Finite Group Databasebeta

← #480681

Subgroups — #480681

Order 512.

1066 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1066 conjugacy classes of subgroups — too many to draw. Only its 194 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2500C211yesmaximal, characteristic
2#17357C212yesmaximal, complemented
3#11554C218yesmaximal, complemented
4#11512C212yesmaximal, complemented
5#11505C212yesmaximal, complemented
6C22×#121C411yescharacteristic
7C22×#117C2211yescharacteristic
8#2920—22no—
9#2500—22no—
10#2500—22no—
11#2500C414yes—
12#2500C2214yes—
13C2×#394—22no—
14C2×#390C411yescharacteristic
15C2×#390C2211yescharacteristic
16C2×#388—22no—
17C2×#384C2211yescharacteristic
18C2×#382C411yescharacteristic
19C2×C4×C22⋊C4C411yescharacteristic
20#1464—22no—
21#1464C2214yescomplemented
22#1463C2214yes—
23#1462—22no—
24#1458—22no—
25#1458C2214yescomplemented
26#1325C2218yescomplemented
27#1318C2218yescomplemented
28C23×D4—21no—
29C23×D4C2×C411yescharacteristic
30C24×C4C2×C411yescharacteristic
31C2×#121—22no—
32C2×#121—41no—
33C2×#121—41no—
34C2×#121—42no—
35C2×#121C2×C414yes—
36C2×#117—42no—
37C2×#117—41no—
38C2×#117—41no—
39C2×#117C2×C412yes—
40C2×#117D412yes—
41C2×#117D412yes—
42C2×#117C2312yes—
43C22×C4⋊C4—22no—
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2311yescharacteristic
47C22×C22⋊C4—21no—
48C22×C22⋊C4—22no—
49C22×C22⋊C4—21no—
50C22×C22⋊C4—22no—
51C22×C22⋊C4D412yes—
52C22×C42C2×C411yescharacteristic
53C22×C42C2×C411yescharacteristic
54C22×C42D412yes—
55#432—24no—
56#432—42no—
57#431—24no—
58#431—42no—
59#423—42no—
60#407—24no—
61C4×M16—42no—
62#394—24no—
63#394—24no—
64#394—24no—
65#394—24no—
66#390—21no—
67#390—21no—
68#390—22no—
69#390C2×C412yes—
70#390C2×C414yes—
71#390C2312yes—
72#388—24no—
73#388—24no—
74#388—24no—
75#388—24no—
76#384—22no—
77#384C2×C412yes—
78#384C2312yes—
79#382—21no—
80#382—21no—
81#382C2×C414yes—
82C4×C22⋊C4—22no—
83C4×C22⋊C4C2×C414yes—
84C2×#95—22no—
85C2×#95—21no—
86#324—24no—
87#324C2318yes—
88#323—24no—
89#323—24no—
90C25—21no—
91C25C22⋊C411yescharacteristic
92C22×D4—24no—
93C22×D4—21no—
94C22×D4—42no—
95C22×D4—41no—
96C22×D4—41no—
97C22×D4—42no—
98C22×D4—42no—
99C22×D4—42no—
100C22×D4—84no—
101C22×D4C22×C412yes—
102C23×C4—21no—
103C23×C4—22no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—42no—
109C23×C4—42no—
110C23×C4C22⋊C412yes—
111C23×C4C22⋊C412yes—
112C23×C4C22⋊C411yescharacteristic
113C23×C4C22×C411yescharacteristic
114C23×C4C22×C412yes—
115C23×C4C2×D412yes—
116C2×M16—44no—
117#121—44no—
118#121—44no—
119#121—44no—
120#121—44no—
121#117—42no—
122#117—42no—
123#117—42no—
124#117—42no—
125#117—42no—
126#117—42no—
127#117—42no—
128#117—42no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—84no—
139C2×C4⋊C4C22⋊C411yescharacteristic
140C2×C4⋊C4C22⋊C411yescharacteristic
141C2×C4⋊C4C22×C412yes—
142C2×C4⋊C4C22×C412yes—
143C2×C4⋊C4C22×C411yescharacteristic
144C2×C4⋊C4C2×D412yes—
145C2×C4⋊C4C2411yescharacteristic
146C2×C22⋊C4—24no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—42no—
153C2×C22⋊C4—42no—
154C2×C22⋊C4—48no—
155C2×C22⋊C4—44no—
156C2×C22⋊C4—41no—
157C2×C22⋊C4—41no—
158C2×C22⋊C4—42no—
159C2×C22⋊C4—84no—
160C2×C22⋊C4C22⋊C412yes—
161C2×C22⋊C4C22⋊C412yes—
