Finite Group Databasebeta

← #233457

Subgroups — #233457

Order 512.

1312 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C211yesmaximal, characteristic
2C2×#3209C211yesmaximal, characteristic
3C2×#3160C211yesmaximal, characteristic
4#21860C211yesmaximal, characteristic
5#21860C211yesmaximal, characteristic
6C2×#1990C211yesmaximal, characteristic
7#10318C212yesmaximal, complemented
8#10314C212yesmaximal, complemented
9#10313C212yesmaximal, complemented
10#10309C212yesmaximal, complemented
11#7849C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C2×#533C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic
15#3230—24no—
16#3230—24no—
17#3209—24no—
18#3209—24no—
19#3160—24no—
20#3160—24no—
21C2×#500C2211yescharacteristic
22C2×#500C2212yes—
23C2×#500C2211yescharacteristic
24C2×#499—22no—
25C2×#498C2211yescharacteristic
26C2×#498C2211yescharacteristic
27C2×#494C2212yes—
28C2×#487C2212yes—
29C2×#487C2212yes—
30C2×#484—22no—
31C2×#477—22no—
32C2×#462—22no—
33C2×#451—22no—
34C2×#431C2211yescharacteristic
35C2×#431C2211yescharacteristic
36C2×#427—22no—
37C22×C4×C8C2211yescharacteristic
38C2×#384C2211yescharacteristic
39#1990—22no—
40#1990C2212yes—
41#1990C2212yes—
42#1990C2212yes—
43#1990C2212yes—
44#1956C2212yes—
45#1956C2212yes—
46#1872—22no—
47#1872—22no—
48#1803—22no—
49#1803—22no—
50C2×#340—22no—
51#1273C2212yes—
52#1273C2212yes—
53C23×Q8—21no—
54C22×Q16—24no—
55C23×C8D412yes—
56#533—22no—
57#533—22no—
58#533—22no—
59#533—22no—
60#533—22no—
61#533—22no—
62#533D414yes—
63#533D414yes—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—22no—
66C2×C4⋊Q8—21no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8D414yes—
71C2×C4⋊Q8C2312yes—
72C2×C4⋊Q8C2311yescharacteristic
73C2×C4⋊Q8C2311yescharacteristic
74C2×C4⋊Q8C2311yescharacteristic
75C2×C4⋊Q8C2311yescharacteristic
76C2×#125—21no—
77C2×#125—21no—
78C2×C22⋊Q8—21no—
79C2×C22⋊Q8—21no—
80C2×C22⋊Q8—21no—
81C2×C22⋊Q8—21no—
82C2×C4×Q8—21no—
83C2×C4×Q8—21no—
84C2×C4×Q8—21no—
85C2×C4×Q8—21no—
86C2×#117—21no—
87C2×#117—21no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—21no—
93C22×C42C2311yescharacteristic
94#500—22no—
95#500—22no—
96#500—22no—
97#500—22no—
98#500—24no—
99#500—24no—
100#499—216no—
101#498—22no—
102#498—22no—
103#498—22no—
104#498—22no—
105#494—24no—
106#494—24no—
107#487—24no—
108#487—24no—
109#487—24no—
110#487—24no—
111#484—216no—
112#477—216no—
113#462—216no—
114#451—216no—
115#431—22no—
116#431—22no—
117#431—22no—
118#431—22no—
119#427—216no—
120C2×#107—24no—
121C2×#107—22no—
122C2×#107—22no—
123C2×#106—22no—
124C2×#106—22no—
125C2×C4⋊C8D412yes—
126C2×C4⋊C8D412yes—
127C2×#103—24no—
128C2×#103—24no—
129C2×#103—24no—
130C2×#103—24no—
131C2×C22⋊C8D412yes—
132C2×C22⋊C8D412yes—
133C2×C4×C8—22no—
134C2×C4×C8—22no—
135C2×C4×C8D412yes—
136C2×C4×C8C2311yescharacteristic
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2312yes—
139C2×C4×C8C2312yes—
140#384—21no—
141#384—21no—
142#384—21no—
143#384C2312yes—
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—216no—
164C2×Q16—84no—
165C2×Q16—84no—
166C22×C8—24no—
167C22×C8—24no—
168C22×C8Q1618yescomplemented
169C22×C8C2×D414yes—
170C22×C8C2×D414yes—
171C22×C8C2×D412yes—
172C22×C8C2×D412yes—
173C4⋊Q8—24no—
174C4⋊Q8—24no—
175C4⋊Q8—24no—
176C4⋊Q8—22no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—22no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—22no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—42no—
192C4⋊Q8—42no—
193C4⋊Q8—42no—
194C4⋊Q8C2×D414yes—
195C4⋊Q8C2×D412yes—
196C4⋊Q8C2×D412yes—
197C4⋊Q8C2×D412yes—
198C4⋊Q8C2×D412yes—
199#125—24no—
200#125—24no—
201#125—42no—
202#125—42no—
203C22⋊Q8—24no—
204C22⋊Q8—24no—
205C22⋊Q8—24no—
206C22⋊Q8—24no—
207C22⋊Q8—42no—
208C22⋊Q8—42no—
209C22⋊Q8—42no—
210C22⋊Q8—42no—
211C4×Q8—24no—
212C4×Q8—24no—
213C4×Q8—24no—
214C4×Q8—24no—
215C4×Q8—42no—
216C4×Q8—42no—
217C4×Q8—42no—
218C4×Q8—42no—
219#117—24no—
220#117—24no—
221#117—42no—
222#117—42no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—82no—
251C2×C4⋊C4—82no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C42—41no—
