Finite Group Databasebeta

← #279288

Subgroups — #279288

Order 512.

599 conjugacy classes of subgroups.

The subgroup lattice has 401 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33196C211yesmaximal, characteristic
2#29219C211yesmaximal, characteristic
3Q8×C4⋊C8C211yesmaximal, characteristic
4C4×#477C211yesmaximal, characteristic
5#10658C212yesmaximal
6#10597C212yesmaximal
7#9898C211yesmaximal, characteristic
8#9602C211yesmaximal, characteristic
9#9216C212yesmaximal
10#8647C212yesmaximal
11#8587C211yesmaximal, characteristic
12C2×#477C2211yescharacteristic
13C2×#475—21no—
14C2×C8×Q8C2212yes—
15#2868C2211yescharacteristic
16C4×C4⋊Q8C2211yescharacteristic
17C42×Q8C2211yescharacteristic
18#2099—22no—
19#1990C2212yes—
20#1988—21no—
21#1988—21no—
22#1988—21no—
23#1987C2211yescharacteristic
24#1948—21no—
25#1948C2212yes—
26#1947C2211yescharacteristic
27#1939—22no—
28#1939C2212yes—
29#1921—21no—
30#1863—21no—
31#1842—21no—
32#1842C2211yescharacteristic
33#1842C2211yescharacteristic
34#1834C2212yes—
35#1803—21no—
36#1803C2212yes—
37#1803C2211yescharacteristic
38C4×#107—21no—
39C4×C4⋊C8—21no—
40C4×C4⋊C8C2211yescharacteristic
41C4×C4⋊C8C2211yescharacteristic
42C4×#103—21no—
43C2×#334C2212yes—
44C2×#333C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8C2311yescharacteristic
53C2×C4×Q8—21no—
54C2×C4×Q8D411yescharacteristic
55C2×C4×Q8D411yescharacteristic
56C2×C4×Q8C2311yescharacteristic
57#477—22no—
58#477Q814yescomplemented
59#475—44no—
60C8×Q8—24no—
61C8×Q8—42no—
62C2×#107—21no—
63C2×#107—21no—
64C2×#106—22no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8D411yescharacteristic
72C2×C4⋊C8D411yescharacteristic
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2311yescharacteristic
76C2×C4⋊C8C2311yescharacteristic
77C2×#103—21no—
78C2×#103—22no—
79C2×#103—21no—
80C2×C4×C8—22no—
81C2×C4×C8—21no—
82C2×C4×C8C2312yes—
83#391—21no—
84#389—21no—
85#389—21no—
86#384—21no—
87#384—21no—
88#384—21no—
89#384C2311yescharacteristic
90#384C2312yes—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4D411yescharacteristic
97C4×C4⋊C4D411yescharacteristic
98C4×C4⋊C4C2311yescharacteristic
99C43—21no—
100C43C2311yescharacteristic
101#340—22no—
102#340—22no—
103#340—22no—
104#340—22no—
105#340—22no—
106#340—21no—
107#340—22no—
108#340—21no—
109#340—21no—
110#340—21no—
111#336—22no—
112#336—21no—
113#336D412yes—
114#334—42no—
115#333—42no—
116C22×Q8—21no—
117C22×Q8—41no—
118C22×Q8C2×D411yescharacteristic
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8C2×D412yes—
124C22×C8C2×D412yes—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—42no—
129C4⋊Q8—42no—
130C4⋊Q8—42no—
131C4⋊Q8—42no—
132C4⋊Q8—82no—
133C4⋊Q8C2×Q812yes—
134C4×Q8—22no—
135C4×Q8—21no—
136C4×Q8—21no—
137C4×Q8—22no—
138C4×Q8—41no—
139C4×Q8—82no—
140C4×Q8SD1614yes—
141C4×Q8C2×Q812yes—
142C4×Q8#4112yes—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4C2×D411yescharacteristic
159C2×C4⋊C4C2×D411yescharacteristic
160C2×C4⋊C4C2×D411yescharacteristic
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—41no—
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42C2411yescharacteristic
175#107—24no—
176#107—42no—
177#106—44no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—21no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—21no—
185C4⋊C8—22no—
186C4⋊C8—21no—
187C4⋊C8—44no—
188C4⋊C8—42no—
189C4⋊C8—42no—
190C4⋊C8—42no—
191C4⋊C8—81no—
192C4⋊C8—82no—
193C4⋊C8C2×Q812yes—
194#103—24no—
195#103—44no—
196#103—42no—
197C4×C8—22no—
198C4×C8—22no—
199C4×C8—44no—
200C4×C8—42no—
