Finite Group Databasebeta

← #276416

Subgroups — #276416

Order 512.

605 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#33198C211yesmaximal, characteristic
2#29206C211yesmaximal, characteristic
3#22821C211yesmaximal, characteristic
4C4×#462C211yesmaximal, characteristic
5#10620C212yesmaximal
6#10479C212yesmaximal
7#9896C211yesmaximal, characteristic
8#9611C212yesmaximal
9#9602C211yesmaximal, characteristic
10#9566C212yesmaximal
11#8586C211yesmaximal, characteristic
12C2×#477—21no—
13C2×#462C2211yescharacteristic
14C2×C4×Q16—21no—
15#2868C2211yescharacteristic
16C4×C4⋊Q8C2211yescharacteristic
17C42×Q8C2211yescharacteristic
18#2058—22no—
19#1990C2212yes—
20#1987—21no—
21#1984C2212yes—
22#1981C2211yescharacteristic
23#1948—21no—
24#1948—21no—
25#1947—21no—
26#1947C2212yes—
27#1947C2211yescharacteristic
28#1928—21no—
29#1916—22no—
30#1878—21no—
31#1844C2212yes—
32#1842C2211yescharacteristic
33#1842C2211yescharacteristic
34#1831—21no—
35#1803—21no—
36#1803C2211yescharacteristic
37#1803C2212yes—
38C8×C4⋊C4C2212yes—
39C4×C4⋊C8—21no—
40C4×C4⋊C8C2211yescharacteristic
41C4×C4⋊C8C2211yescharacteristic
42C4×#103—21no—
43C2×#334C2212yes—
44C2×#332C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C22×Q16—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8C2311yescharacteristic
54C2×C4×Q8—21no—
55C2×C4×Q8D411yescharacteristic
56C2×C4×Q8D411yescharacteristic
57C2×C4×Q8C2311yescharacteristic
58#477—44no—
59#462—22no—
60#462—22no—
61C4×Q16—44no—
62C2×#107—22no—
63C2×#107—21no—
64C2×#106—22no—
65C2×#106—22no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8D411yescharacteristic
71C2×C4⋊C8D411yescharacteristic
72C2×C4⋊C8C2312yes—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2311yescharacteristic
75C2×C4⋊C8C2311yescharacteristic
76C2×#103—21no—
77C2×#103—22no—
78C2×#103—21no—
79C2×#103—21no—
80C2×#103—21no—
81C2×C4×C8—21no—
82C2×C4×C8—22no—
83C2×C4×C8C2312yes—
84#391—21no—
85#389—21no—
86#389—21no—
87#384—21no—
88#384—21no—
89#384—21no—
90#384C2311yescharacteristic
91#384C2312yes—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4D411yescharacteristic
98C4×C4⋊C4D411yescharacteristic
99C4×C4⋊C4C2311yescharacteristic
100C43—21no—
101C43C2311yescharacteristic
102#340—22no—
103#340—22no—
104#340—22no—
105#340—21no—
106#340—22no—
107#340—21no—
108#336—22no—
109#336—22no—
110#336—22no—
111#336—21no—
112#336D412yes—
113#334—24no—
114#332—24no—
115C22×Q8—21no—
116C22×Q8—41no—
117C22×Q8C2×D411yescharacteristic
118C2×Q16—24no—
119C2×Q16—42no—
120C2×Q16—42no—
121C22×C8—21no—
122C22×C8—22no—
123C22×C8—21no—
124C22×C8—22no—
125C22×C8C2×D412yes—
126C22×C8C2×D412yes—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8—42no—
131C4⋊Q8—42no—
132C4⋊Q8—81no—
133C4⋊Q8—81no—
134C4⋊Q8—82no—
135C4⋊Q8#4112yes—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—21no—
139C4×Q8—21no—
140C4×Q8—21no—
141C4×Q8—21no—
142C4×Q8—41no—
143C4×Q8—42no—
144C4×Q8—42no—
145C4×Q8SD1614yes—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4C2×D411yescharacteristic
162C2×C4⋊C4C2×D411yescharacteristic
163C2×C4⋊C4C2×D411yescharacteristic
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—41no—
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42C2411yescharacteristic
178#107—42no—
179#107—44no—
180#106—82no—
181#106—82no—
182C4⋊C8—22no—
183C4⋊C8—21no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—22no—
190C4⋊C8—42no—
191C4⋊C8—42no—
192C4⋊C8—42no—
193C4⋊C8—81no—
194C4⋊C8Q1614yescomplemented
195C4⋊C8#4112yes—
196#103—24no—
197#103—44no—
198#103—42no—
199#103—42no—
200#103—42no—
201C4×C8—22no—
202C4×C8—22no—