255C2×C42—41no—
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D412yes—
260C2×C42C2×D412yes—
261C2×C42C2×D412yes—
262C2×C42#4112yes—
263C2×C42C2411yescharacteristic
264#107—28no—
265#107—28no—
266#107—216no—
267#106—28no—
268#106—28no—
269C4⋊C8—24no—
270C4⋊C8—24no—
271#103—216no—
272#103—216no—
273#103—84no—
274#103—84no—
275C22⋊C8—24no—
276C22⋊C8—24no—
277C4×C8—24no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—22no—
283C4×C8—22no—
284C4×C8—22no—
285C4×C8—22no—
286C24#12711yescharacteristic
287C2×Q8—22no—
288C2×Q8—24no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—42no—
297C2×Q8—44no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—44no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—44no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—88no—
309C2×Q8—82no—
310C2×Q8—82no—
311C2×Q8—88no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—44no—
322C22×C4—42no—
323C22×C4—82no—
324C22×C4—82no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4—82no—
328C22×C4#12012yes—
329C22×C4#12012yes—
330C22×C4#12012yes—
331C22×C4#12112yes—
332C22×C4#12112yes—
333C22×C4#12112yes—
334C22×C4#12711yescharacteristic
335C22×C4#12711yescharacteristic
336C22×C4#12711yescharacteristic
337C22×C4C2×Q1614yes—
338C22×C4#13714yes—
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342Q16—816no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—216no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—24no—
354C2×C8C2×Q1614yes—
355C2×C8C2×Q1614yes—
356C4⋊C4—22no—
357C4⋊C4—24no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—42no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—42no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—82no—
386C4⋊C4—82no—
387C4⋊C4—82no—
388C22⋊C4—44no—
389C22⋊C4—44no—
390C42—44no—
391C42—44no—
392C42#12012yes—
393C42#12012yes—
394C42#12012yes—
395C42#12112yes—
396C42#12112yes—
397C42#12112yes—
398C42C22×D411yescharacteristic
399C42C22×D411yescharacteristic
400C42C22×D411yescharacteristic
401C42C2×#4111yescharacteristic
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23#39212yes—
407C23#49412yes—
408C23#49712yes—
409C23C2×#12711yescharacteristic
410Q8—44no—
411Q8—48no—
412Q8—44no—
413Q8—44no—
414Q8—44no—
415Q8—88no—
416Q8—82no—
417Q8—82no—
418Q8—82no—
419Q8—84no—
420Q8—82no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—88no—
435C2×C4—82no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—82no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—164no—
449C2×C4#39212yes—
450C2×C4#39212yes—
451C2×C4#39212yes—
452C2×C4#45118yes—
453C2×C4#46218yes—
454C2×C4#49412yes—
455C2×C4#49712yes—
456C2×C4C2×#12011yescharacteristic
457C2×C4C2×#12011yescharacteristic
458C2×C4C2×#12011yescharacteristic
459C2×C4C2×#12111yescharacteristic
460C2×C4C2×#12111yescharacteristic
461C2×C4C2×#12111yescharacteristic
462C2×C4C22×Q1612yes—
463C2×C4C2×#13712yes—
464C2×C4#57612yes—
465C2×C4#57712yes—
466C2×C4#57712yes—
467C2×C4#57812yes—
468C8—24no—
469C8—24no—
470C8—44no—
471C8—44no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22#201614yes—
479C22C2×#39211yescharacteristic
480C22C2×#49411yescharacteristic
481C22C2×#49711yescharacteristic
482C22#315411yescharacteristic
483C22#315411yescharacteristic
484C22#315511yescharacteristic
485C22#315611yescharacteristic
486C4—22no—
487C4—22no—
488C4—84no—
489C4—82no—
490C4—82no—
491C4—82no—
492C4—82no—
493C4—162no—
494C4#201614yes—
495C4C2×#45112yes—
496C4#301212yes—
497C4#301312yes—
498C4C2×#46212yes—
499C4#304512yes—
500C4#304612yes—
501C2—24no—
502C2#1966311yescharacteristic
503C2#1966411yescharacteristic
504C2#1970511yescharacteristic
505C2#1970611yescharacteristic
506C2C2×#201611yescharacteristic
507C2#2342611yescharacteristic
508C2#2342711yescharacteristic
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. #10314C2semidirect
9. #10313C2semidirect
10. #10309C2semidirect
168. C22×C8Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#32302
C2×#32092
C2×#31602
#218602
C2×#19902
#103182
#103142
#103132
#103092
#78492