Finite Group Databasebeta

← #129341

Subgroups — #129341

Order 512.

1044 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 1044 conjugacy classes of subgroups — too many to draw. Only its 130 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×#2326C211yesmaximal, characteristic
2C2×#1325C212yesmaximal, complemented
3C2×#1318C212yesmaximal
4#8303C212yesmaximal, complemented
5C22×#124C411yescharacteristic
6C22×C4×Q8C2211yescharacteristic
7C22×C4×D4C411yescharacteristic
8#2326—24no—
9#2326—24no—
10C2×#391C411yescharacteristic
11C2×#390C2211yescharacteristic
12C2×#388—22no—
13C2×#385—22no—
14C2×#384C2211yescharacteristic
15C2×#376C411yescharacteristic
16#1817—22no—
17C2×#324—22no—
18C2×#324C2214yes—
19#1325—24no—
20#1318—24no—
21C23×Q8C2×C411yescharacteristic
22C23×D4C2×C411yescharacteristic
23C24×C4—21no—
24C2×#124—22no—
25C2×#124—22no—
26C2×#124—22no—
27C2×#124—22no—
28C2×#124—22no—
29C2×#124C814yescomplemented
30C2×C4×Q8—22no—
31C2×C4×Q8—22no—
32C2×C4×Q8—42no—
33C2×C4×Q8—42no—
34C2×C4×D4—22no—
35C2×C4×D4—22no—
36C2×C4×D4—42no—
37C2×C4×D4—42no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4D412yes—
41C22×C4⋊C4C2311yescharacteristic
42C22×C22⋊C4—22no—
43C22×C22⋊C4—21no—
44C22×C22⋊C4—22no—
45C22×C42C2×C411yescharacteristic
46C22×C42C2×C411yescharacteristic
47C22×C42C2×C411yescharacteristic
48C22×C42D412yes—
49C2×C22⋊C8—24no—
50#391—22no—
51#391—22no—
52#390—22no—
53#390D414yes—
54#388—216no—
55#385—216no—
56#384—22no—
57#384D414yes—
58#376—22no—
59#376—22no—
60C2×#95—21no—
61C2×#95—22no—
62C2×#95—21no—
63#336—44no—
64#324—24no—
65#324—24no—
66#324—44no—
67C25—21no—
68C22×Q8—21no—
69C22×Q8—21no—
70C22×Q8—21no—
71C22×Q8—21no—
72C22×Q8—41no—
73C22×Q8—41no—
74C22×Q8—41no—
75C22×Q8—41no—
76C22×Q8C2×C812yes—
77C22×Q8M1612yes—
78C22×D4—21no—
79C22×D4—21no—
80C22×D4—21no—
81C22×D4—41no—
82C22×D4—41no—
83C22×D4—41no—
84C22×D4—41no—
85C22×D4C22⋊C412yes—
86C22×D4C2×C812yes—
87C22×D4M1612yes—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—21no—
91C23×C4—24no—
92C23×C4—42no—
93C23×C4—42no—
94C23×C4—84no—
95C23×C4C22⋊C412yes—
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22×C411yescharacteristic
99C23×C4C2×D412yes—
100C22×C8—44no—
101#124—28no—
102#124—28no—
103#124—28no—
104#124—28no—
105C4×Q8—48no—
106C4×Q8—48no—
107C4×D4—48no—
108C4×D4—48no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—28no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—84no—
119C2×C4⋊C4C22⋊C412yes—
120C2×C4⋊C4C2×D412yes—
121C2×C4⋊C4C2×D412yes—
122C2×C22⋊C4—28no—
123C2×C22⋊C4—28no—
124C2×C22⋊C4—28no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—28no—
127C2×C22⋊C4—42no—
128C2×C22⋊C4—42no—
129C2×C22⋊C4—84no—
130C2×C42—22no—
131C2×C42—22no—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42—21no—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—24no—
138C2×C42—21no—
139C2×C42—24no—
140C2×C42—42no—
141C2×C42—41no—
142C2×C42—41no—
143C2×C42C22⋊C412yes—
144C2×C42C22⋊C412yes—
145C2×C42C2×C812yes—
146C2×C42M1612yes—
147C22⋊C8—44no—
148C22⋊C8—44no—
149#95—24no—
150#95—24no—
151#95—28no—
152C24—22no—
153C24—22no—
154C24—22no—
155C24—44no—
156C24—44no—
157C24—41no—
158C24—41no—
159C24—41no—
160C24—41no—
161C24C2×C22⋊C411yescharacteristic
162C2×Q8—22no—
163C2×Q8—22no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×D4—22no—
179C2×D4—22no—
180C2×D4—22no—
181C2×D4—42no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4C22⋊C814yes—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—42no—
216C22×C4—48no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—44no—
221C22×C4—42no—
222C22×C4—44no—
223C22×C4—42no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—88no—
229C22×C4—88no—
230C22×C4—88no—
231C22×C4C22⋊C814yes—
232C22×C4#9912yes—
233C22×C4#9912yes—
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4#12014yes—
237C22×C4C22×C811yescharacteristic
238C22×C4C2×M1611yescharacteristic
239C22×C4C2×M1611yescharacteristic
240C22×C4C2×M1611yescharacteristic
241C2×C8—84no—
242C2×C8—88no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—48no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—88no—
249C22⋊C4—232no—
250C22⋊C4—48no—
251C22⋊C4—88no—
252C42—24no—
253C42—24no—
254C42—22no—
255C42—22no—
256C42—22no—
257C42—48no—
258C42—42no—
259C42—42no—
260C42C22⋊C814yes—
261C23—22no—
262C23—22no—
263C23—24no—
264C23—44no—
265C23—48no—
266C23—44no—
267C23—48no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—82no—
275C23—84no—
276C23—82no—
277C23—84no—
278C23—82no—
279C23—82no—
280C23#37911yescharacteristic
281C23C2×C22⋊C811yescharacteristic
282C23#40711yescharacteristic
283C23C2×#9911yescharacteristic
284C23C2×#9911yescharacteristic
285C23#41311yescharacteristic
286C23#41311yescharacteristic
287Q8—44no—
288Q8—44no—
289Q8—44no—
290Q8—44no—
291D4—44no—
292D4—44no—
293D4—44no—
294D4—44no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—44no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—44no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—82no—
336C2×C4—816no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4C2×C22⋊C811yescharacteristic
342C2×C4C2×C22⋊C811yescharacteristic
343C2×C4#40711yescharacteristic
344C2×C4#40711yescharacteristic
345C8—164no—
346C22—22no—
347C22—22no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—22no—
353C22—22no—
354C22—48no—
355C22—44no—
356C22—84no—
357C22—84no—
358C22—84no—
359C22—84no—
360C22—82no—
361C22—82no—
362C22—84no—
363C22#145911yescharacteristic
364C22#145911yescharacteristic
365C22#146511yescharacteristic
366C22#146511yescharacteristic
367C22#177911yescharacteristic
368C22#178111yescharacteristic
369C22#178511yescharacteristic
370C22#212812yes—
371C22#213312yes—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—44no—
377C4—44no—
378C4—44no—
379C4—42no—
380C4—42no—
381C4—82no—
382C4—82no—
383C2—21no—
384C2—21no—
385C2—21no—
386C2—21no—
387C2—84no—
388C2#830111yescharacteristic
389C2#1148012yes—
390C2#1153412yes—
391C2#1269711yescharacteristic
392C2#1270211yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C22×C8
Frattini, cosocleC22×C4⋊C4C23
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1325C2semidirect
4. #8303C2semidirect
29. C2×#124C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23262
C2×#13252
C2×#13182
#83032