Finite Group Databasebeta

← #203158

Subgroups — #203158

Order 512.

645 conjugacy classes of subgroups.

The subgroup lattice has 399 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21994C211yesmaximal, characteristic
2#21844C211yesmaximal, characteristic
3#17734C211yesmaximal, characteristic
4#17486C211yesmaximal, characteristic
5#17324C211yesmaximal, characteristic
6#9591C214yesmaximal, complemented
7#9589C214yesmaximal, complemented
8#8097C211yesmaximal, characteristic
9#7902C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#423C2211yescharacteristic
16C4×#117C2211yescharacteristic
17#1994—21no—
18#1994—21no—
19#1986C2214yes—
20#1950C2214yes—
21#1855—22no—
22#1849—21no—
23#1838—21no—
24#1826—22no—
25#1559C414yescomplemented
26#1558C412yescomplemented
27#1556C412yescomplemented
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—
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#431D412yes—
73#431D412yes—
74#431C2312yes—
75C2×#108—24no—
76#423—21no—
77#423—21no—
78#423—21no—
79#423C2×C412yes—
80#423D411yescharacteristic
81#423D411yescharacteristic
82#423C2312yes—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2312yes—
87#407—21no—
88#407D412yes—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C4×M16—21no—
92C4×M16C2312yes—
93C2×#97—21no—
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96C4×C4⋊C4—21no—
97C4×C4⋊C4C2312yes—
98C4×C22⋊C4—21no—
99C4×C22⋊C4—21no—
100#376—21no—
101C43—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—44no—
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—22no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C42—22no—
170C2×C42—41no—
171C2×C42—41no—
172C2×C42—41no—
173C2×C42C22×C411yescharacteristic
174C2×C42C22×C411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42C2411yescharacteristic
185#108—84no—
186C4⋊C8—22no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—41no—
197C4⋊C8—41no—
198C4⋊C8C22×C414yes—
199C4⋊C8C22×C414yes—
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204C22⋊C8—22no—
205C22⋊C8—22no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208#97—22no—
209#97—22no—
210#97—22no—
211#97—22no—
212C4×C8—22no—
213C4×C8—22no—
214C4×C8—21no—
215C4×C8—21no—
216C4×C8C2×D412yes—
217C4×C8C2×D412yes—
218#95—22no—
219#95—41no—
220C24#12411yescharacteristic
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8—82no—
226C2×Q8—82no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—81no—
234C22×C4#11711yescharacteristic
235C22×C4#11711yescharacteristic
236C22×C4C4×D411yescharacteristic
237C22×C4C4×D411yescharacteristic
238C22×C4C4×D411yescharacteristic
239C22×C4C4×D411yescharacteristic
240C22×C4#12111yescharacteristic
241C22×C4#12111yescharacteristic
242C22×C4#12311yescharacteristic
243C22×C4#12311yescharacteristic
244C22×C4#12411yescharacteristic
245C22×C4#12411yescharacteristic
246C22×C4#12411yescharacteristic
247C22×C4#12611yescharacteristic
248C22×C4#12611yescharacteristic
249C22×C4C22×D411yescharacteristic
250C22×C4C2×#4111yescharacteristic
251C22×C4C2×#4111yescharacteristic
252M16—24no—
253M16—24no—
254M16—44no—
255M16—82no—
256M16—82no—
257C2×C8—24no—
258C2×C8—24no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—81no—
267C2×C8—81no—
268C2×C8—82no—
269C4⋊C4—24no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C22⋊C4—44no—
284C22⋊C4—42no—
285C22⋊C4—42no—
286C42—44no—
287C42—42no—
288C42—42no—
289C42—42no—
290C42—42no—
291C42—82no—
292C42#11711yescharacteristic
293C42#11711yescharacteristic
294C42C4×D411yescharacteristic
295C42C4×D411yescharacteristic
296C42C4×D411yescharacteristic
297C42C4×D411yescharacteristic
298C42#12111yescharacteristic
299C42#12111yescharacteristic
300C42#12311yescharacteristic
301C42#12311yescharacteristic
302C42#12611yescharacteristic
303C42#12611yescharacteristic
304C42C23×C411yescharacteristic
305C42C22×D411yescharacteristic
306C42C2×#4111yescharacteristic
307C42C2×#4111yescharacteristic
308C23—22no—
309C23—41no—
310C23—41no—
311C23#38811yescharacteristic
312C23#38811yescharacteristic
313C23C2×#12411yescharacteristic
314Q8—44no—
315Q8—82no—
316Q8—82no—
317Q8—82no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4#38811yescharacteristic
345C2×C4#38811yescharacteristic
346C2×C4#38811yescharacteristic
347C2×C4#38811yescharacteristic
348C2×C4#38811yescharacteristic
349C2×C4#38811yescharacteristic
350C2×C4C2×#11711yescharacteristic
351C2×C4C2×C4×D411yescharacteristic
352C2×C4C2×C4×D411yescharacteristic
353C2×C4C2×#12111yescharacteristic
354C2×C4C2×#12311yescharacteristic
355C2×C4C2×#12611yescharacteristic
356C2×C4#57712yes—
357C2×C4#57812yes—
358C8—42no—
359C8—42no—
360C8—44no—
361C8—44no—
362C8—42no—
363C8—42no—
364C8—82no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—82no—
374C22C2×#38811yescharacteristic
375C22#313712yes—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—84no—
386C4—82no—
387C4—82no—
388C4#294712yes—
389C4#295612yes—
390C4#305711yescharacteristic
391C4#305911yescharacteristic
392C2—22no—
393C2—41no—
394C2—41no—
395C2#2107111yescharacteristic
396C2#2108011yescharacteristic
397C2#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. #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
#219942
#218442
#177342
#174862
#173242
#95912
#95892
#80972
#79022