Finite Group Databasebeta

← #120897

Subgroups — #120897

Order 512.

535 conjugacy classes of subgroups.

The subgroup lattice has 397 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21944C211yesmaximal, characteristic
2#8115C211yesmaximal, characteristic
3#8108C211yesmaximal, characteristic
4#8107C211yesmaximal, characteristic
5#8055C211yesmaximal, characteristic
6#8049C211yesmaximal, characteristic
7#7983C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C2×#407C2211yescharacteristic
12#2280C2211yescharacteristic
13#2280C2211yescharacteristic
14C2×#380C2211yescharacteristic
15#1854—22no—
16#1853—21no—
17#1852—21no—
18#1809—22no—
19#1306C414yescomplemented
20#1305—22no—
21#1305—21no—
22#1305—21no—
23#1298—22no—
24#1297—22no—
25#1289—22no—
26C24×C4C2311yescharacteristic
27C22×M16—21no—
28C22×M16—21no—
29C22×M16—21no—
30C22×M16Q812yes—
31C2×#117—21no—
32C2×#117—21no—
33C22×C4⋊C4—21no—
34C22×C22⋊C4—21no—
35C2×#107—22no—
36C2×#106—22no—
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#407—21no—
47#407—21no—
48#407—21no—
49#407C2×C412yes—
50#407C2×C412yes—
51#407D412yes—
52#407Q812yes—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8D412yes—
61#381—21no—
62#381C2×C412yes—
63#380—21no—
64#380—21no—
65#380—21no—
66#380—21no—
67#380—21no—
68#380—21no—
69#380D412yes—
70#380D411yescharacteristic
71#380D411yescharacteristic
72#379—21no—
73#379—21no—
74C4×C22⋊C4—21no—
75C4×C22⋊C4—21no—
76C4×C22⋊C4—21no—
77C4×C22⋊C4—21no—
78C2×#95—21no—
79#340—22no—
80#340—22no—
81C25#4111yescharacteristic
82C23×C4—41no—
83C23×C4—82no—
84C23×C4C22×C411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C23×C4C2×D411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C23×C4C2×D411yescharacteristic
89C23×C4C2×Q811yescharacteristic
90C23×C4C2×Q811yescharacteristic
91C23×C4#4111yescharacteristic
92C23×C4#4111yescharacteristic
93C23×C4#4111yescharacteristic
94C23×C4#4111yescharacteristic
95C23×C4#4111yescharacteristic
96C23×C4#4111yescharacteristic
97C23×C4#4111yescharacteristic
98C2×M16—22no—
99C2×M16—22no—
100C2×M16—22no—
101C2×M16—22no—
102C2×M16—22no—
103C2×M16—22no—
104C2×M16—41no—
105C2×M16—41no—
106C2×M16—82no—
107C2×M16—82no—
108C2×M16—82no—
109C2×M16—82no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—41no—
113C22×C8—41no—
114#117—82no—
115#117—82no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—82no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—41no—
127C2×C22⋊C4—41no—
128C2×C22⋊C4—41no—
129C2×C22⋊C4—41no—
130C2×C22⋊C4—41no—
131C2×C22⋊C4—82no—
132C2×C42—41no—
133C2×C42—41no—
134#107—82no—
135#106—82no—
136C22⋊C8—22no—
137C22⋊C8—22no—
138C22⋊C8—22no—
139C22⋊C8—22no—
140C22⋊C8—22no—
141C22⋊C8—22no—
142C22⋊C8—22no—
143C22⋊C8—41no—
144C22⋊C8—41no—
145C22⋊C8—41no—
146C22⋊C8—41no—
147C22⋊C8—41no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150C22⋊C8—41no—
151C22⋊C8C4⋊C414yes—
152#95—22no—
153#95—22no—
154#95—21no—
155#95—21no—
156#95—22no—
157#95—41no—
158#95—41no—
159C24—41no—
160C24—41no—
161C24—81no—
162C24—81no—
163C24#11711yescharacteristic
164C24#12111yescharacteristic
165C24#12311yescharacteristic
166C24#12411yescharacteristic
167C24#12511yescharacteristic
168C24#12511yescharacteristic
169C24#12611yescharacteristic