203C4×C8—42no—
204C4×C8—82no—
205#95—21no—
206#95—21no—
207#95—21no—
208#95—41no—
209#95—41no—
210C2×Q8—21no—
211C2×Q8—21no—
212C2×Q8—22no—
213C2×Q8—21no—
214C2×Q8—21no—
215C2×Q8—21no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—82no—
222C2×Q8—82no—
223C2×Q8—82no—
224C2×Q8C2×SD1612yes—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—42no—
232C22×C4#12111yescharacteristic
233C22×C4#12111yescharacteristic
234C22×C4#12111yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4C22×D411yescharacteristic
237C22×C4C22×D411yescharacteristic
238C22×C4C2×#4111yescharacteristic
239Q16—48no—
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—42no—
249C2×C8—41no—
250C2×C8—42no—
251C2×C8—41no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—41no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8C2×Q1614yes—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—24no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—42no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C4⋊C4—81no—
279C4⋊C4—82no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C4⋊C4—82no—
283C4⋊C4—82no—
284C4⋊C4C2×SD1612yes—
285C4⋊C4AGL(1,ℤ/8)12yes—
286C42—21no—
287C42—21no—
288C42—21no—
289C42—21no—
290C42—24no—
291C42—22no—
292C42—41no—
293C42—42no—
294C42—41no—
295C42—41no—
296C42—41no—
297C42—41no—
298C42—42no—
299C42—81no—
300C42—82no—
301C42C2×SD1612yes—
302C42C2×Q1612yes—
303C42#13512yes—
304C42#13712yes—
305C42C2×#4111yescharacteristic
306C42C2×#4111yescharacteristic
307C422−1+411yescharacteristic
308C422−1+411yescharacteristic
309C23C2×#12111yescharacteristic
310Q8—22no—
311Q8—22no—
312Q8—41no—
313Q8—41no—
314Q8—42no—
315Q8—42no—
316Q8—81no—
317Q8—81no—
318Q8—164no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—82no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—81no—
349C2×C4#46212yes—
350C2×C4#46212yes—
351C2×C4#46512yes—
352C2×C4#46512yes—
353C2×C4#47112yes—
354C2×C4#52511yescharacteristic
355C2×C4#53611yescharacteristic
356C2×C4#54711yescharacteristic
357C2×C4#54811yescharacteristic
358C2×C4#55211yescharacteristic
359C2×C4#55611yescharacteristic
360C2×C4C22×SD1611yescharacteristic
361C2×C4C22×Q1611yescharacteristic
362C2×C4C2×#13511yescharacteristic
363C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
364C2×C4C2×#13711yescharacteristic
365C2×C4#57511yescharacteristic
366C2×C4#57611yescharacteristic
367C2×C4#57711yescharacteristic
368C8—24no—
369C8—42no—
370C8—42no—
371C8—82no—
372C8—82no—
373C8—81no—
374C22#241511yescharacteristic
375C22C2×#46211yescharacteristic
376C22#304511yescharacteristic
377C22C2×#46511yescharacteristic
378C22#305511yescharacteristic
379C22C2×#47111yescharacteristic
380C22#306911yescharacteristic
381C4—21no—
382C4—21no—
383C4—22no—
384C4—22no—
385C4—41no—
386C4—42no—
387C4—42no—
388C4—42no—
389C4—82no—
390C4—81no—
391C4—162no—
392C4—161no—
393C4#167214yes—
394C4#310111yescharacteristic
395C4#310611yescharacteristic
396C4#322711yescharacteristic
397C4#323311yescharacteristic
398C4#333811yescharacteristic
399C4#335611yescharacteristic
400C4#336311yescharacteristic
401C4#337511yescharacteristic
402C2C2×#167211yescharacteristic
403C2#1917211yescharacteristic
404C2#1940811yescharacteristic
405C2#1948311yescharacteristic
406C2#2468211yescharacteristic
407C2#2470111yescharacteristic
408C2#2472311yescharacteristic
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
194. C4⋊C8Q16semidirect

Maximal subgroups

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

subgroupindex
#331982
#292062
#228212
C4×#4622
#106202
#104792
#98962
#96112
#96022
#95662
#85862