Finite Group Databasebeta

← #122166

Subgroups — #122166

Order 512.

772 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2605C211yesmaximal, characteristic
2#10851C211yesmaximal, characteristic
3C2×#1300C211yesmaximal, characteristic
4#8022C211yesmaximal, characteristic
5#8018C211yesmaximal, characteristic
6C2×#1288C211yesmaximal, characteristic
7C2×#1285C211yesmaximal, characteristic
8C2×#432C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×C4×M16C2211yescharacteristic
11#2605—22no—
12#2605—22no—
13#2605—22no—
14#2605C414yescomplemented
15C2×#400—21no—
16C2×#395—21no—
17C2×#391—21no—
18C2×#391—21no—
19C2×#387—21no—
20C2×#387—21no—
21#2285C2211yescharacteristic
22C2×#383C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×#376C2211yescharacteristic
25C2×#339—21no—
26C2×#339—21no—
27#1300—22no—
28#1294—21no—
29#1294—21no—
30#1294—21no—
31#1294—21no—
32#1288—22no—
33#1285—22no—
34C23×Q8—21no—
35C22×M16—21no—
36C2×#117—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C42—21no—
42C22×C42D411yescharacteristic
43C22×C42D411yescharacteristic
44C22×C42C2311yescharacteristic
45#432—21no—
46#432—21no—
47#432—21no—
48#432—21no—
49#432—21no—
50#432—21no—
51#432Q812yescomplemented
52#432Q812yescomplemented
53C2×C4⋊C8—21no—
54C2×C4⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C4×M16—21no—
58C4×M16—21no—
59C4×M16—21no—
60C4×M16Q812yescomplemented
61C2×#97—21no—
62C2×#97—21no—
63C2×#97—21no—
64C2×C4×C8—21no—
65#400—24no—
66#400—24no—
67#395—24no—
68#395—24no—
69#391—24no—
70#391—24no—
71#391—24no—
72#391—24no—
73#387—24no—
74#387—24no—
75#387—24no—
76#387—24no—
77#384Q812yescomplemented
78#383—21no—
79#383—21no—
80#383C2×C412yes—
81#383D412yes—
82#382—21no—
83#380—21no—
84#380—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4C2×C412yes—
90C4×C4⋊C4D412yes—
91#376—21no—
92#376—21no—
93#376C2×C412yes—
94#376D412yes—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95D411yescharacteristic
104C2×#95D411yescharacteristic
105C2×#95D411yescharacteristic
106C2×#95D411yescharacteristic
107#339—42no—
108#339—42no—
109C22×Q8—24no—
110C22×Q8—44no—
111C22×Q8—44no—
112C22×Q8—81no—
113C22×Q8—81no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C2×M16—22no—
124C2×M16—22no—
125C2×M16—22no—
126C2×M16—41no—
127C2×M16—41no—
128C2×M16—41no—
129C22×C8—41no—
130C22×C8—41no—
131C22×C8—41no—
132#117—41no—
133#117—41no—
134#117—41no—
135#117—41no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—81no—
148C2×C4⋊C4—81no—
149C2×C22⋊C4—41no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—21no—
153C2×C42—21no—
154C2×C42—44no—
155C2×C42—41no—
156C2×C42—41no—
157C2×C42—41no—
158C2×C42—41no—
159C2×C42—41no—
160C2×C42C22⋊C412yes—
161C2×C42C22⋊C412yes—
162C2×C42C22×C411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×Q811yescharacteristic
167C2×C42C2×Q811yescharacteristic
168C2×C42C2×Q811yescharacteristic
169C2×C42C2×Q811yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—41no—
177C4⋊C8—41no—
178C22⋊C8—22no—
179C22⋊C8—22no—
180C22⋊C8—41no—
181C22⋊C8—41no—
182#97—22no—
183#97—22no—
184#97—22no—
185#97—41no—
186#97—41no—
187#97—41no—
188C4×C8—22no—
189C4×C8—41no—
190#95—24no—
191#95—24no—
192#95—24no—
193#95—24no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—21no—
203#95—21no—
204#95—21no—
205#95—21no—
206#95—41no—
207#95—41no—
