Finite Group Databasebeta

← #203045

Subgroups — #203045

Order 512.

662 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21844C211yesmaximal, characteristic
2#20547C211yesmaximal, characteristic
3#17803C211yesmaximal, characteristic
4C2×#1542C211yesmaximal, characteristic
5#17464C211yesmaximal, characteristic
6#9584C214yesmaximal, complemented
7#9583C214yesmaximal, complemented
8#8082C211yesmaximal, characteristic
9#7869C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C4×#117C2211yescharacteristic
17#1994C2214yes—
18#1986—22no—
19#1826—22no—
20#1770—22no—
21#1767C2214yes—
22#1752—22no—
23#1572—22no—
24#1571C2212yes—
25#1570C2212yes—
26#1564—22no—
27#1542C414yescomplemented
28#1542C2212yes—
29#1542C2212yes—
30C2×#333C412yescomplemented
31C2×#332C412yescomplemented
32#1485—22no—
33#1481C2214yes—
34#1479—22no—
35#1301C2214yes—
36#1276C2212yes—
37#1276C2212yes—
38C23×C8—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533C2×C412yes—
46#533C2312yes—
47C2×C4⋊Q8C2×C411yescharacteristic
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×#125—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—21no—
52C2×C4×Q8—21no—
53C2×C4×Q8—21no—
54C2×#117—21no—
55C2×#117—21no—
56C22×C42C2311yescharacteristic
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431—21no—
68#431C2×C412yes—
69#431D412yes—
70#431D412yes—
71#431C2312yes—
72#431C2311yescharacteristic
73#431C2311yescharacteristic
74#423—21no—
75#423C2312yes—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8C2×C412yes—
80C2×C4⋊C8C2×C412yes—
81C2×C4⋊C8C2×C412yes—
82C2×C4⋊C8C2312yes—
83C2×C4⋊C8C2312yes—
84#407—21no—
85#407—21no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C4×M16—21no—
89C4×M16—21no—
90C2×#97—24no—
91C2×C4×C8—21no—
92C2×C4×C8—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4D412yes—
95C4×C22⋊C4—21no—
96C4×C22⋊C4D412yes—
97#376—21no—
98C43C2312yes—
99#339—28no—
100#336—24no—
101#335—24no—
102#334—24no—
103#333C2×C414yes—
104#333C2×C414yes—
105#332C2×C414yes—
106#332C2×C414yes—
107Q8⋊C8—28no—
108C22×Q8—21no—
109C22×Q8—21no—
110C23×C4C2×D411yescharacteristic
111C23×C4#4111yescharacteristic
112C23×C4#4111yescharacteristic
113C2×M16—22no—
114C2×M16—41no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—82no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8C22×C412yes—
128C4⋊Q8C22×C412yes—
129#125—22no—
130#125—22no—
131#125—41no—
132#125—41no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8—41no—
138C22⋊Q8—41no—
139C22⋊Q8—41no—
140C22⋊Q8—41no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—41no—
146C4×Q8—41no—
147C4×Q8—41no—
148C4×Q8—41no—
149#117—22no—
150#117—22no—
151#117—44no—
152#117—41no—
153#117—41no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—41no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—41no—
166C2×C42C22×C411yescharacteristic
167C2×C42C22×C411yescharacteristic
168C2×C42C2×D414yes—
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×C42#4111yescharacteristic
178C2×C42C2411yescharacteristic
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—41no—
188C4⋊C8C22×C414yes—
189C4⋊C8C22×C412yes—
190C4⋊C8C22×C412yes—
191C4⋊C8C2×D414yes—
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195C22⋊C8—21no—
196C22⋊C8—21no—
197C22⋊C8—22no—
198C22⋊C8—41no—
199C22⋊C8—41no—
200C22⋊C8—41no—
201#97—41no—
202#97—41no—
203#97—84no—
204C4×C8—22no—
205C4×C8—22no—
206C4×C8—22no—
207C4×C8—21no—
208C4×C8—21no—
209C4×C8—22no—
