Finite Group Databasebeta

← #104106

Subgroups — #104106

Order 512.

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

normal subgroups only

This group has 658 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#25902C211yesmaximal, characteristic
3#15230C211yesmaximal, characteristic
4#9908C211yesmaximal, characteristic
5#8340C211yesmaximal, characteristic
6#7576C211yesmaximal, characteristic
7#7028C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#487—21no—
10#2857C2211yescharacteristic
11C2×#389C2211yescharacteristic
12C2×#382—21no—
13#2100—21no—
14#2087—22no—
15#2057—21no—
16#2024—22no—
17#1990—21no—
18#1872—22no—
19#1852—21no—
20#1824—21no—
21#1797C2211yescharacteristic
22#1797C2211yescharacteristic
23C4×C22⋊C8C2211yescharacteristic
24C2×#332—22no—
25#1446—22no—
26#1441—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⋊C4D411yescharacteristic
42C22×C4⋊C4D411yescharacteristic
43C22×C4⋊C4D412yes—
44C22×C42—21no—
45C22×C42C2311yescharacteristic
46#487—44no—
47C2×#107—22no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8D412yes—
50C2×#103—24no—
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—22no—
72#340—22no—
73#340—22no—
74#340—24no—
75#336—21no—
76#336—21no—
77#336D412yes—
78#332—44no—
79C22×Q8—21no—
80C22×Q8—44no—
81C22×Q8—41no—
82C22×Q8—41no—
83C22×Q8—42no—
84C22×Q8—42no—
85C22×Q8—82no—
86C23×C4—22no—
87C23×C4—21no—
88C23×C4—22no—
89C23×C4—21no—
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C22×C8—22no—
94C22×C8—22no—
95C22×C8—41no—
96C4⋊Q8—24no—
97C4⋊Q8—24no—
98C4⋊Q8—42no—
99C4⋊Q8—42no—
100C4⋊Q8—42no—
101C4⋊Q8—44no—
102C4⋊Q8—44no—
103C22⋊Q8—44no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
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—44no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—44no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4Q1612yes—
129C2×C22⋊C4—41no—
130C2×C42—24no—
131C2×C42—42no—
132C2×C42—42no—
133C2×C42—41no—
134C2×C42—41no—
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D411yescharacteristic
138C2×C42#4111yescharacteristic
139#107—82no—
140C4⋊C8—42no—
141C4⋊C8—42no—
142#103—48no—
143C22⋊C8—42no—
144C22⋊C8—41no—
145C22⋊C8—41no—
146C22⋊C8—42no—
147C4×C8—42no—
148#95—22no—
149#95—22no—
150#95—42no—
151#95—48no—
152#95—41no—
153#95—41no—
154#95—41no—
155#95—41no—
156#95—84no—
157C24#12011yescharacteristic
158C2×Q8—44no—
159C2×Q8—44no—
160C2×Q8—42no—
161C2×Q8—42no—
162C2×Q8—42no—
163C2×Q8—82no—
164C2×Q8—82no—
165C2×Q8—88no—
166C2×Q8—88no—
167C2×Q8—82no—
168C2×Q8—81no—
169C2×Q8—81no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—22no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—44no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—41no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—41no—
192C22×C4—44no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4#12011yescharacteristic
201C22×C4#12011yescharacteristic
202C22×C4#12111yescharacteristic
203C22×C4#12111yescharacteristic
204C22×C4#12111yescharacteristic
205C22×C4#12711yescharacteristic
206C22×C4C2×Q1611yescharacteristic
207C22×C4#13511yescharacteristic
208C22×C4#13512yes—
209C22×C4#13711yescharacteristic
210C22×C4#13711yescharacteristic
211C22×C4#13712yes—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—81no—
219C2×C8—81no—
220C2×C8—81no—
221C4⋊C4—24no—
222C4⋊C4—24no—
223C4⋊C4—22no—
224C4⋊C4—22no—
225C4⋊C4—42no—
226C4⋊C4—44no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—41no—
230C4⋊C4—44no—
231C4⋊C4—41no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—81no—
236C4⋊C4—81no—
237C4⋊C4—81no—
238C4⋊C4—82no—
239C4⋊C4—82no—
240C4⋊C4—82no—
241C22⋊C4—82no—
242C42—22no—
243C42—22no—
244C42—41no—
245C42—41no—
246C42—44no—
247C42—44no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23#39211yescharacteristic
253C23#45111yescharacteristic
254C23#45212yes—
255C23#45211yescharacteristic
256C23#45412yes—
257Q8—84no—
258Q8—88no—
259Q8—84no—
260Q8—84no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—48no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—84no—
291C2×C4—82no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—84no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4#45111yescharacteristic
307C2×C4#45211yescharacteristic
308C2×C4#45412yes—
309C2×C4#45612yes—
310C2×C4B(4,𝔽2)12yes—
311C2×C4#46412yes—
312C2×C4#46711yescharacteristic
313C2×C4#47011yescharacteristic
314C2×C4#49511yescharacteristic
315C2×C4#49711yescharacteristic
316C8—82no—
317C8—82no—
318C8—161no—
319C22—21no—
320C22—21no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22#163612yes—
328C22#163812yes—
329C22#200111yescharacteristic
330C22#200311yescharacteristic
331C22#200311yescharacteristic
332C22#200511yescharacteristic
333C22#200811yescharacteristic
334C22#201511yescharacteristic
335C22#202111yescharacteristic
336C4—22no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—22no—
341C4—21no—
342C4—41no—
343C4—84no—
344C4—82no—
345C4—84no—
346C4—161no—
347C4—161no—
348C4—161no—
349C4#160312yes—
350C4#160512yes—
351C2—22no—
352C2—22no—
353C2#992511yescharacteristic
354C2#995511yescharacteristic
355C2#1006111yescharacteristic
356C2#1009311yescharacteristic
357C2#1024911yescharacteristic
358C2#1029611yescharacteristic
359C2#1032711yescharacteristic
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
#259022
#152302
#99082
#83402
#75762
#70282