Finite Group Databasebeta

← #103616

Subgroups — #103616

Order 512.

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

normal subgroups only

This group has 604 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#29969C211yesmaximal, characteristic
2#25813C211yesmaximal, characteristic
3C2×#2057C211yesmaximal, characteristic
4#9879C211yesmaximal, characteristic
5#8342C211yesmaximal, characteristic
6#7667C211yesmaximal, characteristic
7#7029C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#500—21no—
10C22×#107—21no—
11#2857C2211yescharacteristic
12C2×#389C2211yescharacteristic
13#2100—21no—
14#2087—22no—
15#2057—22no—
16#2057—21no—
17#2057—22no—
18#2057—22no—
19#1990—21no—
20#1852—22no—
21#1852—21no—
22#1824—21no—
23#1797C2211yescharacteristic
24#1797C2211yescharacteristic
25C4×C22⋊C8C2211yescharacteristic
26C2×#340—21no—
27C2×#336C2211yescharacteristic
28C2×#333—22no—
29#1280—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#500—44no—
47C2×#107—24no—
48C2×#107—24no—
49C2×#107—22no—
50C2×#107—44no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8D412yes—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8D412yes—
56C2×C4×C8—21no—
57#391—21no—
58#389—21no—
59#389—21no—
60#389—21no—
61#389—21no—
62#389—21no—
63#389—21no—
64#384D411yescharacteristic
65#380—21no—
66#380—21no—
67C2×#95—21no—
68#340—24no—
69#340—22no—
70#340—24no—
71#340—22no—
72#340—22no—
73#336—21no—
74#336—21no—
75#336—21no—
76#336—21no—
77#336D412yes—
78#333—44no—
79C22×Q8—22no—
80C22×Q8—21no—
81C22×Q8—44no—
82C22×Q8—41no—
83C22×Q8—82no—
84C22×Q8—82no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—41no—
88C23×C4C2×D411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C23×C4#4111yescharacteristic
91C22×C8—24no—
92C22×C8—22no—
93C22×C8—22no—
94C22×C8—22no—
95C22×C8—42no—
96C22×C8—41no—
97C22×C8—41no—
98C22×C8—41no—
99C4⋊Q8—24no—
100C4⋊Q8—24no—
101C4⋊Q8—42no—
102C4⋊Q8—42no—
103C4⋊Q8—42no—
104C4⋊Q8—44no—
105C4⋊Q8—44no—
106C22⋊Q8—82no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—21no—
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—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—44no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—82no—
131C2×C4⋊C4Q1612yes—
132C2×C22⋊C4—41no—
133C2×C42—42no—
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42C2×D411yescharacteristic
138#107—48no—
139#107—48no—
140#107—82no—
141C4⋊C8—42no—
142C4⋊C8—42no—
143C22⋊C8—22no—
144C22⋊C8—41no—
145C22⋊C8—41no—
146C22⋊C8—42no—
147C22⋊C8Q1614yes—
148C4×C8—42no—
149#95—41no—
150#95—41no—
151#95—41no—
152#95—41no—
153C24#12111yescharacteristic
154C2×Q8—24no—
155C2×Q8—44no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×Q8—82no—
159C2×Q8—82no—
160C2×Q8—82no—
161C2×Q8—81no—
162C2×Q8—81no—
163C2×Q8—816no—
164C2×Q8—82no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—44no—
174C22×C4—41no—
175C22×C4—42no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4—82no—
182C22×C4—82no—
183C22×C4—81no—
184C22×C4—81no—
185C22×C4—81no—
186C22×C4—81no—
187C22×C4#12011yescharacteristic
188C22×C4#12011yescharacteristic
189C22×C4#12011yescharacteristic
190C22×C4#12111yescharacteristic
191C22×C4#12111yescharacteristic
192C22×C4#12711yescharacteristic
193C22×C4C2×Q1611yescharacteristic
194C22×C4C2×Q1612yes—
195C22×C4#13511yescharacteristic
196C22×C4#13711yescharacteristic
197C22×C4#13711yescharacteristic
198C22×C4#13712yes—
199C2×C8—24no—
200C2×C8—24no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—44no—
205C2×C8—44no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—81no—
211C2×C8—81no—
212C2×C8—81no—
213C4⋊C4—24no—
214C4⋊C4—24no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—44no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—44no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—81no—
230C4⋊C4—82no—
231C4⋊C4—82no—
232C4⋊C4—82no—
233C22⋊C4—82no—
234C42—22no—
235C42—22no—
236C42—41no—
237C42—41no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23#39211yescharacteristic
243C23#46212yes—
244C23#46711yescharacteristic
245C23#47011yescharacteristic
246C23#47212yes—
247Q8—88no—
248Q8—84no—
249Q8—84no—
250Q8—162no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—44no—
269C2×C4—42no—
270C2×C4—44no—
271C2×C4—82no—
272C2×C4—88no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—162no—
288C2×C4#45111yescharacteristic
289C2×C4#45112yes—
290C2×C4#45111yescharacteristic
291C2×C4#45211yescharacteristic
292C2×C4#45211yescharacteristic
293C2×C4#45612yes—
294C2×C4B(4,𝔽2)12yes—
295C2×C4#47212yes—
296C2×C4#49411yescharacteristic
297C2×C4#49711yescharacteristic
298C8—44no—
299C8—44no—
300C8—82no—
301C8—161no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—22no—
310C22#167612yes—
311C22#169612yes—
312C22#200111yescharacteristic
313C22#200511yescharacteristic
314C22#200911yescharacteristic
315C22#201611yescharacteristic
316C22#201811yescharacteristic
317C22#201811yescharacteristic
318C22#202111yescharacteristic
319C4—22no—
320C4—21no—
321C4—21no—
322C4—22no—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—84no—
327C4—82no—
328C4—161no—
329C4—161no—
330C4—161no—
331C4#160312yes—
332C4#160512yes—
333C2—22no—
334C2—22no—
335C2#992511yescharacteristic
336C2#995711yescharacteristic
337C2#998611yescharacteristic
338C2#1014211yescharacteristic
339C2#1016511yescharacteristic
340C2#1019011yescharacteristic
341C2#1027911yescharacteristic
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
#299692
#258132
C2×#20572
#98792
#83422
#76672
#70292