Finite Group Databasebeta

← #116094

Subgroups — #116094

Order 512.

808 conjugacy classes of subgroups.

The subgroup lattice has 430 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#13954C211yesmaximal, characteristic
5#8033C211yesmaximal, characteristic
6C2×#1285C211yesmaximal, characteristic
7#7904C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12C2×#389—22no—
13C2×#382C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15C2×C4×C4⋊C4C2211yescharacteristic
16#1993—21no—
17#1988—21no—
18#1986—22no—
19#1986C414yescomplemented
20#1981—21no—
21#1960—21no—
22#1872—22no—
23#1872—22no—
24#1805—22no—
25#1763—22no—
26C2×#339—22no—
27#1444—22no—
28C4⋊#95—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⋊C4—24no—
51C22×C4⋊C4D412yes—
52C22×C4⋊C4D412yes—
53C22×C42—21no—
54C22×C42D412yes—
55C22×C42C2311yescharacteristic
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63#432—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8D412yes—
66C2×#103—24no—
67C2×#103—24no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C4×M16—21no—
71C4×M16—21no—
72C4×M16—21no—
73C4×M16C2×C412yes—
74C2×#97—21no—
75C2×#97—21no—
76C2×#97D412yes—
77C2×C4×C8—21no—
78#389—48no—
79#382—21no—
80#382—21no—
81#382—21no—
82#382—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4C2×C412yes—
90C4×C4⋊C4D412yes—
91C2×#95—21no—
92C2×#95—24no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—21no—
96#340—24no—
97#340—24no—
98#339—28no—
99C22×Q8—21no—
100C22×Q8—21no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4C2×D411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C2×M16—24no—
108C2×M16—22no—
109C2×M16—42no—
110C2×M16—41no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—41no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8C4⋊C412yescomplemented
121C4⋊Q8C4⋊C412yescomplemented
122#125—22no—
123#125—22no—
124#125—41no—
125#125—41no—
126C22⋊Q8—22no—
127C22⋊Q8—22no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—41no—
131C22⋊Q8—41no—
132C22⋊Q8—41no—
133C22⋊Q8—41no—
134C4×Q8—22no—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—41no—
139C4×Q8—41no—
140C4×Q8—41no—
141C4×Q8—41no—
142#117—22no—
143#117—22no—
144#117—41no—
145#117—41no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—48no—
163C2×C4⋊C4—48no—
164C2×C4⋊C4—84no—
165C2×C22⋊C4—41no—
166C2×C42—22no—
167C2×C42—41no—
168C2×C42—44no—
169C2×C42—44no—
170C2×C42—41no—
171C2×C42—41no—
172C2×C42C22⋊C414yes—
173C2×C42C22×C411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×Q811yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—41no—
188C4⋊C8—41no—
189#103—48no—
190#103—84no—
191C22⋊C8—41no—
192C22⋊C8—41no—
193C22⋊C8—41no—
194C22⋊C8—41no—
195#97—22no—
196#97—22no—
197#97—22no—
198#97—41no—
199#97—41no—
200#97—41no—
201C4×C8—22no—
202C4×C8—41no—
203#95—22no—
204#95—22no—
205#95—48no—
206#95—41no—
207#95—41no—
208#95—41no—
209#95—41no—
210#95—41no—
211#95—41no—
212C24#12011yescharacteristic
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—44no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—42no—
219C2×Q8—42no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—42no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4C4×D411yescharacteristic
247C22×C4#12011yescharacteristic
248C22×C4#12011yescharacteristic
249C22×C4#12111yescharacteristic
250C22×C4#12111yescharacteristic
251C22×C4#12111yescharacteristic
252C22×C4#12111yescharacteristic
253C22×C4#12111yescharacteristic
254C22×C4C22⋊Q811yescharacteristic
255C22×C4C22⋊Q811yescharacteristic
256C22×C4C22⋊Q811yescharacteristic
257C22×C4#12311yescharacteristic
258C22×C4#12411yescharacteristic
259C22×C4#12411yescharacteristic
260C22×C4#12411yescharacteristic
261C22×C4#12711yescharacteristic
262M16—24no—
263M16—42no—
264M16—44no—
265C2×C8—44no—
266C2×C8—42no—
267C2×C8—44no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C4⋊C4—28no—
273C4⋊C4—28no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—44no—
291C4⋊C4—82no—
292C4⋊C4—816no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C42—44no—
296C42—42no—
297C42—42no—
298C42—82no—
299C42—82no—
300C42—82no—
301C42C2×C4⋊C411yescharacteristic
302C42#11711yescharacteristic
303C42C4×D411yescharacteristic
304C42C4×D411yescharacteristic
305C42#12111yescharacteristic
306C42#12111yescharacteristic
307C42#12111yescharacteristic
308C42C22⋊Q811yescharacteristic
309C42C22⋊Q811yescharacteristic
310C42C22⋊Q811yescharacteristic
311C42#12311yescharacteristic
312C42#12311yescharacteristic
313C42#12311yescharacteristic
314C42#12311yescharacteristic
315C42#12611yescharacteristic
316C42#12611yescharacteristic
317C23—21no—
318C23—21no—
319C23—22no—
320C23#38611yescharacteristic
321C23#39211yescharacteristic
322C23#39311yescharacteristic
323C23#45512yes—
324C23#45612yes—
325Q8—44no—
326Q8—44no—
327Q8—82no—
328Q8—82no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
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—22no—
345C2×C4—28no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—48no—
351C2×C4—48no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—84no—
367C2×C4—84no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4#38011yescharacteristic
372C2×C4#38511yescharacteristic
373C2×C4#38511yescharacteristic
374C2×C4#38811yescharacteristic
375C2×C4#38811yescharacteristic
376C2×C4#39011yescharacteristic
377C2×C4#39411yescharacteristic
378C2×C4#39411yescharacteristic
379C2×C4#39611yescharacteristic
380C2×C4#39611yescharacteristic
381C2×C4#39711yescharacteristic
382C2×C4#39911yescharacteristic
383C2×C4#45312yes—
384C2×C4#45612yes—
385C8—44no—
386C8—82no—
387C8—82no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22#143911yescharacteristic
397C22#188511yescharacteristic
398C22#188611yescharacteristic
399C22#200211yescharacteristic
400C22#200511yescharacteristic
401C22#200712yes—
402C22#201012yes—
403C22#202812yes—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—48no—
415C4—84no—
416C4—82no—
417C4—82no—
418C4—162no—
419C4#185912yes—
420C4#203912yes—
421C2—21no—
422C2—21no—
423C2—22no—
424C2#1000812yes—
425C2#1001112yes—
426C2#1105511yescharacteristic
427C2#1107811yescharacteristic
428C2#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
19. #1986C4semidirect
120. C4⋊Q8C4⋊C4semidirect
121. C4⋊Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#218662
#213942
C2×#19862
#139542
#80332
C2×#12852
#79042