Finite Group Databasebeta

← #6431651

Subgroups — #6431651

Order 512.

1104 conjugacy classes of subgroups.

The subgroup lattice has 367 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36701C211yesmaximal, characteristic
2#23373C213yesmaximal
3#23267C213yesmaximal
4#23266C213yesmaximal
5#23261C213yesmaximal
6#23259C213yesmaximal
7#22376C213yesmaximal
8#22374C213yesmaximal
9#22306C213yesmaximal
10#22299C213yesmaximal
11#21678C213yesmaximal
12#2845C411yescharacteristic
13#2839C413yes—
14#2839C413yes—
15#2839C413yes—
16#2839C413yes—
17#2822C413yes—
18#2402C2213yes—
19#2402C2213yes—
20#2402C2213yes—
21#2402C2213yes—
22C4×#125C2213yes—
23#1940C2213yes—
24#1940C2213yes—
25#1939C2213yes—
26#1939C2213yes—
27#1939C2213yes—
28#1939C2213yes—
29#1939C2213yes—
30#1939C2213yes—
31#1939C2213yes—
32#1939C2213yes—
33#1917C2213yes—
34#1917C2213yes—
35#1917C2213yes—
36#1917C2213yes—
37#1917C2213yes—
38#1917C2213yes—
39#1917C2213yes—
40#1917C2213yes—
41#1916C2213yes—
42#1916C2213yes—
43#1916C2213yes—
44#1916C2213yes—
45#1916C2213yes—
46#1916C2213yes—
47#1916C2213yes—
48#1916C2213yes—
49#1916C2213yes—
50#1916C2213yes—
51#1916C2213yes—
52#1916C2213yes—
53#1843C2211yescharacteristic
54#1842C2211yescharacteristic
55#1840C2211yescharacteristic
56#1840C2211yescharacteristic
57#1832C2213yes—
58#1771C2213yes—
59#1771C2213yes—
60#1771C2213yes—
61#1771C2213yes—
62#1771C2213yes—
63#1771C2213yes—
64#1770C2213yes—
65C8×C4⋊C4C2213yes—
66C8×C4⋊C4C2213yes—
67C8×C4⋊C4C2213yes—
68#1768C2213yes—
69#1768C2213yes—
70#1746C2213yes—
71#1744C2213yes—
72#1726C2211yescharacteristic
73C2×#125C2×C413yes—
74C2×C4⋊C8C2313yes—
75C2×C4⋊C8C2313yes—
76C2×C4⋊C8C2313yes—
77C2×C4⋊C8C2313yes—
78C2×C4⋊C8C2313yes—
79C2×C4⋊C8C2313yes—
80C2×C4⋊C8C2313yes—
81C2×C4⋊C8C2313yes—
82C2×C4⋊C8C2313yes—
83C2×C4⋊C8C2313yes—
84C2×#97C2313yes—
85C2×#97C2313yes—
86C2×#97C2313yes—
87C2×#97C2313yes—
88C2×C4×C8C2313yes—
89C2×C4×C8C2313yes—
90C2×C4×C8C2313yes—
91C2×C4×C8C2313yes—
92C2×C4×C8C2313yes—
93C2×C4×C8C2313yes—
94#400C2×C413yes—
95#400C2×C413yes—
96#400C2×C413yes—
97#400C2×C413yes—
98#400C2×C413yes—
99#400C2×C413yes—
100#400C2×C413yes—
101#400C2×C413yes—
102#400C2×C413yes—
103#400C2×C413yes—
104#400C2×C413yes—
105#400C2×C413yes—
106#400C2×C413yes—
107#400C2×C413yes—
108#400C2×C413yes—
109#400C2×C413yes—
110#398C2×C413yes—
111#398C2×C413yes—
112#398C2×C413yes—
113#398C2×C413yes—
114#398C2×C413yes—
115#398C2×C413yes—