162C2×C22⋊C4C2×D412yes—
163C2×C22⋊C4C2×D412yes—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—42no—
171C2×C42—42no—
172C2×C42C22⋊C412yes—
173C2×C42C22⋊C412yes—
174C2×C42C22×C412yes—
175C2×C42C22×C412yes—
176C2×C42C2×D412yes—
177C2×C42C2×D412yes—
178C4⋊C8—44no—
179C4⋊C8—44no—
180C22⋊C8—28no—
181C22⋊C8—44no—
182#97—44no—
183C4×C8—44no—
184#95—24no—
185#95—24no—
186#95—24no—
187C24—21no—
188C24—21no—
189C24—22no—
190C24—41no—
191C24—41no—
192C24—42no—
193C24—42no—
194C24—44no—
195C24—44no—
196C24—42no—
197C24—42no—
198C24—41no—
199C24—41no—
200C24C2×C22⋊C411yescharacteristic
201C24C2×C22⋊C412yes—
202C2×D4—24no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4—48no—
208C2×D4—42no—
209C2×D4—84no—
210C2×D4—84no—
211C2×D4—88no—
212C2×D4—88no—
213C2×D4—88no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—48no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—44no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4#9912yes—
250C22×C4#9911yescharacteristic
251C22×C4#9911yescharacteristic
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C412yes—
257C22×C4C23×C411yescharacteristic
258C22×C4C22×D412yes—
259M16—88no—
260C2×C8—48no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—84no—
270C4⋊C4—84no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—48no—
276C22⋊C4—48no—
277C22⋊C4—48no—
278C22⋊C4—44no—
279C22⋊C4—44no—
280C22⋊C4—84no—
281C22⋊C4—84no—
282C42—22no—
283C42—22no—
284C42—22no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—24no—
291C42—42no—
292C42—42no—
293C42—42no—
294C23—21no—
295C23—21no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—42no—
307C23—44no—
308C23—44no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—84no—
315C23—84no—
316C23—82no—
317C23—82no—
318C23—82no—
319C23—88no—
320C23—88no—
321C23—82no—
322C23—82no—
323C23—82no—
324C23C2×#9911yescharacteristic
325C23C2×#9911yescharacteristic
326C23C22×C22⋊C411yescharacteristic
327D4—84no—
328D4—84no—
329D4—816no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—48no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—84no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4C2×#9912yes—
370C2×C4C2×#9912yes—
371C2×C4#41312yes—
372C2×C4#56812yes—
373C8—88no—
374C22—24no—
375C22—24no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—82no—
386C22—82no—
387C22—84no—
388C22—84no—
389C22—84no—
390C22—84no—
391C22—82no—
392C22—82no—
393C22—82no—
394C22—164no—
395C22—164no—
396C22#212812yes—
397C22#213212yes—
398C22#287911yescharacteristic
399C22C22×#9911yescharacteristic
400C22C2×#41311yescharacteristic
401C4—22no—
402C4—22no—
403C4—22no—
404C4—22no—
405C4—44no—
406C4—42no—
407C4—42no—
408C4—42no—
409C4—42no—
410C4—82no—
411C4—82no—
412C2—21no—
413C2—21no—
414C2—22no—
415C2—22no—
416C2—82no—
417C2—162no—
418C2#1739211yescharacteristic
419C2C2×#212811yescharacteristic
420C2C2×#213211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×#121C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17357C2semidirect
3. #11554C2semidirect
4. #11512C2semidirect
5. #11505C2semidirect
21. #1464C22semidirect
25. #1458C22semidirect
26. #1325C22semidirect
27. #1318C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25002
#173572
#115542
#115122
#115052