Finite Group Databasebeta

← #287777

Subgroups — #287777

Order 512.

649 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28689C211yesmaximal, characteristic
2#28511C211yesmaximal, characteristic
3#28284C211yesmaximal, characteristic
4#23149C211yesmaximal, characteristic
5C2×#1676C212yesmaximal
6#10483C212yesmaximal
7#10279C212yesmaximal
8#9691C211yesmaximal, characteristic
9#9601C211yesmaximal, characteristic
10#9216C212yesmaximal
11C4×#333C211yesmaximal, characteristic
12C2×#500C2212yes—
13C2×#477C2212yes—
14C2×#477C2212yes—
15C2×#477C2212yes—
16C2×C8×Q8C2212yes—
17#2868C2211yescharacteristic
18C4×C4⋊Q8C2211yescharacteristic
19C4×C4⋊Q8C2211yescharacteristic
20#2095—22no—
21#2083—22no—
22#2068—22no—
23#1990C2212yes—
24#1984C2212yes—
25#1981—21no—
26#1956—21no—
27#1947—21no—
28#1947—21no—
29#1947—21no—
30#1939—21no—
31#1916—22no—
32#1845—21no—
33#1842C2211yescharacteristic
34#1803C2212yes—
35C8×C4⋊C4—21no—
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2211yescharacteristic
39#1676—28no—
40C2×#335C2212yes—
41C2×#333C2212yes—
42C2×#333C2211yescharacteristic
43C2×#333C2212yes—
44C2×#333C2211yescharacteristic
45#1296C2211yescharacteristic
46#1296C2211yescharacteristic
47#1277C2212yes—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8D411yescharacteristic
51C2×C4⋊Q8C2312yes—
52C2×C4⋊Q8C2311yescharacteristic
53C2×C4×Q8—21no—
54C2×C4×Q8C2312yes—
55#500—24no—
56#500—42no—
57#477—24no—
58#477—24no—
59#477—24no—
60#477—42no—
61#477—42no—
62#477—42no—
63C8×Q8—24no—
64C8×Q8—42no—
65C2×#107—22no—
66C2×#107—22no—
67C2×#107—22no—
68C2×#107—22no—
69C2×#107—22no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8D411yescharacteristic
74C2×C4⋊C8D411yescharacteristic
75C2×C4⋊C8C2312yes—
76C2×C4⋊C8C2311yescharacteristic
77C2×C4⋊C8C2311yescharacteristic
78C2×C4⋊C8C2311yescharacteristic
79C2×C4⋊C8C2311yescharacteristic
80C2×#103—22no—
81C2×#103—22no—
82C2×#103—22no—
83C2×#103—22no—
84C2×C4×C8—22no—
85C2×C4×C8—21no—
86C2×C4×C8—21no—
87C2×C4×C8C2312yes—
88#391—21no—
89#391D412yes—
90#389—21no—
91#389—21no—
92#389—21no—
93#389—21no—
94#384—21no—
95#384—21no—
96#384—21no—
97#384—21no—
98#384D411yescharacteristic
99#384C2311yescharacteristic
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C43C2311yescharacteristic
105#340—22no—
106#340—22no—
107#340—22no—
108#336—22no—
109#336—22no—
110#336—21no—
111#336—21no—
112#336D412yes—
113#335—24no—
114#333—24no—
115#333—22no—
116#333—42no—
117#333D414yes—
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8C2×D412yes—
121C22×C8—21no—
122C22×C8—21no—
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—22no—
126C22×C8C2×D412yes—
127C22×C8C2×D412yes—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—21no—
131C4⋊Q8—22no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—82no—
137C4⋊Q8—82no—
138C4⋊Q8Q1614yes—
139C4⋊Q8C2×D412yes—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—82no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4C2×D412yes—
163C2×C42—22no—
164C2×C42—41no—
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42C2411yescharacteristic
172#107—44no—
173#107—44no—
174#107—44no—
175#107—44no—
176#107—44no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8—21no—
184C4⋊C8—22no—
185C4⋊C8—44no—
186C4⋊C8—42no—
187C4⋊C8—81no—
188C4⋊C8Q1614yes—
189C4⋊C8C2×D412yes—
190C4⋊C8C2×D412yes—
191C4⋊C8#4112yes—
192#103—44no—
193#103—44no—
194#103—44no—
