Finite Group Databasebeta

← #104170

Subgroups — #104170

Order 512.

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

normal subgroups only

This group has 666 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#29967C211yesmaximal, characteristic
2#25813C211yesmaximal, characteristic
3#15230C211yesmaximal, characteristic
4#9711C211yesmaximal, characteristic
5#8340C211yesmaximal, characteristic
6#7576C211yesmaximal, characteristic
7#7028C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#500—21no—
10#2857C2211yescharacteristic
11C2×#389C2211yescharacteristic
12C2×#382—21no—
13#2099—21no—
14#2088—22no—
15#2058—21no—
16#2057—22no—
17#1990—21no—
18#1853—21no—
19#1852—22no—
20#1824—21no—
21#1797C2211yescharacteristic
22#1797C2211yescharacteristic
23C4×C22⋊C8C2211yescharacteristic
24C2×#332—22no—
25C2×#328—22no—
26#1446—22no—
27#1441—22no—
28C4⋊#95—21no—
29C4×#95C2211yescharacteristic
30#1271—22no—
31C23×Q8—21no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—44no—
37C2×C4⋊Q8D411yescharacteristic
38C2×C22⋊Q8—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4D411yescharacteristic
42C22×C4⋊C4D412yes—
43C22×C4⋊C4D411yescharacteristic
44C22×C42—21no—
45C22×C42C2311yescharacteristic
46#500—44no—
47C2×#107—24no—
48C2×#106—22no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8D412yes—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C4×C8—21no—
55#391—21no—
56#389—21no—
57#389—21no—
58#389—21no—
59#389—21no—
60#389—21no—
61#389—21no—
62#384D411yescharacteristic
63#382—44no—
64#380—21no—
65#380—21no—
66C2×#95—24no—
67C2×#95—21no—
68C2×#95—21no—
69C2×#95—24no—
70C2×#95D412yes—
71#340—24no—
72#340—22no—
73#340—22no—
74#340—22no—
75#336—21no—
76#336—21no—
77#336D412yes—
78#332—44no—
79#328—44no—
80C22×Q8—21no—
81C22×Q8—44no—
82C22×Q8—41no—
83C22×Q8—41no—
84C22×Q8—42no—
85C22×Q8—42no—
86C22×Q8—82no—
87C23×C4—22no—
88C23×C4—21no—
89C23×C4—22no—
90C23×C4—21no—
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C22×C8—22no—
95C22×C8—22no—
96C22×C8—41no—
97C4⋊Q8—24no—
98C4⋊Q8—24no—
99C4⋊Q8—42no—
100C4⋊Q8—42no—
101C4⋊Q8—42no—
102C4⋊Q8—44no—
103C4⋊Q8—44no—
104C22⋊Q8—42no—
105C22⋊Q8—42no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—44no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—44no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4Q1612yes—
131C2×C22⋊C4—41no—
132C2×C42—24no—
133C2×C42—42no—
134C2×C42—42no—
135C2×C42—41no—
136C2×C42—41no—
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42#4111yescharacteristic
141#107—48no—
142#106—82no—
143C4⋊C8—42no—
144C4⋊C8—42no—
145C22⋊C8—24no—
146C22⋊C8—41no—
147C22⋊C8—41no—
148C22⋊C8—42no—
149C4×C8—42no—
150#95—22no—
151#95—22no—
152#95—42no—
153#95—48no—
154#95—41no—
155#95—41no—
156#95—41no—
157#95—41no—
158#95—84no—
159C24#12011yescharacteristic
160C2×Q8—44no—
161C2×Q8—44no—
162C2×Q8—42no—
163C2×Q8—82no—
164C2×Q8—82no—
165C2×Q8—81no—
166C2×Q8—81no—
167C2×Q8—88no—
168C2×Q8—88no—
169C2×Q8—82no—
170C2×Q8—81no—
171C2×Q8—81no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—22no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—44no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—41no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—44no—
193C22×C4—42no—
194C22×C4—41no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—42no—
202C22×C4#12011yescharacteristic
203C22×C4#12011yescharacteristic
204C22×C4#12111yescharacteristic
205C22×C4#12111yescharacteristic
206C22×C4#12111yescharacteristic
207C22×C4#12711yescharacteristic
208C22×C4C2×Q1611yescharacteristic
209C22×C4#13511yescharacteristic
210C22×C4#13512yes—
211C22×C4AGL(1,ℤ/8)12yes—
212C22×C4#13711yescharacteristic
213C22×C4#13711yescharacteristic
214C2×C8—24no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—81no—
221C2×C8—81no—
222C2×C8—81no—
223C4⋊C4—24no—
224C4⋊C4—24no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—42no—
228C4⋊C4—44no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—41no—
232C4⋊C4—44no—
233C4⋊C4—41no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—81no—
240C4⋊C4—82no—
241C4⋊C4—82no—
242C4⋊C4—81no—
243C4⋊C4—81no—
244C22⋊C4—82no—
245C42—22no—
246C42—22no—
247C42—41no—
248C42—41no—
249C42—44no—
250C42—44no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23#39211yescharacteristic
256C23#44912yes—
257C23#45111yescharacteristic
258C23#45211yescharacteristic
259C23#45612yes—
260Q8—84no—
261Q8—88no—
262Q8—84no—
263Q8—162no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—48no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—84no—
294C2×C4—82no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—84no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4#45111yescharacteristic
310C2×C4#45211yescharacteristic
311C2×C4#45412yes—
312C2×C4#45612yes—
313C2×C4B(4,𝔽2)12yes—
314C2×C4#46312yes—
315C2×C4#46711yescharacteristic
316C2×C4#47011yescharacteristic
317C2×C4#49511yescharacteristic
318C2×C4#49611yescharacteristic
319C8—44no—
320C8—82no—
321C8—161no—
322C22—21no—
323C22—21no—
324C22—22no—
325C22—22no—
326C22—22no—
327C22—21no—
328C22—21no—
329C22—22no—
330C22#164812yes—
331C22#165012yes—
332C22#200111yescharacteristic
333C22#200311yescharacteristic
334C22#200511yescharacteristic
335C22#200511yescharacteristic
336C22#200811yescharacteristic
337C22#201211yescharacteristic
338C22#202111yescharacteristic
339C4—22no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—22no—
344C4—21no—
345C4—41no—
346C4—84no—
347C4—82no—
348C4—84no—
349C4—161no—
350C4—161no—
351C4—161no—
352C4#160312yes—
353C4#160512yes—
354C2—22no—
355C2—22no—
356C2#992511yescharacteristic
357C2#994711yescharacteristic
358C2#1007711yescharacteristic
359C2#1010111yescharacteristic
360C2#1024911yescharacteristic
361C2#1028611yescharacteristic
362C2#1033011yescharacteristic
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
#299672
#258132
#152302
#97112
#83402
#75762
#70282