Finite Group Databasebeta

← #375506

Subgroups — #375506

Order 512.

1197 conjugacy classes of subgroups.

The subgroup lattice has 373 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35668C211yesmaximal, characteristic
2#20399C211yesmaximal, characteristic
3#20355C211yesmaximal, characteristic
4#11388C212yesmaximal
5#11347C212yesmaximal
6#10893C212yesmaximal
7#10889C212yesmaximal
8#10750C212yesmaximal
9#10741C212yesmaximal
10C2×#544C2211yescharacteristic
11C2×#519C2212yes—
12C2×#519C2211yescharacteristic
13C22×#101C2212yes—
14#2873C2212yes—
15#2568C412yescomplemented
16#2567C412yescomplemented
17#2565C412yescomplemented
18#2565C412yescomplemented
19#2453—22no—
20#2451—22no—
21#2438—24no—
22#2317C2211yescharacteristic
23#2310C2211yescharacteristic
24#2296C2211yescharacteristic
25C2×#391C2211yescharacteristic
26#2285C2211yescharacteristic
27C4×#101—22no—
28C4×#101—22no—
29#1742C2212yes—
30#1740—24no—
31#1740C2212yes—
32#1740C2212yes—
33#1736C2214yes—
34#1733C2214yes—
35C2×#339C2214yes—
36C2×H(3,ℤ/4)C2214yes—
37C23×Q8C2311yescharacteristic
38C22×M16C2314yes—
39#544—44no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8C2×C412yes—
42C2×#125—21no—
43C2×#125C2×C412yes—
44C2×C22⋊Q8—22no—
45#519—28no—
46#519—24no—
47#519—24no—
48#519—44no—
49C2×C4×Q8—44no—
50C2×C4×Q8C2×C412yes—
51C2×C4×Q8C2×C412yes—
52C2×C4×Q8D412yes—
53C2×C4×Q8D412yes—
54C2×#117—24no—
55C2×#117D414yes—
56C2×#117C2314yes—
57C2×#117C2311yescharacteristic
58C2×#117C2311yescharacteristic
59C2×#117C2311yescharacteristic
60C2×#117C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C2×#101—24no—
64C2×#101—24no—
65C2×#101Q818yescomplemented
66#405—44no—
67C4×M16—24no—
68#400—24no—
69#398—24no—
70#398—24no—
71#395—24no—
72#391—22no—
73#391—22no—
74#391—24no—
75#391C2×C412yes—
76#391C2×C412yes—
77#391C2×C412yes—
78#391C2×C412yes—
79#389—21no—
80#389—21no—
81#389—24no—
82#389—22no—
83#387—24no—
84#387—24no—
85#385—22no—
86#385—22no—
87#384C2×C411yescharacteristic
88#384C2×C412yes—
89#384C2×C411yescharacteristic
90#382C2×C411yescharacteristic
91#382C2×C412yes—
92#382C2×C411yescharacteristic
93#381—22no—
94#380—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4C2×C412yes—
97C4×C4⋊C4C2×C412yes—
98C2×#95—22no—
99#343—22no—
100#343—22no—
101#342—24no—
102#342—24no—
103#340—44no—
104#339—28no—
105#339—44no—
106H(3,ℤ/4)—28no—
107C22×Q8—22no—
108C22×Q8—24no—
109C22×Q8—24no—
110C22×Q8C22×C412yes—
111C22×Q8C2×D412yes—
112C22×Q8C2×Q814yes—
113C22×Q8#4114yes—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4C2×D412yes—
117C23×C4C2411yescharacteristic
118C2×M16—24no—
119C2×M16—24no—
120C2×M16—24no—
121C2×M16—24no—
122C2×M16—24no—
123C2×M16C2×Q818yes—
124C22×C8—24no—
125C4⋊Q8—44no—
126C4⋊Q8—44no—
127#125—44no—
128#125—44no—
129C22⋊Q8—48no—
130C4×Q8—44no—
131C4×Q8—44no—
132C4×Q8—44no—
133C4×Q8—44no—
134C4×Q8—416no—
135C4×Q8—416no—
136#117—24no—
137#117—24no—
138#117—24no—
139#117—24no—
140#117—216no—
141#117—24no—
142#117—24no—
143#117—24no—
144#117—48no—
145#117—48no—
146#117#4118yes—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4C22⋊C414yes—
161C2×C4⋊C4C22⋊C414yes—
162C2×C4⋊C4C22×C412yes—
163C2×C4⋊C4C22×C411yescharacteristic
164C2×C4⋊C4C22×C411yescharacteristic
165C2×C4⋊C4C22×C411yescharacteristic
166C2×C4⋊C4C22×C411yescharacteristic
167C2×C4⋊C4C2×D411yescharacteristic
168C2×C4⋊C4C2×D411yescharacteristic
169C2×C22⋊C4—24no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4C2×D412yes—
