Finite Group Databasebeta

← #415530

Subgroups — #415530

Order 512.

580 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#11320C212yesmaximal
2#11306C212yesmaximal
3#11304C212yesmaximal
4#11300C212yesmaximal
5#10873C211yesmaximal, characteristic
6#10872C211yesmaximal, characteristic
7#10868C211yesmaximal, characteristic
8#10867C211yesmaximal, characteristic
9#10865C211yesmaximal, characteristic
10#10860C211yesmaximal, characteristic
11#10707C211yesmaximal, characteristic
12C22×#108C2211yescharacteristic
13C22×#108C2211yescharacteristic
14C22×#108C2211yescharacteristic
15C2×#427C2211yescharacteristic
16C2×#405C2211yescharacteristic
17C2×#405C2211yescharacteristic
18C22×#97C2211yescharacteristic
19C2×#344C2212yes—
20C2×#344C2212yes—
21C2×#344C2212yes—
22#1742C2212yes—
23#1738C2212yes—
24C2×#341C2212yes—
25C2×#341C2211yescharacteristic
26#1735C2211yescharacteristic
27#1733C2212yes—
28C2×#339C2211yescharacteristic
29C2×#339C2211yescharacteristic
30C2×#339C2212yes—
31C2×#338C2212yes—
32C2×H(3,ℤ/4)C2212yes—
33#1729C2212yes—
34#1728C2212yes—
35C22×M16C2312yes—
36C22×M16C2312yes—
37C22×M16C2312yes—
38C22×M16C2311yescharacteristic
39C22×M16C2311yescharacteristic
40C23×C8C2311yescharacteristic
41C23×C8C2311yescharacteristic
42C2×#117C2312yes—
43C2×#117C2311yescharacteristic
44C2×#117C2311yescharacteristic
45C22×C42C2311yescharacteristic
46C2×#108—21no—
47C2×#108—21no—
48C2×#108—21no—
49C2×#108—21no—
50C2×#108—22no—
51C2×#108—22no—
52C2×#108—21no—
53C2×#108—21no—
54C2×#108—22no—
55C2×#108—22no—
56C2×#108—22no—
57C2×#108—22no—
58#427—22no—
59#427—22no—
60C2×#107—21no—
61C2×#106—21no—
62#405—22no—
63#405—21no—
64#405—21no—
65#405—21no—
66#405—21no—
67#405—22no—
68C2×#97—22no—
69C2×#97—22no—
70C2×#97—21no—
71C2×#97—21no—
72C2×#97—21no—
73C2×#97—21no—
74C2×C4×C8—21no—
75C2×C4×C8—21no—
76#344—24no—
77#344—42no—
78#344—42no—
79#343—24no—
80#341—42no—
81#341—42no—
82#341Q814yes—
83#340—22no—
84#339—22no—
85#339—22no—
86#339—42no—
87#339—42no—
88#338—42no—
89H(3,ℤ/4)—42no—
90#336—24no—
91#336—42no—
92C23×C4—21no—
93C23×C4C2411yescharacteristic
94C2×M16—21no—
95C2×M16—21no—
96C2×M16—21no—
97C2×M16—21no—
98C2×M16—21no—
99C2×M16—21no—
100C2×M16—21no—
101C2×M16—21no—
102C2×M16—21no—
103C2×M16—21no—
104C2×M16—22no—
105C2×M16—22no—
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—22no—
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×M16C2×Q812yes—
123C2×M16C2×Q812yes—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8C2×Q812yes—
136C22×C8#4111yescharacteristic
137C22×C8#4111yescharacteristic
138C22×C8#4111yescharacteristic
139C22×C8#4111yescharacteristic
140C22×C8#4112yes—
141C22×C8#4112yes—
142C22×C8#4111yescharacteristic
143C22×C8#4111yescharacteristic
144C22×C8#4112yes—
145C22×C8#4112yes—
146#117—21no—
147#117—21no—
148#117—21no—
149#117—21no—
150#117—21no—
151#117—21no—
152#117—21no—
153#117—22no—
154#117—22no—
155#117—22no—
156#117—22no—
157#117#4112yes—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4#4112yes—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—22no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—22no—
172C2×C42#4112yes—
173#108—42no—
174#108—42no—
175#108—42no—
176#108—42no—
177#108—42no—
178#108—42no—
179#107—42no—
180#106—42no—
181#97—42no—
182#97—42no—
183#97—42no—
184#97—41no—
185#97—41no—
186C4×C8—42no—
187C4×C8—41no—
188C4×C8—41no—
189C242−1+411yescharacteristic
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—42no—
196C22×C4—82no—
197C22×C4C22×Q811yescharacteristic
198C22×C4C2×#4111yescharacteristic
199C22×C4C2×#4111yescharacteristic
200C22×C4C2×#4111yescharacteristic
201C22×C4C2×#4111yescharacteristic
