Finite Group Databasebeta

← #415492

Subgroups — #415492

Order 512.

824 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21172C211yesmaximal, characteristic
2#21171C211yesmaximal, characteristic
3C2×#1847C211yesmaximal, characteristic
4#20427C211yesmaximal, characteristic
5#11331C212yesmaximal
6#11330C212yesmaximal
7#11239C212yesmaximal
8#11238C212yesmaximal
9#10858C211yesmaximal, characteristic
10#10857C211yesmaximal, characteristic
11#10845C211yesmaximal, characteristic
12C2×#428C2211yescharacteristic
13C2×#428C2212yes—
14C2×#428C2211yescharacteristic
15C22×#107C2212yes—
16C22×#106C2212yes—
17C2×#424C2212yes—
18C2×#423C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C2×C4×M16C2211yescharacteristic
21C22×#97C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C4×#117C2211yescharacteristic
25#1949—22no—
26#1949—22no—
27#1948—22no—
28#1947—22no—
29#1847—22no—
30#1847—22no—
31#1768—22no—
32#1767—22no—
33C2×#343C2212yes—
34C2×#343C2212yes—
35C2×#343C2212yes—
36C2×#343C2212yes—
37C2×#341C2212yes—
38C2×#341C2212yes—
39#1736C2212yes—
40#1736C2211yescharacteristic
41#1736C2211yescharacteristic
42#1729C2212yes—
43C22×M16C2312yes—
44C22×M16C2311yescharacteristic
45C22×M16C2311yescharacteristic
46C22×M16C2312yes—
47C23×C8C2312yes—
48C2×#117C2311yescharacteristic
49C2×#117C2311yescharacteristic
50C22×C4⋊C4—22no—
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2311yescharacteristic
54C22×C42C2311yescharacteristic
55#428—42no—
56#428—42no—
57#428—42no—
58#428—42no—
59C2×#107—24no—
60C2×#107—22no—
61C2×#107—21no—
62C2×#107—21no—
63C2×#107—42no—
64C2×#107—42no—
65C2×#106—24no—
66C2×#106—22no—
67C2×#106—21no—
68C2×#106—21no—
69C2×#106—42no—
70C2×#106—42no—
71#424—24no—
72#424Q818yes—
73#423—22no—
74#423—22no—
75#423Q814yes—
76#423Q814yes—
77C2×C4⋊C8—24no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C4×M16—22no—
81C4×M16—22no—
82C2×#97—22no—
83C2×#97—22no—
84C2×#97—21no—
85C2×#97D412yes—
86C2×#97D412yes—
87C2×C4×C8—21no—
88#384—21no—
89#384—21no—
90#384—22no—
91#382—21no—
92#382—21no—
93#382—22no—
94C4×C4⋊C4D412yes—
95C4×C4⋊C4D412yes—
96C4×C22⋊C4—22no—
97#376—21no—
98C2×#95—22no—
99C43—21no—
100#343—42no—
101#343—42no—
102#343—42no—
103#343—42no—
104#341—24no—
105#341—24no—
106#340—24no—
107#340—22no—
108#340—22no—
109#336—24no—
110C23×C4—22no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4#4112yes—
114C23×C4C2411yescharacteristic
115C2×M16—21no—
116C2×M16—21no—
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×M16—22no—
130C2×M16—22no—
131C2×M16C2×Q812yes—
132C2×M16C2×Q812yes—
133C2×M16C2×Q814yes—
134C2×M16C2×Q814yes—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8C2×D414yes—
144C22×C8C2×Q814yes—
145C22×C8#4114yes—
146#117—22no—
147#117—22no—
148#117—22no—
149#117C2×Q814yes—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
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—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—84no—
167C2×C4⋊C4C2×D411yescharacteristic
168C2×C4⋊C4C2×D411yescharacteristic
169C2×C4⋊C4C2×D412yes—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—21no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42C2×D412yes—
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×Q812yes—
180C2×C42C2×Q812yes—
181C2×C42#4112yes—
182C2×C42#4112yes—
