Finite Group Databasebeta

← #203004

Subgroups — #203004

Order 512.

753 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1994C211yesmaximal, characteristic
2C4×#431C211yesmaximal, characteristic
3#17809C211yesmaximal, characteristic
4#17718C211yesmaximal, characteristic
5#17464C211yesmaximal, characteristic
6#9580C214yesmaximal, complemented
7#9579C214yesmaximal, complemented
8#8081C211yesmaximal, characteristic
9#7867C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#423C2211yescharacteristic
16C2×C43C2211yescharacteristic
17#1994—21no—
18#1994—21no—
19#1994C2214yes—
20C4×C4⋊C8C2212yes—
21C4×C4⋊C8C2212yes—
22C4×C22⋊C8—22no—
23#1744—21no—
24C2×#339—22no—
25C42×C8—21no—
26#1574—22no—
27#1571C2212yes—
28#1570C2212yes—
29#1565—22no—
30#1554C414yescomplemented
31#1548C412yescomplemented
32#1546C412yescomplemented
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—
57C22×C42—21no—
58C22×C42—21no—
59C22×C42D412yes—
60C22×C42C2311yescharacteristic
61#432—21no—
62#432—21no—
63#432—21no—
64#432C2×C412yes—
65#431—21no—
66#431—21no—
67#431—21no—
68#431—21no—
69#431—21no—
70#431—21no—
71#431C2×C412yes—
72#431C2×C412yes—
73#431D412yes—
74#431D412yes—
75#431C2312yes—
76#431C2312yes—
77#423—21no—
78#423—21no—
79#423—21no—
80#423—21no—
81#423C2×C412yes—
82#423C2312yes—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8C2312yes—
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88#407—21no—
89#407—21no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C4×M16—21no—
93C4×M16—21no—
94C2×C4×C8—21no—
95C2×C4×C8—21no—
96C2×C4×C8—22no—
97C2×C4×C8—22no—
98C43—21no—
99C43—21no—
100C43D411yescharacteristic
101C43D411yescharacteristic
102C43C2312yes—
103#339—24no—
104#339—24no—
105#336—22no—
106#336—22no—
107#335—24no—
108#334—24no—
109#333C2×C418yescomplemented
110#332C2×C418yescomplemented
111Q8⋊C8—24no—
112Q8⋊C8—24no—
113C22×Q8—21no—
114C22×Q8—21no—
115C23×C4—22no—
116C23×C4C2×D411yescharacteristic
117C23×C4#4111yescharacteristic
118C23×C4#4111yescharacteristic
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—41no—
122C2×M16—41no—
123C2×M16—44no—
124C22×C8—22no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—41no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8C22×C412yes—
135C4⋊Q8C22×C412yes—
136#125—22no—
137#125—22no—
138#125—41no—
139#125—41no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8—41no—
145C22⋊Q8—41no—
146C22⋊Q8—41no—
147C22⋊Q8—41no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—41no—
153C4×Q8—41no—
154C4×Q8—41no—
155C4×Q8—41no—
156#117—22no—
157#117—22no—
158#117—41no—
159#117—41no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C42—28no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—22no—
172C2×C42—24no—
173C2×C42—22no—
174C2×C42—21no—
175C2×C42—22no—
176C2×C42—42no—
177C2×C42—44no—
178C2×C42—41no—
179C2×C42C22×C411yescharacteristic
180C2×C42C22×C411yescharacteristic
181C2×C42C2×D412yes—
182C2×C42C2×D412yes—
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42#4111yescharacteristic
188C2×C42#4111yescharacteristic
189C2×C42#4111yescharacteristic
190C2×C42#4111yescharacteristic
191C2×C42#4111yescharacteristic
192C2×C42C2411yescharacteristic
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—41no—
201C4⋊C8—41no—
202C4⋊C8—41no—
203C4⋊C8C22×C414yes—
204C4⋊C8C22×C414yes—
205C4⋊C8C2×D412yes—
206C4⋊C8C2×D412yes—
207C22⋊C8—22no—
208C22⋊C8—22no—
209C22⋊C8—22no—
210C22⋊C8—22no—
211C22⋊C8—22no—
212C22⋊C8—22no—
213C22⋊C8—22no—
214C22⋊C8—41no—
215#97—22no—
