Finite Group Databasebeta

← #352595

Subgroups — #352595

Order 512.

577 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33198C211yesmaximal, characteristic
2#33073C211yesmaximal, characteristic
3#30304C211yesmaximal, characteristic
4#20573C211yesmaximal, characteristic
5#9906C211yesmaximal, characteristic
6#9905C211yesmaximal, characteristic
7#9614C212yesmaximal
8#9613C212yesmaximal
9#8812C211yesmaximal, characteristic
10#8655C212yesmaximal
11#8653C212yesmaximal
12C2×#477—21no—
13C2×#446C2212yes—
14#2868C2211yescharacteristic
15#2859C2211yescharacteristic
16C42×Q8C2211yescharacteristic
17#2083—21no—
18#2083—21no—
19#2057—22no—
20#1987—21no—
21#1948C2211yescharacteristic
22#1947—21no—
23#1947C2211yescharacteristic
24#1947C2212yes—
25#1920—21no—
26#1918—21no—
27#1865—21no—
28#1863—21no—
29#1845C2212yes—
30#1842C2211yescharacteristic
31#1834C2212yes—
32#1831C2211yescharacteristic
33#1803—21no—
34#1803C2211yescharacteristic
35#1803C2211yescharacteristic
36#1803C2212yes—
37#1771—22no—
38C8×C4⋊C4C2212yes—
39#1767C2211yescharacteristic
40C4×C4⋊C8—21no—
41#1745—21no—
42#1744C2211yescharacteristic
43C2×#335C2212yes—
44C2×#334C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×#125—21no—
53C2×C4×Q8D411yescharacteristic
54C2×C4×Q8D411yescharacteristic
55C2×C4×Q8C2311yescharacteristic
56#477—44no—
57#446—24no—
58#446Q818yescomplemented
59C2×#107—22no—
60C2×#107—21no—
61C2×#107—22no—
62C2×#107—22no—
63C2×#106—21no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8C2311yescharacteristic
68C2×C4⋊C8C2312yes—
69C2×C4⋊C8C2311yescharacteristic
70C2×C4⋊C8C2312yes—
71C2×#103—22no—
72C2×#103—21no—
73C2×#103—21no—
74C2×#97—22no—
75C2×#97—21no—
76C2×C4×C8—22no—
77C2×C4×C8C2312yes—
78#389—21no—
79#389—21no—
80#389—21no—
81#389—21no—
82#384—21no—
83#384—21no—
84#384—21no—
85#384C2311yescharacteristic
86#384C2312yes—
87#384C2311yescharacteristic
88#382—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4D411yescharacteristic
93C4×C4⋊C4D411yescharacteristic
94C4×C4⋊C4C2311yescharacteristic
95C43—21no—
96C43C2311yescharacteristic
97#340—22no—
98#340—22no—
99#340—22no—
100#340—21no—
101#340—21no—
102#340—22no—
103#340—21no—
104#340—21no—
105#336—22no—
106#336—21no—
107#336—21no—
108#336—21no—
109#336D412yes—
110#336D411yescharacteristic
111#336D411yescharacteristic
112#335—24no—
113#334—24no—
114C22×Q8—41no—
115C22×Q8C2×D411yescharacteristic
116C22×C8—22no—
117C22×C8—21no—
118C22×C8—21no—
119C22×C8—22no—
120C22×C8C2×D412yes—
121C22×C8C2×D412yes—
122C4⋊Q8—42no—
123C4⋊Q8—42no—
124C4⋊Q8—81no—
125C4⋊Q8—81no—
126C4⋊Q8—82no—
127#125—82no—
128C4×Q8—22no—
129C4×Q8—22no—
130C4×Q8—22no—
131C4×Q8—41no—
132C4×Q8—41no—
133C4×Q8C2×Q814yes—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4C2×D411yescharacteristic
152C2×C4⋊C4C2×D411yescharacteristic
153C2×C4⋊C4C2×D411yescharacteristic
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—21no—
159C2×C42—21no—
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42C2411yescharacteristic
167#107—42no—
168#107—44no—
169#107—82no—
170#107—82no—
171#106—42no—
172C4⋊C8—28no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—42no—
179C4⋊C8—42no—