201#95—21no—
202#95—21no—
203#95—21no—
204#95—41no—
205#95—41no—
206C2×Q8—21no—
207C2×Q8—21no—
208C2×Q8—21no—
209C2×Q8—22no—
210C2×Q8—41no—
211C2×Q8—81no—
212C2×Q8—81no—
213C2×Q8—82no—
214C2×Q8—82no—
215C2×Q8—82no—
216C2×Q8C22⋊Q812yes—
217C2×Q8C2×SD1612yes—
218C2×Q8#13712yes—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4#12111yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4C22×D411yescharacteristic
231C22×C4C22×D411yescharacteristic
232C22×C4C2×#4111yescharacteristic
233C2×C8—22no—
234C2×C8—22no—
235C2×C8—22no—
236C2×C8—22no—
237C2×C8—22no—
238C2×C8—22no—
239C2×C8—22no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—42no—
243C2×C8—41no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—41no—
247C2×C8—42no—
248C2×C8—41no—
249C2×C8—42no—
250C2×C8—42no—
251C4⋊C4—21no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—21no—
255C4⋊C4—24no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—42no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—81no—
272C4⋊C4—82no—
273C4⋊C4—82no—
274C4⋊C4—82no—
275C4⋊C4—82no—
276C4⋊C4—82no—
277C4⋊C4C22⋊Q812yes—
278C4⋊C4C2×SD1612yes—
279C42—22no—
280C42—22no—
281C42—21no—
282C42—21no—
283C42—21no—
284C42—21no—
285C42—24no—
286C42—22no—
287C42—41no—
288C42—42no—
289C42—41no—
290C42—41no—
291C42—42no—
292C42—81no—
293C42—82no—
294C42C2×SD1612yes—
295C42#13512yes—
296C42#13512yes—
297C42#13712yes—
298C42C22×Q811yescharacteristic
299C42C2×#4111yescharacteristic
300C42H(5,𝔽2)11yescharacteristic
301C422−1+411yescharacteristic
302C23C2×#12111yescharacteristic
303Q8—22no—
304Q8—22no—
305Q8—22no—
306Q8—82no—
307Q8—82no—
308Q8—164no—
309Q8#47514yes—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—82no—
335C2×C4—81no—
336C2×C4—82no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—82no—
340C2×C4#46412yes—
341C2×C4#46412yes—
342C2×C4#46512yes—
343C2×C4#46512yes—
344C2×C4#47212yes—
345C2×C4C2×C22⋊Q811yescharacteristic
346C2×C4#53711yescharacteristic
347C2×C4#54611yescharacteristic
348C2×C4D4×Q811yescharacteristic
349C2×C4#55211yescharacteristic
350C2×C4#55411yescharacteristic
351C2×C4C22×SD1611yescharacteristic
352C2×C4C2×#13511yescharacteristic
353C2×C4C2×#13511yescharacteristic
354C2×C4C2×#13711yescharacteristic
355C2×C4C2×#13711yescharacteristic
356C2×C4#57511yescharacteristic
357C2×C4#57711yescharacteristic
358C2×C4#57811yescharacteristic
359C8—42no—
360C8—42no—
361C8—42no—
362C8—82no—
363C8—81no—
364C8—82no—
365C22#241711yescharacteristic
366C22C2×#46411yescharacteristic
367C22#304611yescharacteristic
368C22C2×#46511yescharacteristic
369C22#305611yescharacteristic
370C22C2×#47211yescharacteristic
371C22#306911yescharacteristic
372C4—24no—
373C4—21no—
374C4—21no—
375C4—22no—
376C4—22no—
377C4—42no—
378C4—41no—
379C4—42no—
380C4—82no—
381C4—81no—
382C4—161no—
383C4—162no—
384C4#167814yes—
385C4C2×#47511yescharacteristic
386C4#307611yescharacteristic
387C4#324611yescharacteristic
388C4#324811yescharacteristic
389C4#330411yescharacteristic
390C4#332811yescharacteristic
391C4#338311yescharacteristic
392C4#338711yescharacteristic
393C2C2×#167811yescharacteristic
394C2#1917511yescharacteristic
395C2#1941511yescharacteristic
396C2#1950311yescharacteristic
397C2#2481811yescharacteristic
398C2#2483811yescharacteristic
399C2#2485511yescharacteristic
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
58. #477Q8semidirect

Maximal subgroups

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

subgroupindex
#331962
#292192
Q8×C4⋊C82
C4×#4772
#106582
#105972
#98982
#96022
#92162
#86472
#85872