Finite Group Databasebeta

← #194144

Subgroups — #194144

Order 512.

1294 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3079C211yesmaximal, characteristic
2#34121C211yesmaximal, characteristic
3#21804C211yesmaximal, characteristic, complemented
4C2×#1987C211yesmaximal, characteristic, complemented
5C2×#1477C211yesmaximal, characteristic
6#8976C214yesmaximal, complemented
7#8974C214yesmaximal, complemented
8#8045C211yesmaximal, characteristic, complemented
9#7901C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12#3079—24no—
13#3079C418yescomplemented
14C2×#477C2212yes—
15C2×#475—21no—
16C2×#476C2212yes—
17C2×#474—21no—
18C2×#431C2211yescharacteristic
19C2×#428—21no—
20C2×#423C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C2×#417—21no—
23#2357—21no—
24#2356—21no—
25#2355C2212yes—
26#2354C2212yes—
27#2353—22no—
28#2352—22no—
29#2298—22no—
30#2297—21no—
31#2285C2211yescharacteristic
32#2283—21no—
33C4×#117—21no—
34C4×#117—21no—
35C2×C4×C4⋊C4C2211yescharacteristic
36#1987—21no—
37#1987—21no—
38#1987C2214yes—
39#1981—22no—
40#1956C2214yes—
41#1787—22no—
42C2×#340—22no—
43#1477—24no—
44C2×Q8⋊C8C2212yes—
45C2×D4⋊C8C2212yes—
46#1296C2214yes—
47#1279C2214yes—
48C22×#41D411yescharacteristic
49C23×Q8—21no—
50C22×M16—21no—
51C23×C8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8C2×C414yes—
59C2×C4⋊Q8C2312yes—
60C4×#41—22no—
61C4×#41—22no—
62C4×#41C2×C414yes—
63C4×#41D414yes—
64C2×C4×Q8D411yescharacteristic
65C2×C4×Q8C2311yescharacteristic
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4D411yescharacteristic
69C2×C4×D4C2311yescharacteristic
70C2×#117—21no—
71C2×#117—22no—
72C2×#117—21no—
73C2×#117D411yescharacteristic
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4D412yes—
77C22×C42—21no—
78C22×C42—21no—
79C22×C42C2311yescharacteristic
80#477—24no—
81#477C2×C418yescomplemented
82#476—24no—
83#476C2×C418yescomplemented
84#475—28no—
85#474—28no—
86#431—22no—
87#431—22no—
88#428—28no—
89C2×#107—22no—
90C2×#106—22no—
91#423—22no—
92#423C2×C414yes—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8C2312yes—
99C2×C4⋊C8C2312yes—
100#417—28no—
101C2×#103—22no—
102C2×#103—28no—
103C2×#102—22no—
104C2×C22⋊C8—21no—
105C2×C4×C8C2312yes—
106#391—22no—
107#390—22no—
108#388—24no—
109#388—24no—
110#387—24no—
111#387—24no—
112#386—24no—
113#384C2312yes—
114#383—22no—
115#382—22no—
116#382—21no—
117#381—22no—
118#380—21no—
119#380—22no—
120#380D412yes—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4D412yes—
128C4×C4⋊C4D411yescharacteristic
129C4×C4⋊C4D411yescharacteristic
130C4×C4⋊C4C2312yes—
131C4×C22⋊C4—24no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—22no—
135#376—22no—
136#376—22no—
137C2×#95—21no—
138C43—22no—
139#340—24no—
140#340—24no—
141Q8⋊C8—24no—
142D4⋊C8—24no—
143C2×#41—22no—
144C2×#41—22no—
145C2×#41—22no—
146C2×#41—22no—
147C2×#41—22no—
148C2×#41C22⋊C414yes—
149C22×Q8—24no—
150C22×Q8—22no—
151C22×Q8—41no—
152C22×Q8—42no—
153C22×Q8—84no—
154C22×Q8C2×D411yescharacteristic
155C22×D4—21no—
156C22×D4C2×D411yescharacteristic
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4—41no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C2×M16—24no—
166C2×M16—42no—
167C2×M16—42no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—44no—
172C22×C8—42no—
173C4⋊Q8—24no—
174C4⋊Q8—24no—
175C4⋊Q8—24no—
176C4⋊Q8—22no—
177C4⋊Q8—22no—
178C4⋊Q8—22no—
179C4⋊Q8—42no—
180C4⋊Q8—42no—
181C4⋊Q8—42no—
182C4⋊Q8C22×C414yes—
183C4×Q8—22no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8C22⋊C414yes—
187C4×Q8C22×C412yes—
188C4×Q8C2×D412yes—
189C4×D4—24no—
190C4×D4—24no—
191C4×D4—24no—
192C4×D4—24no—
193C4×D4—22no—
194C4×D4—21no—
195C4×D4—21no—
196C4×D4C22⋊C414yes—
197C4×D4C22×C412yes—
198C4×D4C2×D412yes—
199#117—22no—
200#117—24no—
201#117—24no—
202#117—44no—
203#117—42no—
204#117—42no—
205#117—44no—
206#117C22⋊C414yes—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D414yes—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—24no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—22no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—22no—
243C2×C42—24no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42C22×C412yes—
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D412yes—
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42#4112yes—
252C2×C42#4112yes—
253C2×C42C2411yescharacteristic
