Finite Group Databasebeta

← #329807

Subgroups — #329807

Order 512.

660 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33207C211yesmaximal, characteristic
2#33148C211yesmaximal, characteristic
3#33108C211yesmaximal, characteristic
4C2×#1939C211yesmaximal, characteristic
5#10654C212yesmaximal, complemented
6#10598C212yesmaximal, complemented
7#10597C212yesmaximal, complemented
8#10544C212yesmaximal, complemented
9#9129C211yesmaximal, characteristic
10#9127C211yesmaximal, characteristic
11#8595C211yesmaximal, characteristic
12C2×#498C2211yescharacteristic
13C2×#483—21no—
14C22×C4⋊C8—21no—
15C22×C4⋊C8C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17#2857C2211yescharacteristic
18#2857C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20#1990C2211yescharacteristic
21#1990C2211yescharacteristic
22#1987C2212yes—
23#1981C2212yes—
24#1948C2212yes—
25#1948C2212yes—
26#1941—21no—
27#1941—21no—
28#1939—21no—
29#1939—21no—
30#1939C2212yes—
31#1939C2212yes—
32#1927—22no—
33#1920—22no—
34#1872—21no—
35#1853—21no—
36#1844C2211yescharacteristic
37#1823—21no—
38#1807—21no—
39#1807—21no—
40#1797C2211yescharacteristic
41#1797C2211yescharacteristic
42C2×#336—21no—
43C2×#334C2212yes—
44C2×#333C2212yes—
45C2×#332C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C23×C8—21no—
50C23×C8D411yescharacteristic
51C23×C8D411yescharacteristic
52C2×C4⋊Q8C2311yescharacteristic
53C2×C4⋊Q8C2311yescharacteristic
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D411yescharacteristic
58C22×C42—21no—
59C22×C42C2311yescharacteristic
60#498—42no—
61#483—44no—
62C2×#106—21no—
63C2×#106—21no—
64C2×#106—21no—
65C2×#106—21no—
66C2×#106—22no—
67C2×#106—22no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—42no—
72C2×C4⋊C8—44no—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2312yes—
76C2×#103—22no—
77C2×#103—22no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C2×C4×C8—42no—
81C2×C4×C8D411yescharacteristic
82C2×C4×C8D411yescharacteristic
83C2×C4×C8C2311yescharacteristic
84C2×C4×C8C2311yescharacteristic
85#391—21no—
86#391—21no—
87#389—21no—
88#389—21no—
89#389—21no—
90#389—21no—
91#384C2311yescharacteristic
92#384C2311yescharacteristic
93#380—21no—
94#380—21no—
95#380—21no—
96#380—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4D412yes—
100C4×C4⋊C4C2312yes—
101C2×#95—21no—
102#340—21no—
103#340—21no—
104#340—21no—
105#340—21no—
106#340—22no—
107#340—22no—
108#340—22no—
109#340—22no—
110#336—22no—
111#336—22no—
112#336—21no—
113#336—21no—
114#334—42no—
115#333—42no—
116#332—42no—
117C22×Q8—21no—
118C22×Q8—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4#4111yescharacteristic
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—41no—
130C22×C8—41no—
131C22×C8—41no—
132C22×C8—42no—
133C22×C8—42no—
134C22×C8—42no—
135C22×C8SD1614yes—
136C22×C8C2×D411yescharacteristic
137C22×C8C2×D411yescharacteristic
138C22×C8C2×D411yescharacteristic
139C22×C8C2×D411yescharacteristic
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C22⋊Q8—42no—
145C22⋊Q8—42no—
146C22⋊Q8—82no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4SD1614yes—
164C2×C4⋊C4C2×D412yes—
165C2×C4⋊C4C2×D412yes—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—42no—
173C2×C42—42no—
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42C2411yescharacteristic
178#106—44no—
179#106—42no—
180#106—42no—
181#106—42no—
182#106—42no—
183#106—44no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—42no—
191C4⋊C8—44no—
192C4⋊C8—44no—
193#103—44no—
194#103—44no—
195C22⋊C8—42no—
196C22⋊C8—42no—
197C4×C8—22no—
198C4×C8—22no—
199C4×C8—41no—
200C4×C8—41no—
201#95—22no—
202#95—22no—
203#95—41no—
204#95—41no—
205#95—41no—
206#95—41no—
207C24#12111yescharacteristic
208C2×Q8—42no—
209C2×Q8—42no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8—42no—
213C2×Q8—81no—
214C2×Q8—81no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4C2×SD1612yes—
240C22×C4C2×SD1612yes—
241C22×C4#13512yes—
242C22×C4#13712yes—
243C22×C4C22×D411yescharacteristic
244C22×C4C22×D411yescharacteristic
245C22×C4C2×#4111yescharacteristic
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—44no—
256C2×C8—44no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—84no—
267C2×C8C2×SD1612yes—
268C2×C8C2×SD1612yes—
269C2×C8#13712yes—
270C4⋊C4—22no—
271C4⋊C4—24no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—42no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—42no—
288C4⋊C4—81no—
289C4⋊C4—81no—
290C4⋊C4—82no—
291C4⋊C4—82no—
292C4⋊C4C2×SD1614yes—
293C22⋊C4—82no—
294C22⋊C4—82no—
295C42—21no—
296C42—21no—
297C42—22no—
298C42—22no—
299C42—42no—
300C42—42no—
301C42—44no—
302C42H(5,𝔽2)12yes—
303C422−1+412yes—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23#46112yes—
309C23#46512yes—
310C23#47112yes—
311C23C2×#12111yescharacteristic
312Q8—82no—
313Q8—82no—
314Q8—162no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—21no—
328C2×C4—24no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—44no—
332C2×C4—48no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—84no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4#46112yes—
352C2×C4#46512yes—
353C2×C4#53411yescharacteristic
354C2×C4#53611yescharacteristic
355C2×C4#53712yes—
356C2×C4#53912yes—
357C2×C4C22×SD1611yescharacteristic
358C2×C4C22×SD1611yescharacteristic
359C2×C4C2×#13511yescharacteristic
360C2×C4C2×#13711yescharacteristic
361C2×C4C2×#13711yescharacteristic
362C2×C4#57511yescharacteristic
363C2×C4#57512yes—
364C8—22no—
365C8—22no—
366C8—22no—
367C8—41no—
368C8—82no—
369C8—82no—
370C8—82no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22#167114yes—
378C22#261811yescharacteristic
379C22C2×#46111yescharacteristic
380C22#303711yescharacteristic
381C22C2×#46511yescharacteristic
382C22#305411yescharacteristic
383C22C2×#47111yescharacteristic
384C22#306711yescharacteristic
385C4—22no—
386C4—22no—
387C4—22no—
388C4—41no—
389C4—44no—
390C4—82no—
391C4—161no—
392C4—161no—
393C4—161no—
394C4—161no—
395C4#318711yescharacteristic
396C4#319511yescharacteristic
397C4#322911yescharacteristic
398C4#323611yescharacteristic
399C4#324612yes—
400C4#325012yes—
401C2—24no—
402C2C2×#167111yescharacteristic
403C2#1913711yescharacteristic
404C2#1926011yescharacteristic
405C2#1928811yescharacteristic
406C2#2975611yescharacteristic
407C2#2977411yescharacteristic
408C2#2979011yescharacteristic
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. #10598C2semidirect
7. #10597C2semidirect
8. #10544C2semidirect

Maximal subgroups

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

subgroupindex
#332072
#331482
#331082
C2×#19392
#106542
#105982
#105972
#105442
#91292
#91272
#85952