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4—81no—
182C22×C4—81no—
183C22×C4—81no—
184C22×C4—81no—
185C22×C4—81no—
186C22×C4—81no—
187C22×C4—81no—
188C22×C4—82no—
189C22×C4—82no—
190C22×C4—82no—
191C22×C4C2×C4⋊C411yescharacteristic
192C22×C4#11711yescharacteristic
193C22×C4C4×D411yescharacteristic
194C22×C4C4×D411yescharacteristic
195C22×C4C4×D411yescharacteristic
196C22×C4C4×Q811yescharacteristic
197C22×C4#12011yescharacteristic
198C22×C4#12111yescharacteristic
199C22×C4#12111yescharacteristic
200C22×C4C22⋊Q811yescharacteristic
201C22×C4C22⋊Q811yescharacteristic
202C22×C4C22⋊Q811yescharacteristic
203C22×C4C22⋊Q811yescharacteristic
204C22×C4C22⋊Q811yescharacteristic
205C22×C4#12311yescharacteristic
206C22×C4#12311yescharacteristic
207C22×C4#12311yescharacteristic
208C22×C4#12311yescharacteristic
209C22×C4#12411yescharacteristic
210C22×C4#12411yescharacteristic
211C22×C4#12411yescharacteristic
212C22×C4#12511yescharacteristic
213C22×C4#12511yescharacteristic
214C22×C4#12511yescharacteristic
215C22×C4#12611yescharacteristic
216C22×C4#12611yescharacteristic
217C22×C4#12611yescharacteristic
218C22×C4C4⋊Q811yescharacteristic
219M16—82no—
220M16—82no—
221M16—82no—
222M16—82no—
223M16—82no—
224M16—82no—
225M16—82no—
226M16—82no—
227C2×C8—24no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—82no—
234C2×C8—82no—
235C2×C8—82no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—82no—
241C4⋊C4—82no—
242C22⋊C4—81no—
243C22⋊C4—81no—
244C22⋊C4—82no—
245C22⋊C4—82no—
246C22⋊C4—82no—
247C22⋊C4—82no—
248C22⋊C4—81no—
249C22⋊C4—81no—
250C22⋊C4—82no—
251C22⋊C4—82no—
252C42—82no—
253C42—82no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—84no—
279C23—84no—
280C23#38211yescharacteristic
281C23#38711yescharacteristic
282C23#38811yescharacteristic
283C23#39411yescharacteristic
284C23#39811yescharacteristic
285C23#40011yescharacteristic
286C23#40011yescharacteristic
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
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—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—82no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—164no—
328C2×C4#38511yescharacteristic
329C2×C4#38711yescharacteristic
330C2×C4#38811yescharacteristic
331C2×C4#38911yescharacteristic
332C2×C4#39311yescharacteristic
333C2×C4#39711yescharacteristic
334C2×C4#39811yescharacteristic
335C2×C4#40011yescharacteristic
336C8—44no—
337C8—82no—
338C8—82no—
339C8—82no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—42no—
361C22—42no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—82no—
367C22—82no—
368C22—82no—
369C22—82no—
370C22#144211yescharacteristic
371C22#204811yescharacteristic
372C22#204911yescharacteristic
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—161no—
381C4—161no—
382C4—162no—
383C4—162no—
384C4#203212yes—
385C2—21no—
386C2—21no—
387C2—41no—
388C2—41no—
389C2—41no—
390C2—41no—
391C2—41no—
392C2—41no—
393C2#1127411yescharacteristic
394C2#1127911yescharacteristic
395C2#1128411yescharacteristic
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
19. #1306C4semidirect

Maximal subgroups

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

subgroupindex
#219442
#81152
#81082
#81072
#80552
#80492
#79832