195#103—82no—
196C4×C8—22no—
197C4×C8—22no—
198C4×C8—44no—
199C4×C8—42no—
200C4×C8—42no—
201#95—22no—
202#95—22no—
203#95—41no—
204#95—41no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8—22no—
208C2×Q8—22no—
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—41no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—81no—
215C2×Q8—81no—
216C2×Q8—82no—
217C2×Q8—82no—
218C2×Q8C2×Q1614yes—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4#12111yescharacteristic
225C22×C4#12111yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4C22×D411yescharacteristic
229C22×C4C22×D411yescharacteristic
230C22×C4C2×#4111yescharacteristic
231C2×C8—22no—
232C2×C8—22no—
233C2×C8—22no—
234C2×C8—22no—
235C2×C8—22no—
236C2×C8—22no—
237C2×C8—22no—
238C2×C8—22no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—42no—
246C2×C8—41no—
247C2×C8—42no—
248C2×C8—41no—
249C2×C8—42no—
250C2×C8—41no—
251C2×C8C2×Q1614yes—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—81no—
272C4⋊C4—81no—
273C4⋊C4—82no—
274C4⋊C4—82no—
275C4⋊C4—82no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—21no—
282C42—22no—
283C42—41no—
284C42—42no—
285C42—42no—
286C42—82no—
287C42C2×Q1612yes—
288C42C2×Q1612yes—
289C42#13512yes—
290C42#13712yes—
291C42C22×D411yescharacteristic
292C42C2×#4111yescharacteristic
293C42H(5,𝔽2)11yescharacteristic
294C42H(5,𝔽2)11yescharacteristic
295C23C2×#12111yescharacteristic
296Q8—24no—
297Q8—24no—
298Q8—42no—
299Q8—42no—
300Q8—82no—
301Q8—82no—
302Q8—162no—
303Q8—162no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—24no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—82no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4#46212yes—
329C2×C4#46212yes—
330C2×C4#46712yes—
331C2×C4#46712yes—
332C2×C4#47212yes—
333C2×C4#53211yescharacteristic
334C2×C4#54011yescharacteristic
335C2×C4D4211yescharacteristic
336C2×C4D4211yescharacteristic
337C2×C4#54611yescharacteristic
338C2×C4#55011yescharacteristic
339C2×C4C22×Q1611yescharacteristic
340C2×C4C22×Q1611yescharacteristic
341C2×C4C2×#13511yescharacteristic
342C2×C4C2×#13711yescharacteristic
343C2×C4#57711yescharacteristic
344C2×C4#57811yescharacteristic
345C2×C4#57811yescharacteristic
346C2×C4#57811yescharacteristic
347C8—24no—
348C8—42no—
349C8—42no—
350C8—42no—
351C8—42no—
352C8—81no—
353C8—82no—
354C22#242811yescharacteristic
355C22C2×#46211yescharacteristic
356C22#304411yescharacteristic
357C22C2×#46711yescharacteristic
358C22#306311yescharacteristic
359C22C2×#47211yescharacteristic
360C22#306811yescharacteristic
361C4—22no—
362C4—21no—
363C4—21no—
364C4—22no—
365C4—44no—
366C4—41no—
367C4—82no—
368C4—82no—
369C4—162no—
370C4—161no—
371C4#167614yescomplemented
372C4#327611yescharacteristic
373C4#328411yescharacteristic
374C4D4×Q1611yescharacteristic
375C4D4×Q1611yescharacteristic
376C4#329911yescharacteristic
377C4#330211yescharacteristic
378C4#339511yescharacteristic
379C4#339811yescharacteristic
380C2C2×#167611yescharacteristic
381C2#1924111yescharacteristic
382C2#1940511yescharacteristic
383C2#1957311yescharacteristic
384C2#2537011yescharacteristic
385C2#2538611yescharacteristic
386C2#2540411yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
371. C4#1676semidirect

Maximal subgroups

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

subgroupindex
#286892
#285112
#282842
#231492
C2×#16762
#104832
#102792
#96912
#96012
#92162
C4×#3332