Finite Group Databasebeta

← #203017

Subgroups — #203017

Order 512.

663 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21329C211yesmaximal, characteristic
2#20470C211yesmaximal, characteristic
3#17809C211yesmaximal, characteristic
4#17734C211yesmaximal, characteristic
5#17464C211yesmaximal, characteristic
6#9584C214yesmaximal, complemented
7#9579C214yesmaximal, complemented
8#8077C211yesmaximal, characteristic
9#7868C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#423C2211yescharacteristic
16C4×#117C2211yescharacteristic
17#1995—21no—
18#1995—21no—
19#1994C2214yes—
20#1767C2212yes—
21C4×C4⋊C8C2212yes—
22#1750—22no—
23#1746—21no—
24#1733—22no—
25#1726—21no—
26#1574—22no—
27#1571C2212yes—
28#1570C2212yes—
29#1565—22no—
30#1559C414yescomplemented
31#1558C412yescomplemented
32#1556C412yescomplemented
33#1542C2214yes—
34#1485—22no—
35#1481C2212yes—
36#1481C2212yes—
37#1479—22no—
38#1301C2214yes—
39#1276C2214yes—
40C22×M16—21no—
41#533—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533C2×C412yes—
48#533C2312yes—
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×#125—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—21no—
54C2×C4×Q8—21no—
55C2×C4×Q8—21no—
56C2×#117—21no—
57C2×#117D412yes—
58C22×C42C2311yescharacteristic
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63#432—21no—
64#432C2×C412yes—
65#432C2×C412yes—
66#432C2×C412yes—
67#431—21no—
68#431—21no—
69#431—21no—
70#431—21no—
71#431D412yes—
72#431D412yes—
73#431C2312yes—
74#431C2312yes—
75#423—21no—
76#423—21no—
77#423—21no—
78#423—21no—
79#423C2×C412yes—
80#423C2312yes—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2312yes—
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86#407—21no—
87#407—21no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C4×M16—21no—
91C4×M16—21no—
92C2×#97—21no—
93C2×#97—22no—
94C2×C4×C8—21no—
95C2×C4×C8—22no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C4×C22⋊C4—21no—
99C4×C22⋊C4—21no—
100#376D411yescharacteristic
101#376D411yescharacteristic
102C43C2312yes—
103#339—24no—
104#339—24no—
105#336—22no—
106#336—22no—
107#335—24no—
108#334—24no—
109#333C2×C418yes—
110#332C2×C418yes—
111Q8⋊C8—24no—
112Q8⋊C8—24no—
113C22×Q8—21no—
114C22×Q8—21no—
115C23×C4C2×D411yescharacteristic
116C23×C4#4111yescharacteristic
117C23×C4#4111yescharacteristic
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—41no—
121C2×M16—41no—
122C2×M16—44no—
123C22×C8—21no—
124C22×C8—22no—
125C22×C8—21no—
126C22×C8—41no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8C22×C412yes—
134C4⋊Q8C22×C412yes—
135#125—22no—
136#125—22no—
137#125—41no—
138#125—41no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C22⋊Q8—41no—
144C22⋊Q8—41no—
145C22⋊Q8—41no—
146C22⋊Q8—41no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—41no—
152C4×Q8—41no—
153C4×Q8—41no—
154C4×Q8—41no—
155#117—24no—
156#117—22no—
157#117—22no—
158#117—42no—
159#117—41no—
160#117—41no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—41no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—41no—
173C2×C42C22×C411yescharacteristic
174C2×C42C22×C411yescharacteristic
175C2×C42C2×D412yes—
176C2×C42C2×D412yes—
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42C2411yescharacteristic
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—41no—
195C4⋊C8—41no—
196C4⋊C8—41no—
197C4⋊C8C22×C414yes—
198C4⋊C8C22×C414yes—
199C4⋊C8C2×D412yes—
200C4⋊C8C2×D412yes—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8—22no—
205C22⋊C8—22no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—41no—
209#97—22no—
210#97—22no—
211#97—22no—
212#97—44no—
213#97—41no—
214C4×C8—22no—
215C4×C8—22no—
216C4×C8—22no—
217C4×C8—22no—
218C4×C8—44no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223C24#12311yescharacteristic
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—44no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—82no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—81no—
238C22×C4#11711yescharacteristic
239C22×C4#11711yescharacteristic
240C22×C4C4×D411yescharacteristic
241C22×C4C4×D411yescharacteristic
242C22×C4C4×D411yescharacteristic
243C22×C4C4×D411yescharacteristic
244C22×C4#12111yescharacteristic
245C22×C4#12111yescharacteristic
246C22×C4#12311yescharacteristic
247C22×C4#12311yescharacteristic
248C22×C4#12311yescharacteristic
249C22×C4#12411yescharacteristic
250C22×C4#12411yescharacteristic
251C22×C4#12611yescharacteristic
252C22×C4#12611yescharacteristic
253C22×C4C22×D411yescharacteristic
254C22×C4C2×#4111yescharacteristic
255C22×C4C2×#4111yescharacteristic
256M16—44no—
257M16—44no—
258M16—82no—
259M16—82no—
260C2×C8—24no—
261C2×C8—24no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—82no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—44no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C22⋊C4—44no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—44no—
291C42—42no—
292C42—42no—
293C42—82no—
294C42#11711yescharacteristic
295C42#11711yescharacteristic
296C42C4×D411yescharacteristic
297C42C4×D411yescharacteristic
298C42C4×D411yescharacteristic
299C42C4×D411yescharacteristic
300C42#12111yescharacteristic
301C42#12111yescharacteristic
302C42#12411yescharacteristic
303C42#12411yescharacteristic
304C42#12611yescharacteristic
305C42#12611yescharacteristic
306C42#13714yes—
307C42C23×C411yescharacteristic
308C42C22×D411yescharacteristic
309C42C2×#4111yescharacteristic
310C42C2×#4111yescharacteristic
311C23—22no—
312C23—41no—
313C23—41no—
314C23#38811yescharacteristic
315C23#38811yescharacteristic
316C23C2×#12311yescharacteristic
317Q8—44no—
318Q8—82no—
319Q8—82no—
320Q8—82no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—24no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4#38811yescharacteristic
351C2×C4#38811yescharacteristic
352C2×C4#38811yescharacteristic
353C2×C4#38811yescharacteristic
354C2×C4#38811yescharacteristic
355C2×C4#38811yescharacteristic
356C2×C4C2×#11711yescharacteristic
357C2×C4C2×C4×D411yescharacteristic
358C2×C4C2×C4×D411yescharacteristic
359C2×C4C2×#12111yescharacteristic
360C2×C4C2×#12411yescharacteristic
361C2×C4C2×#12611yescharacteristic
362C2×C4C2×#13712yes—
363C2×C4#57512yes—
364C8—44no—
365C8—44no—
366C8—42no—
367C8—42no—
368C8—82no—
369C8—82no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—82no—
379C22C2×#38811yescharacteristic
380C22#309512yes—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—44no—
391C4—82no—
392C4—82no—
393C4—82no—
394C4C4×#13712yes—
395C4#295512yes—
396C4#303911yescharacteristic
397C4#304011yescharacteristic
398C2—22no—
399C2—41no—
400C2—41no—
401C2#2106111yescharacteristic
402C2#2108011yescharacteristic
403C2#2110011yescharacteristic
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. #9579C2semidirect
30. #1559C4semidirect
31. #1558C4semidirect
32. #1556C4semidirect

Maximal subgroups

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

subgroupindex
#213292
#204702
#178092
#177342
#174642
#95842
#95792
#80772
#78682