Finite Group Databasebeta

← #233413

Subgroups — #233413

Order 512.

1352 conjugacy classes of subgroups.

The subgroup lattice has 515 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic
2C2×#3170C211yesmaximal, characteristic
3C2×#3165C211yesmaximal, characteristic
4#21860C211yesmaximal, characteristic
5#21322C211yesmaximal, characteristic
6C2×#1990C211yesmaximal, characteristic
7#10318C212yesmaximal, complemented
8#10318C212yesmaximal, complemented
9#10313C212yesmaximal, complemented
10#10309C212yesmaximal, complemented
11#7848C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C2×#533C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic
15#3209—24no—
16#3209—24no—
17#3170—24no—
18#3170—24no—
19#3165—24no—
20#3165—24no—
21C2×#501—22no—
22C2×#500C2211yescharacteristic
23C2×#500C2212yes—
24C2×#498C2212yes—
25C2×#494C2211yescharacteristic
26C2×#494C2212yes—
27C2×#487C2211yescharacteristic
28C2×#487C2211yescharacteristic
29C2×#487C2212yes—
30C2×#467—22no—
31C2×#462—22no—
32C2×#451—22no—
33C2×C4×Q16—22no—
34C2×#431C2211yescharacteristic
35C2×#428—22no—
36C2×C4×M16C2211yescharacteristic
37C22×C4×C8C2211yescharacteristic
38C2×#384C2211yescharacteristic
39#1992—22no—
40#1990—22no—
41#1990C2212yes—
42#1990C2212yes—
43#1990C2212yes—
44#1990C2212yes—
45#1990C2212yes—
46#1956C2212yes—
47#1872—22no—
48#1803—22no—
49#1736—22no—
50C2×#340—22no—
51#1273C2212yes—
52#1273C2212yes—
53C23×Q8—21no—
54C22×Q16—22no—
55C22×Q16—24no—
56C22×M16D412yes—
57C23×C8D412yes—
58#533—22no—
59#533—22no—
60#533—22no—
61#533—22no—
62#533—22no—
63#533—22no—
64#533D414yes—
65#533D414yes—
66C2×C4⋊Q8—21no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8—21no—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8D414yes—
73C2×C4⋊Q8C2311yescharacteristic
74C2×C4⋊Q8C2311yescharacteristic
75C2×C4⋊Q8C2311yescharacteristic
76C2×C4⋊Q8C2312yes—
77C2×C4⋊Q8C2311yescharacteristic
78C2×#125—21no—
79C2×#125—21no—
80C2×C22⋊Q8—21no—
81C2×C22⋊Q8—21no—
82C2×C22⋊Q8—21no—
83C2×C22⋊Q8—21no—
84C2×C4×Q8—21no—
85C2×C4×Q8—21no—
86C2×C4×Q8—21no—
87C2×C4×Q8—21no—
88C2×#117—21no—
89C2×#117—21no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C42C2311yescharacteristic
96#501—216no—
97#500—24no—
98#500—24no—
99#500—22no—
100#500—22no—
101#498—24no—
102#498—24no—
103#494—22no—
104#494—22no—
105#494—24no—
106#494—24no—
107#487—22no—
108#487—22no—
109#487—22no—
110#487—22no—
111#487—24no—
112#487—24no—
113#467—216no—
114#462—216no—
115#451—216no—
116C4×Q16—216no—
117#431—22no—
118#431—22no—
119#428—216no—
120C2×#107—24no—
121C2×#107—22no—
122C2×#106—24no—
123C2×C4⋊C8D412yes—
124C2×#103—22no—
125C2×#103—22no—
126C2×#103—24no—
127C2×#103—24no—
128C2×C22⋊C8D412yes—
129C4×M16—22no—
130C4×M16—22no—
131C2×#97D412yes—
132C2×C4×C8—22no—
133C2×C4×C8—22no—
134C2×C4×C8D412yes—
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137C2×C4×C8C2312yes—
138C2×C4×C8C2312yes—
139#384—21no—
140#384—21no—
141#384—21no—
142#384C2312yes—
143#340—24no—
144#340—24no—
145#340—24no—
146#340—24no—
147#340—24no—
148C22×Q8—24no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—22no—
154C22×Q8—42no—
155C22×Q8—41no—
156C22×Q8—82no—
157C22×Q8—82no—
158C23×C4—41no—
159C23×C4—41no—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C2×Q16—28no—
164C2×Q16—216no—
165C2×Q16—44no—
166C2×Q16—44no—
167C2×Q16—84no—
168C2×Q16—84no—
169C2×M16—24no—
170C2×M16—24no—
171C2×M16Q1618yescomplemented
172C22×C8—24no—
173C22×C8—24no—
174C22×C8—42no—
175C22×C8C2×D412yes—
176C22×C8C2×D414yes—
177C22×C8C2×D414yes—
178C22×C8C2×D412yes—
179C4⋊Q8—24no—
180C4⋊Q8—24no—
181C4⋊Q8—24no—
182C4⋊Q8—22no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—22no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—22no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—42no—
198C4⋊Q8—42no—
199C4⋊Q8—42no—
200C4⋊Q8C2×D414yes—
201C4⋊Q8C2×D412yes—
202C4⋊Q8C2×D412yes—
203C4⋊Q8C2×D412yes—
204C4⋊Q8C2×D412yes—
205#125—24no—
206#125—24no—
207#125—42no—
208#125—42no—
209C22⋊Q8—24no—
210C22⋊Q8—24no—
211C22⋊Q8—24no—
212C22⋊Q8—24no—
213C22⋊Q8—42no—
214C22⋊Q8—42no—
215C22⋊Q8—42no—
216C22⋊Q8—42no—
217C4×Q8—24no—
218C4×Q8—24no—
219C4×Q8—24no—
220C4×Q8—24no—
221C4×Q8—42no—
222C4×Q8—42no—
223C4×Q8—42no—
224C4×Q8—42no—
225#117—24no—
226#117—24no—
