Finite Group Databasebeta

← #203099

Subgroups — #203099

Order 512.

625 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17823C211yesmaximal, characteristic
2#17739C211yesmaximal, characteristic
3#17501C211yesmaximal, characteristic
4#17501C211yesmaximal, characteristic
5#17334C211yesmaximal, characteristic
6#9587C214yesmaximal, complemented
7#9585C214yesmaximal, complemented
8#8064C211yesmaximal, characteristic
9#7887C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13#2920C2211yescharacteristic
14#2920C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17#1574C2214yes—
18#1572—22no—
19#1569—22no—
20#1561C414yes—
21#1556C412yes—
22#1556C412yes—
23#1544C2214yes—
24#1493—22no—
25#1493C2212yes—
26#1493C2212yes—
27#1491—22no—
28#1491—22no—
29#1485—22no—
30#1485C2214yes—
31#1455—22no—
32#1453—22no—
33#1452C2212yes—
34#1450C2212yes—
35#1299C2214yes—
36#1278C2212yes—
37#1278C2212yes—
38C22×M16—21no—
39C22×M16—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533C2×C412yes—
47#533C2312yes—
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×#125—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—21no—
53C2×C4×Q8—21no—
54C2×C4×Q8—21no—
55C2×#117—21no—
56C22×C42C2311yescharacteristic
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432C2×C412yes—
63#432C2×C412yes—
64#432D412yes—
65#431—21no—
66#431—21no—
67#431—21no—
68#431—21no—
69#431—21no—
70#431C2312yes—
71#431C2312yes—
72#431C2312yes—
73#423C2×C412yes—
74#423C2×C412yes—
75#423D412yes—
76#423C2312yes—
77#407—21no—
78#407—21no—
79#407—21no—
80#407—21no—
81C4×M16—21no—
82C4×M16—21no—
83C4×M16—21no—
84C4×M16—21no—
85C4×M16—21no—
86C4×M16—21no—
87C4×M16—21no—
88C4×M16—21no—
89C4×M16—21no—
90C4×M16D412yes—
91C4×M16D412yes—
92C4×M16C2312yes—
93C2×#97—21no—
94C2×#97—21no—
95C2×C4×C8—21no—
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98#335—24no—
99#334—24no—
100#332C2×C418yes—
101#332C2×C418yes—
102Q8⋊C8—28no—
103Q8⋊C8—24no—
104Q8⋊C8—24no—
105#322—22no—
106#322—24no—
107C82—22no—
108C22×Q8—21no—
109C22×Q8—21no—
110C23×C4#4111yescharacteristic
111C23×C4#4111yescharacteristic
112C23×C4#4111yescharacteristic
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—22no—
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—44no—
121C2×M16—44no—
122C2×M16—41no—
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—41no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8C22×C412yes—
133C4⋊Q8C22×C412yes—
134#125—22no—
135#125—22no—
136#125—41no—
137#125—41no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C22⋊Q8—41no—
143C22⋊Q8—41no—
144C22⋊Q8—41no—
145C22⋊Q8—41no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—41no—
151C4×Q8—41no—
152C4×Q8—41no—
153C4×Q8—41no—
154#117—22no—
155#117—22no—
156#117—41no—
157#117—41no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C42—41no—
166C2×C42C22×C411yescharacteristic
167C2×C42C22×C411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42C2411yescharacteristic
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8C22×C414yes—
184C4⋊C8C22×C414yes—
185C4⋊C8C2×D414yes—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—41no—
193C22⋊C8—41no—
194#97—22no—
195#97—22no—
196#97—22no—
197#97—22no—
198#97—22no—
199#97—22no—
200#97—41no—
201#97—41no—
202C4×C8—22no—
203C4×C8—22no—
204C4×C8—22no—
205C4×C8—22no—
206C4×C8—21no—
207C4×C8—21no—
208C4×C8—41no—
209C4×C8—41no—
210C4×C8C2×D412yes—
211C4×C8C2×D412yes—
212C24#12611yescharacteristic
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8—82no—
218C2×Q8—82no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4#11711yescharacteristic
225C22×C4#11711yescharacteristic
226C22×C4C4×D411yescharacteristic
227C22×C4C4×D411yescharacteristic
228C22×C4C4×D411yescharacteristic
229C22×C4C4×D411yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4#12311yescharacteristic
233C22×C4#12311yescharacteristic
234C22×C4#12411yescharacteristic
235C22×C4#12411yescharacteristic
236C22×C4#12611yescharacteristic
237C22×C4#12611yescharacteristic
238C22×C4#12611yescharacteristic
239C22×C4C2×#4111yescharacteristic
240C22×C4C2×#4111yescharacteristic
241C22×C4C2×#4111yescharacteristic
242M16—24no—
243M16—44no—
244M16—44no—
245M16—42no—
246M16—44no—
247M16—82no—
248M16—82no—
249M16—82no—
250M16—82no—
251C2×C8—24no—
252C2×C8—22no—
253C2×C8—24no—
254C2×C8—24no—
255C2×C8—24no—
256C2×C8—22no—
257C2×C8—42no—
258C2×C8—44no—
259C2×C8—42no—
260C2×C8—42no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C42—42no—
276C42—42no—
277C42#11711yescharacteristic
278C42#11711yescharacteristic
279C42C4×D411yescharacteristic
280C42C4×D411yescharacteristic
281C42C4×D411yescharacteristic
282C42C4×D411yescharacteristic
283C42#12111yescharacteristic
284C42#12111yescharacteristic
285C42#12311yescharacteristic
286C42#12311yescharacteristic
287C42#12411yescharacteristic
288C42#12411yescharacteristic
289C42C23×C411yescharacteristic
290C42C22×D411yescharacteristic
291C42C22×D411yescharacteristic
292C42C2×#4111yescharacteristic
293C23—41no—
294C23—41no—
295C23—41no—
296C23#38811yescharacteristic
297C23#38811yescharacteristic
298C23C2×#12611yescharacteristic
299Q8—44no—
300Q8—44no—
301Q8—82no—
302Q8—82no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4#38811yescharacteristic
326C2×C4#38811yescharacteristic
327C2×C4#38811yescharacteristic
328C2×C4#38811yescharacteristic
329C2×C4#38811yescharacteristic
330C2×C4#38811yescharacteristic
331C2×C4C2×#11711yescharacteristic
332C2×C4C2×C4×D411yescharacteristic
333C2×C4C2×C4×D411yescharacteristic
334C2×C4C2×#12111yescharacteristic
335C2×C4C2×#12311yescharacteristic
336C2×C4C2×#12411yescharacteristic
337C8—24no—
338C8—42no—
339C8—44no—
340C8—44no—
341C8—44no—
342C8—44no—
343C8—82no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—82no—
354C22C2×#38811yescharacteristic
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—82no—
365C4—82no—
366C4#295612yes—
367C4#295612yes—
368C4#306912yes—
369C2—41no—
370C2—41no—
371C2—41no—
372C2#2107111yescharacteristic
373C2#2107711yescharacteristic
374C2#2109911yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#388
prime radicalC24#126
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
6. #9587C2semidirect
7. #9585C2semidirect

Maximal subgroups

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

subgroupindex
#178232
#177392
#175012
#173342
#95872
#95852
#80642
#78872