180C4⋊C8C2×Q814yes—
181#103—42no—
182#103—42no—
183#103—44no—
184#97—28no—
185#97—81no—
186C4×C8—82no—
187C4×C8C2×Q814yes—
188C4×C8#4114yes—
189#95—21no—
190#95—21no—
191#95—21no—
192#95—41no—
193#95—41no—
194C2×Q8—21no—
195C2×Q8—21no—
196C2×Q8—21no—
197C2×Q8—21no—
198C2×Q8—21no—
199C2×Q8—82no—
200C2×Q8—82no—
201C2×Q8—82no—
202C2×Q8C4⋊Q812yes—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4#12111yescharacteristic
211C22×C4#12111yescharacteristic
212C22×C4#12111yescharacteristic
213C22×C4#12111yescharacteristic
214C22×C4C22×D411yescharacteristic
215C22×C4C22×D411yescharacteristic
216C22×C4C2×#4111yescharacteristic
217C2×C8—22no—
218C2×C8—22no—
219C2×C8—22no—
220C2×C8—22no—
221C2×C8—24no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—41no—
226C2×C8—41no—
227C2×C8—42no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—42no—
233C2×C8C22⋊Q814yes—
234C2×C8C22⋊Q814yes—
235C4⋊C4—21no—
236C4⋊C4—24no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C4⋊C4—42no—
249C4⋊C4—41no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—81no—
253C4⋊C4—81no—
254C4⋊C4—82no—
255C4⋊C4—82no—
256C4⋊C4—82no—
257C4⋊C4—82no—
258C4⋊C4—82no—
259C4⋊C4—82no—
260C4⋊C4C4⋊Q812yes—
261C4⋊C4AGL(1,ℤ/8)12yes—
262C4⋊C4#13712yes—
263C42—24no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—22no—
271C42—22no—
272C42—22no—
273C42—42no—
274C42—42no—
275C42—41no—
276C42—41no—
277C42—42no—
278C42—42no—
279C42C22×Q812yes—
280C42C2×#4112yes—
281C23C2×#12111yescharacteristic
282Q8—22no—
283Q8—22no—
284Q8—41no—
285Q8—41no—
286Q8—164no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—81no—
312C2×C4—82no—
313C2×C4—81no—
314C2×C4—82no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—82no—
320C2×C4#47212yes—
321C2×C4#47312yes—
322C2×C4C2×C22⋊Q812yes—
323C2×C4C2×C4⋊Q811yescharacteristic
324C2×C4#53211yescharacteristic
325C2×C4#53312yes—
326C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
327C2×C4C2×#13711yescharacteristic
328C2×C4#57511yescharacteristic
329C2×C4#57511yescharacteristic
330C2×C4#57611yescharacteristic
331C2×C4#57711yescharacteristic
332C2×C4#57812yes—
333C8—24no—
334C8—24no—
335C8—44no—
336C8—82no—
337C8—81no—
338C8—81no—
339C22#285711yescharacteristic
340C22#304411yescharacteristic
341C22#304511yescharacteristic
342C22#305511yescharacteristic
343C22#305611yescharacteristic
344C22C2×#47211yescharacteristic
345C22C2×#47311yescharacteristic
346C4—22no—
347C4—22no—
348C4—22no—
349C4—41no—
350C4—42no—
351C4—41no—
352C4—42no—
353C4—42no—
354C4—42no—
355C4—161no—
356C4—161no—
357C4—162no—
358C4#308212yes—
359C4#308312yes—
360C4#316511yescharacteristic
361C4#316611yescharacteristic
362C4#317111yescharacteristic
363C4#317211yescharacteristic
364C2#1947611yescharacteristic
365C2#1947711yescharacteristic
366C2#1948411yescharacteristic
367C2#1948511yescharacteristic
368C2#3320611yescharacteristic
369C2#3321811yescharacteristic
370C2#3322211yescharacteristic
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. #446Q8semidirect

Maximal subgroups

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

subgroupindex
#331982
#330732
#303042
#205732
#99062
#99052
#96142
#96132
#88122
#86552
#86532