Finite Group Databasebeta

← #129756

Subgroups — #129756

Order 512.

968 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 968 conjugacy classes of subgroups — too many to draw. Only its 120 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×#2546C211yesmaximal, characteristic
2C2×#1346C214yesmaximal
3#10763C211yesmaximal, characteristic
4#8303C211yesmaximal, characteristic
5C22×C4⋊Q8—21no—
6C22×C4⋊Q8C411yescharacteristic
7C22×C4×Q8C2211yescharacteristic
8C22×#101—21no—
9#2546—28no—
10C2×#398C412yes—
11C2×#398C2212yes—
12C2×#391—21no—
13C2×#389—21no—
14C2×#389—21no—
15C2×C4×C4⋊C4C411yescharacteristic
16#1817—21no—
17C2×#343C2212yes—
18#1728—21no—
19C2×#324—21no—
20C2×#324C2212yes—
21#1346—28no—
22C23×Q8—21no—
23C23×Q8D411yescharacteristic
24C22×M16—22no—
25C2×C4⋊Q8—28no—
26C2×C4⋊Q8—24no—
27C2×C4⋊Q8—42no—
28C2×C4⋊Q8—42no—
29C2×C4⋊Q8—44no—
30C2×C4⋊Q8—42no—
31C2×C4⋊Q8—42no—
32C2×C4×Q8—24no—
33C2×C4×Q8—42no—
34C2×C4×Q8—42no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4C2×C412yes—
41C22×C4⋊C4D411yescharacteristic
42C22×C4⋊C4C2311yescharacteristic
43C22×C42—21no—
44C22×C42C2×C411yescharacteristic
45C22×C42D411yescharacteristic
46C22×C42D411yescharacteristic
47C2×#101—24no—
48C2×#101—42no—
49C2×#101—42no—
50C2×C22⋊C8—22no—
51#398—24no—
52#398—24no—
53#398—24no—
54#398D418yes—
55#391—28no—
56#389—28no—
57#389—28no—
58C4×C4⋊C4—24no—
59C2×#95—22no—
60C2×#95—22no—
61C2×#95C2×C412yes—
62#343—24no—
63#336—42no—
64#336—42no—
65#324—24no—
66#324—42no—
67C22×Q8—24no—
68C22×Q8—22no—
69C22×Q8—22no—
70C22×Q8—22no—
71C22×Q8—22no—
72C22×Q8—41no—
73C22×Q8—41no—
74C22×Q8—42no—
75C22×Q8—42no—
76C22×Q8—42no—
77C22×Q8—42no—
78C22×Q8—42no—
79C22×Q8—42no—
80C22×Q8D812yes—
81C22×Q8SD1612yes—
82C23×C4—22no—
83C23×C4—22no—
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22⋊C411yescharacteristic
88C23×C4C22×C411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C2×M16—24no—
92C2×M16—42no—
93C2×M16—42no—
94C2×M16—42no—
95C2×M16—42no—
96C2×M16—42no—
97C22×C8—42no—
98C22×C8—42no—
99C4⋊Q8—416no—
100C4⋊Q8—416no—
101C4⋊Q8—416no—
102C4×Q8—416no—
103C2×C4⋊C4—28no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—28no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—28no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—44no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—44no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—44no—
126C2×C4⋊C4C22⋊C414yes—
127C2×C4⋊C4C2×D414yes—
128C2×C42—22no—
129C2×C42—22no—
130C2×C42—22no—
131C2×C42—24no—
132C2×C42—22no—
133C2×C42—22no—
134C2×C42—22no—
135C2×C42—22no—
136C2×C42—41no—
137C2×C42—41no—
138C2×C42—42no—
139C2×C42—42no—
140#101—48no—
141C22⋊C8—42no—
142C22⋊C8—42no—
143#95—28no—
144#95—22no—
145#95—28no—
146#95C22⋊C414yes—
147C24C2×C22⋊C411yescharacteristic
148C2×Q8—24no—
149C2×Q8—24no—
150C2×Q8—24no—
151C2×Q8—24no—
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—44no—
155C2×Q8—44no—
156C2×Q8—44no—
157C2×Q8—44no—
158C2×Q8—48no—
159C2×Q8—48no—
160C2×Q8—48no—
161C2×Q8—84no—
162C2×Q8—84no—
163C2×Q8—84no—
164C2×Q8—84no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—24no—
176C22×C4—28no—
177C22×C4—24no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—24no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—44no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4#9912yes—
205C22×C4#9912yes—
206C22×C4#10212yes—
207C22×C4#10212yes—
208C22×C4#10412yes—
209C22×C4C2×C22⋊C412yes—
210C22×C4#12012yes—
211C22×C4#12012yes—
212C22×C4C2×D811yescharacteristic
213C22×C4C2×SD1611yescharacteristic
214C22×C4AGL(1,ℤ/8)12yes—
215M16—44no—
216M16—44no—
217M16—82no—
218M16—82no—
219C2×C8—44no—
220C2×C8—82no—
221C2×C8—82no—
222C2×C8—82no—
223C2×C8—84no—
224C4⋊C4—216no—
225C4⋊C4—44no—
226C4⋊C4—44no—
227C4⋊C4—44no—
228C4⋊C4—44no—
229C4⋊C4—48no—
230C4⋊C4—44no—
231C4⋊C4—416no—
232C4⋊C4—48no—
233C4⋊C4—48no—
234C42—28no—
235C42—44no—
236C42—44no—
237C42—48no—
238C23—22no—
239C23—22no—
240C23#37911yescharacteristic
241C23C2×#9911yescharacteristic
242C23C2×#9911yescharacteristic
243C23C2×#10211yescharacteristic
244C23#41811yescharacteristic
245C23C2×#10411yescharacteristic
246C23#42111yescharacteristic
247Q8—48no—
248Q8—48no—
249Q8—816no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—28no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—48no—
275C2×C4—48no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—44no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—81no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—81no—
299C2×C4#32714yes—
300C2×C4#35614yes—
301C2×C4#44712yes—
302C2×C4#45012yes—
303C8—84no—
304C8—162no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22C2×#32711yescharacteristic
314C22#150711yescharacteristic
315C22#151011yescharacteristic
316C22#151911yescharacteristic
317C22#178111yescharacteristic
318C22#178611yescharacteristic
319C22#178911yescharacteristic
320C22#212312yes—
321C22C2×#35612yes—
322C4—22no—
323C4—24no—
324C4—22no—
325C4—24no—
326C4—48no—
327C4—42no—
328C4—42no—
329C4—42no—
330C4—84no—
331C4—81no—
332C2—21no—
333C2—21no—
334C2—21no—
335C2—21no—
336C2#832811yescharacteristic
337C2#1178212yes—
338C2#1188312yes—
339C2#1269211yescharacteristic
340C2#1270011yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#25462
C2×#13462
#107632
#83032