183#107—42no—
184#107—42no—
185#107—44no—
186#107—48no—
187#106—44no—
188#106—42no—
189#106—42no—
190#106—48no—
191C4⋊C8—216no—
192C4⋊C8—28no—
193C4⋊C8—28no—
194#97—24no—
195#97—24no—
196#97—42no—
197C4×C8—42no—
198#95—22no—
199#95—22no—
200#95—42no—
201#95—42no—
202C24C2×#4111yescharacteristic
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—84no—
216C22×C4—82no—
217C22×C4—82no—
218C22×C4#12112yes—
219C22×C4#12112yes—
220C22×C4#12514yes—
221C22×C4#12514yes—
222C22×C4C22×D411yescharacteristic
223C22×C4C22×D411yescharacteristic
224C22×C4C22×Q811yescharacteristic
225C22×C4C22×Q811yescharacteristic
226C22×C4C22×Q812yes—
227C22×C4C2×#4111yescharacteristic
228C22×C4C2×#4111yescharacteristic
229C22×C4C2×#4112yes—
230C22×C4H(5,𝔽2)12yes—
231C22×C4H(5,𝔽2)12yes—
232M16—24no—
233M16—24no—
234M16—24no—
235M16—24no—
236M16—24no—
237M16—24no—
238M16—42no—
239M16—42no—
240M16—42no—
241M16—42no—
242M16—42no—
243M16—42no—
244C2×C8—24no—
245C2×C8—24no—
246C2×C8—24no—
247C2×C8—24no—
248C2×C8—24no—
249C2×C8—24no—
250C2×C8—24no—
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—24no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—42no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—42no—
262C2×C8C22⋊Q818yes—
263C4⋊C4—22no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C4⋊C4—84no—
279C4⋊C4—84no—
280C4⋊C4C4⋊Q814yes—
281C22⋊C4—24no—
282C22⋊C4—42no—
283C42—24no—
284C42—42no—
285C42—82no—
286C42—82no—
287C42C22⋊Q814yes—
288C42C22⋊Q814yes—
289C42#12312yes—
290C42#12312yes—
291C42C4⋊Q812yes—
292C42C4⋊Q812yes—
293C23—22no—
294C23—22no—
295C23C2×#12111yescharacteristic
296C23C2×#12511yescharacteristic
297C23C2×#12511yescharacteristic
298C23#53312yes—
299C23#54012yes—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—82no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—84no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—162no—
320C2×C4—162no—
321C2×C4—162no—
322C2×C4—162no—
323C2×C4#39814yes—
324C2×C4#39814yes—
325C2×C4C2×C22⋊Q811yescharacteristic
326C2×C4C2×C22⋊Q811yescharacteristic
327C2×C4C2×C22⋊Q812yes—
328C2×C4C2×#12311yescharacteristic
329C2×C4C2×C4⋊Q811yescharacteristic
330C2×C4C2×C4⋊Q811yescharacteristic
331C2×C4#53211yescharacteristic
332C2×C4#53511yescharacteristic
333C2×C4#53511yescharacteristic
334C2×C4#53512yes—
335C2×C4#55312yes—
336C2×C4#55312yes—
337C2×C4#55412yes—
338C2×C4#55412yes—
339C2×C4#55412yes—
340C2×C4#55412yes—
341C2×C4Q8212yes—
342C2×C4Q8212yes—
343C8—28no—
344C8—44no—
345C8—42no—
346C8—42no—
347C8—44no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—24no—
353C22C2×#39811yescharacteristic
354C22#250712yes—
355C22#250712yes—
356C22#261911yescharacteristic
357C22#285711yescharacteristic
358C22#306812yes—
359C22#306912yes—
360C4—24no—
361C4—24no—
362C4—44no—
363C4—81no—
364C4—81no—
365C4—162no—
366C4—161no—
367C4—161no—
368C4#248712yes—
369C4#256611yescharacteristic
370C4#256611yescharacteristic
371C4#256711yescharacteristic
372C4#256711yescharacteristic
373C4#256712yes—
374C2—22no—
375C2—22no—
376C2#1519611yescharacteristic
377C2#2367512yes—
378C2#2982211yescharacteristic
379C2#2982311yescharacteristic
380C2#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
#211722
#211712
C2×#18472
#204272
#113312
#113302
#112392
#112382
#108582
#108572
#108452