Finite Group Databasebeta

← #203141

Subgroups — #203141

Order 512.

657 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21994C211yesmaximal, characteristic
2#21845C211yesmaximal, characteristic
3#17734C211yesmaximal, characteristic
4C2×#1493C211yesmaximal, characteristic
5#17323C211yesmaximal, characteristic
6#9591C214yesmaximal, complemented
7#9589C214yesmaximal, complemented
8#8093C211yesmaximal, characteristic
9#7898C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#2283C2211yescharacteristic
17#1995—21no—
18#1995—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#1493—21no—
30#1493—21no—
31#1493C2212yes—
32#1493C2212yes—
33C2×Q8⋊C8—21no—
34C2×Q8⋊C8—21no—
35#1452—21no—
36#1451—21no—
37#1451C2212yes—
38#1450—21no—
39C8×M16—21no—
40C8×M16C2212yes—
41#1303C2214yes—
42#1279C2212yes—
43#1279C2212yes—
44C22×M16—21no—
45C23×C8—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533C2×C412yes—
53#533C2312yes—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#125—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×#117—21no—
62C22×C42C2311yescharacteristic
63#432—21no—
64#432—21no—
65#432—21no—
66#432—21no—
67#432—21no—
68#432C2×C412yes—
69#432C2×C412yes—
70#432C2×C412yes—
71#431—21no—
72#431—21no—
73#431—21no—
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⋊C8C2312yes—
86#407—21no—
87#407D412yes—
88C2×C22⋊C8—21no—
89C4×M16—21no—
90C4×M16C2312yes—
91C2×#97—21no—
92C2×C4×C8—21no—
93C2×C4×C8—21no—
94C2×C4×C8—21no—
95C2×C4×C8D411yescharacteristic
96C2×C4×C8D411yescharacteristic
97C2×C4×C8D411yescharacteristic
98C2×C4×C8D411yescharacteristic
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101#383—21no—
102#382—21no—
103#381—21no—
104#380—21no—
105C4×C4⋊C4C2312yes—
106C4×C22⋊C4—21no—
107#376—21no—
108#341—24no—
109#339—24no—
110#339—24no—
111#333C2×C418yes—
112#332C2×C418yes—
113Q8⋊C8—22no—
114Q8⋊C8—22no—
115Q8⋊C8—22no—
116Q8⋊C8—22no—
117#322—22no—
118#322—22no—
119#322—22no—
120C82—22no—
121C22×Q8—21no—
122C22×Q8—21no—
123C23×C4C2×D411yescharacteristic
124C23×C4#4111yescharacteristic
125C23×C4#4111yescharacteristic
126C2×M16—22no—
127C2×M16—22no—
128C2×M16—41no—
129C2×M16—82no—
130C2×M16C2×D412yes—
131C2×M16C2×D412yes—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—22no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—44no—
140C22×C8—41no—
141C4⋊Q8—21no—
142C4⋊Q8—21no—
143C4⋊Q8—21no—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8C22×C412yes—
148C4⋊Q8C22×C412yes—
149#125—22no—
150#125—22no—
151#125—41no—
152#125—41no—
153C22⋊Q8—22no—
154C22⋊Q8—22no—
155C22⋊Q8—22no—
156C22⋊Q8—22no—
157C22⋊Q8—41no—
158C22⋊Q8—41no—
159C22⋊Q8—41no—
160C22⋊Q8—41no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—41no—
166C4×Q8—41no—
167C4×Q8—41no—
168C4×Q8—41no—
169#117—22no—
170#117—22no—
171#117—41no—
172#117—41no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—41no—
182C2×C42—22no—
183C2×C42—41no—
184C2×C42C22×C411yescharacteristic
185C2×C42C22×C411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42#4111yescharacteristic
190C2×C42#4111yescharacteristic
191C2×C42#4111yescharacteristic
192C2×C42#4111yescharacteristic
193C2×C42#4111yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42C2411yescharacteristic
196#108—84no—
197C4⋊C8—22no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—41no—
206C4⋊C8—41no—
207C4⋊C8C22×C414yes—
208C4⋊C8C22×C414yes—
209C22⋊C8—22no—
210C22⋊C8—22no—
