Finite Group Databasebeta

← #203143

Subgroups — #203143

Order 512.

655 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21371C211yesmaximal, characteristic
2#21306C211yesmaximal, characteristic
3#17734C211yesmaximal, characteristic
4#17479C211yesmaximal, characteristic
5#17327C211yesmaximal, characteristic
6#9591C214yesmaximal, complemented
7#9589C214yesmaximal, complemented
8#8094C211yesmaximal, characteristic
9#7900C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16#2285C2211yescharacteristic
17#1986—22no—
18#1986C2212yes—
19#1986C2212yes—
20#1950C2212yes—
21#1950C2212yes—
22#1855—22no—
23#1778—22no—
24#1733—22no—
25#1559C414yescomplemented
26#1558C412yescomplemented
27#1556C412yescomplemented
28#1542C2214yes—
29#1497—21no—
30#1497—21no—
31#1493C2214yes—
32#1479—22no—
33#1453—22no—
34#1451—21no—
35#1451C2212yes—
36C8×M16—21no—
37C8×M16C2212yes—
38#1303C2214yes—
39#1279C2214yes—
40C22×M16D412yes—
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×#117—21no—
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#431—21no—
72#431D411yescharacteristic
73#431D411yescharacteristic
74#431D412yes—
75#431C2312yes—
76C2×#108—22no—
77C2×#108—22no—
78#423C2×C412yes—
79#423C2312yes—
80C2×C4⋊C8—21no—
81C2×C4⋊C8C2312yes—
82#407—21no—
83#407—21no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C4×M16—21no—
87C4×M16—21no—
88C4×M16—21no—
89C4×M16—21no—
90C4×M16D412yes—
91C4×M16C2311yescharacteristic
92C4×M16C2311yescharacteristic
93C2×#97—21no—
94C2×#97—21no—
95C2×C4×C8—21no—
96C2×C4×C8C2312yes—
97#384—21no—
98#382—21no—
99#380—21no—
100#380—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4C2312yes—
103#341—22no—
104#341—22no—
105#339—24no—
106#339—24no—
107#333C2×C418yes—
108#332C2×C418yes—
109Q8⋊C8—24no—
110Q8⋊C8—24no—
111#322—22no—
112#322—22no—
113#322—22no—
114C82—22no—
115C22×Q8—21no—
116C22×Q8—21no—
117C23×C4C2×D411yescharacteristic
118C23×C4#4111yescharacteristic
119C23×C4#4111yescharacteristic
120C2×M16—22no—
121C2×M16—22no—
122C2×M16—24no—
123C2×M16—22no—
124C2×M16—42no—
125C2×M16C2×D412yes—
126C2×M16C2×D412yes—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136C4⋊Q8C22×C412yes—
137C4⋊Q8C22×C412yes—
138#125—22no—
139#125—22no—
140#125—41no—
141#125—41no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8—41no—
147C22⋊Q8—41no—
148C22⋊Q8—41no—
149C22⋊Q8—41no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—41no—
155C4×Q8—41no—
156C4×Q8—41no—
157C4×Q8—41no—
158#117—22no—
159#117—22no—
160#117—41no—
161#117—41no—
162#117—44no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C42—22no—
175C2×C42—41no—
176C2×C42C22×C411yescharacteristic
177C2×C42C22×C411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42C2411yescharacteristic
188#108—44no—
189#108—44no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8C22×C414yes—
198C4⋊C8C22×C414yes—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—21no—
204C22⋊C8—21no—
205C22⋊C8—22no—
206C22⋊C8—41no—
207C22⋊C8—41no—
208#97—22no—
209#97—22no—
210#97—22no—
211#97—22no—
212#97—21no—
213#97—21no—
214#97—41no—
215C4×C8—22no—
216C4×C8—21no—
217C4×C8—22no—
218C4×C8—21no—
219C4×C8—22no—
220C4×C8—41no—
221C4×C8C2×D412yes—
222C4×C8C2×D412yes—
223#95—22no—
224#95—41no—
225C24#12411yescharacteristic
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—44no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—82no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—81no—
239C22×C4#11711yescharacteristic
240C22×C4#11711yescharacteristic
241C22×C4C4×D411yescharacteristic
242C22×C4C4×D411yescharacteristic
243C22×C4C4×D411yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4#12111yescharacteristic
246C22×C4#12111yescharacteristic
247C22×C4#12311yescharacteristic
248C22×C4#12311yescharacteristic
249C22×C4#12411yescharacteristic
250C22×C4#12411yescharacteristic
251C22×C4#12411yescharacteristic
252C22×C4#12611yescharacteristic
253C22×C4#12611yescharacteristic
254C22×C4C22×D411yescharacteristic
255C22×C4C2×#4111yescharacteristic
256C22×C4C2×#4111yescharacteristic
257M16—22no—
258M16—22no—
259M16—22no—
260M16—44no—
261M16—44no—
262M16—42no—
263M16—42no—
264M16#13714yes—
265C2×C8—22no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—44no—
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—44no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—82no—
290C4⋊C4—82no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—82no—
294C42—42no—
295C42—42no—
296C42—44no—
297C42#11711yescharacteristic
298C42#11711yescharacteristic
299C42C4×D411yescharacteristic
300C42C4×D411yescharacteristic
301C42C4×D411yescharacteristic
302C42C4×D411yescharacteristic
303C42#12111yescharacteristic
304C42#12111yescharacteristic
305C42#12311yescharacteristic
306C42#12311yescharacteristic
307C42#12611yescharacteristic
308C42#12611yescharacteristic
309C42C23×C411yescharacteristic
310C42C22×D411yescharacteristic
311C42C2×#4111yescharacteristic
312C42C2×#4111yescharacteristic
313C23—21no—
314C23—21no—
315C23—41no—
316C23—41no—
317C23#38811yescharacteristic
318C23#38811yescharacteristic
319C23C2×#12411yescharacteristic
320Q8—44no—
321Q8—82no—
322Q8—82no—
323Q8—82no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
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—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4#38811yescharacteristic
350C2×C4#38811yescharacteristic
351C2×C4#38811yescharacteristic
352C2×C4#38811yescharacteristic
353C2×C4#38811yescharacteristic
354C2×C4#38811yescharacteristic
355C2×C4C2×#11711yescharacteristic
356C2×C4C2×C4×D411yescharacteristic
357C2×C4C2×C4×D411yescharacteristic
358C2×C4C2×#12111yescharacteristic
359C2×C4C2×#12311yescharacteristic
360C2×C4C2×#12611yescharacteristic
361C2×C4C2×#13712yes—
362C2×C4#57512yes—
363C8—24no—
364C8—42no—
365C8—42no—
366C8—44no—
367C8—44no—
368C8—42no—
369C8—82no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—81no—
379C22—81no—
380C22C2×#38811yescharacteristic
381C22#313911yescharacteristic
382C22#314011yescharacteristic
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—82no—
393C4—82no—
394C4—162no—
395C4C4×#13712yes—
396C4#295512yes—
397C4#305612yes—
398C2—21no—
399C2—21no—
400C2—41no—
401C2—41no—
402C2#2107011yescharacteristic
403C2#2108011yescharacteristic
404C2#2109411yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#388
prime radicalC24#124
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. #9591C2semidirect
7. #9589C2semidirect
25. #1559C4semidirect
26. #1558C4semidirect
27. #1556C4semidirect

Maximal subgroups

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

subgroupindex
#213712
#213062
#177342
#174792
#173272
#95912
#95892
#80942
#79002