227#117—42no—
228#117—42no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—41no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—82no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—41no—
260C2×C42—41no—
261C2×C42—41no—
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42#4112yes—
269C2×C42C2411yescharacteristic
270#107—216no—
271#107—28no—
272#106—216no—
273C4⋊C8—24no—
274#103—28no—
275#103—28no—
276#103—216no—
277#103—84no—
278C22⋊C8—24no—
279#97—24no—
280C4×C8—24no—
281C4×C8—21no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—22no—
286C4×C8—22no—
287C4×C8—22no—
288C4×C8—22no—
289C24#12711yescharacteristic
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—24no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—42no—
300C2×Q8—44no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—44no—
308C2×Q8—42no—
309C2×Q8—44no—
310C2×Q8—42no—
311C2×Q8—88no—
312C2×Q8—82no—
313C2×Q8—82no—
314C2×Q8—88no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—44no—
325C22×C4—42no—
326C22×C4—82no—
327C22×C4—82no—
328C22×C4—82no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4#12012yes—
332C22×C4#12012yes—
333C22×C4#12012yes—
334C22×C4#12112yes—
335C22×C4#12112yes—
336C22×C4#12112yes—
337C22×C4#12711yescharacteristic
338C22×C4#12711yescharacteristic
339C22×C4#12711yescharacteristic
340C22×C4C2×Q1614yes—
341C22×C4#13714yes—
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345Q16—416no—
346Q16—816no—
347M16—216no—
348C2×C8—22no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—24no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—48no—
359C2×C8C2×Q1618yes—
360C2×C8#13714yes—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—42no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—42no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—82no—
391C4⋊C4—82no—
392C4⋊C4—82no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C42—44no—
396C42—44no—
397C42#12012yes—
398C42#12012yes—
399C42#12012yes—
400C42#12112yes—
401C42#12112yes—
402C42#12112yes—
403C42C22×D411yescharacteristic
404C42C22×D411yescharacteristic
405C42C22×D411yescharacteristic
406C42C2×#4111yescharacteristic
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23#39212yes—
412C23#49412yes—
413C23#49712yes—
414C23C2×#12711yescharacteristic
415Q8—44no—
416Q8—44no—
417Q8—44no—
418Q8—48no—
419Q8—44no—
420Q8—88no—
421Q8—82no—
422Q8—82no—
423Q8—82no—
424Q8—82no—
425Q8—84no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—88no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—82no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—82no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—164no—
454C2×C4#39212yes—
455C2×C4#39212yes—
456C2×C4#39212yes—
457C2×C4#45118yes—
458C2×C4#46218yes—
459C2×C4#49412yes—
460C2×C4#49712yes—
461C2×C4C2×#12011yescharacteristic
462C2×C4C2×#12011yescharacteristic
463C2×C4C2×#12011yescharacteristic
464C2×C4C2×#12111yescharacteristic
465C2×C4C2×#12111yescharacteristic
466C2×C4C2×#12111yescharacteristic
467C2×C4C22×Q1612yes—
468C2×C4C2×#13712yes—
469C2×C4C2×#13712yes—
470C2×C4#57512yes—
471C2×C4#57612yes—
472C2×C4#57712yes—
473C8—28no—
474C8—24no—
475C8—42no—
476C8—44no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22#201614yes—
484C22C2×#39211yescharacteristic
485C22C2×#49411yescharacteristic
486C22C2×#49711yescharacteristic
487C22#315211yescharacteristic
488C22#315311yescharacteristic
489C22#315411yescharacteristic
490C22#315511yescharacteristic
491C4—22no—
492C4—22no—
493C4—84no—
494C4—82no—
495C4—82no—
496C4—82no—
497C4—82no—
498C4—162no—
499C4#201614yes—
500C4C2×#45112yes—
501C4#300712yes—
502C4#301212yes—
503C4C2×#46212yes—
504C4#303712yes—
505C4#304512yes—
506C2—24no—
507C2#1963311yescharacteristic
508C2#1963411yescharacteristic
509C2#1968011yescharacteristic
510C2#1968211yescharacteristic
511C2C2×#201611yescharacteristic
512C2#2342611yescharacteristic
513C2#2342911yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#127
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
7. #10318C2semidirect
8. #10318C2semidirect
9. #10313C2semidirect
10. #10309C2semidirect
171. 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×#31702
C2×#31652
#218602
#213222
C2×#19902
#103182
#103132
#103092
#78482