Finite Group Databasebeta

← #116096

Subgroups — #116096

Order 512.

736 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21866C211yesmaximal, characteristic
2#21394C211yesmaximal, characteristic
3C2×#1986C211yesmaximal, characteristic
4#13955C211yesmaximal, characteristic
5#8033C211yesmaximal, characteristic
6C2×#1285C211yesmaximal, characteristic
7#7904C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12C2×#382C2211yescharacteristic
13C2×C4×C4⋊C4C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1993—21no—
16#1988—21no—
17#1986—22no—
18#1986C414yescomplemented
19#1981—21no—
20#1960—21no—
21#1872—22no—
22#1872—22no—
23#1805—22no—
24#1763—22no—
25C2×#339—22no—
26#1442—22no—
27C4⋊#95—22no—
28#1434—22no—
29#1295—22no—
30#1285—22no—
31#1279—22no—
32C22×M16—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533C2×C412yes—
40#533Q812yes—
41C2×C4⋊Q8D411yescharacteristic
42C2×C4⋊Q8D411yescharacteristic
43C2×#125—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8—21no—
48C2×#117—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4D412yes—
51C22×C4⋊C4D412yes—
52C22×C42—21no—
53C22×C42D412yes—
54C22×C42C2311yescharacteristic
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8D412yes—
65C2×#103—24no—
66C2×#103—24no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69C4×M16—21no—
70C4×M16—21no—
71C4×M16—21no—
72C4×M16C2×C412yes—
73C2×#97—21no—
74C2×#97—21no—
75C2×#97D412yes—
76C2×C4×C8—21no—
77#382—21no—
78#382—21no—
79#382—21no—
80#382—21no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4C2×C412yes—
88C4×C4⋊C4D412yes—
89C2×#95—21no—
90C2×#95—24no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—24no—
95#340—24no—
96#340—24no—
97#339—28no—
98C22×Q8—21no—
99C22×Q8—21no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C2×M16—24no—
107C2×M16—22no—
108C2×M16—42no—
109C2×M16—41no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—41no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8C4⋊C412yes—
120C4⋊Q8C4⋊C412yes—
121#125—22no—
122#125—22no—
123#125—41no—
124#125—41no—
125C22⋊Q8—22no—
126C22⋊Q8—22no—
127C22⋊Q8—22no—
128C22⋊Q8—22no—
129C22⋊Q8—41no—
130C22⋊Q8—41no—
131C22⋊Q8—41no—
132C22⋊Q8—41no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—41no—
138C4×Q8—41no—
139C4×Q8—41no—
140C4×Q8—41no—
141#117—22no—
142#117—22no—
143#117—41no—
144#117—41no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—44no—
161C2×C22⋊C4—41no—
162C2×C42—22no—
163C2×C42—41no—
164C2×C42—44no—
165C2×C42—44no—
166C2×C42—41no—
167C2×C42—41no—
168C2×C42C22⋊C414yes—
169C2×C42C22×C411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×Q811yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42#4111yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—41no—
184C4⋊C8—41no—
185#103—48no—
186#103—84no—
187C22⋊C8—41no—
188C22⋊C8—41no—
189C22⋊C8—41no—
190C22⋊C8—41no—
191#97—22no—
192#97—22no—
193#97—22no—
194#97—41no—
195#97—41no—
196#97—41no—
197C4×C8—22no—
198C4×C8—41no—
199#95—22no—
200#95—22no—
201#95—48no—
202#95—41no—
203#95—41no—
204#95—41no—
205#95—41no—
206#95—41no—
207#95—41no—
208#95—84no—
209C24#12011yescharacteristic
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—44no—
213C2×Q8—41no—
214C2×Q8—41no—
215C2×Q8—42no—
216C2×Q8—42no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—42no—
224C22×C4—44no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—42no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4C2×C22⋊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×C4C22⋊Q811yescharacteristic
252C22×C4C22⋊Q811yescharacteristic
253C22×C4C22⋊Q811yescharacteristic
254C22×C4#12311yescharacteristic
255C22×C4#12411yescharacteristic
256C22×C4#12411yescharacteristic
257C22×C4#12411yescharacteristic
258C22×C4#12711yescharacteristic
259M16—24no—
260M16—42no—
261M16—44no—
262C2×C8—44no—
263C2×C8—42no—
264C2×C8—44no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—44no—
288C4⋊C4—82no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C42—44no—
292C42—42no—
293C42—42no—
294C42—82no—
295C42—82no—
296C42—82no—
297C42C2×C4⋊C411yescharacteristic
298C42#11711yescharacteristic
299C42C4×D411yescharacteristic
300C42C4×D411yescharacteristic
301C42#12111yescharacteristic
302C42#12111yescharacteristic
303C42#12111yescharacteristic
304C42C22⋊Q811yescharacteristic
305C42C22⋊Q811yescharacteristic
306C42C22⋊Q811yescharacteristic
307C42#12311yescharacteristic
308C42#12311yescharacteristic
309C42#12311yescharacteristic
310C42#12311yescharacteristic
311C42#12611yescharacteristic
312C42#12611yescharacteristic
313C23—21no—
314C23—21no—
315C23—22no—
316C23#38611yescharacteristic
317C23#39211yescharacteristic
318C23#39311yescharacteristic
319C23#45512yes—
320C23#45612yes—
321Q8—44no—
322Q8—44no—
323Q8—82no—
324Q8—82no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—84no—
354C2×C4—84no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—84no—
363C2×C4—84no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4#38011yescharacteristic
368C2×C4#38511yescharacteristic
369C2×C4#38511yescharacteristic
370C2×C4#38811yescharacteristic
371C2×C4#38811yescharacteristic
372C2×C4#39011yescharacteristic
373C2×C4#39411yescharacteristic
374C2×C4#39411yescharacteristic
375C2×C4#39611yescharacteristic
376C2×C4#39611yescharacteristic
377C2×C4#39711yescharacteristic
378C2×C4#39911yescharacteristic
379C2×C4#45412yes—
380C2×C4#45512yes—
381C8—44no—
382C8—82no—
383C8—82no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22#143911yescharacteristic
393C22#188511yescharacteristic
394C22#188611yescharacteristic
395C22#200311yescharacteristic
396C22#200411yescharacteristic
397C22#200712yes—
398C22#201012yes—
399C22#202812yes—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—84no—
411C4—84no—
412C4—82no—
413C4—82no—
414C4—162no—
415C4#186012yes—
416C4#204012yes—
417C2—21no—
418C2—21no—
419C2—22no—
420C2#1000912yes—
421C2#1001012yes—
422C2#1105511yescharacteristic
423C2#1107911yescharacteristic
424C2#1108311yescharacteristic
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
18. #1986C4semidirect

Maximal subgroups

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

subgroupindex
#218662
#213942
C2×#19862
#139552
#80332
C2×#12852
#79042