202C22×C4C2×#4111yescharacteristic
203C22×C4C2×#4111yescharacteristic
204C22×C4C2×#4111yescharacteristic
205C22×C4C2×#4111yescharacteristic
206C22×C4C2×#4111yescharacteristic
207C22×C4C2×#4111yescharacteristic
208C22×C4H(5,𝔽2)11yescharacteristic
209C22×C4H(5,𝔽2)11yescharacteristic
210C22×C4H(5,𝔽2)11yescharacteristic
211M16—22no—
212M16—22no—
213M16—22no—
214M16—22no—
215M16—41no—
216M16—41no—
217M16—41no—
218M16—41no—
219M16—42no—
220M16—42no—
221M16—42no—
222M16—42no—
223M16—42no—
224M16—42no—
225M16—42no—
226M16—42no—
227M16—42no—
228M16—42no—
229M16—42no—
230M16—42no—
231C2×C8—22no—
232C2×C8—22no—
233C2×C8—22no—
234C2×C8—22no—
235C2×C8—22no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8#12611yescharacteristic
254C2×C8#12611yescharacteristic
255C2×C8#12612yes—
256C2×C8#12612yes—
257C2×C8#12612yes—
258C2×C8#12611yescharacteristic
259C2×C8#12611yescharacteristic
260C2×C8#12612yes—
261C2×C8#12612yes—
262C2×C8#12612yes—
263C2×C8#12612yes—
264C2×C8#12612yes—
265C2×C8#12612yes—
266C2×C8#12612yes—
267C2×C8#12612yes—
268C2×C8#12612yes—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C22⋊C4—22no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—41no—
285C42—41no—
286C42—41no—
287C42—41no—
288C42—41no—
289C42—42no—
290C42—42no—
291C23—42no—
292C23#54211yescharacteristic
293C23#54211yescharacteristic
294C23#54211yescharacteristic
295C23#55211yescharacteristic
296C23#55211yescharacteristic
297C23#55211yescharacteristic
298C23#55711yescharacteristic
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—84no—
308C2×C4—82no—
309C2×C4C2×#12611yescharacteristic
310C2×C4C2×#12611yescharacteristic
311C2×C4C2×#12611yescharacteristic
312C2×C4C2×#12611yescharacteristic
313C2×C4C2×#12611yescharacteristic
314C2×C4C2×#12611yescharacteristic
315C2×C4C2×#12611yescharacteristic
316C2×C4#52911yescharacteristic
317C2×C4#52911yescharacteristic
318C2×C4#52911yescharacteristic
319C2×C4#53311yescharacteristic
320C2×C4#53311yescharacteristic
321C2×C4#53311yescharacteristic
322C2×C4#54011yescharacteristic
323C2×C4#54011yescharacteristic
324C2×C4#54011yescharacteristic
325C2×C4#55311yescharacteristic
326C2×C4#55311yescharacteristic
327C2×C4#55311yescharacteristic
328C2×C4#55411yescharacteristic
329C2×C4#55411yescharacteristic
330C2×C4#55411yescharacteristic
331C2×C4#55511yescharacteristic
332C2×C4#55511yescharacteristic
333C2×C4#55511yescharacteristic
334C2×C4#55911yescharacteristic
335C2×C4#55911yescharacteristic
336C2×C4#55911yescharacteristic
337C8—42no—
338C8—42no—
339C8—42no—
340C8—82no—
341C8—82no—
342C8—82no—
343C8#40112yes—
344C8#40112yes—
345C8#40112yes—
346C8#40112yes—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22#249511yescharacteristic
355C22#249511yescharacteristic
356C22#249511yescharacteristic
357C22#251111yescharacteristic
358C22#251111yescharacteristic
359C22#251111yescharacteristic
360C22#264411yescharacteristic
361C4—82no—
362C4—82no—
363C4—82no—
364C4—162no—
365C4C2×#40111yescharacteristic
366C4#249711yescharacteristic
367C4#249711yescharacteristic
368C4#249711yescharacteristic
369C4#251311yescharacteristic
370C4#251311yescharacteristic
371C4#251311yescharacteristic
372C4#286311yescharacteristic
373C2—21no—
374C2—21no—
375C2—21no—
376C2—21no—
377C2—21no—
378C2—21no—
379C2#1519811yescharacteristic
380C2#2783412yes—
subgroupG/H
centerC2×C4#555
commutator, Frattini, cosocleC23×C4C24
socleC22#2495
prime radicalC242−1+4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ 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
#113202
#113062
#113042
#113002
#108732
#108722
#108682
#108672
#108652
#108602
#107072