Finite Group Databasebeta

← #416776

Subgroups — #416776

Order 512.

806 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21143C211yesmaximal, characteristic
2#20800C211yesmaximal, characteristic
3#20764C211yesmaximal, characteristic
4C2×#1834C211yesmaximal, characteristic
5#11290C214yesmaximal
6#11216C214yesmaximal
7#10857C211yesmaximal, characteristic
8#10857C211yesmaximal, characteristic
9C4×#341C211yesmaximal, characteristic
10C2×#428C2214yes—
11C2×#424C2212yes—
12C2×#423C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#2285C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C2×#384C2211yescharacteristic
19#1939—22no—
20#1939—22no—
21#1847—22no—
22#1845—21no—
23#1844—21no—
24#1836—22no—
25#1834—22no—
26#1834—22no—
27#1834—21no—
28#1834—21no—
29C2×#343C2214yes—
30C2×#343C2214yes—
31C2×#341C2212yes—
32C2×#341C2212yes—
33#1736C2214yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36#1729C2212yes—
37C22×M16C2312yes—
38C22×M16C2312yes—
39C22×M16C2312yes—
40C23×C8C2311yescharacteristic
41C23×C8C2311yescharacteristic
42C2×#117C2312yes—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C42C2311yescharacteristic
50#428—28no—
51#428—44no—
52C2×#107—24no—
53C2×#106—24no—
54#424—44no—
55#423—22no—
56#423—22no—
57C2×C4⋊C8—24no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C4×M16—22no—
61C4×M16—22no—
62C4×M16Q814yes—
63C4×M16Q814yes—
64C2×#97—21no—
65C2×#97—21no—
66C2×C4×C8—21no—
67C2×C4×C8—21no—
68C2×C4×C8—22no—
69C2×C4×C8—22no—
70C2×C4×C8D412yes—
71C2×C4×C8D412yes—
72#384—21no—
73#384—21no—
74#384—21no—
75#384—21no—
76#384—21no—
77#384—21no—
78#384—21no—
79#384—21no—
80#384D411yescharacteristic
81#384D411yescharacteristic
82#382D412yes—
83#380—21no—
84#380—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87#343—28no—
88#343—44no—
89#341—24no—
90#341—24no—
91#340—24no—
92#340—24no—
93#340—24no—
94#340—24no—
95#336—24no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4#4111yescharacteristic
99C23×C4#4111yescharacteristic
100C23×C4C2411yescharacteristic
101C2×M16—22no—
102C2×M16—22no—
103C2×M16—22no—
104C2×M16—22no—
105C2×M16—22no—
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—24no—
110C2×M16—22no—
111C2×M16—22no—
112C2×M16—24no—
113C2×M16C2×Q814yes—
114C2×M16C2×Q814yes—
115C2×M16#4114yes—
116C2×M16#4114yes—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—21no—
122C22×C8—21no—
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8C2×D412yes—
129C22×C8C2×D412yes—
130#117—24no—
131#117—24no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—82no—
153C2×C4⋊C4—82no—
154C2×C4⋊C4—82no—
155C2×C4⋊C4—82no—
156C2×C4⋊C4C2×D412yes—
157C2×C4⋊C4C2×D412yes—
158C2×C22⋊C4—22no—
159C2×C42—22no—
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×Q812yes—
165C2×C42C2×Q812yes—
166C2×C42#4112yes—
167C2×C42#4112yes—
168#107—48no—
169#106—48no—
170C4⋊C8—48no—
171C4⋊C8—42no—
172C4⋊C8—42no—
173#97—24no—
174#97—24no—
175C4×C8—24no—
176C4×C8—24no—
177C4×C8—24no—
178C4×C8—24no—
179#95—22no—
180#95—22no—
181C24C2×#4111yescharacteristic
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—82no—
193C22×C4—82no—
