Finite Group Databasebeta

← #414224

Subgroups — #414224

Order 512.

806 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23745C211yesmaximal, characteristic
2#23553C211yesmaximal, characteristic
3C2×#1992C211yesmaximal, characteristic
4#20436C211yesmaximal, characteristic
5#11383C211yesmaximal, characteristic
6#11354C211yesmaximal, characteristic
7#11249C212yesmaximal
8#11242C212yesmaximal
9#10896C212yesmaximal
10#10865C211yesmaximal, characteristic
11#10827C212yesmaximal
12C22×#125C2211yescharacteristic
13C22×#108C2212yes—
14C2×#428C2212yes—
15C2×#424C2211yescharacteristic
16C2×#405C2211yescharacteristic
17C2×#405C2211yescharacteristic
18C22×#97C2211yescharacteristic
19#2296C2211yescharacteristic
20C2×#384C2211yescharacteristic
21#2100—22no—
22#2099—22no—
23#2059—22no—
24#2059—22no—
25#1992—22no—
26#1992—22no—
27#1771—22no—
28#1770—22no—
29C2×#343C2212yes—
30C2×#343C2212yes—
31C2×#343C2212yes—
32C2×#343C2212yes—
33C2×#341C2212yes—
34#1736C2212yes—
35#1736C2212yes—
36#1736C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39#1729C2211yescharacteristic
40#1728C2211yescharacteristic
41#1728C2211yescharacteristic
42#1728C2211yescharacteristic
43C22×M16C2312yes—
44C22×M16C2311yescharacteristic
45C22×M16C2311yescharacteristic
46C22×M16C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×#125—24no—
50C2×#125—22no—
51C2×#125—21no—
52C2×#125—21no—
53C2×#125—42no—
54C2×#117C2311yescharacteristic
55C2×#117C2311yescharacteristic
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C42C2311yescharacteristic
61C2×#108—24no—
62C2×#108—24no—
63C2×#108—42no—
64#428—24no—
65#428—42no—
66C2×#107—22no—
67C2×#106—22no—
68#424—22no—
69#424Q814yes—
70C2×C4⋊C8—22no—
71#405—22no—
72#405—42no—
73#405Q814yes—
74C2×#97—21no—
75C2×#97—21no—
76C2×#97—42no—
77C2×#97D412yes—
78C2×#97D412yes—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81#389D412yes—
82#389D412yes—
83#385—22no—
84#384—21no—
85#384—21no—
86#384—22no—
87C4×C4⋊C4—22no—
88#343—24no—
89#343—24no—
90#343—24no—
91#343—42no—
92#341—24no—
93#340—24no—
94#340—22no—
95#340—22no—
96#340—24no—
97#340—24no—
98#340—24no—
99#336—22no—
100#336—21no—
101#336—21no—
102#336—21no—
103#336—21no—
104#336—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4#4112yes—
108C23×C4C2411yescharacteristic
109C2×M16—21no—
110C2×M16—21no—
111C2×M16—22no—
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—22no—
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—22no—
122C2×M16—22no—
123C2×M16—22no—
124C2×M16—22no—
125C2×M16—22no—
126C2×M16C2×Q812yes—
127C2×M16C2×Q812yes—
128C2×M16#4114yes—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—21no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—22no—
137C22×C8C2×D412yes—
138C22×C8C2×D411yescharacteristic
139C22×C8C2×D411yescharacteristic
140C22×C8C2×Q812yes—
141C22×C8C2×Q812yes—
142C22×C8#4112yes—
143C22×C8#4112yes—
144#125—28no—
145#125—44no—
146#125—44no—
147#125—44no—
148#117—21no—
149#117—21no—
150#117—22no—
151#117—22no—
152#117C2×Q812yes—
153#117C2×Q812yes—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—84no—
174C2×C4⋊C4C2×D412yes—
175C2×C4⋊C4C2×D412yes—
176C2×C4⋊C4#4114yes—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C42—21no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—21no—
