Finite Group Databasebeta

← #102807

Subgroups — #102807

Order 512.

651 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 651 conjugacy classes of subgroups — too many to draw. Only its 108 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#30293C211yesmaximal, characteristic
2#25856C211yesmaximal, characteristic
3#25813C211yesmaximal, characteristic
4#11384C211yesmaximal, characteristic
5#9679C211yesmaximal, characteristic
6#9679C211yesmaximal, characteristic
7#7016C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#500—21no—
10C2×#487—21no—
11#2857C2211yescharacteristic
12#2857C2211yescharacteristic
13#2084—21no—
14#2084—21no—
15#2058—21no—
16#2058—21no—
17#2057—22no—
18#2043—22no—
19#1990—21no—
20#1956—21no—
21#1852—22no—
22#1807—22no—
23#1787—22no—
24C4×C22⋊C8C2211yescharacteristic
25C4×C22⋊C8C2211yescharacteristic
26C2×#340—22no—
27C2×#336C2211yescharacteristic
28C2×#328—22no—
29C2×#328—22no—
30C4⋊#95C2211yescharacteristic
31C23×Q8—21no—
32C23×C8D412yes—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—44no—
37C2×C4⋊Q8D412yes—
38C2×C4⋊Q8D411yescharacteristic
39C2×C4⋊Q8D411yescharacteristic
40C2×C22⋊Q8—21no—
41C2×C22⋊Q8—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C42C2311yescharacteristic
46#500—44no—
47#487—44no—
48C2×#107—24no—
49C2×#106—22no—
50C2×#106—22no—
51C2×#103—24no—
52C2×#103—24no—
53C2×C22⋊C8D412yes—
54C2×C22⋊C8D412yes—
55C2×C4×C8—21no—
56C2×C4×C8—21no—
57#391—21no—
58#391—21no—
59#389—21no—
60#389—21no—
61#389—21no—
62#389—21no—
63#384D411yescharacteristic
64#384D411yescharacteristic
65#380—21no—
66#380—21no—
67#380—21no—
68#380—21no—
69C2×#95—21no—
70C2×#95—21no—
71#340—24no—
72#340—22no—
73#340—22no—
74#340—44no—
75#336—21no—
76#336—21no—
77#336—21no—
78#336—21no—
79#328—44no—
80#328—44no—
81C22×Q8—22no—
82C22×Q8—21no—
83C22×Q8—21no—
84C22×Q8—44no—
85C22×Q8—41no—
86C22×Q8—42no—
87C22×Q8—42no—
88C22×Q8—82no—
89C23×C4—41no—
90C23×C4—41no—
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C22×C8—22no—
95C22×C8—22no—
96C22×C8—22no—
97C22×C8—22no—
98C22×C8—42no—
99C22×C8—42no—
100C22×C8Q1614yes—
101C4⋊Q8—24no—
102C4⋊Q8—24no—
103C4⋊Q8—42no—
104C4⋊Q8—42no—
105C4⋊Q8—42no—
106C4⋊Q8—42no—
107C4⋊Q8—44no—
108C4⋊Q8—44no—
109C22⋊Q8—42no—
110C22⋊Q8—42no—
111C22⋊Q8—42no—
112C22⋊Q8—42no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—44no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C22⋊C4—41no—
137C2×C22⋊C4—41no—
138C2×C42—42no—
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42#4111yescharacteristic
143#107—48no—
144#106—82no—
145#106—82no—
146#103—48no—
147#103—84no—
148C22⋊C8—22no—
149C22⋊C8—22no—
150C22⋊C8—22no—
151C22⋊C8SD1614yes—
152C4×C8—42no—
153C4×C8—42no—
154#95—41no—
155#95—41no—
156#95—41no—
157#95—41no—
158#95—42no—
159#95—42no—
160C24#12711yescharacteristic
161C2×Q8—24no—
162C2×Q8—44no—
163C2×Q8—42no—
164C2×Q8—44no—
165C2×Q8—44no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—82no—
169C2×Q8—81no—
170C2×Q8—81no—
171C2×Q8—81no—
172C2×Q8—81no—
173C2×Q8—88no—
174C2×Q8—88no—
175C2×Q8—82no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—44no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—82no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4—81no—
194C22×C4—81no—
195C22×C4—81no—
196C22×C4—81no—
197C22×C4—81no—
198C22×C4—81no—
199C22×C4#12011yescharacteristic
200C22×C4#12011yescharacteristic
201C22×C4#12011yescharacteristic
202C22×C4#12111yescharacteristic
203C22×C4#12111yescharacteristic
204C22×C4#12111yescharacteristic
205C22×C4C2×SD1612yes—
206C22×C4C2×Q1612yes—
207C22×C4#13512yes—
208C22×C4AGL(1,ℤ/8)12yes—
209C22×C4#13712yes—
210C22×C4#13712yes—
211C2×C8—24no—
212C2×C8—24no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—44no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—82no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—44no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—44no—
232C4⋊C4—44no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—82no—
242C4⋊C4—82no—
243C4⋊C4—82no—
244C22⋊C4—81no—
245C22⋊C4—81no—
246C22⋊C4—81no—
247C22⋊C4—81no—
248C42—22no—
249C42—22no—
250C42—41no—
251C42—41no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23#39211yescharacteristic
257C23#49211yescharacteristic
258C23#49411yescharacteristic
259C23#49511yescharacteristic
260C23#49611yescharacteristic
261C23#49711yescharacteristic
262C23#49711yescharacteristic
263Q8—48no—
264Q8—84no—
265Q8—88no—
266Q8—162no—
267Q8—162no—
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—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—84no—
288C2×C4—84no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
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—161no—
307C2×C4—161no—
308C2×C4—161no—
309C2×C4—161no—
310C2×C4#44812yes—
311C2×C4#44912yes—
312C2×C4#45112yes—
313C2×C4#45511yescharacteristic
314C2×C4#45611yescharacteristic
315C2×C4B(4,𝔽2)12yes—
316C2×C4#46112yes—
317C2×C4#46212yes—
318C2×C4#46312yes—
319C8—44no—
320C8—44no—
321C8—82no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22#170711yescharacteristic
335C22#171411yescharacteristic
336C22#171811yescharacteristic
337C22#172311yescharacteristic
338C22#200711yescharacteristic
339C22#200811yescharacteristic
340C22#200911yescharacteristic
341C22#201211yescharacteristic
342C22#201311yescharacteristic
343C22#201611yescharacteristic
344C22#202111yescharacteristic
345C4—22no—
346C4—22no—
347C4—22no—
348C4—22no—
349C4—22no—
350C4—84no—
351C4—162no—
352C4—161no—
353C4—161no—
354C4—161no—
355C4—161no—
356C4#160212yes—
357C4#160512yes—
358C2—21no—
359C2—21no—
360C2—21no—
361C2—21no—
362C2#994711yescharacteristic
363C2#994911yescharacteristic
364C2#996711yescharacteristic
365C2#1019711yescharacteristic
366C2#1020511yescharacteristic
367C2#1022411yescharacteristic
368C2#1031411yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC22×C42C23
prime radicalC24#127
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
#302932
#258562
#258132
#113842
#96792
#70162