no—
284C42—21no—
285C42—21no—
286C42—22no—
287C42—22no—
288C42—41no—
289C42—41no—
290C42—41no—
291C42—41no—
292C42—42no—
293C42—42no—
294C42C2×#4112yes—
295C422−1+412yes—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23#46112yes—
301C23#46812yes—
302C23#47112yes—
303C23C2×#12111yescharacteristic
304Q8—162no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—44no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—84no—
334C2×C4—84no—
335C2×C4—81no—
336C2×C4—82no—
337C2×C4—81no—
338C2×C4—82no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—82no—
343C2×C4—81no—
344C2×C4—164no—
345C2×C4#46112yes—
346C2×C4#46812yes—
347C2×C4#52511yescharacteristic
348C2×C4#53611yescharacteristic
349C2×C4#54712yes—
350C2×C4#55212yes—
351C2×C4C22×SD1611yescharacteristic
352C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
353C2×C4C2×#13711yescharacteristic
354C2×C4C2×#13711yescharacteristic
355C2×C4C2×#13711yescharacteristic
356C2×C4#57511yescharacteristic
357C2×C4#57712yes—
358C8—24no—
359C8—42no—
360C8—44no—
361C8—82no—
362C8—81no—
363C8—81no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22#169114yes—
371C22#241311yescharacteristic
372C22C2×#46111yescharacteristic
373C22#304311yescharacteristic
374C22C2×#46811yescharacteristic
375C22#305611yescharacteristic
376C22C2×#47111yescharacteristic
377C22#306811yescharacteristic
378C4—22no—
379C4—22no—
380C4—22no—
381C4—41no—
382C4—42no—
383C4—81no—
384C4—81no—
385C4—81no—
386C4—161no—
387C4—161no—
388C4—162no—
389C4#311311yescharacteristic
390C4#311411yescharacteristic
391C4#323611yescharacteristic
392C4#323811yescharacteristic
393C4#333512yes—
394C4#334412yes—
395C2—24no—
396C2C2×#169111yescharacteristic
397C2#1916511yescharacteristic
398C2#1943211yescharacteristic
399C2#1946911yescharacteristic
400C2#2458011yescharacteristic
401C2#2460911yescharacteristic
402C2#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. #10654C2semidirect
6. #10601C2semidirect
7. #9618C2semidirect
8. #9575C2semidirect

Maximal subgroups

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

subgroupindex
#331482
#279482
#226032
C2×#19482
#106542
#106012
#96182
#95752
#91272
#88292
#87242