Finite Group Databasebeta

← #213708

Subgroups — #213708

Order 512.

1084 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35316C211yesmaximal, characteristic
2#21851C212yesmaximal
3#9136C214yesmaximal, complemented
4#9127C214yesmaximal, complemented
5#8728C212yesmaximal, complemented
6#8342C212yesmaximal, complemented
7C2×#536C412yes—
8C22×C4⋊Q8C2211yescharacteristic
9C2×C4×#41C2211yescharacteristic
10C2×#432C2212yes—
11C22×C4⋊C8C2212yes—
12#2857C2214yes—
13#2621C414yes—
14#2547—22no—
15#2546—22no—
16#2486—22no—
17#2486—22no—
18#2310C412yes—
19#2285C2211yescharacteristic
20#1985—22no—
21#1955—22no—
22#1869—22no—
23#1869—22no—
24#1797C2214yes—
25#1797C2214yes—
26C2×#333C2212yes—
27C2×#333C2214yes—
28C2×#333C2212yes—
29#1296C2212yes—
30#1296C2214yes—
31#1296C2212yes—
32C22×#41C2×C411yescharacteristic
33C23×Q8C2×C412yes—
34C23×C8—22no—
35#536—48no—
36C2×C4⋊Q8—22no—
37C2×C4⋊Q8—22no—
38C2×C4⋊Q8—41no—
39C2×C4⋊Q8—42no—
40C2×C4⋊Q8—41no—
41C2×C4⋊Q8C2×C412yes—
42C2×C4⋊Q8C2312yes—
43C2×#124—22no—
44C2×#123—24no—
45C2×C22⋊Q8C2×C418yes—
46C4×#41—41no—
47C4×#41—42no—
48C4×#41—41no—
49C2×C4×Q8D412yes—
50C2×C4×D4D412yes—
51C2×C4×D4D412yes—
52C2×C4×D4D412yes—
53C2×#117C2×C412yes—
54C2×#117C2×C411yescharacteristic
55C2×#117C2×C412yes—
56C22×C4⋊C4—22no—
57C22×C42C2311yescharacteristic
58C22×C42C2312yes—
59#432—42no—
60#432—42no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—42no—
63C2×C4⋊C8—42no—
64C2×C4⋊C8C2314yes—
65C2×C4⋊C8C2314yes—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—24no—
68C2×C22⋊C8—24no—
69C2×#97—22no—
70#398—24no—
71#398—24no—
72#397—24no—
73#397—24no—
74#391—24no—
75#391C2×C418yes—
76#389—24no—
77#389—24no—
78#384C2312yes—
79#382C2×C412yes—
80#381—22no—
81#381—22no—
82#380—24no—
83#380—24no—
84#380—22no—
85C4×C4⋊C4—22no—
86#336—24no—
87#336—24no—
88#333—42no—
89#333—44no—
90#333—42no—
91C2×#41—24no—
92C2×#41—41no—
93C2×#41—41no—
94C2×#41—42no—
95C22×Q8—22no—
96C22×Q8—42no—
97C22×Q8—42no—
98C22×Q8—44no—
99C22×Q8—44no—
100C22×Q8C22⋊C411yescharacteristic
101C22×Q8C22×C414yes—
102C22×D4C22⋊C412yes—
103C22×D4C22⋊C411yescharacteristic
104C23×C4—22no—
105C23×C4C22×C411yescharacteristic
106C23×C4C22×C412yes—
107C23×C4C2×D412yes—
108C23×C4C2×D412yes—
109C23×C4C2×D412yes—
110C22×C8—24no—
111C22×C8—24no—
112C22×C8—44no—
113C22×C8—84no—
114C4⋊Q8—22no—
115C4⋊Q8—24no—
116C4⋊Q8—22no—
117C4⋊Q8—42no—
118C4⋊Q8—42no—
119C4⋊Q8—44no—
120C4⋊Q8—44no—
121C4⋊Q8—48no—
122#124—48no—
123#123—416no—
124C22⋊Q8—416no—
125C4×Q8—42no—
126C4×Q8—42no—
127C4×D4—42no—
128C4×D4—42no—
129C4×D4—42no—
130C4×D4—42no—
131C4×D4—42no—
132C4×D4—42no—
133#117—41no—
134#117—41no—
135#117—42no—
136#117—42no—
137#117—44no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—44no—
143C2×C4⋊C4—44no—
144C2×C4⋊C4C22⋊C412yes—
145C2×C4⋊C4C22⋊C412yes—
146C2×C4⋊C4C22⋊C412yes—
147C2×C4⋊C4C22×C414yes—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—28no—
150C2×C22⋊C4C22⋊C411yescharacteristic
151C2×C22⋊C4C22⋊C411yescharacteristic
