Finite Group Databasebeta

← #164819

Subgroups — #164819

Order 512.

833 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36446C211yesmaximal, characteristic
2#22022C212yesmaximal, complemented
3#7722C214yesmaximal, complemented
4#7682C212yesmaximal, complemented
5#7649C212yesmaximal
6#7635C211yesmaximal, characteristic
7#7601C212yesmaximal
8#7567C211yesmaximal, characteristic
9C2×#557C411yescharacteristic
10C2×D4×Q8C411yescharacteristic
11#2862C2211yescharacteristic
12#2862C2212yes—
13#2568C411yescharacteristic
14#2568C412yes—
15#2547C411yescharacteristic
16#2547C412yes—
17#2489—24no—
18#2489—22no—
19Q8×C22⋊C4—22no—
20C4×#124C2211yescharacteristic
21C4×#124C2212yes—
22C42×Q8C2211yescharacteristic
23#1984—22no—
24#1939—22no—
25#1917—24no—
26#1839C2214yes—
27#1831C2214yes—
28#1767C2212yes—
29#1744C2212yes—
30#1272C2214yes—
31#1272C2214yes—
32#1272C2212yes—
33#1272C2212yes—
34#1270C2211yescharacteristic
35#1270C2211yescharacteristic
36#1270C2212yes—
37C23×Q8—21no—
38#557—44no—
39D4×Q8—44no—
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×C4⋊Q8C2×C412yes—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2×C412yes—
44C2×#125—22no—
45C2×#125—21no—
46C2×#124C2×C411yescharacteristic
47C2×#124C2×C412yes—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—21no—
50C2×C4×Q8—21no—
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×Q8C2×C412yes—
53C2×C4×Q8C2×C411yescharacteristic
54C2×C4×Q8D412yes—
55C2×C4×D4C2×C411yescharacteristic
56C2×C4×D4C2×C412yes—
57C2×C4⋊C8—22no—
58C2×C4⋊C8—24no—
59C2×#97—22no—
60C2×C4×C8—22no—
61#400—22no—
62#400—24no—
63#397—22no—
64#397—24no—
65#391—22no—
66#391—24no—
67#391—22no—
68#391—24no—
69#391C2×C412yes—
70#391C2×C412yes—
71#391C2×C412yes—
72#391C2312yes—
73#391C2311yescharacteristic
74#390C2311yescharacteristic
75#388—22no—
76#388—24no—
77#382—22no—
78#382—24no—
79#382C2×C411yescharacteristic
80#382C2×C412yes—
81#380—22no—
82#380—24no—
83C4×C4⋊C4C2×C411yescharacteristic
84C4×C4⋊C4C2×C412yes—
85C4×C4⋊C4D414yes—
86C4×C4⋊C4D412yes—
87C4×C22⋊C4—22no—
88C4×C22⋊C4—24no—
89C43C2312yes—
90C43C2311yescharacteristic
91#336—24no—
92#336—24no—
93#336—24no—
94#336—22no—
95#336C2312yes—
96#336C2314yes—
97#336C2312yes—
98C22×Q8—22no—
99C22×Q8—21no—
100C22×Q8—48no—
101C22×Q8—44no—
102C22×Q8C22⋊C411yescharacteristic
103C22×Q8C22×C411yescharacteristic
104C22×Q8C22×C412yes—
105C22×D4C22×C411yescharacteristic
106C23×C4—22no—
107C23×C4—21no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—24no—
111C4⋊Q8—44no—
112C4⋊Q8—42no—
113C4⋊Q8—44no—
114C4⋊Q8—42no—
115#125—48no—
116#125—44no—
117#124—22no—
118#124—24no—
119#124—41no—
120#124—42no—
121C22⋊Q8—48no—
122C22⋊Q8—44no—
123C4×Q8—24no—
124C4×Q8—44no—
125C4×Q8—44no—
126C4×Q8—42no—
127C4×Q8—44no—
128C4×D4—44no—
129C4×D4—42no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4C22⋊C412yes—
139C2×C4⋊C4C22⋊C411yescharacteristic
140C2×C4⋊C4C22⋊C412yes—
141C2×C4⋊C4C22⋊C414yes—
142C2×C4⋊C4C22×C411yescharacteristic
143C2×C4⋊C4C22×C412yes—
144C2×C4⋊C4C22×C411yescharacteristic
145C2×C22⋊C4—21no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—22no—
150C2×C42—22no—
151C2×C42—21no—
152C2×C42C22×C412yes—
153C2×C42C22×C411yescharacteristic
