Finite Group Databasebeta

← #115472

Subgroups — #115472

Order 512.

1004 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic
2C2×#1997C211yesmaximal, characteristic
3#21322C211yesmaximal, characteristic
4C2×#1956C211yesmaximal, characteristic
5C2×#1493C211yesmaximal, characteristic
6#8043C211yesmaximal, characteristic
7C2×#1278C211yesmaximal, 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
19C2×#423C2211yescharacteristic
20C22×#103—22no—
21C2×C4×M16C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23C2×#384C2211yescharacteristic
24#1997—22no—
25#1997—22no—
26#1992—22no—
27#1990—21no—
28#1990—21no—
29#1956—22no—
30#1956—22no—
31C2×#343—22no—
32#1736—22no—
33#1493—22no—
34#1493—22no—
35C2×Q8⋊C8—22no—
36#1296—22no—
37#1278—22no—
38C23×Q8D412yes—
39C22×Q16—24no—
40C22×M16—21no—
41C22×M16D412yes—
42C23×C8—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533C2×C412yes—
50#533D412yes—
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⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8C2×C412yes—
62C2×C4⋊Q8D412yes—
63C2×C4⋊Q8D411yescharacteristic
64C2×C4⋊Q8D411yescharacteristic
65C2×#125—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×#117—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C42C2311yescharacteristic
77#494—24no—
78#494—24no—
79#487—24no—
80#487—24no—
81#462—28no—
82#462—28no—
83#451—28no—
84#451—28no—
85#431—21no—
86#431—21no—
87#431C2×C412yes—
88#431D412yes—
89#423—21no—
90#423—21no—
91#423—21no—
92#423D412yes—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8D412yes—
96C2×#103—24no—
97C2×#103—28no—
98C2×#103—48no—
99C2×C22⋊C8—21no—
100C4×M16—21no—
101C4×M16—21no—
102C4×M16—21no—
103C4×M16—21no—
104C2×#97—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8D412yes—
113#384—21no—
114#384—21no—
115#384—21no—
116#384D412yes—
117#343—28no—
118#340—24no—
119#340—24no—
120Q8⋊C8—28no—
121C22×Q8—22no—
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—21no—
127C22×Q8—22no—
128C22×Q8—44no—
129C22×Q8—44no—
130C22×Q8C22⋊C414yes—
131C23×C4—22no—
132C23×C4—41no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4#4111yescharacteristic
136C2×Q16—216no—
137C2×Q16—84no—
138C2×Q16—84no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—41no—
146C2×M16—42no—
147C2×M16—42no—
148C2×M16Q1614yescomplemented
149C22×C8—24no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—42no—
154C22×C8—41no—
155C22×C8—41no—
156C22×C8—41no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—21no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—41no—
172C4⋊Q8—41no—
173C4⋊Q8—41no—
174C4⋊Q8—41no—
175C4⋊Q8—41no—
176C4⋊Q8—41no—
177C4⋊Q8—41no—
178C4⋊Q8—41no—
179C4⋊Q8C22⋊C412yes—
180C4⋊Q8C22⋊C412yes—
181#125—22no—
182#125—22no—
183#125—41no—
184#125—41no—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8—41no—
190C22⋊Q8—41no—
191C22⋊Q8—41no—
192C22⋊Q8—41no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8—41no—
198C4×Q8—41no—
199C4×Q8—41no—
200C4×Q8—41no—
201#117—22no—
202#117—22no—
203#117—41no—
204#117—41no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
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×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—82no—
233C2×C22⋊C4—41no—
234C2×C42—41no—
235C2×C42C22×C411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42#4111yescharacteristic
245C2×C42#4111yescharacteristic
246C2×C42#4111yescharacteristic
247C4⋊C8—22no—
248C4⋊C8—22no—
249C4⋊C8—22no—
250C4⋊C8—41no—
251C4⋊C8—41no—
252C4⋊C8C22⋊C414yes—
253#103—216no—
254#103—416no—
255C22⋊C8—22no—
256C22⋊C8—22no—
257#97—41no—
258#97—41no—
259C4×C8—22no—
260C4×C8—22no—
261C4×C8—22no—
262C4×C8—22no—
263C4×C8—41no—
264C4×C8—41no—
265C4×C8—41no—
266C4×C8—41no—
267C24#12111yescharacteristic
268C2×Q8—24no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—24no—
272C2×Q8—48no—
273C2×Q8—44no—
274C2×Q8—44no—
275C2×Q8—48no—
276C2×Q8—42no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—44no—
282C2×Q8—42no—
283C2×Q8—44no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—84no—
287C2×Q8—84no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C4×D411yescharacteristic