116#398C2×C413yes—
117#398C2×C413yes—
118#398C2×C413yes—
119#398C2×C413yes—
120#398C2×C413yes—
121#398C2×C413yes—
122#398C2×C413yes—
123#398C2×C413yes—
124#398C2×C413yes—
125#398C2×C413yes—
126#389C2×C413yes—
127#389C2×C413yes—
128#387C2×C413yes—
129#387C2×C413yes—
130#387C2×C413yes—
131#387C2×C413yes—
132#382C2×C413yes—
133C4×C4⋊C4C2313yes—
134C4×C4⋊C4C2313yes—
135C4×C4⋊C4C2313yes—
136C4×C4⋊C4C2313yes—
137C4×C4⋊C4C2313yes—
138C4×C4⋊C4C2313yes—
139C4×C4⋊C4C2313yes—
140C4×C4⋊C4C2313yes—
141C4×C4⋊C4C2313yes—
142C4×C4⋊C4C2313yes—
143#376C2311yescharacteristic
144#376C2311yescharacteristic
145#376C2311yescharacteristic
146#376C2311yescharacteristic
147C43C2311yescharacteristic
148#336C2313yes—
149#336C2313yes—
150#336C2313yes—
151#336C2313yes—
152#336C2313yes—
153#336C2313yes—
154#336C2313yes—
155#336C2313yes—
156#336C2313yes—
157#336C2313yes—
158#336C2313yes—
159#336C2313yes—
160#336C2313yes—
161#336C2313yes—
162#336C2313yes—
163#336C2313yes—
164#336C2313yes—
165#336C2313yes—
166#336C2313yes—
167#336C2313yes—
168C22×C8C2413yes—
169C22×C8C2413yes—
170C22×C8C2413yes—
171C22×C8C2411yescharacteristic
172C22×C8C2413yes—
173C22×C8C2413yes—
174#125—212no—
175C2×C4⋊C4C22×C413yes—
176C2×C4⋊C4C22×C413yes—
177C2×C4⋊C4C22×C413yes—
178C2×C4⋊C4C22×C413yes—
179C2×C4⋊C4C22×C413yes—
180C2×C4⋊C4C22×C413yes—
181C2×C4⋊C4C22×C413yes—
182C2×C4⋊C4C22×C413yes—
183C2×C4⋊C4C22×C413yes—
184C2×C4⋊C4C22×C413yes—
185C2×C4⋊C4C22×C413yes—
186C2×C4⋊C4C22×C413yes—
187C2×C4⋊C4C22×C413yes—
188C2×C4⋊C4C22×C413yes—
189C2×C4⋊C4C22×C413yes—
190C2×C4⋊C4C22×C413yes—
191C2×C4⋊C4C22×C413yes—
192C2×C4⋊C4C22×C413yes—
193C2×C4⋊C4C22×C413yes—
194C2×C4⋊C4C22×C413yes—
195C2×C42C22×C413yes—
196C2×C42C22×C411yescharacteristic
197C2×C42C2413yes—
198C2×C42C2413yes—
199C2×C42C2413yes—
200C2×C42C2413yes—
201C2×C42C2413yes—
202C4⋊C8—26no—
203C4⋊C8—26no—
204C4⋊C8—43no—
205C4⋊C8—43no—
206C4⋊C8—43no—
207C4⋊C8—43no—
208C4⋊C8—43no—
209C4⋊C8—43no—
210C4⋊C8—43no—
211C4⋊C8—43no—
212#97—26no—
213#97—26no—
214#97—43no—
215#97—43no—
216C4×C8—26no—
217C4×C8—43no—
218C4×C8—43no—
219C4×C8—43no—
220C4×C8—43no—
221C4×C8—43no—
222#95C22×C413yes—
223#95C22×C411yescharacteristic
224#95C22×C413yes—
225#95C22×C413yes—
226#95C22×C411yescharacteristic
227#95C22×C411yescharacteristic
228#95C22×C413yes—
229#95C22×C411yescharacteristic
230#95C22×C413yes—
231#95C22×C413yes—
232#95C22×C413yes—
233#95C22×C413yes—
234#95C22×C413yes—
