Finite Group Databasebeta

← #117765

Subgroups — #117765

Order 512.

712 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20765C211yesmaximal, 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×C4×M16C2211yescharacteristic
12#2285C2211yescharacteristic
13C2×#383C2211yescharacteristic
14C2×#382C2211yescharacteristic
15#1836—22no—
16#1835—22no—
17#1823—21no—
18#1803—22no—
19#1787—21no—
20#1763—22no—
21#1736—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—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×#103—24no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—21no—
65C4×M16—21no—
66C4×M16—21no—
67C4×M16C2×C412yes—
68C4×M16D412yes—
69C2×#97—21no—
70C2×#97—21no—
71C2×#97D412yes—
72C2×C4×C8D412yes—
73#384D412yes—
74#383—21no—
75#383—21no—
76#383—21no—
77#383—21no—
78#382—21no—
79#382—21no—
80#382—21no—
81#382—21no—
82#382—21no—
83#380—21no—
84#380—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C2×#95—24no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—24no—
93C2×#95D412yes—
94#340—24no—
95#340—28no—
96Q8⋊C8—28no—
97C22×Q8—21no—
98C22×Q8—21no—
99C22×Q8—21no—
100C22×Q8—21no—
101C22×Q8—21no—
102C22×Q8—44no—
103C22×Q8C4⋊C412yes—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C2×M16—22no—
112C2×M16—22no—
113C2×M16—42no—
114C2×M16—42no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—41no—
118C4×Q8—22no—
119C4×Q8—22no—
120C4×Q8—22no—
121C4×Q8—22no—
122C4×Q8—22no—
123C4×Q8—22no—
124C4×Q8—22no—
125C4×Q8—22no—
126C4×Q8—41no—
127C4×Q8—41no—
128C4×Q8—41no—
129C4×Q8—41no—
130C4×Q8—41no—
131C4×Q8—41no—
132C4×Q8—41no—
133C4×Q8—41no—
134#117—44no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4C4⋊C412yes—
153C2×C22⋊C4—41no—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—41no—
157C2×C42—41no—
158C2×C42—41no—
159C2×C42—41no—
160C2×C42—41no—
161C2×C42C22×C411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×Q811yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C4⋊C8—22no—
174C4⋊C8—41no—
175C4⋊C8—41no—
176C4⋊C8—41no—
177#103—48no—
178C22⋊C8—22no—
179C22⋊C8—41no—
180C22⋊C8—41no—
181C22⋊C8—41no—
182#97—22no—
183#97—22no—
184#97—22no—
185#97—22no—
186#97—22no—
187#97—41no—
188C4×C8—22no—
189C4×C8C22⋊C414yes—
190#95—22no—
191#95—22no—
192#95—48no—
193#95—41no—
194#95—41no—
195#95—41no—
196#95—41no—
197#95—41no—
198#95—41no—
199#95—41no—
200#95—41no—
201#95—41no—
202#95—41no—
203#95—84no—
204C24#12011yescharacteristic
205C2×Q8—24no—
206C2×Q8—24no—
207C2×Q8—22no—
208C2×Q8—22no—
209C2×Q8—44no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—44no—
213C2×Q8—44no—
214C2×Q8—44no—
215C2×Q8—41no—
216C2×Q8—41no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—44no—
241C22×C4C2×C4⋊C411yescharacteristic
242C22×C4C4×D411yescharacteristic
243C22×C4C4×D411yescharacteristic
244C22×C4#12011yescharacteristic
245C22×C4#12011yescharacteristic
246C22×C4#12111yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4#12111yescharacteristic
249C22×C4#12111yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4C22⋊Q811yescharacteristic
253C22×C4C22⋊Q811yescharacteristic
254C22×C4#12311yescharacteristic
255C22×C4#12311yescharacteristic
256C22×C4#12311yescharacteristic
257C22×C4#12411yescharacteristic
258C22×C4#12711yescharacteristic
259M16—44no—
260M16—44no—
261C2×C8—24no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—44no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—82no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—44no—
271C4⋊C4—44no—
272C4⋊C4—44no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—82no—
280C4⋊C4—82no—
281C22⋊C4—82no—
282C42—42no—
283C42—42no—
284C42—42no—
285C42—42no—
286C42—82no—
287C42C2×C22⋊C411yescharacteristic
288C42#11711yescharacteristic
289C42C4×D411yescharacteristic
290C42C4×D411yescharacteristic
291C42#12111yescharacteristic
292C42#12111yescharacteristic
293C42C22⋊Q811yescharacteristic
294C42C22⋊Q811yescharacteristic
295C42C22⋊Q811yescharacteristic
296C42C22⋊Q811yescharacteristic
297C42#12311yescharacteristic
298C42#12311yescharacteristic
299C42#12411yescharacteristic
300C42#12411yescharacteristic
301C42#12611yescharacteristic
302C42#12611yescharacteristic
303C23—22no—
304C23—22no—
305C23#38511yescharacteristic
306C23#39211yescharacteristic
307C23#39411yescharacteristic
308C23#45412yes—
309C23#45512yes—
310Q8—24no—
311Q8—24no—
312Q8—44no—
313Q8—42no—
314Q8—42no—
315Q8—44no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—24no—
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—42no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—82no—
346C2×C4—84no—
347C2×C4—84no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—88no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4#38011yescharacteristic
359C2×C4#38511yescharacteristic
360C2×C4#38611yescharacteristic
361C2×C4#38811yescharacteristic
362C2×C4#38811yescharacteristic
363C2×C4#39011yescharacteristic
364C2×C4#39311yescharacteristic
365C2×C4#39411yescharacteristic
366C2×C4#39611yescharacteristic
367C2×C4#39611yescharacteristic
368C2×C4#39711yescharacteristic
369C2×C4#39911yescharacteristic
370C8—44no—
371C8—82no—
372C8—82no—
373C22—24no—
374C22—24no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22#143911yescharacteristic
380C22#186012yes—
381C22#200311yescharacteristic
382C22#200411yescharacteristic
383C22#200912yes—
384C22#201012yes—
385C22#204012yes—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—44no—
399C4—82no—
400C4—162no—
401C4—162no—
402C4—162no—
403C2—22no—
404C2—22no—
405C2#1002112yes—
406C2#1002412yes—
407C2#1107911yescharacteristic
408C2#1110011yescharacteristic
409C2#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
#207652
#207402
C2×#14912
#143462
#80302
#80182
C2×#12882