Finite Group Databasebeta

← #99101

Subgroups — #99101

Order 512.

1060 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13756C211yesmaximal, characteristic
2#10712C212yesmaximal
3#7025C212yesmaximal, complemented
4#7025C212yesmaximal, complemented
5C22×#117C411yescharacteristic
6C22×#117C411yescharacteristic
7C23×C42C2211yescharacteristic
8C22×C22⋊C8—22no—
9C2×#336—22no—
10C2×#336C2214yes—
11C4⋊#95C411yescharacteristic
12C4⋊#95C411yescharacteristic
13C4⋊#95C2211yescharacteristic
14C4⋊#95C2211yescharacteristic
15C22⋊#95—22no—
16C4×#95—22no—
17C24×C4C2×C411yescharacteristic
18C24×C4D412yes—
19C23×C8—24no—
20C2×#117—22no—
21C2×#117—22no—
22C2×#117—22no—
23C2×#117—42no—
24C2×#117—42no—
25C2×#117—42no—
26C2×#117—42no—
27C2×#117C814yescomplemented
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C22⋊C4—21no—
33C22×C22⋊C4—21no—
34C22×C42—21no—
35C22×C42—21no—
36C22×C42—22no—
37C22×C42—22no—
38C22×C42—42no—
39C22×C42—42no—
40C22×C42C2×C411yescharacteristic
41C22×C42D412yes—
42C22×C42C2311yescharacteristic
43C2×C22⋊C8—28no—
44C2×C22⋊C8—44no—
45C2×C22⋊C8—44no—
46C2×#95—22no—
47C2×#95—22no—
48C2×#95—22no—
49C2×#95—22no—
50#336—24no—
51#336—24no—
52#336—44no—
53C25C22⋊C411yescharacteristic
54C23×C4—22no—
55C23×C4—22no—
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—21no—
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—24no—
63C23×C4—24no—
64C23×C4—22no—
65C23×C4—22no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—24no—
71C23×C4—24no—
72C23×C4—42no—
73C23×C4—42no—
74C23×C4C22⋊C412yes—
75C23×C4C22⋊C411yescharacteristic
76C23×C4C2×C812yes—
77C23×C4M1612yes—
78C23×C4C22×C411yescharacteristic
79C23×C4C2×D412yes—
80C22×C8—28no—
81C22×C8—44no—
82C22×C8—44no—
83C22×C8—44no—
84C22×C8—44no—
85C22×C8—44no—
86C22×C8—44no—
87#117—48no—
88#117—48no—
89#117—48no—
90#117—48no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4C2×C812yes—
104C2×C4⋊C4C2×C812yes—
105C2×C4⋊C4M1612yes—
106C2×C4⋊C4M1612yes—
107C2×C22⋊C4—24no—
108C2×C22⋊C4—24no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—42no—
113C2×C42—24no—
114C2×C42—24no—
115C2×C42—24no—
116C2×C42—21no—
117C2×C42—21no—
118C2×C42—24no—
119C2×C42—41no—
120C2×C42—41no—
121C2×C42—42no—
122C2×C42—41no—
123C2×C42—41no—
124C2×C42—42no—
125C2×C42—48no—
126C2×C42—48no—
127C2×C42—42no—
128C2×C42—42no—
129C2×C42—42no—
130C2×C42#4112yes—
131C2×C42#4112yes—
132C22⋊C8—416no—
133#95—44no—
134#95—44no—
135#95—44no—
136#95—42no—
137#95—42no—
138C24—21no—
139C24—21no—
140C24—21no—
141C24—21no—
142C24—22no—
143C24—22no—
144C24—22no—
145C24—22no—
146C24C22⋊C812yes—
147C24#9912yes—
148C24#10112yes—
149C24C2×C22⋊C411yescharacteristic
150C22×C4—24no—
151C22×C4—24no—
152C22×C4—24no—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—24no—
156C22×C4—24no—
157C22×C4—24no—
158C22×C4—24no—
159C22×C4—24no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—22no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—28no—
179C22×C4—28no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—48no—
191C22×C4—48no—
192C22×C4—48no—
193C22×C4—48no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—44no—
197C22×C4—44no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4C22⋊C812yes—
215C22×C4#9912yes—
216C22×C4#10012yes—
217C22×C4#10414yes—
218C22×C4#11711yescharacteristic
219C22×C4#11711yescharacteristic
220C22×C4#12314yes—
221C22×C4C22×C811yescharacteristic
222C22×C4C2×M1611yescharacteristic
223C22×C4C2×M1611yescharacteristic
224C22×C4C2×M1611yescharacteristic
225C22×C4#13514yes—
226C22×C4AGL(1,ℤ/8)12yes—
227C22×C4#13712yes—
228C2×C8—48no—
229C2×C8—48no—
230C2×C8—48no—
231C2×C8—84no—
232C2×C8—84no—
233C2×C8—84no—
234C2×C8—84no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C22⋊C4—48no—
244C22⋊C4—48no—
245C42—42no—
246C42—42no—
247C42—48no—
248C42—42no—
249C42—42no—
250C42—48no—
251C42—48no—
252C23—24no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—24no—
260C23—24no—
261C23—24no—
262C23—24no—
263C23—24no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—41no—
274C23#32414yes—
275C23#38111yescharacteristic
276C23C2×C22⋊C811yescharacteristic
277C23#40711yescharacteristic
278C23C2×#9911yescharacteristic
279C23C2×#9911yescharacteristic
280C23C2×#10011yescharacteristic
281C23C2×#10111yescharacteristic
282C23#41612yes—
283C23#41712yes—
284C23C2×#10412yes—
285C23#42112yes—
286C2×C4—28no—
287C2×C4—28no—
288C2×C4—28no—
289C2×C4—28no—
290C2×C4—22no—
291C2×C4—24no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—24no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—24no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—416no—
320C2×C4—416no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4#32314yes—
346C2×C4#43111yescharacteristic
347C2×C4#43111yescharacteristic
348C2×C4#43211yescharacteristic
349C2×C4#43211yescharacteristic
350C2×C4#48214yes—
351C2×C4#48514yes—
352C8—88no—
353C22—28no—
354C22—24no—
355C22—44no—
356C22—44no—
357C22—42no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—44no—
362C22—42no—
363C22—42no—
364C22C2×#32311yescharacteristic
365C22C2×#32411yescharacteristic
366C22#146211yescharacteristic
367C22#146311yescharacteristic
368C22#147612yes—
369C22#147712yes—
370C22#148812yes—
371C22#148912yes—
372C22#150112yes—
373C22#150412yes—
374C22#151312yes—
375C22#151612yes—
376C22#152511yescharacteristic
377C22#153112yes—
378C22#153611yescharacteristic
379C22#154311yescharacteristic
380C22#155212yes—
381C22#156011yescharacteristic
382C22#181711yescharacteristic
383C22#181911yescharacteristic
384C22#182111yescharacteristic
385C22#182512yes—
386C22#182612yes—
387C4—24no—
388C4—24no—
389C4—24no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—44no—
395C4—82no—
396C4—82no—
397C4—82no—
398C4—82no—
399C2—44no—
400C2#756311yescharacteristic
401C2#757912yes—
402C2#759711yescharacteristic
403C2#761312yes—
404C2#763111yescharacteristic
405C2#764511yescharacteristic
406C2#869811yescharacteristic
407C2#870812yes—
408C2#871512yes—
409C2#872111yescharacteristic
410C2#872811yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7025C2semidirect
4. #7025C2semidirect
27. C2×#117C8semidirect

Maximal subgroups

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

subgroupindex
#137562
#107122
#70252