235#95C22×C413yes—
236#95C22×C413yes—
237#95C22×C413yes—
238#95C22×C413yes—
239#95C22×C413yes—
240#95C22×C413yes—
241#95C22×C413yes—
242#95C22×C413yes—
243#95C22×C413yes—
244#95C22×C413yes—
245#95C22×C413yes—
246#95C22×C413yes—
247#95C22×C413yes—
248#95C22×C413yes—
249#95C22×C413yes—
250C22×C4C23×C413yes—
251C22×C4C23×C413yes—
252C22×C4C23×C413yes—
253C22×C4C23×C413yes—
254C22×C4C23×C413yes—
255C22×C4C23×C413yes—
256C22×C4C23×C413yes—
257C22×C4C23×C413yes—
258C22×C4C23×C413yes—
259C22×C4C23×C413yes—
260C22×C4C2511yescharacteristic
261C2×C8—23no—
262C2×C8—23no—
263C2×C8—23no—
264C2×C8—23no—
265C2×C8—23no—
266C2×C8—23no—
267C2×C8—23no—
268C2×C8—23no—
269C2×C8—23no—
270C2×C8—23no—
271C2×C8—23no—
272C2×C8—23no—
273C2×C8—23no—
274C2×C8—23no—
275C2×C8—23no—
276C2×C82−1+416yes—
277C4⋊C4—26no—
278C4⋊C4—26no—
279C4⋊C4—26no—
280C4⋊C4—26no—
281C4⋊C4—26no—
282C4⋊C4—26no—
283C4⋊C4—43no—
284C4⋊C4—43no—
285C4⋊C4—43no—
286C4⋊C4—43no—
287C4⋊C4—43no—
288C4⋊C4—43no—
289C4⋊C4—43no—
290C4⋊C4—43no—
291C4⋊C4—43no—
292C4⋊C4—43no—
293C4⋊C4—43no—
294C4⋊C4—43no—
295C4⋊C4—43no—
296C4⋊C4—43no—
297C42—23no—
298C42—23no—
299C42—23no—
300C42—23no—
301C42—23no—
302C42—23no—
303C42—23no—
304C42—23no—
305C42—23no—
306C42—23no—
307C42—43no—
308C42#13114yes—
309C23C24×C411yescharacteristic
310C2×C4—23no—
311C2×C4—23no—
312C2×C4—23no—
313C2×C4—23no—
314C2×C4—23no—
315C2×C4—23no—
316C2×C4—23no—
317C2×C4—23no—
318C2×C4—23no—
319C2×C4—23no—
320C2×C4—23no—
321C2×C4—23no—
322C2×C4—23no—
323C2×C4—23no—
324C2×C4—23no—
325C2×C4—23no—
326C2×C4—23no—
327C2×C4—23no—
328C2×C4—23no—
329C2×C4—23no—
330C2×C4—23no—
331C2×C4—23no—
332C2×C4—23no—
333C2×C4—23no—
334C2×C4—23no—
335C2×C4—23no—
336C2×C4—23no—
337C2×C4—23no—
338C2×C4—23no—
339C2×C4C2×#13116yes—
340C2×C4C2×2−1+413yes—
341C2×C4#58513yes—
342C8—26no—
343C8—43no—
344C8—43no—
345C8—43no—
346C8—43no—
347C8—41no—
348C22C4×2−1+413yes—
349C22C22×#13111yescharacteristic
350C22#357313yes—
351C4—26no—
352C4—43no—
353C4—43no—
354C4—43no—
355C4—43no—
356C4—43no—
357C4—43no—
358C4—43no—
359C4—43no—
360C4—43no—
361C4#356313yes—
362C4#356711yescharacteristic
363C2#3453011yescharacteristic
364C2#6033413yes—
365C2#6034613yes—
subgroupG/H
center, Frattini, cosocleC22×C4C25
commutator, socle, prime radicalC23C24×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#367012
#233732
#232672
#232662
#232612
#232592
#223762
#223742
#223062
#222992
#216782