Finite Group Databasebeta

← #120868

Subgroups — #120868

Order 512.

563 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21940C211yesmaximal, characteristic
2#8107C211yesmaximal, characteristic
3C2×#1305C211yesmaximal, characteristic
4#8057C211yesmaximal, characteristic
5#8016C211yesmaximal, characteristic
6#7983C211yesmaximal, characteristic
7#7925C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12#2280C2211yescharacteristic
13C2×#380C2211yescharacteristic
14C2×#380C2211yescharacteristic
15#1852—22no—
16#1852—22no—
17#1808—22no—
18#1305—22no—
19#1305—22no—
20#1298C414yescomplemented
21#1294—22no—
22#1289—22no—
23#1282—22no—
24C24×C4C2311yescharacteristic
25C22×M16—21no—
26C22×M16—21no—
27C22×M16Q812yes—
28C23×C8—21no—
29C2×#117—21no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—21no—
32C22×C22⋊C4—21no—
33C22×C22⋊C4—21no—
34C2×#107—24no—
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#407—21no—
46#407C2×C412yes—
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⋊C8C2×C412yes—
56C2×C22⋊C8C2×C412yes—
57C2×C22⋊C8D412yes—
58C2×C22⋊C8Q812yes—
59#381—21no—
60#380—21no—
61#380—21no—
62#380—21no—
63#380—21no—
64#380—21no—
65#380—21no—
66#380—21no—
67#380D412yes—
68#380D412yes—
69#380D412yes—
70#379—21no—
71C4×C22⋊C4—21no—
72C4×C22⋊C4—21no—
73C2×#95—21no—
74C2×#95—21no—
75#340—24no—
76C25#4111yescharacteristic
77C23×C4—41no—
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—22no—
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—22no—
108C22×C8—41no—
109C22×C8—82no—
110#117—82no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—82no—
117C2×C4⋊C4—82no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—41no—
122C2×C22⋊C4—41no—
123C2×C22⋊C4—41no—
124C2×C22⋊C4—41no—
125C2×C22⋊C4—82no—
126C2×C22⋊C4—82no—
127C2×C42—41no—
128#107—84no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—22no—
133C22⋊C8—22no—
134C22⋊C8—22no—
135C22⋊C8—22no—
136C22⋊C8—41no—
137C22⋊C8—41no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140C22⋊C8—41no—
141C22⋊C8—41no—
142C22⋊C8—41no—
143C22⋊C8—41no—
144C22⋊C8C4⋊C414yes—
145#95—22no—
146#95—22no—
147#95—22no—
148#95—41no—
149#95—41no—
150#95—41no—
151C24—22no—
152C24—22no—
153C24—81no—
154C24—81no—
155C24C4×Q811yescharacteristic
156C24#12111yescharacteristic
157C24#12311yescharacteristic
158C24#12311yescharacteristic
159C24#12511yescharacteristic
160C24#12611yescharacteristic
161C24#12611yescharacteristic
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
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×C4—82no—
184C22×C4—82no—
185C22×C4C2×C4⋊C411yescharacteristic
186C22×C4#11711yescharacteristic
187C22×C4#11711yescharacteristic
188C22×C4C4×D411yescharacteristic
189C22×C4C4×D411yescharacteristic
190C22×C4C4×D411yescharacteristic
191C22×C4#12011yescharacteristic
192C22×C4#12111yescharacteristic
193C22×C4#12111yescharacteristic
194C22×C4C22⋊Q811yescharacteristic
195C22×C4C22⋊Q811yescharacteristic
196C22×C4C22⋊Q811yescharacteristic
197C22×C4C22⋊Q811yescharacteristic
198C22×C4C22⋊Q811yescharacteristic
199C22×C4#12311yescharacteristic
200C22×C4#12311yescharacteristic
201C22×C4#12311yescharacteristic
202C22×C4#12411yescharacteristic
203C22×C4#12411yescharacteristic
204C22×C4#12411yescharacteristic
205C22×C4#12411yescharacteristic
206C22×C4#12511yescharacteristic
207C22×C4#12511yescharacteristic
208C22×C4#12511yescharacteristic
209C22×C4#12511yescharacteristic
210C22×C4#12611yescharacteristic
211C22×C4#12611yescharacteristic
212C22×C4C4⋊Q811yescharacteristic
213M16—82no—
214M16—82no—
215M16—82no—
216M16—82no—
217M16—82no—
218M16—82no—
219C2×C8—24no—
220C2×C8—44no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—82no—
225C2×C8—82no—
226C2×C8—82no—
227C2×C8—82no—
228C2×C8—82no—
229C4⋊C4—44no—
230C4⋊C4—44no—
231C4⋊C4—82no—
232C4⋊C4—82no—
233C4⋊C4—82no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—82no—
237C22⋊C4—82no—
238C22⋊C4—82no—
239C22⋊C4—82no—
240C22⋊C4—82no—
241C22⋊C4—82no—
242C42—82no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—84no—
266C23—84no—
267C23#38711yescharacteristic
268C23#38711yescharacteristic
269C23#38911yescharacteristic
270C23#39411yescharacteristic
271C23#39711yescharacteristic
272C23#40011yescharacteristic
273C23#40011yescharacteristic
274C23#47212yes—
275C23#47312yes—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
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—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—164no—
314C2×C4—164no—
315C2×C4#38211yescharacteristic
316C2×C4#38511yescharacteristic
317C2×C4#38811yescharacteristic
318C2×C4#38811yescharacteristic
319C2×C4#39311yescharacteristic
320C2×C4#39811yescharacteristic
321C2×C4#39811yescharacteristic
322C2×C4#40011yescharacteristic
323C8—44no—
324C8—82no—
325C8—82no—
326C8—82no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—42no—
346C22—42no—
347C22—42no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22—82no—
352C22—82no—
353C22—82no—
354C22—82no—
355C22#144211yescharacteristic
356C22#195212yes—
357C22#204511yescharacteristic
358C22#204711yescharacteristic
359C22#207812yes—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—162no—
368C4—162no—
369C4—162no—
370C4#203212yes—
371C2—21no—
372C2—22no—
373C2—21no—
374C2—22no—
375C2—41no—
376C2—41no—
377C2—41no—
378C2—41no—
379C2#1126511yescharacteristic
380C2#1127011yescharacteristic
381C2#1127911yescharacteristic
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
20. #1298C4semidirect

Maximal subgroups

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

subgroupindex
#219402
#81072
C2×#13052
#80572
#80162
#79832
#79252