Finite Group Databasebeta

← #414280

Subgroups — #414280

Order 512.

770 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23670C212yesmaximal
2C2×#1944C211yesmaximal, characteristic
3#20436C211yesmaximal, characteristic
4#11331C211yesmaximal, characteristic
5#11271C211yesmaximal, characteristic
6#11239C211yesmaximal, characteristic
7#11238C211yesmaximal, characteristic
8#11204C212yesmaximal
9#10896C212yesmaximal
10#10845C211yesmaximal, characteristic
11#10827C212yesmaximal
12C2×#428C2212yes—
13C22×#107C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C2×#424C2211yescharacteristic
16C2×#423C2212yes—
17C2×#405C2211yescharacteristic
18C2×#405C2211yescharacteristic
19C22×#97C2211yescharacteristic
20#2296C2211yescharacteristic
21C2×#389C2212yes—
22#2088—22no—
23#2088—22no—
24#2087—22no—
25#2086—22no—
26#1944—22no—
27#1944—22no—
28#1771—22no—
29#1770—22no—
30C2×#343C2212yes—
31C2×#343C2212yes—
32C2×#343C2212yes—
33C2×#343C2212yes—
34C2×#341C2211yescharacteristic
35C2×#341C2211yescharacteristic
36C2×#341C2212yes—
37#1736C2212yes—
38#1736C2212yes—
39#1736C2212yes—
40C2×#340C2212yes—
41#1729C2211yescharacteristic
42#1728C2211yescharacteristic
43C22×M16C2312yes—
44C22×M16C2311yescharacteristic
45C22×M16C2311yescharacteristic
46C22×M16C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×#117C2311yescharacteristic
50C2×#117C2311yescharacteristic
51C22×C4⋊C4—22no—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C42C2311yescharacteristic
55#428—24no—
56#428—42no—
57C2×#107—22no—
58C2×#107—22no—
59C2×#107—42no—
60C2×#106—22no—
61C2×#106—24no—
62C2×#106—22no—
63#424—22no—
64#424Q814yes—
65#423—24no—
66#423—42no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70#405—22no—
71#405—42no—
72#405Q814yes—
73C2×#97—21no—
74C2×#97—21no—
75C2×#97—42no—
76C2×#97D412yes—
77C2×#97D412yes—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80#389—22no—
81#389—22no—
82#389—22no—
83#389—22no—
84#389D412yes—
85#389D412yes—
86#385—21no—
87#385—21no—
88C4×C4⋊C4—22no—
89C2×#95—22no—
90#343—24no—
91#343—24no—
92#343—24no—
93#343—42no—
94#341—24no—
95#341—22no—
96#341—22no—
97#340—22no—
98#340—22no—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—22no—
103#340—22no—
104#340—22no—
105#336—22no—
106#336—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4#4111yescharacteristic
110C23×C4#4111yescharacteristic
111C23×C4C2411yescharacteristic
112C2×M16—21no—
113C2×M16—21no—
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×M16—22no—
127C2×M16—22no—
128C2×M16—22no—
129C2×M16C2×Q812yes—
130C2×M16C2×Q812yes—
131C2×M16#4114yes—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8C2×D412yes—
141C22×C8C2×D411yescharacteristic
142C22×C8C2×D411yescharacteristic
143C22×C8C2×Q812yes—
144C22×C8C2×Q812yes—
145C22×C8#4112yes—
146C22×C8#4112yes—
147#117—21no—
148#117—21no—
149#117—22no—
150#117—22no—
151#117C2×Q812yes—
152#117C2×Q812yes—
153C2×C4⋊C4—22no—
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—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—84no—
169C2×C4⋊C4C2×D411yescharacteristic
170C2×C4⋊C4C2×D411yescharacteristic
171C2×C4⋊C4C2×D412yes—
172C2×C4⋊C4#4114yes—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C42—21no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—21no—
179C2×C42C2×D412yes—
180C2×C42C2×D412yes—
181#107—44no—
182#107—44no—
183#106—28no—
184#106—44no—
185C4⋊C8—28no—
186C4⋊C8—44no—
187C4⋊C8—44no—
188#97—24no—
189#97—42no—
190#97—42no—
191#97—42no—
192C4×C8—24no—
193C4×C8—42no—
194#95—22no—
195#95—22no—
196#95—42no—
197#95—42no—
198C24C2×#4111yescharacteristic
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—24no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—82no—
213C22×C4—82no—
214C22×C4—82no—
215C22×C4—82no—
216C22×C4#12112yes—
217C22×C4#12112yes—
218C22×C4C22×D411yescharacteristic
219C22×C4C22×D411yescharacteristic
220C22×C4C22×Q811yescharacteristic
221C22×C4C22×Q811yescharacteristic
222C22×C4C2×#4111yescharacteristic
223C22×C4C2×#4111yescharacteristic
224C22×C4C2×#4112yes—
225C22×C4C2×#4112yes—
226C22×C4H(5,𝔽2)12yes—
227C22×C42−1+412yes—
228M16—24no—
229M16—24no—
230M16—24no—
231M16—24no—
232M16—42no—
233M16—42no—
234M16—42no—
235M16—42no—
236M16—42no—
237M16—42no—
238M16—42no—
239M16—42no—
240C2×C8—22no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—22no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—24no—
247C2×C8—24no—
248C2×C8—24no—
249C2×C8—24no—
250C2×C8—22no—
251C2×C8—22no—
252C2×C8—24no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—42no—
257C2×C8—41no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8C22⋊Q812yes—
263C2×C8C22⋊Q812yes—
264C2×C8C22⋊Q814yes—
265C4⋊C4—22no—
266C4⋊C4—24no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—24no—
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#55212yes—
298C23#55312yes—
299C2×C4—24no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—44no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—84no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—84no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—162no—
323C2×C4—162no—
324C2×C4C2×C22⋊Q811yescharacteristic
325C2×C4C2×C22⋊Q811yescharacteristic
326C2×C4#52511yescharacteristic
327C2×C4#52912yes—
328C2×C4C2×C4⋊Q811yescharacteristic
329C2×C4#53211yescharacteristic
330C2×C4#53211yescharacteristic
331C2×C4#53312yes—
332C2×C4#53411yescharacteristic
333C2×C4#53611yescharacteristic
334C2×C4#53912yes—
335C2×C4#54412yes—
336C2×C4#54612yes—
337C2×C4#54812yes—
338C2×C4#55212yes—
339C2×C4#55312yes—
340C2×C4#55412yes—
341C2×C4#55712yes—
342C8—24no—
343C8—22no—
344C8—22no—
345C8—42no—
346C8—42no—
347C8—44no—
348C8—82no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22#253611yescharacteristic
356C22#254311yescharacteristic
357C22#259012yes—
358C22#260612yes—
359C22#285711yescharacteristic
360C22#306812yes—
361C22#306912yes—
362C4—42no—
363C4—42no—
364C4—44no—
365C4—84no—
366C4—81no—
367C4—81no—
368C4—162no—
369C4#239611yescharacteristic
370C4#255012yes—
371C4#255812yes—
372C4#256511yescharacteristic
373C4#257911yescharacteristic
374C4#262711yescharacteristic
375C2—22no—
376C2—22no—
377C2#1516611yescharacteristic
378C2#2822112yes—
379C2#2859411yescharacteristic
380C2#2859511yescharacteristic
381C2#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
#236702
C2×#19442
#204362
#113312
#112712
#112392
#112382
#112042
#108962
#108452
#108272