Finite Group Databasebeta

← #415699

Subgroups — #415699

Order 512.

792 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1941C211yesmaximal, characteristic
2#21034C212yesmaximal
3C4×#424C211yesmaximal, characteristic
4#11330C211yesmaximal, characteristic
5#11271C211yesmaximal, characteristic
6#11249C212yesmaximal
7#11242C212yesmaximal
8#11238C211yesmaximal, characteristic
9#11238C211yesmaximal, characteristic
10#10913C212yesmaximal
11#10864C211yesmaximal, characteristic
12C22×#108C2212yes—
13C2×#428C2212yes—
14C22×#106C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C2×#424C2211yescharacteristic
17C2×#424C2211yescharacteristic
18C2×C4×M16C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20C2×#389C2212yes—
21C4×#117C2211yescharacteristic
22#1941—22no—
23#1941—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×#106—22no—
62C2×#106—22no—
63C2×#106—24no—
64C2×#106—22no—
65C2×#106—42no—
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⋊C4D412yes—
83C4×C4⋊C4D412yes—
84C4×C22⋊C4—22no—
85#376—21no—
86C2×#95—22no—
87C43—21no—
88#343—24no—
89#343—24no—
90#343—42no—
91#343—42no—
92#341—22no—
93#341—22no—
94#340—24no—
95#340—22no—
96#340—22no—
97#340—22no—
98#340—22no—
99#340—22no—
100#340—22no—
101#336—22no—
102#336—22no—
103#336—22no—
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—22no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—22no—
137C22×C8C2×D412yes—
138C22×C8C2×D412yes—
139C22×C8C2×Q812yes—
140C22×C8C2×Q812yes—
141C22×C8#4112yes—
142C22×C8#4112yes—
143#117—21no—
144#117—21no—
145#117—22no—
146#117—22no—
147#117C2×Q812yes—
148#117C2×Q812yes—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
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—42no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—84no—
163C2×C4⋊C4C2×D412yes—
164C2×C4⋊C4C2×D411yescharacteristic
165C2×C4⋊C4C2×D411yescharacteristic
166C2×C4⋊C4#4114yes—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—21no—
169C2×C42—21no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—21no—
175C2×C42C2×D412yes—
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178#108—48no—
179#107—28no—
180#106—28no—
181#106—28no—
182#106—44no—
183C4⋊C8—28no—
184C4⋊C8—28no—
185#97—42no—
186C4×C8—42no—
187C4×C8—42no—
188C4×C8—42no—
189#95—22no—
190#95—22no—
191#95—42no—
192#95—42no—
193C24C2×#4111yescharacteristic
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—24no—
199C22×C4—22no—
200C22×C4—24no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—82no—
208C22×C4—82no—
209C22×C4—82no—
210C22×C4—82no—
211C22×C4#12112yes—
212C22×C4#12112yes—
213C22×C4C22×D411yescharacteristic
214C22×C4C22×D411yescharacteristic
215C22×C4C22×Q811yescharacteristic
216C22×C4C22×Q811yescharacteristic
217C22×C4C2×#4111yescharacteristic
218C22×C4C2×#4111yescharacteristic
219C22×C4C2×#4112yes—
220C22×C4C2×#4112yes—
221C22×C4H(5,𝔽2)12yes—
222C22×C42−1+412yes—
223M16—24no—
224M16—24no—
225M16—24no—
226M16—24no—
227M16—42no—
228M16—42no—
229M16—42no—
230M16—42no—
231M16—42no—
232M16—42no—
233M16—42no—
234M16—42no—
235C2×C8—22no—
236C2×C8—22no—
237C2×C8—22no—
238C2×C8—22no—
239C2×C8—24no—
240C2×C8—24no—
241C2×C8—24no—
242C2×C8—24no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—24no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—42no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8C22⋊Q814yes—
256C2×C8C22⋊Q814yes—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—24no—
260C4⋊C4—22no—
261C4⋊C4—24no—
262C4⋊C4—22no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—84no—
270C4⋊C4—84no—
271C4⋊C4C4⋊Q812yes—
272C4⋊C4C4⋊Q812yes—
273C22⋊C4—22no—
274C22⋊C4—22no—
275C22⋊C4—42no—
276C42—22no—
277C42—22no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—21no—
282C42—24no—
283C42—44no—
284C42—41no—
285C42—41no—
286C42—44no—
287C23—22no—
288C23—22no—
289C23C2×#12111yescharacteristic
290C23#52912yes—
291C23#53311yescharacteristic
292C23#53311yescharacteristic
293C23#53912yes—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—84no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—84no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—162no—
317C2×C4—162no—
318C2×C4#47114yes—
319C2×C4C2×C22⋊Q811yescharacteristic
320C2×C4C2×C22⋊Q811yescharacteristic
321C2×C4#52511yescharacteristic
322C2×C4#52512yes—
323C2×C4C2×C4⋊Q811yescharacteristic
324C2×C4#53211yescharacteristic
325C2×C4#53211yescharacteristic
326C2×C4#53411yescharacteristic
327C2×C4#53611yescharacteristic
328C2×C4#53712yes—
329C2×C4#55112yes—
330C2×C4#55212yes—
331C2×C4#55312yes—
332C2×C4#55412yes—
333C2×C4#55412yes—
334C2×C4#55612yes—
335C2×C4#55712yes—
336C2×C4#55712yes—
337C8—24no—
338C8—24no—
339C8—42no—
340C8—42no—
341C8—44no—
342C8—82no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22#239211yescharacteristic
352C22#258812yes—
353C22#259012yes—
354C22#261811yescharacteristic
355C22#285711yescharacteristic
356C22C2×#47112yes—
357C22#306712yes—
358C4—24no—
359C4—42no—
360C4—42no—
361C4—42no—
362C4—82no—
363C4—84no—
364C4—162no—
365C4#256411yescharacteristic
366C4#256511yescharacteristic
367C4#257911yescharacteristic
368C4#258111yescharacteristic
369C4#258412yes—
370C4#260112yes—
371C2—22no—
372C2—22no—
373C2#1520611yescharacteristic
374C2#2371812yes—
375C2#2979012yes—
376C2#3322011yescharacteristic
377C2#3322111yescharacteristic
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×#19412
#210342
C4×#4242
#113302
#112712
#112492
#112422
#112382
#109132
#108642