Finite Group Databasebeta

← #104841

Subgroups — #104841

Order 512.

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

normal subgroups only

This group has 635 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#17070C211yesmaximal, characteristic
2#11324C211yesmaximal, characteristic
3#11207C212yesmaximal
4C2×#1271C212yesmaximal
5#7025C211yesmaximal, characteristic
6C22×#107—21no—
7C2×#398—22no—
8C2×#389C2212yes—
9C2×#384C2211yescharacteristic
10C2×#340—21no—
11C2×#340—22no—
12C2×#340—22no—
13C2×#336C2212yes—
14C2×#336C2211yescharacteristic
15#1441—22no—
16#1441—21no—
17#1441—21no—
18C4⋊#95C2211yescharacteristic
19#1271—24no—
20C23×C8—21no—
21C23×C8—22no—
22C22×C4⋊C4—21no—
23C22×C4⋊C4—22no—
24C22×C4⋊C4—22no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4—21no—
27C22×C4⋊C4D412yes—
28C22×C4⋊C4D412yes—
29C22×C4⋊C4D411yescharacteristic
30C22×C4⋊C4D411yescharacteristic
31C22×C42C2311yescharacteristic
32C2×#107—42no—
33C2×#107—42no—
34C2×#107—42no—
35#398—44no—
36#398—44no—
37#389—22no—
38#389—22no—
39#389—22no—
40#389—22no—
41#384—42no—
42C2×#95—22no—
43C2×#95—22no—
44C2×#95—22no—
45C2×#95—22no—
46C2×#95—21no—
47C2×#95—21no—
48#340—44no—
49#340—42no—
50#340—44no—
51#336—22no—
52#336—21no—
53#336—22no—
54#336D412yes—
55C23×C4—22no—
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—22no—
59C23×C4—41no—
60C23×C4C2×D412yes—
61C23×C4C2×D411yescharacteristic
62C22×C8—22no—
63C22×C8—22no—
64C22×C8—22no—
65C22×C8—24no—
66C22×C8—41no—
67C22×C8—41no—
68C22×C8—41no—
69C22×C8—41no—
70C22×C8—42no—
71C22×C8—42no—
72C22×C8—42no—
73C22×C8—42no—
74C22×C8—42no—
75C22×C8—42no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—44no—
104C2×C4⋊C4—44no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—81no—
112C2×C4⋊C4—81no—
113C2×C4⋊C4D814yes—
114C2×C4⋊C4SD1612yes—
115C2×C4⋊C4Q1612yes—
116C2×C42—41no—
117C2×C42—41no—
118C2×C42C2×D411yescharacteristic
119C2×C42#4111yescharacteristic
120C2×C42#4112yes—
121#107—48no—
122#95—44no—
123#95—44no—
124#95—42no—
125#95—42no—
126#95—42no—
127#95—42no—
128#95—81no—
129#95—81no—
130C24#12011yescharacteristic
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—24no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—82no—
165C22×C4—81no—
166C22×C4—81no—
167C22×C4—82no—
168C22×C4#12111yescharacteristic
169C22×C4#12112yes—
170C22×C4#12312yes—
171C22×C4#12411yescharacteristic
172C22×C4C2×D812yes—
173C22×C4C2×SD1611yescharacteristic
174C22×C4C2×Q1611yescharacteristic
175C22×C4#13512yes—
176C22×C4AGL(1,ℤ/8)12yes—
177C22×C4#13711yescharacteristic
178C22×C4#13711yescharacteristic
179C22×C4#13712yes—
180C2×C8—24no—
181C2×C8—42no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—44no—
188C2×C8—44no—
189C2×C8—44no—
190C2×C8—82no—
191C2×C8—82no—
192C2×C8—82no—
193C2×C8—82no—
194C4⋊C4—24no—
195C4⋊C4—24no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—44no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—81no—
206C4⋊C4—81no—
207C4⋊C4—82no—
208C4⋊C4—81no—
209C4⋊C4—81no—
210C4⋊C4—82no—
211C4⋊C4—82no—
212C4⋊C4—84no—
213C4⋊C4—84no—
214C4⋊C4—82no—
215C4⋊C4—82no—
216C4⋊C4—82no—
217C4⋊C4—82no—
218C42—41no—
219C42—41no—
220C42—42no—
221C23#39411yescharacteristic
222C23#44712yes—
223C23#44811yescharacteristic
224C23#45111yescharacteristic
225C23#45212yes—
226C23#45311yescharacteristic
227C23#45412yes—
228C23#45612yes—
229C23#45611yescharacteristic
230C23B(4,𝔽2)11yescharacteristic
231C23#45811yescharacteristic
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—84no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—161no—
283C2×C4—162no—
284C2×C4—161no—
285C2×C4#46111yescharacteristic
286C2×C4#46211yescharacteristic
287C2×C4#46612yes—
288C2×C4#46812yes—
289C2×C4#48012yes—
290C2×C4#48212yes—
291C2×C4#48711yescharacteristic
292C2×C4#49111yescharacteristic
293C8—44no—
294C8—84no—
295C22#159611yescharacteristic
296C22#159812yes—
297C22#160312yes—
298C22#160511yescharacteristic
299C22#161012yes—
300C22#161111yescharacteristic
301C22#161411yescharacteristic
302C22#161512yes—
303C22#161911yescharacteristic
304C22#162712yes—
305C22#163411yescharacteristic
306C22#163612yes—
307C22#164712yes—
308C22#164912yes—
309C22#165312yes—
310C22#165711yescharacteristic
311C22#166011yescharacteristic
312C22#166312yes—
313C22#203312yes—
314C22#203812yes—
315C22#203911yescharacteristic
316C22#204311yescharacteristic
317C22#204411yescharacteristic
318C4—22no—
319C4—22no—
320C4—22no—
321C4—22no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—82no—
327C4—82no—
328C4—84no—
329C4—162no—
330C4—162no—
331C2#773811yescharacteristic
332C2#776011yescharacteristic
333C2#776312yes—
334C2#779711yescharacteristic
335C2#779812yes—
336C2#783111yescharacteristic
337C2#1039712yes—
338C2#1040611yescharacteristic
339C2#1041912yes—
340C2#1042911yescharacteristic
341C2#1043011yescharacteristic
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
#170702
#113242
#112072
C2×#12712
#70252