Finite Group Databasebeta

← #104108

Subgroups — #104108

Order 512.

722 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 722 conjugacy classes of subgroups — too many to draw. Only its 100 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#29968C211yesmaximal, characteristic
2#25902C211yesmaximal, characteristic
3#15143C211yesmaximal, characteristic
4#9908C211yesmaximal, characteristic
5#8340C211yesmaximal, characteristic
6#7573C211yesmaximal, characteristic
7#7028C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#487—21no—
10#2857C2211yescharacteristic
11C2×#395—22no—
12C2×#389C2211yescharacteristic
13C2×#384—21no—
14#2100—21no—
15#2088—22no—
16#2058—21no—
17#2024—22no—
18#1990—21no—
19#1872—22no—
20#1853—21no—
21#1823—21no—
22#1797C2211yescharacteristic
23#1797C2211yescharacteristic
24C4×C22⋊C8C2211yescharacteristic
25C2×#332—22no—
26#1444—22no—
27C4⋊#95—21no—
28C4×#95C2211yescharacteristic
29#1294—22no—
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⋊C4—24no—
42C22×C4⋊C4D411yescharacteristic
43C22×C4⋊C4D411yescharacteristic
44C22×C4⋊C4D412yes—
45C22×C42—21no—
46C22×C42C2311yescharacteristic
47#487—44no—
48C2×#106—22no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8D412yes—
51C2×#103—24no—
52C2×C22⋊C8—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C4×C8—21no—
56#395—44no—
57#395—44no—
58#391—21no—
59#389—21no—
60#389—21no—
61#389—21no—
62#389—21no—
63#389—21no—
64#389—21no—
65#384—44no—
66#384D411yescharacteristic
67#380—21no—
68#380—21no—
69C2×#95—21no—
70C2×#95—24no—
71C2×#95—21no—
72C2×#95D412yes—
73#340—22no—
74#340—22no—
75#340—22no—
76#340—24no—
77#336—21no—
78#336—21no—
79#336D412yes—
80#332—44no—
81C22×Q8—21no—
82C22×Q8—44no—
83C22×Q8—41no—
84C22×Q8—41no—
85C22×Q8—42no—
86C22×Q8—42no—
87C22×Q8—82no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—21no—
91C23×C4—21no—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C22×C8—22no—
96C22×C8—22no—
97C22×C8—41no—
98C4⋊Q8—24no—
99C4⋊Q8—24no—
100C4⋊Q8—42no—
101C4⋊Q8—42no—
102C4⋊Q8—42no—
103C4⋊Q8—44no—
104C4⋊Q8—44no—
105C22⋊Q8—44no—
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—48no—
127C2×C4⋊C4—48no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—84no—
134C2×C4⋊C4Q1612yes—
135C2×C22⋊C4—41no—
136C2×C42—24no—
137C2×C42—42no—
138C2×C42—42no—
139C2×C42—41no—
140C2×C42—41no—
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42#4111yescharacteristic
145#106—82no—
146C4⋊C8—42no—
147C4⋊C8—42no—
148#103—48no—
149C22⋊C8—42no—
150C22⋊C8—41no—
151C22⋊C8—41no—
152C22⋊C8—42no—
153C4×C8—42no—
154#95—24no—
155#95—22no—
156#95—48no—
157#95—41no—
158#95—41no—
159#95—41no—
160#95—41no—
161#95SD1614yes—
162C24#12011yescharacteristic
163C2×Q8—44no—
164C2×Q8—44no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—82no—
169C2×Q8—82no—
170C2×Q8—88no—
171C2×Q8—88no—
172C2×Q8—82no—
173C2×Q8—81no—
174C2×Q8—81no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—24no—
183C22×C4—22no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—44no—
193C22×C4—41no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—44no—
197C22×C4—41no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—42no—
205C22×C4#12011yescharacteristic
206C22×C4#12011yescharacteristic
207C22×C4#12111yescharacteristic
208C22×C4#12111yescharacteristic
209C22×C4#12111yescharacteristic
210C22×C4#12711yescharacteristic
211C22×C4C2×SD1612yes—
212C22×C4C2×Q1611yescharacteristic
213C22×C4#13511yescharacteristic
214C22×C4AGL(1,ℤ/8)12yes—
215C22×C4#13711yescharacteristic
216C22×C4#13711yescharacteristic
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—81no—
224C2×C8—81no—
225C2×C8—81no—
226C4⋊C4—24no—
227C4⋊C4—24no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—42no—
231C4⋊C4—44no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—41no—
235C4⋊C4—44no—
236C4⋊C4—41no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—81no—
241C4⋊C4—81no—
242C4⋊C4—81no—
243C4⋊C4—82no—
244C4⋊C4—816no—
245C4⋊C4—82no—
246C4⋊C4—81no—
247C4⋊C4—81no—
248C22⋊C4—82no—
249C42—22no—
250C42—22no—
251C42—41no—
252C42—41no—
253C42—44no—
254C42—44no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23#39211yescharacteristic
260C23#45012yes—
261C23#45111yescharacteristic
262C23#45211yescharacteristic
263C23#45412yes—
264Q8—84no—
265Q8—88no—
266Q8—84no—
267Q8—84no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—28no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—48no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—84no—
298C2×C4—82no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4—81no—
302C2×C4—82no—
303C2×C4—84no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4#45111yescharacteristic
314C2×C4#45211yescharacteristic
315C2×C4#45612yes—
316C2×C4#45612yes—
317C2×C4B(4,𝔽2)12yes—
318C2×C4#46012yes—
319C2×C4#46711yescharacteristic
320C2×C4#47011yescharacteristic
321C2×C4#49211yescharacteristic
322C2×C4#49611yescharacteristic
323C8—82no—
324C8—82no—
325C8—161no—
326C22—21no—
327C22—21no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—21no—
332C22—21no—
333C22—22no—
334C22#163512yes—
335C22#163712yes—
336C22#200111yescharacteristic
337C22#200311yescharacteristic
338C22#200511yescharacteristic
339C22#200511yescharacteristic
340C22#200811yescharacteristic
341C22#201411yescharacteristic
342C22#202111yescharacteristic
343C4—22no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—22no—
348C4—21no—
349C4—48no—
350C4—41no—
351C4—82no—
352C4—84no—
353C4—161no—
354C4—161no—
355C4—161no—
356C4#160312yes—
357C4#160512yes—
358C2—22no—
359C2—22no—
360C2#992511yescharacteristic
361C2#995411yescharacteristic
362C2#1006011yescharacteristic
363C2#1009311yescharacteristic
364C2#1025311yescharacteristic
365C2#1029911yescharacteristic
366C2#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
#299682
#259022
#151432
#99082
#83402
#75732
#70282