211C22⋊C8—22no—
212C22⋊C8—22no—
213C22⋊C8—22no—
214C22⋊C8—22no—
215#97—22no—
216#97—22no—
217#97—22no—
218#97—22no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—22no—
224C4×C8—22no—
225C4×C8—21no—
226C4×C8—21no—
227C4×C8C2×D411yescharacteristic
228C4×C8C2×D411yescharacteristic
229C4×C8C2×D411yescharacteristic
230C4×C8C2×D411yescharacteristic
231#95—22no—
232#95—41no—
233#95—41no—
234#95—41no—
235C24#12411yescharacteristic
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—44no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—82no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4#11711yescharacteristic
250C22×C4#11711yescharacteristic
251C22×C4C4×D411yescharacteristic
252C22×C4C4×D411yescharacteristic
253C22×C4C4×D411yescharacteristic
254C22×C4C4×D411yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4#12111yescharacteristic
257C22×C4#12311yescharacteristic
258C22×C4#12311yescharacteristic
259C22×C4#12411yescharacteristic
260C22×C4#12411yescharacteristic
261C22×C4#12411yescharacteristic
262C22×C4#12611yescharacteristic
263C22×C4#12611yescharacteristic
264C22×C4C22×D411yescharacteristic
265C22×C4C2×#4111yescharacteristic
266C22×C4C2×#4111yescharacteristic
267M16—24no—
268M16—24no—
269M16—44no—
270M16—44no—
271M16—82no—
272C2×C8—24no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—24no—
276C2×C8—24no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—82no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—44no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—42no—
305C22⋊C4—82no—
306C42—42no—
307C42—42no—
308C42—42no—
309C42—42no—
310C42#11711yescharacteristic
311C42#11711yescharacteristic
312C42C4×D411yescharacteristic
313C42C4×D411yescharacteristic
314C42C4×D411yescharacteristic
315C42C4×D411yescharacteristic
316C42#12111yescharacteristic
317C42#12111yescharacteristic
318C42#12311yescharacteristic
319C42#12311yescharacteristic
320C42#12611yescharacteristic
321C42#12611yescharacteristic
322C42C23×C411yescharacteristic
323C42C22×D411yescharacteristic
324C42C2×#4111yescharacteristic
325C42C2×#4111yescharacteristic
326C23—22no—
327C23—22no—
328C23—41no—
329C23#38811yescharacteristic
330C23#38811yescharacteristic
331C23C2×#12411yescharacteristic
332Q8—44no—
333Q8—82no—
334Q8—82no—
335Q8—82no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4#38811yescharacteristic
363C2×C4#38811yescharacteristic
364C2×C4#38811yescharacteristic
365C2×C4#38811yescharacteristic
366C2×C4#38811yescharacteristic
367C2×C4#38811yescharacteristic
368C2×C4C2×#11711yescharacteristic
369C2×C4C2×C4×D411yescharacteristic
370C2×C4C2×C4×D411yescharacteristic
371C2×C4C2×#12111yescharacteristic
372C2×C4C2×#12311yescharacteristic
373C2×C4C2×#12611yescharacteristic
374C2×C4#57712yes—
375C2×C4#57812yes—
376C8—22no—
377C8—22no—
378C8—41no—
379C8—41no—
380C8—44no—
381C8—44no—
382C8—42no—
383C8—42no—
384C8—82no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—44no—
393C22#193212yes—
394C22C2×#38811yescharacteristic
395C22#313712yes—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—82no—
406C4—82no—
407C4—162no—
408C4#294712yes—
409C4#295512yes—
410C4#305711yescharacteristic
411C4#305911yescharacteristic
412C2—22no—
413C2—22no—
414C2—41no—
415C2#2107011yescharacteristic
416C2C2×#193211yescharacteristic
417C2#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
#218452
#177342
C2×#14932
#173232
#95912
#95892
#80932
#78982