Finite Group Databasebeta

← #120892

Subgroups — #120892

Order 512.

569 conjugacy classes of subgroups.

The subgroup lattice has 369 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21933C211yesmaximal, characteristic
2#8104C211yesmaximal, characteristic
3#8057C211yesmaximal, characteristic
4#8049C211yesmaximal, characteristic
5#8012C211yesmaximal, characteristic
6#7942C211yesmaximal, characteristic
7#7929C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12#2280C2211yescharacteristic
13#2280C2211yescharacteristic
14C2×C4×C22⋊C4C2211yescharacteristic
15#1851—22no—
16#1850—22no—
17#1798—22no—
18#1305—22no—
19#1298—22no—
20#1297—22no—
21#1294—22no—
22#1284C414yescomplemented
23#1282—22no—
24C24×C4C2311yescharacteristic
25C22×M16—21no—
26C22×M16—21no—
27C22×M16Q812yes—
28C23×C8—21no—
29C2×#117—21no—
30C2×#117—21no—
31C22×C22⋊C4—21no—
32C22×C42—21no—
33C2×C4⋊C8—24no—
34#407—21no—
35#407—21no—
36#407—21no—
37#407—21no—
38#407—21no—
39#407—21no—
40#407—21no—
41#407—21no—
42#407—21no—
43#407—21no—
44#407—21no—
45#407C2×C412yes—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8D412yes—
57C2×C22⋊C8D412yes—
58#381—21no—
59#381C2×C412yes—
60#380—21no—
61#380—21no—
62#380D412yes—
63#380Q812yes—
64#379—21no—
65#379—21no—
66C4×C22⋊C4—21no—
67C4×C22⋊C4—21no—
68C4×C22⋊C4—21no—
69C4×C22⋊C4—21no—
70C4×C22⋊C4—21no—
71C4×C22⋊C4—21no—
72C4×C22⋊C4C2×C412yes—
73C4×C22⋊C4D412yes—
74C2×#95—21no—
75#336—24no—
76C25#4111yescharacteristic
77C23×C4—41no—
78C23×C4—82no—
79C23×C4C22×C411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C23×C4C2×Q811yescharacteristic
85C23×C4C2×Q811yescharacteristic
86C23×C4#4111yescharacteristic
87C23×C4#4111yescharacteristic
88C23×C4#4111yescharacteristic
89C23×C4#4111yescharacteristic
90C23×C4#4111yescharacteristic
91C23×C4#4111yescharacteristic
92C23×C4#4111yescharacteristic
93C2×M16—22no—
94C2×M16—22no—
95C2×M16—22no—
96C2×M16—41no—
97C2×M16—41no—
98C2×M16—41no—
99C2×M16—82no—
100C2×M16—82no—
101C2×M16—82no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8—41no—
106C22×C8—41no—
107C22×C8—41no—
108C22×C8—82no—
109#117—82no—
110#117—82no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—44no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—41no—
122C2×C42—22no—
123C2×C42—22no—
124C2×C42—41no—
125C2×C42—41no—
126C2×C42—82no—
127C4⋊C8—84no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—22no—
133C22⋊C8—22no—
134C22⋊C8—41no—
135C22⋊C8—41no—
136C22⋊C8—41no—
137C22⋊C8—41no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140C22⋊C8—41no—
141C22⋊C8—41no—
142C22⋊C8—41no—
143C22⋊C8—41no—
144#95—22no—
145#95—22no—
146#95—22no—
147#95—41no—
148#95—41no—
149#95C4⋊C414yes—
150C24—42no—
151C24—42no—
152C24—81no—
153C24C4×D411yescharacteristic
154C24C22⋊Q811yescharacteristic
155C24#12311yescharacteristic
156C24#12311yescharacteristic
157C24#12511yescharacteristic
158C24#12611yescharacteristic
159C24#12611yescharacteristic
160C22×C4—24no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—44no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—81no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4—81no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4—81no—
178C22×C4—82no—
179C22×C4—82no—
180C22×C4C2×C4⋊C411yescharacteristic
181C22×C4#11711yescharacteristic
182C22×C4#11711yescharacteristic
183C22×C4C4×D411yescharacteristic
184C22×C4C4×D411yescharacteristic
185C22×C4C4×Q811yescharacteristic
186C22×C4#12011yescharacteristic
187C22×C4#12111yescharacteristic
188C22×C4#12111yescharacteristic
189C22×C4#12111yescharacteristic
190C22×C4C22⋊Q811yescharacteristic
191C22×C4C22⋊Q811yescharacteristic
192C22×C4C22⋊Q811yescharacteristic
193C22×C4C22⋊Q811yescharacteristic
194C22×C4#12311yescharacteristic
195C22×C4#12311yescharacteristic
196C22×C4#12311yescharacteristic
197C22×C4#12411yescharacteristic
198C22×C4#12411yescharacteristic
199C22×C4#12411yescharacteristic
200C22×C4#12411yescharacteristic
201C22×C4#12511yescharacteristic
202C22×C4#12511yescharacteristic
203C22×C4#12511yescharacteristic
204C22×C4#12511yescharacteristic
205C22×C4#12611yescharacteristic
206C22×C4#12611yescharacteristic
207C22×C4C4⋊Q811yescharacteristic
208M16—82no—
209M16—82no—
210M16—82no—
211M16—82no—
212M16—82no—
213M16—82no—
214C2×C8—44no—
215C2×C8—44no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—82no—
221C2×C8—82no—
222C2×C8—82no—
223C2×C8—82no—
224C4⋊C4—82no—
225C4⋊C4—82no—
226C22⋊C4—24no—
227C22⋊C4—44no—
228C22⋊C4—42no—
229C22⋊C4—44no—
230C22⋊C4—82no—
231C22⋊C4—82no—
232C22⋊C4—82no—
233C22⋊C4—82no—
234C22⋊C4—82no—
235C42—44no—
236C42—82no—
237C42—82no—
238C42—82no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—88no—
261C23#38511yescharacteristic
262C23#38711yescharacteristic
263C23#38811yescharacteristic
264C23#39711yescharacteristic
265C23#39811yescharacteristic
266C23#40011yescharacteristic
267C23#40011yescharacteristic
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—88no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4#38211yescharacteristic
307C2×C4#38711yescharacteristic
308C2×C4#38811yescharacteristic
309C2×C4#38911yescharacteristic
310C2×C4#39311yescharacteristic
311C2×C4#39411yescharacteristic
312C2×C4#39811yescharacteristic
313C2×C4#40011yescharacteristic
314C8—82no—
315C8—82no—
316C8—82no—
317C8—82no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—44no—
338C22—44no—
339C22—82no—
340C22—82no—
341C22—82no—
342C22—82no—
343C22—82no—
344C22#144211yescharacteristic
345C22#186112yes—
346C22#208012yes—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—84no—
355C4—84no—
356C4—162no—
357C4#203212yes—
358C2—22no—
359C2—22no—
360C2—41no—
361C2—41no—
362C2—41no—
363C2—41no—
364C2—41no—
365C2#1127511yescharacteristic
366C2#1127911yescharacteristic
367C2#1128111yescharacteristic
subgroupG/H
centerC2×C4#393
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
socleC22#1442
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
22. #1284C4semidirect

Maximal subgroups

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

subgroupindex
#219332
#81042
#80572
#80492
#80122
#79422
#79292