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D412yes—
186#108—48no—
187#107—44no—
188#106—44no—
189C4⋊C8—28no—
190#97—24no—
191#97—42no—
192#97—42no—
193#97—42no—
194C4×C8—24no—
195C4×C8—42no—
196#95—22no—
197#95—22no—
198C24C2×#4111yescharacteristic
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—24no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—82no—
212C22×C4—82no—
213C22×C4—84no—
214C22×C4#12112yes—
215C22×C4#12112yes—
216C22×C4C22×D411yescharacteristic
217C22×C4C22×D411yescharacteristic
218C22×C4C22×Q811yescharacteristic
219C22×C4C22×Q811yescharacteristic
220C22×C4C2×#4111yescharacteristic
221C22×C4C2×#4111yescharacteristic
222C22×C4C2×#4112yes—
223C22×C4C2×#4112yes—
224C22×C4H(5,𝔽2)12yes—
225C22×C42−1+412yes—
226M16—24no—
227M16—24no—
228M16—24no—
229M16—24no—
230M16—42no—
231M16—42no—
232M16—42no—
233M16—42no—
234M16—42no—
235M16—42no—
236M16—42no—
237M16—42no—
238C2×C8—22no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—22no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—24no—
245C2×C8—24no—
246C2×C8—24no—
247C2×C8—24no—
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—24no—
251C2×C8—41no—
252C2×C8—42no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8C22⋊Q812yes—
261C2×C8C22⋊Q812yes—
262C2×C8C22⋊Q814yes—
263C4⋊C4—22no—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C4⋊C4—24no—
267C4⋊C4—22no—
268C4⋊C4—48no—
269C4⋊C4—48no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C4⋊C4—84no—
279C4⋊C4—84no—
280C4⋊C4C4⋊Q814yes—
281C22⋊C4—22no—
282C22⋊C4—22no—
283C22⋊C4—42no—
284C42—22no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—41no—
291C42—41no—
292C23—22no—
293C23—22no—
294C23C2×#12111yescharacteristic
295C23#53311yescharacteristic
296C23#53311yescharacteristic
297C23#55112yes—
298C23#55612yes—
299C2×C4—24no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—84no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—84no—
318C2×C4—82no—
319C2×C4—84no—
320C2×C4—162no—
321C2×C4—162no—
322C2×C4C2×C22⋊Q811yescharacteristic
323C2×C4C2×C22⋊Q811yescharacteristic
324C2×C4#52511yescharacteristic
325C2×C4#52912yes—
326C2×C4C2×C4⋊Q811yescharacteristic
327C2×C4#53211yescharacteristic
328C2×C4#53211yescharacteristic
329C2×C4#53312yes—
330C2×C4#53411yescharacteristic
331C2×C4#53611yescharacteristic
332C2×C4#53912yes—
333C2×C4#54412yes—
334C2×C4#54612yes—
335C2×C4#54812yes—
336C2×C4#55112yes—
337C2×C4#55412yes—
338C2×C4#55612yes—
339C2×C4#55712yes—
340C8—24no—
341C8—22no—
342C8—22no—
343C8—44no—
344C8—42no—
345C8—42no—
346C8—82no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22#253411yescharacteristic
354C22#254511yescharacteristic
355C22#258812yes—
356C22#260512yes—
357C22#285711yescharacteristic
358C22#306812yes—
359C22#306912yes—
360C4—44no—
361C4—42no—
362C4—42no—
363C4—82no—
364C4—84no—
365C4—162no—
366C4#239611yescharacteristic
367C4#255012yes—
368C4#256012yes—
369C4#256411yescharacteristic
370C4#258111yescharacteristic
371C4#262711yescharacteristic
372C2—22no—
373C2—22no—
374C2#1516411yescharacteristic
375C2#2813912yes—
376C2#2867311yescharacteristic
377C2#2867411yescharacteristic
378C2#3322312yes—
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#237452
#235532
C2×#19922
#204362
#113832
#113542
#112492
#112422
#108962
#108652
#108272