Finite Group Databasebeta

← #480675

Subgroups — #480675

Order 512.

1036 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1036 conjugacy classes of subgroups — too many to draw. Only its 202 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×#2503C211yesmaximal, characteristic
2C2×#1463C212yesmaximal
3#11551C218yesmaximal, complemented
4#11511C212yesmaximal, complemented
5#11506C212yesmaximal, complemented
6C22×C22⋊Q8C411yescharacteristic
7C23×C4⋊C4C2211yescharacteristic
8C2×#407—22no—
9#2503—22no—
10#2503—22no—
11#2503C414yes—
12#2503C2214yes—
13C2×#400—22no—
14C2×#391C411yescharacteristic
15C2×#391C2211yescharacteristic
16C2×#382C411yescharacteristic
17C2×#382C2211yescharacteristic
18C2×#381—22no—
19C2×C4×C22⋊C4C411yescharacteristic
20#1463—22no—
21#1463—22no—
22#1463C2214yes—
23#1463C2214yes—
24C2×#324—22no—
25C2×#324C2214yes—
26#1324C2218yescomplemented
27#1319C2218yescomplemented
28C23×Q8C2×C411yescharacteristic
29C24×C4C2×C411yescharacteristic
30C24×C4D412yes—
31C22×M16—42no—
32C2×C22⋊Q8—22no—
33C2×C22⋊Q8—42no—
34C2×C22⋊Q8—41no—
35C2×C22⋊Q8—41no—
36C2×C22⋊Q8C2×C414yes—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—41no—
39C22×C4⋊C4—42no—
40C22×C4⋊C4—41no—
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C412yes—
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4D412yes—
45C22×C4⋊C4D412yes—
46C22×C4⋊C4D412yes—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2311yescharacteristic
49C22×C22⋊C4—21no—
50C22×C22⋊C4—21no—
51C22×C42C2×C411yescharacteristic
52C22×C42C2×C411yescharacteristic
53#407—24no—
54#407—24no—
55#407—42no—
56#407—42no—
57C2×C22⋊C8—24no—
58C2×C22⋊C8—42no—
59#400—24no—
60#400—24no—
61#400—24no—
62#400—24no—
63#391—21no—
64#391—21no—
65#391—22no—
66#391C2×C412yes—
67#391C2×C414yes—
68#391C2312yes—
69#382—21no—
70#382—21no—
71#382—22no—
72#382C2×C412yes—
73#382C2×C414yes—
74#382C2312yes—
75#381—24no—
76#381—24no—
77#381—24no—
78#381—24no—
79C4×C22⋊C4—22no—
80C4×C22⋊C4C2×C414yes—
81C2×#95—21no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86#324—24no—
87#324—24no—
88#324—24no—
89#324C2318yes—
90C25C22⋊C411yescharacteristic
91C22×Q8—21no—
92C22×Q8—42no—
93C22×Q8—42no—
94C22×Q8—41no—
95C22×Q8—41no—
96C22×Q8C22×C412yes—
97C23×C4—22no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—42no—
109C23×C4—42no—
110C23×C4—48no—
111C23×C4C22⋊C412yes—
112C23×C4C22⋊C411yescharacteristic
113C23×C4C22⋊C412yes—
114C23×C4C22⋊C412yes—
115C23×C4C22⋊C412yes—
116C23×C4C22×C411yescharacteristic
117C23×C4C22×C412yes—
118C23×C4C2×D412yes—
119C23×C4C2×D412yes—
120C23×C4C2×D412yes—
121C2×M16—44no—
122C2×M16—44no—
123C2×M16—88no—
124C22×C8—44no—
125C22⋊Q8—44no—
126C22⋊Q8—44no—
127C22⋊Q8—44no—
128C22⋊Q8—44no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4C22⋊C411yescharacteristic
150C2×C4⋊C4C22⋊C411yescharacteristic
151C2×C4⋊C4C22⋊C412yes—
152C2×C4⋊C4C22⋊C412yes—
153C2×C4⋊C4C22×C412yes—
154C2×C4⋊C4C22×C411yescharacteristic
155C2×C4⋊C4C22×C412yes—
156C2×C4⋊C4C2×D412yes—
157C2×C4⋊C4C2×D412yes—
158C2×C4⋊C4C2×D412yes—
159C2×C4⋊C4C2411yescharacteristic
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—44no—
163C2×C22⋊C4—42no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—42no—
169C2×C42—42no—
170C2×C42C22×C412yes—
171C2×C42C22×C412yes—
172C22⋊C8—28no—
173C22⋊C8—44no—
174C22⋊C8—44no—
175C22⋊C8—44no—
176#95—24no—
177#95—24no—
178#95—24no—
179#95—24no—
180#95—24no—
181#95—24no—
182#95—24no—
183#95—24no—
184#95—24no—
185C24—24no—
186C24—41no—
187C24—41no—
188C24—42no—
189C24#9912yes—
190C24#9911yescharacteristic
191C24#9911yescharacteristic
192C24C2×C22⋊C411yescharacteristic
193C24C2×C22⋊C412yes—
194C2×Q8—24no—
195C2×Q8—42no—
196C2×Q8—42no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—84no—
201C2×Q8—84no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—48no—
236C22×C4—44no—
237C22×C4—48no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—44no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—84no—
258C22×C4—84no—
259C22×C4—84no—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C412yes—
265C22×C4C23×C411yescharacteristic
266C22×C4C22×D412yes—
267M16—88no—
268M16—88no—
269C2×C8—48no—
270C2×C8—88no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—48no—
284C22⋊C4—48no—
285C22⋊C4—44no—
286C22⋊C4—44no—
287C42—24no—
288C42—42no—
289C42—42no—
290C42—42no—
291C23—21no—
292C23—21no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—24no—
298C23—24no—
299C23—22no—
300C23—24no—
301C23—41no—
302C23—42no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—42no—
307C23—48no—
308C23—48no—
309C23#35414yes—
310C23C2×#9912yes—
311C23C2×#9912yes—
312C23C2×#9911yescharacteristic
313C23C2×#9911yescharacteristic
314C23C22×C22⋊C411yescharacteristic
315Q8—84no—
316Q8—84no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—84no—
364C2×C4—84no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4#41312yes—
370C2×C4#56812yes—
371C8—88no—
372C22—24no—
373C22—24no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—21no—
380C22—21no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—44no—
386C22—44no—
387C22—44no—
388C22—48no—
389C22—44no—
390C22C2×#35412yes—
391C22C2×#35414yes—
392C22#213312yes—
393C22#287911yescharacteristic
394C22C22×#9911yescharacteristic
395C22C2×#41311yescharacteristic
396C4—22no—
397C4—22no—
398C4—42no—
399C4—42no—
400C4—42no—
401C4—42no—
402C4—42no—
403C4—82no—
404C4—82no—
405C4—82no—
406C4—82no—
407C2—21no—
408C2—21no—
409C2—22no—
410C2—22no—
411C2—24no—
412C2—42no—
413C2#1739211yescharacteristic
414C2C22×#35411yescharacteristic
415C2C2×#213311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11551C2semidirect
4. #11511C2semidirect
5. #11506C2semidirect
26. #1324C22semidirect
27. #1319C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25032
C2×#14632
#115512
#115112
#115062