Finite Group Databasebeta

← #105178

Subgroups — #105178

Order 512.

623 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 623 conjugacy classes of subgroups — too many to draw. Only its 136 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#17076C211yesmaximal, characteristic
2#11324C212yesmaximal
3#11323C211yesmaximal, characteristic
4#7025C211yesmaximal, characteristic
5#7025C212yesmaximal
6C22×#107—22no—
7C22×#106—21no—
8C2×#400—21no—
9C2×#384C2211yescharacteristic
10C2×#340—21no—
11C2×#340—22no—
12C2×#336C2211yescharacteristic
13C2×#336C2212yes—
14#1444—22no—
15#1442—22no—
16#1441—21no—
17C4⋊#95C2211yescharacteristic
18C4⋊#95C2212yes—
19C23×C8—21no—
20C23×C8—22no—
21C22×C4⋊C4—21no—
22C22×C4⋊C4—22no—
23C22×C4⋊C4—21no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4D412yes—
26C22×C4⋊C4D412yes—
27C22×C4⋊C4D411yescharacteristic
28C22×C4⋊C4D411yescharacteristic
29C22×C42C2311yescharacteristic
30C2×#107—44no—
31C2×#107—44no—
32C2×#107—44no—
33C2×#106—42no—
34C2×#106—42no—
35C2×#106—42no—
36#400—42no—
37#400—42no—
38#384—42no—
39C2×#95—21no—
40C2×#95—22no—
41C2×#95—22no—
42C2×#95—22no—
43C2×#95—22no—
44C2×#95—21no—
45C2×#95—22no—
46#340—42no—
47#340—44no—
48#336—21no—
49#336—22no—
50#336Q812yes—
51#336Q814yes—
52C23×C4—21no—
53C23×C4—21no—
54C23×C4—22no—
55C23×C4—22no—
56C23×C4—41no—
57C23×C4C2×D411yescharacteristic
58C23×C4C2×D412yes—
59C22×C8—22no—
60C22×C8—24no—
61C22×C8—22no—
62C22×C8—24no—
63C22×C8—22no—
64C22×C8—24no—
65C22×C8—41no—
66C22×C8—42no—
67C22×C8—41no—
68C22×C8—42no—
69C22×C8—41no—
70C22×C8—42no—
71C22×C8—42no—
72C22×C8—41no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—81no—
101C2×C4⋊C4—81no—
102C2×C4⋊C4—82no—
103C2×C4⋊C4—81no—
104C2×C4⋊C4D812yes—
105C2×C4⋊C4Q1612yes—
106C2×C42—41no—
107C2×C42—41no—
108C2×C42C2×Q811yescharacteristic
109C2×C42C2×Q812yes—
110C2×C42#4111yescharacteristic
111#107—416no—
112#106—48no—
113#95—42no—
114#95—44no—
115#95—42no—
116#95—42no—
117#95—42no—
118#95—42no—
119#95—42no—
120#95—42no—
121#95—82no—
122#95—81no—
123C24#12011yescharacteristic
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—42no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—81no—
158C22×C4—82no—
159C22×C4—81no—
160C22×C4—82no—
161C22×C4C22⋊Q811yescharacteristic
162C22×C4C22⋊Q812yes—
163C22×C4C4⋊Q812yes—
164C22×C4C4⋊Q811yescharacteristic
165C22×C4C2×D811yescharacteristic
166C22×C4C2×Q1611yescharacteristic
167C22×C4#13512yes—
168C22×C4#13512yes—
169C22×C4AGL(1,ℤ/8)11yescharacteristic
170C22×C4AGL(1,ℤ/8)12yes—
171C22×C4#13711yescharacteristic
172C22×C4#13712yes—
173C2×C8—24no—
174C2×C8—28no—
175C2×C8—42no—
176C2×C8—44no—
177C2×C8—42no—
178C2×C8—44no—
179C2×C8—44no—
180C2×C8—42no—
181C2×C8—44no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—44no—
185C2×C8—42no—
186C2×C8—44no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—44no—
191C4⋊C4—44no—
192C4⋊C4—42no—
193C4⋊C4—44no—
194C4⋊C4—44no—
195C4⋊C4—81no—
196C4⋊C4—81no—
197C4⋊C4—82no—
198C4⋊C4—82no—
199C4⋊C4—81no—
200C4⋊C4—81no—
201C4⋊C4—82no—
202C4⋊C4—82no—
203C4⋊C4—84no—
204C4⋊C4—84no—
205C42—41no—
206C42—42no—
207C42—41no—
208C23#39511yescharacteristic
209C23#44711yescharacteristic
210C23#44912yes—
211C23#45111yescharacteristic
212C23#45212yes—
213C23#45311yescharacteristic
214C23#45312yes—
215C23#45612yes—
216C23#45611yescharacteristic
217C23B(4,𝔽2)11yescharacteristic
218C23#45811yescharacteristic
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—44no—
246C2×C4—44no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—84no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—162no—
270C2×C4—161no—
271C2×C4—161no—
272C2×C4#47411yescharacteristic
273C2×C4#47711yescharacteristic
274C2×C4#47812yes—
275C2×C4#47912yes—
276C2×C4#49912yes—
277C2×C4#50011yescharacteristic
278C2×C4#50111yescharacteristic
279C2×C4#50112yes—
280C8—44no—
281C8—48no—
282C22#159511yescharacteristic
283C22#160012yes—
284C22#160312yes—
285C22#160511yescharacteristic
286C22#160912yes—
287C22#161011yescharacteristic
288C22#161411yescharacteristic
289C22#161812yes—
290C22#162112yes—
291C22#162411yescharacteristic
292C22#163011yescharacteristic
293C22#163212yes—
294C22#164712yes—
295C22#164812yes—
296C22#165212yes—
297C22#165511yescharacteristic
298C22#166011yescharacteristic
299C22#166012yes—
300C22#205711yescharacteristic
301C22#205912yes—
302C22#206011yescharacteristic
303C22#206012yes—
304C22#206211yescharacteristic
305C4—21no—
306C4—21no—
307C4—21no—
308C4—21no—
309C4—22no—
310C4—22no—
311C4—22no—
312C4—22no—
313C4—82no—
314C4—82no—
315C4—162no—
316C4—162no—
317C4—162no—
318C2#774211yescharacteristic
319C2#775111yescharacteristic
320C2#778312yes—
321C2#779312yes—
322C2#780211yescharacteristic
323C2#783311yescharacteristic
324C2#1056711yescharacteristic
325C2#1057212yes—
326C2#1057711yescharacteristic
327C2#1057912yes—
328C2#1058311yescharacteristic
subgroupG/H
center, socle, prime radicalC24#120
commutator, Frattini, cosocleC22×C42C23
LCSG ▷ C22×C42 ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ 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
#170762
#113242
#113232
#70252