Finite Group Databasebeta

← #203161

Subgroups — #203161

Order 512.

653 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21969C211yesmaximal, characteristic
2#21845C211yesmaximal, characteristic
3C2×#1542C211yesmaximal, characteristic
4#17501C211yesmaximal, characteristic
5#17327C211yesmaximal, characteristic
6#9591C214yesmaximal, complemented
7#9589C214yesmaximal, complemented
8#8096C211yesmaximal, characteristic
9#7895C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16#2283C2211yescharacteristic
17#1995—22no—
18#1986C2214yes—
19#1950—22no—
20#1950C2214yes—
21#1826—22no—
22#1810—22no—
23#1542C414yescomplemented
24#1542C2212yes—
25#1542C2212yes—
26C2×#333C412yescomplemented
27C2×#332C412yescomplemented
28#1493C2214yes—
29#1491—21no—
30#1491—21no—
31#1485—22no—
32#1453—22no—
33#1451—21no—
34#1451C2212yes—
35C8×M16—21no—
36C8×M16C2212yes—
37#1303C2212yes—
38#1303C2212yes—
39#1279C2214yes—
40C22×M16—21no—
41C23×C8—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533C2×C412yes—
49#533C2312yes—
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×#125—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57C2×#117—21no—
58C22×C42C2311yescharacteristic
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431—21no—
68#431—21no—
69#431C2×C412yes—
70#431C2311yescharacteristic
71#431C2311yescharacteristic
72C2×#108—24no—
73#423D412yes—
74#423C2312yes—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8C2×C412yes—
79C2×C4⋊C8C2×C412yes—
80C2×C4⋊C8C2×C412yes—
81C2×C4⋊C8C2312yes—
82#407—21no—
83#407D412yes—
84C2×C22⋊C8—21no—
85C4×M16—21no—
86C4×M16—21no—
87C4×M16—21no—
88C4×M16D411yescharacteristic
89C4×M16D411yescharacteristic
90C4×M16D412yes—
91C4×M16C2312yes—
92C2×#97—21no—
93C2×C4×C8—21no—
94C2×C4×C8C2312yes—
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—28no—
104#333C2×C414yes—
105#333C2×C414yes—
106#332C2×C414yes—
107#332C2×C414yes—
108Q8⋊C8—24no—
109Q8⋊C8—24no—
110#322—22no—
111#322—22no—
112#322—22no—
113C82—22no—
114C22×Q8—21no—
115C22×Q8—21no—
116C23×C4C2×D411yescharacteristic
117C23×C4#4111yescharacteristic
118C23×C4#4111yescharacteristic
119C2×M16—21no—
120C2×M16—22no—
121C2×M16—21no—
122C2×M16—22no—
123C2×M16—44no—
124C2×M16C2×D414yes—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—82no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8C22×C412yes—
138C4⋊Q8C22×C412yes—
139#125—22no—
140#125—22no—
141#125—41no—
142#125—41no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C22⋊Q8—41no—
148C22⋊Q8—41no—
149C22⋊Q8—41no—
150C22⋊Q8—41no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—41no—
156C4×Q8—41no—
157C4×Q8—41no—
158C4×Q8—41no—
159#117—22no—
160#117—22no—
161#117—41no—
162#117—41no—
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×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C42—22no—
173C2×C42—41no—
174C2×C42C22×C411yescharacteristic
175C2×C42C22×C411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42C2411yescharacteristic
186#108—84no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8C22×C414yes—
196C4⋊C8C22×C412yes—
197C4⋊C8C22×C412yes—
198C22⋊C8—22no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—21no—
202C22⋊C8—21no—
203C22⋊C8—41no—
204C22⋊C8—41no—
205#97—21no—
206#97—21no—
207#97—22no—
208#97—22no—
209#97—41no—
210C4×C8—22no—
211C4×C8—21no—
212C4×C8—22no—
213C4×C8—21no—
214C4×C8—22no—
215C4×C8—41no—
216C4×C8C2×D412yes—
217C4×C8C2×D412yes—
218#95—22no—
219#95—41no—
220#95—41no—
221#95—41no—
222C24#12411yescharacteristic
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—82no—
228C2×Q8—82no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4#11711yescharacteristic
237C22×C4#11711yescharacteristic
238C22×C4C4×D411yescharacteristic
239C22×C4C4×D411yescharacteristic
240C22×C4C4×D411yescharacteristic
241C22×C4C4×D411yescharacteristic
242C22×C4#12111yescharacteristic
243C22×C4#12111yescharacteristic
244C22×C4#12311yescharacteristic
245C22×C4#12311yescharacteristic
246C22×C4#12411yescharacteristic
247C22×C4#12411yescharacteristic
248C22×C4#12411yescharacteristic
249C22×C4#12611yescharacteristic
250C22×C4#12611yescharacteristic
251C22×C4C22×D411yescharacteristic
252C22×C4C2×#4111yescharacteristic
253C22×C4C2×#4111yescharacteristic
254M16—24no—
255M16—24no—
256M16—24no—
257M16—44no—
258M16—42no—
259M16—42no—
260M16—42no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—22no—
267C2×C8—44no—
268C2×C8—44no—
269C2×C8—42no—
270C2×C8—82no—
271C2×C8—82no—
272C2×C8—82no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—44no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—82no—
289C42—42no—
290C42—42no—
291C42—42no—
292C42—42no—
293C42#11711yescharacteristic
294C42#11711yescharacteristic
295C42C4×D411yescharacteristic
296C42C4×D411yescharacteristic
297C42C4×D411yescharacteristic
298C42C4×D411yescharacteristic
299C42#12111yescharacteristic
300C42#12111yescharacteristic
301C42#12311yescharacteristic
302C42#12311yescharacteristic
303C42#12611yescharacteristic
304C42#12611yescharacteristic
305C42C23×C411yescharacteristic
306C42C22×D411yescharacteristic
307C42C2×#4111yescharacteristic
308C42C2×#4111yescharacteristic
309C23—22no—
310C23—22no—
311C23—41no—
312C23#38811yescharacteristic
313C23#38811yescharacteristic
314C23C2×#12411yescharacteristic
315Q8—44no—
316Q8—82no—
317Q8—82no—
318Q8—82no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4#38811yescharacteristic
346C2×C4#38811yescharacteristic
347C2×C4#38811yescharacteristic
348C2×C4#38811yescharacteristic
349C2×C4#38811yescharacteristic
350C2×C4#38811yescharacteristic
351C2×C4C2×#11711yescharacteristic
352C2×C4C2×C4×D411yescharacteristic
353C2×C4C2×C4×D411yescharacteristic
354C2×C4C2×#12111yescharacteristic
355C2×C4C2×#12311yescharacteristic
356C2×C4C2×#12611yescharacteristic
357C2×C4#57712yes—
358C2×C4#57812yes—
359C8—24no—
360C8—42no—
361C8—44no—
362C8—42no—
363C8—42no—
364C8—44no—
365C8—82no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—44no—
374C22#193212yes—
375C22C2×#38811yescharacteristic
376C22#313712yes—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—82no—
387C4—82no—
388C4—162no—
389C4#294911yescharacteristic
390C4#295011yescharacteristic
391C4#295712yes—
392C4#306312yes—
393C2—22no—
394C2—22no—
395C2—41no—
396C2#2107211yescharacteristic
397C2C2×#193211yescharacteristic
398C2#2108811yescharacteristic
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
23. #1542C4semidirect
26. C2×#333C4semidirect
27. C2×#332C4semidirect

Maximal subgroups

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

subgroupindex
#219692
#218452
C2×#15422
#175012
#173272
#95912
#95892
#80962
#78952