152C2×C22⋊C4C22⋊C412yes—
153C2×C22⋊C4C22⋊C412yes—
154C2×C42—21no—
155C2×C42—22no—
156C2×C42—21no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42C22×C411yescharacteristic
163C2×C42C22×C412yes—
164C2×C42C2×D412yes—
165C2×C42C2×D412yes—
166C2×C42C2×D412yes—
167C2×C42C2411yescharacteristic
168C4⋊C8—24no—
169C4⋊C8—24no—
170C4⋊C8—24no—
171C4⋊C8—24no—
172C4⋊C8—44no—
173C22⋊C8—44no—
174C22⋊C8—84no—
175C22⋊C8—84no—
176#97—44no—
177#95—24no—
178#95—28no—
179#95—28no—
180C24C2×C22⋊C412yes—
181C24C2×C22⋊C411yescharacteristic
182#41—48no—
183#41—48no—
184C2×Q8—22no—
185C2×Q8—24no—
186C2×Q8—24no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8—44no—
190C2×Q8—44no—
191C2×Q8—44no—
192C2×Q8—44no—
193C2×Q8—44no—
194C2×Q8—88no—
195C2×Q8—88no—
196C2×D4—24no—
197C2×D4—24no—
198C2×D4—41no—
199C2×D4—42no—
200C2×D4—41no—
201C2×D4#10014yes—
202C22×C4—22no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—42no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4C2×C22⋊C412yes—
228C22×C4C2×C22⋊C412yes—
229C22×C4C2×C22⋊C412yes—
230C22×C4C2×C22⋊C412yes—
231C22×C4C2×C22⋊C411yescharacteristic
232C22×C4C23×C411yescharacteristic
233C22×C4C22×D412yes—
234C2×C8—44no—
235C2×C8—44no—
236C2×C8—44no—
237C2×C8—44no—
238C2×C8—44no—
239C2×C8—44no—
240C2×C8—88no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—44no—
245C4⋊C4—48no—
246C4⋊C4—416no—
247C4⋊C4—44no—
248C4⋊C4—44no—
249C4⋊C4—44no—
250C4⋊C4—44no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—44no—
256C22⋊C4—416no—
257C42—22no—
258C42—22no—
259C42—21no—
260C42—21no—
261C42—22no—
262C42—22no—
263C42—22no—
264C42—22no—
265C42—22no—
266C42—22no—
267C42—44no—
268C42H(5,𝔽2)11yescharacteristic
269C42H(5,𝔽2)12yes—
270C42H(5,𝔽2)11yescharacteristic
271C23—21no—
272C23—22no—
273C23—21no—
274C23—22no—
275C23—24no—
276C23—24no—
277C23—24no—
278C23C2×#10014yes—
279C23#41312yes—
280C23C22×C22⋊C411yescharacteristic
281Q8—44no—
282Q8—84no—
283Q8—84no—
284Q8—88no—
285D4—44no—
286D4—48no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—48no—
318C2×C4—88no—
319C2×C4C2×#10012yes—
320C2×C4#51811yescharacteristic
321C2×C4#51811yescharacteristic
322C2×C4#53412yes—
323C2×C4#53412yes—
324C2×C4#57714yes—
325C2×C4#57814yes—
326C8—84no—
327C8—84no—
328C22—28no—
329C22—24no—
330C22—22no—
331C22—24no—
332C22—42no—
333C22—42no—
334C22—42no—
335C22—42no—
336C22—44no—
337C22#230811yescharacteristic
338C22C22×#10011yescharacteristic
339C22C2×#41311yescharacteristic
340C22#289814yes—
341C4—21no—
342C4—21no—
343C4—22no—
344C4—42no—
345C4—42no—
346C4—42no—
347C4—42no—
348C4—84no—
349C4—84no—
350C4#320012yes—
351C4#320114yes—
352C4#320212yes—
353C2—44no—
354C2—42no—
355C2#1777912yes—
356C2#1778012yes—
357C2#2175211yescharacteristic
358C2#2177112yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9136C2semidirect
4. #9127C2semidirect
5. #8728C2semidirect
6. #8342C2semidirect

Maximal subgroups

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

subgroupindex
#353162
#218512
#91362
#91272
#87282
#83422