Finite Group Databasebeta

← #115589

Subgroups — #115589

Order 512.

968 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic
2#21885C211yesmaximal, characteristic
3#21324C211yesmaximal, characteristic
4C2×#1987C211yesmaximal, characteristic
5C2×#1986C211yesmaximal, characteristic
6#8014C211yesmaximal, characteristic
7C2×#1283C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C22×C4⋊Q8C2211yescharacteristic
10#3209—22no—
11#3209—22no—
12#3209—22no—
13#3209C414yescomplemented
14C2×#494—21no—
15C2×#487—21no—
16C2×#462—22no—
17C2×#451—22no—
18C2×#431C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C2×C4×M16C2211yescharacteristic
21#2285C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1993—22no—
24#1990—22no—
25#1987—22no—
26#1987—22no—
27#1987—21no—
28#1986—22no—
29#1986—22no—
30#1981—21no—
31#1892—22no—
32#1803—22no—
33#1736—22no—
34C2×#340—22no—
35C2×#339—22no—
36#1294—22no—
37#1283—22no—
38C23×Q8—21no—
39C22×Q16—24no—
40C22×M16D412yes—
41C23×C8D412yes—
42#533—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533C2×C412yes—
49#533D412yes—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8C2×C412yes—
60C2×C4⋊Q8D411yescharacteristic
61C2×C4⋊Q8D411yescharacteristic
62C2×C4⋊Q8D412yes—
63C2×C4⋊Q8D412yes—
64C2×#125—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×#117—21no—
70C2×#117—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76#494—24no—
77#494—24no—
78#487—24no—
79#487—24no—
80#462—28no—
81#462—28no—
82#451—28no—
83#451—28no—
84#431—21no—
85#431—21no—
86#431C2×C412yes—
87#431D412yes—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×#103—24no—
96C2×#103—24no—
97C2×C22⋊C8D412yes—
98C4×M16—21no—
99C4×M16—21no—
100C4×M16—21no—
101C4×M16—21no—
102C2×#97—21no—
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105#384D412yes—
106#382—21no—
107#380—21no—
108#380—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4D412yes—
115C2×#95—21no—
116#340—24no—
117#340—28no—
118#340—28no—
119#339—28no—
120C22×Q8—22no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—41no—
126C22×Q8—41no—
127C22×Q8—41no—
128C22×Q8—84no—
129C23×C4—41no—
130C23×C4—41no—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C2×Q16—216no—
135C2×Q16—88no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16Q1614yescomplemented
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—44no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—21no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—21no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—41no—
164C4⋊Q8—41no—
165C4⋊Q8—41no—
166C4⋊Q8—41no—
167C4⋊Q8—41no—
168C4⋊Q8—41no—
169C4⋊Q8C22⋊C414yes—
170C4⋊Q8C22⋊C412yes—
171C4⋊Q8C22⋊C412yes—
172#125—22no—
173#125—22no—
174#125—41no—
175#125—41no—
176C22⋊Q8—22no—
177C22⋊Q8—22no—
178C22⋊Q8—22no—
179C22⋊Q8—22no—
180C22⋊Q8—41no—
181C22⋊Q8—41no—
182C22⋊Q8—41no—
183C22⋊Q8—41no—
184C4×Q8—22no—
185C4×Q8—22no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8—41no—
189C4×Q8—41no—
190C4×Q8—41no—
191C4×Q8—41no—
192#117—22no—
193#117—22no—
194#117—41no—
195#117—41no—
196#117—42no—
197#117—42no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C42—22no—
226C2×C42—41no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42—41no—
230C2×C42—41no—
231C2×C42C22×C411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42#4111yescharacteristic
240C2×C42#4111yescharacteristic
241C2×C42#4111yescharacteristic
242C2×C42#4111yescharacteristic
243C4⋊C8—22no—
244C4⋊C8—22no—
245C4⋊C8—41no—
246C4⋊C8—41no—
247C4⋊C8—41no—
248C4⋊C8—41no—
249#103—216no—
250#103—84no—
251C22⋊C8—22no—
252C22⋊C8C22⋊C414yes—
253#97—41no—
254#97—41no—
255C4×C8—22no—
256C4×C8—22no—
257C4×C8—41no—
258C4×C8—41no—
259#95—22no—
260#95—41no—
261#95—41no—
262#95—41no—
263C24#12711yescharacteristic
264C2×Q8—24no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—44no—
268C2×Q8—48no—
269C2×Q8—42no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—84no—
277C2×Q8—84no—
278C2×Q8—84no—
279C2×Q8—84no—
280C2×Q8—84no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—82no—
