Finite Group Databasebeta

← #121030

Subgroups — #121030

Order 512.

570 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21948C211yesmaximal, characteristic
2#8115C211yesmaximal, characteristic
3#8012C211yesmaximal, characteristic
4#7943C211yesmaximal, characteristic
5C2×#1284C211yesmaximal, characteristic
6#7929C211yesmaximal, characteristic
7#7913C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12#2280C2211yescharacteristic
13C2×#381C2211yescharacteristic
14C2×C4×C22⋊C4C2211yescharacteristic
15#1855—22no—
16#1855—22no—
17#1810—22no—
18#1306—22no—
19#1294—22no—
20#1284—22no—
21#1284—22no—
22#1282—22no—
23#1280C414yescomplemented
24C24×C4C2311yescharacteristic
25C22×M16—21no—
26C22×M16—21no—
27C22×M16—21no—
28C23×C8—21no—
29C2×#117Q812yes—
30C22×C22⋊C4—21no—
31C22×C22⋊C4—21no—
32C22×C42—21no—
33C2×#108—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#407D412yes—
45#407D412yes—
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⋊C8C2×C412yes—
55C2×C22⋊C8C2×C412yes—
56C2×C22⋊C8D412yes—
57C2×C22⋊C8Q812yes—
58#381—21no—
59#381—21no—
60#381—21no—
61#381—21no—
62#381—21no—
63#380—21no—
64#380—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⋊C4D412yes—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95C2×C412yes—
75#341—24no—
76C25#4111yescharacteristic
77C23×C4—22no—
78C23×C4—41no—
79C23×C4—82no—
80C23×C4C22×C411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C23×C4C2×Q811yescharacteristic
86C23×C4C2×Q811yescharacteristic
87C23×C4#4111yescharacteristic
88C23×C4#4111yescharacteristic
89C23×C4#4111yescharacteristic
90C23×C4#4111yescharacteristic
91C23×C4#4111yescharacteristic
92C23×C4#4111yescharacteristic
93C23×C4#4111yescharacteristic
94C2×M16—22no—
95C2×M16—22no—
96C2×M16—41no—
97C2×M16—41no—
98C2×M16—41no—
99C2×M16—41no—
100C2×M16—82no—
101C2×M16—82no—
102C2×M16—82no—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—41no—
108C22×C8—41no—
109C22×C8—82no—
110#117—82no—
111C2×C4⋊C4—22no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—44no—
115C2×C22⋊C4—41no—
116C2×C22⋊C4—41no—
117C2×C22⋊C4—41no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—82no—
121C2×C42—22no—
122C2×C42—22no—
123C2×C42—41no—
124C2×C42—82no—
125#108—84no—
126C22⋊C8—22no—
127C22⋊C8—22no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—22no—
133C22⋊C8—22no—
134C22⋊C8—22no—
135C22⋊C8—41no—
136C22⋊C8—41no—
137C22⋊C8—41no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140C22⋊C8—41no—
141C22⋊C8C4⋊C414yes—
142#95—22no—
143#95—22no—
144#95—22no—
145#95—41no—
146#95—41no—
147#95—41no—
148#95—41no—
149#95—41no—
150C24—42no—
151C24—42no—
152C24—42no—
153C24#11711yescharacteristic
154C24C22⋊Q811yescharacteristic
155C24#12311yescharacteristic
156C24#12411yescharacteristic
157C24#12411yescharacteristic
158C24#12511yescharacteristic
159C24#12611yescharacteristic
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—44no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—44no—
171C22×C4—81no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4—81no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4—81no—
178C22×C4—81no—
179C22×C4—82no—
180C22×C4—82no—
181C22×C4—82no—
182C22×C4—82no—
183C22×C4C2×C4⋊C411yescharacteristic
184C22×C4#11711yescharacteristic
185C22×C4C4×D411yescharacteristic
186C22×C4C4×D411yescharacteristic
187C22×C4C4×D411yescharacteristic
188C22×C4C4×Q811yescharacteristic
189C22×C4#12011yescharacteristic
190C22×C4#12111yescharacteristic
191C22×C4#12111yescharacteristic
192C22×C4#12111yescharacteristic
193C22×C4C22⋊Q811yescharacteristic
194C22×C4C22⋊Q811yescharacteristic
195C22×C4C22⋊Q811yescharacteristic
196C22×C4C22⋊Q811yescharacteristic
197C22×C4#12311yescharacteristic
198C22×C4#12311yescharacteristic
199C22×C4#12311yescharacteristic
200C22×C4#12311yescharacteristic
201C22×C4#12411yescharacteristic
202C22×C4#12411yescharacteristic
203C22×C4#12511yescharacteristic
204C22×C4#12511yescharacteristic
205C22×C4#12511yescharacteristic
206C22×C4#12511yescharacteristic
207C22×C4#12611yescharacteristic
208C22×C4#12611yescharacteristic
209C22×C4#12611yescharacteristic
210C22×C4C4⋊Q811yescharacteristic
211M16—44no—
212M16—44no—
213M16—82no—
214M16—82no—
215M16—82no—
216M16—82no—
217C2×C8—24no—
218C2×C8—44no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—82no—
223C2×C8—82no—
224C2×C8—82no—
225C2×C8—82no—
226C2×C8—82no—
227C4⋊C4—82no—
228C22⋊C4—24no—
229C22⋊C4—42no—
230C22⋊C4—44no—
231C22⋊C4—82no—
232C22⋊C4—82no—
233C22⋊C4—82no—
234C22⋊C4—82no—
235C22⋊C4—82no—
236C42—82no—
237C42—82no—
238C42—82no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
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#38211yescharacteristic
262C23#38711yescharacteristic
263C23#38811yescharacteristic
264C23#39311yescharacteristic
265C23#39711yescharacteristic
266C23#39811yescharacteristic
267C23#40011yescharacteristic
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
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—82no—
298C2×C4—82no—
299C2×C4—88no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—164no—
307C2×C4#38511yescharacteristic
308C2×C4#38711yescharacteristic
309C2×C4#38811yescharacteristic
310C2×C4#38911yescharacteristic
311C2×C4#39411yescharacteristic
312C2×C4#39811yescharacteristic
313C2×C4#40011yescharacteristic
314C2×C4#40011yescharacteristic
315C8—44no—
316C8—82no—
317C8—82no—
318C8—82no—
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—44no—
340C22—82no—
341C22—82no—
342C22—82no—
343C22—82no—
344C22#144211yescharacteristic
345C22#183312yes—
346C22#193212yes—
347C22#203212yes—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—84no—
357C4—162no—
358C4—162no—
359C2—22no—
360C2—22no—
361C2—22no—
362C2—41no—
363C2—41no—
364C2—41no—
365C2—41no—
366C2#1125911yescharacteristic
367C2#1127911yescharacteristic
368C2#1128911yescharacteristic
subgroupG/H
center, socleC22#1442
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
23. #1280C4semidirect

Maximal subgroups

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

subgroupindex
#219482
#81152
#80122
#79432
C2×#12842
#79292
#79132