Finite Group Databasebeta

← #104110

Subgroups — #104110

Order 512.

576 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 96 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
1#29969C211yesmaximal, characteristic
2#27905C211yesmaximal, characteristic
3#15242C211yesmaximal, characteristic
4#9908C211yesmaximal, characteristic
5#8723C211yesmaximal, characteristic
6#7576C211yesmaximal, characteristic
7#7028C211yesmaximal, characteristic
8C2×#483—22no—
9#2857C2211yescharacteristic
10C2×#389C2211yescharacteristic
11C2×#384C2211yescharacteristic
12C2×#382—21no—
13#2100—21no—
14#2096—22no—
15#2087—22no—
16#2057—21no—
17#1956—21no—
18#1852—21no—
19#1844—21no—
20#1824—21no—
21#1797C2211yescharacteristic
22#1797C2211yescharacteristic
23C4×C22⋊C8C2211yescharacteristic
24#1445—22no—
25#1441—22no—
26C4⋊#95—21no—
27C4×#95C2211yescharacteristic
28#1296—22no—
29#1294—22no—
30#1271—22no—
31C2×C4⋊Q8D411yescharacteristic
32C2×C22⋊Q8—21no—
33C22×C4⋊C4—21no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4D411yescharacteristic
37C22×C4⋊C4D412yes—
38C22×C4⋊C4D411yescharacteristic
39C22×C42—21no—
40C22×C42C2311yescharacteristic
41#483—48no—
42C2×#107—22no—
43C2×#106—24no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8D412yes—
46C2×#103—24no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C4×C8—21no—
51#391—21no—
52#389—21no—
53#389—21no—
54#389—21no—
55#389—21no—
56#389—21no—
57#389—21no—
58#384—21no—
59#384—21no—
60#384—21no—
61#384—21no—
62#384D411yescharacteristic
63#382—42no—
64#382—42no—
65#380—21no—
66#380—21no—
67C2×#95—24no—
68C2×#95—21no—
69C2×#95—21no—
70C2×#95—24no—
71C2×#95D412yes—
72#340—22no—
73#340—22no—
74#340—22no—
75#336—21no—
76#336—21no—
77#336D412yes—
78C22×Q8—21no—
79C23×C4—21no—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—21no—
83C23×C4C2×D411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C22×C8—22no—
87C22×C8—22no—
88C22×C8—41no—
89C4⋊Q8—42no—
90C22⋊Q8—44no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—44no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—44no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
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—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4SD1612yes—
122C2×C22⋊C4—41no—
123C2×C42—24no—
124C2×C42—42no—
125C2×C42—42no—
126C2×C42—41no—
127C2×C42—41no—
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42#4111yescharacteristic
132#107—82no—
133#106—48no—
134C4⋊C8—42no—
135C4⋊C8—42no—
136#103—48no—
137C22⋊C8—24no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140C22⋊C8—42no—
141C4×C8—42no—
142#95—22no—
143#95—22no—
144#95—42no—
145#95—48no—
146#95—41no—
147#95—41no—
148#95—41no—
149#95—41no—
150#95—84no—
151C24#12011yescharacteristic
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—42no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—22no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—41no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—44no—
183C22×C4—42no—
184C22×C4—44no—
185C22×C4#12011yescharacteristic
186C22×C4#12011yescharacteristic
187C22×C4#12111yescharacteristic
188C22×C4#12111yescharacteristic
189C22×C4#12111yescharacteristic
190C22×C4#12711yescharacteristic
191C22×C4C2×SD1611yescharacteristic
192C22×C4#13511yescharacteristic
193C22×C4#13512yes—
194C22×C4AGL(1,ℤ/8)11yescharacteristic
195C22×C4#13711yescharacteristic
196C22×C4#13712yes—
197C2×C8—24no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—81no—
204C2×C8—81no—
205C2×C8—81no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—44no—
209C4⋊C4—42no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—81no—
216C4⋊C4—81no—
217C4⋊C4—82no—
218C4⋊C4—81no—
219C4⋊C4—82no—
220C4⋊C4—82no—
221C4⋊C4—81no—
222C4⋊C4—81no—
223C4⋊C4—82no—
224C4⋊C4—82no—
225C4⋊C4—81no—
226C4⋊C4—81no—
227C4⋊C4—81no—
228C4⋊C4—81no—
229C22⋊C4—82no—
230C42—22no—
231C42—22no—
232C42—41no—
233C42—41no—
234C42—44no—
235C42—44no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23#39211yescharacteristic
241C23#45011yescharacteristic
242C23#45211yescharacteristic
243C23#45212yes—
244C23#45312yes—
245Q8—84no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—42no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—84no—
277C2×C4—84no—
278C2×C4—84no—
279C2×C4—82no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—82no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—82no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4#44811yescharacteristic
292C2×C4#44911yescharacteristic
293C2×C4#45412yes—
294C2×C4#45612yes—
295C2×C4#45812yes—
296C2×C4#46412yes—
297C2×C4#46511yescharacteristic
298C2×C4#46811yescharacteristic
299C2×C4#49511yescharacteristic
300C2×C4#49711yescharacteristic
301C8—44no—
302C8—82no—
303C8—161no—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—21no—
310C22—21no—
311C22—22no—
312C22#163012yes—
313C22#163112yes—
314C22#200011yescharacteristic
315C22#200211yescharacteristic
316C22#200311yescharacteristic
317C22#200511yescharacteristic
318C22#200911yescharacteristic
319C22#201511yescharacteristic
320C22#202211yescharacteristic
321C4—22no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—22no—
326C4—21no—
327C4—41no—
328C4—84no—
329C4—82no—
330C4—162no—
331C4—161no—
332C4—161no—
333C4—161no—
334C4#160912yes—
335C4#161112yes—
336C2—22no—
337C2—22no—
338C2#992711yescharacteristic
339C2#996111yescharacteristic
340C2#1005511yescharacteristic
341C2#1009111yescharacteristic
342C2#1024711yescharacteristic
343C2#1029611yescharacteristic
344C2#1032811yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C22×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#299692
#279052
#152422
#99082
#87232
#75762
#70282