Finite Group Databasebeta

← #103604

Subgroups — #103604

Order 512.

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

normal subgroups only

This group has 622 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#29968C211yesmaximal, characteristic
2#25856C211yesmaximal, characteristic
3C2×#2057C211yesmaximal, characteristic
4#9650C211yesmaximal, characteristic
5#8340C211yesmaximal, characteristic
6#7667C211yesmaximal, characteristic
7#7029C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#487—21no—
10C22×#107—21no—
11#2857C2211yescharacteristic
12C2×#389C2211yescharacteristic
13#2100—21no—
14#2088—22no—
15#2058—21no—
16#2057—22no—
17#2057—22no—
18#2043—22no—
19#1956—21no—
20#1853—21no—
21#1823—21no—
22#1797C2211yescharacteristic
23#1797C2211yescharacteristic
24#1787—22no—
25C4×C22⋊C8C2211yescharacteristic
26C2×#340—21no—
27C2×#336C2211yescharacteristic
28C2×#332—22no—
29C2×#324—22no—
30#1272—22no—
31C23×Q8—21no—
32C23×C8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—44no—
37C2×C4⋊Q8D412yes—
38C2×C4⋊Q8D411yescharacteristic
39C2×C22⋊Q8—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4D411yescharacteristic
44C22×C4⋊C4D411yescharacteristic
45C22×C42C2311yescharacteristic
46#487—44no—
47C2×#107—24no—
48C2×#107—44no—
49C2×#106—22no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8D412yes—
52C2×#103—24no—
53C2×#103—24no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8D412yes—
57C2×C4×C8—21no—
58#391—21no—
59#389—21no—
60#389—21no—
61#389—21no—
62#389—21no—
63#389—21no—
64#389—21no—
65#384D411yescharacteristic
66#380—21no—
67#380—21no—
68C2×#95—21no—
69#340—24no—
70#340—22no—
71#340—22no—
72#340—22no—
73#336—21no—
74#336—21no—
75#336—21no—
76#336—21no—
77#336D412yes—
78#332—44no—
79#324—44no—
80C22×Q8—22no—
81C22×Q8—21no—
82C22×Q8—44no—
83C22×Q8—41no—
84C22×Q8—42no—
85C22×Q8—42no—
86C22×Q8—82no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—41no—
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4#4111yescharacteristic
93C22×C8—24no—
94C22×C8—22no—
95C22×C8—22no—
96C22×C8—22no—
97C22×C8—42no—
98C22×C8—41no—
99C22×C8—41no—
100C22×C8—41no—
101C4⋊Q8—24no—
102C4⋊Q8—24no—
103C4⋊Q8—42no—
104C4⋊Q8—42no—
105C4⋊Q8—42no—
106C4⋊Q8—44no—
107C4⋊Q8—44no—
108C22⋊Q8—82no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—22no—
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—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—44no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4Q1612yes—
135C2×C22⋊C4—41no—
136C2×C42—42no—
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D411yescharacteristic
141#107—48no—
142#106—82no—
143C4⋊C8—42no—
144C4⋊C8—42no—
145#103—48no—
146#103—84no—
147C22⋊C8—22no—
148C22⋊C8—22no—
149C22⋊C8—41no—
150C22⋊C8—41no—
151C22⋊C8—42no—
152C4×C8—42no—
153#95—41no—
154#95—41no—
155#95—41no—
156#95—41no—
157C24#12111yescharacteristic
158C2×Q8—24no—
159C2×Q8—44no—
160C2×Q8—42no—
161C2×Q8—44no—
162C2×Q8—44no—
163C2×Q8—42no—
164C2×Q8—82no—
165C2×Q8—81no—
166C2×Q8—81no—
167C2×Q8—88no—
168C2×Q8—88no—
169C2×Q8—81no—
170C2×Q8—81no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—44no—
180C22×C4—41no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—41no—
184C22×C4—42no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—42no—
189C22×C4—82no—
190C22×C4—82no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4#12011yescharacteristic
194C22×C4#12011yescharacteristic
195C22×C4#12011yescharacteristic
196C22×C4#12111yescharacteristic
197C22×C4#12111yescharacteristic
198C22×C4#12711yescharacteristic
199C22×C4C2×Q1611yescharacteristic
200C22×C4#13511yescharacteristic
201C22×C4#13512yes—
202C22×C4AGL(1,ℤ/8)12yes—
203C22×C4#13711yescharacteristic
204C22×C4#13711yescharacteristic
205C2×C8—24no—
206C2×C8—42no—
207C2×C8—44no—
208C2×C8—44no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—81no—
217C2×C8—81no—
218C2×C8—81no—
219C4⋊C4—24no—
220C4⋊C4—24no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—44no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—44no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—81no—
236C4⋊C4—82no—
237C4⋊C4—82no—
238C4⋊C4—81no—
239C4⋊C4—81no—
240C22⋊C4—82no—
241C42—22no—
242C42—22no—
243C42—41no—
244C42—41no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23#39211yescharacteristic
250C23#46312yes—
251C23#46711yescharacteristic
252C23#47011yescharacteristic
253C23#47112yes—
254Q8—48no—
255Q8—84no—
256Q8—84no—
257Q8—84no—
258Q8—162no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—42no—
279C2×C4—44no—
280C2×C4—88no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—81no—
284C2×C4—82no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—162no—
296C2×C4#44912yes—
297C2×C4#45111yescharacteristic
298C2×C4#45111yescharacteristic
299C2×C4#45211yescharacteristic
300C2×C4#45211yescharacteristic
301C2×C4#45612yes—
302C2×C4B(4,𝔽2)12yes—
303C2×C4#47312yes—
304C2×C4#49511yescharacteristic
305C2×C4#49611yescharacteristic
306C8—44no—
307C8—82no—
308C8—82no—
309C8—161no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—22no—
318C22#168312yes—
319C22#170312yes—
320C22#200111yescharacteristic
321C22#200511yescharacteristic
322C22#200811yescharacteristic
323C22#201211yescharacteristic
324C22#201911yescharacteristic
325C22#202011yescharacteristic
326C22#202111yescharacteristic
327C4—22no—
328C4—21no—
329C4—21no—
330C4—22no—
331C4—21no—
332C4—21no—
333C4—41no—
334C4—82no—
335C4—84no—
336C4—161no—
337C4—161no—
338C4—161no—
339C4#160312yes—
340C4#160512yes—
341C2—22no—
342C2—22no—
343C2#992511yescharacteristic
344C2#994711yescharacteristic
345C2#998911yescharacteristic
346C2#1013811yescharacteristic
347C2#1017511yescharacteristic
348C2#1018311yescharacteristic
349C2#1027511yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC22×C42C23
prime radicalC24#121
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
#258562
C2×#20572
#96502
#83402
#76672
#70292