210C4×C8—41no—
211#95—22no—
212#95—22no—
213C24#12311yescharacteristic
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—44no—
219C2×Q8—82no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—81no—
227C22×C4#11711yescharacteristic
228C22×C4#11711yescharacteristic
229C22×C4C4×D411yescharacteristic
230C22×C4C4×D411yescharacteristic
231C22×C4C4×D411yescharacteristic
232C22×C4C4×D411yescharacteristic
233C22×C4#12111yescharacteristic
234C22×C4#12111yescharacteristic
235C22×C4#12311yescharacteristic
236C22×C4#12311yescharacteristic
237C22×C4#12311yescharacteristic
238C22×C4#12411yescharacteristic
239C22×C4#12411yescharacteristic
240C22×C4#12611yescharacteristic
241C22×C4#12611yescharacteristic
242C22×C4C22×D411yescharacteristic
243C22×C4C2×#4111yescharacteristic
244C22×C4C2×#4111yescharacteristic
245M16—42no—
246M16—42no—
247M16—82no—
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—24no—
251C2×C8—24no—
252C2×C8—42no—
253C2×C8—44no—
254C2×C8—42no—
255C2×C8—44no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—82no—
259C2×C8—82no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—44no—
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—44no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C42—24no—
277C42—24no—
278C42—44no—
279C42—42no—
280C42—42no—
281C42—82no—
282C42#11711yescharacteristic
283C42#11711yescharacteristic
284C42C4×D411yescharacteristic
285C42C4×D411yescharacteristic
286C42C4×D411yescharacteristic
287C42C4×D411yescharacteristic
288C42#12111yescharacteristic
289C42#12111yescharacteristic
290C42#12411yescharacteristic
291C42#12411yescharacteristic
292C42#12611yescharacteristic
293C42#12611yescharacteristic
294C42C23×C411yescharacteristic
295C42C22×D411yescharacteristic
296C42C2×#4111yescharacteristic
297C42C2×#4111yescharacteristic
298C23—22no—
299C23—22no—
300C23—41no—
301C23#38811yescharacteristic
302C23#38811yescharacteristic
303C23C2×#12311yescharacteristic
304Q8—44no—
305Q8—82no—
306Q8—82no—
307Q8—82no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4#38811yescharacteristic
334C2×C4#38811yescharacteristic
335C2×C4#38811yescharacteristic
336C2×C4#38811yescharacteristic
337C2×C4#38811yescharacteristic
338C2×C4#38811yescharacteristic
339C2×C4#44414yes—
340C2×C4C2×#11711yescharacteristic
341C2×C4C2×C4×D411yescharacteristic
342C2×C4C2×C4×D411yescharacteristic
343C2×C4C2×#12111yescharacteristic
344C2×C4C2×#12411yescharacteristic
345C2×C4C2×#12611yescharacteristic
346C2×C4#57712yes—
347C2×C4#57812yes—
348C8—44no—
349C8—42no—
350C8—42no—
351C8—44no—
352C8—82no—
353C8—82no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—44no—
362C22#193812yes—
363C22C2×#38811yescharacteristic
364C22#309712yes—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—84no—
374C4—82no—
375C4—82no—
376C4#193812yes—
377C4#294911yescharacteristic
378C4#295011yescharacteristic
379C4C2×#44412yes—
380C4#304612yes—
381C2—22no—
382C2—22no—
383C2—41no—
384C2#2106611yescharacteristic
385C2#2108011yescharacteristic
386C2C2×#193811yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#388
prime radicalC24#123
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. #9584C2semidirect
7. #9583C2semidirect
27. #1542C4semidirect
30. C2×#333C4semidirect
31. C2×#332C4semidirect

Maximal subgroups

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

subgroupindex
#218442
#205472
#178032
C2×#15422
#174642
#95842
#95832
#80822
#78692