Finite Group Databasebeta

← #134078

Subgroups — #134078

Order 512.

1076 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 1076 conjugacy classes of subgroups — too many to draw. Only its 94 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×#2857C211yesmaximal, characteristic
2#11512C212yesmaximal, complemented
3#11506C212yesmaximal, complemented
4#8555C212yesmaximal
5C22×C4⋊Q8C411yescharacteristic
6C22×C22⋊Q8C411yescharacteristic
7C22×C22⋊Q8C2211yescharacteristic
8C23×C42C411yescharacteristic
9#2857—24no—
10#2857—24no—
11C2×#391C411yescharacteristic
12C2×#391C2211yescharacteristic
13C2×#389—22no—
14C2×#384C2211yescharacteristic
15C2×#380—22no—
16#1821—22no—
17#1463C2214yes—
18#1462—22no—
19#1325—24no—
20#1319—24no—
21C23×Q8C2×C411yescharacteristic
22C23×Q8C2×C411yescharacteristic
23C24×C4—21no—
24C24×C4C2×C411yescharacteristic
25C2×C4⋊Q8—22no—
26C2×C4⋊Q8—22no—
27C2×C4⋊Q8—44no—
28C2×C22⋊Q8—22no—
29C2×C22⋊Q8—22no—
30C2×C22⋊Q8—22no—
31C2×C22⋊Q8—22no—
32C2×C22⋊Q8—42no—
33C2×C22⋊Q8—42no—
34C2×C22⋊Q8—42no—
35C2×C22⋊Q8—42no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4D412yes—
41C22×C4⋊C4C2311yescharacteristic
42C22×C22⋊C4—21no—
43C22×C22⋊C4D412yes—
44C22×C42—21no—
45C22×C42—22no—
46C22×C42—22no—
47C22×C42—44no—
48C22×C42C2×C411yescharacteristic
49C22×C42C2×C411yescharacteristic
50#407—24no—
51#391—22no—
52#391—22no—
53#391—22no—
54#391D414yes—
55#389—216no—
56#384—22no—
57#384D414yes—
58#380—216no—
59C2×#95—22no—
60C2×#95—22no—
61#343—44no—
62#324—24no—
63#324—24no—
64#323—44no—
65C25C22⋊C411yescharacteristic
66C22×Q8—21no—
67C22×Q8—21no—
68C22×Q8—21no—
69C22×Q8—41no—
70C22×Q8—41no—
71C22×Q8—42no—
72C22×Q8—42no—
73C22×Q8—42no—
74C22×Q8—42no—
75C22×Q8—41no—
76C22×Q8—41no—
77C22×Q8C22⋊C412yes—
78C23×C4—21no—
79C23×C4—28no—
80C23×C4—24no—
81C23×C4—22no—
82C23×C4—21no—
83C23×C4—22no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—44no—
89C23×C4—44no—
90C23×C4—42no—
91C23×C4—42no—
92C23×C4C22⋊C412yes—
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22×C411yescharacteristic
95C23×C4C2×D412yes—
96C2×M16—44no—
97C4⋊Q8—48no—
98C4⋊Q8—44no—
99C4⋊Q8—44no—
100C22⋊Q8—48no—
101C22⋊Q8—48no—
102C22⋊Q8—48no—
103C22⋊Q8—48no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—28no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—28no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—48no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—48no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4C22⋊C412yes—
120C2×C4⋊C4C2×D412yes—
121C2×C4⋊C4C2×D412yes—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—42no—
126C2×C22⋊C4—42no—
127C2×C22⋊C4—42no—
128C2×C42—21no—
129C2×C42—24no—
130C2×C42—24no—
131C2×C42—21no—
132C2×C42—48no—
133C2×C42—44no—
134C2×C42—44no—
135C2×C42—44no—
136C2×C42—42no—
137C2×C42C22⋊C412yes—
138C2×C42C22⋊C412yes—
139C22⋊C8—44no—
140C22⋊C8—44no—
141#95—28no—
142#95—28no—
143C24—24no—
144C24—21no—
145C24—21no—
146C24—41no—
147C24—42no—
148C24—41no—
149C24#9912yes—
150C24C2×C22⋊C411yescharacteristic
151C2×Q8—24no—
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—42no—
155C2×Q8—42no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—42no—
160C2×Q8—42no—
161C2×Q8—42no—
162C2×Q8—42no—
163C2×Q8—84no—
164C2×Q8—84no—
165C2×Q8—84no—
166C2×Q8—84no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—216no—
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—21no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—42no—
190C22×C4—48no—
191C22×C4—48no—
192C22×C4—432no—
193C22×C4—48no—
194C22×C4—48no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—44no—
203C22×C4—42no—
204C22×C4—44no—
205C22×C4—42no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4#9912yes—
213C22×C4C2×C22⋊C411yescharacteristic
214C22×C4C2×C22⋊C411yescharacteristic
215C22×C4#12014yes—
216M16—88no—
217C2×C8—84no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—48no—
221C4⋊C4—42no—
222C4⋊C4—416no—
223C4⋊C4—42no—
224C4⋊C4—48no—
225C4⋊C4—416no—
226C22⋊C4—48no—
227C22⋊C4—48no—
228C42—24no—
229C42—22no—
230C42—42no—
231C42—48no—
232C42—42no—
233C42#10014yes—
234C23—22no—
235C23—22no—
236C23—21no—
237C23—21no—
238C23—24no—
239C23—24no—
240C23—22no—
241C23—22no—
242C23—42no—
243C23—42no—
244C23—48no—
245C23—42no—
246C23—42no—
247C23—48no—
248C23—42no—
249C23—41no—
250C23—41no—
251C23#35614yes—
252C23#37911yescharacteristic
253C23C2×#9911yescharacteristic
254C23C2×#9911yescharacteristic
255Q8—84no—
256Q8—84no—
257Q8—84no—
258Q8—84no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—216no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—24no—
268C2×C4—21no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—22no—
272C2×C4—432no—
273C2×C4—48no—
274C2×C4—48no—
275C2×C4—48no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4C2×#10012yes—
300C2×C4#41312yes—
301C8—164no—
302C22—24no—
303C22—24no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—24no—
311C22—44no—
312C22—44no—
313C22—48no—
314C22—44no—
315C22—44no—
316C22—42no—
317C22—42no—
318C22#178111yescharacteristic
319C22#178311yescharacteristic
320C22#178511yescharacteristic
321C22#212612yes—
322C22C2×#35612yes—
323C4—28no—
324C4—22no—
325C4—22no—
326C4—44no—
327C4—42no—
328C4—44no—
329C4—82no—
330C4—82no—
331C4—82no—
332C2—21no—
333C2—22no—
334C2—22no—
335C2—21no—
336C2—44no—
337C2#854911yescharacteristic
338C2#1269511yescharacteristic
339C2#1270011yescharacteristic
subgroupG/H
center, socleC22#1781
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11512C2semidirect
3. #11506C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
#115122
#115062
#85552