Finite Group Databasebeta

← #415700

Subgroups — #415700

Order 512.

792 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1942C211yesmaximal, characteristic
2#21034C212yesmaximal
3C4×#424C211yesmaximal, characteristic
4#11331C211yesmaximal, characteristic
5#11270C211yesmaximal, characteristic
6#11249C212yesmaximal
7#11242C212yesmaximal
8#11239C211yesmaximal, characteristic
9#11239C211yesmaximal, characteristic
10#10913C212yesmaximal
11#10864C211yesmaximal, characteristic
12C22×#108C2212yes—
13C2×#428C2212yes—
14C22×#107C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C2×#424C2211yescharacteristic
17C2×#424C2211yescharacteristic
18C2×C4×M16C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20C2×#389C2212yes—
21C4×#117C2211yescharacteristic
22#1942—22no—
23#1942—22no—
24#1921—22no—
25#1920—22no—
26#1919—22no—
27#1918—22no—
28#1768—22no—
29C4×C4⋊C8—22no—
30C2×#343C2212yes—
31C2×#343C2212yes—
32C2×#343C2212yes—
33C2×#343C2212yes—
34C2×#341C2211yescharacteristic
35C2×#341C2211yescharacteristic
36#1736C2212yes—
37#1736C2212yes—
38#1736C2212yes—
39C2×#340C2212yes—
40#1729C2211yescharacteristic
41#1729C2211yescharacteristic
42#1728C2212yes—
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
55C2×#108—24no—
56C2×#108—24no—
57C2×#108—42no—
58#428—24no—
59#428—24no—
60C2×#107—22no—
61C2×#107—22no—
62C2×#107—24no—
63C2×#107—22no—
64C2×#107—42no—
65C2×#106—22no—
66#424—22no—
67#424—22no—
68#424Q814yes—
69#424Q814yes—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C4×M16—42no—
73C2×#97—21no—
74C2×C4×C8—21no—
75C2×C4×C8—42no—
76C2×C4×C8D412yes—
77C2×C4×C8D412yes—
78#389—22no—
79#389—22no—
80#389—22no—
81#389—22no—
82C4×C4⋊C4D411yescharacteristic
83C4×C4⋊C4D411yescharacteristic
84C4×C4⋊C4D411yescharacteristic
85C4×C4⋊C4D411yescharacteristic
86C4×C22⋊C4—21no—
87C4×C22⋊C4—21no—
88#376—21no—
89C2×#95—22no—
90C43—21no—
91#343—24no—
92#343—24no—
93#343—42no—
94#343—42no—
95#341—22no—
96#341—22no—
97#340—24no—
98#340—22no—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—22no—
103#340—22no—
104#336—22no—
105#336—22no—
106#336—22no—
107#336—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4#4111yescharacteristic
111C23×C4#4111yescharacteristic
112C23×C4C2411yescharacteristic
113C2×M16—21no—
114C2×M16—21no—
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×M16—22no—
130C2×M16C2×Q812yes—
131C2×M16C2×Q812yes—
132C2×M16#4114yes—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8C2×D412yes—
142C22×C8C2×D412yes—
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—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—84no—
167C2×C4⋊C4C2×D411yescharacteristic
168C2×C4⋊C4C2×D411yescharacteristic
169C2×C4⋊C4C2×D411yescharacteristic
170C2×C4⋊C4C2×D411yescharacteristic
171C2×C4⋊C4#4114yes—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—21no—
174C2×C42—21no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186#108—48no—
187#107—28no—
188#107—28no—
189#107—44no—
190#106—28no—
191C4⋊C8—28no—
192C4⋊C8—28no—
193#97—42no—
194C4×C8—42no—
195C4×C8—42no—
196C4×C8—42no—
197#95—21no—
198#95—21no—
199#95—21no—
200#95—21no—
201#95—42no—
202#95—42no—
203C24C2×#4111yescharacteristic
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—24no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—82no—
221C22×C4—82no—
222C22×C4—82no—
223C22×C4—82no—
224C22×C4#12111yescharacteristic
225C22×C4#12111yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4C22×D411yescharacteristic
229C22×C4C22×D411yescharacteristic
230C22×C4C22×Q811yescharacteristic
231C22×C4C22×Q811yescharacteristic
232C22×C4C2×#4111yescharacteristic
233C22×C4C2×#4111yescharacteristic
234C22×C4C2×#4112yes—
235C22×C4C2×#4112yes—
236C22×C4H(5,𝔽2)12yes—
237C22×C42−1+412yes—
238M16—24no—
239M16—24no—
240M16—24no—
241M16—24no—
242M16—42no—
243M16—42no—
244M16—42no—
245M16—42no—
246M16—42no—
247M16—42no—
248M16—42no—
249M16—42no—
250C2×C8—22no—
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—24no—
255C2×C8—24no—
256C2×C8—24no—
257C2×C8—24no—
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—24no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—42no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8C22⋊Q814yes—
271C2×C8C22⋊Q814yes—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—22no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—22no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—84no—
285C4⋊C4—84no—
286C4⋊C4C4⋊Q812yes—
287C4⋊C4C4⋊Q812yes—
288C22⋊C4—22no—
289C22⋊C4—22no—
290C22⋊C4—42no—
291C42—22no—
292C42—22no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—21no—
297C42—22no—
298C42—22no—
299C42—42no—
300C42—42no—
301C42—41no—
302C42—41no—
303C42—42no—
304C42—42no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23C2×#12111yescharacteristic
310C23#52912yes—
311C23#53311yescharacteristic
312C23#53311yescharacteristic
313C23#53912yes—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—44no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—84no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—84no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—162no—
347C2×C4—162no—
348C2×C4#47212yes—
349C2×C4#47312yes—
350C2×C4C2×C22⋊Q811yescharacteristic
351C2×C4C2×C22⋊Q811yescharacteristic
352C2×C4#52511yescharacteristic
353C2×C4#52512yes—
354C2×C4C2×C4⋊Q811yescharacteristic
355C2×C4#53211yescharacteristic
356C2×C4#53211yescharacteristic
357C2×C4#53411yescharacteristic
358C2×C4#53611yescharacteristic
359C2×C4#53712yes—
360C2×C4#55112yes—
361C2×C4#55212yes—
362C2×C4#55312yes—
363C2×C4#55412yes—
364C2×C4#55412yes—
365C2×C4#55612yes—
366C2×C4#55712yes—
367C2×C4#55712yes—
368C8—24no—
369C8—24no—
370C8—42no—
371C8—42no—
372C8—44no—
373C8—82no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22#239211yescharacteristic
383C22#258812yes—
384C22#259012yes—
385C22#261811yescharacteristic
386C22#285711yescharacteristic
387C22C2×#47211yescharacteristic
388C22C2×#47311yescharacteristic
389C22#306711yescharacteristic
390C22#306711yescharacteristic
391C4—22no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—42no—
396C4—42no—
397C4—81no—
398C4—84no—
399C4—81no—
400C4—162no—
401C4#256411yescharacteristic
402C4#256511yescharacteristic
403C4#257911yescharacteristic
404C4#258111yescharacteristic
405C4#258412yes—
406C4#260112yes—
407C2—22no—
408C2—22no—
409C2#1520611yescharacteristic
410C2#2371611yescharacteristic
411C2#2371711yescharacteristic
412C2#2978811yescharacteristic
413C2#2978911yescharacteristic
414C2#3322111yescharacteristic
415C2#3322211yescharacteristic
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
C2×#19422
#210342
C4×#4242
#113312
#112702
#112492
#112422
#112392
#109132
#108642