305C22×C4C4×D411yescharacteristic
306C22×C4#12011yescharacteristic
307C22×C4#12011yescharacteristic
308C22×C4#12011yescharacteristic
309C22×C4#12011yescharacteristic
310C22×C4#12011yescharacteristic
311C22×C4#12111yescharacteristic
312C22×C4#12111yescharacteristic
313C22×C4#12111yescharacteristic
314C22×C4#12111yescharacteristic
315C22×C4#12111yescharacteristic
316C22×C4#12311yescharacteristic
317C22×C4#12311yescharacteristic
318C22×C4#12311yescharacteristic
319C22×C4#12411yescharacteristic
320C22×C4#12711yescharacteristic
321C22×C4C2×Q1612yes—
322C22×C4#13512yes—
323C22×C4#13712yes—
324C22×C4#13712yes—
325Q16—816no—
326M16—24no—
327M16—24no—
328M16—42no—
329M16—42no—
330M16—44no—
331M16—82no—
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—42no—
336C2×C8—44no—
337C2×C8—44no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—81no—
345C2×C8—81no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—44no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—44no—
368C4⋊C4—82no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4—81no—
372C4⋊C4—81no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C42—42no—
376C42—42no—
377C42C2×C22⋊C411yescharacteristic
378C42C2×C22⋊C411yescharacteristic
379C42C4×D411yescharacteristic
380C42C4×D411yescharacteristic
381C42#12011yescharacteristic
382C42#12011yescharacteristic
383C42#12011yescharacteristic
384C42#12111yescharacteristic
385C42#12111yescharacteristic
386C42#12111yescharacteristic
387C42#12111yescharacteristic
388C42#12111yescharacteristic
389C42#12111yescharacteristic
390C42#12311yescharacteristic
391C42#12411yescharacteristic
392C42#12711yescharacteristic
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23#38611yescharacteristic
398C23#39211yescharacteristic
399C23#39411yescharacteristic
400C23#46212yes—
401C23#46412yes—
402Q8—48no—
403Q8—44no—
404Q8—48no—
405Q8—44no—
406Q8—84no—
407Q8—82no—
408Q8—82no—
409Q8—84no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—84no—
426C2×C4—82no—
427C2×C4—81no—
428C2×C4—82no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—84no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—164no—
437C2×C4#37911yescharacteristic
438C2×C4#38611yescharacteristic
439C2×C4#38611yescharacteristic
440C2×C4#38611yescharacteristic
441C2×C4#39011yescharacteristic
442C2×C4#39011yescharacteristic
443C2×C4#39211yescharacteristic
444C2×C4#39211yescharacteristic
445C2×C4#39211yescharacteristic
446C2×C4#39411yescharacteristic
447C2×C4#39411yescharacteristic
448C2×C4#39411yescharacteristic
449C2×C4C4×Q1612yes—
450C2×C4#44112yes—
451C2×C4#45112yes—
452C2×C4#45112yes—
453C2×C4#45212yes—
454C2×C4#45212yes—
455C2×C4#45312yes—
456C2×C4#45512yes—
457C2×C4#46212yes—
458C2×C4#46412yes—
459C2×C4#46712yes—
460C2×C4#47012yes—
461C2×C4#47112yes—
462C2×C4#47312yes—
463C2×C4#49411yescharacteristic
464C2×C4#49511yescharacteristic
465C2×C4#49711yescharacteristic
466C2×C4#49711yescharacteristic
467C8—44no—
468C8—44no—
469C8—44no—
470C8—82no—
471C22—21no—
472C22—21no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—21no—
478C22—21no—
479C22#143811yescharacteristic
480C22#187812yes—
481C22#200211yescharacteristic
482C22#200411yescharacteristic
483C22#201511yescharacteristic
484C22#201611yescharacteristic
485C22#201911yescharacteristic
486C22#202011yescharacteristic
487C22#205612yes—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—21no—
492C4—82no—
493C4—84no—
494C4—82no—
495C4—162no—
496C4#179212yes—
497C4#187812yes—
498C4#190812yes—
499C4#196911yescharacteristic
500C4#197111yescharacteristic
501C4#200112yes—
502C4#201511yescharacteristic
503C4#201611yescharacteristic
504C4#203712yes—
505C4#205212yes—
506C2—22no—
507C2—22no—
508C2#1014711yescharacteristic
509C2#1015511yescharacteristic
510C2#1016311yescharacteristic
511C2#1017111yescharacteristic
512C2#1099711yescharacteristic
513C2#1102111yescharacteristic
514C2#1102411yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
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
148. C2×M16Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
C2×#19972
#213222
C2×#19562
C2×#14932
#80432
C2×#12782