Finite Group Databasebeta

← #121865

Subgroups — #121865

Order 512.

549 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21949C211yesmaximal, characteristic
2#8016C211yesmaximal, characteristic
3#7983C211yesmaximal, characteristic
4#7946C211yesmaximal, characteristic
5#7927C211yesmaximal, characteristic
6#7925C211yesmaximal, characteristic
7#7915C211yesmaximal, characteristic
8C2×#407C2211yescharacteristic
9C2×#407C2211yescharacteristic
10C2×#407C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12#2280C2211yescharacteristic
13#2280C2211yescharacteristic
14C2×#380C2211yescharacteristic
15#1855—22no—
16#1855—22no—
17#1811C414yescomplemented
18#1294—22no—
19#1289—22no—
20#1284—22no—
21#1282—22no—
22#1282—22no—
23#1280—22no—
24C24×C4C2311yescharacteristic
25C22×M16—21no—
26C22×M16—21no—
27C22×M16C2×C412yes—
28C23×C8—21no—
29C2×#117—21no—
30C2×#117—21no—
31C22×C4⋊C4—21no—
32C22×C22⋊C4—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#407C2×C412yes—
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⋊C8D412yes—
56C2×C22⋊C8D412yes—
57C2×C22⋊C8Q812yes—
58#381—21no—
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#380Q812yes—
68#379—21no—
69#379—21no—
70C4×C22⋊C4—21no—
71C4×C22⋊C4—21no—
72C4×C22⋊C4—21no—
73C4×C22⋊C4D412yes—
74C2×#95D412yes—
75#341—24no—
76C25#4111yescharacteristic
77C23×C4—22no—
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—22no—
97C2×M16—41no—
98C2×M16—41no—
99C2×M16—82no—
100C2×M16—82no—
101C2×M16—82no—
102C22×C8—22no—
103C22×C8—41no—
104C22×C8—41no—
105C22×C8—41no—
106C22×C8—41no—
107C22×C8—82no—
108C22×C8C22⋊C414yes—
109#117—82no—
110#117—82no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—82no—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—41no—
122C2×C22⋊C4—41no—
123C2×C22⋊C4—41no—
124C2×C22⋊C4—82no—
125C2×C42—22no—
126C2×C42—41no—
127#108—84no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—22no—
133C22⋊C8—22no—
134C22⋊C8—22no—
135C22⋊C8—22no—
136C22⋊C8—22no—
137C22⋊C8—22no—
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—22no—
148#95—41no—
149#95—41no—
150C24—41no—
151C24—41no—
152C24—42no—
153C24—81no—
154C24C4×Q811yescharacteristic
155C24C22⋊Q811yescharacteristic
156C24#12311yescharacteristic
157C24#12311yescharacteristic
158C24#12411yescharacteristic
159C24#12611yescharacteristic
160C24#12611yescharacteristic
161C22×C4—42no—
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—44no—
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×C4C2×C22⋊C411yescharacteristic
182C22×C4#11711yescharacteristic
183C22×C4#11711yescharacteristic
184C22×C4C4×D411yescharacteristic
185C22×C4C4×D411yescharacteristic
186C22×C4C4×Q811yescharacteristic
187C22×C4#12111yescharacteristic
188C22×C4#12111yescharacteristic
189C22×C4C22⋊Q811yescharacteristic
190C22×C4C22⋊Q811yescharacteristic
191C22×C4C22⋊Q811yescharacteristic
192C22×C4C22⋊Q811yescharacteristic
193C22×C4C22⋊Q811yescharacteristic
194C22×C4#12311yescharacteristic
195C22×C4#12311yescharacteristic
196C22×C4#12311yescharacteristic
197C22×C4#12311yescharacteristic
198C22×C4#12311yescharacteristic
199C22×C4#12311yescharacteristic
200C22×C4#12411yescharacteristic
201C22×C4#12511yescharacteristic
202C22×C4#12511yescharacteristic
203C22×C4#12511yescharacteristic
204C22×C4#12511yescharacteristic
205C22×C4#12611yescharacteristic
206C22×C4#12611yescharacteristic
207C22×C4#12711yescharacteristic
208C22×C4C4⋊Q811yescharacteristic
209M16—44no—
210M16—44no—
211M16—82no—
212M16—82no—
213M16—82no—
214M16—82no—
215C2×C8—24no—
216C2×C8—44no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—82no—
221C2×C8—82no—
222C2×C8—82no—
223C2×C8—82no—
224C2×C8—82no—
225C4⋊C4—82no—
226C4⋊C4—82no—
227C4⋊C4—82no—
228C4⋊C4—82no—
229C22⋊C4—44no—
230C22⋊C4—44no—
231C22⋊C4—81no—
232C22⋊C4—81no—
233C22⋊C4—81no—
234C22⋊C4—81no—
235C22⋊C4—82no—
236C22⋊C4—82no—
237C22⋊C4—82no—
238C22⋊C4—82no—
239C42—82no—
240C42—82no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—22no—
246C23—21no—
247C23—21no—
248C23—22no—
249C23—22no—
250C23—41no—
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#39111yescharacteristic
270C23#39711yescharacteristic
271C23#39711yescharacteristic
272C23#39911yescharacteristic
273C23#40011yescharacteristic
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
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—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—164no—
312C2×C4#38111yescharacteristic
313C2×C4#38711yescharacteristic
314C2×C4#38711yescharacteristic
315C2×C4#39011yescharacteristic
316C2×C4#39811yescharacteristic
317C2×C4#39811yescharacteristic
318C2×C4#39911yescharacteristic
319C2×C4#40011yescharacteristic
320C8—44no—
321C8—82no—
322C8—82no—
323C8—82no—
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—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—42no—
344C22—42no—
345C22—42no—
346C22—42no—
347C22—81no—
348C22—82no—
349C22—82no—
350C22—81no—
351C22—82no—
352C22—82no—
353C22#144511yescharacteristic
354C22#208111yescharacteristic
355C22#208211yescharacteristic
356C22#209812yes—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—162no—
365C4—162no—
366C4—162no—
367C4#182812yes—
368C2—21no—
369C2—22no—
370C2—21no—
371C2—41no—
372C2—41no—
373C2—41no—
374C2—41no—
375C2—41no—
376C2#1135111yescharacteristic
377C2#1136211yescharacteristic
378C2#1137411yescharacteristic
subgroupG/H
center, socleC22#1445
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
17. #1811C4semidirect

Maximal subgroups

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

subgroupindex
#219492
#80162
#79832
#79462
#79272
#79252
#79152