Finite Group Databasebeta

← #287776

Subgroups — #287776

Order 512.

675 conjugacy classes of subgroups.

The subgroup lattice has 397 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28687C211yesmaximal, characteristic
2#28511C211yesmaximal, characteristic
3#28255C211yesmaximal, characteristic
4#23149C211yesmaximal, characteristic
5C2×#1672C212yesmaximal
6#10482C212yesmaximal
7#10273C212yesmaximal
8#9691C211yesmaximal, characteristic
9#9602C211yesmaximal, characteristic
10#9216C212yesmaximal
11C4×#332C211yesmaximal, characteristic
12C2×#498C2212yes—
13C2×#477C2212yes—
14C2×#477C2212yes—
15C2×#462—21no—
16C2×#462C2212yes—
17C2×C8×Q8C2212yes—
18#2868C2211yescharacteristic
19C4×C4⋊Q8C2211yescharacteristic
20C4×C4⋊Q8C2211yescharacteristic
21#2095—22no—
22#2084—22no—
23#2056—22no—
24#1990C2212yes—
25#1984C2212yes—
26#1956—21no—
27#1948—21no—
28#1948—21no—
29#1939—21no—
30#1916—22no—
31#1845—21no—
32#1842C2211yescharacteristic
33#1803—21no—
34#1803C2212yes—
35C8×C4⋊C4—21no—
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2211yescharacteristic
39#1672—28no—
40C2×#334C2212yes—
41C2×#333C2212yes—
42C2×#333C2212yes—
43C2×#332C2211yescharacteristic
44C2×#332C2211yescharacteristic
45#1296C2211yescharacteristic
46#1296C2211yescharacteristic
47#1277C2212yes—
48C22×Q16—22no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8D411yescharacteristic
52C2×C4⋊Q8C2312yes—
53C2×C4⋊Q8C2311yescharacteristic
54C2×C4×Q8—21no—
55C2×C4×Q8C2312yes—
56#498—24no—
57#498—42no—
58#477—24no—
59#477—24no—
60#477—42no—
61#477—42no—
62#462—24no—
63#462—42no—
64#462—44no—
65C8×Q8—24no—
66C8×Q8—42no—
67C2×#107—22no—
68C2×#107—22no—
69C2×#106—22no—
70C2×#106—22no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8D411yescharacteristic
75C2×C4⋊C8D411yescharacteristic
76C2×C4⋊C8C2311yescharacteristic
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2311yescharacteristic
79C2×C4⋊C8C2311yescharacteristic
80C2×C4⋊C8C2311yescharacteristic
81C2×#103—22no—
82C2×#103—22no—
83C2×#103—22no—
84C2×#103—22no—
85C2×C4×C8—22no—
86C2×C4×C8—21no—
87C2×C4×C8—21no—
88C2×C4×C8C2312yes—
89#391—21no—
90#391D412yes—
91#389—21no—
92#389—21no—
93#389—21no—
94#389—21no—
95#384—21no—
96#384—21no—
97#384—21no—
98#384—21no—
99#384D411yescharacteristic
100#384C2311yescharacteristic
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C43C2311yescharacteristic
106#340—22no—
107#340—22no—
108#340—22no—
109#336—22no—
110#336—22no—
111#336—21no—
112#336—21no—
113#336D412yes—
114#334—24no—
115#333—24no—
116#333—42no—
117#332—22no—
118#332D414yes—
119C22×Q8—21no—
120C22×Q8—22no—
121C22×Q8C2×D412yes—
122C2×Q16—44no—
123C2×Q16—44no—
124C2×Q16—82no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—22no—
129C22×C8—21no—
130C22×C8C2×D412yes—
131C22×C8C2×D412yes—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—21no—
135C4⋊Q8—22no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—82no—
141C4⋊Q8—82no—
142C4⋊Q8SD1614yes—
143C4⋊Q8C2×D412yes—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—42no—
149C4×Q8—42no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4C2×D412yes—
168C2×C42—22no—
169C2×C42—41no—
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42C2411yescharacteristic
177#107—44no—
178#107—44no—
179#106—44no—
180#106—44no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—21no—
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—22no—
189C4⋊C8—44no—
190C4⋊C8—42no—
191C4⋊C8—81no—
192C4⋊C8Q1614yescomplemented
193C4⋊C8C2×D412yes—
194C4⋊C8C2×D412yes—
195C4⋊C8#4112yes—
