Finite Group Databasebeta

← #98549

Subgroups — #98549

Order 512.

1150 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1150 conjugacy classes of subgroups — too many to draw. Only its 138 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×#2542C211yesmaximal, characteristic
2#8723C212yesmaximal, complemented
3#8560C212yesmaximal, complemented
4#7640C212yesmaximal
5C22×C4⋊Q8C411yescharacteristic
6C22×C22⋊Q8—22no—
7C22×C4×D4C411yescharacteristic
8#2542—24no—
9#2542—24no—
10C2×#399—22no—
11C2×#390C411yescharacteristic
12C2×#390C2211yescharacteristic
13C2×#384C411yescharacteristic
14C2×#384C2211yescharacteristic
15C2×C4×C4⋊C4C2211yescharacteristic
16#1797C2214yes—
17C2×#324—22no—
18#1296—24no—
19#1280—22no—
20#1270—24no—
21C23×Q8—21no—
22C23×D4C2×C411yescharacteristic
23C24×C4—21no—
24C2×C4⋊Q8—22no—
25C2×C4⋊Q8—22no—
26C2×C4⋊Q8—42no—
27C2×C4⋊Q8—42no—
28C2×C22⋊Q8—216no—
29C2×C22⋊Q8—48no—
30C2×C22⋊Q8—44no—
31C2×C22⋊Q8—44no—
32C2×C4×D4—22no—
33C2×C4×D4—22no—
34C2×C4×D4—42no—
35C2×C4×D4—42no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4D412yes—
43C22×C22⋊C4—21no—
44C22×C22⋊C4—22no—
45C22×C22⋊C4—22no—
46C22×C42C2×C411yescharacteristic
47C22×C42C2×C411yescharacteristic
48C22×C42C2311yescharacteristic
49C2×C4⋊C8—24no—
50C2×C22⋊C8—24no—
51#399—216no—
52#390—22no—
53#390—22no—
54#390—22no—
55#390D414yes—
56#384—22no—
57#384—22no—
58#384—22no—
59#384D414yes—
60C4×C4⋊C4—22no—
61C4×C4⋊C4—22no—
62C2×#95D412yes—
63#336—24no—
64#324—44no—
65C25—21no—
66C22×Q8—24no—
67C22×Q8—44no—
68C22×Q8—42no—
69C22×Q8—42no—
70C22×Q8—44no—
71C22×D4—24no—
72C22×D4—21no—
73C22×D4—41no—
74C22×D4—41no—
75C22×D4—42no—
76C22×D4C22⋊C412yes—
77C23×C4—24no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—42no—
82C23×C4—44no—
83C23×C4—42no—
84C23×C4—44no—
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22⋊C412yes—
88C23×C4C22×C411yescharacteristic
89C23×C4C2×D412yes—
90C22×C8—44no—
91C4⋊Q8—48no—
92C4⋊Q8—48no—
93C22⋊Q8—432no—
94C22⋊Q8—432no—
95C4×D4—48no—
96C4×D4—48no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—28no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—28no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—44no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—44no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—44no—
118C2×C4⋊C4C22⋊C412yes—
119C2×C4⋊C4C22⋊C412yes—
120C2×C4⋊C4C22⋊C412yes—
121C2×C4⋊C4D814yes—
122C2×C4⋊C4SD1614yes—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—28no—
125C2×C22⋊C4—28no—
126C2×C22⋊C4—42no—
127C2×C22⋊C4—44no—
128C2×C22⋊C4—42no—
129C2×C22⋊C4—44no—
130C2×C22⋊C4—42no—
131C2×C22⋊C4—42no—
132C2×C22⋊C4—44no—
133C2×C42—22no—
134C2×C42—22no—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—42no—
140C2×C42—42no—
141C2×C42C2×D412yes—
142C2×C42C2×D412yes—
143C4⋊C8—48no—
144C22⋊C8—44no—
145C22⋊C8—44no—
146#95—24no—
147C24—24no—
148C24—22no—
149C24—42no—
150C24—44no—
151C24—41no—
152C24—41no—
153C24—42no—
154C24—42no—
155C24C2×C22⋊C411yescharacteristic
156C2×Q8—416no—
157C2×Q8—48no—
158C2×Q8—88no—
159C2×Q8—84no—
160C2×Q8—84no—
161C2×D4—24no—
162C2×D4—22no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—48no—
167C2×D4—48no—
168C2×D4—42no—
169C2×D4#10114yes—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—48no—
187C22×C4—416no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—88no—
204C22×C4—84no—
205C22×C4—84no—
206C22×C4#10012yes—
207C22×C4#10012yes—
208C22×C4#10214yes—
209C22×C4#10214yes—
210C22×C4C2×C22⋊C411yescharacteristic
211C22×C4C2×C22⋊C411yescharacteristic
212C22×C4#12014yes—
213C22×C4C2×D812yes—
214C22×C4C2×SD1612yes—
215C22×C4AGL(1,ℤ/8)14yes—
216C2×C8—84no—
217C2×C8—84no—
218C2×C8—84no—
219C4⋊C4—28no—
220C4⋊C4—28no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—416no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—48no—
229C4⋊C4—48no—
230C4⋊C4—48no—
231C22⋊C4—48no—
232C22⋊C4—416no—
233C22⋊C4—48no—
234C22⋊C4—48no—
235C42—24no—
236C42—24no—
237C42—24no—
238C42—42no—
239C42—42no—
240C42—42no—
241C23—22no—
242C23—21no—
243C23—21no—
244C23—24no—
245C23—44no—
246C23—44no—
247C23—48no—
248C23—48no—
249C23—44no—
250C23—42no—
251C23—42no—
252C23—44no—
253C23—82no—
254C23—82no—
255C23—84no—
256C23—84no—
257C23—81no—
258C23—82no—
259C23—81no—
260C23#37911yescharacteristic
261C23C2×#10011yescharacteristic
262C23C2×#10011yescharacteristic
263C23C2×#10212yes—
264C23#41812yes—
265Q8—816no—
266D4—44no—
267D4—48no—
268D4—44no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—22no—
274C2×C4—24no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—24no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—816no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4#33114yes—
310C2×C4#41012yes—
311C2×C4C2×#10112yes—
312C2×C4#44714yes—
313C2×C4#45014yes—
314C8—164no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—44no—
324C22—48no—
325C22—84no—
326C22—88no—
327C22—84no—
328C22—82no—
329C22—82no—
330C22—84no—
331C22C2×#33111yescharacteristic
332C22#154512yes—
333C22#155511yescharacteristic
334C22#178211yescharacteristic
335C22#178311yescharacteristic
336C22#178411yescharacteristic
337C22#178612yes—
338C22#179612yes—
339C4—22no—
340C4—22no—
341C4—24no—
342C4—24no—
343C4—24no—
344C4—42no—
345C4—44no—
346C4—82no—
347C4—82no—
348C4—82no—
349C2—21no—
350C2—22no—
351C2—21no—
352C2—84no—
353C2#833911yescharacteristic
354C2#837612yes—
355C2#838812yes—
356C2#852411yescharacteristic
357C2#855511yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC23×D4C2×C4
LCSG ▷ C23×C4 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #8723C2semidirect
3. #8560C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
#87232
#85602
#76402