154C2×C42C22×C412yes—
155C2×C42C22×C411yescharacteristic
156C2×C42C2×D412yes—
157C2×C42C2×D412yes—
158C2×C42C2×D414yes—
159C2×C42C2×D414yes—
160C2×C42C2411yescharacteristic
161C4⋊C8—44no—
162C4⋊C8—84no—
163#97—82no—
164C4×C8—44no—
165#95—22no—
166#95—24no—
167#95—24no—
168#95—24no—
169#95—24no—
170#95—22no—
171#95—24no—
172#95C22⋊C412yes—
173#95C22⋊C414yes—
174C24—21no—
175C2×Q8—24no—
176C2×Q8—21no—
177C2×Q8—22no—
178C2×Q8—21no—
179C2×Q8—22no—
180C2×Q8—48no—
181C2×Q8—42no—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×Q8—42no—
185C2×Q8—44no—
186C2×Q8—42no—
187C2×Q8—44no—
188C2×Q8—816no—
189C2×Q8#10014yes—
190C2×D4—21no—
191C2×D4—41no—
192C2×D4—41no—
193C2×D4#13112yes—
194C22×C4—22no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—48no—
200C22×C4—44no—
201C22×C4—42no—
202C22×C4—44no—
203C22×C4C2×C22⋊C412yes—
204C22×C4C2×C22⋊C414yes—
205C22×C4C2×C22⋊C411yescharacteristic
206C22×C4C2×C22⋊C412yes—
207C22×C4C2×C22⋊C412yes—
208C22×C4C2×C22⋊C411yescharacteristic
209C22×C4C23×C411yescharacteristic
210C22×C4C22×D412yes—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—44no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—44no—
217C2×C8—44no—
218C2×C8—42no—
219C2×C8—42no—
220C4⋊C4—24no—
221C4⋊C4—22no—
222C4⋊C4—24no—
223C4⋊C4—44no—
224C4⋊C4—42no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—44no—
228C4⋊C4—42no—
229C4⋊C4—44no—
230C4⋊C4—42no—
231C4⋊C4—44no—
232C4⋊C4—42no—
233C4⋊C4—44no—
234C4⋊C4—42no—
235C22⋊C4—44no—
236C22⋊C4—48no—
237C22⋊C4—44no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C42—22no—
242C42—21no—
243C42—22no—
244C42—21no—
245C42—22no—
246C42—24no—
247C42—22no—
248C42—24no—
249C42—24no—
250C42—22no—
251C42—24no—
252C42—44no—
253C42—41no—
254C42—44no—
255C42—42no—
256C23—22no—
257C23—41no—
258C23—41no—
259C23—42no—
260C23—42no—
261C23C22×C22⋊C411yescharacteristic
262Q8—28no—
263Q8—82no—
264Q8—81no—
265Q8—82no—
266Q8—81no—
267Q8—88no—
268Q8—84no—
269D4—42no—
270D4—42no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—24no—
276C2×C4—22no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—24no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—42no—
298C2×C4—88no—
299C2×C4—84no—
300C2×C4C2×#10014yes—
301C2×C4C2×#10012yes—
302C2×C4#42114yes—
303C2×C4C2×#13111yescharacteristic
304C2×C4#56811yescharacteristic
305C2×C4#57712yes—
306C2×C4#57812yes—
307C8—82no—
308C8—84no—
309C8—82no—
310C22—42no—
311C22—44no—
312C22—81no—
313C22—82no—
314C22—81no—
315C22#288011yescharacteristic
316C22#288411yescharacteristic
317C22C22×#10011yescharacteristic
318C22#289812yes—
319C22C2×#42112yes—
320C4—21no—
321C4—21no—
322C4—22no—
323C4—22no—
324C4—24no—
325C4—41no—
326C4—42no—
327C4—42no—
328C4—44no—
329C4—82no—
330C4—81no—
331C4—81no—
332C4—82no—
333C4—81no—
334C4#153314yes—
335C2—82no—
336C2#1742511yescharacteristic
337C2#1752312yes—
338C2#1760112yes—
339C2C2×#153311yescharacteristic
340C2#1777311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22022C2semidirect
3. #7722C2semidirect
4. #7682C2semidirect

Maximal subgroups

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

subgroupindex
#364462
#220222
#77222
#76822
#76492
#76352
#76012
#75672