Finite Group Databasebeta

← #203174

Subgroups — #203174

Order 512.

627 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21993C211yesmaximal, characteristic
2#21845C211yesmaximal, characteristic
3#17718C211yesmaximal, characteristic
4#17486C211yesmaximal, characteristic
5#17324C211yesmaximal, characteristic
6#9591C214yesmaximal, complemented
7#9589C214yesmaximal, complemented
8#8093C211yesmaximal, characteristic
9#7894C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#423C2211yescharacteristic
16#2283C2211yescharacteristic
17#1995—21no—
18#1995—21no—
19#1986C2214yes—
20#1950C2214yes—
21#1855—22no—
22#1848—21no—
23#1837—21no—
24#1826—22no—
25#1554C414yescomplemented
26#1548C412yescomplemented
27#1546C412yescomplemented
28#1542C2214yes—
29#1493C2212yes—
30#1493C2212yes—
31#1487—22no—
32#1481—22no—
33#1454—22no—
34#1453—22no—
35#1451C2212yes—
36C8×M16C2212yes—
37#1303C2214yes—
38#1279C2212yes—
39#1279C2212yes—
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×C42C2311yescharacteristic
58#432—21no—
59#432—21no—
60#432—21no—
61#432C2×C412yes—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431—21no—
68#431—21no—
69#431C2×C412yes—
70#431C2×C412yes—
71#431D412yes—
72#431D412yes—
73#431C2312yes—
74C2×#108—24no—
75#423—21no—
76#423—21no—
77#423—21no—
78#423C2×C412yes—
79#423D411yescharacteristic
80#423D411yescharacteristic
81#423C2312yes—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8C2312yes—
86#407—21no—
87#407D412yes—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C4×M16—21no—
91C4×M16C2312yes—
92C2×C4×C8—21no—
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95#383—21no—
96#382—21no—
97#381—21no—
98#380—21no—
99C4×C4⋊C4C2312yes—
100C4×C22⋊C4—21no—
101#376—21no—
102#341—24no—
103#339—24no—
104#339—24no—
105#333C2×C418yes—
106#332C2×C418yes—
107Q8⋊C8—24no—
108Q8⋊C8—24no—
109#322—22no—
110#322—24no—
111C82—22no—
112C22×Q8—21no—
113C22×Q8—21no—
114C23×C4C2×D411yescharacteristic
115C23×C4#4111yescharacteristic
116C23×C4#4111yescharacteristic
117C2×M16—22no—
118C2×M16—22no—
119C2×M16—41no—
120C2×M16—82no—
121C2×M16C2×D412yes—
122C2×M16C2×D412yes—
123C22×C8—22no—
124C22×C8—21no—
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—22no—
156#117—22no—
157#117—41no—
158#117—41no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C42—22no—
169C2×C42—41no—
170C2×C42C22×C411yescharacteristic
171C2×C42C22×C411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42C2411yescharacteristic
182#108—84no—
183C4⋊C8—22no—
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—41no—
194C4⋊C8—41no—
195C4⋊C8C22×C414yes—
196C4⋊C8C22×C414yes—
197C22⋊C8—22no—
198C22⋊C8—22no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8—22no—
205#97—22no—
206#97—22no—
207C4×C8—22no—
208C4×C8—22no—
209C4×C8—22no—
210C4×C8—22no—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8C2×D412yes—
214C4×C8C2×D412yes—
215#95—22no—
216#95—41no—
217#95—41no—
218#95—41no—
219C24#12411yescharacteristic
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8—82no—
225C2×Q8—82no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—42no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—42no—
232C22×C4—81no—
233C22×C4#11711yescharacteristic
234C22×C4#11711yescharacteristic
235C22×C4C4×D411yescharacteristic
236C22×C4C4×D411yescharacteristic
237C22×C4C4×D411yescharacteristic
238C22×C4C4×D411yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4#12111yescharacteristic
241C22×C4#12311yescharacteristic
242C22×C4#12311yescharacteristic
243C22×C4#12411yescharacteristic
244C22×C4#12411yescharacteristic
245C22×C4#12411yescharacteristic
246C22×C4#12611yescharacteristic
247C22×C4#12611yescharacteristic
248C22×C4C22×D411yescharacteristic
249C22×C4C2×#4111yescharacteristic
250C22×C4C2×#4111yescharacteristic
251M16—24no—
252M16—24no—
253M16—44no—
254M16—82no—
255M16—82no—
256C2×C8—24no—
257C2×C8—24no—
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—81no—
266C2×C8—81no—
267C2×C8—82no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—44no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—82no—
284C42—42no—
285C42—42no—
286C42—42no—
287C42—42no—
288C42#11711yescharacteristic
289C42#11711yescharacteristic
290C42C4×D411yescharacteristic
291C42C4×D411yescharacteristic
292C42C4×D411yescharacteristic
293C42C4×D411yescharacteristic
294C42#12111yescharacteristic
295C42#12111yescharacteristic
296C42#12311yescharacteristic
297C42#12311yescharacteristic
298C42#12611yescharacteristic
299C42#12611yescharacteristic
300C42C23×C411yescharacteristic
301C42C22×D411yescharacteristic
302C42C2×#4111yescharacteristic
303C42C2×#4111yescharacteristic
304C23—22no—
305C23—41no—
306C23—41no—
307C23#38811yescharacteristic
308C23#38811yescharacteristic
309C23C2×#12411yescharacteristic
310Q8—44no—
311Q8—82no—
312Q8—82no—
313Q8—82no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4#38811yescharacteristic
341C2×C4#38811yescharacteristic
342C2×C4#38811yescharacteristic
343C2×C4#38811yescharacteristic
344C2×C4#38811yescharacteristic
345C2×C4#38811yescharacteristic
346C2×C4C2×#11711yescharacteristic
347C2×C4C2×C4×D411yescharacteristic
348C2×C4C2×C4×D411yescharacteristic
349C2×C4C2×#12111yescharacteristic
350C2×C4C2×#12311yescharacteristic
351C2×C4C2×#12611yescharacteristic
352C2×C4#57712yes—
353C2×C4#57812yes—
354C8—42no—
355C8—42no—
356C8—44no—
357C8—44no—
358C8—42no—
359C8—42no—
360C8—82no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—82no—
370C22C2×#38811yescharacteristic
371C22#313712yes—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—82no—
382C4—82no—
383C4—162no—
384C4#294712yes—
385C4#295712yes—
386C4#305711yescharacteristic
387C4#305911yescharacteristic
388C2—22no—
389C2—41no—
390C2—41no—
391C2#2107211yescharacteristic
392C2#2108011yescharacteristic
393C2#2109011yescharacteristic
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. #1554C4semidirect
26. #1548C4semidirect
27. #1546C4semidirect

Maximal subgroups

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

subgroupindex
#219932
#218452
#177182
#174862
#173242
#95912
#95892
#80932
#78942