196#103—44no—
197#103—44no—
198#103—44no—
199#103—44no—
200C4×C8—22no—
201C4×C8—22no—
202C4×C8—44no—
203C4×C8—42no—
204C4×C8—42no—
205#95—22no—
206#95—22no—
207#95—41no—
208#95—41no—
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—41no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—82no—
223C2×Q8—82no—
224C2×Q8C2×SD1614yes—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4#12111yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4#12111yescharacteristic
233C22×C4#12111yescharacteristic
234C22×C4C22×D411yescharacteristic
235C22×C4C22×D411yescharacteristic
236C22×C4C2×#4111yescharacteristic
237Q16—88no—
238C2×C8—22no—
239C2×C8—22no—
240C2×C8—22no—
241C2×C8—22no—
242C2×C8—22no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—22no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—42no—
253C2×C8—41no—
254C2×C8—42no—
255C2×C8—41no—
256C2×C8—42no—
257C2×C8—41no—
258C2×C8C2×Q1614yes—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—81no—
279C4⋊C4—81no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C4⋊C4—82no—
283C4⋊C4—82no—
284C4⋊C4—82no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—21no—
289C42—22no—
290C42—41no—
291C42—42no—
292C42—42no—
293C42—82no—
294C42C2×SD1612yes—
295C42C2×Q1612yes—
296C42#13512yes—
297C42#13712yes—
298C42C22×D411yescharacteristic
299C42C2×#4111yescharacteristic
300C42H(5,𝔽2)11yescharacteristic
301C42H(5,𝔽2)11yescharacteristic
302C23C2×#12111yescharacteristic
303Q8—24no—
304Q8—24no—
305Q8—42no—
306Q8—42no—
307Q8—42no—
308Q8—42no—
309Q8—81no—
310Q8—81no—
311Q8—162no—
312Q8—162no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—24no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—82no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—81no—
337C2×C4#46212yes—
338C2×C4#46212yes—
339C2×C4#46512yes—
340C2×C4#46512yes—
341C2×C4#47112yes—
342C2×C4#53211yescharacteristic
343C2×C4#54011yescharacteristic
344C2×C4D4211yescharacteristic
345C2×C4D4211yescharacteristic
346C2×C4#54611yescharacteristic
347C2×C4#55011yescharacteristic
348C2×C4C22×SD1611yescharacteristic
349C2×C4C22×Q1611yescharacteristic
350C2×C4C2×#13511yescharacteristic
351C2×C4C2×#13711yescharacteristic
352C2×C4#57711yescharacteristic
353C2×C4#57811yescharacteristic
354C2×C4#57811yescharacteristic
355C2×C4#57811yescharacteristic
356C8—24no—
357C8—42no—
358C8—42no—
359C8—42no—
360C8—42no—
361C8—81no—
362C8—82no—
363C22#242811yescharacteristic
364C22C2×#46211yescharacteristic
365C22#304411yescharacteristic
366C22C2×#46511yescharacteristic
367C22#306311yescharacteristic
368C22C2×#47111yescharacteristic
369C22#306811yescharacteristic
370C4—22no—
371C4—21no—
372C4—21no—
373C4—22no—
374C4—44no—
375C4—41no—
376C4—82no—
377C4—82no—
378C4—162no—
379C4—161no—
380C4#167214yescomplemented
381C4#327611yescharacteristic
382C4D4×SD1611yescharacteristic
383C4#328311yescharacteristic
384C4D4×Q1611yescharacteristic
385C4#329811yescharacteristic
386C4#330311yescharacteristic
387C4#339511yescharacteristic
388C4#339811yescharacteristic
389C2C2×#167211yescharacteristic
390C2#1924111yescharacteristic
391C2#1940311yescharacteristic
392C2#1957111yescharacteristic
393C2#2537011yescharacteristic
394C2#2538511yescharacteristic
395C2#2540711yescharacteristic
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
192. C4⋊C8Q16semidirect
380. C4#1672semidirect

Maximal subgroups

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

subgroupindex
#286872
#285112
#282552
#231492
C2×#16722
#104822
#102732
#96912
#96022
#92162
C4×#3322