293C22×C4—84no—
294C22×C4—81no—
295C22×C4—81no—
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C4×D411yescharacteristic
298C22×C4C4×D411yescharacteristic
299C22×C4#12011yescharacteristic
300C22×C4#12011yescharacteristic
301C22×C4#12011yescharacteristic
302C22×C4#12011yescharacteristic
303C22×C4#12011yescharacteristic
304C22×C4#12011yescharacteristic
305C22×C4#12111yescharacteristic
306C22×C4#12111yescharacteristic
307C22×C4#12111yescharacteristic
308C22×C4#12111yescharacteristic
309C22×C4#12111yescharacteristic
310C22×C4#12111yescharacteristic
311C22×C4#12111yescharacteristic
312C22×C4#12111yescharacteristic
313C22×C4#12411yescharacteristic
314C22×C4C2×Q1612yes—
315C22×C4#13512yes—
316C22×C4#13712yes—
317C22×C4#13712yes—
318Q16—816no—
319M16—24no—
320M16—24no—
321M16—42no—
322M16—42no—
323C2×C8—24no—
324C2×C8—24no—
325C2×C8—44no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—44no—
329C2×C8—42no—
330C2×C8—44no—
331C2×C8—82no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—44no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—44no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—82no—
359C22⋊C4—42no—
360C22⋊C4—42no—
361C22⋊C4—81no—
362C22⋊C4—81no—
363C42—42no—
364C42—42no—
365C42—44no—
366C42—82no—
367C42—82no—
368C42C2×C22⋊C411yescharacteristic
369C42C2×C22⋊C411yescharacteristic
370C42C4×D411yescharacteristic
371C42C4×D411yescharacteristic
372C42#12011yescharacteristic
373C42#12011yescharacteristic
374C42#12111yescharacteristic
375C42#12111yescharacteristic
376C42#12111yescharacteristic
377C42#12111yescharacteristic
378C42#12311yescharacteristic
379C42#12311yescharacteristic
380C42#12311yescharacteristic
381C42#12311yescharacteristic
382C42#12411yescharacteristic
383C42#12711yescharacteristic
384C23—22no—
385C23—22no—
386C23#39011yescharacteristic
387C23#39211yescharacteristic
388C23#39211yescharacteristic
389C23#49411yescharacteristic
390C23#49511yescharacteristic
391C23#49711yescharacteristic
392C23#49711yescharacteristic
393Q8—44no—
394Q8—48no—
395Q8—44no—
396Q8—82no—
397Q8—84no—
398Q8—84no—
399Q8—84no—
400Q8—81no—
401Q8—81no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—84no—
420C2×C4—84no—
421C2×C4—82no—
422C2×C4—84no—
423C2×C4—84no—
424C2×C4—82no—
425C2×C4—84no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4#37911yescharacteristic
432C2×C4#38611yescharacteristic
433C2×C4#38611yescharacteristic
434C2×C4#38611yescharacteristic
435C2×C4#38611yescharacteristic
436C2×C4#39011yescharacteristic
437C2×C4#39211yescharacteristic
438C2×C4#39211yescharacteristic
439C2×C4#39411yescharacteristic
440C2×C4#39411yescharacteristic
441C2×C4#39411yescharacteristic
442C2×C4#39411yescharacteristic
443C2×C4C4×Q1612yes—
444C2×C4#44112yes—
445C2×C4#45112yes—
446C2×C4#45112yes—
447C2×C4#45212yes—
448C2×C4#45212yes—
449C2×C4#45512yes—
450C2×C4#45612yes—
451C2×C4#46212yes—
452C2×C4#46212yes—
453C2×C4#46412yes—
454C2×C4#46412yes—
455C2×C4#46712yes—
456C2×C4#47012yes—
457C8—44no—
458C8—44no—
459C8—82no—
460C22—21no—
461C22—21no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—21no—
467C22—21no—
468C22#143811yescharacteristic
469C22#196911yescharacteristic
470C22#197111yescharacteristic
471C22#200712yes—
472C22#200812yes—
473C22#201511yescharacteristic
474C22#201511yescharacteristic
475C22#201611yescharacteristic
476C22#201611yescharacteristic
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—84no—
484C4—82no—
485C4—84no—
486C4—82no—
487C4—162no—
488C4#187812yes—
489C4#187812yes—
490C4#188212yes—
491C4#200112yes—
492C4#200411yescharacteristic
493C4#200511yescharacteristic
494C4#205612yes—
495C2—21no—
496C2—21no—
497C2—21no—
498C2—21no—
499C2#1019812yes—
500C2#1019912yes—
501C2#1100411yescharacteristic
502C2#1101711yescharacteristic
503C2#1102411yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#127
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
13. #3209C4semidirect
141. C2×M16Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
#218852
#213242
C2×#19872
C2×#19862
#80142
C2×#12832