Finite Group Databasebeta

← #117763

Subgroups — #117763

Order 512.

756 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20801C211yesmaximal, characteristic
2#20740C211yesmaximal, characteristic
3C2×#1491C211yesmaximal, characteristic
4#14346C211yesmaximal, characteristic
5#8030C211yesmaximal, characteristic
6#8018C211yesmaximal, characteristic
7C2×#1288C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#428—22no—
12C2×C4×M16C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×#383C2211yescharacteristic
15C2×#382C2211yescharacteristic
16#1847—22no—
17#1846—22no—
18#1823—21no—
19#1803—22no—
20#1787—21no—
21#1763—22no—
22#1491—22no—
23#1491C414yescomplemented
24C2×Q8⋊C8—22no—
25#1446—21no—
26#1445—21no—
27#1441—22no—
28#1434—22no—
29#1295—22no—
30#1294—22no—
31#1288—22no—
32C23×Q8D411yescharacteristic
33C22×M16—21no—
34C2×C4×Q8—21no—
35C2×C4×Q8—21no—
36C2×C4×Q8—21no—
37C2×C4×Q8—21no—
38C2×C4×Q8—21no—
39C2×C4×Q8—21no—
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8—21no—
43C2×C4×Q8—21no—
44C2×C4×Q8C2×C412yes—
45C2×C4×Q8Q812yes—
46C2×#117—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4D411yescharacteristic
49C22×C4⋊C4D412yes—
50C22×C42—21no—
51C22×C42C2311yescharacteristic
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432C2×C412yes—
60#428—48no—
61C2×#107—24no—
62C2×#106—24no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×#103—24no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—21no—
68C4×M16—21no—
69C4×M16—21no—
70C4×M16C2×C412yes—
71C4×M16D412yes—
72C2×#97—21no—
73C2×#97—21no—
74C2×#97D412yes—
75C2×C4×C8D412yes—
76#384D412yes—
77#383—21no—
78#383—21no—
79#383—21no—
80#383—21no—
81#382—21no—
82#382—21no—
83#382—21no—
84#382—21no—
85#382—21no—
86#380—21no—
87#380—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C2×#95—24no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—24no—
96C2×#95D412yes—
97#340—24no—
98Q8⋊C8—28no—
99C22×Q8—21no—
100C22×Q8—21no—
101C22×Q8—21no—
102C22×Q8—21no—
103C22×Q8—21no—
104C22×Q8—44no—
105C22×Q8C4⋊C412yes—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C2×M16—24no—
114C2×M16—24no—
115C2×M16—22no—
116C2×M16—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—41no—
120C4×Q8—22no—
121C4×Q8—22no—
122C4×Q8—22no—
123C4×Q8—22no—
124C4×Q8—22no—
125C4×Q8—22no—
126C4×Q8—22no—
127C4×Q8—22no—
128C4×Q8—41no—
129C4×Q8—41no—
130C4×Q8—41no—
131C4×Q8—41no—
132C4×Q8—41no—
133C4×Q8—41no—
134C4×Q8—41no—
135C4×Q8—41no—
136#117—44no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4C4⋊C412yes—
155C2×C22⋊C4—41no—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42—41no—
159C2×C42—41no—
160C2×C42—41no—
161C2×C42—41no—
162C2×C42—41no—
163C2×C42C22×C411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×Q811yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175#107—48no—
176#106—48no—
177C4⋊C8—22no—
178C4⋊C8—41no—
179C4⋊C8—41no—
180C4⋊C8—41no—
181#103—48no—
182C22⋊C8—22no—
183C22⋊C8—41no—
184C22⋊C8—41no—
185C22⋊C8—41no—
186#97—22no—
187#97—22no—
188#97—22no—
189#97—22no—
190#97—22no—
191#97—41no—
192C4×C8—22no—
193C4×C8C22⋊C414yes—
194#95—22no—
195#95—22no—
196#95—48no—
197#95—41no—
198#95—41no—
199#95—41no—
200#95—41no—
201#95—41no—
202#95—41no—
203#95—41no—
204#95—41no—
205#95—41no—
206#95—41no—
207#95—84no—
208C24#12011yescharacteristic
209C2×Q8—24no—