216#97—41no—
217C4×C8—22no—
218C4×C8—22no—
219C4×C8—22no—
220C4×C8—22no—
221C4×C8—22no—
222C4×C8—22no—
223C4×C8—44no—
224C4×C8—44no—
225C24#12311yescharacteristic
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—44no—
231C2×Q8—82no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—28no—
237C22×C4—22no—
238C22×C4—42no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4#11711yescharacteristic
244C22×C4#11711yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4C4×D411yescharacteristic
247C22×C4C4×D411yescharacteristic
248C22×C4C4×D411yescharacteristic
249C22×C4#12111yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12311yescharacteristic
252C22×C4#12311yescharacteristic
253C22×C4#12311yescharacteristic
254C22×C4#12411yescharacteristic
255C22×C4#12411yescharacteristic
256C22×C4#12611yescharacteristic
257C22×C4#12611yescharacteristic
258C22×C4C22×D411yescharacteristic
259C22×C4C2×#4111yescharacteristic
260C22×C4C2×#4111yescharacteristic
261M16—44no—
262M16—44no—
263M16—82no—
264M16—82no—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—82no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C42—24no—
291C42—28no—
292C42—24no—
293C42—24no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—44no—
298C42—44no—
299C42—42no—
300C42—42no—
301C42—42no—
302C42#11711yescharacteristic
303C42#11711yescharacteristic
304C42C4×D411yescharacteristic
305C42C4×D411yescharacteristic
306C42C4×D411yescharacteristic
307C42C4×D411yescharacteristic
308C42#12111yescharacteristic
309C42#12111yescharacteristic
310C42#12411yescharacteristic
311C42#12411yescharacteristic
312C42#12611yescharacteristic
313C42#12611yescharacteristic
314C42#13714yes—
315C42C23×C411yescharacteristic
316C42C22×D411yescharacteristic
317C42C2×#4111yescharacteristic
318C42C2×#4111yescharacteristic
319C23—22no—
320C23—22no—
321C23—41no—
322C23#38811yescharacteristic
323C23#38811yescharacteristic
324C23C2×#12311yescharacteristic
325Q8—44no—
326Q8—82no—
327Q8—82no—
328Q8—82no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—24no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—416no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4#38811yescharacteristic
359C2×C4#38811yescharacteristic
360C2×C4#38811yescharacteristic
361C2×C4#38811yescharacteristic
362C2×C4#38811yescharacteristic
363C2×C4#38811yescharacteristic
364C2×C4#44314yes—
365C2×C4C2×#11711yescharacteristic
366C2×C4C2×C4×D411yescharacteristic
367C2×C4C2×C4×D411yescharacteristic
368C2×C4C2×#12111yescharacteristic
369C2×C4C2×#12411yescharacteristic
370C2×C4C2×#12611yescharacteristic
371C2×C4C2×#13712yes—
372C2×C4#57512yes—
373C8—44no—
374C8—44no—
375C8—42no—
376C8—42no—
377C8—82no—
378C8—82no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—44no—
387C22#193712yes—
388C22C2×#38811yescharacteristic
389C22#309512yes—
390C4—28no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—44no—
399C4—82no—
400C4—82no—
401C4#193212yes—
402C4#193712yes—
403C4C4×#13712yes—
404C4C2×#44312yes—
405C4#303911yescharacteristic
406C4#304011yescharacteristic
407C2—22no—
408C2—22no—
409C2—41no—
410C2#2106111yescharacteristic
411C2C2×#193211yescharacteristic
412C2C2×#193711yescharacteristic
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. #9580C2semidirect
7. #9579C2semidirect
30. #1554C4semidirect
31. #1548C4semidirect
32. #1546C4semidirect
109. #333C2×C4semidirect
110. #332C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19942
C4×#4312
#178092
#177182
#174642
#95802
#95792
#80812
#78672