194C22×C4—82no—
195C22×C4—82no—
196C22×C4#12111yescharacteristic
197C22×C4#12111yescharacteristic
198C22×C4#12111yescharacteristic
199C22×C4#12111yescharacteristic
200C22×C4#12512yes—
201C22×C4#12512yes—
202C22×C4#12512yes—
203C22×C4#12512yes—
204C22×C4C22×D411yescharacteristic
205C22×C4C22×D411yescharacteristic
206C22×C4C22×Q811yescharacteristic
207C22×C4C22×Q811yescharacteristic
208C22×C4C2×#4111yescharacteristic
209C22×C4C2×#4111yescharacteristic
210C22×C4C2×#4112yes—
211C22×C4C2×#4112yes—
212C22×C4H(5,𝔽2)12yes—
213C22×C42−1+412yes—
214M16—24no—
215M16—24no—
216M16—24no—
217M16—24no—
218M16—24no—
219M16—42no—
220M16—42no—
221M16—44no—
222M16—44no—
223M16#12518yes—
224C2×C8—24no—
225C2×C8—24no—
226C2×C8—24no—
227C2×C8—24no—
228C2×C8—22no—
229C2×C8—22no—
230C2×C8—22no—
231C2×C8—22no—
232C2×C8—22no—
233C2×C8—22no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—44no—
254C4⋊C4—44no—
255C4⋊C4—82no—
256C4⋊C4—82no—
257C4⋊C4—82no—
258C4⋊C4—82no—
259C4⋊C4—82no—
260C4⋊C4—82no—
261C4⋊C4—82no—
262C4⋊C4—82no—
263C22⋊C4—44no—
264C42—42no—
265C42—42no—
266C42C22⋊Q812yes—
267C42C22⋊Q812yes—
268C42C22⋊Q812yes—
269C42C22⋊Q812yes—
270C42#12312yes—
271C42#12312yes—
272C42C4⋊Q812yes—
273C42C4⋊Q812yes—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23C2×#12111yescharacteristic
279C23C2×#12511yescharacteristic
280C23C2×#12511yescharacteristic
281C23#55612yes—
282C23#55912yes—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—164no—
304C2×C4—164no—
305C2×C4#39812yes—
306C2×C4#39812yes—
307C2×C4#39812yes—
308C2×C4#39812yes—
309C2×C4#47112yes—
310C2×C4#47112yes—
311C2×C4#47212yes—
312C2×C4#47312yes—
313C2×C4C2×C22⋊Q811yescharacteristic
314C2×C4C2×C22⋊Q811yescharacteristic
315C2×C4C2×#12311yescharacteristic
316C2×C4C2×#12512yes—
317C2×C4C2×C4⋊Q811yescharacteristic
318C2×C4#53211yescharacteristic
319C2×C4#53211yescharacteristic
320C2×C4#53312yes—
321C2×C4#53511yescharacteristic
322C2×C4#53611yescharacteristic
323C2×C4#54412yes—
324C2×C4#54412yes—
325C2×C4#54612yes—
326C2×C4#54812yes—
327C2×C4#55212yes—
328C2×C4#55312yes—
329C2×C4#55512yes—
330C2×C4#55612yes—
331C8—28no—
332C8—44no—
333C8—42no—
334C8—42no—
335C8—82no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22C2×#39811yescharacteristic
345C22#254411yescharacteristic
346C22#254511yescharacteristic
347C22#257012yes—
348C22#257112yes—
349C22C2×#47111yescharacteristic
350C22C2×#47111yescharacteristic
351C22C2×#47211yescharacteristic
352C22C2×#47311yescharacteristic
353C4—22no—
354C4—22no—
355C4—82no—
356C4—82no—
357C4—162no—
358C4—162no—
359C4#208914yes—
360C4#209014yes—
361C4#243912yes—
362C4#244212yes—
363C4#249911yescharacteristic
364C4#250011yescharacteristic
365C4#266011yescharacteristic
366C4#285911yescharacteristic
367C2—22no—
368C2—22no—
369C2#1523111yescharacteristic
370C2C2×#208911yescharacteristic
371C2C2×#209011yescharacteristic
372C2#2863111yescharacteristic
373C2#2863211yescharacteristic
374C2#2866911yescharacteristic
375C2#2867011yescharacteristic
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
#211432
#208002
#207642
C2×#18342
#112902
#112162
#108572
C4×#3412