254#107—28no—
255#106—28no—
256C4⋊C8—24no—
257C4⋊C8—22no—
258C4⋊C8—22no—
259C4⋊C8—22no—
260C4⋊C8—42no—
261C4⋊C8—42no—
262C4⋊C8C22×C414yes—
263#103—28no—
264#103—88no—
265#102—28no—
266C22⋊C8—42no—
267C4×C8—22no—
268C4×C8—22no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—21no—
274#95—21no—
275#95—22no—
276#95—22no—
277#95—22no—
278C24—21no—
279C24#12011yescharacteristic
280#41—28no—
281#41—28no—
282#41—44no—
283#41—44no—
284C2×Q8—21no—
285C2×Q8—24no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—48no—
291C2×Q8—42no—
292C2×Q8—44no—
293C2×Q8—84no—
294C2×Q8—88no—
295C2×Q8—88no—
296C2×Q8C2×C22⋊C412yes—
297C2×D4—24no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4C2×C22⋊C412yes—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—42no—
319C22×C4—44no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—82no—
328C22×C4—82no—
329C22×C4—84no—
330C22×C4C2×C22⋊C412yes—
331C22×C4C4×D412yes—
332C22×C4C4×D412yes—
333C22×C4#12011yescharacteristic
334C22×C4#12011yescharacteristic
335C22×C4#12011yescharacteristic
336C22×C4#12112yes—
337C22×C4#12112yes—
338C22×C4#12312yes—
339C22×C4#13512yes—
340C22×C4#13512yes—
341C22×C4#13712yes—
342C22×C4#13712yes—
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346M16—48no—
347C2×C8—24no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—44no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—44no—
354C2×C8—84no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—22no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4C2×C22⋊C412yes—
376C4⋊C4C2×C22⋊C412yes—
377C4⋊C4#13714yes—
378C22⋊C4—24no—
379C22⋊C4—22no—
380C22⋊C4—22no—
381C22⋊C4—24no—
382C22⋊C4—48no—
383C22⋊C4—44no—
384C22⋊C4—84no—
385C42—22no—
386C42—22no—
387C42—24no—
388C42—24no—
389C42—48no—
390C42—42no—
391C42—42no—
392C42—42no—
393C42—42no—
394C42—44no—
395C42—84no—
396C42C2×C22⋊C412yes—
397C42C4×D412yes—
398C42C4×D412yes—
399C42#12112yes—
400C42#12112yes—
401C42#12312yes—
402C42C23×C411yescharacteristic
403C42C22×D411yescharacteristic
404C42C2×#4111yescharacteristic
405C42C2×#4111yescharacteristic
406C23—22no—
407C23—22no—
408C23—22no—
409C23—41no—
410C23—41no—
411C23—42no—
412C23—42no—
413C23#38612yes—
414C23#45212yes—
415C23#45212yes—
416C23C2×#12011yescharacteristic
417Q8—22no—
418Q8—22no—
419Q8—48no—
420Q8—41no—
421Q8—41no—
422Q8—84no—
423Q8—88no—
424D4—22no—
425D4—22no—
426D4—48no—
427D4—41no—
428D4—41no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—48no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—84no—
459C2×C4—88no—
460C2×C4—88no—
461C2×C4—84no—
462C2×C4#38612yes—
463C2×C4#38612yes—
464C2×C4#38612yes—
465C2×C4#41614yes—
466C2×C4#41914yes—
467C2×C4#45212yes—
468C2×C4#45212yes—
469C2×C4#48514yes—
470C2×C4#48514yes—
471C2×C4C22×C22⋊C411yescharacteristic
472C2×C4C2×C4×D411yescharacteristic
473C2×C4C2×C4×D411yescharacteristic
474C2×C4C2×#12111yescharacteristic
475C2×C4C2×#12111yescharacteristic
476C2×C4C2×#12311yescharacteristic
477C2×C4C2×#13511yescharacteristic
478C2×C4C2×#13511yescharacteristic
479C2×C4C2×#13711yescharacteristic
480C2×C4C2×#13711yescharacteristic
481C2×C4C2×#13712yes—
482C2×C4#57512yes—
483C8—44no—
484C8—82no—
485C8—82no—
486C22—24no—
487C22—24no—
488C22—22no—
489C22—22no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—82no—
494C22—82no—
495C22#187314yes—
496C22C2×#38611yescharacteristic
497C22C2×#45211yescharacteristic
498C22C2×#45211yescharacteristic
499C22#300712yes—
500C22#302112yes—
501C4—22no—
502C4—21no—
503C4—22no—
504C4—21no—
505C4—44no—
506C4—42no—
507C4—42no—
508C4—42no—
509C4—84no—
510C4—82no—
511C4—84no—
512C4#187314yes—
513C4C2×#41611yescharacteristic
514C4C2×#41911yescharacteristic
515C4C4×#13712yes—
516C4#303911yescharacteristic
517C4#304011yescharacteristic
518C4C2×#48511yescharacteristic
519C4C2×#48511yescharacteristic
520C2—24no—
521C2—82no—
522C2#1879112yes—
523C2#1879312yes—
524C2C2×#187311yescharacteristic
525C2#2092711yescharacteristic
526C2#2094511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ 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
3. #21804C2semidirect
4. C2×#1987C2semidirect
6. #8976C2semidirect
7. #8974C2semidirect
8. #8045C2semidirect
9. #7901C2semidirect
13. #3079C4semidirect
81. #477C2×C4semidirect
83. #476C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30792
#341212
#218042
C2×#19872
C2×#14772
#89762
#89742
#80452
#79012