210C2×Q8—24no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—44no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—44no—
217C2×Q8—44no—
218C2×Q8—44no—
219C2×Q8—41no—
220C2×Q8—41no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—44no—
244C22×C4—42no—
245C22×C4C2×C4⋊C411yescharacteristic
246C22×C4C4×D411yescharacteristic
247C22×C4C4×D411yescharacteristic
248C22×C4#12011yescharacteristic
249C22×C4#12011yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4#12111yescharacteristic
255C22×C4#12111yescharacteristic
256C22×C4C22⋊Q811yescharacteristic
257C22×C4C22⋊Q811yescharacteristic
258C22×C4#12311yescharacteristic
259C22×C4#12311yescharacteristic
260C22×C4#12311yescharacteristic
261C22×C4#12411yescharacteristic
262C22×C4#12711yescharacteristic
263M16—28no—
264M16—28no—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—44no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—82no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—44no—
275C4⋊C4—42no—
276C4⋊C4—44no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—41no—
282C4⋊C4—42no—
283C4⋊C4—41no—
284C4⋊C4—82no—
285C4⋊C4—82no—
286C22⋊C4—82no—
287C42—42no—
288C42—42no—
289C42—42no—
290C42—42no—
291C42—82no—
292C42C2×C22⋊C411yescharacteristic
293C42#11711yescharacteristic
294C42C4×D411yescharacteristic
295C42C4×D411yescharacteristic
296C42#12111yescharacteristic
297C42#12111yescharacteristic
298C42C22⋊Q811yescharacteristic
299C42C22⋊Q811yescharacteristic
300C42C22⋊Q811yescharacteristic
301C42C22⋊Q811yescharacteristic
302C42#12311yescharacteristic
303C42#12311yescharacteristic
304C42#12411yescharacteristic
305C42#12411yescharacteristic
306C42#12611yescharacteristic
307C42#12611yescharacteristic
308C23—22no—
309C23—22no—
310C23#38511yescharacteristic
311C23#39211yescharacteristic
312C23#39411yescharacteristic
313C23#45312yes—
314C23#45612yes—
315Q8—24no—
316Q8—24no—
317Q8—44no—
318Q8—42no—
319Q8—42no—
320Q8—44no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—24no—
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—44no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—82no—
351C2×C4—84no—
352C2×C4—84no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—88no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4#38011yescharacteristic
364C2×C4#38511yescharacteristic
365C2×C4#38611yescharacteristic
366C2×C4#38811yescharacteristic
367C2×C4#38811yescharacteristic
368C2×C4#39011yescharacteristic
369C2×C4#39311yescharacteristic
370C2×C4#39411yescharacteristic
371C2×C4#39611yescharacteristic
372C2×C4#39611yescharacteristic
373C2×C4#39711yescharacteristic
374C2×C4#39911yescharacteristic
375C8—28no—
376C8—82no—
377C8—82no—
378C22—24no—
379C22—24no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22#143911yescharacteristic
385C22#185912yes—
386C22#200211yescharacteristic
387C22#200511yescharacteristic
388C22#200912yes—
389C22#201012yes—
390C22#203912yes—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—44no—
404C4—82no—
405C4—162no—
406C4—162no—
407C4—162no—
408C2—22no—
409C2—22no—
410C2#1002012yes—
411C2#1002512yes—
412C2#1107811yescharacteristic
413C2#1110011yescharacteristic
414C2#1113011yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
23. #1491C4semidirect

Maximal subgroups

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

subgroupindex
#208012
#207402
C2×#14912
#143462
#80302
#80182
C2×#12882