208#95C22⋊C412yes—
209#95C22⋊C412yes—
210#95C22⋊C412yes—
211#95C22⋊C412yes—
212C24#12011yescharacteristic
213C2×Q8—48no—
214C2×Q8—48no—
215C2×Q8—84no—
216C2×Q8—88no—
217C2×Q8—88no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—42no—
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4#12011yescharacteristic
259C22×C4#12011yescharacteristic
260C22×C4#12011yescharacteristic
261C22×C4C22⋊Q811yescharacteristic
262C22×C4C22⋊Q811yescharacteristic
263C22×C4C22⋊Q811yescharacteristic
264C22×C4C22⋊Q811yescharacteristic
265C22×C4C22⋊Q811yescharacteristic
266C22×C4C22⋊Q811yescharacteristic
267C22×C4#12411yescharacteristic
268C22×C4#12411yescharacteristic
269C22×C4#12411yescharacteristic
270C22×C4C4⋊Q811yescharacteristic
271C22×C4C4⋊Q811yescharacteristic
272C22×C4C4⋊Q811yescharacteristic
273M16—24no—
274M16—42no—
275M16—42no—
276M16—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—82no—
283C2×C8—82no—
284C4⋊C4—48no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—48no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—82no—
291C4⋊C4—82no—
292C4⋊C4—84no—
293C22⋊C4—81no—
294C22⋊C4—81no—
295C42—42no—
296C42—42no—
297C42—44no—
298C42—44no—
299C42—81no—
300C42—81no—
301C42—81no—
302C42—81no—
303C42C4×Q811yescharacteristic
304C42C4×Q811yescharacteristic
305C42C4×Q811yescharacteristic
306C42C4×Q811yescharacteristic
307C42C22⋊Q811yescharacteristic
308C42C22⋊Q811yescharacteristic
309C42C22⋊Q811yescharacteristic
310C42C22⋊Q811yescharacteristic
311C42C22⋊Q811yescharacteristic
312C42C22⋊Q811yescharacteristic
313C42#12411yescharacteristic
314C42#12411yescharacteristic
315C42#12411yescharacteristic
316C42C4⋊Q811yescharacteristic
317C42C4⋊Q811yescharacteristic
318C42C4⋊Q811yescharacteristic
319C23—22no—
320C23—22no—
321C23#37911yescharacteristic
322C23#39311yescharacteristic
323C23#39511yescharacteristic
324C23#45311yescharacteristic
325C23#45411yescharacteristic
326C23#45511yescharacteristic
327C23#45611yescharacteristic
328Q8—816no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4#39111yescharacteristic
387C2×C4#39111yescharacteristic
388C2×C4#39111yescharacteristic
389C2×C4#39111yescharacteristic
390C2×C4#39111yescharacteristic
391C2×C4#39111yescharacteristic
392C2×C4#39311yescharacteristic
393C2×C4#39311yescharacteristic
394C2×C4#39311yescharacteristic
395C2×C4#39511yescharacteristic
396C2×C4#39511yescharacteristic
397C2×C4#39511yescharacteristic
398C8—44no—
399C8—82no—
400C8—82no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22#144611yescharacteristic
410C22#179211yescharacteristic
411C22#179311yescharacteristic
412C22#202711yescharacteristic
413C22#202811yescharacteristic
414C22#203112yes—
415C22#203112yes—
416C22#206011yescharacteristic
417C22#206111yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—44no—
431C4—44no—
432C4—82no—
433C4—82no—
434C4—82no—
435C4—82no—
436C4—162no—
437C2—21no—
438C2—21no—
439C2—21no—
440C2—21no—
441C2#1037812yes—
442C2#1037912yes—
443C2#1138711yescharacteristic
444C2#1138711yescharacteristic
445C2#1139211yescharacteristic
subgroupG/H
center, socleC23#393
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
14. #2605C4semidirect
51. #432Q8semidirect
52. #432Q8semidirect
60. C4×M16Q8semidirect
77. #384Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26052
#108512
C2×#13002
#80222
#80182
C2×#12882
C2×#12852