173C2×C22⋊C4C2×D412yes—
174C2×C42—24no—
175C2×C42—24no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—44no—
179C2×C42C22⋊C414yes—
180C2×C42C22⋊C414yes—
181C2×C42C22×C412yes—
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C411yescharacteristic
184C2×C42C22×C412yes—
185C2×C42C2×D412yes—
186#101—216no—
187#97—44no—
188C4×C8—44no—
189#95—24no—
190#95—24no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—24no—
196#95—24no—
197#95—24no—
198C24C22×D411yescharacteristic
199C2×Q8—24no—
200C2×Q8—24no—
201C2×Q8—24no—
202C2×Q8—24no—
203C2×Q8—28no—
204C2×Q8—28no—
205C2×Q8—28no—
206C2×Q8—28no—
207C2×Q8—44no—
208C2×Q8—84no—
209C2×Q8C4×Q818yes—
210C2×Q8C22⋊Q818yes—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—44no—
222C22×C4—42no—
223C22×C4—81no—
224C22×C4—82no—
225C22×C4C2×C22⋊C414yes—
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C412yes—
228C22×C4C2×C22⋊C412yes—
229C22×C4C2×C22⋊C412yes—
230C22×C4#12414yes—
231C22×C4C4⋊Q814yes—
232C22×C4C23×C411yescharacteristic
233C22×C4C22×D411yescharacteristic
234C22×C4C22×Q812yes—
235C22×C4C2×#4112yes—
236C22×C4C2×#4114yes—
237C22×C4H(5,𝔽2)14yes—
238M16—28no—
239M16—28no—
240M16—44no—
241M16—44no—
242C2×C8—28no—
243C2×C8—44no—
244C2×C8—44no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—24no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—28no—
253C4⋊C4—44no—
254C4⋊C4—44no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—44no—
258C4⋊C4—44no—
259C4⋊C4—44no—
260C4⋊C4—416no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—44no—
264C4⋊C4—82no—
265C4⋊C4—84no—
266C4⋊C4—82no—
267C22⋊C4—24no—
268C22⋊C4—24no—
269C22⋊C4—28no—
270C22⋊C4—42no—
271C22⋊C4—44no—
272C22⋊C4—42no—
273C42—24no—
274C42—24no—
275C42—22no—
276C42—22no—
277C42—24no—
278C42—28no—
279C42—44no—
280C42—44no—
281C42—416no—
282C42—42no—
283C42—42no—
284C42—44no—
285C23—41no—
286C23—41no—
287C23C22×C22⋊C411yescharacteristic
288C23#52512yes—
289C23C2×#12411yescharacteristic
290C23C2×C4⋊Q811yescharacteristic
291C23#53412yes—
292Q8—216no—
293Q8—216no—
294Q8—164no—
295C2×C4—28no—
296C2×C4—24no—
297C2×C4—28no—
298C2×C4—24no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—44no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—48no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—84no—
324C2×C4—84no—
325C2×C4—84no—
326C2×C4—162no—
327C2×C4#39118yes—
328C2×C4C2×C4×Q812yes—
329C2×C4C4×#4112yes—
330C2×C4#51812yes—
331C2×C4C2×C22⋊Q812yes—
332C2×C4#53212yes—
333C2×C4#53412yes—
334C2×C4#55114yes—
335C2×C4#55114yes—
336C2×C4#55414yes—
337C2×C4#55414yes—
338C8—48no—
339C22—21no—
340C22—22no—
341C22—21no—
342C22—22no—
343C22—42no—
344C22—41no—
345C22—41no—
346C22C2×#39111yescharacteristic
347C22#229711yescharacteristic
348C22#230811yescharacteristic
349C22#256112yes—
350C22#258312yes—
351C4—28no—
352C4—42no—
353C4—42no—
354C4—42no—
355C4—42no—
356C4—42no—
357C4—82no—
358C4—81no—
359C4—88no—
360C4—81no—
361C4—82no—
362C4—81no—
363C4#236814yes—
364C4#256414yes—
365C2—21no—
366C2—22no—
367C2—22no—
368C2—21no—
369C2#1415911yescharacteristic
370C2#2178311yescharacteristic
371C2#2178511yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #2568C4semidirect
16. #2567C4semidirect
17. #2565C4semidirect
18. #2565C4semidirect
65. C2×#101Q8semidirect

Maximal subgroups

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

subgroupindex
#356682
#203992
#203552
#113882
#113472
#108932
#108892
#107502
#107412